Quantum Frontiers of Atoms and Molecules

CHEMISTRY RESEARCH AND APPLICATIONS

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CHEMISTRY RESEARCH AND APPLICATIONS

QUANTUM FRONTIERS OF ATOMS AND MOLECULES

MIHAI V. PUTZ EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Quantum frontiers of atoms and molecules / editor, Mihai V. Putz. p. cm. Includes index. ISBN 978-1-61324-867-6 (eBook) 1. Chemical bonds--Mathematical models. 2. Dirac equation. 3. Quantum chemistry. I. Putz, Mihai V. QD461.Q36 2009 541'.28--dc22 2010001746

Published by Nova Science Publishers, Inc. † New York

CONTENTS Foreword

ix

Chapter 1

Fulfilling Dirac’s Promise on Quantum Chemical Bond Mihai V. Putz

Chapter 2

Duality within the Structure of Complementarity: Right Where It Has No Place to Be Constantin Antonopoulos

21

Chapter 3

Complementarity Out of Context: Essay on the Rationality of Bohr’s Thought Constantin Antonopoulos

41

Chapter 4

Molecular Integrals over Slater-Type Orbitals. From Pioneers to Recent Developments P.E. Hoggan, M.B. Ruiz and T. Özdoğan

61

Chapter 5

Tunneling Dynamics and Its Signatures in Coupled Systems S. Ghosh and S.P. Bhattacharyya

91

Chapter 6

Theoretical Calculation of the Low Laying Electronic States of the Molecular Ion CsH+ with Spin-Orbit Effects M. Korek and H. Jawhari

111

Chapter 7

Theoretical Explanation of Light Amplifying by Polyethylene Foil Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski and Bratislav Tosic

141

Chapter 8

Anharmonic Effects in Normal Mode Vibrations: Their Role in Biological Systems Attila Bende

157

Chapter 9

Emergent Properties in Bohmian Chemistry Jan C.A. Boeyens

191

1

vi

Contents

Chapter 10

The Algebraic Chemistry of Molecules and Reactions Cynthia Kolb Whitney

217

Chapter 11

Quantum and Electrodynamic Versatility of Electronegativity and Chemical Hardness Mihai V. Putz

251

Chapter 12

Physics and Chemistry of Carbon in the Light of Shell-Nodal Atomic Model G.P. Shpenkov

277

Chapter 13

Molecular Modeling of the Peanut Lectin - Carbohydrate Interaction by Means of the Hybrid QM/MM Method Alexei N. Pankratov, Nikolay A. Bychkov and Olga M. Tsivileva

325

Chapter 14

Electron Density Distributions of Heterocycles: A Shortcoming of the Resonance Model Ricardo A. Mosquera, Marcos Mandado, Laura Estévez and Nicolás Otero

343

Chapter 15

Electromerism in Small Molecule Activation by Metal Centers of Biological Relevance Radu Silaghi-Dumitrescu

367

Chapter 16

Structural Modelling of Nano-Carbons and Composites Mihai Popescu and Florinel Sava

387

Chapter 17

Nanostructure Design—between Science and Art Mircea V. Diudea

425

Chapter 18

Quantifying Structural Complexity of Graphs: Information Measures in Mathematical Chemistry Matthias Dehmer, Frank Emmert-Streib, Yury Robertovich Tsoy and Kurt Varmuza

479

Chapter 19

Topological Indices of Nanostructures Ali Reza Ashrafi

499

Chapter 20

On Uniform Representation of Proteins by Distance Matrix M. Randić, M. Vračko, M. Novič and D. Plavšić

521

Chapter 21

Timisoara Spectral – Structure Activity Relationship (SpectralSAR) Algorithm: From Statistical and Algebraic Fundamentals to Quantum Consequences Mihai V. Putz and Ana-Maria Putz

539

Chapter 22

On Plots in QSAR/QSPR Methodologies Emili Besalú, Jesus Vicente de Julián Ortiz and Lionello Pogliani

589

Contents

vii

Chapter 23

Application of the Fuzzy Logic Theory to QSPR-QSAR Studies Pablo R. Duchowicz and Eduardo A. Castro

607

Chapter 24

Modeling the Toxicity of Alcohols. Topological Indices versus Van Der Waals Molecular Descriptors Dan Ciubotariu, Vicentiu Vlaia, Ciprian Ciubotariu, Tudor Olariu and Mihai Medeleanu

629

Index

669

FOREWORD ATOMS AND MOLECULES AS THE QUANTUM FRONTIERS OF LIFE Elements of Nature: elements of Life: atoms and molecules. From the earlier time of rational modeling of the Universe, the atomistic vision (Leucippus & Democritus) gave the mechanistic vision upon which all objects are created by unitary elementary entities; they eventually later found the autonomic conceptualization by Leibniz’s monads — the veritable frontiers, two-sided coins, of Creation. On one side they realize the first step from the nonorganized vacuum towards the matter structure, while on the other hand preserve the entangled connection and unity between all the created/observed (and beyond) world. From the epistemological point of view, the Atom at the base – frontier of micro-to-macro Universe marks a drastic unification of the ancient beliefs and myths of the Elements of Nature: water, earth, fire, air – all of them being in one way or another fluidization, sublimation, exaltation or aeration of atomic systems in various conjugations or composites; moreover, Plato’s geometric elements attributing the octahedrons to air, cubes to earth, tetrahedrons to fire, icosahedrons to water, while dodecahedrons build the constellation and paradise – appears as a further atomic combination, yet in a more organized paradigm. Atoms therefore — the first frontier of microcosm, the most pre-eminent physical system in which the fermions (electrons) live in a boson environment (atoms as a whole) — so simple in principle to describe and speculate upon, so difficult to model and quantify. The reason is because of being material and also immaterial, since it spans in principle the whole Universe by its wavefunction. Moreover, all living Nature behaves like its most simple benchmark: Hydrogen. We put under the Hydrogenic wave-function: life and death follows its localized-delocalized curvatures, all events encountered as the Hydrogen wave-function shape; acceleration of particles and bodies, the maximum metastable-equilibrium — the climax of life, followed by the smooth descent toward the zero ground of being, leaving nevertheless space for believers regarding the eventual unification of minus and plus infinites, such that all of Nature works in cycles, with the satisfaction of energy conservation. This is Nature — this is Physics; but maybe it is not enough. Just when more than one electron is stabilized in an atomic system — the atomic world diversifies in such manner that the multiple can spring out of Monads, the Elements become now the Periodic Elements, the Chemistry arises. As such the Chem listens by the words of his father Noah in bringing new life in the Universe by dissemination, and cross-fertilization of the earth’s seeds, as such the cheos gives the fluids of Life, and archeos

x

Mihai V. Putz

the archetypes of living, the paradigms of Nature. Living with more than one electron in an orbital, the appearance of duality, of spinning, that’s Chemistry. As a name enough obscure, partly mythical from the dark side of Egypt, partly theological from the sacral combination it provides (El-Chemia: Alchemy) towards the molecular world. With this we arrive at the bonding mystery; it is called in general the chemical bond, but stays as the frontier of the bigger world to come: the Bios. The biological component of Nature may be approached through the generalization of the chemical bonding idea, or the resonance between the ionic and covalent bindings, to the ligand-receptor one. Such picture is fully feasible after rooting in the hard and soft acids and bases theory in which terms any chemical reaction can in principle be formulated. On the other hand, for the bio-, eco-, and pharmaco-logical involved systems the chemical bond appears unstable respecting the isolated systems, due to the environment damping; in other words, a toxicological action (meaning a chemical binding with a toxic effect) is diminishing in time as a consequence of metabolic actions (intern factors) and of the environment (extern factors). This way, it raises the challenge in formulating a theory of the chemical bond variable in time at the biomolecular level, specialized on the in vitro enzymic case, and then extended to the in vivo situations in order to better cover the record of the (non)toxic or therapeutic actions in organisms and the environment. Recent attention was jointly focused on eco-, bio-, as well as on pharmacosciences. In this stage new pharmacophoric reactivity indices may be formulated with the help of quantum molecular topology combined with the graph theory (due to its versatility to be iteratively computed from atoms to molecules) producing a direct quantification of the toxic or therapeutic effect for a given chemical. The Hydrogenic atoms, the many-electron atoms, and the (bio-)molecules: the Physics, Chemistry, and Biology — and the wholeness they cover in a rheological manner with the help of the quanta, fields and particles, in reciprocal transformation, combination, excitation — the thin red line of the Universe. Life’s frontiers —the quantum-verse, the true nature of the uni-verse! With the actual ever-expanding developments of the edge technology with direct impact on life and environment, a lucid review of the main foreground conceptual realms of electronic matter at the level of atoms and molecules is by this volume unfolded aiming to offer a unitary perspective of the quantum principles as applied to many-electronic states, either in isolate and interacting context, to chemical bond and bonding as well to the relation between the physical-chemical structure and chemical-biological reactivity and activity manifestations of nature. The volume widely gives larger and deeper coverage of the electronic matters through the physical, chemical and biological ordered systems, with their increasing complexity through gradually presenting the matter structure from the physical to chemical to biological manifestations in an inter-disciplinary cross-fertilization manner. With equilibrated contents provided by important scientists worldwide with a valuable impact on quantum fundaments and applications, the book presents and reviews the avant-garde contributions for the XXI century. In fact, the present volume steps aside to serve for the unification of the physicalchemical-biological manifestation of atoms in molecules and in nanostructures by means of expanding the quantum frontiers by conjunction with either relativity or topological or information or graph theories as well. The book successfully balances among the physical, chemical and biological sides of the quantum theory and of its applications emphasizing both conceptual and computational sides while experiment is addressed only for reference; in this regard the book is more focused on why rather than how quantum effects are produced with

Foreword

xi

the inner belief that this way the second issue is self-contained in the first one. The book is addressed to a large audience as well as to advanced research wisely combining the epistemological, heuristic and philosophic aspects of quantum manifestation of matter in atoms, molecules and of their combination in complex nano- and bio- structures. On the other hand, the book likes to show how far the quantum theory furnishes the background and the framework in which the simple as well as most complex electronic structures may unfold and evolve in an interacting environment. Overall, searching the unity of the manifestation forms of the chemical bonding at various levels of matter organization had become a very active interdisciplinary field in the last years, being one of the main goals in the frame of the nanosciences. The quantum paradigm of bonding unification through the formulation of a minimal set of concepts and quantities having as much universal multi-electronic relevance as possible represents a real challenge for the conceptualization and prescription of the viable applicative directions of the nanosystems, from atoms to biomolecules. For that we are certain that the present edited volume will have a special role for making further advancement in improving the scientific knowledge in this priority domain of research. And last, but not least, Editor and Authors like to sincerely thank all the NOVA team involved in the present challenging editorial venture, and to NOVA Vice-president Nadya Columbus especially, for kind assistance and patience throughout all stages of publication. Mihai V. PUTZ (Volume Editor) Chemistry Department West University of Timişoara Romania

In: Quantum Frontiers of Atoms and Molecules Editor: Mihai V. Putz, pp. 1-19

ISBN: 978-1-61668-158-6 © 2010 Nova Science Publishers, Inc.

Chapter 1

FULFILLING DIRAC’S PROMISE ON QUANTUM CHEMICAL BOND Mihai V. Putz* Laboratory of Computational and Structural Physical Chemistry, Chemistry Department, West University of Timişoara, Str. Pestalozzi No.16, Timisoara, RO-300115, Romania

Abstract One of the most fundamental issues of quantum chemistry, the forming and electronic description of bonding, is here unfolded at the level of Dirac’s theory, in conjunction with the density functional concept of chemical action or, equivalently, with the beloved electronegativity for practical applications of encountering atoms in molecules. The resulting chemical bonding equation allows for the first time the geometrical identification of molecular region along bonding in which the parallel and anti-parallel spin-electrons coexist in antibonding and bonding states, respectively.

1. Introduction After the Nobel Prize 1933 jointly awarded to the “productive” contribution on atomic theory for Schrödinger and Dirac’s theories of electrons, there was created the impression that at least the chemistry world of phenomena will be entirely explained [1-17]. This is not minor, because of the huge importance the Chemistry of atoms and molecules plays in life and in metabolism [18]. Yet, soon it became clearer that the chemical systems are not as simple as the physical models intend to imply; for instance, no spherical molecule exists to transfer upon the principles of special orthogonal groups; or even more complicated the chemical systems consist of many—but numerable—electrons, therefore not amendable with the thermodynamic physical limit ( N → ∞ ); or more subtle, the chemical bond is supporting *

E-mail addresses: [email protected], [email protected]. Tel: +40-256-592633; Fax. +40-256-592620,

2

Mihai V. Putz

both the parallel and anti-parallel spins in different states, inversely known as anti-bonding and bonding states, emphasizing that the first instance is firstly involved in the chemical reactivity through belonging to the so-called valence state [3-5]. Although the molecular orbital theory attempted to explain this intriguing behavior by the anti-symmetry rule of a given state’s spin-electron wave function, i.e. the total Pauli antisymmetry wave function may be obtained either by convoluting symmetrical spin with antisymmetrical coordinate wave functions—for anti-bonding or anti-symmetrical spin with symmetrical coordinate wave function—for bonding states, this remains only as a qualitative description with no practical effect on geometrical description of anti-bonding parallel spins’ coexisting along the bonding [11-13]. One can say that since electrons are waves they virtually occupy all space of bonding with an anti-symmetrical coordinate wave that eventually does not interfere with that symmetrical one coming from the bonding wave; very true, but also this seems a qualitative argument [8]. Therefore, one should be next interested in describing the geometrical locus along the chemical bond where the bonding and anti-bonding waves behave as within resonators (since they are stationary in the formed molecule), and eventually explaining why they do not interfere (i.e., they are orthogonal states belonging to separate Hilbert spaces); and all these with the help of Dirac’s theory directly since the spin problem is aiming to be responded, and not by employing the Schrödinger indirect argumentum of coordinate symmetry [9]. To this end, the present work likes to step forward in elucidating the mystery of “the chemical bond” through combining the fundamental Dirac equation with spinorial solution, generalized from the Schrödinger equation, while being combined with the recently advanced chemical action description of bonding [19,20].

2. Dirac-Schrödinger Equivalence There is a known fact that the basic Dirac equation unfolds as the temporal generalized operatorial form [1,21-23]

i=∂ t [Ψ ] = Hˆ Dir [Ψ ]

(1)

in a very similar shape with the Schrödinger one, however with the Dirac Hamiltonian specialization: 0 Hˆ Dir = Hˆ Dir + vˆ( x)

(2a)

with the free particle and applied potential components:

G Gˆ 0 Hˆ Dir = −i=cαˆ ⋅ ∇ + mc 2 βˆ ,

(2b)

vˆ( x) = V ( x) βˆ ,

(2c)

Fulfilling Dirac’s Promise on Quantum Chemical Bond

3

respectively, with m - the particle mass, c - the light velocity, = - the Planck constant, while

G the introduced special operators αˆ , βˆ assume the Dirac 4D representation:

⎡0

αˆ k = ⎢ ⎣σˆ k

σˆ k ⎤

, k = 1,2,3 , 0 ⎥⎦

⎡1ˆ

0⎤ ˆ⎥ ⎣0 − 1⎦

βˆ = ⎢

(3a)

(3b)

in terms of bi-dimensional Pauli and unitary matrices (operators)

⎡1 0⎤ ⎡0 1 ⎤ ⎡0 − i ⎤ ⎡1 0 ⎤ , σˆ 1 = ⎢ , σˆ 2 = ⎢ , σˆ 3 = ⎢ ⎥ ⎥ ⎥ ⎥ ⎣0 1 ⎦ ⎣1 0 ⎦ ⎣i 0 ⎦ ⎣0 − 1⎦

σˆ 0 = 1ˆ = ⎢

(4)

and with the wave function featuring the so called spinorial (bi-dimensional) equivalent formulation

⎡ϕ ⎤ − =i E⋅t [Ψ ] = ⎢ ⎥ e ⎣φ ⎦ ⎧⎡ϕ ⎤ − =i E ⋅t , E > 0 anti − bonding states ⎪⎢ ⎥ e ⎪⎣ 0 ⎦ =⎨ i ⎪⎡ 0 ⎤ e + = E ⋅t , E < 0 bonding states ⎪⎢φ ⎥ ⎩⎣ ⎦

⎡ − =i E ⋅t ⎤ ϕe = ⎢ i ⎥ (5) ⎢ + = E ⋅t ⎥ ⎢⎣φe ⎥⎦ However, there also arises the question whether the general Dirac equation (1) may be reduced or transformed so that to represent the eigen-equation for the electronic states for a given quantum system. For this, through closely analyzing the form of Eq. (1) with all its contribution, one may resume the free motion Dirac operator to the working form [8]

∂ ∂ ∂ ∂ Dˆ = −i=Aˆ σˆ 0 + i=σˆ 1 + i=σˆ 2 + i=σˆ 3 + Cˆ ∂t ∂x1 ∂x 2 ∂x3 and employing it to the stationary operatorial equation:

(6a)

4

Mihai V. Putz

⎡ Ψ e i (kx−ωt ) ⎤ ⎡ζ ⎤ 0 = Dˆ [Ψ ] = Dˆ ⎢ 1 −i (kx−ωt ) ⎥ = Dˆ ⎢ ⎥ ⎣ξ ⎦ ⎣Ψ2 e ⎦

(6b)

with the oscillatory phase written so that to be in accordance with the prescription of eq. (5) for the Planck energy-frequency identification:

E = =ω

(7)

In these conditions, one notes that for the time and coordinate derivatives yields:

−ζ ∂ [Ψ ] = iω ⎡⎢ ⎤⎥ , ∂t ⎣ξ ⎦

(8a)

−ζ ∂ [Ψ ] = −ik k ⎡⎢ ⎤⎥ ∂xk ⎣ξ ⎦

(8b)

which reduce the above stationary condition (6) to the form

(=ωAˆ1ˆ + =k σˆ 1ˆ)⎡⎢−ξζ ⎤⎥ + Cˆ ⎡⎢ζξ ⎤⎥ = 0 k

k

⎣

⎦

(9)

⎣ ⎦

Choosing now appropriately (that stands for the optimization procedure) the matrices

⎡ 0 0 ⎤ ˆ ⎡0 2 m ⎤ Aˆ = ⎢ ⎥ , C = ⎢0 0 ⎥ ⎣1 0⎦ ⎣ ⎦

(10)

the last form (9) further rearranges as

⎡=k k σˆ k ⎢ ˆ ⎣ =ω1

2m1ˆ ⎤ ⎡− ζ ⎤ ⎥⎢ ⎥ = 0 =k k σˆ k ⎦ ⎣ ξ ⎦

(11a)

leaving with the system:

⎧⎪− =k k σˆ k ζ + 2m1ˆξ = 0 ⎨ ⎪⎩− =ω1ˆζ + =k k σˆ k ξ = 0 Now, since solving the first equation of the system (11b) in one variable, say

(11b)

Fulfilling Dirac’s Promise on Quantum Chemical Bond

ξ=

=k k σˆ k 1ˆ p σˆ 1ˆ ζ = k k ζ 2m 2m

5

(12a)

and substituting into the second one, there is obtained:

( p σˆ )( p σˆ ) p 2σˆ 2 p2 ˆ 1ζ E1ˆζ = k k k k ζ = k k ζ = 2m 2m 2m

(12b)

where the above Planck relationship was supplemented by the companion de Broglie one for the momentum,

=k k = p k

(13)

σˆ k2 = 1ˆ

(14)

while the Pauli matrices basic property

applies, as may be immediately verified from their realization of eq. (4). Nevertheless, the eq. (12a) represents in fact the eigen-equation for the free motion, supporting the latent generalization to the bounded state, either in anti-bonding or bonding existence

Eζ = εζ , Eξ = εξ

(15)

This is an interesting result because abolished many odd perception about Dirac equation and its meaning; actually, there follows that: •

•

•

Dirac equation is formally related with the temporal Schrödinger one, while producing the same eigen-problems, thus describing in essence the same nature of the electronic motion; The spin information modeled by the bi-dimensional spinors is not necessarily a relativistic effect (beside completing the 2+2=4 relativistic framework dimension of the Dirac equation) but merely a quantum one since fulfilling the eigen-value problems, each separately; The two spinors of the Dirac equation may be associated with the bonding (for negative energies) and anti-bonding (for positive energies) of a system, being thus suited for physically modeling of the chemical bond, beside the common interpretation of negative/positive spectrum of free positronic/electronic energies in the Dirac Sea.

However, before effectively pursuit to the chemical bonding description based on Dirac equation one needs some background of the recent non-orbitalic quantum modeling of the chemical bond.

6

Mihai V. Putz

3. Quantum Chemical Bond 3.1. Binding Functions and the Chemical Bond Employing the dimensional quantum-relativity relationship

< energy > ⋅ < distance >~ Joule ⋅ meter ~ = ⋅ c

(16)

recently, there were introduced the chemical binding functions [19]:

⎧1, λ → 0 f α (λ , C A ) = 1 − ΩλC A = ⎨ ⎩− ∞ , λ → ∞

(17a)

⎧1, λ → 0 f β (λ , C A ) = exp(− ΩλC A ) = ⎨ ⎩0, λ → ∞

(17b)

called as the anti-bonding and bonding functions, for the reason grounded on their asymptotical behavior, respectively; the introduced Ω-factor accounts for assumed dimensionless nature of functions (17), being adequately settled as:

Ω= and where

1 = 0.506773 ⋅ 10 −3 J −1 m −1 =c

(18)

λ stands for the localization distance, while C A stays for the chemical action

[19,20]

G G G C A = ∫ ρ (x )V ( x )dx ≅ χ [ρ ]

(19)

o

expressed in A (ångstrom) and eV (electron-volts), respectively. Note that in eq. (19) the equivalence between chemical action and electronegativity χ [ρ ] , as electronic density functionals, was assumed based on their similar nature in convoluting the applied potential with the concerned electronic density ρ [24], although electronegativity is currently unfolded as a generalization of the chemical action, being dependent on it, showing a more complex density functional expression at various localization levels [25, 26]. Nevertheless, the bonding functions (17) reciprocally combine within a paradigmatic AB molecule, with a coordinate system centered in A, to provide the electronic pair-localization region within the bond length RAB by means of the binding equations (see Figure 1) [19],

(

)

(

)

(I) : f αA λI , χ A = f βB RAB − λI , χ B ,

(20a)

Fulfilling Dirac’s Promise on Quantum Chemical Bond

(

)

(

(II) : f αB RAB − λII , χ B = f βA λII , χ A as the interval

)

7 (20b)

λII − λI or as the single point λII = λI for the hetero- and homo- bonding

systems, i.e. having different or identical isolated electronegativities

χ A and χ B ,

respectively.

Figure 1. Geometrical loci of the sigma-bonding (σ-B in blue), anti-bonding (⎤-B in red), nobonding (∅-B in orange), and pi-bonding (π-B in green) for chemical binding from equal electronegativity influences of two systems A and B throughout equations (20) with binding functions (17) through constants and parametric settings as = = c = 1 , χ A = χ B = 1 , RAB = 1.

In each of above equations (20) the binding “points” I and II appear as the informational crossing (transfer) between the anti-bonding and bonding functions of both A and B systems driven by their electronegativities; however, they fix the all types of involved bonding regions (see Figures 1 and 2) as follows [19]: •

the sigma-bonding region (σ-B in Figures 1 and 2) is uniquely defined and has no “nodes” or discontinuities; it is delimited by the area under bonding crossing functions inside of the pairing interval bordered by the projection of the points I and II along the bond; it corresponds to the consecrated bonding obtained by the composed wave-function density Ψ A + ΨB

•

2

in the conventional molecular orbital

(MO) theory [4]; the anti-bonding region (⎤-B in Figures 1 and 2) is represented by the two parts spanning the space from the systems A and B until the sigma-bonding limit, being defined by the area under bonding functions f βA and f βB but outside of the interval

8

Mihai V. Putz fixed by the projection of points I and II on the bond length; it corresponds to the 2

anti-bonding state density ΨA − ΨB with separated parallel electronic spins in MO •

•

theory [4]; the no-bonding region (∅-B in Figures 1 and 2) is composed of two parts, one in each binding side respecting sigma-bonding, being formed by the area delimited by all the binding functions of (17) around the binding points I and II, outside of their projected interval on the bond length, while not intersecting between them and with the bond length; the pi-bonding region (π-B in Figures 1 and 2) spans the bond length entirely without crossing it, thus having nodes on it, being resulted from the area defined by all the binding functions of (17) around the binding points I and II, partially outside and partially inside (with a common point inside the projected interval of the points I and II on bond; therefore, this region is compatible with the consecrated pi-bond type of the MO theory.

Figure 2. Geometrical loci of the bonding regions as in Figure 1 for chemical binding from different electronegativity influences of two systems A and B throughout equations (20) with binding functions (17) through constants and parametric settings as = = c = 1 , χ A = 2 χ B = 2 , RAB = 1 .

Note that the difference between the equal and different electronegativity influences on bonding in Figures 1 and 2 is reflected in sigma-bonding shift towards the more electronegative bonding component, while slightly enlarging the spanning interval of projection of points I and II on bond in Figure 2; moreover, the bond of Figure 2 is accompanied by a slightly decreasing of the sigma bonding apex value on binding probability, being now about 0.5 respecting the recorded 0.6 value in Figure 1. This may lead with the

Fulfilling Dirac’s Promise on Quantum Chemical Bond

9

meaningful consequence in describing the covalency (in Figure 1) and ionicity (in Figure 2) characters of chemical bonding, in terms of quantum tunneling of the sigma bonding region: • •

covalent binding is characterized by a symmetric higher and thinner well of electrons, being those more localized on middle of bond; ionic binding features a dissymmetric taller and thicker well of electrons with more delocalized pairing electrons towards the more electronegative component.

The bonding regions may appear as the consequence of equilibrium between binding functions (17) that caries the density functional information either as chemical action or as electronegativity. As a consequence, all above identified binding regions are defined within positive (0, 1) realm of binding functions (17) allowing the natural probabilistic interpretation for their inside. Nevertheless, there remains to explore their influence on bonding within the Dirac equation framework, and how the Dirac equation in generals influences the chemical bonding phenomenology when the spin (or spinors) are involved. This will be addressed in the sequel.

3.2. Chemical Bond by Dirac Equation The above spinorial identification as bonding and anti-bonding, see eq. (5), may be now combined with the introduced bonding and anti-bonding functions (17) so that to create the actual working binding spinor:

⎡ ⎛ λχ ⎞ ⎛ i ⎞ ⎤ ⎢ ⎜1 − =c ⎟ exp⎜ − = E ⋅ t ⎟ ⎥ [Ψ ] = ⎢ ⎝ λχ⎠ ⎝ i ⎠ ⎥ ⎞ ⎛ ⎞ ⎢exp⎛⎜ − exp⎜ + E ⋅ t ⎟⎥ ⎟ ⎢⎣ ⎝ =c ⎠ ⎝ = ⎠⎥⎦

(21)

where the previous chemical action dependence was here reconsidered as the more generalized (density functional) electronegativity. Next, we impose the condition the spinor (21) fulfilling the Dirac equation (1); for this we separately express the involved terms, while self-understanding the presence of the (bidimensional) unitary and other Dirac operators on both spinorial upper and down components so that the implicit total dimension of the wave-function to be completed to four-dimensional space: •

the time derivative Dirac term is directly computed as:

⎡ ⎛ λχ ⎞⎛ i ⎞ ⎛ i ⎞⎤ ⎢i=⎜1 − =c ⎟⎜ − = E ⎟ exp⎜ − = E ⋅ t ⎟⎥ ⎠⎝ ⎠ ⎝ ⎠⎥ i=∂ t [Ψ ] = ⎢ ⎝ λχ i i ⎛ ⎞ ⎛ ⎞ ⎢ i= E exp⎜ − ⎟ exp⎜ + E ⋅ t ⎟ ⎥⎥ ⎢⎣ = ⎝ =c ⎠ ⎝ = ⎠⎦

10

Mihai V. Putz

⎡ ⎛ λχ ⎞ ⎛ i ⎞ ⎤ ⎢ E ⎜1 − =c ⎟ exp⎜ − = E ⋅ t ⎟ ⎥ ⎝ ⎠ ⎝ ⎠ ⎥ =⎢ λχ i ⎛ ⎞ ⎛ ⎞ ⎢− E exp⎜ − exp⎜ + E ⋅ t ⎟⎥ ⎟ ⎢⎣ ⎝ = ⎠⎥⎦ ⎝ =c ⎠ •

(22)

the space coordinate Dirac derivative needs the pre-requisite of simple derivative

∂ k λ = ∂ k λk λk =

λk λ

(23a)

providing the yield:

⎤ ⎡ ⎛ λ χ⎞ ⎛ i ⎞ − i=cαˆ k ⎜ − k ⎟ exp⎜ − E ⋅ t ⎟ ⎥ ⎢ ⎝ = ⎠ ⎝ =cλ ⎠ ⎥ − i=cαˆ k ∂ k [Ψ ] = ⎢ ⎢− i=cαˆ k ⎛ − λk χ ⎞ exp⎛ − λχ ⎞ exp⎛ + i E ⋅ t ⎞⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎥ ⎢⎣ ⎝ =c ⎠ ⎝ = ⎠⎦ ⎝ =cλ ⎠ ⎤ ⎡ χ ⎛ i ⎞ i λk exp⎜ − E ⋅ t ⎟ ⎥ ⎢ ˆ σk ⎤ λ ⎝ = ⎠ ⎥ ⎢ ⎛ λχ ⎞ ⎛ i ⎞⎥ 0 ⎥⎦ ⎢ χ i λ exp⎜ − ⎟ exp⎜ + E ⋅ t ⎟⎥ ⎢⎣ λ k ⎝ =c ⎠ ⎝ = ⎠⎦

⎡0 =⎢ ⎣σˆ k

⎡ χ ⎛ λχ ⎞ ⎛ i ⎞⎤ ⎢i λ σˆ k λk exp⎜ − =c ⎟ exp⎜ + = E ⋅ t ⎟⎥ ⎝ ⎠ ⎝ ⎠⎥ =⎢ χ i ⎛ ⎞ ⎥ ⎢ i σˆ k λk exp⎜ − E ⋅ t ⎟ ⎥⎦ ⎢⎣ λ ⎝ = ⎠ •

(mc

(23b)

the free + potential term:

2

(

⎡ mc + V ( x) + V ( x) βˆ [Ψ ] = ⎢ 0 ⎣

)

2

⎡ ⎛ λχ ⎞ ⎛ i ⎞ ⎤ 1− exp⎜ − E ⋅ t ⎟ ⎥ ⎜ ⎟ ⎢ ⎤ ⎝ 0 =c ⎠ ⎝ = ⎠ ⎥ ⎥⎢ 2 − mc + V ( x) ⎦ ⎢exp⎛ − λχ ⎞ exp⎛ + i E ⋅ t ⎞⎥ ⎜ ⎟⎥ ⎢⎣ ⎜⎝ =c ⎟⎠ ⎝ = ⎠⎦

)

(

)

⎡ ⎛ λχ ⎞ ⎛ i ⎞ ⎤ 2 ⎢ mc + V ( x ) ⎜1 − =c ⎟ exp⎜ − = E ⋅ t ⎟ ⎥ ⎝ ⎠ ⎝ ⎠ ⎥ =⎢ λχ i ⎛ ⎞ ⎛ ⎞ ⎢− mc 2 + V ( x ) exp⎜ − exp⎜ + E ⋅ t ⎟⎥ ⎟ ⎢⎣ ⎝ =c ⎠ ⎝ = ⎠⎥⎦

(

(

)

)

With these, the Dirac equation (1) now provides the system of equations:

(24)

Fulfilling Dirac’s Promise on Quantum Chemical Bond λχ

11

− E ⋅t χ ⎛ λχ ⎞ − = E ⋅t ⎛ λχ ⎞ − = E ⋅t = i σˆ k λk e =c e = + 1ˆ(mc 2 + V ( x) )⎜1 − 1ˆ E ⎜1 − , ⎟e ⎟e λ =c ⎠ =c ⎠ ⎝ ⎝

i

− 1ˆ E e

−

λχ =c

e

i E ⋅t =

i

i

(25a)

λχ

i i − − E ⋅t E ⋅t χ 2 c = = = ˆ = i σˆ k λk e − 1(mc + V ( x) )e e λ

(25b)

Next, through getting out from the second equation (25b) the term containing the covariant product: λχ

i

− χ σˆ k λk = 1ˆ(mc 2 + V ( x) − E )e =c λ

(26a)

it is then replaced in the first equation (25a) to obtain:

(mc

2

λχ −2 ⎛ λχ + V ( x) − E )⎜⎜1 − + e =c =c ⎝

⎞ ⎟=0 ⎟ ⎠

(26b)

whose solutions expresses the energy conservation

E = mc 2 + V ( x)

(27)

and the Dirac adapted bonding equation

−e

−2

λχ =c

= 1−

λχ =c

(28a)

Yet, due to the negative sign of the left hand side of eq. (28) one may infer that it is just one solution of a quadratic equation, say

e

−4

λχ =c

≅ 1− 2

λχ =c

(28b)

providing the second accompanied solution

e

−2

λχ =c

≅ 1−

λχ =c

(28c)

However, there is noted the formal difference between eqs. (28a) and (28c) not only because of sign but also due to the approximate nature of the second, coming from the form (28b) in short range of binding distance regime λ ≅ 0 . Nevertheless, the appearance of two

12

Mihai V. Putz

(±) forms of Dirac chemical bonding equation is in accordance with the manifestation of the Dirac positive/negative manifestation of energies respecting the electronic/positronic motions within the Dirac Seas, respectively. Still, for the chemical bond description the difference in sign allows for further mixing of the bonding equations for a paradigmatic AB molecule generating more bonding points so modeling in more detail the bonding with spins in bonding and anti-bonding states. Therefore, the actual working Dirac binding functions are: •

The Dirac anti-bonding function remains the same as given within density kernel approach by eq. (17a):

f αDir (λ , χ ) = 1 − •

λχ

(29)

=c

The Dirac bonding function is modified respecting the previous one given by eq. (17b) while being two-folded:

λχ ⎞ ⎛ f βDir ⎟, ( + ) (λ , χ ) = exp⎜ − 2 =c ⎠ ⎝

(30a)

λχ ⎞ ⎛ f βDir ⎟ ( − ) (λ , χ ) = − exp⎜ − 2 =c ⎠ ⎝

(30b)

Now, the bonding geometric loci are determined, for the molecule AB, by the system of equations: −B B (I) : f αA (λI , χ A ) = f βDir ( + ) (RAB − λI , χ ) ,

(31a)

−A A (II) : f αB (RAB − λII , χ B ) = f βDir ( + ) (λ II , χ ) ,

(31b)

−A A (III) : f αA (λIII , χ A ) = f βDir ( − ) (λIII , χ ) ,

(31c)

(

)

(

−B (IV) : f αB RAB − λIV , χ B = f βDir RAB − λIV , χ B ( −)

)

(31d)

which is regarded as Dirac generalization of the previous one of eqs. (20) by means of the last two equations which quantifies the “interference” effect of the anti-bonding and the negative bonding functions belonging to the same atom in the to be transferred towards a virtual bonding partner. The representations of Figures 3-4 show how the Dirac binding functions and equations (29)-(31) provides more insight in modeling of chemical bonding respecting the previous density functional ones of Figures 1 and 2.

Fulfilling Dirac’s Promise on Quantum Chemical Bond

13

The differences comes from two basic facts: the bonding function (17b) takes through Dirac equation two forms, i.e. it degenerates into one positive and other negative, see eqs. (30a) and (30b), respectively, while having also the modified argumentum. Instead, the antibonding equation (17b) is Dirac preserved either as in form and multiplicity. Due to this fact, depending on the electronegativity differences between the bonding partners the anti-bonding spin state may be located in various locations between the mixed (positive) bonding-antibonding crossing points I and II, eqs. (30a) and (30b), and the self (negative) bonding – antibonding crossing points III and IV, eqs. (30b) and (30c). Even more, for equal electronegativity three types of parallel spin (antibonding) separation may arise as illustrated by the Figures 3a, 3b and 3d, i.e. as being delocalized outside, precisely localized at the edge and delocalized inside of the sigma-bonding region, respectively, while the Figure 3c illustrates the case when the bonding pairing is precisely localized on bond. Actually, following the bonding points delivered by the system (31) one has, for the equal electronegativity cases of Figures 3a-3d, the following configurations respecting the electronegativity equal values, respectively: Figure 3a:

χ A = χ B = 1 : IV II < I < III N N<  ↑

(32a)

↑

↑↓

providing the anti-bonding (parallel) spins delocalized outside of the region with delocalized anti-parallel pairing electrons; Figure 3b:

χ A = χ B = 2 : IV =N  I <

II = N III ↑

↑↓

(32b)

↑

providing the precise localization of the anti-bonding (parallel) spins at the margins of the region with delocalized anti-parallel pairing electrons; Figure 3c:

χ A = χ B = 2.22 :  I < III IV <

II N

= ↑

↑↓

(32c)

↑

providing the precise localization of the anti-parallel pairing electrons at the half of the bond length, being outside of it the delocalization of the anti-bonding (parallel) spins; Figure 3d:

χ A = χ B = 3: N I < III < IV II 


↑↓

(32d)

↑

providing the limited delocalization of the anti-bonding (parallel) spins at the margins of regions with delocalized anti-parallel pairing electrons.

14

Mihai V. Putz

Figure 3a. Geometrical loci of the bonding regions as in Figure 1 for chemical binding from equal electronegativity influences of two systems A and B throughout equations (31) with Dirac binding functions (29) and (30) through constants and parametric settings as = = c = 1 , χ A = χ B = 1 , RAB = 1 .

Figure 3b. Geometrical loci of the bonding regions as in Figure 1 for chemical binding from equal electronegativity influences of two systems A and B throughout equations (31) with Dirac binding functions (29) and (30) through constants and parametric settings as = = c = 1 , χ A = χ B = 2 ,

RAB = 1 .

Fulfilling Dirac’s Promise on Quantum Chemical Bond

15

Figure 3c. Geometrical loci of the bonding regions as in Figure 1 for chemical binding from equal electronegativity influences of two systems A and B throughout equations (31) with Dirac binding functions (29) and (30) through constants and parametric settings as = = c = 1 , χ A = χ B = 2.2 ,

RAB = 1 .

Figure 3d. Geometrical loci of the bonding regions as in Figure 1 for chemical binding from equal electronegativity influences of two systems A and B throughout equations (31) with Dirac binding functions (29) and (30) through constants and parametric settings as = = c = 1 , χ A = χ B = 3 , RAB = 1 .

16

Mihai V. Putz

Figure 4. Geometrical loci of the bonding regions as in Figure 2 for chemical binding from different electronegativity influences of two systems A and B throughout equations (31) with Dirac binding functions (29) and (30) through constants and parametric settings as = = c = 1 , χ A = 2 χ B = 2 ,

RAB = 1 .

Worth observing that the case of the equal electronegativities presents the so called critical electronegativity fulfilling the equation (31c), for instance, at the half of the bond length:

⎞ ⎞ ⎛R − A ⎛ RAB f αA ⎜ AB , χ ⎟ = f βDir ,χ⎟ ( −) ⎜ ⎠ ⎝ 2 ⎠ ⎝ 2

(33)

which give the information of precisely localization of pairing of anti-parallel electronic spins. As well, the precise localization of the anti-bonding parallel spins of eq. (32b), namely at the critical distances

λI = λIV = 0.445571 [ RAB ] ,

(34a)

λII = λIII = 0.554429 [ RAB ]

(34b)

on the actual bond length scale, are candidate for universal application for the given (known) equal electronegativity as stipulated by the case (32b), in adequate units. However, for both cases of eqs. (33) and (34), one providing localization of pairing antiparallel spins at half RAB / 2 of bond length for critical electronegativity, and the other the critical localization of the parallel spins of the anti-bonding states for equal electronegativities

Fulfilling Dirac’s Promise on Quantum Chemical Bond

17

equaling the twice of current units, respectively, furnishes important practical result when one likes to control the magnetic properties of quantum material composed by two aggregates. The Figure 4 is nothing than the extension of the Figure 3a in ionic manner as was the case for Figure 2 different from Figure 1. Still, for the present case worth remarking that due to the Dirac treatment, that is by the involvement of the negative bonding function of (30b) type, the actual model for anti-bonding state, although delocalized in Figures 3a and 4 is still finite respecting the infinite asymptotic space coverage in Figures 1 and 2. The final remark regards the height of bonding probability that is more and more contracted for the increase equal electronegativity cases in Figures 3a-3d, while for different electronegativities the appreciable contraction of width of the sigma bonding region is recorded, see Figures 2 and 4. Overall, the inclusion of the binding functions of eqs. (17) into the Dirac spinor of eq. (21) leaves with the so called Dirac binding functions (30) that enlarge through the binding equations (31) the cases of bonding and anti-bonding spin states, while being able to identify the specific situation when either parallel or anti-parallel spin states are precisely localized along the bond lengths, with presumed higher importance in designing and controlling of atoms-in-molecules spin based reactivity, and of nano-composites [27].

4. Conclusion Although containing a great amount of quantum and relativistic information about the electronic existence, origin, and motion, the Dirac equation is usually used to correct the atomic and molecular energies, eventually through the density functional Dirac-Kohn-Sham equation employed the Dirac-Coulomb Hamiltonian (called as the no-pair approximation) [17]:

[cαGˆ ⋅ pGˆ + (mc with

2

]

)

G G G G + V (r ) βˆ + Vxc (ρ (r ) ) ϕ j (r ) = ε jϕ j (r )

(35)

G Z Z Z G ρ (r ' ) G V (r ) = ∫ G G dr '−∑ G A G + ∑ G A BG + ... r − r' A RA − r A ≠ B RA − RB

(36a)

G δE [ρ (r )] G Vxc (ρ (r ) ) = xc G δρ (r )

(36b)

G

occ

G

G

ρ (r ) = ∑ n jϕ +j (r )ϕ j (r )

(36c)

j

where n j stay for occupancy numbers and where the exchange-correlation energy may be appropriately chose according with the desired approximation framework [28].

18

Mihai V. Putz

The present approach goes beyond the energetic description of bonding while evaluating the geometrical (physical) phenomenology of bonding employing the Dirac equation for the binding functions abstracted from the density kernel approximation [19]:

G G λχ B [ρ ( r )] ⎛ λχ A [ρ ( r )] ⎞ exp⎜ ⎟ ≅ 1− =c c ⎝  =

  ⎠  bonding function of atom A

(37)

anti −bonding function of atom B

Remarkably, when functions (37) are considered as bonding and anti-bonding parts of the Dirac spinor and plugged into the Dirac equation they provide, beside the energy conservation – as the control correct result, also the quantum relativistic (Dirac) binding equations (28) are obtained; their appearance differs from earlier binding functions (17) only in bonding terms (30) which are now two-folded degenerate, in accordance with positive-negative Dirac energetic Seas [1,10,21]; yet, the difference exists also in the bonding argument of eqs. (28) that is doubled respecting the previous non-relativistic one of (17b). These special features of Dirac treatment of binding provide a diversity of situations, depending on reciprocal electronegativity influences of adducts, however, including the situations of precisely localization of the bonding pair and of the anti-bonding non-pairing electrons along the bond length. This may lead to important practical application in chemical reactivity driving by spin, in nano-structures’ composites, as well as in bio-materials, throughout controlling the binding and electronic bonding electrons by electronegativity. Overall, there is also proof that the electronegativity concept and its realization may serve as the main ingredient in bonding modeling, being related with the most intimate quantum-relativistic structures of matter [29]. These ideas are very fruitful and should be further unfolded and applied in forthcoming works aiming to model Chemical Bonding and Reactivity.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

Dirac, P.A.M. The Quantum theory of the electron. Proc. Roy. Soc. A 1928, 117, 610-624. Bethe, H.A.; Salpeter, E.E. Quantum Mechanics of One- and Two-Electron Atoms; Springer-Verlag, Berlin, 1957. Slater, J.C. Quantum Theory of Atomic Structure, Vol. 2; McGraw-Hill, New York, 1960. McWeeney, R.C.; Sutcliffe, B.T. Methods of Molecular Quantum Mechanics, Academic Press, New York, 1969. Wilson, S. Electron Correlation in Molecules; Clarendon Press, Oxford, 1984. Lindgren, I.; Morrison, J. Atomic Many-Body Theory; Springer-Verlag, Berlin, 1982. Malli, G.L. (Ed.) Relativistic Effects in Atoms, Molecules, and Solids; Plenum Press, New York, 1983. Boeyens, J.C.A. New Theories for Chemistry, Elsevier, Amsterdam, 2005. Hoffman, E. O. (Ed.) Progress in Quantum Chemistry Research, Nova Science Publisher, New York, 2007.

Fulfilling Dirac’s Promise on Quantum Chemical Bond [10] [11] [12] [13]

[14] [15]

[16]

[17] [18] [19] [20] [21] [22] [23] [24]

[25]

[26] [27] [28] [29]

19

Grant, I.P.; Quiney, H.M. Foundations of the relativistic theory of atomic and molecular structure. Adv. At. Mol. Phys. 1988, 23, 37-86. Pyykkö, P. Relativistic Effects in Structural Chemistry. Chem. Rev. 1988, 88, 563594. Ziegler, T.; Snijders, J.G.; Baerends, E.J. Relativistic effects on bonding. J. Chem. Phys. 1981, 74, 1271-1284. Schwarz, W.H.E. Fundamentals of relativistic effects in chemistry, in Theoretical Models of Chemical Bond. Part 2. The Concept of the Chemical Bond, Ed. Z.B. Maksić. Springer, Berlin, 1990, p. 593-643. Buenker, R.J.; Chandra, P. Application of configuration interaction for the study of relativistic effects in atoms and molecules. Pure Appl. Chem. 1988, 60, 167-173. Hess, B.A. Applicability of the non-pair equation with free-particle projection operators to atomic and molecular structure calculations. Phys. Rev. A 1985, 32, 756763. Buenker, R.J.; Chandra, P.; Hess, B.A. Matrix representation of the relativistic kinetic energy operator: two-component variational procedure for the treatment of many-electron atoms and molecules. Chem. Phys. 1984, 84, 1-9. Liu, W.; Dolg, M. Benchmark calculations for lanthanide atoms: Calibration of ab initio and density-functional methods. Phys. Rev. A 1998, 57, 1721-1728. Putz, M.V. (Ed.) Chemical Bond and Bonding, special issue of Int. J. Mol. Sci. 20072009, http://www.mdpi.com/journal/ijms/special_issues/bond_bonding. Putz, M.V. Chemical action and chemical bonding, J. Mol. Struct. THEOCHEM 2009, 900, 64-70. Putz, M.V. Levels of a unified theory of chemical interaction, Int. J. Chem. Model. 2009, 1, 141-147. Schweber, S.S. An Introduction to Relativistic Quantum Field Theory; Harper and Row, New York, 1964. Jauch, J.M.; Rohrlich, F. The Theory of Photons and Electrons, 2nd ed., Springer, New York, 1976. Harriman, J.E. Theoretical Foundations of Electronic Spin Resonance; Academic Press, New York, 1978. Putz, M.V. Contributions within Density Functional Theory with Applications in Chemical Reactivity Theory and Electronegativity, Disertation. Com, Parkland, Florida, 2003. Putz, M.V. Systematic Formulation for Electronegativity and Hardness and Their Atomic Scales within Density Functional Softness Theory, Int. J. Quantum Chem. 2006, 106, 361-386. Putz, M.V. Absolute and Chemical Electronegativity and Hardness, Nova Science Publisher, New York, 2008. Putz, M.V. (Ed.) Advances in Quantum Chemical Bonding Structures, Transworld Research Network, Kerala 2008. Putz, M.V. Density functionals of chemical bonding, Int. J. Mol. Sci. 2008, 9, 10501095. Putz, M.V. Can Quantum-Mechanical Description of Chemical Bond Be Considered Complete?, in Quantum Chemistry Research Trends, Mikas P. Kaisas (Ed.), Nova Science Publishers Inc., New York, (2007), pp.3-5.

In: Quantum Frontiers of Atoms and Molecules Editor: Mihai V. Putz, pp. 21-40

ISBN: 978-1-61668-158-6 © 2010 Nova Science Publishers, Inc.

Chapter 2

DUALITY WITHIN THE STRUCTURE OF COMPLEMENTARITY: RIGHT WHERE IT HAS NO PLACE TO BE Constantin Antonopoulos* Department of Applied Mathematics and Physics National Technological University of Athens, 28 Oktovriou (Patision) 42,10682 Athens, Greece

Abstract Bohr defines as complementary a pair of concepts “united in the classical mode of description”. But waves and particles are not united in this mode of description. They are separated in it. And from now on it only gets worse: Waves and particles are mutually exclusive even in Classical Mechanics. And Classical Mechanics does not contain the quantum. In other words, the incompatibility between waves and particles is self-sufficient and fact-independent and therefore such as cannot even be attributed to the quantum. But then again Complementarity (CTY) must at all costs be attributed to the quantum. In consequence, the sort of incompatibility afforded by Wave-Particle Duality (WPD) is the wrong sort for CTY, which is dependent on the quantum. Once this is realized, CTY is in search of a new foundation. This foundation is not really new at all and is offered in abundance in Bohr’s conception of the wholeness of the quantum of action. CTY is what happens to our structural conceptual scheme, when it is applied to the atom. Once CTY is properly derived in this way, Bohr’s ideas are further utilized to effect a reduction of WPD to the all pervasive, underlying Wholeness. It will then follow that WPD is a derivative instance in Bohr’s doctrine and not the primitive axiom that almost everyone assumes it to be.

1. Why a Reappraisal Is Necessary The closing decade of the twentieth century and perhaps the one immediately preceding it have witnessed a literal explosion of philosophers’ interest in Bohr’s quantum philosophy, *

E-mail address: [email protected]

22

Constantin Antonopoulos

considered now as a purely philosophical subject in its own right, and pursued after increasingly complex pathways of contemporary semantics and epistemology. Accounts exemplary in finesse and subtlety, by far transcending the clumsy, older presentations of semi-instructed physicists, appear all around us, capable for the first time of doing real justice to “the depth of Bohr’s thought” (Hooker states [p.174] that he has had no cause to change his earlier views on Bohr, as presented in his all-time classic monograph, see Hooker, 1972. I mention this because I will be quoting essential passages from this monograph and very little from this present work; see Hooker, 1994, 155-194). Yet despite the number, the extent, the competence and the sophistication of the works recently written on Bohr’s obscure genius and its products, the issue of the relation between Duality and Complementarity (CTY hereafter), if any, seems to be left untouched on the whole and pretty much where it stood back in the forties (with possibly one other exception; see D. Murdoch’s work of 1987). Since 1984 I have made it clear [in a paper first presented in front of a live audience, comprising the results of my doctoral thesis, in a conference held in Athens, titled Forms of Physical Determinism; the paper proposed a compact argument for deriving energy-time CTY on the basis of discontinuity alone, and “without the intervention of duality”, see Antonopoulos 1985, 95; See also E. Marquit’s reference to this work, Marquit 1988, 193] that I consider Wave-Particle Duality (WPD hereafter) an inelegant, facile, confused and, what is of essence, an incoherent method for deriving the relations of mutual exclusion demanded by CTY. It was therefore a matter of particular personal interest for me to observe, whether the above noted sophistication of recent Bohrian scholars has at last branded the continuing survival of Duality in Bohr’s system as the anachronism it is, or whether, competence, depth and subtlety notwithstanding, has otherwise resigned itself to a venerable tradition. The answer to this question was not particularly satisfying to me, so I presently intend to do something about it. The line of authors who have explicitly rejected Duality as a basis of CTY is sadly not a long one and the fingers of one’s one hand suffice to do the numbering. I have actually found two of them: Max Born and Adolf Grünbaum. Max Born makes it clear that, one: he fully endorses Bohr’s CTY; two: that he rejects the idea of waves altogether, claiming that “waves are waves of probability. They determine the supply of particles, that is to say, their distribution in space and time” [Born, 1951, 157; the author’s italics]. Hence, there are no real waves for the particles to be complementary to. Yet he otherwise speaks most approvingly of CTY [Born, 1969, 98]. To him the relationship of CTY must therefore pertain to and obtain between pairs of concepts other than the “wave” and the “particle”. Which are my sentiments exactly I nevertheless doubt whether the reasons for Born’s dismissal of Duality as a basis, or even an instance of CTY, share much in common with my own. In all probability this is but another case of two people having reasoned to the same conclusion from different premises. But with A. Grünbaum it is not so. Here a good deal is shared: <> [Grünbaum, 1957, 717; italics in the original].

Duality within the Structure of Complementarity…

23

This is much closer to the complementary scheme of things I have in mind. Yet the position I am about to defend is far stronger than Grünbaum’s. In fact, it is quite extreme: “waves” and “particles” are not just a left over from classical methods and habits, and hence totally unsuitable for a concept as unclassical as that of CTY, though left over and all the rest they certainly are. Waves and particles are a logical impossibility as a putative basis of CTY. It is not without interest to see how, if only by stark contrast to the ‘extreme’ thesis I am going to defend, WPD fares in recent, hence presumably well informed (?) literature: “Complementarity appears to have been introduced by Bohr in response to the waveparticle duality, which becomes the wave-particle complementarity in his framework” [Plotnitsky, 1994, 68]. To complete this untroubled, and slightly museum-piece picture of things with what this author is really aiming at, there is then introduced the irrationality lurking in the concept of WPD: The author (predictably) infers that “wave-particle complementarity is thus defined in anti-epistemological terms”. [ibid.] and is, to the same extent, the scientific counterpart of contemporary French obscurantism, involving Derrida, deconstructionism and a curtailed form of Hegelian dialectics, from which French philosophy finds it hard to at last emancipate itself, the author quoted very probably included. Having acknowledged subtlety and finesse in the texts of contemporary Bohrians does not necessarily include all Bohrians nor all texts. It does not include the passage just cited nor, for that matter, the one to now be, although its author, Henry Folse, is undoubtedly a very well informed and very attentive reader of Bohr’s text on nearly all other topics and occasions. But even to a scholar of so high a fidelity to the original, as Folse is, the idea that WPD may not be what is really behind CTY, has never occurred; or, perhaps because of it. For suggesting, as Grünbaum has, that CTY is something other than WPD, or that it is inconsistent with it, as I have, is not high fidelity. It is heresy. In consequence, Folse writes that: •

“by 1925, when matrix mechanics appeared, both wave and particle representations seemed necessary to describe the full range of phenomena in which atomic systems and radiation were to be observed” [Folse, 1985, 83; author’s italics]. or that • “by 1925 Bohr came to believe that the key to the new framework which would resolve the inconsistencies between the classical and the quantum ideas was to work with both particle and wave pictures” [86; author’s italics]. or that • “what was needed [for Bohr] was not a victory for the particle picture or the wave picture [for] by 1925 he was convinced that neither was to be discarded in favor of the other” [Ibid.; slightly rephrased]. Ergo, wave-particle CTY, of which there are plenty more samples in the relevant section of Folse’s (otherwise excellent) book, some preceding the ones I quoted, some succeeding them and some in between, convincingly backed up by bulky historical evidence too direct to question, such as remembrances of Bohr by Heisenberg at al., private correspondence, notes taken in a hurry, indeed everything it takes to make Grünbaum and myself (and perhaps Max Born a trifle) weep bitter tears at our heretic presumption.

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So I guess it is high time we got some of our own back and cause, if not bitter tears as such, then at least a modicum of worry to the crowded orthodoxy, relying on evidence drawn not from letters, nor from recollections, nor from ill remembered phrases and flashes of memory nor, finally, from Bohr’s still confused ideas of 1925, but from the first official statement of CTY as it is specified in the first published volume of Bohr’s work: <> [Bohr, 1934, 19]. Here then: since CTY is the combination of features united in the classical mode of description, if CTY obtains between waves and particles, waves and particles are united in the classical mode of description. This routine definition of CTY, known to all to be as commonplace as the ABC of the doctrine, should give Plotnitsky and Folse something to really think about; for it is not waves and particles which are united in the classical mode of description and separated in the quantum. If anything, waves and particles are (quasi)‘united’ in the quantum mode of description and separated in the classical [And even there the (quasi)‘unity’ is only pretentious; complementarists do say that they are properties of one and the same micro-entity but the unrelenting classical opposition between waves (large) and particles (small) not only is retained full force in QM. It is actually utilized to derive CTY]. In consequence, if there is one possibility which this important passage excludes, this is the possibility of wave-particle CTY. On the other hand, what this common, but at the same time unique, passage establishes as the proper pairs of complementary concepts are none other than those which Grünbaum has already specified above, viz. the two pairs of classical conjugate concepts, whose products (not unexpectedly) yield a unit of physical action: Et and pq. The very ones which also enter Heisenberg’s Uncertainty Relations (UR hereafter), in many ways Bohr’s CTY expressed quantitatively. For it is these, it goes without saying, which are united in the classical mode of description and separated in the quantum. Not waves and particles. In consequence, treating WPD as the basis, or an instance, of CTY creates a definite and insurmountable hermeneutic impossibility. Bohr’s own words directly forbid it. If they are adhered to, WPD is the last thing in the world that fits the description. Folse is too alert a reader of Bohr’s text (quite unlike Plotnitsky and his un-welcome improvisations) to have missed a point of such importance. But he receives it rather differently than I: “It is clear that in this original statement of Bohr’s new viewpoint, the complementary relationship holds between space-time coordination and the principle of causality, both of which were combined in the classical framework. Only later does Bohr speak of complementarity between wave picture and particle picture” [Folse, 1985, 114.]. If “later” in this connection is supposed to mean “in his mature years” it won’t do. Maturity to the extent of self-contradiction is no development or enrichment of the doctrine. Bohr may have included wave-particle CTY “later”, if Folse so wishes. But it is just as true, if not indeed much more so, that the CTY between the classical conjugate concepts he has never retracted or abandoned and only sheer folly would lead him to do a thing like that. CTY has the exact same age that Heisenberg’s UR have, and in Heisenberg’s UR the concepts which are mutually exclusive are still today the action products Et and pq. Hence, E with t and p with q are still as complementary as ever. Are also the wave and the particle, which were “later” added to complete the doctrine, of a like nature? If so, then on

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the basis of Bohr’s own definition, CTY should now obtain between concepts which are both: united and separated in the classical mode of description, since the classical concepts E,t and p,q correspond to the former account, the wave and the particle to the latter. In consequence, unless Folse is to explicitly reject classical concept CTY for the sake of wave-particle CTY, an idea whose pronouncement alone seems suicidally absurd, since the latter is said to support the former, we shall inevitably have to make room for both of them. In other words allow for CTIES between both, concepts united and concepts separated in the classical theory. Some definition of CTY this is. I don’t know whether to call it a contradiction, an ugly patchwork or just an empty formula which licenses everything. Let there be no mistake about that. Bohr’s own words, as explicitly voiced in the quoted, inconvenient passage, make wave-particle CTY exegetically impossible. And this, until now, only as far as exegesis goes. But there is more. For the exegetic impossibility implicitly contains a logical one, thus far concealed by the focus of emphasis. So let us reveal it, if only in moderation, for not all the story should be told at so early a stage. I have claimed that wave-particle CTY fails to satisfy the standards of Bohr’s definition, because waves and particles are not united, as the definition requires, they are separated in classical mechanics. Separated they certainly are. But then again, without any help from the quantum. For nothing of this sort exists in classical mechanics. Nevertheless, wave and particle are as incompatible as ever, even in classical mechanics. Waves (large) and particles (small) are self-sufficiently incompatible. So why need the quantum to separate them? The incompatibility between “large” and “small” is factindependent and therefore cannot even relate to the quantum. But then again, as I’m sure Folse would be the first to agree, CTY has to relate to the quantum. How then can it be based on WPD, which –by definition– can not? Complementarists, who make a habit of easy, sweeping solutions, will not find an easy way out of this one. Nor, strangely, will any body else.

2. The Relevance of the Classical Framework Probing deeper into the logical structure of Bohr’s relevant passage we can discern not only which actual concepts should qualify as members of CTY; we can, through Bohr’s thrifty but pregnant words, also discern why. Why, that is, Bohr thinks that such concepts can still complement one another in QM, despite their separation therein. They can, indeed they must, because they are united in the classical mode of description. Hence, the prototype sets as candidates for the relation of CTY such concepts only, which are firstly shown, united in the classical mode of description. (But waves and particles are not united in this mode of description. Still, people say that they are ‘complementary’.) The demand that concepts complementary in QM must first be shown to be united in Cl.M (classical mechanics) is not perchance a weak, dispensable analogy, to be dealt with or waived aside on the physical aphorism, that QM has already undone Cl.M, so what’s the point of calling them complementary in the first place. For the demand to treat them as complementary in QM, iff united in Cl.M, is not itself a physical argument at all. It is a transcendental one. The pressures to call them complementary are not factual. They are conceptual. The interference of the quantum may well undermine the physics of the situation, which is in any case contingent and therefore transitory. But the conceptual aspects are not as

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easily removable because of this and Bohr was the last person ready to remove them [Bohr emphatically denies that the classical concepts can be replaced by different concepts in QM; whence, of course, their resulting CTY. See for greater details my essay, Antonopoulos, 1987, passim]. To put the point more forcefully, Et and pq are not complementary in QM just because they are united in Cl.M. Far more than merely establishing the unity of Et and pq Cl.M. is itself established on the presupposition of such unity; for the unity spoken of antedates scientific systemization and is the rock bottom, transcendental foundation of its very possibility. Whence the Kantian dimension of the point [Kant’s Transcendental Aesthetic, treating space and time, and Transcendental Analytic, treating causality, combine to furnish the transcendental substratum, whose joint fulfillment alone will confer upon a provisional entity the status of an event. Compare this with Favreholdt’s description of Bohrian epistemology: “(All of this) fits well with Bohr’s insistence that space, time and causality are necessary forms of sensibility in human knowledge”, see Favreholdt, 1994, 81; for Bohr’s transcendentalism, see also Honner, 1994, 142]. This, then, is the reason why momentum and position must be considered complementary: Because they constitute equally necessary components of one and the same phenomenon of motion; as Folse elegantly puts it, “Motion is change of position in time” [Op.cit., 57], hence, the idea of motion without simultaneous reference to both of these constituents, p and q, borders on conceptual impossibility. Whence, of course, the very roots of the representational, intuitive, epistemological and even logical problems related with QM and CTY, which demand their separation. However, in spite of their physical separation in QM, position and momentum still remain inerasable conceptual correlatives, for their correlation was other than physical to begin with. Hence, when split apart by the quantum of action, they should still be considered complementary, because, even so, the two of them belong to one another, if motion of any kind is to ever result. And in belonging to one another, they therefore complement one another. (NB: Mutual complementation is also necessary to CTY!) Here now is why energy and time should be considered as complementary. They are because the following principle, elementary in its simplicity, still continues to obtain in spite of everything: P.1: An object can only occupy a particular energy state at some time, or over a time, if at all; which, in turn, has very little to do with QM or even Cl.M, for that matter. For it is itself but a special case of a more general and fundamental principle, which underlies it: P: An object occupies a state at some time, if at all (energy states included). I hold this to be an important conceptual truth for its alternatives lead to absurdity. To show this, I will restate P in the following, Kantian form; P': What happens, happens within time, therefore at a time (or over a time). That is to say, cannot happen outside time (or outside the flow of time). To then suppose it can, may be magic or miracle but it certainly isn’t physics. And, in any case, cannot be referring to what we normally call an event. Since, therefore, P, P.1 and P' leave little room for coherent alternatives, energy states as much as anything else must needs be associable with a time [Take, for instance, the following assertion by D.M.Mackay: “Energy must always be associated with a tract of time”, see

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Mackay, 1958, 112; italics the author’s; what is so special about this assertion is that it comes in the text just after the author has derived the energy-time uncertainty! Even that result is not powerful enough to erase the transcendental correlatedness of the two concepts]. And hence, even if energy and time are no longer directly associable at the physical level, due to the quantum, they nevertheless retain their formerly established conceptual link and so their very correlatedness. For the very arguments establishing such link are not themselves empirical and contingent. They are transcendental. And this kind of link lingers on even in the face of adverse empirical conditions, revealing its true origin. The time of its occurrence belongs to an event and, conversely, the very notion of timing something can in turn only refer to an event. Hence, the two complement one another and should be regarded as complementary in all the cases, where the physical conditions obtaining prevent their hitherto joint application; which brings us directly to Bohr’s second string of definitions: <> [Op. cit, 10]. In other words, such and only such pairs of concepts qualify as complementary, which, in the presence of any one member of the pair that is empirically realized, the absent member is still as necessary for a wholesome, cogent representation of reality, as the present member is; which is only sound, plain sense. One must never forget that, as can be verified from Bohr’s own words, mutual exclusion is only a necessary condition for CTY. Not a sufficient one. The other necessary condition trivially being, I would suppose, the capacity of the alternatingly excluded member of the pair to complement its complementary. The epistemological insanity lurking within QM is that it precisely seeks to drive a wedge and set apart such pairs of concepts, which make good sense only when taken together. This is the nature of the conceptual pressures exerted upon us by the fundamental structure of the classical framework, retrodictively demanding its toll even in the face of the quantum. Can I put the point in plain, human, unpretentious terms? A concept A is complementary with a concept B, iff it would have been preferable for human knowledge to have obtained A together with B, rather than without it. Normally, this is what CTY is all about. Now let us try and apply all these rather commonplace facts about CTY to the case of waves and particles and see what follows. Waves are large (extended) and particles are small (local). And assigning them both simultaneously to one and the same thing would result to a silly contradiction. Is it then preferable for human knowledge to be able to obtain the wave together with the particle, rather than without it? For this is what their alleged CTY would boil down to. What is the fundamental epistemological problem raised by QM and its complementary account? It is, for example, that it renders a classical position incompatible with a classical momentum, perennially frustrating Einstein, who has branded the theory incomplete because of this very reason [Einstein, Podolsky, Rosen, 1935, 777-780]. Was Einstein complaining about quantum incompleteness because QM forbids the co-existence (or the ‘co-verifiability’) of waves and particles? Was that what Einstein’s demand for completeness was about? Here are some further, impressive ‘similarities’(?) between classical concept CTY and wave-particle CTY, ‘similarities’ which, I’m sure, all concerned have noticed long before I have. To start with, energy and time or position and momentum are attributes which can be jointly assigned to a macro-system. Wave and particle are not attributes which can be jointly

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assigned to a macro-system, except at the pain of blatant contradiction. Who can deny such similarity? Accordingly, for energy and time or position and momentum the descriptive problem raised by CTY, or by QM as such, is that they cannot be observed together, when all concerned feel they should. So, one must conclude, the problem would be removed if they were observed together. But the descriptive problem raised by wave-particle CTY is not that these two cannot be observed together. They cannot but, so far, no one has called this a problem. The problems here would start, if wave and particle could be observed together, though, thankfully, they cannot. For some pairs of concepts, it would seem, their CTY is a deep frustration. But for some others, their CTY is pure salvation! Something here has gone terribly wrong, terribly not just due to the magnitude of the mistake but, indeed, due mainly to how easy it was to avoid it. In view of these remarks I trust no one will invoke experiments to save the day and argue that, since it is experiments which force upon us the “two natures”, we have no choice but to comply. At this point experiments are irrelevant nor have I built my case on them. Experiments can at best establish the existence of waves and particles. Not their complementarity. CTY is a conceptual doctrine and the citing of brute experimental facts, however persistent or inexplicable, is but a lame excuse for calling them complementary, thus making a virtue of necessity. The deplorable logic of “experiments are like that, so what we can do?” does not per se necessitate the application of CTY. How about doing nothing? In my view, to argue in this way is to just take hold of a problem and, through its sheer longevity and inexplicability, decide abruptly to turn it into its own solution, because it is a problem. Duality experiments do not establish the complementarity of waves and particles. They just establish their observation. And he is surely a poor reader of Bohr, who conflates these two. If they also establish a contradiction in the world at least behind the scenes [See also Folse, 1985, 83-4], then, at best, this is what CTY is supposed to handle. Not therefore what CTY is supposed to be. Indeed, for CTY to be able to handle the contradiction, it must itself be distinct from the contradiction. Otherwise, and were the contradiction also a ‘CTY’, the problems plaguing the contradiction would eo ipso become the problems now plaguing CTY; which is exactly what they have become, only spreading more and more butter on the bread of Bohr’s adversaries. And then, rather than solving the problem of WPD, CTY would itself become a part of this problem. That alone is excellent reason for wanting to separate between CTY and WPD. Hence, WPD experiments do not establish wave-particle CTY. Then what does? There are some who find it consoling to suppose that “wave” belongs to “particle” because, they say, some of the concepts which even I have admitted as authentically belonging to one another, viz. p with q and E with t, are themselves connected, some with the wave and some with the particle [This confused contention is based on a tragically erroneous reading of de Broglie’s p=h/λ, mistaking p, or mv, as the particle property, and so (correctly) taking λ as the remaining, i.e. the wave property; but p is no longer the particle property in this relation. p is the magnitude which is here determined by the wave, by simply assigning to λ a unique value; then the fraction h/λ can also receive a definite value; but a unique value to λ entails the use of the wave picture, if to be assigned, indeed the picture of a harmonic wave, which makes recourse to the ‘particle’ impossible a priori, at least if complementarists themselves are to be believed. So, if p is the particle property, p has to define in absence of the particle, whose property it is! For a lengthier demonstration of the point see

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Antonopoulos, 2004]. So if, they say, p belongs with q, so must “wave” belong with “particle”, which is respectively connected to them. I honestly hope that no serious arguer will attempt this circle against me. To assume that “waves” belong together with “particles” simply because p and q also do, is to take for granted the very thing I have already called into question. For I have been arguing that this is precisely what the relation between wave and particle does not do and cannot do; namely, come even close to comparing with the relation between p and q or that between E and t in the first place. To therefore suggest against me that the wave is complementary to the particle, because these two ‘correspond’ with p and q or E and t, which latter are complementary, is to freely assume that they do so correspond and this is the last thing I am presently inclined to allow. In a nut shell; wave and particle are contradictory concepts, exactly as contradictory as “yes” is with “no”. But energy with time and position with momentum are not contradictory concepts. So the whole nonsense of ‘associating’ waves and particles with energies and times or positions and momenta is as sound and rational a practice as urging us to treat as contradictory, concepts which we have ourselves admitted that are not contradictory. Other than that, Plotnitsky sees no problem in calling waves complementary with particles, and then proceeds full speed to the understanding of CTY in “anti-epistemological” terms. Were CTY as incoherent as all that, it would definitely be anti-epistemological and, besides, every single one of the things, which its opponents have claimed it, is for nearly seventy years now. Sadly, this practice is adopted also by Folse, who latter should at least know better [In the Editorial to the 1994 collection, see Folse, Faye, eds., 1994; Folse, if he is indeed the co-author of the piece (I suppose he must be), speaks as if it is still evident to him that CTY≈WPD. There is, however, a slight revision: “The connection between wave-particle dualism and the complementarity of the dynamic and kinematic properties remains a problematic issue for the analysis of Bohr’s philosophy and for the interpretation of QM”, see Editorial, 1994, xvi. I would say it does! But I think I may have an answer to it, see Section 5]. Well, if this is orthodoxy, I’d choose heresy any day. An exasperation shared by all heretics, I’m sure.

3. On the Two Kinds of Incompatibility If something is behind the mutual exclusion of the classical conjugate concepts, Et and pq, and this cannot be WPD, then what can it be? Bohr is pretty clear about what it is. All we need to do is listen: Complementarity is a term suited to embrace the features of individuality of quantum phenomena. [Bohr, 1958, 39] “The fundamental postulate of the indivisibility of the quantum of action [...] forces us to adopt a mode of description designated as complementary” [Bohr, 1934, 10]. Quantum theory is characterized by the emphasis on the feature of wholeness connected with the quantum of action. [Bohr, 1963, 53]

Bohr stresses here that what is behind CTY of the classical conjugate concepts is not some ‘duality’ or other but, indeed, and very directly so, the indivisibility of the elementary quantum; that is to say, its “wholeness”. Further exegetic efforts disclose remarkable persistence on this point:

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Constantin Antonopoulos The essence of quantum theory may be expressed in the quantum postulate, which attributes to any atomic process an essential discontinuity or, rather, individuality. This postulate implies a renunciation of the causal, space-time description of atomic phenomena [Bohr, 1934, 43].

Once again it is the individuality of the processes, or their discontinuity, which is behind the failure to coordinate the causal and the spatiotemporal description. CTY has appeared in these passages several times already and still the word “duality” is spectacularly absent from the text. And there is more. Bohr is well known to have attempted to generalize CTY over the (philosophical) domains of Action, Mind and Life, where he claims that CTIES comparable to those of QM can be detected. It is interesting to see why he thinks so in these other three domains, where quantum wholeness is not present: In general philosophical perspective [...] in other fields of knowledge, we are confronted with situations reminding us of the situation in quantum physics. Thus, the integrity of living organisms and the characteristics of conscious individuals present features of wholeness, the account of which implies a typically (!) complementary mode of description. [Bohr, 1963, 7]

So, after all, a comparable wholeness is present in these other three domains, analogously inviting application of the doctrine. Wholeness, it should be stressed, which implies a typically complementary mode of description. How much more typically holistic does this mode of description have to be, to at last detach itself from Duality? I think that after seventy five years of CTY complementarists must simply accept the facts as they are and stop foisting on Bohr the dualities of QM, or those of the Copenhagen Interpretation (a caricature of Bohr’s real doctrines, if there ever was one), or those of their own experiences with WPD, in case they are also physicists. The man speaks differently. The man himself never said “complementarity is a description based on duality”, as P.K.Feyerabend said in his name some fort five years ago [1958, 94] and as Plotnitsky or Folse plus very many others still say today. It is only natural to assume that, if Bohr thought CTY was based on Duality, this is what he would have said right from the start, just as Feyerabend has, rather than unendingly repeat that it is based on indivisibility and wholeness instead. But I still have an ace up my sleeve, to determine once and for all whether it is the indivisibility of the quantum that is behind CTY, or whether it is its ‘duality’ which really is. Either way, concerning the quantum of action one thing is certain. It is the quantum which is behind CTY and nothing but the quantum, independently of what other properties it is deemed to possess. I trust no complementarist will deny me that. This, besides, is what the quantum uncertainties, ΔEΔt, ΔpΔq≥h, also say. For h→0 both uncertainties would vanish. Suppose then that it is essentially WPD which is behind the mutual exclusion of the four conjugated parameters, i.e. the two pairs of classical concepts. However, the wave and the particle are mutually exclusive even in classical mechanics. And classical mechanics does not contain the quantum. How then can CTY, which according to all quoted Bohrian passages is dependent on the quantum, be the synonym of a pair of concepts, wave and particle, which are mutually exclusive without the presence of the quantum? For wave and particle exclude one another even in classical mechanics and therefore do so independently of the quantum. Some CTY this is! When I have emphatically announced in the opening section of this essay that it is logically impossible for WPD to be the basis of CTY I was not exaggerating. It is indeed no less than logically impossible to so be.

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The sole way that this inevitable conclusion can be avoided is to then argue that WPD is the quantum. Whereupon CTY, by being dependent on the quantum, would eo ipso be dependent on WPD. Accounts and interpretations of this sort are anything but hard to come by and, it should be mentioned, are generally considered as the ‘cultivated’ or the ‘sophisticated’ version of the ordinary and vulgar version of Duality. Needless to say, they are nothing of the kind. They are as incoherent and confused as any other, less sophisticated and far more vulgar presentation of wave-particle CTY available. Suppose we examine this contention closer: Originally, wave and particle are two things; in fact, two incompatible things; the quantum of action just one. How then are we to understand the claim that WPD and the quantum are one? Is this supposed to mean that the wave and the particle cease to be two incompatible things, becoming one, or that the quantum ceases to be one, becoming two incompatible things? Isn’t all this asking of us to swallow unresistingly more and more logical curiosities as we go along, rather than less and less, as it would be if the answers given us were only clear, convincing and reasonable? Identifications have their own strict rules. What we have here as candidates for identification are a pair of mutually exclusive concepts, on the one hand, and a (so far) unique concept, on the other, in whose frame the former two are alleged to be joined; but treating a pair of concepts as mutually exclusive presupposes treating them as not being able to be joined in the frame of a third concept, if it presupposes anything. Else there is no point in calling them mutually exclusive in the first place. In consequence, any identification of the sort presently proposed is impossible to implement without severe loss of identity of either of the participants. Either the two hitherto incompatible concepts will lose their identity, and cease being mutually exclusive (to no benefit of wave-particle CTY!), or else they will retain their identity, warranting mutual exclusion, and then, if WPD is the quantum, for the quantum to partake in the identity, the quantum will have to be mutually exclusive to ... itself. Which conclusion is more incoherent and absurd than even the vulgar versions of wave-particle CTY it seeks to replace. If complementarists are to be taken seriously they must give up such symptoms of exasperating double talk and start talking some sense. WPD cannot constitute the foundation of CTY because it yields the wrong sort of mutual exclusion. But then, if there is such a thing as the wrong sort, there must also be the right sort as well. In other words, there must be two kinds of mutual exclusion, that is to say, two kinds of incompatibility, only one of which is suitable for CTY. This brings us to the title heading of this section. Consider, then, the propositions “A>B” and “A=B”. These are certainly incompatible. Consider next the proposition “your money or your life!” as pronounced by the proverbial bandit. Are these two cases of incompatibility anything at all like one another? They are not. They are diametric opposites. The former, logical incompatibility is a consequence of definitions and is therefore necessary (as Leibniz would say). The latter, factual incompatibility is a mere consequence of obtaining facts and is therefore contingent (as Leibniz would say). Contingent, that is to say, upon a fact (e.g. the quantum) [A preliminary formulation of this distinction I have given in Antonopoulos, 1988, 323; Its complete and comprehensive formulation can be found in Antonopoulos, 1994, 187-9, its application to the EPR-Bohr debate in Antonopoulos, 1996, 1998 and its application to the problem of nonlocality in Antonopoulos, 1997a, 1999 and, partly, 2005]. This we may entitle “the prohibitive fact” [Antonopoulos, 1994, 188] (e.g. again the quantum). Hence, on the whole, there is fact-dependent and there is fact-

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independent incompatibility. And Wave-Particle Duality is of the latter sort. Not of the former. This, I believe, suffices to make all the mess perfectly clear as it suffices, I hope, to make clear, why it is a mess. WPD, in expressing a logical incompatibility, which its own defenders go out of their way in establishing (!), is therefore fact-independent and, as such, cannot even relate to the quantum. At least not coherently, though otherwise it certainly can. And is therefore unsuitable not just for CTY but for QM itself on the whole. Most people assume that WPD is the mystery of QM, the sole quantum mystery there is, the rest of them, if any, presumably reducible to its narrower or at worst its wider frame. Three physicists, for instance, one of whom, R.Feynman, is quite well known and popular, introduce the reader to their discussion of Duality (i.e. the double slit experiment) in the following way: “We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics” [Feynman, Leighton & Sands, 1965, 1-1; italics the authors’]. That it is a mystery, I can see as much as the next man. That it is a quantum mystery, I cannot. Since WPD incorporates a logical and therefore a fact-independent type of incompatibility, hence a type of incompatibility that is given a priori, I simply see no way in sight of how to even relate it with the quantum, which is a fact, and only given a posteriori; and so with QM.

4. Wholeness and the Proper Logic for Complementarity Having regarded the distinction between logical and factual incompatibility of paramount importance for the proper understanding of QM and CTY and, most definitely, of the true place of Duality in either of the two former, I have been systematically scanning the quantum literature for traces of its presence in other authors. The following two accounts, which really come close, are the best that I have been able to come up with: “Bohr believes that while it has seemed to us at the macro-level of classical physics that the conditions were in general satisfied for the joint applicability of all classical concepts, we have discovered this century that this is not accurate and that the conditions required for the applicability of some classical concepts are actually incompatible with those required for the applicability of other classical concepts. This is the burden of the doctrine (B4). [=Complementarity] This conclusion is necessitated by the discovery of the quantum of action and only because of its existence. It is not therefore a purely conceptual discovery that could have been made a priori through a more critical analysis of classical concepts. It is a discovery of the factual absence of the conditions required for the joint applicability of certain classical concepts” [Hooker, 1972, 137; italics the author’s]. The distinction between the two types of incompatibility previously spoken of is certainly presupposed in this passage. For where there is a factual absence (in italics) forbidding joint applicability, there must also be a logical one, to which the former is implicitly contrasted. But Hooker did not pursue the issue further. In fact, he gave it up altogether later on and, indeed, repudiated it! [Hooker wrote to me: «These remarks... lead me to be cautious about your insistence “that wave-particle duality must be abolished in Bohr’s philosophy as well as

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in quantum theory itself”», Hooker, letter dated 15 August 1983, upon receiving a draft of my doctoral thesis. Of course, my rejection was based on the distinction under consideration. In subsequent communication he even questions the overall validity of the distinction. In his letter of 18th December 1989 he tells me that regarding “formal” as opposed to “factual” aspects of the problem at hand “naturalists like me (him) cannot make a sharp distinction between the two kinds of truth”. Well, up there he has! Let’s only hope first thoughts are best thoughts]. The second passage faintly reflecting the same logical differences between CTY and Duality, which I have been insisting upon, belongs to G.Holton: “Position and momentum are not mutually exclusive notions since both are needed to specify the state of the system. But they are complementary in the restricted sense that they cannot both at the same time be ascertained with arbitrary precision [...] In contrast (!), the wave-particle aspects of matter are complementary and mutually exclusive” [Holton, 1970, 1050; italics the author’s]. Position and momentum are not mutually exclusive notions, that is to say, they are not really incompatible, and so it takes a fact to make them so – “in a restricted sense”. “By contrast”, wave and particle are complementary and mutually exclusive. That is to say, logically incompatible right from day one and, therefore, independently of the quantum (which detail escapes Holton’s attention). Hence, so far as the distinction between incompatibilities goes, this passage is practically identical with my point. But here all affinity stops. Holton sees no obstacle in speaking of restricted as opposed to unrestricted forms of CTY and accommodates them both under the same broad conceptual roof. But if it makes no difference, whether two concepts are restrictedly or unrestrictedly incompatible for being commonly called complementary, it makes no difference either, whether two concepts are restrictedly or unrestrictedly incompatible in the first place. So the difference is denied right within the very reasoning which detects it. This is why, besides, this potentially shrewd remark remains entirely idle in the work referred to [Antonopoulos, 1988]. Having designed the appropriate logical instrument for CTY, the sole available for permitting it to relate to the quantum, I shall now proceed to its application to indicate at least structurally what I take the logic of CTY to be. However, I will do so in purely abstract and formal terms, perhaps too abstract and too formal at that, but such as can surely make CTY accessible to any fellow philosophers, namely, the people I am presently addressing. Yet staying close to at least the spirit of Bohr’s words and that of QT I will still make indivisibility the centre of my arguments. Abstraction of the sort I have in mind will involve a slight modification of terminology. In physical terms I have spoken of CTY as presupposing a fact-dependent type of incompatibility. Since I will now discuss purely abstract (numerical) entities I will add the following, already implied provision. Fact-dependent incompatibility is conditional incompatibility. Conditional, that is, on the very fact it is dependent upon. Since all fact-dependent incompatibility is eo ipso conditional, the modification is minimal and valid. Consider therefore a simple equation of the form A+B=x, where all three variables, A B and x, are to be receiving values exclusively from the field of natural numbers, i.e. positive integers without zero. (Some say because zero is not a number. I would say because zero is not a positive number.) Then for every value ascription to x such that x>1, it is always possible for both the other two variables of the equation, A and B, to receive equally precise and definite values themselves. E.g. for the ascription x=7, one can correlatively assign A=2 and B=5 in satisfaction of the equation, or any other mutually constrained values he chooses,

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values the availability of which increases in direct proportion to the value assigned to x; and decreases in comparable manner. So then consider what happens for the limiting ascription, x=1. Now it will no longer be possible for the other two variables, A and B, to both receive a precise and definite value. There is now but a single value available, 1, and this latter cannot be distributed among the two variables, A and B. It will therefore be assigned as a whole either to A or to B but not to both. (Observe how naturally talk of wholes enters this description.) If it is assigned as a whole to A, and this is the only way it can be assigned, then we get: A=1, B=indefinite –since zero is not a possible natural ascription. And if it is assigned as a whole to B, and this is the only way it can be assigned, we get: B=1, A=indefinite –for the same reason. Due to the indivisibility of 1 in this particular number field, the two variables simultaneously referred to it are thereby driven to mutual exclusion. This, I submit is (the logic of) Complementarity. The incompatibility of simultaneous and definite value assignments for A and B is conditional on the value assignment to x. For any value ascription x>1 A and B are compatible. For x=1 they are not. We have not assumed an incompatibility between the variables A and B themselves. Their incompatibility is extrinsic to their nature, not intrinsic to it as is the case with Duality. For any value assignment x>1 it disappears as much as it will, if we change the entire number field from that of the naturals to that of the fractional numbers, now permitting simultaneous value assignments even in the case that x=1.(This would correspond to a return to the classical ontology.) Mutual exclusion is in this case ascription-dependent, fully corresponding to the factdependent incompatibility required by the indivisibility of the elementary quantum, of which 1 as a natural, is presently the formal representation; that is to say, ascription-dependent and not a priori. And, it goes without saying, it is this very possibility which allows us to link the resulting variable incompatibility with the indivisibility of the elementary entity, to which they are simultaneously referred and applied. The whole logic of the scheme is: Two into one won’t go. (This is a logical truth.) Incidentally, and since there has been a fleeting reference to Einstein’s demand for quantum completeness, which is something essential to the cogency of CTY, the structure of the foregoing model makes it perfectly clear that the disjunctive (and definite) value ascriptions available are the best possible ones allowed by the nature of the mathematical phenomenon at hand and, respectively, by the nature of the comparably indivisible quantum. The fact that either value ascription is available, prior to actually ascribing any, will not go to show that they are both assignable simultaneously.

5. The Reduction of Duality The structure of the numerical model on the basis of which I have designed the logic appropriate for CTY, not only for staying true to the actual words of Bohr, naming indivisibility as the raison d’être of CTY (an exegetic requirement), but also for warranting the possibility of even relating CTY with indivisibility, (a logical requirement), is a structure utterly opposed to anything we can extract from WPD. A priori incompatibility cannot relate to the world, and therefore cannot relate to whatever exists in the world, which is predicated as indivisible; in other words, the quantum. This alone is reason enough to dismiss WPD as a possible basis for CTY or a possible basis for anything. Having demonstrated this

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unsuitability and having set the record straight on the basis of Bohr’s so poorly understood doctrines, I would consider my obligations fulfilled and my investigation concluded. So far as I’m concerned they are. But so far as Bohr is they aren’t. For Bohr has an explanation for WPD, deriving it as the consequence of deeper laying realities, that is to say, of Indivisibility. This, come to think of it, is anything but surprising and certainly anything but incongruous. So I will stretch the limits of this enquiry somewhat further as an ultimate test of the unity and the consistency of his thought. The aim of this last section is therefore to indicate the relevance of Indivisibility (or Wholeness) to the emergence of WPD. I will commence with my own version and proceed with Bohr’s. Here then, in a nutshell, is my version: • •

AXIOM: If an indivisible thing is successfully predicated, the predication will embrace the entire thing. (This is an analytic truth.) THEOREM: An indivisible thing can only manifest undivided aspects of itself.

It would then follow that alternative predications of an indivisible thing, in being impossible to contain in their rightful place each, will embrace the indivisible thing in its entirety, its “wholeness”, and hence conflict with one another in ways which would never obtain, if this thing was subdivisible and therefore such as to allow its predications to be kept separate, and thus distinct and combinable. Wholeness will not permit its partial predications, which for an ordinary object could certainly be distinct and therefore consistently attributable. Hence, any pair of alternative, successful predications, A and B, of an indivisible entity will successively apply to its entirety, now representing the entity as all-A, and therefore as no-B, or as all-B, and therefore as no-A, which two, given the appropriate context, will most certainly conflict. The appropriate context is, in this case, the Part-Whole relation, which is, essentially, what the Axiom and Theorem, above formulated, are all about. If no parts of an indivisible thing can ever be obtained, the “would-have-been” part (for any other, subdivisible entity) will now emerge as the whole thing instead. And then the two (holistic) predications will have to conflict, for the “would-have-been” part, peacefully coexistent with another of its kind, will now extend to the entire thing, blocking the other part out; and conversely. The question may arise, whether it is at all legitimate to introduce parts in the case of indivisible things, even if permanently behind the scenes, and derive the conflict on their implicit presence, which is precisely what I’ve done. Indivisible things should have no parts, one might point out. This is not so. It is true that something simple has to be indivisible, e.g. the unit of my complementary model. But the converse is not likewise true; something indivisible does not have to be simple. It may be possessed of parts, the difference being that they’d have to be inseparable parts. Nor are we in want of concrete examples. The good old proposition is such an indivisible, though structural whole. The properties of truth and falsehood emerge with the entire proposition as such. They do not accompany any of the propositional terms, when separately regarded, not even in analogy. The proposition is a whole which, as metaphysical tradition has so correctly grasped, is greater than the sum of its parts. So also in my argument each resulting ‘part’ is greater than itself, becoming the whole. If that is denied, no predication of an indivisible entity will ever be successful and then we’d simply have no problem in the first place.

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A quick regress to several of the Bohrian passages I have quoted, and to many which I have not, directly reveal that in his general epistemology Bohr conceives of classical mechanics as the methodology of subdividing, analyzing and “dissecting” its objects. Of course, if the objects turn out to be indivisible, rather than being themselves dissected, they will dissect us, that is to say, the structural conceptual scheme which we habitually apply to our objects. And will divide it in two, rendering it complementary. (My rudimentary ‘atomic’ model shows exactly how this is brought about.) Were the question to ever arise, and it has, whether Bohr adopts any Kantian-like category of the understanding as an ‘a priori’ formula for future experience, that would surely have to be the Category of Divisibility. We can only make sense of divisible, analyzable, dissectible phenomena. We are habitual ‘analyzers’ [This is, incidentally, Henrgy Bergson’s view of the understanding as well, incapable of grasping the duration, i.e. the wholeness of organisms and our own uninterrupted flow of consciousness. I have already quoted one Bohrian passage to this end but there are countless to choose from, see Antonopoulos, 1997c, where all are gathered]. Hence, when confronted with indivisible phenomena we cannot but persist in our “customary ways” (a favorite Bohrian motto) of analysis and subdivision, ways that now are bound to be frustrated. One consequence of this practice is, of course, CTY. Our structural, i.e. composite conceptual scheme will not be accommodated by a structureless, simple reality, symbolized by “1” in my model. The scheme can only apply to it complementarily (Two into one won’t go); the other consequence of this practice is what else? the wave - particle duality. Indivisible entities, when differently dissected will either retreat and become inaccessible or else emerge by manifesting undivided aspects of themselves, which, given the conditions of dissection currently obtaining, namely, our specific macro-inspired experiments, are the sole permitted to be realized in the setting. To overcome dissection and stay true to their nature atomic phenomena must make up for the severed part of their identity by ‘replacing’ it, as it were, by whatever other properties the dissecting environment allows them to manifest. So in a dissecting A-environment they will emerge as if all-A and in a B-dissecting environment as if all-B, which are evidently inconsistent. Depending on the dissection –or predication– chosen, the indivisible thing will respond accordingly, if it will respond at all. And will emerge transformed, manifesting properties which it can be independently shown to not autonomously possess. The contradictions resulting from atomic phenomena, when too closely looked at (dissected) are not original. They are nothing but the combined result of their indivisibility and our own attempts at subdividing them. The contradiction is not inherent in the world itself. The contradiction is of our own making, when due to our persistent macro-habits we do to an object something contrary to its own nature. This is what Wave-Particle Duality is all about for Bohr, if it is anything, and, it should be stressed by the by, there is really nothing subjectivist in this situation. The thing is indivisible, so it behaves indivisibly, so it reacts indivisibly, so it interacts indivisibly, so it is observed indivisibly. When others see “the human subject” in all this, I see nothing but an entity always staying true to its independently asserted nature (So none of the usual, subjectivist fairy tales please! – See also the next Chapter). Bohr’s way of reducing WPD to the underlying quantum wholeness is similar to mine and is distinguished from it only in the respect of introducing indivisible interactions, when I have found it more convenient to speak of directly indivisible entities instead. I cannot say whether this deviation contains elements of substance in it or is just an idiosyncratic peculiarity. Be that as it may, Bohr proceeds in his reduction by the following steps:

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“The finite magnitude of the quantum of action prevents altogether a sharp distinction between the phenomenon and the agency by which it is observed” [Bohr, 1934, 11]. Object-device inseparability is, as is well known, the opening stage of Bohrian wholeness, directly connected to the indivisibility of the quantum itself. [Also Hooker, 1972, 195] Once this is laid down, the next step consists in dragging to surface the potential consequences that such inseparability is expected to produce, when we are warned of “the impossibility of any sharp separation between the behavior of atomic objects and the interaction with measuring instruments” [Bohr, 1958, 39]. From this premise Bohr proceeds to obtain immediately below that “under these circumstances an essential element of ambiguity is involved in ascribing conventional physical attributes to atomic objects, as is at once evident in the dilemma regarding the corpuscular and wave properties of electrons and photons, where we have to do with contrasting pictures, each referring to an essential aspect of empirical evidence” [ibid.]. All three passages now jointly entail that such contrasting pictures emerge during experiments because of the impossibility of separating between atomic object and measuring instrument for the duration of the experiment. That is to say, Duality exists because Wholeness exists first. Any denial of the proposed connection between the two, intending to posit WPD as fundamental and irreducible in Bohr’s system of interdependent axioms, must now take its case with these revealing and restrictive passages. And also take its case with those that follow: <> [Ibid.]. The whole conception from beginning to end is epitomized in the following, comprehensive way: (a) Phenomena like individual, atomic processes, due to their very nature, are essentially determined by the interaction, between the objects in question and the measuring instruments. [Hence] (b) No result of an experiment concerning [such] a phenomenon can be interpreted as giving information about independent properties of the objects but is inherently connected with a definite situation in the description of which the measuring instruments interacting with the objects also enter essentially. [So] (c) This fact gives the straightforward (!) explanation of the apparent contradictions which appear when results about atomic objects are tentatively combined into a selfcontained picture of the object [Bohr, 1958, 25-6]. What Bohr is saying, therefore, though it is clear enough to be plain to all, is roughly this: If during an indivisible interaction atomic object and measuring instrument form an inseparable whole, in the sense of becoming indistinguishable, then they are one thing. But if object and instrument become one thing in a given measurement and another thing in a different measurement, then one and the same thing will appear as different to itself as the two measurements are to one another. This corresponds with my own alternative predications, extending to the entirety of the atomic object whereupon, by being uneven to one another,

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will compel the indivisible object, capable only of displaying undivided aspects of itself, to appear uneven to itself. Bohr implies the exact same thing: “In the study of atomic phenomena we are presented with the situation where the repetition of an experiment with the same arrangement may lead to different recordings, and experiments with different arrangements may give results which at first sight seem contradictory” [Bohr, 1958, 18]. Hence, in view of all the preceding, namely, my analysis and that of Bohr’s, it is simply impossible for Duality to be anything else save a created situation. It nowhere follows from any of the previous considerations that Duality is a foundation of the doctrine. In this vein I would also call the reader’s attention to the following point. The reduction of WPD to indivisibility was here not the inductive procedure which many suppose it to be, beginning from collected and contradictory experimental results, and inferring therefrom that the devices of observation must have something to do with the contradictions. This is the eternal (and pernicious) route which takes us from Duality to ‘wholeness’ between observational results and the devices by means of which they were acquired; which reasoning essentially continues to posit WPD as the sole physical basis behind quantum effects. The reduction of Duality to wholeness was here furnished as a deductive process, commencing from wholeness and advancing to the emergence of the contradictory predications of it, where wholeness itself was taken as an independently ascertainable property of the indivisible object, entailing the ‘contradictions’ when treated in disregard of its true nature. Whether or not we will obtain inconsistent predications thereafter depends solely on the Theorem, that an atomic thing will only manifest undivided aspects of itself. Psychologically, we had to see it first to become aware of it. But the logical order is strictly one way in either derivation, mine or Bohr’s. Wholeness precedes and Duality follows, which is what a proper reduction consists of. This, I submit, is all there is behind Bohr’s account of wave-particle duality and this is its true status in his doctrines; namely, but a second order consequence of a presupposed wholeness and of our own attempts at subdividing it. It is therefore a derivative notion and not a primitive axiom at all, as it has been so frequently asserted all the way to the very present. It is, so to speak, a side product of indivisibility, not an ontological property inherently subsisting in the things themselves, and as a side product it should be regarded both in Bohr’s philosophy and QM as such. In this light Duality is divested of any special privilege in delivering all the other, subordinate propositions of the doctrine, least of all of Complementarity. There is but one such protagonist in Bohr’s doctrines and its name is not “wave-particle duality” at all. It has a far simpler, one-word name: Wholeness.

References [1] Antonopoulos, C. “Discontinuous Alterations of State and the Question of Determinism”. Determinism in Physics, Gutenberg, Athens 1985. [2] Antonopoulos, C. “The Uncertainty Relations of Energy and Time and the Conflict between Discontinuity and Duality”. Microphysical Reality and Quantum Formalism, Kluwer, Dordrecht 1988, vol.2.

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[3] Antonopoulos, C. “Innate Ideas, Categories and Objectivity”. Philosophia Naturalis, 26, 2, 1989. [4] Antonopoulos, C. “Indivisibility and Duality; A Contrast”. Physics Essays, 7, 2, 1994. [5] Antonopoulos, C. “Bohr on Nonlocality; The Facts and the Fiction”. Philosophia Naturalis, 33, 2, 1996. [6] Antonopoulos, C. “A Schism in Quantum Physics or How Locality Can Be Salvaged”. Philosophia Naturalis, 34, 1, 1997. [7] Antonopoulos, C. “Time as Non-Observational Knowledge; How to Straighten Out ΔEΔt≥h”. International Studies in the Philosophy of Science, 11, 2, 1997. [8] Antonopoulos, C. “(In Refutation of ) Complementary Conceptual Schemes; The Objective Metaphysics of Complementarity”. Idealistic Studies, 27, 2, 1997. [9] Antonopoulos, C. “The Remaining Alternative of Bell’s Theorem”. Physics Essays, 12, 1, 1999. [10] Antonopoulos, C. “Reciprocity, Complementarity and Minimal Action”. Annales de la Fondation Louis de Broglie, 29, 3, 2004. [11] Antonopoulos C. “Investigating Incompatibility: How to Reconcile Complementarity with EPR”. Annales de la Fondation Louis de Broglie, 30, 1, 2005. [12] Bohr, N. Atomic Theory and the Description of Nature. Cambridge University Press, 1934. [13] Bohr, N. Atomic Physics and Human Knowledge. Wiley & Sons, New York, 1958. [14] Bohr, N. Essays 1958-62 on Atomic Physics and Human Knowledge. R.Clay & Co, Suffolk 1963. [15] Born, M. The Restless Universe. Dover Publications, New York 1951. [16] Born, M. Physics in My Generation. Springer&Verlag, New York 1969. [17] Einstein, A., Podoslky, B., Rosen, N. “Can Quantum Mechanical Description of Physical Reality Be Consired Complete?”. Physical Review, 45, May 1935. [18] Favreholdt, D. “Niels Bohr and Realism”. Niels Bohr and Contemporary Philosophy, Faye, J. & Folse, H. eds., Kluwer, Dordrecht 1994. [19] Feynman, R., Leighton, R., Sands, M. The Feynman Lectures on Physics. AddisonWesley, California 1965. [20] Feyerabend, P.K. “Complementarity”. Proceedings of the Aristotelian Society, supplementary vol. xxxii, 1958. [21] Folse, H. The Philosophy of Niels Bohr. North Holland, Amsterdam 1985. [22] Folse, H. “Bohr’s Framework of Complementarity and the Realism Debate”. Niels Bohr and Contemporary Philosophy, Faye, J., Folse, H., eds. Kluwer, Dordrecht, 1994. [23] Grünbaum, A. “Complementarity in Quantum Physics and its Philosophical Generalization”. The Journal of Philosophy, lvi 23, 1957. [24] Holton, G. “The Roots of Complementarity”. Daedalus, 99, 1970. [25] Honner, J. “Description and Deconstruction: Niels Bohr and Modern Philosophy”. Niels Bohr and Contemporary Philosophy, 1994. [26] Hooker, C.A. “The Nature of Quantum Mechanical Reality”. Paradigms and Paradoxes, University of Pittsburgh Press, 1972. [27] Hooker, C.A. “Bohr and the Crisis of Empirical Intelligibility: An Essay on the Depth of Bohr’s Thought and Our Philosophical Ignorance”. Niels Bohr and Contemporary Philosophy, 1994.

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[28] Mackay, D.M. “Complementarity”. Proceedings of the Aristotelian Society, supplementary vol., xxxii, 1958. [29] Marquit, E. “Determinism in Physics, 1985: Book Review”. Foundations of Physics Letters, 1, 2, 1988. [30] Murdoch, D. Niels Bohr’s Philosophy of Physics. Cambridge University Press, 1987. [31] Plotnitsky, A. Complementarity. London 1994.

In: Quantum Frontiers of Atoms and Molecules Editor: Mihai V. Putz, pp. 41-60

ISBN: 978-1-61668-158-6 © 2010 Nova Science Publishers, Inc.

Chapter 3

COMPLEMENTARITY OUT OF CONTEXT: ESSAY ON THE RATIONALITY OF BOHR’S THOUGHT Constantin Antonopoulos* Department of Applied Mathematics and Physics National Technological University of Athens, 28 Oktovriou (Patision) 42,10682 Athens, Greece

Abstract It has been suggested by several people that Bohr’s Complementarity (CTY) is a form of context-dependence. In defense of the realism and objectivism of Bohr’s doctrine, I provide the following refutations: A. Position and momentum are not the same word in different contexts. They are different words in the same context, = a single instance of motion. Hence, are complementary in a single context rather than so across two. This refutes a contextualist reading of CTY. B. CTY does not resemble Kuhnian paradigms, unless it is identified with Duality. But the incompatibility between waves and particles is self-sufficient, hence such that makes the quantum redundant. Hence CTY, which is dependent on the quantum, is unlike Duality; hence unlike paradigms. C. The foundation of CTY is shown to be Indivisibility, that is to say, Atomism. An atomic singularity, thus conceived, splits in two and renders complementary a pair of classical concepts, E with t and p with q, on the basis of the analytic truth that two into one won’t go, except of course complementarily. For this kind of singularity Bohr is shown to speak as a realist. He says it “—in principle—resists an analysis”, implying that we try and fail. The failure however is due to the thing, not to the analysis. Bohr is therefore a metaphysical realist and, as an atomist, a metaphysical realist twice over.

1. Complementarity Contextualized The view that Bohr is a nonrealist, due mainly to the ideas of older physicists inadequately instructed in matters philosophical, has recently dropped out of fashion. It was a deserving fate but there are still some pockets of fierce resistance left in this country [see *

E-mail address: [email protected]

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Note 1] and abroad. The roles have slightly changed, however. Bohr is no longer labeled as an out and out “idealist”. He is now depicted as a contextualist, hence still a nonrealist. One version of the point is delivered by his grandson, Christian: In order to prepare for my comparison between Complementarism and Anti-Realism let me attempt an Anti-Realist’s approach to these problems: According to Post-Tractatus Wittgenstein, problems (philosophical problems, it is) in principle arise because of the rules of language usage being violated. In the case of man’s free will the problem is that it— dependent (roughly) on whether we look into our selves or study other beings—seems consistent with experience to assume both that we do have, and that we don’t have a free will; with the two contentions being of course mutually inconsistent.— And the cure for this apparent inconsistency is the understanding that we though using the same word, scil. will, still are not using it with the same meaning, because we have changed the context. [...] Now let’s see how the same problems are attacked with Bohr’s complementarity analysis. A case for complementarity arises, if I understand Bohr correctly, when our use of two (or more) words, vocables, terms, etc., involves us in inconsistencies and ambiguities, because these words, though intimately related in ordinary usage, as a consequence of the different conditions for their use actually belong in different contexts. [Chr. Bohr, correspondence, received 18 April 97. All italics mine. Other than that, the author’s text is quoted unchanged.] This, therefore, is an identification of CTY with context-dependence spreading, as it seems, throughout the entire conceptual matrix. But it is anything but a solitary proposal; not by any means. For that matter, it stands in the company of a good deal of influential names in the issue at hand, past and present, such as those of N.R. Hanson, P.K. Feyerabend, Th.Kuhn (as interpreted by others, however; cf. C.Glymour), Gonzalo Munevar (editor of the Kluwer volume dedicated to Feyerabend [see Note 2]), and last (in my list, at least) C.Glymour himself [see Note 3]. Nor would I ever venture to launch an attack against “contextual CTY” drawing merely from the limited confines of my personal correspondence with Chr. Bohr, however close to the man himself he might otherwise have been, without cross referring it with available literature. From the list of complementarian contextualists above offered I would only wish to exempt a single name as a potential target of my criticisms that of N.R.Hanson. I have always interpreted Hanson far more realistically than most and probably far more than Hanson himself intended. I have, in fact, received Hanson’s transitory—and expendable— quantum quasi-contextualism as but a more elaborate, semantic version of underlying realities, which to me have always signaled a deep going macro-micro domain dualism; this is a view which I myself espouse and which Bohr also had. Hanson’s Patterns of Discovery may abound with (seemingly) contextualist talk, but one that is not particularly difficult to de-contextualize: “Classical mechanics as a whole is said to be but a special case of quantum mechanics. They are apparently clusters of statements in the same language. Statements and languages however do not work in this way [Hanson, 1961, 151]. Finite and transfinite arithmetic, Euclidean and non-Euclidean geometries, the language of time and the language of space, the language of mind and the language of body, these show themselves to be different and discontinuous on just this principle: What can be meaningfully said in one case may express nothing intelligible in the other” [Op.cit., 152]. I myself entertain not the slightest doubt, as neither does he, that the rift between quantum and macro phenomena is only evaded by the mathematical scheme, in other words by the quantitative cover up, convenienced by the numerical convergence of classical and

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quantum laws in the case of large numbers, a technique, on the whole, especially designed to patch up the irreducible qualitative discrepancies between the two classes of events. But my own reaction is, in due accordance, that the asymptotic nature of the two languages results from the nature of the phenomena themselves and is to that extent only derivative. It is the phenomena that manifest the primordial discrepancy; not the languages, which but follow. The fact that physicists are professionally inclined to nourish dreams of integration and unification of the micro with the macro within some grand comprehensive scheme or other is no evidence of the essential, besides the numerical, continuity between the two realms. Hooker describes this (classical) presupposition, coming natural to Einsteinians, with the following words: “All complex objects consist of definite structures of the fundamental elements which are their constituents” [Hooker, 1972, 70.]. Where, Hooker notes, it is the word “definite” which carries the primary ontological weight. But this untroubled assumption can be shown to falter even within a mathematical frame, provided it is suitably devised. Here, then, is the frame in question, manifesting –by analogy–within the confines of a single rudimentary model all the requisite properties, starting from atomicity upwards and all the way to CTY, the quantum uncertainties (UR hereafter), Bohrian holism as such and the micro-macro dualism previously referred to plus, as a bonus, the completeness of the quantum mechanical description of micro phenomena, a consequence of course of the repeatedly noted micro-macro dualism. Consider an elementary equation of the form A+B=x, where all variables, A, B and x are to receive values exclusively from the field of naturals (for obvious reasons), that is to say, of positive integers, not including zero. (Some say because zero is not a number. I would say because zero is not a positive number.) Once the previous condition is posited, then what follows is that for any value ascription to x, ranging from 2 to infinity, we shall at all times be capable of assigning definite and simultaneous bound values to variables A and B. But for the unique value ascription, x=1, this is no longer possible. For x=1 and therefore for A+B=1, we can either ascribe the entire, non-distributive value, 1, to A, whereupon A=1 and B=indeterminate or, conversely, ascribe it to B, whereupon, B=1 and A=indeterminate. (Zero is not a permissible value ascription.) This, I will argue in the appropriate places, is Bohr’s doctrine of wholeness. In my numerical model the wholeness is that of 1 as the smallest natural in existence; the atom at the basis of the natural sequence. This wholeness takes us almost immediately to the “mutual exclusion” imported within the components of the schema |A+B|, because the schema itself is structural, whereas 1 in this field is simple and devoid of structure and cannot be divided in consistency with the premises of the argument. We shall later on see Bohr referring to phenomena comparable to 1 as a natural, when aiming at defending the completeness of the complementary description, by stressing that they “in principle resist such an analysis”; which is a realist, and a dualist, thesis, for if atomic phenomena resist analysis, as macro objects do not, both classes of phenomena turn out to behave exactly as independently predicated; which is much closer to hard core realism, than to any other option, because reality behaves as independently predicated. But the rift between macro and micro, in the frame of the model the dualism between x=1 or x>1, is none the less irreducible, and in this case even mathematically irreducible, though to no noticeable hindrance to a realist account of the situation. “If S can be used to express an intelligible statement in one context”, says Hanson, “but not in another, it would be natural to conclude that the languages involved in these different contexts were different and

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discontinuous” [Hanson, 1961, 151]. What is unnatural, however, is to conclude that therefore the linguistic phenomenon is thereby elemental and primitive, with no traceable ontological roots to anchor it upon. Mutual exclusion of value ascriptions in this model results from alternating conceptions of reality: x>1 or x= 1, the “macro” (=sub-divisible) and the atomic, respectively. And atomicity is a predicate of reality, not of complementary “languages”. Bohr is pretty clear about this: The fundamental postulate of the indivisibility of the quantum of action forces us to adopt a mode of description designated as complementary in the sense that any given application of classical concepts precludes the simultaneous use of other classical concepts which in a different connection are equally necessary for the elucidation of the phenomena [Bohr, 1934, 10]. And, I take it, the quantum of action is a property of the world. Not the property of a language, whatever that might come to mean in this weird connection and however such a language might have then been adopted. Bohr himself has no particular qualms, concerning how this language is adopted. It is forced upon us by the restrictions of reality, that is to say, indivisibility, affecting the current operation of our descriptions. My own numerical model of CTY reflects just this sort of dependence. The mutual exclusion between the related variables, A and B, in the context of my model results from the predication of atomicity to 1 and this predication is independent of the schema |A+B| per se, which latter is the “language”. There is nowhere discernible within the confines of the schema, that it must needs receive values from the field of naturals and at exclusion of other possibilities. It could just as well receive values from the fractionals, depending on what universe of discourse we are after representing, and then no mutual exclusion of simultaneous and definite value ascriptions would ever come up. The provision that the values be drawn from the naturals is extrinsic to the schema. And is therefore the foundation for its subsequent complementary reaction, when the limiting case x=1 commands it. “...=1” is not a “vocable” in any coherent sense here, as Bohr the grandson has suggested to me. [C. Bohr, April 97.]. It is the schema ⁄A+B ⁄ that is if anything has to be. And “...=1” is what “forces us”, to use Bohr’s own words, to adopt a mode of account designated as complementary. A and B in themselves considered are nothing of the sort. Just try x=2, 3, 4 or any other natural up to infinity. A and B will never enter a relation of mutual exclusion, because A and B are not mutually exclusive to one another and independently of selected assignments. They are rendered so for the specific –and unique- ascription x=1. But otherwise are cosatisfiable. |A+B| can handle all the cases except the simple. The quantum is not just a miniature of the macro. To this extent, therefore, Hanson is right: “How can intelligible empirical assertions become unintelligible just because quantum numbers get smaller? Conversely, how can unintelligible clusters of symbols become meaningful just because quantum numbers get larger?” [Op.cit. 152]. Well, how can A+B>1 yield coherent answers but A+B=1 not? They do because they refer to incompatibly predicated aspects of reality; aspects which, as I am at pains to stress, are both there. The atomic and the macro. It is not that the meaning of the variables, A or B, has changed or even that the meaning of the entire schema |A+B| as a whole has, now being intelligible now unintelligible. It is that the conditions of its application to reality have changed, those being now atomic, now divisible; hence, now jointly, now disjunctively permitting definite value ascriptions. |A+B| responds to their alternation without changing its meaning. It just responds to them by rearranging its conditions of internal relatability; which is what CTY is all about.

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Were we to suppose any different, mutual exclusion itself would be jeopardized or falsely attributed; to postulate a change of meaning in the schema |A+B| as we alternate between x=1 and x>1 value assignments is exactly the same as postulating that, if books A and B cannot fit together within a drawer of a size smaller than their sum of surfaces, they have eo ipso changed their shape! And this is patently absurd. Not just because this supposition is evidently false, as it clearly is. But because it is even compatible with the supposition that, in having changed their shapes thus, the books could now very possibly fit in the said drawer. And this is surely contradictory to the assumption. The books are no different. The drawer is. Similarly, the self identity of the schema |A+B| stays rigid across whatever number field we might consider, natural or fractional. |A+B| can receive values either from the field of naturals or the field of fractionals. Yet its internal identity stays rigid across its transfer from one field to the other. The only transformation apparent, comes with the limiting ascription x=1 in the field of the naturals. And this ascription introduces a dichotomy not between the two number fields, as it were, but one within the self same number field instead, that of the naturals alone. Which latter, I would suppose, should feature here as a single context. Perhaps complementarian contextualists could explain to us how, in this case, |A+B| manifests properties of mutual exclusion in a single context, that of the naturals for x=1, and no such thing when |A+B| travels across one number field to another, for any natural ascription x>1. Concepts which behave complementarily in a single context for the ascription x=1, still behave identically across two distinct contexts, the naturals as opposed to the fractionals, for any natural value ascription greater than 1. Hence, mutual exclusion is due to a factor other than change of context. But not other than the ascription x=1 of a context. I submit therefore that Hanson’s view above quoted is dualistic and realistic. The unintelligibility, as quantum numbers get smaller, results from not having any more numbers at our disposal. Reality itself is running out on us. Until, for x=1, the tank is empty. And the intelligibility is restored, as quantum numbers get larger, because now we have as many of them as we care to count up to. “The monads [=1] have no windows”, Leibniz warns us [Monadology, prop.7] But the rest of things open theirs to us. What Hanson understandably refers to as fluctuations of intelligibility are essentially fluctuations of available reality. Whatever happens to our concepts thereafter is purely conditional on just how much reality there is. This is a ‘contextual’ account of Bohr’s correspondence principle and, by extension, of his CTY, that I can quite comfortably live with. Other such, I certainly cannot; Time then to rebut them.

2. Complementarity: A Stumbling Block on the Way to Context C. Bohr began his contextualist narrative of CTY by briefly mentioning the word “will” as potentially emerging in distinct, hence uncommunicating, contexts, pointing to Wittgenstein as the ultimate source for this suspicion [see Note 4]. I will briefly indicate my initial reply, for isolating whatever elements will be requisite for the quantum integration of the point. First, let me remind to all concerned, the exact position of context-dependence: The meaning of the same word differs in different contexts. Now then; either “free–caused” or “mind–body” are pairs of words semantically connected or else words not semantically connected. Suppose first they are not connected. Then they are simply two different words,

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signifying two different concepts. Then they will trivially not have the same meaning, since they are different words to begin with. And this rules out all contextual treatment ex hypothesi. So suppose they are semantically connected instead, as they clearly seem to be. But then this semantic connection of theirs can be none other than their very opposition. For if “free” is just taken to mean “not caused”, and “mental” to mean “not bodily”, and vice versa, i.e. the only provision at our disposal for semantically connecting the two pairs of words (otherwise we fall back to the previous horn: different words), then, unless I’m horribly mistaken, such pairs of terms have now been turned into straightforward contradictories. And contradictory descriptions must have a unique reference, if to be at all contradictory. If you say “this car is moving” and I say “this car is at rest” but point at different cars, we are certainly not disagreeing. Again there is no profit of sorts for the contextualist here. In fact, my former impasse still persists. Either these pairs of concepts are unconnected, whereupon their difference in meaning is semantically trivial and not susceptible to a contextualist analysis, or else they are connected, but are then contradictories and hence such, as designating and belonging to a unique context, antithetically predicated; The context being, a single act. Now to quantum CTY; the contextualist motto is still what it always has been (or is it?); Different meanings (in different contexts) of the same word. But the quantum or, for that matter, the classical terms “position” and “momentum” are certainly not the same word. They are two different words. Why should we ever expect a meaning “variance” between words which are different in any case, and which C.Glymour somewhat (self) forgetfully conflates? “Meaning holism has the consequence that if beliefs about a collection of sentences change, so do the meanings of the terms of the sentences. This phenomenon is sometimes called meaning variance” [Glymour, 1992, 124, the author’s italics]. Then, extending this to p and q, which should normally be elements of a single collection, he concludes: “Just as no world of experience combines different conceptual schemes, no reality we can experience, even indirectly through our experiments, combines precise position and precise momentum.” [Glymour, 1992, 128, my italics]. “Just as...?” Just as what? When we speak of meaning variance, what exactly is it, which the variance is a variance of? A different word? A different word does not yield “variance of meaning”, in the sense here demanded. It just yields a difference of meaning because it is a different word; period. And so far as I can see momentum and position are different words of a collection. Not the same word in two collections. For although, perhaps, different conceptual schemes are not combinable, on whose authority are momentum and position elements of different conceptual schemes? Are they not, then, constituents of a single phenomenon of motion? Classically, the velocity, U, is given by U= ds/dt, where ds in the limiting case is position, derived by means of velocity and, indeed, conversely, since velocity is itself a derivative of position. To compare these two with different conceptual schemes is, in a word, nonsensical. As constituents of a single phenomenon of motion, they evidently belong to the same context. “Motion is change of position in time”, as H.Folse puts it concisely [Folse, 1985, 57]. This, if it is evidence of anything, is evidence that they belong, by definition, to one and the same context. And if the mutual determinations of position and momentum get in each other’s way at all in QM, this is because they belong to the same context in the first place. For were they unrelated instead, their determinations would be independent of one another and hence attained in combinable separation (see below), in utter contrast to Glymour’s different conceptual schemes.

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This situation, though it does lead directly to their CTY, rather than leading to their claimed context dependence at the same time, obeys instead laws of logic and semantics which are the very negation of Glymour’s ‘analogy’. Let us consider mass and position instead. Mass is a concept definable in contexts differing, or even unrelated, to contexts appropriate for defining position. The two of them are, for that matter, as distinct, unconnected and divergent as we care to imagine. The question of where an object is, is the last question relevantly entering issues of specifying its contained quantity of matter (or inertia). The two of them belong, one is entitled to declare, to totally alien contexts. Then, perhaps, C.Glymour would care to enlighten us just how it happens that, wide disparateness of context notwithstanding, position and mass are still not complementary quantities in QM! And how, instead, position and momentum, which are clearly constituents of a single phenomenon of motion, and are therefore closely bound to one another within the confines of a single context instead, do happen to so be. Well, to me the answer is simple. CTY has nothing to do with context dependence and, I venture to propose, a lot to do with context independence. A question, then, for some people at Pittsburgh University: How can two contextually unrelated concepts be compatible, and definable with arbitrary precision, while two contextually connected ones, and both immersed within a single context, motion, be complementary? If I have done my quantum homework, as some contextualists no doubt also should, it is actually because position and mass are thus unrelated, that they are thereby jointly, rather than disjunctively, definable within the frame of the theory. Because, in other words, their mutual determinations are effectively isolatable, never interfering in any form with one another. Hence, position and mass are both known and they are both known simply because they have nothing at all in common. Their contextual diversity is not their CTY. It is their compatibility. On the contrary, position and momentum cannot both be known because, unluckily, they have a lot in common. Their contextual identity is their CTY. It’s not therefore as if CTY is a tacit form of context-dependence or, even, vice versa. If anything, to call a pair of concepts complementary, in Bohr’s sense, is to call these concepts context– insensitive. Which is a point going beyond the confines of the present approach, and generalizable over all relevant cases. The very notion of CTY rules out context–dependence. ‘Complementarian’ contextualists have got the truths all backwards; again.

3. The Duality Route to Context Gonzalo Munevar and P.K.Feyrabend have developed between themselves a parallel version of subjecting complementary relationships to context–dependence. This, Munevar explains, was on Feyerabend’s incitation. [Munevar, 1995,6.] Well, surely that won’t make things better for either of them! Here the scheme of things, unlike the cautious formulation earlier preferred by Chr. Bohr, assumes all the features of grandiose ambition, reaching to giddy heights when, together with complementary conceptual schemes of different cultures, we are announced of complementary conceptual schemes between humans and bats! [Munevar, 1995, 6-8.] The term “CTY” here mainly serves the purpose of stressing the equality in epistemic status of the disjunctive world accounts. Possibly, even, the equal epistemic status between human and bat cognition: <
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produce information in one frame that is not logically, conceptually, theoretically, or mathematically equivalent to any produced in another, even if it is presumably about the same aspect of “reality”>> [Munevar, 1995, 6; italics the author’s.]. This, I would imagine, is a big step away from momentum-position or energy-time CTY. We no longer have to do with CTIES established between concepts collectively belonging to a certain conceptual scheme and rift apart by indivisibility, or do with indivisibility itself for that matter. Rather than dealing with disjunctiveness between concepts constitutive of a conceptual scheme, such as the products Et and pq formerly of classical mechanics, we are now dealing with one conceptual scheme disjunctive to another. Whence, of course, the irresistible temptation on the author’s part to involve Kuhnian Paradigms in the discussion, not an uncommon supposition in the matter at hand, as we shall soon see, though at least one wished that he wouldn’t have to drag “bat cognition” together with them just to drive the point home [Munevar, 1995, 6]. The analogy supporting all this, or in any case what Munevar takes an analogy to be, is to be located, he claims, to quantum experiments disclosing corpuscular and/or undulatory properties of micro-systems: “As Bohr pointed out, one experimental arrangement that enables us to see the electron behave as a wave complements, but also rules out, another arrangement that enables us to see the electron behave as a particle. The wave and the particle descriptions are thus complementary –there is no sense in which they are equivalent. The same may obtain between descriptions produced in different frames” [Munevar, 1995, 6; the italicized words directly connect CTY with context]. So there is really no escape, is there? After ninety years of complementary theorizing and still Bohrian scholars, methodic or perfunctory, aficionado or free lance, subtle or populist, exhaust what little imagination they can muster, to take us back to the ‘CTY’ of waves and particles. Contextualists regard themselves as avant guarde progressivists in life, art, politics, and philosophy as well as in science. But on this matter they just never learn; from Feyerabend’s early formulation of the thesis, “CTY is a description based on duality” [Feyerabend, 1958, 94], repeated a few years later (“...duality as well as the idea of CTY based upon it”. [Feyerabend, 1962, 223]), we proceed to more recent times with an occasional, but not too decisive, and always back stepping, improvement: <> [Folse, 1985, 114]. Well, close enough. But back steps are, it seems, the rule in the pathway to CTY. Thus, even more recently we are once again assured that “CTY is introduced by Bohr in response to the wave-particle duality, which becomes wave-particle complementarity in his framework” [Plotnitsky, 1994, 68]. The message transmitted has now become all the more ominous for the handful of objectivist complementarists still in life (counting Folse and myself) and the even fewer rationalist ones (just counting myself, presumably). According to this author “waveparticle CTY is, thus, defined in anti-epistemological terms” [ibid.] and hence serves almost as the scientific Trojan horse that throws the gates wide to contemporary French obscurantism, Derrida, deconstructionism and a curtailed version of Hegelian dialectics, from which last French philosophy simply finds it impossible to emancipate itself, the author included, it would seem.

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Bohr, surely, has on occasion spoken of wave-particle CTY, though mostly as a rather degenerate instance of the concept and, chiefly, as a derivative situation in QM, rather than as the primitive axiom which these authors take it to be. But, and that’s what matters, he has also spoken very differently: “The term Complementarity, which is already coming into use, may be suited to remind us of the fact that it is the combination of features which are united in the classical mode of description but appear separated in the quantum theory” [Bohr, 1934, 19]. But it is not waves and particles which are united in the classical mode of description and separated in the quantum. It is the conjugated classical concepts, whose products, Et and pq, yield a unit of action. To say nothing of the fact that, on the very words of most complementarists of the persuasion in question, wave and particle are just about ‘united’ in the quantum mode, besides being separated in the classical and, I would add, everywhere else as well. Once again contextualists have got the truth backwards. There has been, I strongly suspect, too much casual reading of Bohr’s text and too much poor logic involved. Let me then demonstrate, and I mean demonstrate, not just indicate, suggest or argue, why wave-particle duality literally cannot be an instance, let alone the basis for CTY. The first step I will take hand in hand with Hooker’s account: “Bohr believes that while it has seemed to us at the macro level of classical physics that the conditions were in general satisfied for the joint [see Note 5] applicability of all classical concepts, we have discovered this century that this is not accurate and that the conditions required for the applicability of some classical concepts are actually incompatible with those required for the applicability of other classical concepts. This is the burden of the doctrine (B4) [Doctrine (B4)=CTY]. This conclusion is necessitated by the discovery of the quantum of action and only because of its existence. [...] It is a discovery of the factual absence of the conditions required for the joint applicability of certain classical concepts” [Hooker, 1972, 137; dark letters for the author’s italics]. In a word, CTY and its quantitative expression by the two UR is conditional on the quantum. (As if it could ever be otherwise.) Precisely as the mutual exclusion of the two variables of my own complementary model, A and B, is conditional on whether the value ascription is x=1 or x>2 &c. And this entails a conditional form of incompatibility; Conditional, in this case, on the quantum of action itself. But the incompatibility of waves and particles is not conditional on anything. Waves (large) and particles (small) are incompatible by definition and hence incompatible in the ways of logic. That is to say, necessarily incompatible. This is why, besides, it is called a “duality” in the first place. And in being necessarily – rather than conditionally – incompatible, waves and particles are also incompatible in classical mechanics. And classical mechanics does not contain the quantum. Stated in the simplest of terms, waves and particles are mutually exclusive without any help from the quantum. Were CTY to be grounded on the incompatibility afforded us by the wave and the particle, CTY would equally have no need of the quantum. This sort of incompatibility is self contained and obtains independently of the quantum. So then would a CTY based upon it, in direct contradiction with the premises of the argument. Wave–particle duality, rather than being the basis of CTY and/or the UR, is in actual fact a situation yielding exactly the sort of incompatibility which renders the quantum redundant. And this is patently absurd. Duality, I submit, yields the wrong sort of incompatibility. Let me rephrase this. The wrong sort, if there ever was one. Duality is an unmanageable idea and the sooner one gets rid of it altogether, the

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better for CTY. In more recent years this is beginning to tentatively dawn upon commentators, at long last: “The connection between wave-particle dualism and the complementarity of the dynamic and the kinematic properties remains a problematic issue for the analysis of Bohr’s philosophy and for the interpretation of QM” [Faye & Folse, Editoditorial, 1994, xvi]. So then does, to put it mildly, the ambitious plan of associating CTY with context– dependence. If Duality was retained, together with its correlative subjectivism (see Munevar’s account earlier cited) the task would have looked more promising. If, that is, we incorrigibly stick to the old ways, always known to die hard, and continue to identify CTY with “the technique, applicable to all domains of experience, where the phenomena depend on the conditions on which they appear” [Petersen, 1968, 242] this would (albeit still erratically) suggest a plausible similarity with Kuhnian Paradigms, capable, if their author is to be believed, of making us see things because we have been operating within this Paradigm, while other researchers, operating within the confines of a different Paradigm, would literally see different things, because secluded in that other Paradigm [Kuhn, 1970, 110ff.]. Either way, the complementary or the paradigmatic, the phenomena experienced would intrinsically depend on the conditions of their observation, either as wave and particle properties, depending on the experimental design chosen but arising from a unique referent, or as noncommunicating, “incommensurable” and Paradigm-dependent visions of a per se inaccessible referent (as in Munevar’s account). If there is a temptation at all, to the task of tracing down parallelisms between CTY and context-dependence, this is it: namely, Duality. This is the sole (good) reason behind the intended comparison. Thus: <<What properties the world exhibits depends on what we ask of it. Niels Bohr called this phenomenon “complementarity”. [...] Changing the experiments we conduct is like changing conceptual schemes or paradigms: we experience a different world>> [Glymour, 1992, 127-28]. This is really most regrettable, when affirmed by so many. That what the world exhibits may have something to do with what we ask of it, is, so far as I can see, at best a necessary condition for the answers we receive. Not a sufficient one. Were it not so, why pray, do we get Complementarity of the classical concepts in the first place, right when we were all so complacently confident beforehand that, asking in the way we had, we would get their compatibility instead? My own answer [1987] is quite simple: We do, because the world refuses to comply with our preferred conceptual instructions and goes its own way. And if that’s not evidence for realism, then I recommend to the authors above to rethink their thoughts on this as carefully as I promise to do mine. But none of this is really necessary. Once Duality is stripped bare of its pretenses, namely that it matches the logical structure of CTY, when it cannot even coherently relate to the quantum, then I suspect that not even contextualists will be all that eager to assimilate such incoherence in their scheme of things. But when it comes to contextualists, one should never be too sure.

4. Incommensurability: Last Resort—and Last Exit Feyrabend claims that: “The combination of duality with the second set of empirical premises introduced (the Einstein-de Broglie relations) shows that the duality [see Note 6]

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between the wave and the particle properties of matter may also be interpreted as a duality between two sets of variables, e.g. position and momentum” [Feyerabend, 1962, 225]. Well, if this is so, I suppose I just have to repeat for positions and momentum what just a few pages ago I said for waves and particles. The incompatibility between waves and particles is self-sufficient and has no need of the quantum. In consequence, if the incompatibility between position and momentum mirrors the former, as Feyerabend asserts, it will likewise have no need of the quantum. Some interpretation of ΔpΔq≥h this is; one that derives ΔpΔq≥h without the quantum; Now to further disasters. Consider the following argument: (a) Position and momentum are either incompatible by virtue of Logic, or, (b), not incompatible by virtue of Logic. Suppose first that (a). Then, if position and momentum are incompatible by virtue of Logic, just as Duality also is, they must be incompatible across the board. But then they must be incompatible even in classical mechanics. And then QM and Cl.M. would not be incommensurable at all, for they would both manifest the same type of incompatibility between position and momentum, now holding across theories. (As Zeno was the first to contend!) Suppose then that (b). Position and momentum are not incompatible by virtue of Logic. Then they must be incompatible due to specific conditions, obtaining in QM but not also in Cl.M. But then their incompatibility in QM is due to reasons other than their meaning. Consequently, position and momentum have not changed their meanings in QM. Therefore QM and Cl.M are not incommensurable theories, despite their evident contrast in all else. Thus, both exhaustive alternatives refute context–dependence. But they do not refute Complementarity. In fact, they confirm it down to its last letter. “The combination of features which are united in the classical mode of description but appear separated in the quantum theory”, is what we have witnessed Bohr demand in [1934, 19] for the establishment of a complementary relationship. And if the classical concepts are merely factually [see Note 7] incompatible in QM –due, of course to the premise that h>0, which could have gone the other way– but compatible otherwise, hence compatible also in CM, then the classical concepts behave exactly as Bohr’s definition of CTY prescribes in both the theories considered. But they behave contrary to what context- dependence prescribes for both theories considered, for when separated in QM by a premise extrinsic to their root core meaning, they eo ipso enter a relation of incompatibility without a change of their meanings, even though now, qua incompatible (NB: factually), they operate in a different physical theory. Hence, though QM and CM are certainly different physical theories due to this, they are anything but incommensurable. Hence, there is no such thing as context-dependence, entailing the incommensurability between CM and QM, and if there is none such in their case, there is none such anywhere at all, the quantum revolution being the strongest case ever for conceptual revolution. I have begun intent on freeing CTY from the unappealing clutches of context-dependence. What this has resulted down to, however, is a refutation of context-dependence per se. In my impasse to Feyerabend and contextualism, above, the alternatives “logical or factual incompatibility” is exhaustive. There is no ‘middle’ type of incompatibility. And the concepts themselves, which are thereby involved, simply manage to preserve their semantic core, whether we treat them as logically or as factually incompatible instead. For if the former, CM and QM are thereby united. And there would then be no incommensurability; And if the latter, then only extrinsically incompatible, whereupon CM and QM would be different theories, as they clearly are via the said incompatibility, but still different without incommensurability.

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“Either-or” therefore, contextualism is rebutted per se. Consider this a bonus of the point, rather than a side effect.

5. The Concept of Action Quantization In view of the mistakes and confusions thus far listed, it is crucial to get our primary assumptions right, before we ever embark on the issue of what CTY actually is. So it is essential to start building from rock bottom upwards. Olivier Darrigol writes: <<Einstein’s form of the quantum condition fitted well with de Broglie’s idea that action played the role of a phase. Langevin called the action variables “the cyclic periods” of the action integral. This denomination implied that Langevin regarded action as a periodic function>> [Darrigol, 1993, 330]. The meaning of this most illuminating account is that in QM the least possible exemplification of a dynamical quantity, E or p, can only be recorded over a period; a wave period. If this period is expressed in terms of the frequency, ν, we obtain E=hν and so Et=h. If it is expressed in terms of the wavelength, λ, we obtain p=h/λ and so pλ=h, as two alternative expressions of minimal action, h. The idea, stated in more precise terms discloses that the basic quantum relation, E=hν or E≈ν, signifies that E, exactly like ν, can no longer be defined at an instant dt→0, as was classically assumed, but only over a period t>0, whose boundary instants {t1,t2} are here given by the frequency of this period, ν. Correspondingly, p also cannot be determined at a point location, dq→0, but only over a distance, whose boundary points {q1q2}are given by the wavelength of this period, λ. It is because E cannot be defined at an instant dt→0, but can only be mapped over a period, that the resulting product of action Et is not arbitrarily reducible to a diminishing value, Et→0, but can only be determined over a limiting period t>0, that it is rendered a quantum of action, h, instead. Clearly, in the case that E could be recorded at an instant, taking that instant as narrow as we might wish, leading to a t→0, the product Et would be a vanishing quantity and no quantum of action would result. But since a t={t1,t2}>0 as opposed to a t→0 is a time latitude, and q={q1q2}>0 a place latitude as opposed to a q→0, it immediately follows that the dynamical quantities E and p cannot be defined within an arbitrarily narrow space-time interval, as was classically assumed, but only over such limiting space-time latitudes, in other words only within accuracies not smaller than periods, instead of instants, and not smaller than distances, instead of points. Hence, E can only be defined over a Δt and p only over a Δq. The quantum uncertainties already announce themselves in the mere unpacking of the concept of quantized action. So let us proceed to derive them. [See also Appendix.] ΔEΔt≥h: If there is such a thing as a shortest time permissible, i.e. a time limit, imposed on the conditions warranting the very manifestation of E, this being a time limit of dimensions Δt={t1t2}or ν, then any subsequent narrowing of this interval, of the order, say, Δt'={t1t2}/2 can only mean that the overall energy determination will only be reciprocally affected and, therefore, reciprocally inaccurate. For if we require at least a time length of dimensions Δt, if we are to determine the energy with an accuracy ΔE=n, where n is sufficiently small to stand for a high E approximation, then, all other conditions being identical, at half the time formerly allowed, i.e. within Δt/2, we can only expect to end up

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with an uncertainty ΔE=2n if the action product itself, of which E and t are the components, is to always remain constant, i.e. h. And so on, reciprocally, for any other diminution. In other words, if energy can never be defined within an instant t→0, and hence is on-ly to be defined over the boundaries {t1,t2}=Δt, as is dictated by the assumption that the action unit which is the product of components E and t is to remain constant (or minimal) at all times, then the optimal definitions of these two action components, E and t, cannot themselves be any better than the said limiting product, Et. And therefore that the joint errors in the definitions of these two action components, E and t, can at best be equal, or if not, then greater than this limiting product Et. Hence, in symbols, ΔEΔt≥Et. But Et=h. So ΔEΔt≥h. ΔpΔq≥h: By strict analogy with energy before, if momentum can only be defined over a distance and therefore not at a point location Δq→0 (whence also Zeno’s paradox), hence only over a distance of dimensions Δq={q1q2}(=λ), it equally follows that any attempt at its definition within spatial boundaries narrower than those specified, will be as inaccurate, as the distance itself employed for its definition is taken shorter. Same as before, if we need at least a distance of dimensions {q1q2}=λ=Δq, if we are to determine the momentum within an accuracy Δp=n, where n is sufficiently small to stand for a high momentum approximation, then, all other conditions being identical, at half the distance formerly allowed, i.e. at {q1q2}/2, we can only expect to end up with an uncertainty Δp=2n, if the action product itself, of which p and q are the components, is to always remain constant, i.e. h. So, in general, if momentum cannot be defined at a space point, but is only to be defined over the spatial boundaries {q1q2}=Δq, as is dictated by the assumption that the product pλ is to remain constant (or minimal) at all times, then the optimal definitions of these action components, p and q, cannot themselves be any better than the limiting product pλ. And then, consequently, the joint inaccuracies in the definitions of p and q can at best be equal, or if not then greater, than this limiting product pλ; hence, in symbols, ΔpΔq≥pλ. But pλ=h; so ΔpΔq≥h. And thus the CTY between energy and time and the corresponding CTY between position and momentum have been derived, simply and, I believe, elegantly, from the mere analysis of the quantization of action alone. The definition of quantized action included, it has all taken me but a page. And CTY (or the UR), in this deduction, were shown to be strictly dependent on the quantum; which is the primary requirement. Let us then embed this result, i.e. the UR thus derived, in the deeper spirit of CTY. The structure of quantized action –its limiting value- demands a delicate balance to be maintained overall, which, once upset by leaning too heavily on either side of the balancing scale (defining either complementary quantity involved too accurately) can only result to leaning too lightly on the other side (total loss of its complementary), because the quantum they jointly comprise (Et or pλ) has itself to stay rigid either way. A sharp instant of duration Δt→0 at a sharp point Δq→0, though themselves ideally accurate, will eo ipso make the definitions of E and p, respectively, impossible, if energies are to be defined only over periods, momenta only over distances. And, conversely, when energies and momentum are to be defined only over periods and distances, respectively, then their determinations will have been attained within margins way too wide and hence way too inaccurate, when compared – as above- with instants of duration Δt→0 and points of dimensions Δq→0; all this because of the inward reciprocity [see Note 8] possessed by the indivisible action block. Expand it at one end, it will eo ipso contract at the other, in directly reciprocal manner, to keep itself a

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constant, undiminishing quantity. This, then, is indivisibility. And this is how such indivisibility forces us to adopt a mode of description designated as complementary. Well, then. Has there been any ‘duality’ assumed? None has. For what I have said, and what Darrigol has said before me, was not that a “particle” is associated with a wave period, but that action so is. Hence, reading Duality into this argument is quite simply a misnomer. What this argument discloses is that the quantities E and p must “stretch out” over periods or distances, if to at all materialize and be determined. And not that particles must so stretch. To read “particles” into the quantities themselves, E and p, is not what the two quantum relations entail but what the two quantum relations preclude. The restriction that energy is to be defined over a period rather than at an instant, says nothing at all about particles. It just says how long it takes to (minimally) define energy. That the energy thus defined is that of a particle is but a gross non sequitur and the last thing that the relation E=hν, i.e. the relation that E≈ν, will ever license. Energy is connected with the frequency, is what the relation itself prescribes, hence by definition not with a particle. And the restriction that momenta be defined over a distance, now related in spatial terms with the said period, rather than at a point, is the very thing which makes reference to a particle literally impossible, for particles can only have a locally concentrated momentum, not one spreading over a distance. Particles, therefore, emerge in these relations from literally nowhere. They are simply imported by an act of fiat, whereas so far as the two quantum relations are concerned, particles only originate in the great outside and it is from there alone, that they are introduced in the action scheme. To persist ‘associating’ the dynamical quantities E and p with the presence of a particle (a phrase usually attributed to de Broglie) is but the product of a classical remnant within relations whose presence alone should be sufficient warning against this very practice. The thought behind it all is, “if there is energy and momentum, there must be a particle around”, a loan from classical preconceptions, if there ever was one. Not even the face of a particle is inscribed in these relations. It is action, not particles, which is periodic, and hence ‘wavelike’. And action as such is not even material, in the sense that particles are. For it is a process; not a thing. Once Duality is shut off from the picture, either on the side of waves or on that of particles, the only backdoor to contextualizing CTY in the ways indicated by Munevar, Feyerabend or Glymour, is shut together with it. Were Duality a coherent way of stating the quantum situation, then the relativism that has become part and parcel of this notion would provide the basis for a fusion between CTY and context-dependence, itself a method of regarding meaning – and fact (cf. Paradigms) – relative to a context, i.e. what Duality is also supposed to do in its usual descriptions. But Duality, as I trust I have established, does very many others things as well, none of them too good for the theory (as several have already surmised). Duality is a quantum myth and a lethal one at that. Hence, context-dependence has really nothing to go by for their planned assimilation, except, that is, that context-dependence may be a myth too, and a lethal one at that, whence the obstacles to their unity are thereby removed. Now to another matter; an inherent part of wave-particle duality reasoning is the one we have extracted from Petersen’s (alleged) definition of CTY, as a method of handling such phenomena, as those which depend on their own observation. Which is a very definite form of relativism (and one, I will not deny, that Bohr has frequently encouraged, when addressing audiences or ‘cultures’ which he had a mind, or a diplomacy, to flatter); but what relativism

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and what observations? In my own –primordial- derivation of the two UR, and their correlated CTY, there were no measurements assumed at all. Only strictly purified quantum axioms and their immediate logical entailments, i.e. the two UR derived as theorems. I have done nothing but unpack the contents of the concept of quantization. I have not said a word about ‘measurements’ or ‘observations’ or any such irrelevancies. But CTY has been derived, simply and directly, from such a concept alone and despite my not once having introduced or even so much as hinted at observations. What I have derived, valid or not, I have derived prior to observations; which is precisely what it takes in the first place, if the resulting theorems, the two UR, are to be subsequently confirmed by experiment, if at all. CTY, therefore, has no need of any ‘measurements’ to come forth, as it had none in my |A+B|=1 model, where mutual exclusion was warranted a priori. To proceed to measurements directly, as so many physicists are professionally prone to do, implying in practice that there can be no uncertainties, or no CTY, unless a measurement is first performed, as if the purely ontological counterparts of these notions are literally absent when no measurement is in process to then spring forth from zero only upon measurement, not only makes the practice of experimenting utterly incomprehensible. What are the measurements about? Themselves? But, in addition, and due precisely to such obscure experimental self-reference, this tendency makes talk of context–dependence and Paradigms almost inevitable. If measured uncertainties only measure measured uncertainties (!), rather than limitations built into reality itself, then surely there is nothing behind them, except a shapeless, amorphous void and not the atom (i.e. the foundation). My reply is really quite simple. I have assumed no measurements. Hence whatever CTY was deduced in the process, was deduced from an ontological premise and that alone. In addition, in my account of CTY and the UR, the quantized products Et and pq (or pλ) are not the same words featuring in different contexts. They are different words featuring in the same context, namely, the one, unique wave period, and thereby the one, unique unit of action they jointly comprise. This is why they are complementary. And this refutes context– dependence not just as a possible reading of Complementarity but as a possible reading of anything. Does the period t, over which E is defined to yield Et, belong to a context other than E? Does the distance λ, over which momentum is defined to yield pλ, belong to a context other than p? Do, that is, concepts operating in a certain context, serve in defining concepts that operate in a different context and still define them? This is the sole way contextualism could ever handle the present situation and it is a way that leads it to but a blatant contradiction. But that is nothing new.

6. Tying Loose Ends There are, I believe, not too many of them in the argument. But there are points unmentioned or points only hinted at, such as dualism, atomism, foundations, and, ultimately, realism, which need to be brought to focus. An atomist is a metaphysicist, twice over. Once, because he is a realist, and this is a metaphysical position, twice because he is an atomist and atoms (since Leibniz) are said to be anything but physical. Kant’s 2d Antinomy between the Composite and the Simple reflects the same thing [402 ff]. Atomists, in addition, are foundationalists. Ancient atomism began as a response to Zeno’s paradoxes, which led infinite divisibility of processes to a literal dead

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end. I shall name but one such paradox, that of the stadium runner (though the arrow paradox is saturated with complementary connotations), to drive the point home. The runner, in order to cross the stadium must take the step which is the most crucial step of all that follow; the first ever. But there is no first step to take, if distances are infinitely divisible [see Note 9]. The first step would be right where he’s standing; to realize this clearly, suppose you ask someone to start counting –from the beginning. He will go “one, two, three, four ...” and so on. No problem. Now ask him to start counting, again from the beginning, but count fractional numbers instead. He cannot. There just is no place, from where to start. It is a logically impossible task to count fractional numbers from the beginning. He will remain as speechless as Zeno’s runner is said to have remained motionless. There is nowhere to start from and, respectively, nowhere to go. Atomism seems the sole escape from this riddle. Atomism is a way to start and therefore atomism is a form of foundationalism. It is however, once atoms are closer examined, a form of dualism as well. Atoms, to circumvent perpetual divisibility, turn out to consistently lack a structure, for structure is a (near) synonym of inward separability; and so of divisibility. And what lacks a structure cannot become the part of a thing possessed of one. Hence, strictly speaking, atoms cannot be the parts of anything. One would think that this undermines the very essence of their introduction, namely, that of offering a foundation. I am of the opposite inclination myself. If atoms were within the structures, which they are supposed to support, then they would themselves become part of the structure in need of the support. And it would be then that the foundation would be shaken, not now. Hence an atomist can hardly avoid being a dualist. And can therefore hardly avoid running across Hanson’s paradox. As (‘quantum’) numbers get smaller, unintelligibility will result, not perchance because we have changed languages, which in a way we have, but because we are running out of reality. And it is this that ‘changes’ the language, rendering it complementary, whereas it was unified previously, and will again become so, as (‘quantum’) numbers start growing larger. As my numerical model of CTY endeavors to disclose, and as the directly quantal derivation of CTY per se affirms, the atom is just too narrow to make room for all our macro attributes in any one attempt at fitting them to it. But it will make room for them disjunctively, which is what CTY is all about. Two into one won’t go –this is an analytic truth– and, despite our efforts, the indivisible will only manifest undivided aspects of itself; that is to say, complementary ones. If we lean too hard towards the way of the prospective attributes E and p, the other two remaining, prospective attributes, t and q, will vanish from the picture, and conversely, because the quantum is indivisible and can respond affirmatively to either pair of them only in distinct succession, if at all. Language does what Hanson notices only because it conforms to the restrictions. Is this nonrealism? On whose irresponsible definition? If the atomic resists accommodating all our macro attributes at any one time, disjunctively manifesting only undivided aspects of itself [see Note 10], in what exact fashion, pray, does it behave differently than its specified nature? And what other inference would be consistent with the premises of the argument, viz. that of indivisibility? In accordance: “In quantum mechanics we are not dealing with an arbitrary renunciation of a more detailed analysis, but with the recognition that such an analysis is in principle excluded“ [Bohr, 1958, 62]. This is because: “Any consistent use of the concept of the quantum of action refers to phenomena resisting such an analysis” [Bohr, 1963, 94].

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How on earth, then, when an object resists its analysis, can this have anything to do with context-dependence or subject-dependence, if it came to that? It can only have to do with the object. Phenomena resisting analysis, our analysis, and frustrating our attempts at subdividing them because they are unanalyzable (NB: in principle), is the closest approximation to hard core realism ever. The quantum resists its analysis as any true atom should in strictest accordance with its own, independent nature. The language of CTY simply preserves and mirrors this effect by inwardly splitting up in classical halves because the quantum it seeks to describe in corresponding halves itself just won’t. Namely, exactly what my numerical model in the naturals succeeds in preserving. For A+B=1, it will be the variables, not the Unit, that will undergo the splitting. The resulting descriptions, which are complementary in a language (that of classical physics) rather than two languages complementary to one another, as some find it convenient or consoling to suppose, have everything to do with the independent predication of the world as atomistic and non-distributive and nothing at all to do with context–dependence, or its ideological aims [see Note 11]. Bohr is, quite simply, a modern day atomist; and so a dualist, a foundationalist and a metaphysical realist. Relativism and context-dependence have little or nothing to do with any of that. Human rights apart, they have little to do with an anything.

Appendix I have presented the derivations of section 5 twice already, in Author, Refs. 3 and 4; Now, I will present an additional proof, nor presented before, which reconfirms them: Assume first that E=hν. Then assume that we know the energy, E, within the margins of an error ΔE. Then, tautologically, ΔE=Δhν and, since ν=1/t, ΔΕ=Δh1/t. Then, since “Δ” cannot concern h, we obtain ΔΕ=hΔ(1/t). Now, it already becomes clear even at this early stage of the argument that Δ(1/t) is an error in the time, derived tautologically from the initial premise, E=hν, leading (tautologically) to ΔE=Δhν. In my initial derivation, t>0 = Δt, since, in accordance with it, the very fact that E cannot be defined in an instant, t→0, but only over an interval t>0, is the raison d’etre of an uncertainty in time. Now it is known that, in accordance with Error Theory, the expression Δ(1/t) is equal with Δt/t2. Therefore, by substituting for Δt/t2 in ΔE=hΔ(1/t), we obtain: ΔE=hΔt/t2. Since, however, in my initial derivation, t>0=Δt, the denominator of the fraction becomes (Δt)2. Hence, ΔE=hΔt/(Δt)2. This, in turn, yields ΔE=h1/Δt and this, finally, ΔEΔt=h. The point of importance is this: Unless it is assumed in this line of reasoning that t>0=Δt, which is the exact same assumption in my initial derivation, ΔEΔt=h will never follow from ΔE=hΔν, itself derived tautologically from E=hν, thus contradicting the theory. In short, if E=hν implicitly entails ΔE=Δhν, then ΔE=Δhν can only be turned into ΔEΔt=h –and indeed how can it not be? – iff t>0=Δt, in other words, iff the t of the primary relation, E=hν, is interpreted as a limiting time latitude for the definition of the energy, exactly as postulated in my first deduction. This second proof of ΔEΔt=h, not only reconfirms the first but, in addition, it furnishes in the process a by and large self-contained deduction of ΔEΔt=h, fully crosschecking with the first.

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Notes 1. Mainly by Dr. V. Karakostas. See his “Nature of Physical Reality in the Light of Quantum NonSeparability” [Oviedo, Spain, 7–13 August 2003] or his “Nonseparability, Potentiality and the Context-Dependence of Quantum Objects”. Dialectica, 2005 (to be published). 2. Beyond Reason: Essays on the Philosophy of Paul Feyerabend. Gonzalo Munevar ed., Kluwer, Dordrecht, 1991. 3. Plus, that is, eight other people signing the text. John Earman and Wesley Salmon are two of them. (See 2d paragraph from the end, Section 3.) 4. The general idea is that “when language-games change, then there is change in concepts, and with the concepts the meanings of words change” [Wittgenstein, 1977, 66.]. 5. The very expression “joint applicability” of the classical concepts, eo ipso excludes waves and particles as its possible recipients, which are –trivially– not jointly applicable in classical mechanics. Only the variables of the products Et and pq are. 6. The correct thing to say, of course, is that the two quantum relations, when submitted to a Fourier treatment, will alternatively represent the system as either an extended, plane wave or as a localizable wave packet. The former allows the determinations of E and p, since a unique wavetrain is employed, the latter, the determinations of t and q, since a superposition of many waves is employed. And thus the classical concepts, E and p, on the one side, t and q, on the other, will through this process assume an incompatibility analogous to that between “one” and “many”. The sole duality, therefore, derivable from this reasoning is that between waves and wave packets. That is to say, of waves alone, though in antithetic wave profile configurations. Particles are absent even here. 7. This is what Hooker has earlier referred to as “the factual absence” for classical compatibility. For a thorough analysis of logical vs factual incompatibility see my essays [2, 1994, pp.187-9 and 4, 223-5, 2005]. 8. I can hardly overemphasize the importance of this word, soon to be abandoned by Bohr for the sake of the dominant term, CTY, [1934, 19], although the former shows exactly how complementary counter happenings at either end of the indivisible cluster result from uneven balance. The two notions, I should think, go hand in hand. 9. “More strikingly, Zeno shows that no body can ever start its journey: it can never take the first step, since there is no first step to take” [Barnes, 1982, 262]. This, of course, is Kant’s problem in the Second Antinomy, that between Composite and Simple and the frustration thereby of constructing the former by repetitive accumulations of the latter. It is anticipated, in a uniquely important degree, by Zeno’s paradox of extension: “How can a line of finite length be divided into infinitely many parts of finite length? And how can a line which is made up of lengthless parts add up to a line which has length?” [Harrison, 1996, 273]. 10. Dimly, Duality is discerned in the distant horizon. What can only manifest undivided aspects of itself, will reasonably manifest such aspects as may conflict with one another, when what should otherwise be only a part of the entity, must

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now extend over its whole profile instead; and may thus conflict with its alternative, indivisibly manifested profile. Duality is not a primitive assumption in this syllogism. It is only a side product of Indivisibility. 11. The aim is, of course, an equalitarian ideology, offered in abundance through ‘humanitarian’ incommensurability. Incommensurable scientific beliefs let alone moral, social, political or aesthetic ones, are impossible to bring to conflict, therefore impossible to select from. They are all just as good (or just as bad, if it came to that). For the essentially ideological character of context-dependence see [Katz, 1978, 364] and, it goes without saying, A. Sokal’s pseudo-paper [Longino, 1977, 119]. But Feyrabend himself has beaten them all to it, openly confessing that his scientific relativism is preferable because more humane [1971, 33; and 1978 passim]. Scientific truth must be silenced, if it conflicts with politics.

References [1] Antonopoulos C. “Innate Ideas, Categories and Objectivity”. Philosophia Naturalis, 26, 2, 1987. [2] Antonopoulos C. “Indivisibility and Duality; A Contrast”. Physics Essays, 7, 2, 1994. [3] Antonopoulos C. “Investigating Incompatibility: How to Reconcile Complementarity With EPR”. Annales de la Fondation Louis de Broglie, 30, 1, 2005. [4] Antonopoulos C. “Making the Quantum of Relevance”. Journal for General Philosophy of Science, 36, 2, 2005. [5] Barnes, J. The Presocratic Philosophers. Routledge and Kegan Paul, London, 1982. [6] Bohr, N. Atomic Theory and the Description of Nature. Cambridge University Press, 1934. [7] Bohr, N. Atomic Physics and Human Knowledge. New York, 1958. [8] Bohr, N. Essays 1958-62 on Atomic Physics and Human Knowledge. Suffolk, 1963. [9] Darrigol, O. “Strangeness and Soundness in de Broglie’s Early Works”. Physis, 30, 1993. [10] Feyerabend, P.K. “Complementarity”. Proceedings of the Aristotelian Society, supplementary volume, xxxii, 1958. [11] Feyerabend, P.K. “Problems in Microphysics”.Frontiers of Science and Philosophy, R.Colodny, ed., University of Pittsburgh Press, 1962. [12] Feyrabend, P.K. “How to Be a Good Empiricist”. The Philosophy of Science, P.H. Nidditch ed., Oxford University Press, 1971. [13] Feyerabend, P.K. Science in a Free Society, NLB 1978. [14] Faye, J., Folse, H. eds. Niels Bohr and Contemporary Philosophy. Dordrecht 1994. [15] Folse, H. The Philosophy of Niels Bohr. Amsterdam, 1985. [16] Hanson, N.R. Patterns of Discovery. Cambridge University Press 1961. [17] Harrison, C. “The Three Arrows of Zeno”. Synthese, 107, 1996. [18] Hooker, C.A. “The Nature of Quantum Mechanical Reality”. Paradigms and Paradoxes, The University of Pittsburgh Press, 1972. [19] Kant, Imm. The Critique of Pure Reason. Transl. N.K.Smith, Macmillan, 1973. [20] Karakostas, V. “The Nature of Physical Reality in the Light of Quantum Nonseparability”. Oviedo, Spain, 7–13 August 2003.

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[21] Karakostas V. “Nonseparability, Potentiality and the Context-Dependence of Quantum Objects”. Dialectica, 2005 (to be published). [22] Katz, J. “Semantics and Conceptual Change”. The Philosophical Review, 88, 1979. [23] Kuhn, Th. The Structure of Scientific Revolutions. The University of Chicago Press, 1970. [24] Leibniz, W. Monadology. The European Philosophers from Descartes to Nietzsche. M.C.Beardsley ed., New York, 1960. [25] Longino, H.E. “Alan Sokal’s ‘Transgressing Boundaries’”. International Studies in the Philosophy of Science, 11, 2, 1997. [26] Mackay, D.M. “Complementarity”. Proceedings of the Aristotelian Society, suppl. volume xxxii, 1958. [27] Munevar, G. “Bohr and Evolutionary Relativism”. Explorations in Knowledge, xii, 2, 1995. [28] Munevar, G., Ed., Beyond Reason: Essays on the Philosophy of Paul Feyerabend. Dordrecht, 1991. [29] Petersen, A. “On the Philosophical Significance of the Correspondence Argument”. Boston Studies in the Philosophy of Science, Volume 1966-68. [30] Rosenfeld, L. “Foundations of Quantum Theory and Complementarity”. Nature, 190, 1961. [31] Plotnitsky, A. Complementarity. London 1994. [32] Salmon Marilee H., Earman, J., Glymour C., Lennox J., Machamer P., McGuire, J.E., Norton J.D., Salmon Wesley C., Schaffner, K.N. Introduction to the Philosophy of Science, Prentice Hall, New Jersey 1992. [33] Wittgenstein, L. On Certainty, transl. by G.E.Anscombe, B.Blackwell, Oxford 1977.

In: Electrostatics: Theory and Applications Editor: Camille L. Bertrand, pp. 61-89

ISBN 978-1-61668-549-2 c 2010 Nova Science Publishers, Inc.

Chapter 4

M OLECULAR I NTEGRALS OVER S LATER -T YPE O RBITALS . F ROM P IONEERS TO R ECENT D EVELOPMENTS ¨ Philip E. Hoggan1,∗, Mar´ıa Bel´en Ruiz2,† and Telhat Ozdo˘ gan3 1 LASMEA, UMR 6602 CNRS, University Blaise Pascal, 24 avenue des Landais, BP 80026, 63171 AUBIERE Cedex, France 2 Department of Theoretical Chemistry of the Friedrich-Alexander-University Erlangen-N¨urnberg, Egerlandstraße 3, D-91058 Erlangen, Germany 3 Department of Physics, Faculty of Arts and Sciences, Rize University, 53100 Rize, Turkey

Abstract It can readily be demonstrated that atomic and molecular orbitals must decay exponentially at long-range. They should also possess cusps when two particles approach each other. Therefore, Slater orbitals are the natural basis functions in quantum molecular calculations. Their use was hindered over the last four decades by integration problems. Consequently, Slater orbitals were replaced by Gaussian expansions in molecular calculations (in spite of their more rapid decay and absent cusps). From the 90s until today considerable effort has been made by several groups to develop efficient algorithms which have fructified in new computer programs for polyatomic molecules. The key ideas of the different methods of integration: one-center expansion, Gauss transform, Fourier transform, use of Sturmians and elliptical co-ordinate methods are reviewed here, together with their advantages and disadvantages, and the latest developments within the field. A recent approximation separating the variables of the Coulomb operator will be described, as well as its usefulness in molecular calculations. Recently, due to the developments of the computer technology and the accuracy of the experiments, there is a renewed interest in the use of Slater orbitals as basis functions for Configuration Interaction (CI) and Hylleraas-CI atomic and molecular calculations, and in density functional and density matrix theories. ∗ E-mail † E-mail

addresses: [email protected] addresses: [email protected]

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Keywords: Slater orbitals, computer programs, Kato conditions, accurate molecular wavefunctions

1.

Introduction

Slater-type orbitals (STO) [1] are the natural basis functions in quantum molecular calculations. Nevertheless, their use has been rather restricted, mostly due to mathematical integration difficulties. Even today there are no simple general algorithms to solve all the integrals appearing in a Hartree-Fock (HF) or Configuration Interaction (CI) molecular calculation, where integrals involving up to four atomic centers may appear. In spite of these difficulties the research on Slater orbitals has always continued. The reason is the requirement for large basis sets of Gaussian orbitals (GTO) and large wave function expansions to perform more accurate calculations of energy and properties of ever larger interesting systems. As a consequence those calculations need enormous computational times. In 1981, in a Congress in Tallahassee about Slater type orbitals, Milan Randic described the situation: ”Gaussian functions are not the first choice in theoretical chemistry. They are used (...) primarily because molecular integrals can be evaluated, not because they posses desirable properties. Today this may be a valid reason for their use, but tomorrow they may be thought of as bastard surrogates, which served their purpose in the transition period, have no longer viable merits and will fall into oblivion” [2]. The use of an expansion of GTOs instead of an STO was then a pragmatic solution and originally intended for solving the problems in the calculation of the first molecules on early mainframe computers. The GTO expansion together with the popular distribution of computer programs like GAUSSIAN have encouraged the use of GTOs for accurate calculations of large systems. The limits are receding with respect size of the systems and dimension of the wave function, i.e. HF calculations of clusters of hundreds of atoms, CI calculations including hundreds of thousands of Slater determinants. In spite of the rapid development of the computer technology and the availability of supercomputers, the computational times are unreasonably long, so that the computational chemist is restricted i.e. to perform numerous test calculations. This motivates the search for basis functions, where fewer would give a good CI, in particular. The possibility of using Slater orbitals, where a minimal basis would consist in one function per atom would provide a forward impulse to theoretical and computational chemistry. Since the difficulties are of a purely mathematical nature, e.g. definite integrations, it would be worthwhile pursuing investigations of Slater orbitals. The purpose of this paper is to explain the key ideas about Slater orbitals for readers outside the field. It is beyond our scope to review the whole work of the all authors in this field, what would deserve a longer treatment. The history of Slater orbitals and the first computer programs using them is exposed and the currently used computer programs are listed. The STO and GTO are defined and compared. The methods used in the literature are explained recalling in the key ideas in which these methods are based. The last developments within the field are reported.

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Early History of the Slater Orbitals

The history of STOs is the history of theoretical chemistry. In 1928 Slater [1] simplified the hydrogen-like orbitals (which are eigenfunctions of the Hamiltonian for a one-electron atom) obtaining the orbitals which bear his name. Curiously Slater called these orbitals at that time Hartree orbitals. Slater orbitals are a simplification of the hydrogen-like orbitals, which are eigenfunctions of the atomic one-electron Schr¨odinger equation. Brief time-line of events in molecular work over Slater type orbitals to date: 1928 Slater and London. 1929 Hylleraas: He atom. 1933 James and Coodlidge: Hylleraas calculations on H2 . 1949 Roothaan LCAO paper. 1950 Boys: first Gaussian expansion of STO published. 1951 Two-center Coulomb Integrals. Roothaan formulae. 1954 Boys and Shavitt ’Automated calculations’. 1958 Tauber: Work on analytic two-center Exchange integrals: Poisson equation. 1962 Scrocco: first publishes STO work, (in Italian) but with a programme. This follows early molecular work in 1951-53. [3, 4]. 1963 Clementi produces tables of optimised single zeta basis sets for atoms. Shavitt B-Functions described. 1970 The Journal of Chemical Physics published work on STO codes by E. Scrocco and R. Stevens. Gaussian 70 prepared for QCPE by J. Pople and R. Ditchfield. 1973 E. J. Baerends: numerical integration over STO used for ADF DFT code. 1978 Filter and Steinborn: Fourier transform work. B-functions and plane-wave expansion of Coulomb operator. 1981 ETO conference in Tallahassee. Weatherford and Jones. 1994 First STOP (Slater Type Orbital Package, QCPE 667 1996) code. Bouferguene and Hoggan. 2001 First SMILES (Slater Molecular Integrals for Large Electronic Systems) code. Fern´andez Rico, L´opez et al. 2008 Gill: Coulomb resolution.

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Very soon with Slater at MIT, researchers broached the problem of evaluating the twoelectron integrals in this basis. During the 1950s the Chicago group led by Mulliken took on the task of evaluating all the molecular integrals. Roothaan treated the Coulomb and Hybrid two-center integrals [5,6], R¨udenberg the exchange integrals [7]1 . Among the many authors who were working around the world on the solution of the necessary integrals one may mention Masao Kotani in Japan [8], who wrote the famous tables of integrals which bear his name and that were widely used. Coulson in Oxford (England) proposed a method to evaluate the three- and four-center integrals [9], L¨owdin in Uppsala [10], and young American scientist called Harris [11] were involved. Work in the early 50s mostly focused on integrals over STO. The interest was to make the first theoretical calculations of some molecules starting with the diatomic systems H2 , N2 . For three-center molecules the problem of integration was encountered (orbital translation). Mulliken and Roothaan called this ”The bottleneck of Quantum Chemistry” [12], Mulliken mentioning it in his Nobel Lecture in 1966, on the molecular orbital method. Boys in Cambridge published his landmark paper [13] containing the evaluation of three- and four-center integrals using Gaussian functions, for which he derived the so-called product theorem: the product of two Gaussian functions located on different centers is a new Gaussian function located on a new center. Thus four-center electron distributions could be reduced to single-center distributions and evaluation was analytically facilitated. Boys regarded his work as an existence theorem. It was to change the course of molecular computations. Note that the product theorem for Slater orbitals leads to complicated infinite sums, making evaluation awkward compared with the simple closed forms for Gaussians. In 1954 Boys, Shavitt et al [14] expanded Slater orbitals into Gaussians to perform quantum mechanical calculations. In 1963 Clementi presented the so-called basis set using Slater orbitals [15]. Later Pople would base his programs on Boys’ pragmatism.

3.

History of the STO Computer Programs

The first (and surely the last) manual calculation of a molecule, the N2 molecule, was done by Scherr in 1956. It was necessary the work of 2 (sometimes it appears 3 ) men for 2 years. Afterwards this calculation was reproduced by the first digital computer calculation [16, 17], taking 35 minutes. In 1962 Shull initiated the Quantum Chemistry Program Exchange (QCPE) at Indiana University. The first automatic computer program was POLYATOM [18] which used nevertheless GTOs with SCF-LCAO. The program was developed at MIT in 1963 when Slater was there. In 1963 the program IBMOL [19] was developed by Clementi and others when he visited the Chicago group. In 1968 a STO code was developed by Scrocco and Tomasi from Pisa. Preliminary work by Scrocco is reported in Italian as early as 1962 [4]. 1 The two-center two-electron integrals are classified according to the centers a, b. Writing them according the charge distributions [Ω(1)|Ω(2)] the Coulomb integrals are [aa|bb], the hybrids [aa|ab] and the exchange integrals [ab|ab]. The most difficult are the exchange integrals because the charge distribution of every electron is located over two centers.

Molecular Integrals over Slater-Type Orbitals

65

This program was also used by Berthier in France. The program ALCHEMY in 1968 was originally developed using Slater orbitals by Clementi and the staff of the IBM laboratory in San Jose [20], afterwards, the new ALCHEMY 2 by Bagus and others used GTOs. The program DERIC [21] by Hagstrom in 1972 perform STO calculations of two-center molecules. In the 80s, the advent of GAUSSIAN [22] saw development in the STO field hibernate somewhat. By the 90s several groups around the world developed new STO computer programs which are now distributed. The program STOP, by Boufergu`ene and Hoggan [23] was published first in 1996. It is based on the single center strategy and was first presented in 1994 at the 8th ICQC in Prague. New versions appeared, the latest (parallel) in 2009. Then in 1998 a program was written using B-functions by Steinborn, Weniger, Homeier et al, in Regensburg [24]. The program SMILES by Fern´andez Rico, L´opez, Ema, and Ram´ırez in Madrid appeared in 1998 and new versions have appeared, the latest in 2004 for the HF and CI calculations of molecules [25]. The program CADPAC [26] in Cambridge uses techniques like density fitting, involving auxiliary Slater type orbital basis sets to perform Hartree-Fock and Density Functional Theory (DFT) calculations with a reduced number of indices in requisite integrals. They aimed to obtain better Nuclear Magnetic Resonance (NMR) chemical shifts on the basis involving nuclear cusps. In the density functional theory field in 2001 the program ADF (Amsterdam Density Functional) [27] begun in 1973 by Baerends et al uses Slater orbitals for their calculations. This much-used package offers a very extensive series of atomic basis sets for input, including most elements. It is a numerical grid strategy and this review will not detail it. The program ATMOL of Bunge et al performs large highly accurate CI calculations on atoms using Slater orbitals [28]. In the first century of the third millennium much interest is concentrated in generating more efficient calculation algorithms, use of non-integer Slater orbitals, numerical solution of integrals when using B-functions and in the electron correlation when using Hylleraas wave functions.

4.

Slater Orbitals & Gaussian Orbitals

It is well known that hydrogen-like orbitals are the solution of the Sch¨odinger equation for a one-electron atom. For helium and atoms with more electrons the Sch¨odinger equation has no analytical solution due to the potential term 1/ri j which correlates the (otherwise) independent electrons. It is assumed that for systems with N ≥ 2 this form of the exponential e−αr will be the asymptote of the formal solution. The hydrogen-like orbitals have nodes, i.e. the 2s orbital is of the form (1 − br) e−αr , and higher quantum number orbitals are similar but STOs are node-less. A related problem appears for Gaussians. In 1928 Slater [1] regarded the hydrogen-like orbitals as polynomials in r which make the calculations messy and proposed the use of single powers of r i.e. linear combinations of hydrogen-like terms. A picture which helps to visualize the differences between Slater and Gaussian orbitals is the representation of the 1s orbital function of both types, see Figure 1.

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Figure 1. Comparison of the shape of a STO and GTO functions. STOs represent well the electron density near the nucleus (cusp) and far from the nucleus (correct asymptotic decay). STOs thus resemble the true orbitals. Conversely, the GTOs have erroneous shape near and far from the nucleus (no cusp). One can observe that far from the nucleus the GTOs tend to zero much faster than STOs. To reproduce a 1s STO using 3 GTOs (the so-called minimal GTO basis) an orbital is obtained with the shape of a Gauss curve, no cusp, see Figure 2. To reproduce a single STO many GTOs are necessary, but the electron cusp at the nucleus is missing. This is one of the reasons of the slow convergence of the wave function solutions to the exact (HF or CI) result. In general, if the basis function is not a formal solution of the Schr¨odinger equation its convergence is slower. That means that more Slater determinants are required to obtain the same result. Thus Slater orbitals show faster convergence when increasing their number. Another advantage of Slater orbitals is the size of the basis, one orbital per electron is of reasonable quality and multiple-zeta basis sets converge fast to the Hartree-Fock limit. therefore, the number of integrals to be evaluated is dramatically smaller. CI is spectacularly more efficient. Finally, conceptually the Slater orbitals give a more intuitive description of the atomic orbitals and of the molecular orbitals (MO) formed with them. The disadvantages of Slater orbitals have been already mentioned: the three- and fourcenter two-electron integrals are the bottleneck. There is no general analytical solution for them, which would be the most effective and fastest way of calculation. Instead there are a number of approximate methods of calculation, involving infinite series, or truncated approximations to the Coulomb operator itself. They will be treated in the next Sections. The radial Slater functions do not represent the bonding region adequately, it being then necessary to add higher angular momentum functions. It is nevertheless possible to use linear combinations restoring radial nodes. This approach is advocated particularly for ADF, where the hydrogen-like basis is obtained by fixing the coefficients for combining Slater functions. Another disadvantage is that some of the two-center integrals since the times of Roothaan and R¨udenberg have been solved for a co-axial conformation of the atomic coordinate systems (the z-axes point to each other) that is not the molecular frame. Therefore rotations and reflections are necessary. These problems have been solved, but it requires

Molecular Integrals over Slater-Type Orbitals

67

Figure 2. Construction of a STO with 3 GTOs. additional calculations [29]. Nowadays, Slater orbitals are used in atomic calculations, especially in highly accurate calculations of atoms using Hylleraas wave functions (with explicit ri j dependence, and also in diatomics. They are used in DFT and in Density matrix theories. Traditionally they have been used in semi-empirical calculations where of course the three- and four-center integrals were neglected. The Gaussian orbitals are generally used in standard quantum mechanical calculations. As explained they are not shaped like analytical orbitals, with no cusp at the nucleus, for that reason they are not good for the calculation of properties where the density at the nucleus has to be well described. Also the radial dependence is not well represented and the number of integrals increases with the dimension of the basis dramatically. The major advantage of GTOs is the existence of a product theorem. Over many years, workers have improved the calculation of the necessary integrals, having achieved a considerable speed-up. For example the Coulomb operator with a Laplace transform enables to calculate three- and four-center integrals like two center integrals. Concluding, the main defect of GTO expansions is the absent cusp which slows the convergence and the large number of integrals to be computed.

5.

Types of Exponentially Decaying Orbital, Based on Eigenfunctions for One-Electron Atoms

In general one calls Slater-type orbitals those with an exponential radial factor of the form rn e−αr , for n a positive integer (or 0). The atom-centered Slater orbitals are defined as: ϕnlm (r) = rn−1 e−αrYlm (θ, φ),

(1)

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68

where n, l, m are the quantum numbers. Ylm (θ, φ) are the spherical harmonics defined using the Condon-Shortley phase: Ylm (θ, φ) =

m

(−1)



2l + 1 (l − m)! 4π (l + m)!

1/2

Plm (cos θ)eimφ ,

(2)

Plm (cos θ) are the associated Legendre functions. The spherical harmonics are eigenfunctions of the angular momentum operator Lˆ 2 and its z-projection Lˆ z . The complex spherical harmonics are used mainly in atoms and in developing theories because it is easier to work out general formulae and derivations with them. The real spherical harmonics are linear combinations of the complex ones. These are used mainly in molecules. Note that they are written using polar coordinates. They can be also straightforwardly converted into Cartesian Slater orbitals by the exchange: x = r sin θ cos φ,

(3)

y = r sin θ sin φ,

(4)

z = r cos θ,

(5)

obtaining in general: χnlm (r) = xnx yny znz rn−1 e−αr .

(6)

Cartesian Slater type orbitals are very seldom used compared with Cartesian Gaussians, that are an almost systematic choice. When the principal quantum number n in Eq. (1) is a non-integer we have the NISTOs (Non Integer Slater Orbitals). The main difficulty when working with these orbitals is during the derivations a binomial has to be used with an non-integer power what leads to an infinite expansion. These orbitals are widely investigated in the present [30]. The additional flexibility of using non-integer quantum numbers brings a lowering in the energy results. There is the possibility to transform also the polar coordinates to elliptical coordinates. Traditionally the Elliptical Slater orbitals have been used as basis functions for two-center molecules [31]- [33]. These orbitals are known to lead to lower energy results, see Ref. [34]. Using ξ = λ1 = ra + rb and η = µ1 = ra − rb : ϕnlm (r) = ξn ηl (ξ2 − 1)m/2 (1 − η2 )m/2 e−αξ eimφ ,

(7)

where ξ, η, φ are the elliptical coordinates. Now we go to orbitals which are linear combinations of Slater orbitals: B-functions [35], hydrogen-like, Sturmians [36]. The B-functions are Bessel functions. The orbitals have some helpful properties like a compact Fourier transform. Written in the form n

Bnlm (r) =

(2n − j − 1)!

∑ 22n+l−1 (n + l)!(n − j)!( j − 1)! (ζr)l+ j−1 e−ζrYlm (θ, φ),

j=1

(8)

Molecular Integrals over Slater-Type Orbitals

69

one can see that they are a linear combination of Slater orbitals. The angular parts are the spherical harmonics. The hydrogen-like orbitals which are solutions of the Schr¨odinger equation for the hydrogen atom have a radial part which is a Laguerre polynomial. The polynomial and the exponent coefficient depend on the atomic number Z and the principal quantum number n:   2Zr l − Zr m 2l+2 χnlm (r) = Nnl Ln−l−1 (9) r e n Yl (θ, φ). n Due to that fact, the hydrogen-like orbitals do not form a complete set (for finite n), they need orbitals of the continuum to be complete. This would be important for the convergence of the solutions. Shull and L¨owdin [37] realized that this was due to the dependence of Z with n that dilates the orbitals and they proposed the following orbitals where these were substituted by adjustable parameters, i.e. usual orbital exponents: 2l+2 χnlm (r) = Nnl Ln−l−1 (2αr) rl e−αrYlm (θ, φ),

(10)

so these orbitals form a complete set. These orbitals were subsequently called Coulomb Sturmians because they fulfill the so-called Sturm-Liouville theorem for eigenfunctions of such differential equations, with central Coulomb attraction. In the Section 7 methods of the literature we will see how these kinds of orbitals have been used.

6.

Types of Integral over Slater Orbitals

Due to the form of the Hamiltonian and of its expectation value we find the following kinds of integrals. First the integrals which appear when using Hartree-Fock and CI wave functions, in general ab initio methods. The integrals are classified according the number of electrons and centers which are linked. We present them in order of difficulty.

6.1.

One-Electron Integrals

These are the one- and two-center overlap integrals ha|bi, kinetic energy integrals ha|bi and two-center nuclear attraction ones ha|1/rb |bi. Other case of one-electron integral is the three-center nuclear attraction, originated from the nuclear attraction operators in the Hamiltonian: ha|1/rc |bi.

6.2.

Two-Electron Integrals

They can be up to four-centers because of the determinant giving the wave-function and thus the four orbitals which form the integral. According to the number of centers: The two-center integrals have been traditionally the most investigated, they have the following nomenclature: The Coulomb integrals where the charge distribution of every electron is located at a center: [aa|bb]. Hybrid integrals, one charge distribution is located at one center and the other over two centers [aa|ab] and their equivalents [bb|ab]. The exchange integral is more difficult, it leads in case of different exponents to an infinite

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¨ P.E. Hoggan, M.B. Ruiz and T. Ozdo˘ gan

sum. Every electron is located in two centers: [ab|ab]. To solve these integrals a change to elliptical co-ordinates is useful. The Coulomb operator in elliptical co-ordinates contains associated Legendre functions of the first and second kind, for which integration is very difficult. In the case of slightly different exponents there are some singularities. In actual calculations, the Coulomb and Hybrid integrals are calculated exactly, numerous methods exist. The exchange integrals are calculated with great accuracy. The three-center integrals are of several types [aa|bc], [ab|ac]. For different exponents there is no general solution. The four-electron integrals are of the type [ab|cd].

6.3.

Three- and Four-Electron Integrals

They appear in the Hylleraas-CI method [38] when using one inter-electronic distance ri j per configuration. For the two-center case they have been solved generally by Budzinski [39]. Three- and higher number of centers have not been solved yet. These can be many-center integrals, as every electron from right and left in the expectation value operator may be in a different center. These integrals are of the type, i.e. the easier [aa|r12 r13 |ab|bb], to the most difficult [ab|r12 r13 |ab|ab]. Four-electron ones [aa|r12 r13 /r14 |bb|ab|bb], and so on. For three- and higher number of centers one would find three- and four electron integrals with as many centers as the molecule has up to 8. These integrals are still not solved. Interest nowadays focusses on the solution of two and three center molecules using explicitly correlated methods.

7.

Methods in the Literature

In this section the main methods of evaluation of the three- and four-center integrals over Slater orbitals from the literature will be explained. The methods are approximate because they consist in transformations, expansions or include numerical integrations. Therefore they are not as effective as analytical integration would be. Nevertheless, by these methods the evaluation of these integrals is possible and the programs are even as competitive as those using Gaussians.

7.1.

Single-Center Expansion

The single-center expansion method requires expanding the Slater orbitals located at different centers at only one of them and then as for atoms to perform the integrations. The translation method consists in selecting an atom as origin then the translation of other orbitals from their atom to the origin. Therefore both methods are essentially the same. To expand one function centered in A at another point B the following expansion:  ∞ Z ϕAi = ∑ ϕAi χB j dτ χB j . (11) j=1

This formula is due to Smeyers [40]. In brackets, the requisite coefficients. The different methods of single-center expansion differ in the way to calculate these coefficients.

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71

This method was first proposed by Barnett and Coulson [9] in 1956 using radial orbitals (s-orbitals) and was called the zeta function method because of expansions in terms of successive derivatives with respect to exponents. The method has similarities with the alpha function method of L¨owdin [10]. Harris and Michels [41] extended the method to angular general orbitals in 1965. This method has been used by Smeyers, Jones, Guseinov, Fern´andez Rico et al, and others. The idea is the translation of an orbital from one point to the other. The translation of a spherical harmonic is a limited expansion, the translation of the radial part is nevertheless an infinite expansion. This situation can be best explained with formula of Guseinov [42]: χn,l,m (ζ, rA ) =

∞ n′ −1

l′

∑ ∑ ∑

n′ =1 l ′ =0 m′ =−l ′

Vnlm,n′ l ′ m′ (ζ, RAB ) χn′ ,l ′ ,m′ (ζ, rB ),

(12)

where V are the coefficients of the expansion. The method is very stable but it requires computation of a lot of terms to obtain sufficient correct decimal digits, therefore this method needs very long computational times.

7.2.

Gaussian Expansion

This is the Boys-Shavitt method [43], which consist in solving some integrals over Slater orbitals expanding them into a finite series of Gaussians: NG

e−αr = ∑ ci e−αi r , 2

(13)

i=1

ci and αi are obtained by minimizing the least squares. This method and some improvements of this method are used at present in the program SMILES [25]. As NG is usually larger than the number of the primitives when using only Gaussian basis sets, the number of integrals to calculate is large. The method is very stable and robust. It requires lengthy computational times to get accurate integral values.

7.3.

Gaussian Transform Method

The Gaussian transform method by Shavitt and Karplus 1965 [44] has been probably the most used method. It consists in the Laplace transform of the exponential function, here exemplified by the simplest one i.e. a 1s orbital: α e−αr = √ 2 π

Z ∞ 0

s−3/2 e−α

2 /(4s)

 2 ds e−sr .

(14)

Every Slater exponential within the integral is transformed into a Gaussian one, for that one has to solve the integrals over s which have a special form. This integral has to be solved numerically. This is the disadvantage of the method.

7.4.

Fourier-Transform Method

The B-functions Eq. (8) proposed by Filter and Steinborn in 1978 [35] have a highly compact Fourier transform. The group of Steinborn has developed this method [24]. The

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¨ P.E. Hoggan, M.B. Ruiz and T. Ozdo˘ gan

Figure 3. Transformation from polar to elliptical coordinates. evaluation of integrals using B-functions leads to some integrals including a Bessel function of first kind which is oscillatory: Z ∞ 0

rn e−αr Jl+1/2 (rx)dr.

(15)

To evaluate these Safouhi [45,46] used the SD-transform, due to Sidi [47], which consists in substituting this integral by a sine integral which has the same behavior. It needs numerical integration.

7.5.

Use of Sturmians

The Sturmians were proposed by Shull and L¨owdin in 1956 [37]. Smeyers used the Sturmians to evaluate three-center nuclear attraction integrals using the one-center expansion [40]. Guseinov 2001 used also them [48]. The Sturmians Eq. (10) satisfy the SturmLiouville theorem:   2αn m 2 m 2 ∇ Sn,l = α − Sn,l . (16) r The so-called Coulomb Sturmians orthogonalise the Coulomb potential in their argument. This generally applies to the attraction term, at least for one-electron functions. Geminals useful for explicit correlation have also been used. A seminal text by Avery gives more details to the interested reader on this subject [36].

7.6.

Elliptic Coordinate Method

The elliptic coordinate method is the transformation of the polar orbital coordinates into elliptical ones λ, µ according to Figure 3. The two coordinate systems pointed to each other so that the elliptical angle φ coincides with polar angle φ. This transformation is: r1a =

R (λ1 + µ1 ), 2

r1b =

R (λ1 − µ1 ), 2

(17)

Molecular Integrals over Slater-Type Orbitals 1 + λ1 µ1 , λ1 + µ1

cos θ1b =

1 − λ1 µ1 , λ1 − µ1

(18)

[(λ21 − 1)(1 − µ21 )]1/2 , λ1 + µ1

sin θ1a =

[(λ21 − 1)(1 − µ21 )]1/2 , λ1 − µ1

(19)

cos θ1a = sin θ1a =

The volume element and the domain change are: Z ∞ 0

r2 dr

73

Z π 0

sin θdθ

Z 2π 0

dφ →

R3 8

Z ∞ 1

dλ1

Z +1 −1

dµ1 (λ21 − µ21 )

Z 2π 0

dφ1 .

(20)

The method has been used by numerous authors: Mulliken, Rieke, Orloff, R¨udenberg, Roothaan, Eyring, Randic, Saika, Yoshimine, Maslen and Trefry, Guseinov, Bosanac, ¨ Randic, Harris, Fernandez Rico, Lopez, Ozdogan and many others. Some types of three¨ electron integrals have been recently solved by Ozdogan and Ruiz using this method [49].

8. 8.1.

General Two-electron Exponential Type Orbital Integrals in Poly-Atomics without Orbital Translations Introduction

Now, the Coulomb resolution will be presented. This is a readily controlled approximation to separating the variables in the 1/r12 which, in recent work by Gill and by Hoggan is shown to spell the end of exponential orbital translations and ensuing integral bottlenecks. This section advocates the use of atomic orbitals which have direct physical interpretation, i.e. hydrogen-like orbitals. They are Exponential Type Orbitals (ETOs). Until 2008, such orbital products on different atoms were difficult to manipulate for the evaluation of two-electron integrals. The difficulty was mostly due to cumbersome orbital translations involving slowly convergent infinite sums. These are completely eliminated using Coulomb resolutions. They provide an excellent approximation that reduces these integrals to a sum of one-electron overlap-like integral products that each involve orbitals on at most two centers. Such two-center integrals are separable in prolate spheroidal coordinates. They are thus readily evaluated. Only these integrals need to be re-evaluated to change basis functions. The above is still valid for three-center integrals. In four- center integrals, the resolutions require translating one potential term per product. This is outlined here and detailed elsewhere. Numerical results are reported for the H2 dimer and CH3 F molecule. The choice between Gaussian and exponential basis sets for molecules is usually made for reasons of convenience at present. In fact, it appears to be constructive to regard them as being complementary, depending on the specific physical property required from molecular electronic structure calculations. As regards exponential type orbitals (ETOs) such as Slater functions, it seems to be difficult to evaluate two-electron integrals because the general three- and four-center integrals evaluated by the usual methods require orbital translations. Some workers avoid the problem using large Gaussian expansions, as in SMILES [50, 51].

¨ P.E. Hoggan, M.B. Ruiz and T. Ozdo˘ gan

74

It would be helpful to devise a separation of the variables of integration. This would eliminate orbital translations, although some other translations remain involving a simple analytic potential. The present work describes a breakthrough in two-electron integral calculations, as a result of Coulomb operator resolutions. This separates the independent variables of the operator and gives rise to simple analytic potentials. The two-center integrals are replaced by sums of overlap-like one-electron integral products. One potential term in these products requires translation in four-center terms, which is significantly simpler to carry out than that of the orbitals. This implies a speed-up for all basis sets, including Gaussians. The improvement is most spectacular for exponential type orbitals. A change of basis set is also facilitated as only these one-electron integrals need to be changed. The Gaussian and exponential type orbital basis sets are, therefore interchangeable in a given program. The timings of exponential type orbital calculations are no longer significantly greater than for a Gaussian basis, when a given accuracy is sought for molecular electronic properties. Numerical values for all two-electron integrals evaluated using Coulomb resolutions as well as total energies will be tabulated for the H2 dimer and CH3 F molecule.

8.2.

Basis Sets

Although the majority of electronic quantum chemistry uses Gaussian expansions of atomic orbitals [13, 43], the present work uses exponential type orbital (ETO) basis sets which satisfy Kato’s conditions for atomic orbitals: they possess a cusp at the nucleus and decay exponentially at long distances from it [52]- [54]. It updates a ‘real chemistry’ interest beginning around 1970 and detailed elsewhere [3, 4, 15, 27, 44, 55, 56]. Slater type orbitals (STOs) [57, 58] are considered here. STOs allow us to use routines from the STOP package [23, 59] directly. The integrals may be evaluated after Gaussian expansion or expressed as overlaps to obtain speed up [60]. Exponents may be optimized solving a secular determinant as in [61].

8.3.

Programming Strategy

Firstly, the ideal ab initio code would rapidly switch from one type of basis function to another. Secondly, the chemistry of molecular electronic structure must be used to the very fullest extent. This implies using atoms in molecules (AIM) and diatomics in molecules (DIM) from the outset, following Bader (in an implementation due to Rico et al [50]) and Tully [62] implemented in our previous work [59, 63], respectively. The natural choice of atomic orbitals, i.e. the Sturmians or hydrogen-like orbitals lend themselves to the AIM approach. To a good approximation, core eigenfunctions for the atomic hamiltonian remain unchanged in the molecule. Otherwise, atom pairs are the natural choice, particularly if the Coulomb resolution recently advocated by Gill is used. This leads us to products of auxiliary overlaps which are either literally one- or two- centered, or have one factor of the product where a simple potential function is translated to one atomic center. The Slater basis set nightmare of the Gegenbauer addition theorem is completely avoided. Naturally, the series of products required for, say a four-center two-electron integral may require 10 or even 20 terms to converge to chemical accuracy, when at least one

Molecular Integrals over Slater-Type Orbitals

75

atom pair is bound but the auxiliaries are easy to evaluate recursively and re-use. Unbound pairs may be treated using approximate methods. Now, the proposed switch in basis set may also be accomplished just by re-evaluating the auxiliary overlaps. Furthermore, the exchange integrals are greatly simplified in that the products of overlaps just involve a two-orbital product instead of a homogeneous density. The resulting cpu-time growth of the calculation is n2 for SCF, rather than n4 . Further gains may be obtained by extending the procedure to post-Hartree-Fock techniques involving explicit correlation, since the r12 −1 integrals involving more than two electrons, that previously soon led to bottlenecks, are also just products of overlaps. This Coulomb resolution is diagonal in Fourier space in some cases.

8.4.

Avoiding ETO Translations for Two-Electron Integrals over Three and Four Centers

Previous work on separation of integration variables is difficult to apply, in contrast to the case for Gaussians [64, 65]. Recent work by Gill et al [66] proposes a resolution of the Coulomb operator, in terms of potential functions φi , which are characterized by examining Poisson’s equation. In addition, they must ensure rapid convergence of the implied sum in the resulting expression for Coulomb integrals J12 as products of ”auxiliaries” i.e. overlap integrals, as detailed in [66]: J12 = hρ(r1 ) φi (r1 )i hφi (r2 ) ρ(r2 )i, with implied summation over i.

(21)

This technique can be readily generalized to exchange and multi-center two-electron integrals. For two-center terms it is helpful to define structure harmonics by Fourier transforms, limiting evaluation to non-zero terms [67]. Note, however, that in four-center integrals, the origin of one of the potential functions only may be chosen to coincide with an atomic (nuclear) position. Define the potential functions [67]: φi = 23/2 φn l (r)Ylm (θ, φ) . Omitting the spherical harmonic term gives radial factors: φn l (r) =

Z +∞ 0

hn (x) jl (rx)dx, with jl (x) denoting the spherical Bessel function.

(22)

Here, hn (x) is the nth member of any set of functions that are complete and orthonormal on the interval [0, +∞), such as the nth order polynomial function (i.e. polynomial factor of an exponential). The choice made in [66] is to use parabolic cylinder functions (see also another application [51]), i.e. functions with the even order Hermite polynomials as a factor. This is not the only possibility and a more natural and convenient choice is based on the Laguerre polynomials Ln (x): Define: √ hn (x) = 2 Ln (2 x)e−x . (23)

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These polynomial functions are easy to use and lead to the following analytical expressions for the first two terms in the potential defined in (22): V00 (r) =

V10 (r) =

√

2[

√ tan−1 (r) , 2 r

tan−1 (r) 2 − ], r (1 + r2 )

(24)

(25)

Furthermore, higher n expressions of Vn0 (r) all resemble (25) (see [67] eq (23)): Vn0 (r) =

n √ 1 sin(2 k tan−1 (r)) 2 (tan−1 (r) + ∑(−1)k ), r k 1

(26)

and analytical expressions of Vnl (r) with non-zero l are also readily obtained by recurrence. These radial potentials can generally be expressed in terms of hypergeometric functions, whether the choice of polynomial is the present one, i.e. Laguerre, or Hermite polynomials, as in [66]. This structure has been used to confirm the results of [67] using a rapid code in C [68]. Spherical harmonics are translated using Talman’s approach [69]. The displaced potential in one factor of the product of ’auxiliaries’, from four-center integrals is readily expanded in two-center overlaps, after applying Euler’s hypergeometric transformation. [70, 71]. The auxiliary overlap integrals hρ(r1 ) φi (r1 )i and hφi (r2 ) ρ(r2 )i will involve densities obtained from atomic orbitals centered on two different atoms in exchange multi-center two-electron integrals. The overlap integrals required in an ETO basis are thus of the type: hψa (r1 ) ψb (r1 ) φi (r1 )i =

µmax

∑ Nµ (n1 , n2 , ni , li , |mi | αβ) s(n1, l1 , m, n2, l2 , αβ),

(27)

µ=0

with: α = ζ1 R and β = ζ2 R. Slater exponents. In three-center overlaps, Nµ is a normalised Racah coefficient [71]. In two-center cases the sum reduces to a single normalisation term, N0 . A Fourier transform approach is also being investigated, extending [67]. The real space core overlaps then take the form:     λ 1 1 (α + β) B j (α − β) , (28) s(n1 , l1 , m, n2 , l2 , α, β) = Dl1 ,l2 ,m ∑ Yiλj Ai 2 2 ij Yiλj is a matrix with integer elements uniquely determined from n, l and m. Dl1 ,l2 ,m is a coefficient that is independent of the principal quantum number. It is obtained upon expanding the product of two Legendre functions in this co-ordinate system. Symmetry conditions imply that only m1 = m2 = m lead to non-zero coefficients:   Z ∞   1 1 Ai exp − (α + β)µ µi dµ, (α + β) = (29) 2 2 1 Bj



 Z 1   1 1 (α − β) = exp − (α − β)ν ν j dν. 2 2 −1

(30)

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Here, recurrence relations on the auxiliary integrals A and B lead to those for the requisite core integrals [72, 73]. These integrals may be pre-calculated and stored. Such integrals appear for two-center exchange integrals and three- and four-center integrals (although just in one factor for three-center Coulomb terms). Note that exchange integrals require distinct orbitals ψa and ψb . In the atomic case, they must have different values for at least one of n, l, m or ζ. In the two-center case, the functions centered at a and b may be the same. The product does not correspond to a single-center density: it is two-centered. Equation (27) then illustrates the relationship to the one-electron two-center overlap integral, although it clearly includes the extra potential term from the Coulomb operator resolution. This assumes tacitly that the potential obtained from the Coulomb operator resolution be centered on one of the atoms. Whilst this choice can be made for one pair in a four-center product, it cannot for the second. There remains a single translation for this potential in one auxiliary of the two in a product representing a four-center integral and none otherwise. This method obviates the need to evaluate infinite series that arise from the orbital translations efficiently. They have been eliminated in the Coulomb operator resolution approach, since only orbitals on two centers remain in the one-electron overlap-like auxiliaries. These can be evaluated with no orbital translation, in prolate spheroidal co-ordinates, or by Fourier transformation [67, 71].

8.5.

Numerical Results of Coulomb Resolutions: Efficiency

First a test system is studied, built up of four hydrogen atoms. The second example is the full RHF calculation of CH3 F using the Coulomb resolutions. Consider the H2 molecule and its dimer/agregates. In an s-orbital basis, all two-center integrals are known analytically, because they can be integrated by separating the variables in prolate spheroidal co-ordinates. A modest s-orbital basis is therefore chosen, simply for accuracy demonstration on a rapid calculation, for which some experimental data could be corroborated. The purpose of this section is to compare evaluations using the Coulomb resolution to the exact values, obtained analytically. The IBM Fortran compiler used is assumed to be reliable to 14 decimals in double precision. The worst values in the Coulomb resolution approximation have 10 correct decimals for two-center integrals with a 25-term sum. Timings are then compared for translation of a Slater type orbital basis to a single center (STOP) [59] with the Poisson equation solution using a DIM (Diatomics in molecules or atom pair) strategy and finally to show that the overlap auxiliary method is by far the fastest approach, for a given accuracy (the choice adopted is a sufficient six decimals, for convenient, reliable output). H2 molecule with interatomic distance of 1.402 atomic units (a.u.). One and two-center Coulomb integrals may be obtained analytically and Coulomb resolution values compare well with them [66]. The two-center exchange integrals are dominated by an exponential of the interatomic distance and thus all have values close to 0.3. The table is not the full set. All index ‘15‘ terms, involving 1sa1 (1) 1sb1 (2) are given, to illustrate symmetry relations. Note that this is by no means the best possible basis set for H2 , since it is limited to

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Table 1. Atomic exchange integrals (6 distinct single center values between pairs of different AOs). AOs (zeta) 1sa1 1.042 999 1sa2 1.599 999 2sa1 1.615 000 2sa2 1.784 059 1sb1 1.042 999 1sb2 1.599 999 2sb1 1.615 000 2sb2 1.784 059

Label 1 2 3 4 5 6 7 8

[a(1)b(2)a′ (2)b′ (1)] 1212 1313 1414 2323 2424 3434 2121 3232

Value 0.720 716 0.585 172 0.610 192 0.557 878 0.607 927 0.602 141 0.720 716 0.557 878

Table 2. Two-center exchange integrals. All pair permutations possible. Some are identical by symmetry. Labels 1515 1516 1517 1518 1527 1528 1538 2525 2516 2517 2518

Value 0.319 902 0.285 009 0.325 644 0.324 917 0.291 743 0.293 736 0.329 543 0.260 034 0.254 814 0.290 533 0.290 149

l = 0 functions (simply to ensure that even the two-center exchange integral has an analytic closed form). The total energy obtained for the isolated H2 molecule is -1.1284436 a.u. as compared to a Hartree-Fock limit estimate of -1.1336296 a.u. Nevertheless, the Van der Waals well, observed at 6.4 au with a depth of 0.057 kcal/mol (from Raman studies) is quite reasonably reproduced [74]. Dimer geometry: rectangular and planar. Distance between two hydrogen atoms of neighboring molecules: 6 a.u. Note that this alone justifies the expression dimer, the geometry corresponds to two almost completely separate molecules. However, the method is applicable in any geometry (for 3 a.u. all three- and four-center integrals evaluated by Coulomb resolution agree with those of STOP to at least 6 decimals). Timings on an IBM RS6000 Power 6 workstation, for the dimer (all four-center integrals in msec): STOP: 12 POISSON: 10 OVERLAP: 2. Total dimer energy: -2.256998 a.u. This corresponds to a well-depth of 0.069 Kcal/mol, which may be considered reasonable in

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Table 3a. Orbital exponents. AO No. 01 02 3-5 06 07 8-10 H

n 1 2 2 1 2 2 1

l 0 0 1 0 0 1 0

m 0 0 m 0 0 m 0

zeta 5.6727 1.6083 1.5679 8.5600 2.5600 2.5200 1.2400

Table 3b. Selected examples of three-center exchange integrals. Integral h2sC 2sF |2sC 1sHa i h2sC 2sF |2sC 1sHa i h2sC 1sF |1sC 1sHa i h2sC 1sF |2sC 1sHa i h1sC 2pzF |2pzC 1sHa i h2sC 2pzF |2pzC 1sHa i h2pzC 1sF |1sC 1sHa i h2pzC 1sF |2sC 1sHa i h1sC 1sF |1sC 1sHa i h1sC 1sF |2sC 1sHa i h1sC 2sF |1sC 1sHa i h1sC 2sF |2sC 1sHa i h2pzC 2sF |1sC 1sHa i h2pzC 2sF |1sC 1sHa i

Value 0.4970 48510 ×10−1 0.8420 56635 ×10−2 0.5737 90540 ×10−3 0.3789 18525 ×10−2 0.1587 58344 ×10−2 0.5258 34208 ×10−2 0.1025 32536 ×10−2 0.6772 76818 ×10−2 0.1099 00118 ×10−6 0.6794 54131 ×10−6 0.1446 31297 ×10−2 0.4235 59085 ×10−2 0.1112 10955 ×10−1 0.6738 14908 ×10−1

Integral h2sF 1sHa |1sF 2sC i h2sF 1sHa |2sF 2sC i h2sF 1sHa |2pzF 2sC i h2sF 1sHa |1sF 2pzC i h2sF 1sHa |2sF 2pzC i h2sF 1sHa |2pzF 2pzC i h1sHa 2sF |1sHa 2sC i h1sHa 2sF |1sHa 2pzC i h1sHa 2pzF |1sHa 2pzC i h1sF 1sHb |2sF 1sC i h1sHb 2sF |1sHb 1sC i h2sC 1sHa |1sC 1sHb i h1sC 1sHa |1sC 1sHb i h2sC 1sHa |2sC 1sHb i

Value 0.1014 05594 ×10−2 0.9341 35949 ×10−2 -0.8442 95091 ×10−2 0.1813 23479 ×10−2 0.1379 64387 ×10−1 -0.1135 01125 ×10−1 0.1252 319411 ×10−1 -0.1591 49899 ×10−2 0.1772 90873 ×10−2 0.2287 77210 ×10−4 0.1939 63837 ×10−2 0.2034 841982 ×10−1 0.7154 932331 ×10−2 0.1137 390852

view of the basis set.

8.6.

Selected Exchange Integrals for the CH3 F Molecule (Evaluated Using the Coulomb Resolution)

Geometry and exponents are those of previous work [75]: Tetrahedral angles, with C-H 2.067 and C-F 2.618 a.u. No symmetry is assumed but geometric relationships are observed, as well as those due to m values, at least to the nano-Hartree accuracy chosen. For illustrative purposes, three-center exchange integrals are tabulated in a real basis. Timings on IBM RS6000 Power 6 workstation for all two-electron integrals: STOP: 1.21 s, OVERLAP: 0.17 s. All the two-electron integrals are identical to better than six significant figures with those obtained using the STOP software package [59]. The factor limiting precision in this study is the accuracy of input. The values of Slater exponents and geometric parameters are required to at least the accuracy demanded of the

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integrals and the fundamental constants are needed to greater precision.

8.7.

Conclusions

A remarkable gain in simplicity is provided by Coulomb operator resolutions [66], that now enables the exponential type orbital translations to be completely avoided in ab initio molecular electronic structure calculations. This breakthrough that Coulomb resolutions represent (in particular with the convenient choice of Laguerre polynomials) in the ETO algorithm strategy stems from a wellcontrolled approximation, analogous to the resolution of the identity. The convergence has been shown to be rapid in all cases [67]. The applications to H2 dimer Van der Waals complexes and CH3 F uses a general code within the STOP package [59]. They show the Coulomb resolution can be used to give fast and accurate results for basis sets of s and p Slater type orbitals. Generalisation is in progress. Numerical vales for the H2 dimer geometry and interaction energy agree well with complete ab initio potential energy surfaces obtained using very large Gaussian basis sets and data from vibrational spectroscopy [74].

9.

Explicitly Correlated Methods for Molecules

The application and development of such methods to determine accurately the ground and excited states, and properties of diatomic and triatomic molecules is very promising and more interesting for the Computational Chemist than the atomic case. There is nowadays a growing interest in this field. Subroutines and programs which perform these calculations are often requested in the community. The investigation of these integrals should be approached within the Molecular Orbital method (MO) [76], because the MO wave function is the simplest wave function for a molecular system. As Coulson [77] discussed, the MO method permits the visualization of electrons and nuclei and interpretation of individual electrons and their orbital exponents better than the wave functions written in elliptical coordinates. The wave functions constructed with elliptical orbitals are of two types, the so-called James-Coodlige [78] wave functions (one-alpha exponent), recently extended to the twoalpha case [33], and Kolos-Wolniewick [79,80] wave functions (with both orbital exponents alpha, and beta ). Both have been applied to the H2 molecule. The elliptical wave functions are the natural representation of a two-center problem but for three-center and larger molecules the use of the MO method becomes necessary. Frost [81] used the MO method and the Correlated Molecular Method (CMO) in H2 calculations. About the extension of the method he wrote: ”The extension of CMO-type wave functions to more complex molecules does not seem feasible at the present time. The new integrals which will be introduced would involve more than two centers if more nuclei were involved and higher atomic orbitals than 1s if more electrons were considered, and their evaluation would be extremely difficult”.

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Recently, impressive calculations using Hylleraas wave functions have been done for H2 , see Table 4. Hylleraas [33], the Iterative Complement Iteration method (ICI) [82], and explicitly correlated Gaussian (ECG) [83] calculations of the hydrogen molecule, Hylleraas calculations on HeH+ and some other species [84] leading to −2.9710784698 a.u. using 9576 configurations and calculations of He2 using 4800 optimized ECG configurations with energy −5.80748359014 a.u. [83] achieved the highest known accuracy in molecules (picohartree accuracy is more than that of chemical measurements, e.g. a micro cm−1 , a nano eV or micro cal/mol. Although one must recall that in the calculation of properties according to Drake [85], only half of the digits of the energy are kept). Note also that input exponents, distances and some fundamental constants may limit accuracy of calculations compared with measurements and that molecules may not be rigid. Eventually, dynamics and the effect of the Born-Oppenheimer approximation should be included. Hylleraas-CI (Hy-CI) was applied in 1976 to LiH molecule by Clary [32] using elliptical STOs. For two-center molecules the three-electron and four-electron integrals occurring in the Hy-CI have been developed by Budzinski [39]. Another type of explicitly correlated wave functions are the ones that use Gaussian orbitals. Clementi et al extended the Hy-CI to molecules using Gaussian orbitals [86], and applied it to the calculation of H3 . The ECG wave function is appropriate also for molecules [83, 87], as the inter-electronic distance r12 is a Gaussian exponent. This leads to results, which are comparable with Hylleraas calculations [83]. The R12 -wave function proposed by Kutzelnigg and Klopper [88, 89] has the merits to fulfill the cusp condition, to use Gaussian functions avoiding the three- and four-center integration problems, and to include precisely r12 , involving electrons 1 and 2, close to the nucleus, where the probability that r12 = 0 is larger, also these electrons are present at any system starting from helium atom. The r12 variation influences energy. The R12 wave function, developed for molecular calculations is nowadays widely used and combined with all kinds of methods. The occurring three- and four-electron integrals are calculated in terms of two-electron ones. Due to the use of a single r12 value, the accuracy achieved for atomic calculations is lower than the accuracy of Hy and Hy-CI calculations. Recent improvements of the method [90]- [92] can achieve microhartree accurate energy results for chemically interesting systems. Table 4. Highly accurate calculations on the H2 molecule with different types of wave functions at R=1.4011 a.u. Authors 1933 James and Coolidge 1960 Kolos and Roothaan 1968 Kolos and Wolniewicz 1995 Wolniewicz 2006 Sims and Hagstrom 2007 Nakatsuji 2008 Cencek and Szalewicz

type w. f. JC KR KW KW JC ICI ECG,opt

Confs. 5

833 7034 6776 4800

Energy (a.u.) -1.1735 -1.17214 -1.174475 -1.17447467 -1.17447593139984 -1.17447571400027 -1.17447571400135

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Short wave function expansions lead to very good results. When a large number of configurations are used (up to 10000) the energy results are beyond pico-hartree accuracy, while the CI wave function would need in the order of millions of configurations.

10.

Highly Accurate Calculations Using STOs

Another problem appearing in these calculations is the digital erosion. For many operations and subtraction numbers of similar value some digits can be lost leading to erroneous results. Quadruple precision avoids this, about 30 decimal digits are correct on our computer. Other possibility is high precision arithmetic software. Some programs are available like Bailey’s MPFUN [93], the Brent and Miller program packages [94, 95]. One example of the use of Slater orbitals in the present are the highly accurate calculations of small molecules using explicitly correlated wave functions i.e. wave functions where the inter-electronic coordinate ri j is included explicitly in the wave function. These are the Hylleraas and Hylleraas-CI wave functions, ICI method, compared with the explicit correlated Gaussians ECG and the R12 method.

11.

Closing Remarks

We conclude with the words of G. Berthier: GTOs are like medicine, you have to use them as long as they are healing, but once they don’t work any more, you much change them, Gaston Berthier, Interview, Paris, 2nd June 1997. Recently, a whole book ”Recent Advances in Computational Chemistry: Molecular Integrals over Slater Orbitals ” was dedicated to a mathematical review of methods of integration over Slater orbitals and Hylleraas wave functions [96].

Acknowledgements The authors would like to thank very much Profs. Milan Randic, Ante Graovac, Roberto Todeschini and Peter Otto for their interest in Slater orbitals.

References [1] Slater J.C. Central fields and Rydberg formulas in wave mechanics. Phys. Rev. 1928 31, 333-343. [2] Randic M. International Conference on ETO Multicenter Molecular Integrals, Tallahassee, 1981, C. A. Weatherford and H. Jones Eds.: Reidel, Dortdrech, 1982, pp. 141-155. [3] Scrocco E. and Salvatti O. Ric. Sci. 1951 21, 1629; ibid 1952 22, 1766; ibid 1953 23, 98;

Molecular Integrals over Slater-Type Orbitals

83

[4] Petrongolo C., Scrocco E. and Tomasi J. Minimal-Basis-Set LCAOSCFMO calculations for the ground state of O3 , NO2 , NOF, and OF2 molecules. J. Chem. Phys. 1968 48, 407-411, and Refs. therein. [5] Roothaan C.C.J. A study of two-center integrals useful in calculations on molecular structure. I. J. Chem. Phys. 1951 19, 1445-1458. [6] Wahl A.C., Cade P.E., and Roothaan C.C.J. Study of two-center integrals useful in calculations on molecular structure. V. General methods for diatomic integrals applicable to digital computers. J. Chem. Phys. 1964 41, 2578-2599. [7] R¨udenberg K. A study of two-center integrals useful in calculations of molecular structure. II. The two-center exchange integrals. J. Chem. Phys. 1951 19, 1459-1477. [8] Kotani M., Amemiya A. and Simose T. Tables of integrals useful for the calculations of molecular energies. Proc. Phys. Math. Soc. Japan 1938 20 extra No. 1, 1-70; Kotani M., and Amemiya A. Tables of integrals useful for the calculations of molecular energies. II. Proc. Phys. Math. Soc. Japan 1940 22 extra No. 1, 1-28. [9] Barnett M.P., and Coulson C.A. The evaluation of integrals occurring in the theory of molecular structure. Parts I & II. Phil. Trans. R. Soc. Lond. A 1951 243, 221-249. [10] L¨owdin P.O. Quantum theory of cohesive properties of solids. Adv. Phys. 1956 5, 1172. [11] Harris F.E. Gaussian wave functions for polyatomic molecules. Rev. Mod. Phys. 1963 35, 558-568. [12] Mulliken R.S, and Roothaan C.C.J. Broken bottlenecks and the future of molecular Quantum Mechanics. Proc. Natl. Acad. Sci. (U.S.) 1959 45, 394-398. [13] Boys S.F. Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system. Proc. Roy. Soc. (London) 1950 A200, 542-554. [14] Shavitt I., and Karplus M. Multicenter integrals in molecular Quantum Mechanics. J. Chem. Phys. 1962 36, 550-551. [15] Clementi E. and Raimondi D.L. Atomic screening constants from SCF functions. J. Chem. Phys. 1963 38, 2686-2689. [16] Scherr C.W. An SCF LCAO MO study of N2 . J. Chem. Phys. 1955 23, 569-578. [17] Smith S.J. and Sutcliffe B.T. The Development of Computational Chemistry in the United Kingdom, in Reviews in Computational Chemistry, D.B. Boy D.B. Lipkowitz K.B. Eds. 1997 10 pp. 271-316. [18] POLYATOM: Newmann D.B., Basch H., Korregay R.L., Snyder L.C., Moskowitz J., Hornback C., and Liebman P. Quantum Chemistry Program Exchange, Indiana University, No. 199; Csizmadia I.G., Harrison M.C., Moscowitz J.W., Sutcliffe B.T. Non-empirical LCAO-MO-SCF-CI calculations on organic molecules with Gaussian type functions. Theoret. Chim. Acta (Berl.) 1966 6, 191-216 .

84

¨ P.E. Hoggan, M.B. Ruiz and T. Ozdo˘ gan

[19] IBMOL: David D.J. CDC 6600 Version. Technical Report of C.C. ENSJF and Lab. de Chimie ENS, Paris 1969; Clementi E., Davis D.R. Electronic structure of large molecular systems. J. Comput. Phys. 1967 1, 223-244; Veillard A. IBMOL: Computation of wave-functions for molecules of general geometry, Version 4; IBM Research Laboratory, San Jose. [20] McLean A.D., Yoshimine M., Lengsfield B.H., Bagus P.S. and Liu B. ALCHEMY II: A Research Tool for Molecular Electronic Structure and Interactions, in Modern Techniques in Computational Chemistry, MOTECC 91, Clementi E. Ed., Elsevier B.V. (Leiden): 1991 pp. 233-353. [21] DERIC: Diatomic Integrals over Slater-Type Orbitals. Hagstrom S.A. Quantum Chemistry Program Exchange, Indiana University No. 252, 1974. [22] GAUSSIAN 70: Hehre W. J., Lathan W.A., Ditchfield R., Newton M.D., and Pople J.A. Gaussian 70, Quantum Chemistry Program Exchange, Program No. 237 1970. [23] STOP: A slater-type orbital package for molecular electronic structure determination. Bouferguene A., Fares M., and Hoggan P.E., Int. J. Quantum Chem. 1996 57, 801810. [24] Homeier H.H.H., Joachim Weniger E., and Steinborn E.O. Programs for the evaluation of overlap integrals with B functions, Comp. Phys. Comm. 1992 72, 269-87; Homeier H.H.H and Steinborn E.O. Programs for the evaluation of nuclear attraction integrals with B functions, Comp. Phys. Comm. 1993 77, 135-151. [25] SMILES: Fern´andez Rico J., L´opez R., Ema I., and Ram´ırez G. Reference program for molecular calculations with Slater-type orbitals. J. Comp. Chem. 1998 19(11), 12841293; ibid Electrostatic potentials and fields from density expansions of deformed atoms in molecules. J. Comp. Chem. 2004 25, 1347-1354. [26] CADPAC: The Cambridge Analytic Derivatives Package. Amos R.D., Alberts I.L., Andrews J.S., Colwell S.M., Handy N.C., Jayatilaka D., Knowles P.J., Kobayashi R., Laidig K.E., Laming G., Lee A.M., Maslen P.E., Murray C.W., Rice J.E., Simandiras E.D., Stone A.J., Su M.D. and Tozer D.J, Cambridge, 1995. [27] ADF: Amsterdam Density Functional, available at http://www.scm.com/. [28] ATMOL: Atomic and Molecular CI calculations. Bunge C.F. et al, 1965-2005. [29] Pinchon D. and Hoggan P.E. Rotation matrices for real spherical harmonics: general rotations of atomic orbitals in space-fixed axes. J. Phys. A: Math. Theor. 2007, 40, 1597-1610 . [30] Ozdogan T. Unified treatment for the evaluation of arbitrary multicenter molecular integrals over Slater-type orbitals with noninteger principal quantum numbers. Int. J. Quantum Chem. 2003 92, 419-427. [31] Hagstrom S.A. and Shull H. The nature of the two-electron chemical bond. III. Natural orbitals for H2 . Rev. Mod. Phys. 1963 35, 624-629.

Molecular Integrals over Slater-Type Orbitals

85

[32] Clary D.C. Variational calculations on many-electron diatomic molecules using Hylleraas-type wave functions. Mol. Phys. 1977 34, 793-811. [33] Sims J.S. and Hagstrom S.A. High precision variational calculations for the BornOppenheimer energies of the ground state of the hydrogen molecule. J. Chem. Phys. 2006 124, 094101 (2006). [34] Ruiz M.B. and Peuker K. Mathematical techniques in molecular calculations using Slater orbitals, in Recent Advances in Computational Chemistry: Molecular Integrals ¨ over Slater Orbitals, Ozdogan T. and Ruiz M.B. Eds., Transworld Research Network: Kerala, India, 2008 pp. 100-144. [35] Filter E. and Steinborn E.O. Extremely compact formulas for molecular two-cenetr one-electron integrals and Coulomb integrals over Slater-type atomic orbitals. Phys. Rev. A 1978 18 1-11. [36] Avery J. Hyperspherical Harmonics and Generalized Sturmians, Kluwer: Boston, 2000; Avery J. Many-center Coulomb Sturmians and Shibuya-Wulfman integrals. Int. J. Quantum Chem. 2004 100, 121-130; Red E. and Weatherford C.A. Derivation of a general formula for the Shibuya-Wulfman matrix. Int. J. Quantum Chem. 2004 100, 208-213. [37] Shull H. and L¨owdin P.O. Superposition of configurations and natural spin orbitals. Applications to the He problem. J. Chem. Phys. 1959 30, 617-626. [38] Sims J.S. and Hagstrom S.A. Combined Configuration-InteractionHylleraas-type wave-function study of the ground state of the beryllium atom. Phys. Rev. A 1971 4, 908-916; Sims J.S. and Hagstrom S.A. One-center ri j integrals over Slater-type orbitals. J. Chem. Phys. 1971 55, 4699-4710. [39] Budzinski J. Evaluation of two-center, three- and four-electron integrals over Slatertype orbitals in elliptical coordinates. Int. J. Quantum Chem. 2004 97, 832-843. [40] Smeyers Y.G. About evaluation of many-center molecular integrals. Theoret. chim. Acta (Berl.) 1966 4, 452-459. [41] Harris F.E. and Michels H.H. Multicenter integrals in Quantum Mechanics. I. Expansion of Slater-type orbitals about a new origin. J. Chem. Phys. 43 1965, S165-S169. [42] Guseinov I.I. Unified analytical treatment of multicenter multielectron integrals of central and noncentral interaction potentials over Slater orbitals using Ψα -ETOs. J. Chem. Phys. 2003 119, 4614-4619. [43] Boys S.F., Cook G.B., Reeves C.M., and Shavitt I. Automatic Fundamental Calculations of Molecular Structure. Nature 1956 178, 1207-1209. [44] Shavitt I. Methods in Computational Physics 2, NY, Eds. Adler, S.Fernbach, M. Rotenberg, Academic Press: NY 1963 pp. 1-45.

86

¨ P.E. Hoggan, M.B. Ruiz and T. Ozdo˘ gan

[45] Berlu L., Safouhi H. and Hoggan P. Fast and accurate evaluation of three-center, twoelectron Coulomb, hybrid and three-center nuclear attraction integrals over Slater-type orbitals using the S transformation. Int. J. Quantum Chem. 2004, 99, 221-235. [46] Safouhi H. and Berlu L. The Fourier transform method and the SD approach for the analytical and numerical treatment of multicenter overlap-like quantum similarity integrals. J. Comp. Phys. 2006 216, 19-36. [47] Levin D., Sidi A. Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series. Appl. Math. Comp. 1981 9, 175-215; Sidi A. The numerical evaluation of very oscillatory infinite integrals by extrapolation. Math. Comp. 1982 38, 517-529. [48] Guseinov I.I. New complete orthonormal sets of exponential-type orbitals and their application to translation of Slater orbitals. Int. J. Quantum Chem. 2002 90, 114-118. ¨ [49] Ozdogan T., and Ruiz M.B. Evaluation of three-center repulsion integrals over Slater orbitals, in preparation. [50] Fern´andez Rico J., L´opez R., Ram´ırez G., Ema I. and Lude˜na E.V. Analytical method for the representation of atoms-in-molecules densities. J. Comp. Chem. 2004 25, 1355-1363. [51] Pinchon D. and Hoggan P.E. Gaussian approximation of exponential type orbitals based on B functions. Int. J. Quantum Chem. 2009 109, 135-148. [52] Kato T. On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Applied Math. 10 151-177. [53] Kato T. Schr¨odinger Operators, Graffi Ed., Springer Verlag: Berlin, 1985 pp. 1-38. [54] Agmon S. Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schr¨odinger Operators, Princeton University: Princeton, NJ, 1982. [55] Stevens R.M. Geometry optimization in the computation of barriers to internal rotation. J. Chem. Phys. 1970 52, 1397-1402. [56] Jones H.W. International Conference on ETO Multicenter Integrals, (Tallahassee, USA 1981), Weatherford C.A. and Jones H.W. Eds., Reidel: Dordrecht 1982; Int. J. Quantum Chem., Special Issue in memory of Jones H.W., Weatherford C.A. and Jones H.W. Eds., 2004 100(2), pp. 63-243. [57] Slater J.C. Atomic shielding constants. Phys. Rev. 1930 36, 57-64. [58] Slater J.C. Analytic atomic wave functions. Phys. Rev. 1932 42, 33-43. [59] Boufergu`ene A. and Hoggan P.E., QCPE, Programme No. 667, 1996. [60] Weatherford C.A., Red E., Joseph D. and Hoggan P.E. Poisson’s equation solution of Coulomb integrals in atoms and molecules. Mol. Phys. 2006 104, 1385-1389.

Molecular Integrals over Slater-Type Orbitals

87

[61] Avery J. and Avery J. Hyperspherical Harmonics and Generalized Sturmians , Kluwer: Boston 2007. [62] Tully J.C. Diatomics-in-molecules potential energy surfaces. I. First-row triatomic hydrides. J. Chem. Phys. 1973 58, 1396-1410. [63] Weatherford C.A., Red E. and Hoggan P.E. Solution of Poisson’s equation using spectral forms. Mol. Phys. 103 (2005) 2169-2172. [64] Cesco J.C., P´erez J.E., Denner C.C., Giubergia G.O. and Rosso A.E. Error bounds for direct evaluation of four-center molecular integrals. Applied Num. Math. 2005 55(2), 173-190, and references therein. [65] Shao Y., White C.A. and Head-Gordon M. Efficient evaluation of the Coulomb force in density-functional theory calculations. J. Chem. Phys. 2001 114, 6572-6577. [66] Vagranov S.A., Gilbert A.T.B., Duplaxes E. and Gill P.M.W. Resolutions of the Coulomb operator. J. Chem. Phys. 2008 128, 201104. [67] Gill P.M.W. and Gilbert A.T.B. Resolutions of the Coulomb Operator. II The Laguerre Generator, Chem. Phys. 2009 356 86-92. [68] Pinchon D. and Hoggan P.E., unpublished code. [69] Talman J.D. Expression for overlap integrals of Slater orbitals. Phys. Rev. A 1993 48, 243-249. [70] Whittaker E.T. and Watson G.N., A Course in Modern Analysis, Cambridge University Press, 4th Ed.: Cambridge, England, 1990. [71] Pinchon D. and Hoggan P.E., Translating Coulomb potentials over Slater type orbitals, in preparation. [72] Guseinov I.I., Ozmen A., Atav U., Uksel H. Computation of overlap integrals over Slater-type orbitals using auxiliary functions. Int. J. Quantum Chem. 1998 67, 199204. [73] Hoggan, P.E. DSc Thesis, 1991, Theoretical study of physisorption and its influence on reactivity. Appendix 2. (in French available at INIST). [74] Hinde R.J. A six-dimensional H2 -H2 potential energy surface for bound state spectroscopy. J. Chem. Phys. 2008 128, 154308. [75] Absi N. and Hoggan P.E. Analytical evaluation of molecular electronic integrals using Poisson’s equation: Exponential-type orbitals and atom pairs. Int. J. Quantum Chem 2006 106, 2881-2888. [76] Mulliken R.S. Electronic structures of polyatomic molecules and valence VI. On the method of Molecular Orbitals. J. Chem. Phys. 1935 3, 375-378.

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¨ P.E. Hoggan, M.B. Ruiz and T. Ozdo˘ gan

[77] Coulson C.A. The energy and screening constants of the hydrogen molecule. Trans. Faraday Soc. 1937 33, 1479-1492. [78] James H.M. and Coolidge A.S. The ground state of the hydrogen molecule. J. Chem. Phys. 1933 1, 825-835. [79] Kolos W. and Roothaan C.C.J. Correlated orbitals for the ground state of the hydrogen molecule. Rev. Mod. Phys. 1968 32, 205-210. 3 + 1 [80] Kolos W. and Wolniewicz L. Potential curves for the χ1 ∑+ g , b ∑u , and C Πu states of the hydrogen molecule. J. Chem. Phys. 1965 43, 2429-2441; ibid 1968 49, 404-410.

[81] Frost A.A. and J. Braunstein J. Hydrogen molecule energy calculation by correlated molecular orbitals. J. Chem. Phys. 1951 19, 1133-1138. [82] Kurokawa Y., Nakashima H. and Nakatsuji H. Free iterative-complement-interaction calculations of the hydrogen molecule. Phys. Rev. A, 2005 72, 062502. [83] Cencek W. and Szalewicz K. Ultra-high accuracy calculations for hydrogen molecule and helium dimer. Int. J. Quantum Chem. 2008 108, 2191-2198. [84] Zhou B.L., Zhu J.M., and Yan Z.C. Ground state energy of HeH+ . Phys. Rev. A 2006 73 064503. [85] Drake G.W.F. High precision theory of atomic helium. Phys. Scr. 1999 T83, 83-92. [86] Frey D., Preiskorn A., Lie G.C., and Clementi E. HYCOIN: Hylleraas Configuration Interaction method using Gaussian functions, in Modern Techniques in Computational Chemistry: MOTECC-90, Clementi E. Ed., ESCOM Science Publ.: Leiden 1990, Chapter 5, pp. 57-97. [87] Rychlewski J. and Komasa J. Explicitly correlated functions in variational calculations, in Explicitly Correlated Wave Functions in Chemistry and Physics, Rychlewski J. Ed., Kluwer Academic Publishers: Netherlands 2004, pp. 91-147. [88] Kutzelnigg W. r12 -dependent terms in the wave function as closed sums of partial wave amplitudes for large l. Theor. Chim. Acta 1985 68, 445-469. [89] Klopper W. and Kutzelnigg W. Møller-Plesset calculations taking care of the correlation cusp. Chem. Phys. Lett. 1987 134, 17-22. [90] Klopper W. and Noga J. Linear R12 terms in Coupled Cluster theory, in Explicitly Correlated Wave Functions in Chemistry and Physics, Rychlewski J. Ed., Kluwer Academic Publishers: Netherlands 2004, pp. 149-183. [91] Cardoen W., Gdanitz R.J., and Simons J. Transition-state energy and geometry, exothermicity, and van der Waals wells on the F + H2 → FH + H ground-state surface calculated at the r12 -ACPF-2 level. J. Phys. Chem. A 2006 110, 564-571. [92] Explicitly Correlated Wave Functions in Chemistry and Physics, Rychlewski J. Ed., Kluwer Academic Publishers: Netherlands, 2004.

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[93] Bailey D.H. High-precision software directory. Available from: http://crd.lbl.gov/˜dhbailey/mpdist/mpdist.html [94] Brent R.P. A Fortran multiple-precision arithmetic package, ACM Transactions on Mathematical Software (TOMS), v.4 n.1, pp.57-70, 1978. [95] Miller A.J. Alan Miller’s Fortran software. Available from: http://users.bigpond.net.au/amiller/ [96] Recent Advances in Computational Chemistry: Molecular Integrals over Slater Or¨ bitals, Ozdogan T. and Ruiz M.B. Eds., Transworld Research Network: Kerala, India, 2008.

In: Electrostatics: Theory and Applications Editor: Camille L. Bertrand, pp. 91-110

ISBN 978-1-61668-549-2 c 2010 Nova Science Publishers, Inc.

Chapter 5

T UNNELING DYNAMICS AND I TS S IGNATURES IN C OUPLED S YSTEMS S. Ghosh and S.P. Bhattacharyya∗ Department of Physical Chemistry Indian Association for the Cultivation of Science Jadavpur, Calcutta 700 032, INDIA

Abstract It has been demonstrated through numerical experiments that tunneling in a symmetric double well may be either suppressed or enhanced by a Morse oscillator coupled to it depending on the form of coupling. An external time varying electric field that causes 0+ − 0− transition in the double well affects the tunneling rate. A well defined minimum in the rate is observed for λ = λc for which maximum energy transfer from the double well to the Morse mode takes place. If the field is chosen to cause 0 → 1 transition in the Morse mode, the dissociation probability is influenced by the tunneling mode, being maximized for a particular value λ = λ′c . In the uncoupled system, field with similar intensity practically fails to cause any dissociation. The tunneling dynamics is analyzed in a situation when the particle has a coordinate dependent mass as is often assumed in the charge transport in hetero-structures. The tunneling time or rate are seen to be very significantly affected by the nature of the coordinate dependence of the tunneling mass.

1.

Introduction

As a purely quantum phenomenon, tunneling is ubiquitous in microscopic systems. It pervades all areas in physics, chemistry and biology and ever since its use in nuclear physics in explaining the decay of α particles from atomic nuclei, the importance of tunneling has been increasingly recognized. The study of electron tunneling in condensed matter physics has led to the Josephson effect and the tunneling diode [1] . The tunneling of proton or hydrogen atom has often been invoked in chemistry to explain unusual features of chemical ∗ E-mail

address: [email protected]

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reaction rates or mechanisms. In fact, the atom tunneling phenomenon has been recognized to be important in science of various types of materials and biology. Starting from atom tunneling reactions in quantum solid hydrogen or in solid or liquid Helium, there have been studies relating to rather unusual aspects of tunneling reactions of organic substances, tunneling insertion reaction of carbenes and heavy particle tunneling. The role of atom tunneling reaction in vitamin-C in the suppression of mutation or of vitamin-E in its antioxidant, pro-oxidant and regeneration reactions have attracted serious attention from biologists. The possible occurrence of spontaneous tunneling elimination of hydrogen molecules from hydrocarbon cations has led to serious questions about our conventional idea of a stable chemical structure [2]. The relevance of tunneling in the interpretation of molecular and crystal structure in very low temperature regimes can hardly be overestimated. It would be appropriate therefore to review briefly how the idea of tunneling was developed and exploited in different fields of science.

2.

Historical Development

A. Tunneling in Physics The theory of α radioactivity proposed by Gamow [3] explained the law of exponential −Γt by solving the Schrodinger decay P(t) = N(t) ¨ equation for the α particle inside the N0 = e nucleus where the attractive nuclear force and coulomb repulsion were assumed to provide an effective barrier confining the α particle. Gamow imposed the ’outgoing wave’ boundary condition at large distances from the center of the nucleus and found that the Schrodinger ¨ equation does not have solutions for the real energies while for complex energies, it had solutions. Gamow interpreted the imaginary part of the energy as the decay width Γ2 and obtained a relation between Γ2 and the energy of the emitted α particle. The use of complex energy meant the use of a non-hermitian hamiltonian and the idea was criticized as quantum mechanics worked with hermitian operators. The same result was later obtained by Bohr by considering states with real energies and working with a hermitian hamiltonian [4]. The idea of resonant tunneling was introduced by Gurney who realized that particles with low energies that match with quasi-stationary energies of the nucleus could easily penetrate the barrier [5]. The importance of the idea in artificial disintegration can hardly be overestimated. The idea of tunneling was soon exploited in other areas of physics. Notably, many attempts were made to relate the dynamics of electron current in the metal semi-conductor systems to the tunneling of electrons in solids. The discovery of transistors in 1947 rekindled interest in the tunneling of electrons in solids the occurrence of which was conclusively proved by L. Esaki (1957) who discovered the tunneling diode [6]. Close on the heels of the discovery of tunneling diode, Giaever found if one or both the metals are superconducting, the voltage-current plots could lead to measurement of energy gaps in superconductors [7]. Josephson discovered that the superconductors separated by a thin layer of insulating oxide provide a system in which a second current (over and the above the Giaever’s current), the so-called supercurrent, exists and that it is caused by tunneling of electrons in pairs [8]

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B. Tunneling in Chemistry Tunneling has a long history in chemical kinetics. Traditionally, curvature in Arrhenius plots of rate constants has been interpreted as a signature of tunneling. It is generally very difficult to observe such curvatures in Arrhenius plots for gas phase chemical reactions while such curvatures are a common-place occurrence in chemical reactions in condensed phases below 100 K. Accurate theoretical calculations, however, indicate that tunneling can contributes significantly to the reaction rates even at room temperature where the Arrhenius plots are very nearly linear [8]. In fact, ’tunneling’ can be taken to be synonymous with a chemical reaction occurring at energies less than the barrier energy (Ea ). The barrier arises on the 3N − 6 dimensional potential energy surface in the Born-Oppenheimer approximation as the nuclei of the N atom reactive system move breaking and making bonds. Traditionally one identifies the one dimensional minimum energy path (x) as the reaction path along which an effective potential energy Ve f f (x) is defined by adding the vibrational energies ε(x) along the reaction path associated with the nuclear motion perpendicular to it [9]. This vibrationally adiabatic approximation reduces the multidimensional problem into a one dimensional problem, for which tunneling has a unique definition. The tunneling probabilities calculated using the idea described above are generally a bit too small due to the neglect of the reaction path curvature. Several alternative approaches have been explored for better representation of the tunneling path, for example, the least action ground state method [10] and the ’tunneling tube method’ [11]. The barrier energy on the reaction −Ea

path can be identified with the activation energy (Ea ) in the Arrhenius rate law k = Ae Kb T which has been recognized as the central law of chemical kinetics. Many attempts have been made to derive the central law of chemical kinetics. The transition state theory (TST) of Eyring [12] was the first attempt in this direction. The TST was derived assuming complete thermodynamic equilibrium wherein the possibility of reverse transition was neglected, which was later taken care of in an ad hoc manner by introducing a transmission coefficient in the pre-exponential factor. Kramers’ [13] treated elementary rate process in the presence of a medium as a generalized Brownian motion in a potential field and showed how the rate constant would be influenced by the viscosity of the medium. Kramers’ results have been derived from the TST applied to an extended system of the reactants and reservoir of oscillators which provide a frictional force and a random force along the reactive mode [14]. Kim and Hynes [15], Truhler etal. [16] introduced additional coordinates describing the dynamics of the solvent mode along with the reagent mode within the framework of TST and derived results equivalent to those obtained by Kramers. Chemical reactions at sufficiently low temperatures are marked by remarkably special features. Goldanski [17], showed that the rate constant of a chemical process has a low temperature limit (non-zero) as T → 0 and introduced the concept of crossover temperature (Tc )which divides the whole temperature interval into over and under barrier regimes. Goldanskii assumed that the tunneling particle moves through the saddle point for obtaining TC . A tunneling particle may not, however move in the traditional classical manner along the adiabatic reaction coordinate through the first order saddle point, but take a short cut wherein it experiences a higher barrier, but a shorter tunneling length. In the absence of highly precise data for the reaction systems, numerical calculations for obtaining the tun-

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neling rate constant become difficult. One is then left with the alternative of relating the special features of tunneling chemical reactions to the changes in the reaction barrier due to the vibrations (intermolecular) [18]. Atom transfer reactions are the most thoroughly investigated reactions [19]. Here the transferring atom moves between the two potential wells. A reversible transfers leads to tunneling level splitting (spectral signature of tunneling) and an irreversible transfer precipitates chemical reactions (signature of tunneling dynamics). If the period of inter-well quantum oscillation is smaller than the relaxation time, coherent transfer takes place while the process is incoherent if the opposite situation is encountered. If the two time scales are comparable, the process is better described by density matrix methods. Ivanov and Kozhushner [20] showed that the time period of oscillation (Ω−1 ) and the particle transfer time (the tunneling time τ) play an important role. If Ωτ << 1, the particle transfer is instantaneous and takes place at predetermined positions of the reagents. If Ωτ >> 1, the transfer takes place at average positions of the reagents and the surrounding molecules. If Ωτ = 1, the tunneling particle adjusts its positions corresponding to those of the reagents and the surroundings. The synchronous motion of the reagent and the surroundings may either hasten or delay the particle transfer rate [22]. In a tunneling reaction we have three subsystems taking part of which the electronic and the intramolecular subsystem constitute the fast variables and the inter subsystem variables constitute the slow variable. Using the Fermi Golden rule, and a modified theory of radiationles transition, an expression for the tunneling rate constant has been obtained [21] ktunneling =

 2π  ~

Aνi ∑hφν f |V (R)|φνi i2 δ(~ν f + ∆E − ~νi ). νf

The transition matrix element delicately depends on the distance between the reagents and that means intermolecular vibrations play a dominant role in shaping the tunneling rate constant. Such vibrations cause variations in the tunneling distance. Vibrations that bring the reagent closer are called promotive modes which are taken into account by invoking either the Einstein or the Debye model of promotive modes [22-24], depending upon the situation. It has been shown that in atom tunneling reactions in the solid phase the low temperature limit of ktunneling exists for nonendothermic processes only. Along with inter-molecular vibrations (which can change inter-reagent separation), medium reorganization and under barrier friction play significant roles in determining temperature and pressure dependence of tunneling rate constants. The effect of reorganization of the medium alone leads to a temperature dependence that is independent of the form of the barrier. We note that the tunneling particle is assumed to interact with phonons only in the initial and final states, but not during the course of tunneling. The interaction of the tunneling particle with phonons in the underbarrier regimes lead to the appearance of frictional effects that depend rather sensitively on the form of the barrier and its modulations, if any.

C. Tunneling in Coupled Systems With the preceeding background in view, our purpose in this chapter has been to investigate typical signatures of tunneling dynamics in several coupled systems. In one of these, the tunneling mode is described by a symmetric double well potential which is coupled to,

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let us say, a bond stretching mode. The latter is represented by a typical Morse potential and the form and strength of the coupling are assumed to vary. We propose to investigate how the mode of coupling with bond stretching mode influences the tunneling dynamics in the double well and vice versa. If the bond stretching mode is externally driven, how does it affect the tunneling dynamics? These questions are addressed in subsequent sections. Normally, the tunneling particle is assumed to have a fixed mass- fixed in space and time. However, in charge transfer and tunneling of electrons through heterostructures, the effective mass of the electron is often assumed to be coordinate dependent. Could the coordinate dependence of the mass of the tunneling particle have a typical signature on the tunneling dynamics? We have investigated the question through numerical experiments. Let us now focus on the general methodology used in our explorations.

3.

The Method

For the calculations, we have used time dependent Fourier grid hamiltonian method [25-19]. The basic framework of the approach adopted in the present series of calculations is described in this section.

A. Dynamics of the Coupled System in the Absence of Driving Let H0 (x, y) be the hamiltonian of a particle of mass m moving on the x − y plane in a potential V0 (x, y), where V0 (x, y) is assumed to be additively separable into a double well potential V0 (x) and a Morse potential V0 (y) . Thus H0 (x, y) = T0 (x) + T0 (y) +V0 (x, y) = T0 (x) +V1 (x) + T0 (y) +V2 (y) = H0 (x) + H0 (y).

(1)

Let us assume that x represents the tunneling coordinate and y stands for a bond stretching coordinate. To be more specific, we may take H0 (x) is taken to represent the Hamiltonian describing the motion of a quantum particle in a asymmetric double well potential V1 (x) while H0 (y) is assumed to represent the motion along a bond stretching coordinate described by an appropriate Morse potential V2 (y) . We may now introduce a coupling between the two modes by using the interaction potential Vint (x, y) so that the total hamiltonian H(x, y) that describes the coupled system can be written as H(x, y) = H0 (x, y) + λVint (x, y).

(2)

λ being the strength of the coupling between the tunneling and bond stretching coordinates. The purpose of the present study has been to investigate how the tunneling dynamics gets affected by the coupling with the bond stretching mode. Let us assume that the eigen functions φi (x) of H0 (x) and χi (y) of H0 (y) are known to start with. The states of the coupled system can be described by superposition of the products of the eigenstates of the uncoupled system [H0 (x) and H0 (y)]. Thus we may write ni n j

|ψ(x, y,t)i = ∑ ∑ ci j (t)|φi (x)χ j (y)i. i

j

(3)

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The product functions generated by the eigenfunction of the uncoupled system described by the hamiltonians.H0 (x) and H0 (y) serve as the basis of the coupled system. The introduction of the coupling term Vint (x, y) would cause the superposition in equation 3 to evolve in time. The combination coefficients ci j ’s are therefore taken to be time dependent. Time evolution equations for the combining coefficients are obtained from the time dependent Schrodinger equation by following the standard procedures and are given by i~

dci j = dt

mx ,my

∑ k,l

ckl hφxk χyl |H0 (x, y)|φxi χyj i

(4)

for i = 1, mx , j = 1, ny We may call it a time dependent configuration interaction type of formalism. To find out the time independent basis functions |φi (x)i and |χ j (y)i we make use of the Fourier grid hamiltonian (FGH) technique [29] and compute the eigenfunctions and eigenvalues of the uncoupled model systems as follows H0 (x)|φi (x) = εxi |φi (x)i,

(5)

where nx

∑ wxpi |x p i∆x,

(6)

H0 (y)χi (y) = εyi |χi (y)i,

(7)

|φi (x)i =

p=1

and

where |χyi i =

ny

∑ wyqi |yq i∆y.

(8)

q=1

wxp,i and wyq,i are the corresponding grid point amplitudes along x and y coordinates, respectively. We note that the coordinates are uniformly discretized, ∆x and ∆y being the uniform grid spacing along the x and y coordinate, respectively. Using equations (6) and (8) the time evolution equations of the combining coefficients ci j (i = 1, nx ; j = 1, ny ) of equation 4 reduce to nx ,ny dci j i~ = ∑ ckl ∑ wxpi wyq j H(x p , yq )wxpk wyql (∆x∆y)2 = ∑ ckl Hi j,kl . dt p,q k,l k,l

(9)

In matrix form equation 9 can be written as ˙ = HC(t). i~C(t)

(10)

The time integration may be done by employing sixth order Runge-Kutta method. When a quantum mechanical particle moves in a double well there is a nonzero probability that the particle tunnels from one well to the other well whatever be its energy relative to the barrier

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top. Let us suppose the particle was initially localized in the left well of the uncoupled symmetric double well. The lowest energy localized states (φL and φR ) can be described by linear combinations of the two lowest energy eigen-states of even (ψ+ 0 ) and odd parity − (ψ0 ), respectively 1 − φL = √ (ψ+ (11) 0 + ψ0 ), 2 1 − φR = √ (ψ+ (12) 0 − ψ0 ), 2 where φL and φR represent the states localized in the left and the right well, respectively. In the absence of any coupling between the tunneling mode and the bond stretching mode the particle tunnels coherently from the left well to the right well. If one takes φL to be the initial state, tunneling probability is obtained by calculating the probability of finding the particle in the right well at energies less than the barrier energy. The corresponding probability PR (t) at any particular instance is given by (the barrier is located at x = 0) PR (t) =

Z ∞ 0

|ψ(x,t)|2 dx.

(13)

In the two dimensional case, the corresponding probability can be computed by using PR (t) =

Z +∞ Z +∞ y=−∞ x=0

|ψ(x, y,t)|2 dxdy.

(14)

In the CI formalism used by us in the FGH basis, equation 14 can be written as ni ,n j

PR (t) =

∑ i, j

|ci j |2

nx

ny

nx −1 2

q=1

∑ ∑ wxpi wyq j ∆x∆y.

(15)

The rate of tunneling can then be obtained from the average slope of the PR (t) - t plot. Alternatively, the tunneling rate can be calculated from the average rate of the change of the hx(t)i with time plot, where hx(t)i is given by hx(t)i = hψ(x, y,t)|x|ψ(x, y,t)i.

(16)

Substituting the expression for hψ(x, y,t)| from equation (4) we get hx(t)i =

∑ c∗i j (t)ck j (t)hφxi |x|φxk i

(17)

i, j,k

nx

=

∑ c∗i j ck j ∑ wxpi x p wxpk .

i, j,k

(18)

p

The computed rate of tunneling is related to the tunneling splitting in the double well [30]. When the double well gets coupled to a bond stretching mode described by a Morse oscillator, we may envisage three probabilities: (a) The bond stretching mode enhances the tunneling rate. We may call it a promoting mode. (b) The bond stretching mode reduces the tunneling rate. We call it a suppressing mode (c) The bond stretching mode does not affect the tunneling rate at all. It acts as a passive mode We anticipate that the type of the effect could depend on the nature of the coupling (see later).

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B. Dynamics of the Coupled System in the Presence of External Driving When a spatially homogeneous external electric field is applied along the x direction the perturbed hamiltonian H(x,t) can be represented as H(x,t) = T (x) +V1 (x) +V (x,t) = H0 (x) +V (x,t),

(19) (20)

where V (x,t) = xeεx sin(ωt).

(21)

ε is the field intensity and e represents the charge carried by an electron. Similarly, with an electric field applied along y direction H(y,t) = H0 (y) +V (y,t),

(22)

V (y,t) = yeεy sin(ωt).

(23)

where

The time evolution equations now turn out to be (with field along x direction) i~

nx ,ny dci j = ∑ ckl ∑ wxpi wyq j (H(x p ,t) + H0 (yq ) + λVint (x p , yq ))wxpk wyql (∆x∆y)2 dt p,q k,l

(24)

for i = 1, nx ; j = 1, ny . These equations can be integrated numerically over ckl ’s at t = 0 are provided.

4.

Results and Discussion

A. Tunneling in the Coupled System in Absence of External Driving We take the model potentials V1 (x) = ax4 − bx2 + c,

(25)

(−β(y−ye )) 2

V(2) (y) = D(1 − e

) .

(26)

The system parameters are listed in table-1. FGH calculations have been done with 151 grid points in each coordinate. The grid lengths along x axis and y axis are 6 a.u and 10 a.u, respectively. When V1 (x) and V2 (y) are coupled through an interaction potential Vint (x, y) and the coupling strength λ. the shape of the two dimensional potential energy surface (PES) gets modified depending on the functional form of V (x, y) and strength λ. Figure 1a shows the two dimensional potential energy surface (PES) when V1 (x) and V2 (y) are uncoupled, i.e. λ = 0. Figure 1b shows the two dimensional PES for Vint (x, y) = λxy, (λ = 0.001). Evidently, for the particular functional form the symmetry of the double well potential gradually gets distorted (along the y axis). The left well becomes deeper than the right well. By increasing the coupling strength the asymmetry between the two wells

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Figure 1. Potential Energy Surfaces for different forms of interaction potentials Vint (x, y). (a) Vint (x, y) = 0; (b) Vint (x, y) = λxy; (c) interaction potential Vint (x, y) = λx2 y2 ; (d) Vint (x, y) = λ(x2 y + xy2 ).

increases and the coherence in the tunneling process is progressively disturbed. Figure 2a shows the plot of the computed tunneling rate versus the coupling strength. The initial wave function ψ(x, y,t = 0) corresponds to the state in which the Morse mode and the tunneling mode are both in the ground state. From figure 2a we can clearly conclude that the tunneling rate decreases almost linearly with increasing coupling strength for the particular functional form of interaction potential Vint (x, y) = λxy. Figures 1c and 1d show the two dimensional PES for the functional forms Vint (x, y) = λx2 y2 and Vint (x, y) = λ(x2 y + y2 x), respectively (with λ = 0.001). In figure 1c the symmetry of the double well potential remains intact but the barrier height of the double well decreases (along y axis). As we go on increasing λ with the functional form Vint (x, y) = λx2 y2 the barrier height gradually decreases and as a consequence the tunneling rate increases almost linearly which is very clearly reflected in figure 2b. If Vint (x, y) is taken to have the form Vint (x, y) = λ(x2 y+y2 x), the PES (figure 1d) is obtained in which the barrier height of the double well potential increases as well as the symmetry of the double well potential is lost. The tunneling rate decreases (figure 2c) as λ increases. These results indicate that the form (Vint (x, y)) and strength (λ) of the interaction potential between the symmetric double well (tunneling mode) and the Morse oscillator (bond stretching mode)are very important in determining the nature and the extent of the influence that the coupling could

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Figure 2. Tunneling rate in the coupled systen in absence of external driving when (a) Vint (x, y) = λxy; (b) Vint (x, y) = λx2 y2 ; (c) V (x, y) = λ(x2 y + xy2 ). have on the tunneling dynamics in the double well.

B. Tunneling Dynamics in the Coupled System in the Presence of an External Electric Field Coupled to the Tunneling Coordinate We apply an external time varying electric field along the tunneling coordinate with an intensity of 0.01 a.u and frequency of 0.00227 a.u. The specific frequency matches with the 0 → 1 transition frequency (ω0→1 )of the double well. We take the product of the lowest even parity state the double well and the ground state of the Morse oscillator, as the initial state; ψ(x, y,t = 0) = φ+ 0 χ1 .

(27)

For the uncoupled composite system, λ = 0 and we see the particle execute a to and fro movement between the two wells. Figure 3a shows the plot of < x > vs t for the uncoupled double well Morse oscillator system. The variation of < x > seems to be very regular and periodic in nature. The corresponding quantum phase space diagram (figure 3b) shows that

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Figure 3. Tunneling dynamics in the presence of an external time varying field coupled to the tunneling coordinate x with Vint (x, y) = λxy (a) < x > versus t profile at λ = 0; (b) ’Quantum phase space’ diagram along λ = 0 for the tunneling mode; (c) < x > versus t profile at λ = 0.0001; (d) ’Quantum phase space’ diagram along λ = 0.0001 for the tunneling coordinate; (e) ’Quantum phase space’ diagram for λ = 0.0001 for the bond stretching mode.

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the particle is symmetrically distributed in the two wells ( the symmetry about < x >= 0 line is maintained). Tunneling rate is calculated from the average slope of < x > versus t plots. When the symmetrical double well system gets coupled to the Morse oscillator with a coupling term Vint (x, y) = λxy the dynamics gets modified. We have shown the variation of < x > with t for λ = 0.0001 for such a system in figure 3c which is substantially different from the corresponding picture of the uncoupled system (figure 3a). However, figure 3c also exhibits regularity and periodicity. The ’quantum phase space’ picture (figure 3d) is still symmetric about the < x >= 0 line, but the area occupied in the phase space diagram has decreased substantially. It indicates transfer of some energy from the tunneling to the bond stretching mode. Figure 3e displays the ’quantum phase space’ structure along y direction (the bond stretching mode). The occupied region of the phase space in figure 3e is rather small which indicates that the energy transfer from the x mode to the y mode (i.e. from the tunneling to the bond stretching or the Morse mode) is also small for λ = 0.0001. However, because of this small energy transfer the tunneling rate gets reduced. We have investigated the dynamics at different values of λ and we have found that the energy transfer from the x mode to the y mode becomes maximum at λ = 0.003. The tunneling rate in turn passes through a minima for λ = 0.003 (figure 4a). Figure 4b shows the variation of < x > with time for λ = 0.003. At this λ value the ’quantum phase space’ diagram for the Morse mode occupies a larger area which again indicates that the energy transfer (figure 4d) from the tunneling to the Morse mode for the given value of λ is higher. The population of the first excited state of the Morse oscillator grows (figure 4d) with oscillations which is reflected in the increase of the amplitude of the oscillation of the average bond length < y > (figure 4e). It is clear therefore that direct excitations by external field coupled to the tunneling mode can influence the bond stretching mode coupled to it.

C. Tunneling Dynamics in the Presence of External Field Coupled to the Bond Stretching Mode If we locally excite the Morse mode with an external time varying electricfield of intensity 0.01 a.u, and frequency ω0→1 (y) (frequency of the 0 → 1 transition in the Morse mode), we do not observe any significant probability of bond dissociation in the uncoupled system. But if we introduce an interaction potential Vint (x, y) = λxy, the dissociation probability slowly rises as λ increases. Table-3 shows the values of the computed dissociation probability and tunneling rate for various values of coupling strength (λ). At λ = 0.004 we observe a dissociation probability of 0.37 which is the maximum for the given intensity. From table-3 it is clear that tunneling is strongly impeded when the tunneling mode gets coupled to a stretching mode that is locally excited. Quantum phase space picture along the tunneling coordinate at λ = 0.004 is depicted in figure 5a. Figure 5b exhibits the corresponding picture for the stretching mode at λ = 0.004. The growth of dissociation probability attains a maximum at λ = 0.004 (Table-3). Figures 5c and 5d show the ’quantum phase space’ diagrams for the stretching mode for λ = 0.001 and λ = 0.006, respectively. In both the cases the area occupied in the phase space is smaller compared to what is observed for λ = 0.004 (figure 5b). It indicates that the maximum energy transfer from the symmetric double well to the Morse mode takes place at λ = 0.004 which is

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Figure 4. Tunneling dynamics in the presence of an external time varying field coupled to the tunneling coordinate with Vint (x, y) = λxy (a) Tunneling rate versus coupling strength λ; (b) < x > versus t profile for λ = 0.003 along the tunneling coordinate; (c) ’Quantum phase space’ diagram for λ = 0.003; (d) Population of the ground and the first excited states of the bond stretching mode for λ = 0.003; (e) < x > versus t profile at λ = 0.003 for the bond stretching mode.

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Figure 5. Tunneling dynamics in the presence of an external time varying field coupled to the bond stretching coordinate with Vint (x, y) = λxy (a) Quantum phase space diagram for λ = 0.004 along the tunneling coordinate; (b) “Quantum phase space” diagram for λ = 0.004 along the bond stretching coordinate; (c) “Quantum phase space” diagram for λ = 0.001 along the bond stretching coordinate; (d) “Quantum phase space” diagram for λ = 0.006 along the bond stretching coordinate; (e) Dissociation probability versus time for λ = 0.004.

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reflected in the relatively high dissociation probability achieved at the given intensity of the external field . Figure 5e shows the plot of the computed dissociation probability against time at λ = 0.004 a.u. The dissociation probability grows fast and attains a relatively high value. The tunneling, on the other hand gets quenched completely. Thus, excitations in the non-tunneling mode in the coupled system can be exploited for controlling tunneling.

D. Tunneling Dynamics of a Particle with Coordinate Dependent Mass Tunneling of electron through heterostructures is complicated by the interaction of the tunneling particles with many centres of scattering. Instead of taking these complexities directly into account in the calculation, it is often expedient to replace the real system by a model one in which the tunneling potential remains unaffected, but the tunneling particle is assumed to have coordinate dependent mass. The question that arises now concerns the signature of the coupling of the tunneling particle with the lattice or equivalently of the coordinate dependent mass [31], on the dynamics of the tunneling. We have carried out a series of experiments within the framework of the basic methodology described in SectionII. The numerical experiments are done with a symmetric double well potential and are subdivided into two classes. 1. The mass of the tunneling particle varies symmetrically along the tunneling coordinate. 2. The mass variation is asymmetric

Figure 6. Pattern of coordinate dependent mass variation (a) mass distribution is gaussian; (b)mass distribution has a minima at the barrier top. Under category 1 three possibilities have been investigated: (a) the tunneling mass is constant m0 everywhere, (b) the tunneling particle has mass m0 at x = 0 where the barrier height is maximum while away from the barrier top, the mass decreases. More specifically, 2 the mass has a gaussian profile along the tunneling coordinate (figure 6a), m(x) = m0 e−βx , and (c) the tunneling mass is m0 at the bottom of the left or right well and it decreases as 2 2 it approaches the barrier top at x = 0 (figure 6b) i.e m(x) = m0 e−β(x −a ) . x = ±a being the location of the well minima. Figure 7a shows how the dynamics appears to be when the tunneling mass (m0 ) is fixed along x. It is perfectly coherent and the particle oscillates back and forth between the right and the left wells. When the tunneling mass is higher in the

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Figure 7. The pattern of tunneling dynamics displayed when (a) tunneling mass is independent of x i.e. m(x) = m0 ; (b) tunneling mass has a maxima at the bottom of the wells; (c) tunneling mass has a maxima on the barrier top.

Figure 8. The mass distribution plot when it is sharply peaked at x = −α. wells and lower in the barrier, the tunneling becomes slower, still remaining coherent (figure 7b). However, if the tunneling mass is lower in the well and increases as it approaches the barrier, the tunneling rate is increased very significantly and the coherent oscillation frequency becomes much larger (figure 7c). In such a situation very significant increase in tunneling rate through heterostructure should be seen. It is clear that mass concentration in the well lowers the tunneling rate while mass concentration in the barrier region enhances tunneling. Under category-2, we have considered an asymmetric distribution of the tunneling mass along the tunneling coordinate by assuming m(x) =

m0 . |x + α|

(28)

The function is peaked at x = −α (figure 8). The larger the value of α the larger is the shift of the peak towords the region x < 0. Figure-8 shows the mass variation pattern along the tunneling coordinate for different values of α. The tunneling time [32] has been computed for each value of α with m0 being assumed to be the proton mass and reported in table-IV. It is clearly seen that tunneling time increases as α increases meaning thereby that the rate of

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tunneling decreases. The asymmetric mass variation of the type displayed in figure 8 along the tunneling coordinate could lead to quenching tunneling. The coupling with the lattice (’environment’) can therefore modulate tunneling current in hetero structures. Table 1. Morse and Symmetric double well potential parameters used in model calculations Parameters a b c D β xe µ

Values (a.u) 0.1 0.12 0.04 0.135 0.731 2.23 250.0

Table 2. Energies of the Morse and the double well oscillators

1st state energy(a.u) 2nd state energy (a.u)

double well 0.02270 0.02496

Morse Oscillator 0.00439 0.01267

Table 3. Computed dissociation probability and tunneling rate in a symmetric double well and Morse oscillator system for various strength of coupling parameters λ 0 0.001 0.002 0.003 0.004 0.005 0.006

Dissociation probability 0.04 0.05 0.1 0.18 0.37 0.05 0.05

Tunneling rate(au−1 ) 0.000769 0 0 0 0 0 0

Table 4. Computed tunneling time for various α Tunneling time 5.0 ∗ 105 1.0 ∗ 106 3.3 ∗ 106

α 0.5 1.0 1.5

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Conclusion

In a tunneling system where the tunneling motion of the particle along the coordinate gets coupled to non-tunneling motion along another coordinate, the coupling leaves its signature on the dynamics in different ways. It can quench tunneling, depending upon the strength and form of coupling. Similarly the tunneling motion can also affect the dynamics along the non-tunneling (e.g. bond stretching) coordinate. Experimentally, little seems to be known about the possible impact that excitation in the tunneling coordinate could have on the bond stretching and the dissociation dynamics of a mode coupled to the tunneling coordinate and vice-versa. Work along these lines could enrich our knowledge about the dynamics of coupled quantum system.

References [1] M. Razavi, Quantum Theory of Tunneling, World Scientific, 2003. [2] R. P. Bell, The Tunneling Effect in Chemistry, Chapman & Hall, London, 1980. [3] G. Gamow, The quantum theory of nuclear disintegrationm, Nature 1928, 122, 805806; Zur Quantentheorie des Atomkevnes, Z. fur. Phys. 1928, 51, 204–212. [4] M. Born, Zur Theorie des Kernzerfalls, Z. fur. Phys. 1928, 58, 306–321. [5] R. W. Gurney, Nuclear levels and artificial disintegration, Nature 1929, 123, 565–566. [6] L. Esaki, Long journey into tunneling, Proc. IEEE 1974, 62, 825–831. [7] I. Giaever, Electron tunneling and superconductivity, Science 1974, 183, 1253–1258. [8] B. D. Josephson, The discovery of tunneling supercurrents, Science 1974, 184, 527– 530. [9] G. C. Schatz,Tunnelling in bimolecular collisions, Chem. Rev. 1987, 87, 81–89; Quantum effects in gas phase bimolecular chemical reactions, Ann. Rev. Phys. Chem. 1988, 39, 317–340. [10] D. G. Truhler, A. D. Isaacson, B. C. Garrett, Theory of chemical reaction dynamics, D. C. clary, Dordrecht Ed.; NATO ASI series. Series C, Mathematical and physical sciences; vol. 170, NATO ASI series., no. 170. CRC Press, Boca Raton, FL, 1985, Vol. 4, pp. 65–137. [11] B. C. Garrett and D. G. Truhlar, A least action variational method for calculating multidimensional tunneling probabilities for chemical reactions, J. Chem. Phys. 1983, 79, 4931–4938. [12] H. Ushiyama and K. Taktsuka, Semiclassical study on multidimensuional effects in tnneling chemical reactions: tunneling paths and tunneling tubes, J. chem. Phys. 1997, 106, 7023–7035.

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[13] H. Eyring, The activated complex in chemical reactions, J. Chem. Phys. 1935, 3, 107– 115. [14] H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 1940, 7, 284–304. [15] P. Hanggi, P. Talkner, M. Borkovec, Reaction-rate theory: fifty years after Kramers, Rev. Mod. Phys. 1990, 62, 251–270. [16] H. J. Kim, J. T. Hynes, A theoretical model for SN1 ionic dissociation in solution. 1. Activation free energetics and transition-state structure, J. Am. Chem. Soc. 1992, 114, 10508–10528. [17] D. G. Truhlar, G. K. Schenter, B. C. Garrett, Inclusion of nonequilibrium continuum solvation effects in variational transition state theory, J. Chem. Phys. 1993, 98, 5756– 5770. [18] V. I. Goldanskii, Role of the tunneling effect in the kinetics of chemical reactions at low temperatures Dokl. Akad. Nauk. Phys. Chem. 1959, 124, 1261–1264. [19] V. A. Benderskii, V. I. Goldanskii, A. A. Ovchinnikov, Effect of molecular motion on low temperature and other anomalously fast chemical reactions in solid phase, Chem. Phys. Lett. 1980, 73, 492–495. [20] T. Miyazaki; Ed. Springer, Atom tunneling phenomena in Physics, Chemistry and Biology, Berlin, 2004, 36. [21] G. K. Ivanov and M. A. Kozhushner, Sov. J. Chem. Phys. 1983, 2, 1299. [22] G. K. Ivanov, M. A. Kozhushner, L. I. Trakhtenberg, Temperature dependence of cryochemical H-tunneling reactions, J. Chem. Phys. 2000, 113, 1992–2002. [23] V. I. Goldanskii, L. I. Trakhtenberg and V. N. Fleurov, Tunneling Phenomena in chemical Physics, Gordon and Breach Science Publishers, New York, 1989. [24] L. I. Trakhtenberg, V. L. Klochikhin, S. Ya. Pshezhetskii, Theory of tunnel transitions of atoms in solids, Chem. Phys. 1982, 69, 121–134. [25] L. I. Trakhtenberg in ref-18, 46–47. [26] C. C. Marston, G. B. Baliant Kurti, The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions, J. Chem. Phys. 1989, 91, 3571–3576. [27] S. Adhikari, P. Dutta and S. P. Bhattacharyya, A time-dependent Fourier grid Hamiltonian method. Formulation and application to the multiphoton dissociation of a diatomic molecule in intense laser field, Chem. Phys. Lett. 1992, 91, 574–579. [28] P. Dutta, S. Adhikari and S. P. Bhattacharyya, Fourier grid Hamiltonian method for bound states of multidimensional systems. Formulation and preliminary applications to model systems, Chem. Phys. Lett. 1993, 212, 677–684.

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[29] S. Adhikari, P. Dutta and S. P. Bhattacharyya, Properties, dynamics, and electronic structure of atoms and molecules applications of a local grid method for modeling chemical dynamics at a mean-field level, Int. J. Quant. Chem. 1996, 59, 109–117. [30] G. B. Balint Kurti, C. L. Ward, R. N. Dixon, A. J. Mulholland, The calculation of product quantum state distributions and partial crsssections in time-dependent molecular collision and photodissocistion theory, Comput. phys. Commun. 1991, 63, 126–134. [31] A. V. Kolesnikov, A. P. Silin, Quantum mechanics with coordinate-dependent mass, Phys. Rev. B 1999, 59, 7596–7599. [32] Kaushik Maji, C. K. Mondal, S. P. Bhattacharyya, Tunneling time and tunneling dynamics, International Reviews in Phys. Chem. 2007, 26, 647–670.

In: Quantum Frontiers of Atoms and Molecules Editor: Mihai V. Putz, pp. 111-140

ISBN: 978-1-61668-158-6 © 2010 Nova Science Publishers, Inc.

Chapter 6

THEORETICAL CALCULATION OF THE LOW LAYING ELECTRONIC STATES OF THE MOLECULAR ION CSH+ WITH SPIN-ORBIT EFFECTS M. Korek* and H. Jawhari Faculty of Science, Physics Department, Beirut Arab University, P.O.Box 11-5020 Riad El Solh, Beirut 1107 2809, Lebanon

Abstract Research studies on ultracold molecules are a current and great challenge in the spectroscopic study of alkali dimers, because of their importance in the cooling and trapping of atoms and molecules, their role in high precision spectroscopy and Bose-Einstein condensation (BEC). Using the high reliability of the ab initio technique combined with the easily amenable phenomenology core polarization concept, the theoretical calculation of the electronic structure of the molecular ion CsH+ has been performed. This ion is treated as a one electron system where the interaction between the outer electron and the atomic core of Cs+ is modulated through non empirical relativistic effective one-electron core potential. The lowest 2 71 electronic states for the ion CsH+ have been calculated for the molecular states Λ(+) and Ω and dissociating into the 16 asymptotes considered, i.e. up to 16 states 2Σ+, 9 states 2Π, 4 states 2Δ, 25 states Ω=1/2, 13 states Ω=3/2 and 4 states Ω=5/2. Some avoided crossings are pointed out for the symmetries 2Σ+, 2Π, Ω=1/2 and Ω=3/2, their positions rAC and the energy difference ΔEAC at these positions have been determined. For 19 bound states, the harmonic vibrational constant ωe, the internuclear distance re and the electronic transition energy with respect to the ground Te have been calculated. Using the canonical functions approach, we calculate in the present work Ev, Bv and Dv of the molecular ion CsH+ up to the vibrational levels v = 19 for 17 electronic states. From the calculated values of Ev for a given vibrational level v and by using a cubic spline interpolation between each 2 consecutive points of the potential energy curves, the rmin and rmax of the turning points have been investigated for a these bound states. Permanent dipole moments M a (r) as well as all non-zero transition dipole moments *

M ab (r)

(a≠b ) have been calculated for each electronic state (a,b) under

E-mail address: [email protected]

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M. Korek and H. Jawhari consideration and in the whole range of r investigated here. The comparison of the present results with those available in literature shows a very good agreement.

1. Introduction Beginning in the late 1980s, methods to use laser light to cool and confine atoms at unprecedented temperatures have made a dramatic impact on atomic physics. The cooling and manipulation of cold molecules is likely opening up new branches of research. As a gas of molecules is cooled, their average velocity is decreased and the spread of their molecular velocities narrowed. This is important not only for studying molecular physics, but also for studying fundamental physics. The internal structure of certain molecules provides an ideal “laboratory” for sensitive measurements of fundamental physical quantities. Ultracold molecules are of a current and great challenge in the spectroscopic study of alkali dimers, because of their importance in the cooling and trapping of atoms [1,2] and molecules [3], their role in high precision spectroscopy [4], Bose-Einstein condensation (BEC) [5], atomic clocks, ultrasensitive isotope detection, quantum information, and processing ultracold collisions. At ultracold temperatures, the collisions of atoms, which may be characterized by s-wave scattering lengths, have received considerable attention because of their importance in cooling and trapping of atoms and molecules [6] and their role in high precision spectroscopy [7]. Collisions of ions involve higher-order partial waves because of the long-range attractive polarization forces and because of the possibility of charge transfer. Ion-atom charge transfer collisions are of great interest both theoretically and experimentally. Another especially promising area will be the study of collisions between ultracold molecules, in a regime where they behave like waves, perhaps giving rise to a new chemistry [8]. They may also allow for the study of collective quantum effects in molecular systems, including BEC [9]. There are proposed experiments to study polar molecular systems in order to measure the electron’s permanent electric dipole moment (EDM), the lifetime of long-lived energy levels, and the effects of the dipole-dipole interactions on the molecular samples properties [10]. Ultracold polar molecules interact with each other via highly anisotropic electric dipoledipole forces providing access to qualitatively new regimes previously unavailable by ultracold homonuclear_nonpolar systems [11-12]. Novel phenomena are expected from ultracold polar molecules as new features in phase diagrams of degenerate states [13] and anisotropic collisions caused by anisotropic dipole-dipole interaction [14]. The sympathetic cooling of these mixtures has led to the achievement of simultaneous quantum degeneracy of bosonic and fermionic species, producing BEC and Fermi-Bose mixtures [15-16]. Theory groups developed the methods to map problems, such as the BEC-BCS crossover, superfluid phases, Bose glasses and spin-charge separation from solid state quantum systems to the pure world of degenerate quantum gases [17-19]. Meanwhile, a growing community developed the technique of atom chips. From this dataset and the experience of three generations of atom chip experiments on hand, a truly next-generation experiment was designed. Furthermore, the interaction of bosons and fermions with attractive and repulsive interactions can be studied without the need of a Feshbach resonance, as the interaction between Rb and Li is repulsive and the interaction between Rb and K is attractive [20]. Recent advances in precision control of the optical spectrum emitted by a femtosecond laser have made a revolutionary impact on the fields of optical frequency metrology (via self-referenced, ultra-broad bandwidth

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frequency combs) and ultrafast optical science (via carrier-envelope phase stabilized pulse trains). These advances have, in effect, provided an entirely new class of optical sources available for experimental investigations which is the Femtosecond lasers+cold atoms/molecules. In the three past decades a limited number of theoretical studies have been performed for the molecular ions AH+(where A is an alkaline atom). The potential energy curves of these molecular ions are well needed to understand ion-atom collision [21] and to serve as input for diatomics-in-diatomics studies of the potential energy surfaces for A+H2 collisions [22]. These systems are interesting for the theory of chemical bonding since each involves a single valence electron. Interest in alkali dimers is closely related to developments in the ultra-cold alkali dimers atom trapping, which are at the root of photoassociation spectroscopy. The charge transfer in collision of neutral alkaline atoms with protons affects the ionization balance in the atmospheres of planets, dwarf stars, and the interstellar medium [23-27]. At low temperature, such as in ultracold experiments, collision energies are much les than 1eV, and radiative charge transfer may become dominant over nonradiative charge transfer. Neutralization of H+ is of interest in the area of plasma fusion as a method of energetic neutral beam injection in fusion reactors. At lower energies, this charge exchange process is used to make metastable [H(2s)] hydrogen used for atomic experiments [28,29] and for the creation of spin polarized proton beam for injection into large accelerators [30]. Due to these results, the lack of the theoretical calculation on a certain number of alkali dimers, and because of its use particularly in the domain of quantum computer to create the qubite [31], we investigated recently theoretical calculation of these molecules and their ions [32-38]. Using an improvement on the ab initio pseudo-potential method [39-45], we investigate in this chapter the lowest 29 electronic states of Λ-representation (neglect spin-orbit effect), the lowest 42 electronic states of Ω-representation (including spin-orbit effect), and the spectroscopic constants of the regularly bound states. Based on the canonical functions approach [46-48], a rovibrational study has been done to calculate the eigenvalue energies Ev, the rotational constants Bv, the centrifugal distortion constants Dv, and the abscissas of the turning points (rmin, rmax) for 17 electronic states. Moreover, the dipole moment functions and the transition dipole moment are calculated for many of the states in the Ω-representation.

2. The Theory A. Ab initio Calculation For the molecular ion CsH+ the energies for the molecular states including the spin-orbit effect Ω=1/2, 3/2, and 5/2 have been obtained from the treatment of the total Hamiltonian Ht=He+WSO where He is the Hamiltonian in the Born-Openheimer approximation for the calculation of the energies for the molecular states labelled 2S+1Λ(+/-) and WSO is the spin-orbit pseudo-potential. The Spin-orbit (SO) effects are considered for Cs while they are neglected for H. The CsH+ ion is treated as a one electron system where the interaction between the outer electron and the atomic core of Cs+ is modulated through non empirical relativistic effective one-electron core potential of the Durand and Barthelat type [42-44]. The electroncore interaction is represented by the effective potential

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M. Korek and H. Jawhari 2

V[r ] = ∑ U [r ]P =0

where ℓ is the orbital angular momentum and Pℓ corresponds to the projection operator on the subspace defined by the Ym spherical harmonies with a given ℓ. U[r] is written as: 2

U [ r ] = ∑ ci r ni e −α r

2

i

i =1

with c, n and α adjusted to fit the energy and wave functions of the valence Hartree-Fock orbitals. Core valence effects including core-polarization and core valence correlation are taken into account by using an ℓ-dependent core-polarization potential of the Foucrault et al. type [49]

1 ∑α k f k . f k 2 k

Vcpp = −

where the index k labels the ionic cores, αk is the static dipole polarizability of the ionic core, fk is the electric field action on the ionic core k due to the valence electrons and the other core. The ℓ-dependent form proposed by Foucrault et al [49] ∞

fk = ∑

m=+

∑ F (r

=0 m = −

ik

, rk )

m > kk < m

with ⏐ℓm>k are spherical harmonic centered on the core k and rkℓ are cut-off parameters. For l

the one-valence-electron atom Cs the parameters rk have been determined in order to reproduce the experimental values of the ionization potential IP, as well as the transition energies for the atomic. In this way rl has been obtained for l = 0, 1, 2 and we have chosen

rk2 = rk3 . [50] The parameters defining the core-polarization potentials and the comparison of the calculated IPs and atomic transition energies with the experimental values of the atom Cs are given in Ref. [51] The core-core interaction is evaluated as the ground state energy for the molecular ion RbH2+ instead of the approximation 1/r which not accurate enough for this species, at least for small values of the internuclear distance. In the present calculation including the spin-orbit effect the total Hamiltonian Ht is diagonalized in the basis of the SΛΣ states yielding the relativistic Ω adiabatic states. The symmetry used in this calculation is being C∞v with a common set of molecular orbitals for all symmetries. Semi-empirical spinorbit pseudo-potentials have been designed for Cs atom [52-53]. The present investigation of the electronic structure including the spin-orbit effect for the molecular ion CsH+ has been performed by using the package CIPSO (Configuration Interaction by Perturbation of a multiconfiguration wave function with Spin-Orbit interaction) of the Laboratoire de Physique

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115

Quantique Toulouse, France, which allows a full CI calculation as well as perturbative CI calculations with SO effects.

B. Vibration – Rotation Calculation In the Rayleigh-Schrödinger perturbation theory the eigenvalue EvJ and the eigenfunction ΨvJ are given respectively by

E vJ = ∑ e n λ n

(1)

ΨvJ (r ) = ∑ Φ n (r )λ n

(2)

n =0

n =0

where r is the internuclear distance, v and J are respectively the vibrational and rotational quantum numbers, λ = J (J + 1) , and e0 = Ev, e1=Bv, e2 =- Dv , …, φ0 is the pure vibrational wave function and φn its rotational corrections. By replacing Eqs.(1) and (2) into the radial Schrödinger equation [54-58]

⎡ d 2 2μ λ⎤ ⎢ 2 + 2 (E vJ − U(r )) − 2 ⎥ ΨvJ (r ) = 0 r ⎦ ⎣ dr

(3)

φ '0' (r ) + [e 0 − U (r )]φ 0 (r ) = 0

(4)

one can write [36]

φ1'' (r ) + [e 0 − U(r )] φ1 (r ) = −[e1 − R (r )] φ 0 (r )

(5-1)

φ2'' (r ) + [e0 − U (r )]φ2 (r ) = −[e1 − R (r )]φ1 (r ) − e2 φ0 (r )

(5-2)

n

φ 'n' (r ) + [e 0 − U(r )] φ n (r ) = R (r ) φ n −1 − ∑ e m φ n −m (r )

(5-n)

m =1

where R(r)=1/r2, the first equation is the pure vibrational Schrödinger equation and the remaining equations are called the rotational Schrödinger equations. One may project Eqs.(5) onto φ0 and find [47]

< φ 0 | φ 0 > e1 =< φ 0 |

1 | φ0 > r2

(6-1)

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M. Korek and H. Jawhari

1 | φ1 > −e1 < φ 0 | φ1 > r2

(6-2)

n −1 1 | φ n −1 > − ∑ e m < φ n − m | φ 0 > 2 m =1 r

(6-n)

< φ 0 | φ 0 > e 2 =< φ 0 | < φ 0 | φ 0 > e n =< φ 0 |

Once e0 is calculated from Eq.(4), e1, e2, e3 … can be obtained by using alternatively Eqs.(5) and (6).

3. The Results Atomic Calculation The electronic structure of the molecular ion CsH+ is studied with and without the spinorbit (SO) coupling. The spin-orbit effects has been studied by using the package CIPSO (Configuration Interaction by Perturbation including Spin-Orbit coupling) of the "laboratoire de physique Quantique Toulouse-France". The values of the cut-off parameters involved in the polarization potentials are given in Table 1 [38,59]. Table 1. Parameters for the polarization potential of the Cs and H atoms α(ao3) 15.117 1.000

Atom Cs H

rko(ao) 2.6915 1.000

rk1(ao) 1.8505 1.000

rk2(ao) 2.8070 1.000

Table 2. Energies of the lowest lying levels of the H-atom Configuration 1s

Term 2 S

2p

2 0

2s

2

3p

2 0

3s

2

3d

2

4p

2 0

4s

2

4d

2

P

S

P

S

D

P

S

D

J Etheoretical(cm-1) 1/2 0.00 1/2 82312.7386 3/2 82324.9802 1/2 82324.9802 1/2 97568.0092 3/2 97568.0092 1/2 97568.0092 3/2 97568.0092 5/2 97568.0092 1/2 102948.0576 3/2 102948.0576 1/2 102948.0576 3/2 102948.0576 5/2 102948.0576 Average relative error

Eexperimental(cm-1) 0.00 82258.9206 82259.2865 82258.9559 97492.2130 97492.3214 97492.2235 97492.3212 97492.3574 102823.8505 102823.8962 102823.8549 102823.8961 102823.9114

δE/E% 0.00 0.07 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.12 0.12 0.12 0.12 0.12 0.09

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117

By using the Gaussian basis set given in Ref. [32] for the cesium atom and the modified basis cc-pv6z [60] for the hydrogen atom we calculate the energy levels up to 4d and 8s respectively for H and Cs-atoms. The comparison of these values, in Tables 2 and 3, to those obtained experimentally [61-63] shows an excellent agreement with an average relative error δE/E=0.09% for the H-atom and 0.01% for the Cs-atom. This agreement confirms the validity and the accuracy of the chosen basis sets in this calculation. Table 3. Energies of the lowest lying levels of the Cs-atom Configuration

Term

6s

2

6p

2 0

5d

2

S

P

D

7s

2

7p

2 0

6d

2

8s

2

S

P

D S

J

Etheoretical(cm-1)

Eexperimental(cm-1)

δE/E%

1/2 1/2 3/2 3/2 5/2

0.00 11172.7488 11728.5025 14480.8502 14579.0300

0.00 11178.2686 11732.3079 14499.2584 14596.8423

0.00 0.04 0.03 0.12 0.12

1/2

18537.3276

18535.529

~0.00

1/2 21837.0533 3/2 22019.7714 3/2 22776.7197 5/2 22818.5012 1/2 24354.4958 Average relative error

21765.35 21946.396 22588.8210 22631.6863 24317.0185

0.32 0.33 0.83 0.82 0.01 0.23

3.3. Spin-Orbit Effects Neglected For the molecular ion CsH+ the potential energy curves (PECs) of the 2Λ(+) states have been investigated in the range 3.0a0≤r≤60a0 of the internuclear distance and dissociating into the 16 asymptotes considered up to the dissociation limit H+(4d 2D3/2,5/2)+ Cs(1S0); i.e. up to 16 states 2Σ+, 9 states 2Π, 4 states 2Δ, as displayed in Table 4. This table also reports the calculated energies at the dissociation limits; the comparison of these values to those obtained experimentally [62-63] shows a very good agreement with an average relative error of 0.06 %. The PECs versus the internuclear distance for the 2Λ(+)-states are plotted respectively in the Figs1-3. Among the calculated 29 PECs 22 states are proved to be attractive. For each bound state the harmonic vibrational constant ωe, the rotational constant Be, the internuclear distance at equilibrium re and the electronic transition energy with respect to the ground state Te are calculated by fitting the energy data around the equilibrium position to a polynomial in terms of the internuclear distance r, these values are given in Table 5. The comparison of our calculated values of re and Be of the ground state with those available in literature shows a good agreement with relative errors 3.6% and 7.5% respectively. Double minima potentials are obtained for the states (14)2Σ+, (16)2Σ+, and (9)2Π, the minima of these potentials are given in Table 5. No comparison for the other results since they are given here for the first time.

118

M. Korek and H. Jawhari

Table 4. Numbering of the various Λ-states of the CsH+ correlated adiabatically to the 16 lowest dissociation limits Cs

H

2

1

2

1 0 2 78268.9598 3 1 82318.8594 4 82324.9802 5 2 89719.5854 6 3 1 92798.8999 7 96806.2874 8 97568.0092 9 4 97568.0092 10 5 2 97568.0092 11 6 100197.3722 12 7 3 101066.5703 13 102623.4556 14 102948.0576 15 8 102948.0576 16 9 4 102948.0576 Average relative error

S0 6s 2S1/2 1 S0 1 S0 2 6p P1/2,3/2 5d 2D3/2,5/2 7s 2S1/2 1 S0 1 S0 1 S0 7p 2P1/2,3/2 6d 2D3/2,5/2 8s 2S1/2 1 S0 1 S0 1 S0

1s S1/2 1 S0 2 2p P1/2,3/2 2s 2S1/2 1 S0 1 S0 1 S0 3s 2S1/2 3p 2P1/2,3/2 3d 2D3/2,5/2 1 S0 1 S0 1 S0 4s 2S1/2 4p 2P1/2,3/2 4d 2D3/2,5/2

Σ+

2

Π

2

Δ

Etheoretical(cm-1)

Eexperimental(cm-1)

δE/E%

0 78268.9598 82259.1035 82258.9559 89724.2480 92817.0101 96804.4888 97492.2235 97492.2672 97492.3393 100124.8328 100879.2135 102585.9783 102823.8549 102823.8733 102823.9037

0.00 0.00 0.07 0.08 ~0.00 0.02 ~0.00 0.08 0.08 0.08 0.07 0.18 0.03 0.12 0.12 0.12 0.06

460000

E(cm-1)

440000

420000

400000 0

10

20

30

40

R(Bohr)

Figure 1. Continued on next page.

50

60

Theoretical Calculation of the Low Laying Electronic States…

119

400000

E(cm-1)

300000

200000

100000

0 0

10

20

30

40

50

60

R(Bohr)

Figure 1. Potential energy curves of the states 2Σ+ of the molecule CsH+. 460000

E(cm-1)

440000

420000

400000 0

10

20

30

40

50

R(Bohr)

Figure 2. Potential energy curves of the states 2Π of the molecule CsH+.

60

120

M. Korek and H. Jawhari Table 5. Calculated spectroscopic constants for the various [(k)2Λ] states of CsH+ [(k)2s+1Λ]

Te(cm-1)

[(1)2Σ+]

0.00

[(2)2Σ+]

re(Å)

204220.06

3.372 3.25a 3.467

2 +

401701.74

2 +

[(4) Σ ] [(5)2Σ+]

ωe(cm-1)

Be(cm-1)

409.50

1.479 1.59a 1.402

4.611

263.99

0.792

414566.28

5.178

129.26

0.628

418211.74

7.854

115.45

0.273

2 +

423296.11

9.699

96.250

0.179

2 +

428459.83

11.65

81.54

0.124

2 +

432489.77

15.185

41.21

0.073

2 +

[(9) Σ ]

433523.86

24.742

6.017

0.029

2 +

435366.04

16.978

46.20

0.058

2 +

[(11) Σ ]

438110.28

19.817

40.05

0.042

[(12)2Σ+]

441472.00 444554.93 Max 444583.4 443451.88

11.932 13.921 At r=14.4 23.834

66.71 62.13

0.117 0.086

32.93

0.029

[(15) Σ ]

444571.45

17.765

72.78

0.053

447113.83 Max 447239.8 447111.94

15.030 At r=16.6 25.870

78.56

0.074

[(16)2Σ+]

7.20

0.025

2

204796.07

3.519

344.80

1.359

2

[(2) Π]

414065.64

3.425

457.16

1.437

[(3)2Π]

424463.77

6.946

53.227

0.344

[(4)2Π]

429363.48

9.583

59.41

0.183

[(6) Π]

438641.40

14.171

36.11

0.083

[(8)2Π]

444031.66

18.80

17.15

0.047

447229.32 Max 447249.3 447118.05

13.270 At r=14 23.338

46.83

0.095

[(9)2Π]

7.14

0.030

[(3) Σ ]

[(6) Σ ] [(7) Σ ] [(8) Σ ] [(10) Σ ]

[(14)2Σ+] 2 +

[(1) Π]

2

460.350

(a) Ref. [64].

These PECs present avoided crossings in quite complex forms (humps and wells) at short and large value of the internuclear distance which are due to either crossings or avoided crossings. The internuclear distance at the avoided crossing RAC with the energy difference ΔEAC between two corresponding states at these points for the different states are given in Table 6. We show avoided crossing between the states (9)2Σ+ and (8)2Σ+ in Figs 4 as illustration.

Theoretical Calculation of the Low Laying Electronic States…

121

460000

-1

E(cm )

440000

420000

400000 0

10

20

30

40

50

60

R(Bohr)

Figure 3. Potential energy curves of the states 2Δ of the molecule CsH+.

Table 6. Some avoided crossing between 2Λ+ states of the molecular ion CsH+ (n+1)State / (n)State

rAC(Bohr)

ΔEAC(cm-1)

(8)2Σ+/(7)2Σ+

12.4

1573.97

(9)2Σ+/(8)2Σ+

10.4

1461.58

(9)2Σ+/(8)2Σ+

18.4

33.14

(12)2Σ+/(11)2Σ+

8.3

543.19

(12)2Σ+/(11)2Σ+

11.5

547.72

(12)2Σ+/(11)2Σ+

20.6

504.07

2 +

2 +

6.4

357.90

2 +

2 +

17.8

473.14

2 +

2 +

32

118.38

2 +

2 +

(16) Σ /(15) Σ

10.6

154.25

(5) 2Π/(4) 2Π

8.2

1886.88

(9) 2Π/(8) 2Π

8.3

1630.83

(15) Σ /(14) Σ (15) Σ /(14) Σ (15) Σ /(14) Σ

122

M. Korek and H. Jawhari 434400 (8)Σ (9)Σ

E(cm-1)

434300

434200

434100

434000 18.1

18.3

18.5

18.7

R(Bohr)

Figure 4. Avoided crossing between (9)2Σ+ and (8)2Σ+states of the molecule CsH+.

3.4. Spin-Orbit Effects Included Energy calculation for the Ω-representation is performed for the states corresponding to the 16 lowest dissociation limits, i.e. up to H+(4d 2D3/2,5/2)+ Cs+(1S0) is the range of internuclear distance r from 3ao to 60ao. Consequently 25 states of Ω=1/2, 13 states of Ω=3/2 and 4 states of Ω=5/2 are correlated adiabatically as shown in Table 7. The PECs of these states are displayed in Figs 7, 8 and 9 respectively. Among these calculated PECs 24 states are proved to be attractive and the other are repulsive. Table 7. Numbering of the various Ω-states of CsH+ correlated adiabatically to the 25 lowest dissociation limits Cs 1

S0 6s 2S1/2 1 S0 1 S0 2 6p P1/2,3/2 5d 2D3/2,5/2 7s 2S1/2 1 S0 1 S0

H+

Ω=1/2

2

1 2 3,4 5 6,7 8,9 10 11 12,13

1s S1/2 1 S0 2 2p P1/2,3/2 2s 2S1/2 1 S0 1 S0 1 S0 2 3s S1/2 3p 2P1/2,3/2

Ω=3/2

Ω=5/2

1 2 3,4

5

1

Theoretical Calculation of the Low Laying Electronic States…

123

Table 7. Continued Cs 1

S0 2 7p P1/2,3/2 6d 2D3/2,5/2 8s 2S1/2 1 S0 1 S0 1 S0

H+

Ω=1/2

Ω=3/2

Ω=5/2

3d 2D3/2,5/2 1 S0 1 S0 1 S0 4s 2S1/2 4p 2P1/2,3/2 4d 2D3/2,5/2

14,15 16,17 18,19 20 21 22,23 24,25

6,7 8 9,10

2

11 12,13

3

4

For each bound state the harmonic vibrational constant ωe, the rotational constant Be, the internuclear distance at equilibrium re and the electronic transition energy with respect to the ground state Te are calculated by fitting the energy data around the equilibrium position to a polynomial in terms of the internuclear distance r, these values are given in Tables 8 and 9 respectively along with the main parents 2Λ(+) of the states Ω=1/2 and Ω=3/2 near the equilibrium positions. It should be noticed that, such identification was not possible for other states since their minima are situated close to the crossings between 2Λ(+) states. 460000

E(cm-1)

440000

420000

400000 0

10

20

30

R(Bohr)

Figure 7. Continued on next page.

40

50

60

124

M. Korek and H. Jawhari 400000

E(cm-1)

300000

200000

100000

0 0

10

20

30

40

50

60

50

60

R(Bohr)

Figure 7. Potential energy curves of the states Ω=1/2 of CsH+. 460000

E(cm-1)

440000

420000

400000 0

10

20

30

R(Bohr)

Figure 8. Continued on next page.

40

Theoretical Calculation of the Low Laying Electronic States…

125

400000

E(cm-1)

300000

200000

100000

0 0

10

20

30

40

50

60

R(Bohr)

Figure 8. Potential energy curves of the states Ω=3/2 of the molecule CsH+. 460000

E(cm-1)

440000

420000

400000 0

10

20

30

40

50

60

R(Bohr)

Figure 9. Potential energy curves of the states Ω=5/2 of CsH+.

Since the spin orbit coupling has no effect on the Σ−state we compare these results with those available in literature without spin orbit. This comparison shows a very good agreement with a relative errors 3.6% and 7.5% respectively for re and Be for the ground state. No comparison for the other results since they are given here for the first time.

126

M. Korek and H. Jawhari

Table 8. Calculated spectroscopic constants for the various (n)Ω=1/2 states of CsH+ n[(k)2s+1Λ]

Te(cm-1)

re(Å)

ωe(cm-1)

Be(cm-1)

(1) [(1)2Σ+]

0.00

3.372 3.25a

460.356

1.479 1.59a

(2) [(2)2Σ+]

204220.06

3.467

409.473

1.402

(3) [(1)2Π]

204795.65

3.519

345.114

1.359

(4) [(3)2Σ+]

401700.63

4.611

263.463

0.792

3.426 At r=3.9 5.177

456.004

1.437

(5) [(4)2Σ+]

414063.89 Max 414752.7 414565.97

129.213

0.628

(6)

414792.88

4.169

480.602

0.974

418211.24

7.855

115.627

0.272

(8) [(6) Σ ]

423222.19

9.696

96.473

0.179

(9) [(3)2Π]

424450.23

7.574

1105.84

0.541

(10) [(7)2Σ+]

428383.70

11.694

78.041

0.123

(11) [(4) Π]

429284.02

10.256

58.061

0.159

(12) [(8)2Σ+]

432489.53

15.182

41.239

0.073

(15) [(10)2Σ+]

435366.45

16.977

46.185

0.058

(16)

438617.06 Max 438026.4 438110.06

12.252 At r=13.7 19.815

26.045

0.082

40.064

0.042

(17) [(6) Π]

438666.78

15.846

95.567

0.066

(18) [(12)2Σ+]

441471.64

11.934

66.424

0.117

(21) [(14) Σ ]

443451.88

23.834

32.939

0.029

(22)

444555.10 Max 444583.4 444026.53 444571.46 Max 444632.2 444080.58

12.624 At r=13.8 17.430 17.765 At r=18.5 30.278

92.079

0.085

58.353 72.777

0.044 0.053

15.702

0.017

(24) [(16) Σ ]

447113.96

15.019

79.332

0.075

(25)

447237.12 Max 447249.2 447225.70 Max 447239.8 447118.11

12.224 At r=13.5 15.187 At r=15.9 23.973

67.025

0.081

225.477

0.042

102.062

0.028

2

(5) [(2) Π]

(7) [(5)2Σ+] 2 +

2

(16) [(11)2Σ+] 2

2 +

(22) [(8)2Π] (23) [(15)2Σ+] (23) 2 +

(25) [(9)2Π] (25) [(9)2Π] (a) Ref.[64].

Theoretical Calculation of the Low Laying Electronic States…

127

Double minima potentials are obtained for the states (5)Ω=1/2, (16)Ω=1/2, (22)Ω=1/2, (23)Ω=1/2 and (13)Ω=3/2, while the (25)Ω=1/2 is a triple well potential state. Table 9. Calculated spectroscopic constants for the various (n) Ω=3/2 states of CsH+ n[(k)2s+1Λ]

Te(cm-1)

re(Å)

ωe(cm-1)

Be(cm-1)

(1) [(1)2Π]

204796.49

3.519

344.896

1.359

2

414067.25

3.424

458.206

1.438

2

(4) [(3) Π]

424505.55

7.170

14.270

0.295

(5)

414565.97

5.177

129.213

0.628

(2) [(2) Π]

(8) [(6)2Π]

438666.91

14.144

36.317

0.084

2

444033.05

18.801

17.144

0.047

2

447229.54 Max 447249.4 447118.05

13.272 At r=13.4 23.337

46.757

0.095

7.246

0.030

(11) [(8) Π] (13) [(9) Π] (13) [(9)2Π]

At the internuclear distance at equilibrium re, the SO splitting for the states (1, 2, 3, 6, 8, 9)2Π have been identified and evaluated, the difference between the lowest and highest energy are 0.84cm-1, 3.36cm-1, 0.13cm-1, 6.52cm-1, 3.84cm-1 and 0.6cm-1 respectively. In the Ω-representation the PECs present avoided crossings in quite complex forms (humps and wells) at short and large value of the internuclear distance which are due to either crossings or avoided crossings of the Λ-states. The internuclear distance at the avoided crossing rAC with the energy difference ΔEAC between two corresponding states at these points are given in Tables 10 and 11. These avoided crossings are drawn, as illustration, in figures 10 and 11. Table 10. Some avoided crossings between Ω=1/2 states of CsH+ . rAC and ΔEAC are respectively the internuclear distance and the energy difference at the avoided crossing between the two corresponding states (n+1)Ω =1/2/(n)Ω=1/2

rAC(Bohr)

ΔEAC(cm-1)

(3)Ω=1/2/(2) Ω=1/2 (5)Ω=1/2/(4) Ω=1/2 (6)Ω=1/2/(5) Ω=1/2 (8)Ω=1/2/(7) Ω=1/2 (9)Ω=1/2/(8) Ω=1/2 (11)Ω=1/2/(10) Ω=1/2

5.1 4.7 7.7 5.2 13.9 17.7 6.1

25.18 169.97 46.92 137.64 109.66 378.17 8.87

Crossing between (n)state/(m)state (2)2Σ+/(1)2Π (3)2Σ+/(2)2Π (4)2Σ+/(2)2Π (5)2Σ+/(3)2Π (6)2Σ+/(3)2Π (7)2Σ+/(4)2Π (8)2Σ+/(5)2Π

(13)Ω=1/2/(12) Ω=1/2

12.1 18.8 18.2

81.59 18.14 52.56

(8)2Σ+/(5)2Π (9)2Σ+/(5)2Π (9)2Σ+/(5)2Π

(14)Ω=1/2/(13) Ω=1/2

Aviod crossing (n)state/(m)state

128

M. Korek and H. Jawhari Table 10. Continued

(n+1)Ω =1/2/(n)Ω=1/2

rAC(Bohr)

ΔEAC(cm-1)

Crossing between (n)state/(m)state

(16)Ω=1/2/(15) Ω=1/2

11.4

65.71

(10)2Σ+/(6)2Π

(17)Ω=1/2/(16) Ω=1/2

29.4

66.70

(11)2Σ+/(6)2Π

8.8

3.26

(11)2Σ+/(7)2Π

11.5

519.22

(12)2Σ+/(11)2Σ+

20.6

502.78

(12)2Σ+/(11)2Σ+

8.3

545.70

(12)2Σ+/(11)2Σ+

11.1

121.37

(12)2Σ+/(7)2Π

12

88.07

(12)2Σ+/(7)2Π

17.7

4.62

(12)2Σ+/(7)2Π

34.5

62.83

(12)2Σ+/(7)2Π

16

44.80

(13)2Σ+/(8)2Π

21.7

18.20

(14)2Σ+/(8)2Π

25.8

6.20

(14)2Σ+/(8)2Π

35

73.10

(14)2Σ+/(8)2Π

32.2

163

56.5

5.5

(15)2Σ+/(8)2Π

26.4

8.5

(16)2Σ+/(9)2Π

31

13.5

(16)2Σ+/(9)2Π

(18)Ω=1/2/(17) Ω=1/2

(19)Ω=1/2/(18) Ω=1/2

(21)Ω=1/2/(20) Ω=1/2 (22)Ω=1/2/(21) Ω=1/2

(23)Ω=1/2/(22) Ω=1/2 (25)Ω=1/2/(24) Ω=1/2

Aviod crossing (n)state/(m)state

(15)2Σ+/(14)2Σ+

Table 11. Some avoided crossings between Ω=3/2 states of CsH+ (n+1)Ω =3/2/(n)Ω=3/2

rAC(Bohr)

ΔEAC(cm-1)

Crossing between (n)state/(m)state

(6)Ω =3/2/(5)Ω=3/2

9

63.07

(4)2Π/(2)2Δ

(7)Ω =3/2/(6)Ω=3/2

8.2

1848.77

(8)Ω =3/2/(7)Ω=3/2

4.6

14.75

(5)2Π/(3)2Δ

(9)Ω =3/2/(8)Ω=3/2

7.4

20.47

(6)2Π/(3)2Δ

(12)Ω =3/2/(11)Ω=3/2

11.8

26.63

(8)2Π/(4)2Δ

Avoid crossing (n)state/(m)state (5)2Π/(4)2Π

Each Ω-state, except for few ones, has more than one main parent Λ-state. This is proved in Tables 12 and 13 by showing the percentage of parent Λ-states over a certain range of the internuclear distance of Ω-states.

Theoretical Calculation of the Low Laying Electronic States…

129

425000 (8)Ω=1/2 (9)Ω=1/2

424800

(6)Σ+ (3)Π

E(cm-1)

424600

424400

424200

424000 13.2

13.4

13.6

13.8

14

14.2

14.4

14.6

R(Bohr)

Figure 10. Avoided crossing between (8)Ω=1/2 and (9)Ω=1/2 is due to crossing between (6)2Σ+ and (3)2Π. 445200 (22)Ω=1/2 (23)Ω=1/2 (14)Σ+ (15)Σ+

E(cm-1)

444800

444400

444000 30

31

32

33

34

35

R(Bohr)

Figure 11. Avoided crossing between (22) Ω=1/2 and (23) Ω=1/2 is due to avoided crossing between (14) 2Σ+ and (15) 2Σ+.

Table 12. Parent states for the potential energy curves of (n)Ω=1/2 (n)

From

to

(1)

3

60.5

(2)

3

5

(3)

3

5

(4)

3

4.7

(5)

3

4.7

(6)

3

7.8

(7)

3

5

(8)

3

5.1

(9)

3

4.7

(10)

3

4.7

% State 100% (1)2Σ+ 3.4% (1)2Π 3.4% (2)2Σ+ 2.9% (2)2Π 2.9% (3)2Σ+ 8.3% (4)2Σ+ 3.4% (3)2Π 3.6% (5)2Σ+ 2.9% (4)2Π 2.9% (6)2Σ+

From

to

5

60.5

5

60.5

4.7

60.5

4.7

7.9

7.8

60.5

5

60.5

5.1

14

4.7

13.9

4.7

17.6

% State

96.5% (2)2Σ+ 96.5% (1)2Π 97% (3)2Σ+ 5.5% (2)2Π 91.6% (2)2Π 96.5% (5)2Σ+ 15.4% (3)2Π 16% (6)2Σ+ 22.4% (4)2Π

From

to

% State

7.9

60.5

91.5% (4)2Σ+

14

60.5

13.9

60.5

17.6

60.5

80.9% (6)2Σ+ 81.1% (3)2Π 74.6% (7)2Σ+

Table 12. Continued (n)

From

to

(11)

3

17.6

(12)

3

6

(13)

3

6

(14)

3

60.5

(15)

3

11.5

(16)

3

4.7

(17)

3

5

(18)

3

8.8

(19)

3

60.5

3

6.6

16

60.5

(20)

% State 25.3% (7)2Σ+ 5.2% (8)2Σ+ 5.2% (5)2Π 100% (9)2Σ+ 14.7% (6)2Π 2.9% (7)2Π 3.4% (10)2Σ+ 10% (11)2Σ+ 100% (12)2Σ+ 5.7% (8)2Π 77.9% (13)2Σ+

From

to

17.6

60.5

6

12

6

12.4

11.5

60.5

4.7

11.3

5

8.7

8.8

60.5

6.6

11.8

% State 74.6% (4)2Π 10.4% (5)2Π 11.1% (8)2Σ+

85.2% (10)2Σ+ 11.4% (10)2Σ+ 6.4% (7)2Π 89.9% (7)2Π

9% (13)2Σ+

From

to

12

60.5

12.4

60.5

11.3

60.5

8.7

60.5

11.8

16

% State

84.3% (8)2Σ+ 83.6% (5)2Π

85.6% (6)2Π 90.1% (11)2Σ+

7.3% (8)2Π

Table 12. Continued (n)

From

to

3

6.7

16

60.5

3

3.7

21.6

60.5

3

5.2

3

13.3

32

44

3

26.4

44

60.5

(21)

(22)

(23)

(24)

(25)

% State 6.4% (13)2Σ+ 77.4% (8)2Π 1.2% (13)2Σ+ 67.6% (8)2Π 3.8% (14)2Σ+ 17.9% (15)2Σ+ 20.8% (9)2Π 41% (15)2Σ+ 28.3% (9)2Π

From

to

6.7

11.5

3.7

5.2

5.2

13.3

13.3

26.4

44

60.5

26.4

31

% State 8.3% (8)2Π

2.6% (9)2Π

14% (9)2Π 22.7% (9)2Π 28.8% (16)2Σ+ 8% (9)2Π

From

to

11.5

16

5.2

21.6

13.3

60.5

26.4

32

31

44

% State 7.8% (13)2Σ+

28.5% (14)2Σ+

82.1% (15)2Σ+ 9.7% (16)2Σ+

22.6% (15)2Σ+

Table 13. Parent states for the potential energy curves of (n)Ω=3/2 (n)

From

to

(1)

3

60.5

(2)

3

60.5

(3)

3

60.5

(4)

3

60.5

(5)

3

9

(6)

3

9

(7)

3

21.8

(8)

3

4.5

(9)

3

7.2

(10)

3

60.5

(11)

3

11.7

(12)

3

11.7

(13)

3

60.5

% State 100% (1)2Π 100% (2)2Π 100% (1)2Δ 100% (3)2Π 10.4% (2)2Δ 10.4% (4)2Π 32.6% (5)2Π 2.6% (5)2Π 7.3% (6)2Π 100% (7)2Π 15.1% (4)2Δ 15.1% (8)2Π 100% (9)2Π

From

to

9

60.5

9

60.5

21.8

60.5

4.5

7.2

7.2

60.5

11.7

60.5

11.7

60.5

% State

89.5% (4)2Π 89.5% (2)2Δ 67.3% (2)2Δ 4.6% (3)2Δ 92.6% (3)2Δ

84.8% (8)2Π 84.8% (4)2Δ

From

to

% State

7.2

60.5

92.7% (6)2Π

134

M. Korek and H. Jawhari

3.5. The Vibration-Rotation Calculation The canonical functions approach [1, 2, 3] enables us to calculate the eigenvalue energy Ev, the rotational constant Bv, the centrifugal distortion constant Dv at any vibrational level even near dissociation. However, this approach fails if avoided crossings between states occur because of the break down of the Born-Oppenheimer approximation at these points. Here for the alkali dimmer CsH+, these constants have been calculated for 8 states in Λ-representation and 9 states in the Ω-representation up to vibrational level v=19. From the cubic spline interpolation between each two consecutive energy values of the PECs, and by using eigenvalue energies Ev, the abscissas of the turning points (rmin, rmax) of the above mentioned states have been calculated. These constants for the states (1, 2, 3)2Σ+ are reported in the Tables 14 to 16, and the other are given in Ref. [59]. No comparison of these values with other results since they are given here for the first time. Table 14. Values for the eigenvalues (Ev), the abscissas of the turning points (rmin, rmax), the rotational constant (Bv), and the centrifugal distortion constant (Dv) for the state (1)2Σ+ of the molecular ion CsH+ v

Ev(cm-1)

rmin(Å)

rmax(Å)

Bv(cm-1)

Dv×10+5(cm-1)

0 1 2 3

214.728 587.566 873.605 1089.894

3.156 3.049 2.997 2.966

3.716 4.131 4.552 5.035

1.423 1.276 1.326 0.971

7.036 9.568 11.934 14.5043

Table 15. Values for the eigenvalues (Ev), the abscissas of the turning points (rmin, rmax), the rotational constant (Bv), and the centrifugal distortion constant (Dv) for the state (2) 2 + Σ of the molecular ion CsH+ v

Ev(cm-1)

rmin(Å)

rmax(Å)

Bv(cm-1)

Dv×10+5(cm-1)

0 1 2 3

190.687 501.900 737.346 921.226

3.240 3.134 3.083 3.051

3.849 4.319 4.795 5.287

1.3260 1.1693 1.0219 0.8949

7.9254 10.5019 11.8595 12.8388

Table 16. Values for the eigenvalues (Ev), the abscissas of the turning points (rmin, rmax), the rotational constant (Bv), and the centrifugal distortion constant (Dv) for the state (3) 2 + Σ of the molecular ion CsH+ v

Ev(cm-1)

rmin(Å)

rmax(Å)

Bv(cm-1)

Dv×10+5(cm-1)

0 1 2 3 4

129.638 377.385 608.215 820.972 1014.498

4.290 4.096 3.981 3.899 3.835

5.014 5.384 5.696 5.999 6.314

0.7785 0.7490 0.7169 0.6819 0.6435

2.9326 3.0982 3.3159 3.5967 3.9675

Theoretical Calculation of the Low Laying Electronic States…

135

Table 16. Continued v

Ev(cm-1)

rmin(Å)

rmax(Å)

Bv(cm-1)

Dv×10+5(cm-1)

5 6 7 8 9 10 11 12

1187.593 1339.148 1468.346 1574.907 1659.412 1723.533 1769.979 1802.086

3.785 3.745 3.714 3.690 3.761 3.658 3.648 3.641

6.655 7.038 7.484 8.018 8.676 9.504 10.564 11.938

0.6011 0.5544 0.5031 0.4475 0.3887 0.3287 0.2700 0.2149

4.4562 5.0841 5.8789 6.8411 7.9306 9.0650 10.1569 11.1810

3.6. Dipole Moment Knowledge of the permanent or transition dipole moment is essential. For the molecular ion CsH+ we calculate the transition electric dipole moment and the permanent electric dipole moment values of the different bound Ω-states as functions of the internuclear distance r. This data provides us information about the most efficient scheme of forming the CsH+ molecular ion. Moreover, our ab initio potentials for the excited states can be used to identify the complex behavior of the transition dipole moment as function of a

r. Permanent dipole moments M a (r) as well as all non-zero transition dipole moments

M ab (r)

M ab =≺ Ψea | μ e (r ) | Ψeb have been calculated for each electronic states (a, b) under consideration and in the whole range of r investigated here. Ψe and Ψe are respectively the electronic wave functions of two a

b

different electronic states and μe(r) is the permanent electronic dipole moment. This dipole moment function has been calculated for most of the Ω-states [59]. In Fig 14 we show, as illustration, the transition dipole moment between the states (22)Ω=1/2 → (21)Ω=1/2. The three peaks at 21.7, 25.8 and 35.0 Bohr respectively, correspond to crossing or avoid crossing between the states (14)2Σ+ and (8)2Π as shown in these figure15, 16 and 17and given in Table 10.

136

M. Korek and H. Jawhari 4

2

1

0 20

22

24

26

28

30

32

34

36

38

40

R(Bohr)

Figure 15. Variation dipole moment between (22) Ω=1/2 → (21) Ω=1/2 states. 445600 (21)Ω=1/2 (22)Ω=1/2 (14)Σ+

445400

E(cm-1)

Re(r)

3

(8)Π

445200

445000

444800 21

21.4

21.8

22.2

R(Bohr)

Figure 16. Crossing between (14)2Σ+ and (8)2Π states.

22.6

Theoretical Calculation of the Low Laying Electronic States…

137

444800 (21)Ω=1/2 (22)Ω=1/2 444700

(14)Σ+ (8)Π

E(cm-1)

444600

444500

444400

444300 24.8

25.2

25.6

26

26.4

26.8

R(Bohr)

Figure 17. Crossing between (14)2Σ+ and (8)2Π states. 444600

(21)Ω=1/2 (22)Ω=1/2

444400

(14)Σ+ (8)Π

E(cm-1)

444200

444000

443800

443600 33

34

35

36

37

R(Bohr)

Figure 18. Crossing between (14)2Σ+ and (8)2Π states.

38

138

M. Korek and H. Jawhari

4. Conclusion Using an ab initio approach the potential energy has been calculated for 16 states 2Σ+, 9 states 2Π, 4 states 2Δ, 25 states Ω=1/2, 13 states Ω=3/2 and 4 states Ω=5/2 of the molecular ion CsH+. For 19 bound states the harmonic vibrational constant ωe, the internuclear distance re and the electronic transition energy with respect to the ground Te have been calculated. Based on the canonical functions methods Ev, Bv and Dv have been calculated up to the vibrational levels v = 19 for 17 electronic states. From the calculated values of Ev for a given vibrational level v and by using a cubic spline interpolation between each two consecutive points of the potential energy curves the rmin and rmax of the turning points have been a

investigated for these bound states. Permanent dipole moments M a (r) as well as all nonb

zero transition dipole moments M a (r) (a≠b ) have been calculated for each electronic states (a,b) under consideration and in the whole range of r investigated here. The comparison of the present results with those available in literature shows a very good agreement.

References [1] [2] [3]

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

John Weiner, Vanderlei S. Bagnato, Sergio Zilio, Paul S. Julienne, Rev. Mod. Phys., 71, 1 (1999). R. Côte´ and A. Dalgarno, Phys. Rev. A 58, 498 (1998). A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, and P. Pillet, Phys., Rev. Lett. 80, 4402 (1998); T. Takekoshi, B.M. Patterson, and R.J. Knize, ibid. 81, 5109 (1998); A. N. Nikolov, E. E. Eyler, X. T. Wang, J. Li, H. Wang, W. C. Stwalley, and P.L. Gould , ibid. 82, 703 (1999); N. Balakrishnan, R.C. Forrey, and A. Dalgarno, ibid. 80, 3224 (1998). R. Côté, A. Dalgarno, and Y. Sun, Phys. Rev. Lett. 74, 3581 (1995). D. J. Heinzen, R. Wynar, P. D. Drummond, and K.V. Kheruntsyan, Phys. Rev. Lett. 84, 5029 (2000). A. N. Nikolov, E. E. Eyler, X. T. Wang, J. Li, H. Wang, W. C. Stwalley, and P. L. Gould Phys. Rev. Lett. 82, 703 (1999). W.C. Stwalley and H. Wang, J. Mol. Spectrosc. 195, 194 (1999); R. Côte´ and A. Dalgarno, ibid. 195, 236 (1999). M. Mark, T. Kraemer, P. Waldburger, J. Herbig, C. Chin, H. -C. Naegerl, R. Grimm. Phys. Rev. Lett. 99, 113201 (2007). H. Bethlem and G. Meijer, Int. Rev. Phys. Chem. 22, 73 (2003). O. Docenko, M. Tamanis, J. Zaharova, and R. Ferbera_A. Pashov, H. Knöckel and E. Tiemann, J. Chem. Phys. 124, 174310 (2006). I. Klincare, J. Zaharova, M. Tamanis, and R. Ferber, A. Zaitsevskii, E. A. Pazyuk, and A. V. Stolyarov, Phys. Rev. A, 76, 032511 (2007). O. Docenko, M. Tamanis, R. Ferber, A. Pashov, H. Knöckel, and E. Tiemann, Eur. Phys. J. D, 31, 205 (2004). J. M. Sage, S. Sainis, T. Bergeman, and D. DeMille, Phys. Rev. Lett. 94, 203001 (2005).

Theoretical Calculation of the Low Laying Electronic States…

139

[14] D. Wang, J. Qi, M. F. Stone, O. Nikolayeva, H. Wang, B. Hattaway, S. D. Gensemer, P.L. Gould, E. E. Eyler, and W. C. Stwalley, Phys. Rev. Lett. 93, 243005 (2004). [15] G. Roati, F. Riboli, G. Modugno, and M. Inguscio, Phys.Rev. Lett. 89, 150403 (2002). [16] G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, M. Inguscio, Science, 294, 1320 (2001); G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, M. Inguscio, Science, 297, 2240 (2002). [17] Roberto Casalbuoni and Giuseppe Nardulli, Rev. Mod. Phys., 76(1):263, (2004). [18] Hui Hu, Xia-Ji Liu, and Peter D. Drummond, Phys. Rev. Lett., 98(7):070403, (2007). [19] D. S. Petrov, G. E. Astrakharchik, D. J. Papoular, C. Salomon, and G. V. Shlyapnikov, Phys. Rev. Lett., 99(13):130407, (2007). [20] Christoph Graf vom Hagen, Thesis for the degree of Doctor of Natural Sciences, Ruperto- Carola University of Heidelberg, Germany (2008). [21] H. Scheidt, G. Speiss, A. Valance, and P. Pradel, J. Phys,. B. 11, 2665 (1978). [22] L. Kahn, P. Baybutt, and D. J. Truhlar, J. Chem., 65, 3826 (1976). [23] T.J. Millar, P.R.A. Farquhar, and K. Willacy, Astron. Astro-phys., Suppl. Ser. 121, 139 (1997). [24] G.E. Langer, C.F. Prosser, and C. Sneden, Astron. J. 100, 216 (1990). [25] M. Samland, Astrophys. J. 496, 155 (1998). [26] A. Natta and C. Giovanardi, Astrophys. J. Lett. 356, 646 (1990). [27] J.K. Watson and D.M. Meyer, Astrophys. J. Lett. 473, L127 (1996). [28] R.R. Lewis and W. L. Williams, Phys. Rev. Lett. 59B, 70 (1975). [29] E.A. Hinds, Phys. Rev. 44, 374 (1980). [30] J.P. Lawrence, G.G. Ohlson and J.L. McKibben, Phys. Lett. 28B, 594 (1969). [31] D. De Mille, Phys. Rev. Lett. 88, 067901 (2002). [32] M. Korek, A .R. Allouche, M. Kobeissi, A. Chaalan, M. Dagher, K. Fakherddin and M. Aubert-Frécon, Chem. Phys. 256, 1 (2000). [33] M .Korek, A .R. Allouche, K. Fakherddine, and A. Chaalan, Can. J. Phys .78, 977 (2000). [34] M. Korek and A .R. Allouche, J .Phys. B: At. Mol. Opt. Phys. 34, 3689 (2001). [35] M Korek, A .R. Allouche, and S .N. Abdul al, Can. J. Phys. 80, 1025 (2002). [36] M. Korek, G. Younes, and A. R. Allouche, Int. J. Quant. Chem. 92, 376 (2003). [37] M. Korek and G .Younes, Int. J. Quant. Chem. 101, 84 (2005). [38] M. Korek, K. Badreddine, A. R. Allouche, Can. J. Phys. In press [39] D. Maynau and J.P. Daudey, Chem. Phys. 81, 273 (1981). [40] W. Müller and W. Meyer, J. Chem. Phys. 80, 3311 (1984). [41] F. Spiegelmann, D. Pavolini, and J .P. Daudey, J. Phys. B: At. Mol. Opt. Phys. 22, 2456 (1989). [42] J.C. Barthelat and Ph. Durand, Gazz. Chem. Ital. 108, 225 (1978). [43] Ph. Durand and J.C. Barthelat, Theor. Chim. Acta, 38, 283 (1975). [44] D. Pavolini, T. Gustavson, F. Spiegelmann and J.P. Daudey, J. Phys. B: At. Mol. Opt. Phys. 22, 1721 (1989). [45] W. R. Wadt and P.J. Hay, J. Chem. Physics. 82, 299 (1985). [46] H. Kobeissi, M Dahger, M. Korek and A. Chaalan, J. Comput. Chem., 4, 218, (1983). [47] M. Korek and H. Kobeissi, J. Comput. Chem., 13, 1103, (1992). [48] M. Korek, Comput. Phys. Commun., 119, 169, (1999). [49] M. Foucrault, Ph. Millié, and J.P. Daudey, J. Chem. Phys. 96, 1257 (1992).

140

M. Korek and H. Jawhari

[50] S. Magnier and Ph. Millié, Phys. Rev. A. 53, 204 (1996). [51] A. R. Allouche, M. Korek, K. Fakherddin, A. Chaalan, M. Dagher, F. Taher and M. Aubert-Frécon, J. Phys. B: At. Mol. Opt. Phys. 33, 2307 (2000). [52] A.R.Allouche, G. Nicolas G, J.C. Barthelat and F. Spiegelmann, J. Chem. Phys. 96, 7646 (1992). [53] S. Rousseau S, A.R. Allouche, and M. Aubert-Frécon, J. Mol. Spectrosc. 203, 235 (2000). [54] M. Korek and H. Kobeissi. Can. J. Phys. 73, 559 (1995). [55] M. Korek. Can J. Phys. 75, 795 (1997). [56] M. Korek and K.Fakhreddine. can. J. Phys. 78, 969 (2000). [57] M. Korek, B. Hamdan and K.Fakhreddine. Physica scripta. 61, 66 (2000). [58] G. Herzberg, Spectra of diatomic molecule, Van Nostrand, Toronto, 1950. [59] H. Jawhari, Master Thesis, Beirut Arab University (2008). [60] See EPAPS Document No. E-JCPSA6-129-605840 for supplementary tables. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html. [61] S. A. Hammoud, Master Thesis, Beirut Arab University (2007) [62] National Institute of standards and Technology (NIST): Tables of Spectra of Hydrogen, Carbon, Nitrogen, and Oxygen Atoms and Ions, C.E. Moore, edited by J.W. Gallagher. CRC Handbook of Chemistry and Physics, Edition 76 (CRC Press, Boca Raton, FL), 336 pp. (1993) [63] National Institute of Standards and Technology, http://physics.nist.gov/ PhysRefData/ASD/ [64] L.Von szentpaly, P. Fuentealba, H. Preuss and H. Stoll, Chem. Phys. Letters, 93, 8, (1982).

In: Quantum Frontiers of Atoms and Molecules Editor: Mihai V. Putz, pp. 141-155

ISBN: 978-1-61668-158-6 © 2010 Nova Science Publishers, Inc.

Chapter 7

THEORETICAL EXPLANATION OF LIGHT AMPLIFYING BY POLYETHYLENE FOIL Vjekoslav Sajfert1a,, Dušan Popov2,b, Stevo Jacimovski3,c and Bratislav Tosic4,d 1

Technical Faculty “Mihajlo Pupin” Zrenjanin, Serbia 2 Universitatea “Politehnica”, Timisoara, Romania 3 Police Academy Belgrade, Serbia 4 Vojvodina Academy of Science and Arts Novi Sad, Serbia

Abstract In connection with the experimental result which stated that polyethylene foil amplifies about three times the penetrated light, we propose two theoretical explanations of this phenomenon. One of them is that several amplified peaks are the consequence of the forming of solitons in a polyethylene chain whose velocities are close to the velocity of sound. Forming of solitons, together with boundary conditions in a polyethylene macromolecules chain, which contain about thirty monomers, lead to the amplification of light. The second explanation requires introduction of homeopolar excitons in polymer macromolecules. Both energy gap of homeopolar excitons and width of homeopolar exciton zone are of the same order of magnitude. It means that transitions in a very wide zone give light quanta which are able to amplify the initial light. In order to avoid some confusion and misunderstandings, we wish to point out the following. Atoms and molecules as the whole are treated classically (transition through potential barriers, for example, etc.). The exception to this rule are phonon theories of crystals where the phonon is considered as a quanta of boson field, i.e., it means that, in the theory of mechanical oscillations, molecules and atoms as the whole are treated quantum mechanically. On the other hand, elementary excitations in crystals such as excitons, vibrons, spin waves, and ferroelectric excitations, etc., which arise from changes of some parts of atoms or molecules are treated quantum mechanically exclusively.

a

E-mail address: [email protected]. E-mail address: [email protected]. Correspondent author. c E-mail address: [email protected]. d E-mail address: [email protected]. b

142

Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski et al. In the analyses of this work, the excitations of an individual molecule subsystem (i.e. the quantum objects) would serve as an explanation of the light amplification by a polymer chain.

1. Introduction In some previous works [1,2] it was experimentally found that the polyethylene foil noticeably amplifies intensity of light, and that amplification is proportional to the foil thickness. It was separated seven lines from mercury lamp and each of them was amplified. The experimental procedure is described in detail in [1], and therefore it will not be repeated here. We shall quote only the results of the experiment. The seven lines of mercury lamp were amplified 3–4 times after passing the light through a polyethylene foil. The wavelengths of amplified peaks were 254, 315, 366, 436, 506, 577 and 624 nanometers. Seven initial lines which are separated, are the more probable consequence of impurities of Hg, since in the foil of absolutely pure Hg exists only one level at λ = 253.7 nm. We shall try to explain the amplification of incident peaks in the frames of two models. One model is based on the idea that solitons, whose velocity is close to the velocity of sound appearing in polyethylene molecular chains, create the conditions for light amplification. The second model requires introducing homeopolar excitons in the system of possible excitations in the polyethylene chain. In this case, amplification of light by polyethylene arises as a consequence of the fact that the width of homeopolar exciton zone is of the same order of amplitude as the energy of excitation of individual polyethylene monomer. The amplifications of peaks were produced by one polyethylene foil whose thickness was 2 mm. The peaks were amplified 3–4 times by this foil. The amplifications produced by two foils (thickness: 4 mm) could not be registered since the devices were not able to register highness of peaks. It means that amplifications sharply increase with respect to the thickness of polyethylene foil. The theoretical explanation of this amplification of light intensity, given in [1], was based on inverse population of electrons. Namely, it was assumed that in the polyethylene chain exist metastable energy levels where electrons gathered and that they coherently transit to ground state. This coherent transition, as in a laser, leads to the energy amplification. In the work [2], the amplification was explained by exciton and soliton transitions. The excitons were not of dipole-dipole type, but of exchange (homeopolar) type, since polyethylene monomers are forming linear polyethylene macromolecules (polyethylene chains) by homeopolar forces whose potentials are of the order of magnitude of excitation energy of an electron in isolated monomer. The behavior of such excitons will be analyzed in this work with the goal to explain amplification of the light.

2. Excitons and Solitons in Infinite Polyethylene Chains Frenkel excitons [3] more often appear in a molecular crystal where dipole-dipole interactions propagate excitations of an isolated molecule produced by visible light. Excitation energy of an isolated molecule lies between 2.5 eV and 5 eV, while energies of

Theoretical Explanation of Light Amplifying by Polyethylene Foil

143

dipole-dipole interactions are 50–100 times lower. In such situation, energy of exciton and energy of visible photons are practically identical. For forming polyethylene linear chain made of monomers C2H4 are responsible exchange forces (covalent connection of monomers) [4,5]. Since potential energies between two electrons are reflexive, the potential energy is of positive sign. On the other hand if electron spins are parallel, the configurationally parts of wave function must be antisymmetric and this leads to negative exchange integrals which are binding monomers in chain. Consequently, if one isolated monomer of chain is excited by quanta of visible light this excitation propagates along the chain in covalent forces field. The matrix elements of covalent potential are of the order of five electronvolts. It means that they are of approximately of same magnitude as excitations of an isolated molecule. It is essential difference with respect to excitons in molecular crystals. In polyethylene chains wideness of exciton zone is practically same as the energy of excitation of an isolated monomer. It means that in polyethylene chains can appear excitons whose energy is two or three times higher than energy of visible light which produces excitons. This amplifying is registered in experiment described in the previous section. The mechanical oscillations interacting with excitons lead to forming of new quasiparticles - solitons [6], which can have higher energy than exciton. Besides, the solitons are qusiparticles of stable form moving superfluidely and having higher luminescence time than excitons. Luminescence time for singlet excitons is about 10

−8

s, for triplet excitons

−3

−2

−1

(corresponding to parallel electron-spins) 10 s [7] and for solitons about 10 s - 10 s [8]. The long times of luminescence means that metastable state are lasting sufficiently long to cause coherent illumination. The upper ideas will be demonstrated on the simple case of two level monomer excitation. The electronic Hamiltonian of monomer chain can be written in the following way:

H = ∑ ε s ans+ ans + n,s

1 2

∑W (s , s , s , s )a nm

1

2

3

n, m s1 , s 2 , s 3 , s 4

4

+ ns 1

+ ams a ans 2 3 ms 4

(1)

where a are Fermi operators of electrons localized in monomer and ε (which are of the order 2.5 -5 eV) are energies of excitations of an isolated monomer. The matrix element W include Coulomb and exchange forces and they are given by

( ) ( ) n e− m Ψ (ξ ) Ψ (ξ )

Wnm = ∫ d 3ξ n d 3ξ m Ψs*1 ξ n Ψs*3 ξ m

2

s4

m

s2

n

(2)

where ξ are internal coordinates of monomers. Because we consider two-level scheme, indices s1, s2 , s3 , s4 must have only two values: 0 - corresponding to ground state and f corresponding to excited state.

144

Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski et al. Here and below, for reasons of simplifying the writing of equations, we will use in the

denominator the following notation: n − m ≡

ξn − ξm . +

+

+

Introducing the operators of monomers excitations P = a f a0 and P = a0 a f we

(

easily concluded that they are closed in electron subspace 10 ,0 f operators P

+

)

(

)

and 00 ,1 f . The

and P are Pauli operators which in the lowest ASQ approximation can be +

substituted by Bose operators B and B . +

Extracting from the Hamiltonian P only quadratic terms in P and substituting P with Bose operators B , we obtain the following excitonic Hamiltonian in ASQ approximation +

+

(ground state terms as well as the terms proportional to P P and PP are omitted):

V ( f , f ,0,0) + V (0,0, f , f ) ⎤ + ⎡ H = ∑ ⎢ε f − ε 0 − V (0,0,0,0) + ⎥ Bn Bn 2 n ⎣ ⎦ +

1 ∑ 2 [Wnm ( f ,0,0, f ) + Wnm (0, f , f ,0)]Bn+ Bm

(3)

n, m

where V =

∑ Wn − m . m

It will be now shown that W ( f , f ,0,0 ) and W (0,0, f , f

while W ( f ,0,0, f

) are positive Coulomb terms

) and W (0, f , f ,0) are negative exchange terms.

The antisymmetric electron pair function is given by:

Ψnm =

[ ( ) ( )

( ) ( )]

1 Ψ f ξ n Ψ0 ξ m − Ψ f ξ m Ψ0 ξ n 2

(4)

and, as a consequence of this fact, the integral of the electron-electron interaction

∫

3

d ξn d

3

* ξm Ψnm

e2 Ψnm n−m

consists from four integrals:

∫

d 3ξ n d 3ξ mΨ nm*

( ) ( )

( ) ( )

e2 1 e2 Ψ nm = { ∫ d 3ξ n d 3ξ mΨ f* ξ n Ψ 0* ξ m Ψ0 ξm Ψ f ξn n−m 2 n−m

( ) ( ) n e− m Ψ (ξ )Ψ (ξ )

+ ∫ d 3 ξ n d 3ξ mΨ 0* ξ n Ψ f* ξ m

2

f

m

0

n

Theoretical Explanation of Light Amplifying by Polyethylene Foil

145

( ) ( ) n e− m Ψ (ξ )Ψ (ξ ) }

− ∫ d 3 ξ n d 3ξ mΨ 0* ξ n Ψ f* ξ m =

2

0

m

f

n

1 [Wnm ( f , f ,0,0) + Wnm (0,0, f , f ) + Wnm ( f ,0,0, f ) + Wnm (0, f , f ,0)] 2

(5)

Introducing notations

Δ = ε f − ε0 D=

V ( f , f ,0,0 ) + V (0,0, f , f ) − V (0,0,0,0) 2 1 − Wnm = [Wnm ( f ,0,0, f ) + Wnm (0, f , f ,0)] 2

(6)

and taking the nearest neighboring approximation ( Wn, n ±1 = W ), the Hamiltonian (3) can be written as (see [5,6]):

H exc = (Δ + D )∑ Bn+ Bn − W ∑ Bn+ (Bn +1 + Bn −1 ) ; W > 0 n

(7)

n

By means of Fourier transformations

Bn =

1 N

∑

eikan Bk

(8)

k

the Hamiltonian (7) goes over to

H exc = ∑ Ek Bk+ Bk

(9)

Eexc (k ) = Δ + D − 2W cos ak

(10)

k

where

Now we can estimate exciton energy in polymer chain with exchange interaction. We will take Δ = 3 eV , D = 1 eV (matrix elements W ( f , f ,0,0 ) are bigger than matrix

elements W (0,0,0,0 ) and W ~2.5 eV. The maximal energy of excitons for ak = π is 9 eV

and it is 3 times higher than energy excitations (energy of photons). This corresponds to the results of quoted experiment. In the case of dipole-dipole interaction we can take D − 2W cos ak = 0.06 eV , so that Eexc (k )max = 3.06 eV . This example points out a

146

Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski et al.

sharp difference between exciton energies in systems with dipole-dipole interactions with respect to those energies in systems with exchange interactions. At the end, we shortly expose the soliton characteristics of this exciton system. To the excitonic Hamiltonian we add the phonon Hamiltonian:

H ph =

1 2M

∑

pn2 +

n

C (un − un −1 )2 ∑ 2 n

(11)

where M is the mass of monomer, C is the Hook's constant of monomer chain, un are displacements of monomers and pn = u are corresponding momenta. The exciton phonon Hamiltonian is given as follows:

∂W + ⎡ ∂D + ⎤ ∂u H ep = ∑ ⎢2 Bn Bn − Bn (Bn +1 + Bn −1 )⎥ a ∂a ⎦ ∂ (na ) n ⎣ ∂a

(12)

Using complete Hamiltonian of the system

H = H exc + H ph + H ep

(13)

and equations of motion for B we obtain

∂W ⎡ ∂D (Bn +1 + Bn −1 )⎤⎥ a ∂u (14) EBn = (Δ + D )Bn − W (Bn +1 + Bn −1 ) + ⎢2 Bn − ∂a ⎣ ∂a ⎦ ∂ (na ) and

∂ 2u n C (un +1 + un −1 − 2un ) = M dt 2

(15)

The equations can be averaged by coherent states

< Ψep eβ n (Bn − Bn )Bn e −β n (Bn − Bn ) Ψep >=< Ψep eβ n (Bn − Bn )Bn+ e −β n (Bn − Bn ) Ψep >= βn +

+

+

Ψep >= nexc > n ph > ;

+

β*n = βn

(16)

and

< Ψep

γn pn ei u

ne

−

γn pn i

Ψep >= − γ n ; γ* = γ

(17)

After that we go over to continual variable na → x and to common variable ξ = x − vt , where v is velocity of soliton. After quoted operations, the equation (15) goes over to:

Theoretical Explanation of Light Amplifying by Polyethylene Foil

(v

2

− s2

147

)ddξγ = 0

(18)

C M

(19)

2

2

where

s2 = a2

is the square velocity of sound in the chain. The boundary condition for this equation is

γ (0 ) = 0 ,

dγ dξ

ξ =0

= M −1 p0

(20)

where p0 is the beginning momentum which produce mechanical oscillations. By the way, one of explanations of appearance of tsunami is based on such beginning conditions. By means of conditions (20) we obtain

dγ M −2 p02 = dξ v 2 − s 2

(21)

The equation (14), written in continuum, after the substitution (21), becomes:

⎡ 2aM −2 p 2 ⎛ ∂V ∂W E − ⎢Δ + D − 2W + 2 2 0 ⎜ − v − s ⎝ ∂a ∂a d 2β ⎢⎣ = ∂W ⎞ ⎛ dξ 2 a 2 ⎜W + a ⎟ ∂a ⎠ ⎝

⎞⎤ ⎟⎥ ⎠⎥⎦

β=0

(22)

If the coefficient in second term of (22) is positive the solution of (22) is

β = c1 cosh Qξ = c1 cosh Q( x − vt )

(23)

where

Q=

⎡ 2aM − 2 p 2 ⎛ ∂V ∂W ⎞⎤ E − ⎢Δ + D − 2W + 2 2 0 ⎜ − ⎟⎥ v − s ⎝ ∂a ∂a ⎠⎦⎥ ⎣⎢ ∂W ⎞ ⎛ a 2 ⎜W + a ⎟ ∂a ⎠ ⎝

(24)

The details of further calculations one can find in ref. [6, pp. 27-31]. Here they will not be quoted. For our aims it is important to say that the wave function of soliton is proportional to reciprocal value of β , i.e.

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Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski et al.

Ψsol =

c2 1 = β cosh Q( x − vt )

The function Ψsol is taken as Ψsol = cosh

−1

(25)

[Q(x − vt )]

since Q contains arbitrary

parameter soliton energy E. Soliton energy Esol can be determined from normalization condition: +∞

1 2 dξ Ψsol = 1 ∫ a −∞

(26)

2 =1 aQ

(27)

which reduces to

wherefrom we obtain

E = Δ + D + 2W +

2aM −2 p02 ⎛ ∂V ∂W − ⎜ v 2 − s 2 ⎝ ∂a ∂a

∂W ⎞ ⎟ + 4a ∂a ⎠

(28)

Soliton wave function (25) is of the stable form and propagates practically superfluidly. Comparing (26) with (10) we see that soliton energy is some higher than the exciton one, but when s → v , i.e. when soliton velocity becomes close to sound velocity then soliton energy becomes noticeably higher than exciton energy. It means that in the soliton model of excitations in polyethylene, amplification can be considered as a consequence of sharp increase of soliton energies when velocity of solitons becomes very close to the velocity of sound.

3. Green’s Function of Homeopolar Excitons The general form of electron Hamiltonian, where overlapping of electron wave functions of neighbor molecules is weak, is given by N

H = ∑ ∑ ε s ans+ ans + n=0 s

+

8 1 N Wn − m ( s1 , s2 , s3 , s4 )ans+ 1 ans 2 ans+ 3 ans 4 ∑ ∑ 2 n, m = 0 s1 , s 2 , s3 , s 4 = 0

Here a and are a electron operators,

(29)

ε s is energy of electron in an isolated monomer

and W are matrix elements of reflexive electron potential

e2 taken over products of n−m

Theoretical Explanation of Light Amplifying by Polyethylene Foil

149

configuration electron functions and spin electron functions. For two electrons forming bonds between two monomers spin functions is even (it corresponds to parallel spins) while the configurationally function must be odd. It gives attractive transition potentials between monomers whose order of magnitude is 5 eV, i.e. of the same order of magnitude as excitation energies of electrons in an isolated monomer. It should be noted that in the case of dipole-dipole interactions (Frenkel excitons) attractive potentials between molecules are of the order of 10-2 eV. Using the well known procedure, taken from the theory of Frenkel excitons for two level schemes of electron excitations [3, 4] and going over to approximate second quantization approach, we can write the Hamiltonian (29) in the nearest-neighboring approximation as: N

N

N

n=0

n=0

n=0

H = Δ ∑ Bn+ Bn + ∑ (Dn , n +1 + Dn , n −1 )Bn+ Bn −∑ Bn+ (J n , n +1Bn +1 + J n , n −1 Bn −1 ) (30) where

Δ = ε f − ε 0 ; Dn, n ±1 =

J n, n ±1 =

Here

Wn , n ±1 ( f , f ,0,0) + Wn , n ±1 (0,0, f , f ) − Wn , n ±1 (0,0,0,0) (31) 2

Wn , n ±1 ( f ,0,0, f ) + Wn , n ±1 (0, f , f ,0) ; D > 0, J > 0 2

ε f denotes energy of excited electron, while ε 0 denotes electron energy in ground

state. Assuming that polyethylene linear chain has N monomers (N is 30-50 monomers) we must introduce the boundary conditions

D0, −1 = DN , N +1 = 0 ; J 0, −1 = J N , N +1 = 0

(32)

The exciton system with Hamiltonian (2) will be analyzed by means of the “real space” Green’s function method. The double-time Green’s functions are

⎧1, n = m (33) Gnm (t ) =<< Bn (t ) | Bm+ (0) >>= θ (t ) < Bn (t ), Bm+ (0) > ; θ (t ) = ⎨ ⎩0 , n ≠ m

[

where

[, ]

is the commutator, <

]

> is the thermal average over the grand canonical

ensemble and θ (t ) is the Heaviside function. Using the equations of motion for Bose operators B and taking into account the boundary conditions (32) we will obtain, after Fourier transformations time-frequency, the system of three difference equations defining Green’s functions,

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Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski et al.

J [Gn +1, m (ω ) + Gn −1, m (ω )] + ρGn , m (ω ) = JG1, m (ω ) + ( ρ + D)G0, m (ω ) =

i δ n, m ; 1 ≤ n ≤ N − 1 2π i δ 0, m ; n = 0 2π

JGN −1, m (ω ) + ( ρ + D)GN , m (ω ) = where

i δ N ,m ; n = N 2π

ρ = E − Δ − 2D

(34)

(35)

(36)

(37)

It can be shown that the system of equations (34) - (36) reduces into unique equation (39) by the substitution: N +1

Gn,m (ω ) = ∑ Aν (m, ω ) Fν (n)

(38)

ν =1

and N

i

∑ (2 J cosϕν + ρ )Aν (m,ω ) Fν (n) = 2π δ ν =0

n,m

; n = 0, 1, 2,…, N

(39)

where

Fν (n) = sin(n + 1)ϕν −

D sin nϕν J

(40)

Parameters φν are the solutions of the transcendental equation 2

D ⎛D⎞ sin( N + 2)ϕν − 2 sin( N + 1)ϕν + ⎜ ⎟ sin Nϕν = 0 J ⎝J⎠

(41)

The Kronecker symbol will be represented as follows N +1

δ n, m = ∑ Fν (n) Φν (m) ; n, m ∈ {0, 1, 2,…, N }

(42)

ν =1

Putting equation (42) into (39) we obtain

Aν ( n , ω ) = Φ ν ( n ) gν ( ω )

(43)

Theoretical Explanation of Light Amplifying by Polyethylene Foil

151

where

gν ( ω ) =

1 i 2π ω − Ων

(44)

After substituting the equations (42) - (44) into equation (39), we get

Ων =

Eν

; Eν = Δ + 2 D − 2 J cos ϕν

(45)

The spectral intensity of Green’s functions:

Gn , m (ω ) =

i N Fν (n) Φν (m) ∑ 2π ν = 0 ω − Ων

(46)

is given by [5, 6] N +1

I n, m (ω ) = ∑ ν =1

Fν (n) Φν (m) ω

e

k BT

δ (ω − Ων )

(47)

−1

By means of the formula (3.19 ) we can determine the correlation function ∞

N +1

Cn , m (t ) =< Bm+ (0) Bn (t ) >= ∫ dω e − iω t I n , m (ω ) = ∑ ν =1

−∞

Fν (n) Φν (m) e

Eν k BT

e

i − Eν t

(48)

−1

and the concentration of excitons N +1

Cn , m (0) =< Bm+ (0) Bn (0) >= ∑ ν =1

Fν (n) Φν (m) e

Eν k BT

(49)

−1

This formula gives the exciton energies in analysed polyethylene chain.

4. The Exciton Wave Function of Polyethylene Chain The equation (45) gives the energies of excitons in polymer chain. Using this formula we can find the transitions whose wavelenghts correspond to experimentally obtained wavelenghts , which are quoted in Introduction. Besides these wavelenghts, we shall determine the probabilities of finding of exciton at given energy level, as well as the probabilities of exciton transition between two energy levels. These probabilities can be find out by means of exciton one particle wave function

152

Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski et al. N +1

| Ψν >= ∑ Aν (n) Bn+ | 0 >

(50)

n =1

On the basis of equations motion and boundary conditions, it is obtained the system of homogeneous diference equations for coefficients Aν(n):

An +1 + An −1 + ρAn = 0 ; 1 ≤ n ≤ N − 1 D⎞ ⎛ A1 + ⎜ ρ + ⎟ A0 = 0 ; n=0 W⎠ ⎝ D⎞ ⎛ AN −1 + ⎜ ρ + ⎟ AN = 0 ; n=N W⎠ ⎝

(51)

which, by the substitution N +1 D ⎡ ⎤ N +1 An = ∑ Aν ⎢sin( n + 1)ϕν − sin nϕν ⎥ = ∑ Aν Fν ( n) J ⎣ ⎦ ν =1 ν =1

(52)

goes over to the unique equation N +1

∑ (2 J cosϕν + ρ )Fν (n) = 0 ; ν

n ∈ (0,1, 2,…, N )

(53)

=1

where Fν (n) is given by (40) and parameters φν are real solutions of the equation (41). Consequently, finally we have N

Bν+ | 0 >= ∑ Fν (n)Bn+ | 0 >

(54)

n=0

Multiplying equation (54) with Φν(m), summing both sides of obtained equality over ν and taking into account (42), we obtain: N +1

Bn+ | 0 >= ∑ Φν (n)Bν+ | 0 >

(55)

ν =1

+

The normalised function Bn 0 is given by

Bν+ | 0 >=

N

1 N

∑ Fν (n) n =0

2

∑ Fν (n)B n =0

+ n

|0>

(56)

Theoretical Explanation of Light Amplifying by Polyethylene Foil

153

+

while normalised function Bν 0 is:

Bn+ | 0 >=

N +1

1 N +1

∑ Φν (n) ν 2

∑ Φν (n)Bν ν

+

|0>

(57)

=1

=1

The dependence of exciton energies with respect to

ϕν , which are obtained from (17) for

values Δ = 3 eV; D = 2.5 eV; J = 2.7 eV, is given in Figure 1.

Figure 1. The energy levels of excitons in polyethilen chain.

Table 1. Experimental and theoretical wavelengths of transitions leading to amplifications

λexp (nm) λtheor (nm) Transition Amplification factor ν →ν ' 254 315 366 436 506 577 624

256 314 363 439 503 574 625

32 → 18 27 → 18 24 → 17 32 → 22 23 → 18 30 → 23 29 → 23

4.46 4.16 3.81 4.46 3.67 4.39 4.33

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Vjekoslav Sajfert, Dušan Popov, Stevo Jacimovski et al.

Experimental and theoretical wavelenghts of peaks as well as the theoretical amplifying of peaks are given in the Table 1. The theoretical values λ are determined as

hc (Eν − Eμ ) ; ν > μ . −1

On the other hand, the probabilities of finding excitons on levels before luminiscence are given Table 2. Table 2. The probabilities of existence at stable levels where Φν =

Pν

Pν =

Φν2

N +1

∑ Φν2

1 31 ∑ Φν (n ) 32 n = 0

Pν (%)

ν =1

P23 P22 P18 P17

0.299281 0.072849 0.265614 0.152059

8.956 0.531 7.055 2.312

Table 3. The transition probabilities of exciton to stable levels

Pν , μ =

Φν Φ μ N +1

∑ Φν ν

2

Pν , μ (%)

=1

P32;18 P30;23 P32;23 P27;18 P32;22 P24;17 P23;18

3.454 7.794 6.289 0,796 0.947 3.652 7.949

Similarly, the transitions probabilities from higher level Eν to the stable levels Eμ quoted in the previous Table 2. are presented in the Table 3.

Conclusion In this work are analyzed two possible mechanisms of light amplification by polyethylene foil, registered in experiments with a mercury lamp. In these experiments the light, after transition through polyethylene foil was about 3 times more intensive than in the cases when the foil is absent. In the first model it was assumed that in the polymer chain solitons are forming whose velocities were close to the velocity of sound and this caused the light amplification.

Theoretical Explanation of Light Amplifying by Polyethylene Foil

155

In the second model was introduced the concept of homeopolar excitons having approximately equal gap and zone width. The analyses by means of Green's function as well as the wave of homeopolar excitons wave function were determined metastable levels of homeopolar excitons and probabilities of transition levels to ground state. Calculated theoretical values of mentioned probabilities were in good agreement with experimental data.

Acknowledgements This work was supported by the Serbian Ministry of Science and Technology: Grant No 141044, by Vojvodina Academy of Sciences and Arts, and by the Provincial Secretariat for Science and Technological Development of the Autonomous Province of Vojvodina (Project 114-451-00615/2007-06).

References [1] [2]

[3] [4] [5] [6] [7] [8]

Janjić J. D., Tošić B. S., Scientific Bulletin of the „Politehnica“ University of Timisoara, ROMANIA, Transactions on Mathematics and Physics 2006, 51, 80. Janjić J. D., Tošić B. S., Sajfert V., "Luminescence Mechanism of Light Amplyfing by Polyethylene, Bulletin of the “Politehnica” University of Timisoara, ROMANIA, Transactions on Mathematics and Physics 2008, 53, 91. Agranovich V.M., JETP 1959, 37, 430. Agranovich V. M., Theory of Excitons; Nauka: Moscow, RUSSIA, 1978. Тyablikov S.V., Methods in the Quantum Theory in Magnetism, Plenum Press: New York, USA, 1967. Tošić B. S., Statistical Physics, Faculty of Natural Sciences, Novi Sad, SERBIA, 1978. (in Serbian) Masterson W. W., Hurley C. N., Chemical Principles and Reactions, Saunders College Publishing, Phyladelphia, USA, 1993; pp. 579. Brown, W.H., Introduction in Organic Chemistry, Saunders College Publishing Phyladelphia, USA, 1997; pp. 428.

In: Quantum Frontiers of Atoms and Molecules Editor: Mihai V. Putz, pp. 157-189

ISBN: 978-1-61668-158-6 © 2010 Nova Science Publishers, Inc.

Chapter 8

ANHARMONIC EFFECTS IN NORMAL MODE VIBRATIONS: THEIR ROLE IN BIOLOGICAL SYSTEMS Attila Bende* Molecular and Biomolecular Physics Department, National Institute for Research and Development of Isotopic and Molecular Technologies, Donath street, No: 65 – 103, Ro-400293, Cluj-Napoca, ROMANIA

Abstract The intra- and intermolecular hydrogen-bond dynamics plays an important role in thermal stability and molecular functionality of biomolecules like DNA base pairs and proteins. The phenomena of thermal relaxation, conformational changes as well as the ultrafast nonradiative decays in molecules, especially in biomolecules are realized among other physical effects also with the help of molecular vibrations. Intermolecular hydrogen- or weak chemical bonds (covalent bond including H atoms) usually present large anharmonic normal mode vibrations. The anharmonic effects of normal mode vibrations for some small molecular model systems (formamide and urea) as well as for guanine-cytosine and adenine-thymine DNA base pairs are presented considering DFT and ab initio second order Møller-Plesset (MP2) theoretical methods. The role of basis set superposition errors in harmonic and anharmonic frequency calculations is briefly discussed. It was observed that anharmonic effects can significantly change the blue- or red-shifted harmonic frequency values. Large anharmonic effects were found in the case of protonated molecular structures, especial for Hoogsteen conformation of guanine-cytosine base pairs.

1. Introduction Hydrogen bonding is ubiquitous in nature and governs a wide array of chemical and biological processes ranging from local structure in molecular liquids to the structure and folding dynamics of proteins [1, 2]. Although the hydrogen bond is well studied, its lowfrequency vibrations—the large-amplitude motions involving stretching and bending along the actual hydrogen-bond coordinates—have been rarely investigated [3]. Information about *

E-mail addresses: [email protected], [email protected]

158

Attila Bende

these vibrations offers exceptional insight into the potential energy surface of the interaction and so further enhances our understanding of the hydrogen bond and its impact on molecular structure and dynamics. The “C=O···H-N” type hydrogen-bond (H-bond) is one of the most frequently occurring Van der Waals (VdW) bonds in the biological systems. They can be found as a main component of the DNA bases pair’s interaction systems or in protein α-helix and β-sheets. In this sense, not only is the cognition of molecular structures important, but also their dynamics, which in essence represent their biological functionality. Vibrational spectroscopy is a powerful tool for getting information about the complex dynamics of atoms in H-bonds. In particular, such calculations provide sensitive information on the potential energy surfaces of these systems [4, 5]. In the case of infrared (IR) spectroscopic studies, the most evident effect of hydrogen bonding are the red shift of the high-frequency X–H···Y stretching mode, its intensity increase and band broadening; the latter is often accompanied by the development of peculiar band-shapes. The large increase of the bandwidth, the band asymmetry, the appearance of subsidiary absorption maxima and minima, such as Evans windows, peculiar isotope and temperature effects and the similarity of features of the line shape when passing from the condensed to the gas phase [6, 7] are the challenge for theories of H-bonds. Computational studies are essential in this context since comparison of calculations to the measured spectroscopic data provides a test of the potential surface [8–11]. While calculations at the level of the harmonic approximation are very useful, they are often not of sufficient accuracy [8, 9, 12–14]. One approach is to apply empirical scaling factors in order to represent the anharmonic effects [15, 16]. However, such an approach, while useful in many cases, has no fundamental (quantum mechanical) basis. Moreover, it does not provide any insights into the nature of the anharmonic part of the potential, which itself is of great interest. Calculations of the spectra of large molecules beyond the harmonic approximation remain a major challenge. The goal of this Chapter is to give an accurate description of intermolecular normal modes and to present different intermolecular interaction effects which could influence the monomer type vibrations, considering the formamide and urea dimer cases and the guaninecytosine DNA base pair molecular system.

2. Theoretical Background and Computational Methods Regarding the theoretical methods, it is well established that for an accurate description of HBs, ab initio techniques which include the electronic correlation level are needed, i.e. with an error bar of less than 0.04 eV (1 kcal/mol) for predicting the HB strength. Thus the observed underestimation of the HB strength by the Hartree-Fock (HF) calculations (where the electron correlation effects are missing) is overcome using correlated methods like second order Møller-Plesset perturbation theory (MP2) or coupled cluster (CC) methods. But it is necessary to use very large high quality basis sets to expand the wave function and to get reliable HB properties. This fact and the necessity of correlated methods to accurately describe HBs make such studies computationally too expensive and only applicable to molecular complexes of at most a few tens of atoms. Therefore strategies to study HBs with similar accuracy to MP2 or higher levels of theory but computationally less expensive are needed. In this vein, density functional theory (DFT) is a method that includes electronic correlation. Unfortunately the accuracy of DFT to describe the HB interaction relies on the

Anharmonic Effects in Normal Mode Vibrations

159

applied functional to approximate the electronic exchange - correlation (XC) contribution. To overcome this problem, first we try to find those XC functionals which could describe HBs with an approximate accuracy as MP2. In the last few years a number of significant works could be found in the literature, which compared the efficiency of different DFT functionals with those obtained using MP2 method. Besides of the accurate electron correlation methods, based on the perturbational, coupled-cluster or Kohn-Sham (KS) scheme, the basis set superposition errors (BSSE) present a very important source of discrepancies. Many papers have indicated that the impact of BSSE on the geometries of weakly bound systems is smaller for DFT methods than those for ab initio methods such as MP2, but their influence even in case of DFT could not be neglected [17–19]. The reason why BSSE is similar in HF and KS methods is that the KS method does not require the high oscillatory basis functions which are needed to describe electron correlation in traditional electronic structure methods, and which are virtually impossible to include at saturation.

2.1. The BSSE Problem The BSSE is a pure “mathematical effect” and it is an important problem to solve when we study a weakly bonded molecular complex. This effect appears only as a result of the use of finite basis sets, because the description of the monomer is actually better within the supermolecule than that which one has for the free monomers by applying the same basis set, so thus leads to incomplete description in the individual monomers. Due to BSSE, the calculated interaction energies show too deep minima and the computed potential energy surface (PES) is distorted. The most important and straightforward a posteriori correction scheme, the so-called “function counterpoise” (CP), or simply the Boys–Bernardi method, was introduced by Jansen and Ross [20] and, independently, by Boys and Bernardi [21] in 1969/1970. In this CP scheme, the monomer energies are recalculated by using the whole supermolecular basis set and these corrected monomers are used in the molecular interaction energy calculations. A conceptually different way to handle the BSSE problem is to apply the “chemical Hamiltonian approach” (CHA) for the case of intermolecular complexes proposed by Mayer [22] in 1983. (For a detailed review on CHA, see Ref. [23].) By using the a priori CHA method one can eliminate the nonphysical terms of the Hamiltonian, which leads to wave functions free from the nonphysical delocalizations caused by BSSE. Using this CHA scheme, several approaches have been developed both at the HF [24–37] and correlated [38– 45] levels of theory to study the structures and interaction energies for different van der Waals and H-bonded systems.

The CP Scheme The simplest definition of the uncorrected interaction energy between two molecules is the difference of the supermolecular energy and the sum of the free monomer energies, each calculated in its own basis set: unc ΔE AB = E AB ( AB ) − E A ( A) − EB (B ) ,

(1)

160

Attila Bende

where EAB(AB), EA(A), and EB(B) denote the total energy of the AB “supermolecule” and the energy of the A and B monomers, respectively. The notations in parentheses indicate that basis sets corresponding to (sub)system A, B, and AB, respectively, were used. To compute unc , we need to use (nearly) complete basis sets the correct value of the interaction energy ΔE AB on the supermolecule and on each monomer, which is usually impossible in practice. CP In the CP scheme, the interaction energy ΔE AB is defined as the difference of the supermolecule and monomer energies, all computed in the same supermolecule basis set: CP ΔE AB = E AB ( AB ) − E A ( AB ) − EB ( AB )

(2)

Using Eqs. (1) and (2), one can define the BSSE content in the interaction energy as unc CP δEBSSE = ΔE AB − ΔE AB

= E A ( AB ) − E A ( A) + EB ( AB ) − EB (B )

(3)

According to Eq. (3), the CP-corrected potential energy surface (PES) of the dimer becomes unc E CP ( AB ) = ΔE AB − δE BSSE

unc = ΔE AB − E A ( AB ) + E A ( A) − E B ( AB ) + E B (B )

(4)

Equation (4) shows that by considering only the intermolecular internal coordinates as optimized parameters one has to calculate three different total energies to determine the CPcorrected PES.

The CHA Scheme In the alternative a priori CHA scheme introduced by Mayer [22, 23] one can omit the BSSE caused terms of the Hamiltonian, which is a conceptually different way of handling the BSSE problem. The CHA procedure permits the supermolecule calculations to remain consistent with those for the free monomer performed in their original basis sets. The basic idea of Mayer’s scheme is that one can divide the Born–Oppenheimer Hamiltonian into two parts:

Hˆ BO = Hˆ CHA + Hˆ BSSE

(5)

where Hˆ CHA is the BSSE-free part of the Hamiltonian and Hˆ BSSE is the “unphysical” part of the Hamiltonian that is responsible for the BSSE. The only difficulty of this scheme is that the resulting “physical” Hamiltonian Hˆ CHA is not Hermitical, so one cannot expect the BSSE-

Anharmonic Effects in Normal Mode Vibrations

161

free Hamiltonian Hˆ CHA to be Hermitical either. Based on this CHA Hamiltonian, Mayer and Vibók developed different SCF-type equations [28]:

Hˆ CHA ΨCHA = ΛΨCHA ,

ECHA / CE =

ΨCHA Hˆ BO ΨCHA ΨCHA ΨCHA

(6)

.

(7)

In the CHA framework [32–35] described by Eqs. (6)–(7) the non-Hermitical CHA Hamiltonian Hˆ CHA is used only to provide the BSSE-free wave function (or a perturbative approximation to it) but the energy should be calculated by using the conventional (Hermitical) Born– Oppenheimer Hamiltonian Hˆ BO , making not trivial the question of how one should calculate the second-order energy correction. The 0th-order Hamiltonian is defined in terms of the CHA Fockian [45]:

Hˆ 0 =

∑ ε ϕˆ ϕ~ˆ p

+ p

− p

,

(8)

p

+ − where ϕˆ p and ϕ~ˆ p are the creation and annihilation operators.

To obtain the first-order wave function in the perturbation theory, the non-Hermitical CHA Hamiltonian could be partitioned as

Hˆ CHA = Hˆ 0 + VˆCHA ,

(9)

where Hˆ defined by Eq. (8), is the Møller–Plesset type unperturbed Hamiltonian, which is 0

also non-Hermitical, and VˆCHA represents the perturbation. At the same time, the perturbation energy could be obtained considering the following partition of the Born–Oppenheimer Hamiltonian Hˆ BO :

Hˆ BO = Hˆ 0 + Vˆ

(10)

Then the energy up to second order could be presented as

E

(2 )

=

Ψ0 Hˆ BO Ψ0 Ψ0 Ψ0

where J2 is the generalized Hylleraas functional:

+ J2 ,

(11)

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Attila Bende

J2 =

[ (

) (

)]

1 2 Re Qˆ Ψ1 Vˆ Ψ0 − Re Ψ1 Hˆ 0 − E0 Ψ1 , Ψ0 Ψ0

(12)

Here Ψ0 is the unperturbed wave function, E0 is the zero order energy ( Hˆ Ψ0 = 0

= E0 Ψ0 ), Ψ1 is the first order wave function of the perturbation, and Qˆ is the projection operator on to the orthogonal complement to Ψ0 . These equations define our working formula at the second-order perturbation level. This formalism is called “CHA/MP2” theory [40].

2.2. Anharmonic Approach of Normal Mode Vibrations Computation of harmonic force field and implicit the harmonic vibrational frequencies many times give results which are very far from the experimentally measured values. The explanation seems to be quite simple. Beyond of harmonic approximation, many other contributions like solvent effects, anharmonic corrections or cluster effects, are not taken into account. Usually, all of these corrections were considered by multiplying the harmonic frequency values with a scaling factor. Unfortunately, the collective contribution of different approximations could not always be fitted into the previous scaling scheme. So, it is very important to study them separately considering different theoretical models which describe these effects. The linear relationship between normal coordinates Q and Cartesian displacement X coordinates are:

Q = L+ M 1 2 X ,

(13)

where, by convention, all the components of Q and X vanish at the reference geometry, M is the diagonal matrix of atomic masses, and L is the matrix of (columwise) eigenvectors of the mass weighted Cartesian force constant matrix M-1/2FM-1/2. The second-derivative matrix over normal coordinates, Φ is

Φ = L+ M −1 2 FM −1 2 L

(14)

and is diagonal when evaluated at the equilibrium geometry with eigenvalues λ proportional to the squares of harmonic vibrational frequencies ω. In this way we can evaluate the third and fourth order energy derivatives with respect to normal coordinates by numerical differentiations of analytical Hessian matrices at geometries displaced by small increments δQ from the reference geometry [46, 47]: 1 ⎛ Φ (δQi ) − Φ jk (− δQi ) Φ ki (δQ j ) − Φ ki (− δQ j ) Φ ij (δQk ) − Φ ij (− δQk ) ⎞⎟ (15) Φ ijk = ⎜ jk + + ⎟ 3 ⎜⎝ 2δQi 2δQ j 2δQk ⎠

Anharmonic Effects in Normal Mode Vibrations

Φ ijkk =

163

Φ ij (δQk ) + Φ ij (− δQk ) − 2Φ ij (0 )

(16)

δQk2

1 ⎛ Φ (δQk ) + Φ ii (− δQk ) − 2Φ ii (0 ) Φ kk (δQi ) + Φ kk (− δQi ) − 2Φkk (0) ⎞ ⎟⎟ (17) + Φ iikk = ⎜⎜ ii δQk2 δQi2 2⎝ ⎠ In case of nonlinear molecules these computations require at most the Hessian matrices at 6N-11 different points, N being the number of atoms in molecules. Expanding the vibrational Hamiltonian Hvib using the vibrational wavefunctions in the framework of the second-order perturbation theory, one can obtain the following expression:

vi H vib vi = ξ 0 +

⎛

1⎞

1 ⎞⎛

⎛

∑ ω ⎜⎝ n + 2 ⎟⎠ + ∑ ξ ⎜⎝ n + 2 ⎟⎠⎜⎝ n i

i

ij

i

i≤ j

j

1⎞ + ⎟ 2⎠

(18)

where the ξ constants are simple functions of the cubic and quartic force constants.

2.3. Density-Fitting Local Perturbation Methods Hyper-accurate quantum chemical techniques, suffer from high-degree polynomial scaling. For example, CCSD scales as O(N6) and MP2 scales as O(N5) where N is some measure of system size. In this sense, to develop new techniques which reduce their highdegree polynomial scaling are very important in order to describe larger molecular systems. In the last decade were developed two important methods (density fitting and local perturbational or coupled cluster approximation) which all together can change the highdegree scaling and reduce them close to the linear dependency. In this way, medium sized molecular systems (60 – 80 atoms) can be treated at triple-zeta basis set quality level.

Density Fitting In the HF theory, calculations of electron repulsion integrals (ERI)

G G ψ p (1)ψ q (1)ψ r (2)ψ s (2 ) pr r qs = pq rs = dr1 dr2 r12 −1 12

∫ ∫

*

*

(19)

are the most expensive computational procedures. The computational effort for ERI valuation and transformation can be reduced by 1–2 orders of magnitude using density fitting (DF) methods [48 – 51]. In this approach the one-electron charge densities in the ERIs, which are binary products of orbitals, are approximated by linear expansions in an auxiliary basis set

| pq ) ≈ | ~ pq ) = DApq | A)

(20)

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Attila Bende

This leads to a decomposition of the 4-index ERIs in terms of 2- and 3-index ERIs, and the O(N4) dependence of the computational cost is reduced to O(N3)

( pq | rs ) ≈ ( ~pq | ~r s ) = DApq J AB DBrs ,

(21)

G G A(1)B(2) J AB = dr1 dr1 r12

(22)

where

∫ ∫

Local Approximation In the LMP2 method the occupied space is spanned by localized molecular orbitals (LMOs), which can be obtained from the canonical orbitals by standard localization procedures as proposed by Boys [52] or Pipek and Mezey[53]. The virtual space is spanned by a basis of non-orthogonal projected atomic orbitals (PAOs),

1 = Tabij Ψijab

(23)

which are obtained from the AO basis functions by projecting out the occupied orbital space

1 =

∑ ∑[ T]

ij ab

Ψijab

(24)

i , j∈P ab∈ i , j

In the following, PAOs will be labeled a,b. Since these functions are inherently local, one can introduce two approximations: First, excitations from a pair of occupied LMOs can be restricted to subsets of PAOs that are spatially close to the two LMOs. The number of functions Nij in each of these subsets (pair domains) is independent of the molecular size, and the number of excitations for each electron pair reduces from Nvirt2 to Nij2. Second, the integrals (ai|bj) for distant orbitals i and j can be approximated by multipole expansions [54] or neglected. The remaining number of non-distant orbital pairs (ij), and therefore the total number of excitations, scales linearly with molecular size.

3. Results and Discussions 3.1. Formamide Dimers Formamide (FA) is the simplest molecule that contains a peptide linkage built by the carbonyl and amino groups; therefore, we can consider the formamide dimer (FA–FA) as the simplest model of the pairing of nucleic acids and the formamide–water (FA–WA) complex as a hydration of proteins, respectively. Geometry structures [3, 55–64] and vibrational spectra [3, 56, 60, 65–68] of the FA–FA and FA–WA dimers have been the subject of many studies using different ab initio [HF and MP2] methods, which give valuable information about the structure and dynamics of the H-bonds in molecular systems.

Anharmonic Effects in Normal Mode Vibrations

165

The standard HF, MP2, and CP-corrected HF/MP2 calculations were performed using the Gaussian 03 computer code [69]. The CHA/CE- and CHA/MP2-type calculations were done by generating the input data (integrals and RHF orbitals) with a slightly modified version of HONDO-8 [70]. In these calculations the CHA/SCF code [25–27] and the CHA/MP2 program of Mayer and Valiron [40, 42] were used. For the frequency calculations based on Wilson’s G-F method, the program written by Beu [71] was applied. We considered six different basis sets: 6-31G, 6-31G**, 6-31G**++, D95V, D95V**, and D95V++**. 6-31G to 6-31G**++ are standard Pople basis sets; D95V to D95V++** are Dunning/Huzinaga valence basis sets. The conventional supermolecule geometries were optimized at both the HF and MP2 levels, applying the analytical gradient method included in the Gaussian 03; the CHA- and CP-corrected geometries were calculated by using a numerical gradient method [inverse parabolic interpolation (IPI) [72]] in internal coordinates including only internal coordinates with intermolecular character (one bond, two angles, and three torsion angles). The reason for this choice is that the CPU time of MP2-CHA program is fairly big. To test the applicability of our numerical gradient method we performed several sample calculations using both the IPI method and the analytical gradient built into Gaussian 03. There is practically no difference between them. For conventional uncorrected cases we also performed similar calculations to check the values of the force constants and harmonic vibrational frequencies. The uncorrected HF and MP2 results for the force constants (in internal coordinates) and for the harmonic vibrational frequencies were obtained by using the standard routines of the Gaussian 03 program. As for the CHA- and CP-corrected calculations, at first the numerical second derivatives of the energies were calculated to obtain the CHA and CP force constants and then the NOMAD program [71] was applied to obtain the appropriate CHA and CP harmonic vibrational frequencies. As we are interested in the BSSE content in the molecular interaction energies, only those components of the force constant matrix were recalculated that correspond to intermolecular internal coordinates. The anharmonic frequencies were obtained using the standard full-CP method implemented in Gaussian 03.

The Geometry Structure Table I shows the optimized geometry parameters for the FA–FA complex (Figure 1), using the conventional (Uncorr.), CHA, and CP schemes at both the HF and MP2 levels. The FA–FA dimer has planar geometry configuration: the global minimum for the dimer is a cyclic structure of C2h symmetry involving two equivalents N-H···O=C intermolecular hydrogen bonds. Two other planar minima has been identified [56, 62, 73] that establish a single N-H·O=C HB building up the linear and zig-zag configurations. In the work we consider just the planar cyclic dimer configuration, for which the BSSE-corrected geometrical parameters are presented. Once this assumption is made, the only variables left are the rHO bond length and two angles, αNHO and αHOC, which could be associated with the in-plane vibration normal modes, while all three torsion angles having intermolecular character are kept constant, their normal modes representing out-of-plane vibrations. The results show that only the rHO bond length has an important BSSE correction (0.06 Å for MP2-CHA/6-31++G**, 0.08 Å for MP2-CHA/D95V++**, and 0.06 Å for MP2-CP/6-31++G** and MP2-CP/D95V++**), which increase the bond size. Furthermore the change in the αNHO and αHOC angle values is

166

Attila Bende

insignificant but their corresponding force constants also include BSSE effects. Unfortunately, the experimental value for rHO bond length presented in Ref. [63] does not have the desired precision, so we could not compare with high precision the uncorrected values and the given BSSE corrected bond lengths. Both the corrected and the uncorrected values are close to the experimental value, 1.9 Å (1.87 Å for MP2/D95V++** and 1.9 Å for MP2/6-31++G** in the uncorrected case; 1.95 Å for the CHA- and CP-corrected cases). However, we can compare our calculated result for the rNO intermolecular distance with the experimental values obtained by Itoh and Shimanouchi [66]. Table 1. Intermolecular coordinate for the FA–FA dimer computed at the HF and second-order Møller–Plesset perturbation theory (Uncorr., CHA, CP) level, using D95V, D95V**, D95V++**, 6-31G, 6-31G**, and 6-31++G** basis sets rHO (Å)a Basis

ΑNHO (Deg.)

RHF

MP2

RHF

MP2

RHF

MP2

Uncorr.

1.904

1.872

129.7

125.9

165.6

169.3

(66)

CHA

1.950

1.936

128.9

125.1

166.3

170.1

CP

1.943

1.960

129.4

125.7

165.8

169.5

D95V**

Uncorr.

2.002

1.862

125.7

122.1

168.9

172.6

(120)

CHA

2.029

1.912

125.0

122.1

169.6

172.6

CP

2.019

1.924

124.9

121.8

169.7

172.8

D95V++**

Uncorr.

2.021

1.874

125.2

120.9

169.4

173.7

(150)

CHA

2.039

1.915

125.2

122.5

169.4

172.1

CP

2.034

1.936

125.2

122.4

169.4

172.2

D95V b

Method

αHOC (Deg.)

6-31G

Uncorr.

1.919

1.912

126.8

122.0

168.0

172.4

(66)

CHA

1.934

1.923

126.8

123.0

168.0

171.3

CP

1.939

1.965

127.6

123.7

167.3

170.7

6-31G**

Uncorr.

1.998

1.887

122.7

119.0

171.5

175.0

(120)

CHA

2.007

1.898

123.6

120.9

170.5

173.0

CP

2.025

1.947

123.6

120.4

170.6

173.6

Uncorr.

2.017

1.899

125.4

121.7

169.2

172.8

CHA

2.034

1.954

125.1

122.6

169.4

171.9

CP

2.038

1.954

125.2

122.5

169.3

172.1

6-31++G** (150) a

Experimental value ≈ 1.9 Å, taken from Ref. [63]. The number of basis functions are given in parenthesis.

b

The calculated lengths are 2.895 Å for the uncorrected case, 2.970 Å for CHA-type, and 2.955 Å for CP-type BSSE-corrected cases, while the X-ray data for the formamide crystal [66] gives 2.935 Å for the rNO intermolecular distance. If we suppose that the intermolecular bond length in the crystal phase is a bit shorter than in the gas phase, it can be considered that the corrected values are very close to the experimental.

Anharmonic Effects in Normal Mode Vibrations

167

Figure 1. The formamide dimer.

Table 2. Interaction energiesa (in kcal/mol) for different FA–FA dimer geometry computed at the HF and second-order Møller–Plesset perturbation theory (Uncorr., CHA, CP) level, using D95V, D95V**, D95V++**, 6-31G, 6-31G**, and 6-31++G** basis sets Basis Method Uncorr.

D95V

D95V**

D96V++**

RHF

MP2

RHF

MP2

RHF

MP2

-17.018

-18.370

-12.912

-16.921

-12.043

-15.903

CHA

-15.279

-14.511

-12.086

-13.971

-11.567

-13.140

CP

-15.185

-14.046

-11.886

-13.552

-11.547

-13.201

Basis Method

6-31G RHF

6-31G** MP2

RHF

MP2

6-31++G** RHF

MP2

Uncorr.

-17.387

-18.799

-14.212

-18.330

-12.385

-15.365

CHA

-15.580

-14.496

-12.576

-13.851

-11.965

-13.228

CP

-14.956

-13.705

-12.032

-13.492

-11.837

-13.399

a

Experimental value ≈ -13.967 kcal/mol; taken from Ref. [63].

In Table 2 we present the calculated intermolecular binding energies, considering the optimized geometry in the given basis and using the given methods (uncorrected, CHA, and CP) and levels of theory (RHF and MP2). The experimental value was obtained using Rydberg electron transfer technique between laser-excited atoms; the molecular system [63] is 606 meV, which corresponds to 13.967 kcal/mol. Our BSSE-corrected results (13.288 kcal/mol for MP2-CHA/6-31++G**, 13.399 kcal/mol for MP2-CP/6-31++G**, 13.140

168

Attila Bende

kcal/mol for MP2-CHA/D95V++**, and 13.201 for MP2-CP/D95V++**) are very close to experimental values, whereas the uncorrected results show more than 1.4 kcal/mol difference. Moreover, we can obtain reasonable binding energy value even if we use the 6-31G and D95V bases without diffuse or polarization functions, applying BSSE correction for uncorrected geometry at the same time.

Harmonic and Anharmonic Frequencies The FA–FA dimer has 30 vibrational normal modes from among which 24 (12 for each monomer) vibrations are characteristic to the monomer-type vibrational motion, while six modes have a pure intermolecular character. Table 3. The uncorrected and CP-corrected harmonic (ν) and diagonal anharmonic (x) frequency of FA–FA dimer computed at MP2 level of theory, using 6-31++G** basis set Nr.

νdim

νCP

νmon

xdim

xCP

xmon

Assign.

intramolecular 1

3770.0 3769.9

3770.3 3770.1

3814.1

-32.5 -32.9

-32.4 -32.8

-40.5

N-N a.

2

3465.8 3424.8

3467.5 3426.4

3660.9

-49.9 -58.9

-49.2 -57.8

-36.1

N-N s.

3

3103.9 3101.6

3104.2 3101.9

3083.9

-33.0 -33.1

-33.0 -33.1

-67.7

C-H

4

1795.5 1773.9

1796.4 1774.4

1794.9

-24.0 -22.3

-24.4 ???

-6.6

C=O

5

1677.0 1666.6

1677.2 1667.9

1654.2

-9.5 -3.9

-9.4 -4.0

-8.5

H-N-H

6

1444.0 1444.8

1446.7 1445.8

1444.0

-3.8 -3.9

-3.9 -3.9

-9.1

O=C-H

7

1367.7 1353.1

1367.4 1352.4

1295.9

-2.9 -2.6

-2.8 -2.6

-5.0

C-N

8

1112.1 1106.0

1111.6 1105.6

1075.2

-0.8 -0.7

-0.8 -0.7

-1.3

C-N-H

9

1060.2 1046.2

1067.7 1056.1

1041.0

0.5 1.1

-1.1 -0.7

-1.3

OofP

10

824.3 785.1

827.8 789.7

628.1

-15.5 -6.5

-20.7 -8.3

-15.9

Torsion

11

622.8 605.7

621.2 604.8

565.6

-0.6 -0.2

-0.6 0.3

1.2

O=C-N

12

413.9 394.5

425.5 404.4

276.1

17.8 28.2

12.9 24.8

-462.7

Torsion

Anharmonic Effects in Normal Mode Vibrations

169

Table 3. Continued Nr.

νdim

νCP

νmon

xdim

xCP

xmon

Assign.

intermolecular 1

47.5

118.3

451.2

47.6

OofP

2

29.6

99.7

2101.0

11.0

OofP

3

9.2

49.0

16800.0

1.5

OofP

4

171.8

176.2

-3.1

-2.3

H…O

5

212.6

217.3

-1.8

-1.2

H…H

6

136.4

136.2

-0.2

-0.1

O…O

Usually the vibration frequencies of molecular complex are evaluated at the harmonic approximation applying the Wilson F-G analysis and using the BSSE-uncorrected Hessians. On the other hand, the vibrational frequencies, in particular there with an intermolecular character, contain considerable anharmonic effects and therefore it is difficult to follow these two important corrections in a distinct way. Considering the monomer-type vibrations (Table 3) it can be found that two different dimer frequencies correspond to the similar monomer vibrations, but their values are usually shifted due to the intermolecular interaction. Taking in to account the full CP-corrected values in the dimer calculations, we found another frequency shift, but in this case due to the BSSE effects. Moreover, the frequency values show an important basis size effect at the MP2 level, which implies the shifts in dimer frequency values will change. The results of the anharmonic frequency corrections show a more complex picture. Because of the large numbers of the anharmonic frequencies in Table 3 only the diagonal elements of anharmonic frequency matrix are presented. The most important effect in the anharmonic values is given by the influence of the adjacent molecule, which generates substantial shifts in the anharmonicity of different monomer normal modes within the dimer system. Although the above-mentioned “cluster” effect is quite uniform, the basis size effects become much more complicated. In the case of hydrogen bond stretching (N–H, C–H) and angle-bending vibrations, changes in the anharmonic frequency are not so important, but the torsion angle and C=O stretching modes show very dissimilar results. Considering the full CP-corrected BSSE-free anharmonic frequency calculations, no major corrections can be found for the monomer-type vibrations, which in practice mean that their effects could be generally neglected for the FA–FA dimer. Regarding the intermolecular normal modes, in addition to the “cluster” and basis size effects, the BSSE corrections become very important, especially for “out of plane” normal modes (see MP2/6-31++G**). After these, considering the collective effects of basis size and BSSE corrections on the intermolecular vibration frequencies, it can be concluded that major corrections are obtained in the cases of the harmonic approximation given by the quality (applying polarization and diffuse basis sets) and the BSSE of the applied basis sets, which is followed by the similar correction of the anharmonic approximation. With respect to the intermolecular normal modes, the out-of-plane vibration with low frequencies usually shows a very dissimilar and unrealistic anharmonic correction, especially for the MP2/6-31++G** CP-corrected case. This phenomenon may be related to many facts: i) in the CP method the PES is very flat, the intermolecular force constants are very small [74] and the numerical

170

Attila Bende

calculations could give significant errors; ii) the role of the well-balanced basis set is very important, therefore we consider that the 6-31++G** does not give us adequate results. For example, the 6-31++G(2d, 2p) basis set could be a more suitable choice, but the available computer capacity does not allow us to perform such full-CP anharmonic calculations.

3.2. Urea Dimer The urea dimer presents to some extent a good similarity with the formamide-formamide system. The difference between these two systems is the existence of the second cyclic system in the urea dimer. In the case of the urea dimer one of the HBs is the weak C-N···H-N hydrogen bond. The planarity (or nonplanarity) of the urea dimer system has been the focus of a number of studies. Masunov and Dannenberg [75, 76] considered different levels of theory (HF, MP2 and DFT with and without BSSE corrections), and the most stable conformation was found to be the non-planar structure (using the MP2 method with the D95++** basis set). However, they accentuate that inclusion of vibrational and thermal corrections in the calculations of the molecular structure might give an effectively planar structure. We consider the papers of Rousseau and Keuleers [77, 78] as very important work in elucidating the urea structure, where detailed descriptions of vibrational spectra of urea both in gas and crystal phase are presented. Their concisely conclusion was that the vibrational analysis of solid urea and of the gas phase of urea are not comparable, which is mostly due to the different planar or non-planar conformation of urea in different states. The goal of this study is to give an accurate description of intermolecular normal modes and to present different intermolecular interaction effects which could influence the monomer type vibrations, considering the cyclic urea dimer case. Accordingly, several DFT functionals were tested by comparing them with the corresponding MP2 results. The BSSE was corrected using the counterpoise (CP) method [20, 21] as implemented in the Gaussian03 package suite [69]. The uncorrected and BSSE-corrected energies, geometries, harmonic frequencies and their anharmonic corrections [79, 80] were calculated for MP2 and DFT levels of theory using D95V, D95V** and D95V**++ basis sets [81]. The combined local and density fitting approximations for MP2 and Coupled-Cluster methods (DF-LMP2 and DF-LCCSD(T)) were taken into account considering Molpro program package suite [82].

The Geometry Structure In Table 4 we list the interaction energies (given in kcal/mol) and intermolecular HB distances (in Å) obtained for the cyclic urea dimer structure with MP2 method and six different DFT XC functionals (both in uncorrected and CP-corrected cases) using D95V**++ basis set. In the last five rows of Table 4 the same interaction energy and intermolecular HB distance obtained with DF-LMP2 method using D95V**++ and cc-pVQZ basis sets are also shown, as well as the interaction energy obtained using DF-LCCSD(T) method. The cyclic urea dimer is bonded by two equivalent C=O···H-N HBs presented in Figure 2.

Anharmonic Effects in Normal Mode Vibrations

171

Table 4. The he ε interaction energies (in kcal/mol) and R intermolecular distances (in Ǻ) in case of cyclic urea dimer structure, optimized at different levels of theory and considering the D95V**++ and cc-pVQZ basis sets Method

Cyclic

MP2

NoCP

a

b

CP BLYP

ε

RO…H

-16.53

1.86

-13.18

1.93

NoCP

-14.15

1.84

CP

-13.52

1.86

NoCP

-15.04

1.84

CP

-14.43

1.85

NoCP

-16.81

1.79

CP

-16.12

1.80

NoCP

-12.33

1.96

CP

-11.76

1.98

NoCP

-20.74

1.75

CP

-20.08

1.75

NoCP

-15.59

1.81

CP

-15.01

1.86

-

-13.26

1.93

-

-2.21

-

DF-LMP2/vqz

-

-14.83

1.86

LMP 2 E disp

/vqz

-

-3.56

-

DF-LCCSD(T)e/vqz

-

-14.29

-

B3LYP PBE HTCH407 KMLYP c

BHLYP

DF-LMP2 LMP 2 E disp d

a

Without counterpoise correction With counterpoise correction c BHandHLYP d cc-pVQZ e The LMP2 optimized geometry was used. b

Figure 2. The urea cyclic dimer.

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Table 5. The harmonic (ν) and anharmonic (a) frequencies (in cm-1) of intramolecular normal modes (H – N: ν1 – ν4 and O=C: ν5,ν6 bond stretching) of cyclic urea dimer obtained at MP2, B3LYP and BHLYP levels of theory with and without BSSE correction and using the D95V++** basis set No

ν1

Meth

MP2 B3LYP BHLYP

ν2

MP2 B3LYP BHLYP

ν3

MP2 B3LYP BHLYP

ν4

MP2 B3LYP BHLYP

ν5

MP2 B3LYP BHLYP

ν6

MP2 B3LYP BHLYP

dim ν NCP

dim ν CP

3782.5 3782.4 3712.1 3712.1 3848.3 3848.3 3752.3 3751.7 3657.4 3675.7 3813.7 3813.4 3649.7 3649.7 3589.9 3589.9 3723.5 3723.5 3457.8 3419.6 3340.0 3297.2 3510.8 3478.8 1798.3 1777.9 1768.6 1743.7 1855.9 1825.3 1693.2 1684.7 1656.9 1651.0 1722.7 1721.6

3784.1 3784.0 3712.6 3712.6 3848.8 3848.8 3759.4 3758.6 3675.9 3675.3 3814.7 3814.4 3651.1 3651.0 3590.2 3590.1 3723.8 3723.7 3509.0 3480.0 3348.9 3307.3 3518.2 3483.2 1801.2 1781.0 1769.4 1744.3 1856.6 1825.6 1693.1 1687.4 1659.1 1653.5 1724.3 1723.3

ν mon 3770.1 3706.8 3841.3 37701 3706.6 3841.1 3641.5 3578.8 3720.1 3638.9 3582.5 3715.1 1853.2 1797.0 1887.4 1823.2 1629.0 1696.2

dim a NCP

dim aCP

3608.7 3608.7 3546.4 3546.4 3688.1 3688.0 3584.7 3584.3 3512.3 3511.7 3652.1 3652.1 3493.6 3493.7 3441.9 3441.8 3580.2 3580.0 3229.3 3149.0 3144.0 3008.5 3322.0 3305.0 1764.9 1729.1 1722.6 1769.3 1813.0 1781.8 1648.3 1650.1 1619.9 1602.4 1681.6 1682.4

3610.5 3610.4 3546.7 3546.7 3688.6 3688.6 3592.6 3592.2 3514.3 3513.3 3653.8 3653.8 3495.3 3495.2 3442.1 3442.0 3580.6 3580.3 3388.8 3370.4 3148.1 2984.4 3307.5 3325.0 1765.6 1740.9 1723.9 1700.5 1800.9 1782.6 1647.3 1646.6 1622.3 1610.6 1683.7 1685.4

a mon

νexp

3596.1 3544.5

3450a 3348b

3658.4 3596.3 3543.9

3444 3435

3684.8 3484.3 3438.9

3349 3345

3580.3 3482.3 3435.3

3331 3330

3575.3 1782.1 1751.5

1683 1683

1841.4 1728.1 1589.4

1625 1627

1659.0

a

At T = 20 ºC At T = -196 ºC

b

The interaction energy and HB distance values show that a good fitting of DFT values with MP2 results are not obvious for any of the selected XC functionals. In the case of cyclic structure one can observe that the best agreement for the intermolecular interaction energies and intermolecular RO...H distances are given by the BHLYP and B3LYP XC functionals. The

Anharmonic Effects in Normal Mode Vibrations

173

interaction energy values are: -15.04 kcal/mol for B3LYP, -15.59 kcal/mol for BHLYP and 16.53 kcal/mol for MP2, while the intermolecular distance for RO...H is: 1.84 Å (B3LYP), 1.85 Å (BHLYP) and 1.86 Å (MP2). At the same time, reasonable values could be also obtained applying the BLYP functional, while in the case of KMLYP, PBE and HTCH407 the energy and geometry parameter results are quite different from the MP2 values. If we compare the same ε and R values at DF-LMP2 level of theory, but obtained with different basis sets (D95V**++ versus cc-pVQZ), one could see that the energy results increase with 1.58 and 2.39 kcal/mol, respectively, while for the R distance we get smaller values with 0.06 and 0.07 Å, respectively. Focusing on the dispersion part of the intermolecular interaction energy, significant dispersion energy growth can be found (1.35 kcal/mol for cyclic dimer and 1.71 kcal/mol for asymmetric case), when the larger cc-pVQZ basis set is used against the D95V**+ one. These represent the major contribution to the total interaction energy increase. In order to consider higher correlation effects, other than those included in the MP2, the interaction energies were computed applying the DF-LCCSD(T) method, using the same cc-pVQZ basis set and taking into account the DF-LMP2 optimized geometry.

Harmonic and Anharmonic Frequencies The cyclic urea dimer has 42 vibrational normal modes from which a number of 36 (18 for each monomer) vibrations are characteristic to monomer-type vibrational motion. The last 6 normal modes with the lowest frequency values have purely intermolecular character. These 18 monomer frequencies of dimer structure are split in a form of frequency pairs (doublets). For the whole theoretical IR spectra, see Figure 3. Compared to the individual monomer lines they present different frequency shifts and their magnitudes depend very much on the dimer molecular symmetry.

Figure 3. The theoretical IR spectra of urea cyclic dimer obtained at MP2-CP, DF-LMP2 and B3LYPCP levels of theory, using D95V**++ basis set.

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In Table 5 were collected the harmonic frequencies (doublets) and their anharmonic corrections for those monomer type normal modes of which vibrational motions are substantially perturbed by the adjoining molecule. They were determined in advance by identifying the frequency value which corresponds to each normal mode vibration and by visualization of their vibrational characters [83]. It was found that, the ν1, ν2 and ν3 intramolecular normal modes are N-H bond stretching vibrations where the H atoms do not take part in intermolecular HB formations. The ν4, ν5 and ν6 are one N-H and two C=O bond stretching vibrations located at the N-H···O=C intermolecular HBs. It can be observed that the doublet frequency splits for ν1, ν2 and ν3 normal modes are almost irrelevant. More precisely, the frequency shifts, which are induced by each monomer on the adjoining molecules, have the same magnitude. Only those normal modes presents different frequency shifts and implicitly larger doublet splits where the perturbation of HB vibration is present (ν4, ν5 and ν6). As it can be seen in Table 5 the dimer frequency shift could be attributed to several effects like: anharmonic corrections, BSSE effects, or intermolecular effects. It should be mentioned that BSSE is not a real physical effect and normally it must be considered together with the intermolecular effects. But, in order to see how important frequency shifts could the BSSE error induce we consider as a separate effects.

Table 6. The harmonic (ν) and anharmonic (a) frequencies (cm-1) of intermolecular normal modes of cyclic urea dimer obtained at MP2, B3LYP and BHLYP levels of theory with and without BSSE correction and using the D95V++** basis set No. νI

νII

νIII

νIV

νV

νVI

Meth. MP2 B3LYP BHLYP MP2 B3LYP BHLYP MP2 B3LYP BHLYP MP2 B3LYP BHLYP MP2 B3LYP BHLYP MP2 B3LYP BHLYP

Cyclic dim ν NCP

dim ν CP

dim a NCP

dim aCP

159.3 157.8 158.4 154.3 152.0 154.7 126.8 134.6 133.4 94.4 93.7 93.7 65.1 67.9 68.6 35.8 45.4 45.8

141.5 150.6 156.0 135.2 137.9 150.7 121.9 127.2 133.4 89.9 81.5 94.1 62.0 50.5 67.6 38.0 46.8 45.0

151.9 155.2 151.5 144.7 147.4 143.9 121.6 135.0 127.0 80.5 93.8 78.7 55.6 66.9 53.8 29.5 44.1 43.9

132.4 150.1 153.1 123.7 137.4 141.3 117.6 132.6 132.6 75.3 80.7 80.7 54.4 53.0 55.9 33.8 45.7 54.4

Anharmonic Effects in Normal Mode Vibrations

175

In the case of N-H bond stretching vibrations, the magnitudes of anharmonic corrections (frequency red-shift) are quite large (≈ 150 - 170 cm-1 for ν1, ν2 and ν3, and >200 cm-1 for ν4, respectively) while in the case of a1, a2 and a3 anharmonic corrections, for both of doublet frequency values, the magnitude of frequency shifts are the same. For a4 correction the size of doublet frequency shifts is different (ex. at MP2 level ν′4-a'4=3457.8 - 3229.3 cm-1 = 228.5 cm-1 and ν"4 - a"4 = 3419.6 - 3149.0 cm-1 = 270.6 cm -1). Regarding to ν5 and ν6 C=O stretching modes one can see that the anharmonic corrections are much smaller than in the HN case, their shifts are about 30-40 cm-1 and the behavior of their frequency split is similar to ν4 mode. The correction scheme for ν 4 and ν 5 is as follows:

ν 4' : ν mon = 3638 .9 cm −1 ⎯dimer ⎯⎯→ν dim ⎯anh. ⎯→ ⎯ ν anh ⎯BSSE ⎯⎯→ν BSSE = 3388 .8 cm −1 -

ν : ν

mon

ν : ν

mon

ν : ν

mon

" 4

' 5

" 5

= 3638 .9 cm

−1

= 1853 .2 cm

−1

= 1853 .2 cm

−1

-

⎯ ⎯⎯→ν

dim

⎯ ⎯⎯→ν

dim

⎯ ⎯⎯→ν

dim

dimer

-

dimer

-

⎯ ⎯→ ⎯ ν

⎯ ⎯⎯→ν BSSE = 3370 .4 cm −1

⎯ ⎯→ ⎯ ν

anh

⎯ ⎯⎯→ν BSSE = 1765 .6 cm −1

⎯ ⎯→ ⎯ ν

anh

⎯ ⎯⎯→ν BSSE = 1740 .9 cm −1

-

-

dimer

+1

anh

anh.

anh.

+2

-

anh.

-

BSSE

BSSE

+0.

BSSE

+1

Considering the scheme of frequency corrections by dimer, anharmonic and BSSE effects presented above, one could be observe that they bring different contributions in the case of the ν 4 monomer frequency value (3638.9 cm-1), where finally we got a double-split frequency pair (ν 4' and ν 4" ) of 3370.4 cm-1 and 3388.8 cm-1 values. Similar situation can be found for the ν 5 monomer frequency value where we have two frequency values with 24.7 cm-1 distance between them. Beside of the monomer-type frequencies, the molecular association induces a group of another six normal mode vibrations which can be called intermolecular normal modes. They can be found in the very-far region (10–250 cm-1) of the molecular IR spectra and show the relative vibrations of two “rigid” urea monomers according to the six degree of freedom which derive from the intermolecular coordinates. The frequency values of these intermolecular normal mode vibrations are shown in Table 6 obtained both at MP2 and DFT (B3LYP and BHLYP) levels of theory. They have strictly intermolecular character and show only anharmonic and BSSE corrections. Scrutinizing the results of normal mode vibrations for the cyclic dimer one can observe that there are three frequencies of which motion take place in the supermolecular plane (νI – νIII) – let calls them “in-plane” vibrations, while the another three intermolecular normal modes (νIV – νVI) show “out-of-plane” vibrations. If we consider together the amount of anharmonic and BSSE corrections one can see that the “inplane” vibrations are more affected by these errors than “out-of-plane” normal modes. Yet analyzing separately for only one mode, it can be seen that they have almost similar magnitude, becoming equally relevant corrections for the intermolecular normal mode vibrations. At the same time, if we consider the off-diagonal vibration couplings of these normal modes, one can find that frequencies which belong to the group of “in-plane” or “outof-plane” their vibrations are strongly coupled inside the group, but much less coupled between the groups. In the case of asymmetric urea dimer the separation of the above mentioned “in-plane” and “out-of-plane” group of vibrations is not so obvious, but similarly

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to the cyclic dimer the BSSE and the anharmonic effects are equally relevant. In both cases of dimer structures the collective effect of BSSE and anharmonic corrections presents about 10 – 15% from the frequency values. The intra – intermolecular normal mode couplings could give us more detailed information about how strongly the intra- and intermolecular normal modes are coupled between them. Since the intra- and intermolecular normal modes have very different vibrational frequency values one should obtain a strong coupling only in some special cases. Analyzing the anharmonic coupling matrix we found that in case of cyclic dimer the νI – νIII intermolecular normal modes are coupled only with ν 4' and ν 4" (ex. x(ν 4" -ν I ) = +6.73 cm-1) intramolecular vibrations, while couplings between ν 5' and ν 5" intra-normal modes and νI – νIII inter-normal modes are almost missing. This could be explained with the fact that molecular vibrations are performed in the same molecular plane and along the same vibrational direction. In the case of asymmetric urea dimer neither for ν 4' and ν 4" , nor for ν 5' and ν 5" so significant anharmonic coupling with the intermolecular normal modes could be found than for the cyclic structure. In this case, the only relevant coupling is x(ν 5" -ν II ) = -3.86

cm-1. If one compares the selected theoretical normal mode frequencies with the experimental data (Table 5), the more appropriate values are obtained for the CP-corrected anharmonic frequencies, both for cyclic and asymmetric structures. At the same time, it can be seen that the dimer approximation is not satisfactory in order to reproduce these experimental data with the desired high accuracy. Thus, other extended theoretical calculation where also solvent effects and larger cluster sizes are taken into account would be absolutely necessary. In addition, other spectroscopic data, like vibrational absorption intensities calculated via atomic polar tensor [84] and comparing them with the measured vibrational intensities would be also useful in order to explain the unusual behavior of urea in gas or solid phase.

3.3. Guanine-Cytosine DNA Base Pair The guanine-cytosine (GC) base pair's intermolecular interaction can be considered as a contribution of two very important HBs: a number of two C=O···H-N and one PN···H-N bonds. Due to the biological manifoldness of DNA base pair conformations (dry-DNA, wetDNA, double helix, super-double helix, etc.) their theoretical characterization is much diversified. A given theoretical model should take into account the backbone and environment effects as well as the influence of the DNA-protein interactions. In virtue of this fact, choosing the correct theoretical method is essential in the investigation of DNA base pairs. In the last five years the substantial computer advance gave for molecular modeling scientists the opportunity to include accurate electron correlation effects in their calculations. Šponer et al. [85] and Podolyan et al. [86] used the well-known MP2 method with different basis sets (mostly including the polarization effects) and they give a very detailed description of the interaction energies and geometry structures of the adenine-thymine and guaninecytosine base pairs. At the same time the normal mode analysis of molecular vibration could give us supplementary information about the efficiency of different methods. In connection with this, several vibrational frequency calculations and experimental measurements were

Anharmonic Effects in Normal Mode Vibrations

177

performed [9, 86, 87, 88, 89] in order to study different interaction effects in normal mode vibrations of base pairs (mostly for guanine-cytosine dimer). For instance, the intermolecular interactions could significantly influence the intramolecular normal mode vibrations such as red-shift or improper blue-shifting [90, 91] of the vibrational frequencies. Furthermore, the intermolecular normal mode vibrations depend very much on the applied method or basis sets, and last but not least on the BSSE effects [3, 74, 92, 93]. The different Watson-Crick and Hoogsteen G-C geometries, harmonic frequencies and their anharmonic corrections [79, 80] were calculated at DFT level of theory considering the PBE exchange-correlation functional implemented in Gaussian03 [69] program package and using the 6-31G basis set. The harmonic frequency values were scaled using the 0.986 factor corresponding to PBE/6-31G type calculations.

The Geometry Structure In Figures 4 and 5 two different conformation (4: Watson-Crick [94] and 5: Hoogsteen [95]) of G-C base pair are presented. Beside of the well-known Watson-Crick base pair configuration the Hoogsteen base pairs have been known for more than 40 years. In Hoogsteen base pairs the N face of guanine is hydrogen bonded to cytosine N side by an H+ proton. Such interactions were postulated in U(A·U) triple helices [96] and it was also found in chemically modified nucleic acids [97, 98]. Isolated Hoogsteen base pairs have been reported in some protein/DNA complexes [99] and occasionally in RNA [100]. Its crystal structure is presented in [101].

Figure 4. The Watson-Crick conformation of Guanine-Cytosine base pair.

Figure 5. The Hoogsteen conformation of Guanine-Cytosine base pair.

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The geometry of the DNA base pair was the subject of many scientific works, among which the theoretical investigation performed by Guerra at al. [102] and van der Wijst [103] are the most detailed ones. They present a systematic comparison of different DFT exchangecorrelation functionals and basis sets as well as they also compare the intermolecular interaction energy values and H-bond distances with the experimental results. The best agreement was obtained for the BP86 and PW91 exchange-correlation functionals, while the widely used B3LYP functional consistently underestimates hydrogen-bond strengths and overestimates hydrogen-bond distances. Nearly same good results can be obtained also with the PBE functional as it found in case of BP86 and PW91 ones. Considering the above mentioned method and basis set, for the intermolecular distances were obtained the following values: a) Watson-Crick: d1(O···H-N) = 2.80 Å, d2(N···H-N) = 2.93 Å and d3(O···H-N) = 2.92 Å, b) Hoogsteen: d1(N···H+-N) = 2.66 Å, d2(O···H-N) = 3.08 Å.

Harmonic and Anharmonic Frequencies The guanine-cytosine binary system in its Watson-Crick configuration has a number of 29 atoms and presents a number of 81 normal mode vibrations from which 6 have intermolecular character. In Figure 6 the IR absorption spectra for two characteristic spectral regions of 500 – 2000 cm-1 and 2750 – 3750 cm-1, respectively, are presented.

Figure 6. The IR absorption spectra for Watson-Crick configuration of guanine-cytosine DNA base pair at harmonic and anharmonic approximation.

The black vertical line shows the normal mode vibrational frequencies in harmonic approximation, while the red lines are frequencies where the anharmonic approximation was also taken into account. The values show a considerable frequency shift due to the anharmonic approximation. These shifts are very pronounced in case of 2750 – 3750 cm-1 spectral region, where the C-H and N-H covalent stretching vibration are located.

Anharmonic Effects in Normal Mode Vibrations

179

Figure 7. The IR absorption spectra for Hoogsteen configuration of guanine-cytosine DNA base pair at harmonic and anharmonic approximation.

Figure 8. The IR absorption spectra for Watson-Crick and Hoogsteen configuration of guanine-cytosine DNA base pair at harmonic approximation.

The guanine-cytosine binary system in its Hoogsteen configuration has a number of 30 atoms and presents a number of 84 normal mode vibrations from which 6 have intermolecular character. In Figure 5 can be observed that the N···H+-N intermolecular bond is stabilized with the help of the H+ proton. The H+-N is not a real covalent bond and therefore its stretching vibration is also different from the usual H-N stretching vibrations. In case of Watson-Crick configuration there are three characteristic stretching vibrations: the first one is a pure H4-N4 vibration (ν5 = 3292.9 cm-1) of O6···H4-N4 intermolecular donor-acceptor complex, while the second and third ones are a combined N1-H1 and N2-H2 symmetric (ν9 = 3085.8 cm-1) and asymmetric (ν10 = 3027.8 cm-1) stretching vibrations of N1-H1···N3 and N2H2···O2 intermolecular donor-acceptor complexes. In case of Hoogsteen configuration there are only two characteristic vibrations which belong to the intermolecular interaction region:

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the first is an N-H stretching vibration (ν10 = 3014.4 cm-1) of N-H···O donor-acceptor complex, while the second one is also an N-H stretching vibration (ν11 = 2053.2 cm-1) but as a component of the N···H+-N unusual intermolecular donor-acceptor complex. If one considers the anharmonic corrections, the similar behavior can be observed as it was found in case of formamide and urea systems. There are large anharmonic shifts for all three ν5, ν9 and ν10 stretching vibrations of Watson-Crick configuration: a5 = 263.1 cm-1, a9 = 294.8 cm-1 and a10 = 362.4 cm-1. In case of Hoogsteen conformation the anharmonic shifts are: a10 = 187.6 cm-1 and a11 = 958.0 cm-1. The a11 anharmonic shift is very large compared with the harmonic frequency values. This can be explained with the unusual nature of N···H+-N intermolecular donor-acceptor complex. Two different type intermolecular normal mode vibrations can be found both in case of Watson-Crick and Hoogsteen configurations. In one of them the vibration motion occurs in the plane defined by the monomer molecules (here we have a number of three normal modes), while in the second case the intermolecular vibrational motions are out-of-plane vibrations. Since the characterization of the out-of-plane vibrations is quite difficult, this will not constitute the subject of our further investigation. Accordingly, the three intermolecular normal mode (in-plane) frequencies are: νI = 101.2 cm-1, νII = 125.5 cm-1 and νIII = 131.0 cm-1 for Watson-Crick conformation and νI = 88.9 cm-1, νII = 109.4 cm-1 and νIII = 170.6 cm-1, for Hoogsteen conformation, respectively. Similar to the previous urea case presented in section 3.2, there are also significant interand intramolecular normal mode couplings. In case of Watson-Crick system the intramolecular ν9 and ν10 stretching vibrations are relatively strong coupled (about 5-6 cm-1) with all three νI, νII and νIII intermolecular normal modes. For Hoogsteen configuration these couplings are stronger: 17.0-21.0 cm-1.

3.4. Adenine-Thymine DNA Base Pair The adenine-thymine binary system (Watson-Crick configuration) has a number of 30 atoms and presents a number of 84 normal mode vibrations from which 6 have intermolecular character. For its geometry structure see Figure 9. Since we are interested in the stretching vibrations (ν4 and ν11) of those intramolecular covalent bonds where H atoms are involved, as well as in those intermolecular vibrations of which motion is situated mostly in the molecular plane (ν78, ν80 and ν82), we present only these specific normal mode vibrations. At the same time, one can identify another group of seven normal modes (ν41, ν42, ν54, ν60, ν61, ν64, and ν69) specific for purine and pyrimidine ring vibrational deformation which can significantly disturb the H-bond vibrations. All these normal mode frequency values in harmonic, BSSE corrected harmonic, and anharmonic approximations are presented in Table 7. Considering the BSSE correction, the most affected normal modes are the ν4 and ν11 N-H stretching vibrations. In these two cases we found 76.6 cm-1 and 429.5 cm-1 frequency increasing, respectively. Similar findings can be obtained in case of νI and νII intermolecular normal modes, where in spite of the fact that BSSE effects not show large frequency shifts, they could represent significant corrections compared with the uncorrected frequency values. Regard to anharmonic corrections, one can observe that large frequency shifts are obtained in the case of N-H covalent-bond stretching vibrations (281.1 cm-1 for ν4 and -160.4 cm-1 for ν11). Considering the ring deformation normal mode

Anharmonic Effects in Normal Mode Vibrations

181

vibrations (ν41, ν42, ν54, ν60, ν61, ν64, and ν69), the anharmonic correction is more important than BSSE effects, but even so, their frequency shifts are less than 20 cm-1. As it was concluded in the previous case of urea dimers, the anharmonic and BSSE corrections could be considered in a very good approximation as additive effects. According to this fact, we present in the fifth column of Table 7 the integral correction (νint) of the anharmonic and BSSE effects. The results show that in some frequency cases we have an opposite contribution of frequency shifts, while in some other cases the anharmonic and BSSE collective corrections increase the magnitude of the frequency shift. Analyzing the anharmonic coupling matrix one observes strong vibrational coupling between ν11 intra- and ν78 intermolecular normal modes (x11,I = 8.2 cm-1) as well as between ν11 intra- and νII intermolecular normal modes (x11,II = 10.4 cm-1). The case of this strong coupling could be explained by the same fact that in the case of cyclic urea dimer namely, by the presence of the same molecular plane for both normal modes and the same vibrational direction for ν11 and νI modes. In other cases there is no significant coupling between the intra- and intermolecular normal modes.

Table 7. The harmonic (ν) and anharmonic (a) frequencies (in cm-1) of some selected intramolecular and intermolecular normal modes in adenine-thymine DNA base pair, obtained at B3LYP level of theory and using D95V basis set (The ν int is the predicted frequency value, considering, together, the BSSE and anharmonic corrections) Nr.

dim ν NCP

dim ν CP

dim a NCP

νint

ν4

3329.3

3405.9

3048.2

3124.8

ν11

2546.7

2976.2

2386.3

2815.8

ν41

1035.2

1035.7

1018.5

1019.0

ν42

1031.4

1024.2

1014.2

1007.0

ν54

742.4

745.9

726.9

730.4

ν60

637.7

630.3

632.8

625.4

ν61

607.4

611.1

601.2

604.9

ν64

550.2

550.4

542.0

542.2

ν69

397.2

404.1

388.8

395.7

νI

119.9

108.2

115.4

103.7

νII

114.6

101.6

110.3

97.3

νIII

66.8

60.3

58.3

51.8

Krishnan et al [104] show that to compare the experimentally observed IR spectra of adenine-thymine base pair with the calculated frequencies is not a simple task. First of all because the Watson-Crick configuration of A-T base pair is not the most stable isomer conformation [105]. Making a detailed theoretical anharmonic frequency analysis they were able to assign the IR-UV double resonance spectra [106] also to a particular isomer which is not the Watson-Crick structure. A direct experimental assignment of N-H stretching vibrations in A-T oligomers in condense phase is very difficult because the N-H stretching

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Attila Bende

vibration spectral region overlap with the water’s O-H stretching spectral region. Reducing the water content of the A-T oligomers does not solve the problem because they do not adopt a well define structure at extremely low water concentration. In order to identify and to characterize the N-H stretching vibrations of A-T base pair oligomers in DNA Heyne et al [107] use the two-color IR pump-probe technique to overcome the above mentioned experimental problem. They found that the adenine ν(NH2) absorbs at 3215 cm-1 and has pronounced anharmonic couplings to the ν(C=O) mode of the thymine and δ(NH2) mode of the adenine.

Figure 9. The adenine-thymine DNA base pair.

4. Conclusion In this Chapter a detailed investigation, including geometry structure, harmonic and anharmonic frequency calculations, on cyclic conformation of formamide and urea dimers, as well as on the guanine-cytosine and adenine-thymine DNA base pairs were presented. Considering the results some general conclusions can be drawn. First of all, in order to describe accurately the geometry structures of the molecules one needs to consider proper ab initio methods, which are able to describe correctly the different intermolecular interaction effects (electrostatic, polarization, induction, dispersion and three- or four-body terms). Including all this effects in our calculation, high level electron correlation methods like perturbation or coupled-cluster methods as well as basis sets which contain successively larger shells of polarization (correlating) functions (d, f, g, etc.) are required. For these calculations one need a huge amount of computer capacity and therefore we can only limit to a small size molecular systems (up to 15-20 atoms). Recently introduced local and densityfitting approximations are very promising tools, they can reduce very much the calculation efforts and in this way the molecular size limit could be extended. These methods are also free from the mathematical artifact called basis set superposition error or BSSE. In order to obtain accurate description for the infrared absorption spectra, the harmonic approximation is not good enough and therefore the anharmonic approximation is also should be included. Including these effects, some normal mode vibrations, like X-H (X=C or N), C=O and N-H+ stretching vibrations, need more detailed investigation. For the N-H and C=O covalent-bond stretching vibrations the anharmonic frequency correction is significant. In case of Hoogsteen conformation of guanine-cytosine base pair the anharmonic approximation is comparable with the harmonic frequency value. For the same N-H and C=O covalent-bond

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stretching vibrations the BSSE corrections are also significant, but only in the case when the normal mode vibrations are in the intermolecular region. Normal mode vibrations which are located in the intermolecular region, the influence of the adjoining molecule (dimer effect) on the vibrational frequency is comparable with the magnitude of the BSSE and anharmonic corrections. Analyzing the magnitude of the anharmonic and BSSE corrections, it was found that their contributions in the harmonic frequency shift could be considered in a very good approximation as an additive effect. The anharmonic coupling between intra- and intermolecular normal modes is significant only when the motion of normal mode vibrations occur in the same molecular plane (the plane defined by those atoms which move during the vibration) and along of the same vibrational direction. Using the classical DFT functionals and applying double-zeta quality basis sets, like 6-31G or D95V, one could obtain a good qualitative description of the anharmonic corrections, but further investigations considering the recently developed DFT or perturbation methods and triple-zeta quality basis sets are needed in order to obtain also a good quantitative description for them. The strong coupling between different intra- and intermolecular normal modes can be considered as an important and biologically relevant effect. In this way the vibrational relaxation processes can take place very easily and some strongly excited localized vibrations can wither away as thermal vibrations.

References [1] [2]

[3]

[4]

[5] [6]

[7] [8]

[9]

Rose G.; Fleming P.; Banavar J.; Martin A. A backbone-based theory of protein folding PNAS, 2006, 103, 16623-16633. Deechongkit S.; Nguyen H.; Dawson P. E.; Gruebele M.; Kelly J. W. Contextdependent contributions of backbone hydrogen bonding to β-sheet folding energetics, Nature, 2004, 430, 101-105. Bende A.; Suhai S. BSSE-Corrected Geometry and Harmonic and Anharmonic Vibrational Frequencies of Formamide–Water and Formamide–Formamide Dimers, Int. J. Quantum Chem., 2005, 103(6), 841-853. Gerber R. B.; Chaban G. M.; Brauer B.; Miller Y., Theory and Applications of Computational Chemistry: The First 40 years, Elsevier, Amsterdam, The Netherlands, 2005, Chap. 9, pp. 165–193. Pratt D. W. High Resolution Spectroscopy in the Gas Phase: Even Large Molecules Have Well-Defined Shapes, Annu. Rev. Phys. Chem., 1998, 49, 481-530. Belhayara K.; Chamma D.; Velcescu A. C.; Rousseau O. H. On the similarity of the IR lineshapes of weak H-bonds in the gas and liquid phases: Quantum combined effects of strong anharmonic coupling, multiple Fermi resonances, weak dampings and rotational structure, J. Mol. Struct., 2007, 833, 65-73. Maréchal Y. IR spectra of carboxylic acids in the gas phase: A quantitative reinvestigation, J. Chem. Phys., 1987, 87, 6344-6353. Chaban G. M.; Jung J. O.; Gerber R. B. Anharmonic Vibrational Spectroscopy of Glycine: Testing of ab Initio and Empirical Potentials, J. Phys. Chem. A, 2000, 104, 10035-10044. Brauer B.; Gerber R. B.; Kabelac M.; Hobza P.; Bakker J. M.; Riziq A. G. A.; de Vries M. S. Vibrational Spectroscopy of the G···C Base Pair: Experiment, Harmonic and

184

[10]

[11]

[12]

[13] [14]

[15]

[16]

[17]

[18] [19]

[20] [21]

[22] [23] [24] [25] [26]

Attila Bende Anharmonic Calculations, and the Nature of the Anharmonic Couplings, J. Phys. Chem. A, 2005, 109, 6974-6984. Carbonniere P.; Barone V. Performances of different density functionals in the computation of vibrational spectra beyond the harmonic approximation, Chem. Phys. Lett., 2004, 399, 226-229. Pele L.; Gerber R. B. On the number of significant mode-mode anharmonic couplings in vibrational calculations: Correlation-corrected vibrational self-consistent field treatment of di-, tri-, and tetrapeptides, J. Chem. Phys., 2008, 128, 165105. Roitberg A.; Gerber R. B.; Elber R.; Ratner M. A. Anharmonic wave functions of proteins: quantum self-consistent field calculations of BPTI, Science, 1995, 268, 13191322. Bowman J. M. The self-consistent-field approach to polyatomic vibrations, Acc. Chem. Res., 1986, 19, 202-208. Rekik N.; Oujia B.; Wójcik M. J. Theoretical infrared spectral density of H-bonds in liquid and gas phases: Anharmonicities and dampings effects, Chem. Phys., 2008, 352, 65-76. Sinha P.; Boesch S. E.; Gu C. M.; Wheeler R. A.; Wilson A. K. Harmonic Vibrational Frequencies: Scaling Factors for HF, B3LYP, and MP2 Methods in Combination with Correlation Consistent Basis Sets, J. Phys. Chem. A, 2004, 108, 9213-9217. Halls M. D.; Velkovski J.; Schlegel H. B. Harmonic frequency scaling factors for Hartree-Fock, S-VWN, B-LYP, B3-LYP, B3-PW91 and MP2 with the Sadlej pVTZ electric property basis set, Theor. Chem. Acc., 2001, 105, 413-421. Wang N. X.; Venkatesh K.; Wilson A. K. Behavior of Density Functionals with Respect to Basis Set. 3. Basis Set Superposition Error, J. Phys. Chem. A, 2006, 110, 779-784. Tuma C.; Boese A. D.; Handy H. C. Predicting the binding energies of H-bonded complexes: A comparative DFT study, Phys. Chem. Chem. Phys., 1999, 1, 3939-3948. Haynes P. D.; Skylaris C.-K.; Mostofi A. A.; Payne M. C. Elimination of basis set superposition error in linear-scaling density-functional calculations with local orbitals optimised in situ, Chem. Phys. Lett., 2006, 422, 345-349. Jansen H. B.; Ros P. Non-empirical molecular orbital calculations on the protonation of carbon monoxide, Chem. Phys. Lett., 1969, 3, 140-143. Boys S. B.; Bernardi F. The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors, Mol. Phys., 1970, 19, 553-566. Mayer I. Towards a “Chemical” Hamiltonian, Int. J. Quantum Chem., 1983, 23, 341363. Mayer I. The chemical Hamiltonian approach for treating the BSSE problem of intermolecular interactions, Int. J. Quantum Chem., 1998, 70, 41-63. Mayer I.; Surján P. R. Monomer geometry relaxation and the basis set superposition error, Chem. Phys. Lett., 1992, 191, 497-499. Mayer I.; Vibók Á. SCF theory of intermolecular interactions without basis set superposition error, Chem. Phys. Lett., 1987, 136, 115-121. Mayer I.; Vibók Á.I ntermolecular SCF method without bsse: the closed-shell case, Chem. Phys. Lett., 1987, 140, 558-564.

Anharmonic Effects in Normal Mode Vibrations

185

[27] Vibók Á.; Mayer I. Intermolecular SCF theory without BSSE: The equations and some applications for small systems, J. Mol. Struct. (Theochem), 1988, 170, 9-17. [28] Mayer I.; Vibók Á.; Halász G. J.; Valiron P. A BSSE-free SCF algorithm for intermolecular interactions. III. Generalization for three-body systems and for using bond functions, Int. J. Quantum Chem., 1996, 57, 1049-1055. [29] Halász G. J.; Vibók Á.; Valiron P.; Mayer I. BSSE-Free SCF Algorithm for Treating Several Weakly Interacting Systems, J. Phys. Chem., 1996, 100, 6332-6335. [30] Halász, G. J.; Vibók Á., Suhai S. A BSSE-free SCF algorithm for intermolecular interactions. IV. Generalization for open-shell systems, Int. J. Quantum Chem., 1998, 68, 151-158. [31] Mayer I.; Turi L. An analytical investigation into the bsse problem, J. Mol. Struct. (Theochem), 1991, 227, 43-65. [32] Mayer I.; Surján P. R. Improved intermolecular SCF theory and the BSSE problem, Int. J. Quantum Chem., 1989, 36, 225-240. [33] Mayer I.; Surján P. R.; Vibók Á. BSSE-free SCF methods for intermolecular interactions, Int. J. Quantum Chem., Quant. Chem. Sym., 1989, 23, 281-290. [34] Mayer I.; Vibók Á. A BSSE-free SCF algorithm for intermolecular interactions, Int. J. Quantum Chem., 1991, 40, 139-148. [35] Vibók Á.; Mayer I. A BSSE-free SCF algorithm for intermolecular interactions. II. Sample calculations on hydrogen-bonded complexes, Int. J. Quantum Chem., 1992, 43, 801-811. [36] Hamza A.; Vibók, Á.; Halász G. J.; Mayer I. BSSE-free SCF theories: a comment, J. Mol. Struct. (Theochem), 2000, 501-502, 427-434. [37] Halász G. J.; Vibók, Á.; Suhai S.; Mayer I. Toward a BSSE-free description of strongly interacting systems, Int. J. Quantum Chem., 2002, 89, 190-197. [38] Mayer I.; Vibók, Á. BSSE-free second-order intermolecular perturbation theory, Mol. Phys., 1997, 92, 503-510. [39] Vibók Á.; Halász G. J.; Mayer I. BSSE-free second order intermolecular perturbation theory II. Sample calculations on hydrogen-bonded complexes, Mol. Phys., 1998, 93, 873-877. [40] Mayer I.; Valiron P. Second order Møller–Plesset perturbation theory without basis set superposition error, J. Chem. Phys., 1998, 109, 3360-3373. [41] Vibók Á.; Halász G. J.; Mayer I., in Electron Correlation and Material Properties; Gonis, A.; Kioussis, N.; Ciftan, M.; Eds.; Kluwer Academic/Plenum: New York, 2004, pp. 263–283. [42] Mayer I.; Valiron P., Program CHA-MP2, Budapest, 1998. [43] Valiron P.; Vibók Á.; Mayer I. Comparison of a posteriori and a priori BSSE correction schemes for SCF intermolecular energies, J. Comp. Chem., 1993, 14, 401-409. [44] Halász G. J.; Vibók Á.; Mayer I. Comparison of basis set superposition error corrected perturbation theories for calculating intermolecular interaction energies, J. Comp. Chem., 1999, 20, 274-283. [45] Mayer I. On the Hylleraas functional for a non-Hermitian unperturbed Hamiltonian, Mol. Phys., 1996, 89, 515-519. [46] Schneider W.; Thiel W. Anharmonic force fields from analytic second derivatives: Method and application to methyl bromide, Chem. Phys. Lett., 1989, 157, 367-373.

186

Attila Bende

[47] Barone V. Anharmonic vibrational properties by a fully automated second-order perturbative approach, J. Chem. Phys., 2005, 122, 014108. [48] Weigend F.; Häser M.; Patzelt H.; Ahlrichs R. RI-MP2: optimized auxiliary basis sets and demonstration of efficiency, Chem. Phys. Lett., 1998, 294, 143-152. [49] Dunlap B. I. Robust and variational fitting, Phys. Chem. Chem. Phys., 2000, 2, 21132116. [50] Weigend F. A fully direct RI-HF algorithm: Implementation, optimised auxiliary basis sets, demonstration of accuracy and efficiency, Phys. Chem. Chem. Phys., 2002, 4, 4285-4291. [51] Weigend F.; Köhn A.; Hättig C. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations, J. Chem. Phys., 2002, 116, 3175-3183. [52] Boys S. F., in Quantum Theory of Atoms, Molecules, and the Solid State, edited by P. O. Löwdin, Academic, New York, 1966, pp. 253. [53] Pipek J.; Mezey P. G. A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions, J. Chem. Phys.,1989, 90, 4916-4926. [54] Hetzer G.; Pulay P.; Werner H.-J. Multipole approximation of distant pair energies in local MP2 calculations, Chem. Phys. Lett., 1998, 290, 143-149. [55] Cabaleiro-Lago E. M.; Ríos, M. A. Development of an intermolecular potential function for interactions in formamide clusters based on ab initio calculations, J. Chem. Phys., 1999, 110, 6782-6791. [56] Cabaleiro-Lago E. M.; Otero, J. R. Ab Initio and density functional theory study of the interaction in formamide and thioformamide dimers and trimers, J. Chem. Phys., 2002, 117, 1621-1632. [57] Martell J. M.; Yu H.; Goddard J. D. Molecular decompositions of acetaldehyde and formamide: theoretical studies using Hartree-Fock, Moller-Plesset and density functional theories, Mol. Phys., 1997, 92, 497-502. [58] Lii J.-H.; Allinger N. L. Directional hydrogen bonding in the MM3 force field: II, J. Comp. Chem., 1998, 19, 1001-1016. [59] Hobza P.; Šponer J. MP2 and CCSD(T) calculations on H-bonded and stacked formamide···formamide and formamidine···formamidine dimers, J. Mol. Struct. (Theochem), 1996, 388, 115-120. [60] Florián L.; Leszczynski J.; Johnson B. G. On the intermolecular vibrational modes of the guanine···cytosine, adenine···thymine and formamide···formamide H-bonded dimers, J. Mol. Struct., 1995, 349, 421-426. [61] Sobolewski A. L. Ab initio study of the potential energy functions relevant for hydrogen transfer in formamide, its dimer and its complex with water, J. Photochem. Photobio. A, 1995, 89, 89-97. [62] Vargas R., Garza J., Frisner R. A., Stern H., Hay B. P., Dixon D. A., J. Phys. Chem. A, 2001, 105, 4963. [63] Defrançois C.; Périquet V.; Carles S.; Schermann J. P.; Adamowicz L. Neutral and negatively-charged formamide, N-methylformamide and dimethylformamide clusters, Chem. Phys., 1998, 239, 475-483. [64] Sathyan N.; Santhanam V.; Sobhanadti J. Ab initio calculations on some binary systems involving hydrogen bonds, J. Mol. Struct. (Theochem), 1995, 333, 179-189.

Anharmonic Effects in Normal Mode Vibrations

187

[65] Wójcik M. J.; Hirakawa A. Y.; Tsuboi M.; Kato S.; Morokuma K. Ab initio MO calculation of force constants and dipole derivatives for the formamide dimer. An estimation of hydrogen-bond force constants, Chem. Phys. Lett., 1983, 100, 523-528. [66] Itoh K.; Shimanouchi T. Vibrational spectra of crystalline formamide, J. Mol. Spectrosc., 1972, 42, 86-99. [67] Lovas F. J.; Suenram R. D.; Fraser G. T. The microwave spectrum of formamide–water and formamide–methanol complexes, J. Chem. Phys., 1988, 88, 722-729. [68] Engdahl A.; Nelander B. Complex formation between water and formamide, J. Chem. Phys., 1993, 99, 4894-4907. [69] Gaussian 03; Revision C.02; Frisch M. J.; Trucks G. W.; Schlegel H. B.; Scuseria G. E.; Robb M. A.; Cheeseman J. R.; Montgomery J. A. Jr.; Vreven T.; Kudin K. N.; Burant J. C.; Millam J. M.; Iyengar S. S.; Tomasi J.; Barone V.; Mennucci B.; Cossi M.; Scalmani G.; Rega N.; Petersson G. A.; Nakatsuji H.; Hada M.; Ehara M.; Toyota K.; Fukuda R.; Hasegawa J.; Ishida M.; Nakajima T.; Honda Y.; Kitao O.; Nakai H.; Klene M.; Li X.; Knox J. E.; Hratchian H. P.; Cross J. B.; Adamo C.; Jaramillo J.; Gomperts R.; Stratmann R. E.; Yazyev O.; Austin A. J.; Cammi R.; Pomelli C.; Ochterski J. W.; Ayala P. Y.; Morokuma K.; Voth G. A.; Salvador P.; Dannenberg J. J.; Zakrzewski V. G.; Dapprich S.; Daniels A. D.; Strain M. C.; Farkas O.; Malick D. K.; Rabuck A. D.; Raghavachari K.; Foresman J. B.; Ortiz J. V.; Cui Q.; Baboul A. G.; Clifford S.; Cioslowski J.; Stefanov B. B.; Liu G.; Liashenko A.; Piskorz P.; Komaromi I.; Martin R. L.; Fox D. J.; Keith T.; Al-Laham M. A.; Peng C. Y.; Nanayakkara A.; Challacombe M.; Gill P. M. W.; Johnson B.; Chen W.; Wong M. W.; Gonzalez C.; Pople J. A.; Gaussian, Inc., Wallingford CT, 2004. [70] Dupuis M.; Farazdel A., HONDO-8, from MOTECC-91; IBM Corporation Center for Scientific & Engineering Computations: Kingston, NY, 1991. [71] Beu T. A., Program NOMAD, Cluj-Napoca, Romania, 1997. [72] Press W. H.; Flannery B. P.; Teukolsky S. A.; Vetterling W. T., Numerical Recipes, Cambridge University Press: Cambridge, U.K., 1986. [73] Bende A.; Vibók Á.; Halász G. J.; Suhai S. BSSE-Free Description of the Formamide Dimers, Int. J. Quantum Chem., 2001, 84, 617-622. [74] Bende A.; Vibók Á.; Halász G. J.; Suhai S. Ab Initio Study of the Ammonia–Ammonia Dimer: BSSE-Free Structures and Intermolecular Harmonic Vibrational Frequencies, Int. J. Quantum Chem., 2004, 99, 585-593. [75] Masunov A.; Dannenberg J. J. Theoretical Study of Urea. I. Monomers and Dimers, J. Phys. Chem. A, 1999, 103, 178-184. [76] Masunov A.; Dannenberg J. J. Theoretical Study of Urea and Thiourea. 2. Chains and Ribbons, J. Phys. Chem. B, 2000, 104, 806-810. [77] Rousseau B.; Van Alsenoy C.; Keuleers R.; Desseyn H. O. Solids Modeled by AbInitio Crystal Field Methods. Part 17. Study of the Structure and Vibrational Spectrum of Urea in the Gas Phase and in Its P21m Crystal Phase, J. Phys. Chem. A, 1998, 102, 6540-6548. [78] Keuleers R.; Desseyn H. O.; Rousseau B.; Van Alsenoy C. Vibrational Analysis of Urea, J. Phys. Chem. A, 1999, 103, 4621-4630. [79] Clabo Jr. D. A.; Allen W. D.; Remington R. B.; Yamaguchi Y.; Schaefer III H. F. A systematic study of molecular vibrational anharmonicity and vibration-rotation

188

[80] [81] [82]

[83]

[84] [85] [86]

[87]

[88]

[89]

[90] [91]

[92]

[93]

[94]

Attila Bende interaction by self-consistent-field higher-derivative methods. Asymmetric top molecules, Chem. Phys., 1988, 123, 187-239. Barone V. Anharmonic vibrational properties by a fully automated second-order perturbative approach, J. Chem. Phys., 2005, 122, 014108. Dunning Jr. T. H.; Hay P. J., in Modern Theoretical Chemistry, Ed. Schaefer III H. F., Plenum, New York, 1976, Vol. 3, 1-28. MOLPRO, version 2008.1, a package of ab initio programs, Werner H.-J.; Knowles P. J.; Lindh R.; Manby F. R; Schütz M.; Celani P.; Korona T.; Mitrushenkov A.; Rauhut G.; Adler T. B.; Amos R. D.; Bernhardsson A.; Berning A.; Cooper D. L.; Deegan M. J. O.; Dobbyn A. J.; Eckert F.; Goll E.; Hampel C.; Hetzer G.; Hrenar T.; Knizia G.; Köppl C.; Liu Y.; Lloyd A. W.; Mata R. A.; May A. J.; McNicholas S. J.; Meyer W.; Mura M. E.; Nicklass A.; Palmieri P.; Pflüger K.; Pitzer R.; Reiher M.; Schumann U.; Stoll H.; Stone A. J.; Tarroni R.; Thorsteinsson T.; Wang M.; Wolf A., see http://www.molpro.net. Molden 4.7, Schaftenaar G.; Noordik J. H. Molden: a pre- and post-processing program for molecular and electronic structures, J. Comput-Aided Mol. Design, 2000, 14, 123134. Jalkanen K. J.; Stephens P. J. Ab initio calculation of force fields and vibrational spectra: 2-oxetanone, J. Phys. Chem., 1999, 95, 5446-5454. Šponer J.; Jurečka P.; Hobza P. Accurate Interaction Energies of Hydrogen-Bonded Nucleic Acid Base Pairs, J. Am. Chem. Soc., 2004, 126, 10142-10151. Podolyan Y.; Nowak M. J.; Lapinski L.; Leszczynski J. Probing ab initio MP2 approach towards the prediction of vibrational infrared spectra of DNA base pairs, J. Mol. Struct., 2005, 744-747, 19-34. Bakker J. M.; Compagnon I.; Meijer G.; von Helden G.; Kabeláč M.; Hobza P.; The mid-IR absorption spectrum of gas-phase clusters of the nucleobases guanine and cytosine de Vries M. S., Phys. Chem. Chem. Phys., 2004, 6, 2810-2815. Gorb L.; Podolyan Y.; Dziekonski P.; Sokalski W. A.; Leszczynski J. Double-Proton Transfer in Adenine-Thymine and Guanine-Cytosine Base Pairs. A Post-Hartree-Fock ab Initio Study, J. Am. Chem. Soc., 2004, 126, 10119-10129. Müller A.; Talbot F.; Leutwyler S. Hydrogen Bond Vibrations of 2-Aminopyridine·2Pyridone, a Watson−Crick Analogue of Adenine·Uracil, J. Am. Chem. Soc., 2002, 124, 14486-14494. Hobza P.; Halvas Z. Blue-Shifting Hydrogen Bonds, Chem. Rev., 2000, 100, 42534264. Hobza P.; Špirko V. Why is the N1–H stretch vibration frequency of guanine shifted upon dimerization to the red and the amino N–H stretch vibration frequency to the blue?, Phys. Chem. Chem. Phys., 2003, 5, 1290-1294. Wang N. X.; Venkatesh K.; Wilson A. K. Behavior of Density Functionals with Respect to Basis Set. 3. Basis Set Superposition Error, J. Phys. Chem. A, 2006, 110, 779-784. Simon S.; Bertran J.; Sodupe M. Effect of Counterpoise Correction on the Geometries and Vibrational Frequencies of Hydrogen Bonded Systems, J. Phys. Chem. A, 2001, 105, 4359-4364. Watson J. D.; Crick F. H. Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid, Nature, 1953, 171 (4356), 737-738.

Anharmonic Effects in Normal Mode Vibrations

189

[95] Hoogsteen K. The structure of crystals containing a hydrogen-bonded complex of 1methylthymine and 9-methyladenine, Acta Crystallogr., 1959, 12, 822-823. [96] Felsenfeld G.; Davies D. R.; Rich, A. Formation Of A Three-Stranded Polynucleotide Molecule, J. Am. Chem. Soc., 1957, 79, 2023-2024. [97] Hakoshima T.; Fukui T.; Ikehara M.; Tomita K.-I. Molecular structure of a double helix that has non-Watson-Crick type base pairing formed by 2-substituted poly(A) and poly(U), PNAS, 1981, 78, 7309-7313. [98] Isaksson J.; Zamaratski E.; Maltseva T. V.; Agback P.; Kumar A.; Chattopadhyaya J. The First Example of a Hoogsteen Basepaired DNA Duplex in Dynamic Equilibrium with a Watson-Crick Basepaired Duplex – A Structural (NMR), Kinetic and Thermodynamic Study, J. Biomol. Struct. Dyn., 2001, 18, 783-806. [99] Patikoglou G. A.; Kim J. L.; Sun L.; Yang S.-H.; Kodadek T.; Burley; S. K. TATA element recognition by the TATA box-binding protein has been conserved throughout evolution, Genes Dev., 1999, 13, 3217-3230. [100] Leontis N. B.; Westhof E. Conserved geometrical base-pairing patterns in RNA, Quart. Rev. Biophys., 1998, 31, 399-445. [101] Abrescia N. G. A.; Thompson A.; Huynh-Dinh T.; Subirana J. A. Crystal structure of an antiparallel DNA fragment with Hoogsteen base pairing, PNAS, 2002, 99, 2806-2811. [102] Guerra C. F.; Bickelhaupt F. M.; Snijders J. G.; Baerends E. J. Hydrogen Bonding in DNA Base Pairs: Reconciliation of Theory and Experiment, J. Am. Chem. Soc., 2000, 122, 4117-4128. [103] van der Wijst T.; Guerra C. F.; Swart M.; Bickelhaupt F. M. Performance of various density functionals for the hydrogen bonds in DNA base pairs, Chem. Phys. Lett., 2006, 426, 415-421. [104] Krishnan G. M.; Kühn O. Identifying adenine–thymine base pairing by anharmonic analysis of the hydrogen-bonded NH stretching vibrations, Chem. Phys. Lett., 2007, 435, 132-135. [105] Kabeláč M.; Hobza P. Potential Energy and Free Energy Surfaces of All Ten Canonical and Methylated Nucleic Acid Base Pairs: Molecular Dynamics and Quantum Chemical ab Initio Studies, J. Phys. Chem. B, 2001, 105, 5804-5817. [106] Plützer C.; Hünig I.; Kleinermanns K.; Nir E.; de Vries M. S. Pairing of Isolated Nucleobases: Double Resonance Laser Spectroscopy of Adenine-Thymine, Chem. Phys. Chem., 2003, 4, 838-842. [107] Heyne K.; Krishnan G. M.; Kühn O. Revealing Anharmonic Couplings and Energy Relaxation in DNA Oligomers by Ultrafast Infrared Spectroscopy, J. Phys. Chem. B, 2008, 112, 7909-7915.

In: Electrostatics: Theory and Applications Editor: Camille L. Bertrand, pp. 191-215

ISBN 978-1-61668-549-2 c 2010 Nova Science Publishers, Inc.

Chapter 9

E MERGENT P ROPERTIES IN B OHMIAN C HEMISTRY Jan C.A. Boeyens Unit for Advanced Study, University of Pretoria, South Africa

Abstract Bohmian mechanics developed from the hydrodynamic interpretation of quantum events. By this interpretation all dynamic variables retain their classical meaning in quantum systems. It is of special significance in chemistry as a discipline which is traditionally based on the point electrons of quantum field theory. It could be more informative to assume a non-dispersive electronic spinor, or wave packet, with divergent and convergent spherical wave components, and with many properties resembling those of a point particle. Complex chemical matter is endowed with three attributes: cohesion, conformation and affinity, which can be reduced to the three fundamental electronic properties of charge, angular momentum and quantum potential, known from the wave structure. The chemical effects of these respective scalar, vector and temporal principles, all manifest as extremum phenomena. The optimal distribution of electronic charge in space appears as Pauli’s exclusion principle. Minimization of orbital angular momentum becomes the generator of molecular conformation. Equilization of electronegativity, the quantum potential of the valence state, dictates chemical affinity. The chemical environment is said to generate three emergent properties: the exclusion principle, molecular structure and the second law of thermodynamics. These concepts cannot be predicted from first fundamental principles. Only by recognition of the emergent properties of chemistry is it possible to simulate chemical behaviour. The exclusion principle controls all forms of chemical cohesion, atomic structure and periodicity; molecular structure underpins vector properties such as conformational rigidity, optical activity, photochemistry and other stereochemical phenomena; while transport properties and chemical reactivity depend on the second law. Simulation of these chemical concepts by constructionist procedures, starting from basic physics, is impossible. The ultimate reason is that complex chemical properties are not represented by quantum-mechanical operators in the same sense as energy and momentum. The Bohmian interpretation, which enables the introduction of simplifying emergent parameters, in analogy with classical procedures, allows the calculation of molecular properties by generalized Heitler-London methods, point-charge simulation and molecular mechanics.

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Jan C.A. Boeyens

Introduction

The editorial challenge to address the quantum frontiers of atoms and molecules in chemistry cuts far deeper than a survey of current activity in quantum chemistry, which for all practical purposes means ab-initio computational chemistry. There is no quantum theory of chemistry: Quantum mechanics originated as a theory to understand radiation and its interaction with sub-atomic matter. It gave birth to the modern science of spectroscopy, in which form it stimulated the development of sophisticated observational techniques that revolutionized physics, chemistry and biology. However, the early promise of a quantum theory of matter in general has not come to fruition. A theory that fails to elucidate the nature of electrons, atoms and molecules can never lead to an understanding of chemistry. It may enable computations of unrivalled complexity, but without a conceptual framework the results have no meaning. As a theory of spectroscopy, quantum mechanics serves to relate measured quantities, such as frequency and wavelength to the dynamic concepts of energy and angular momentum, by means of differential operators e.g.: −i~∂/∂t → E,

i~∂/∂ϕ → L.

Although chemically more useful information on molecular structure and electronegativity is also embedded in the state functions, there are no known operators whereby to extract this information directly. The current computational alternative fails for the same reason. Minimization of the scalar quantity, energy, can never generate three-dimensional structure, which is a vector concept. The failure to rationalize chemical behaviour is not a failure of quantum theory, but rather, a failure of the traditional interpretation of the theory. The ruling interpretation of quantum theory, known as the Copenhagen interpretation, incorporates a number of features totally at variance with chemistry. It defines quantum objects, including photons, electrons, atoms and molecules, as structureless point particles without extension. It reduces a continuously varying density, such as the electronic charge distribution in atoms, molecules and crystals to a probabilistic function. It offers no rationale for the occurrence of stationary states, apart from a postulate. Non-local effects are forbidden and the doctrine teaches more about measurement problems and quantum uncertainty than about chemical interaction. The unfortunate reality is that Schr¨odinger’s description of elementary matter as wave structures, which could not be reconciled with the Copenhagen orthodoxy, was ruled unacceptable and re-interpreted in terms of the quantum jumps and probability density of the particle model. This compromise resulted in the awkward concept of wave-particle duality, familiar to all modern chemists, but understood by none. The challenge that we are facing here is to retrace our steps to the point where the time-honoured concepts of classical chemistry merge in a natural way with the ideas of wave mechanics and start rebuilding a theory that ′′ stimulate(s) the mutual understanding of the various branches of chemistry and its neighbouring sciences′′ , realizing that ′′ the main stumbling block for the development of a theory of large and complex molecular systems is not computational but conceptual′′ [1].

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I propose to start with an outline of the essential concepts of cohesion, conformation and affinity; show how these relate to the more fundamental concepts of space, matter and number; and the derived concepts of interaction, waves and periodicity. Next we examine the Schr¨odinger formalism, the Bohmian interpretation and the application of wave mechanics, supported by higher-level emergent properties. The photoelectric effect, probably the most effective argument to have established the particle nature of photons is shown to be explained more convincingly by the transactional-wave model of electromagnetic interaction.

2.

The Fundamental Concepts

Any object or event observed in Nature can always be considered as the product of more primitive events or the interaction between more primitive entities. It is quite natural, in this reductionist spirit, to ascribe the actions and features of a living organism to some lower-level activity of biological cells, which in turn is driven by intracellular chemical interactions. The molecular building blocks of biological cells are assumed to consist of atoms, held together by electromagnetic forces, while the sub-atomic nucleons interact with strong interaction, and so on, ad infinitum. Not really. The reduction has to stop somewhere. As progressively smaller entities are implicated at the more primitive levels, it is reasonable to conclude that the cascade ends in a void, traditionally assumed to be the same as space, or the vacuum. At this point it is easy to get distracted by the interminable philosophical dispute about the possible existence or non-existence of a void. As a practical alternative I prefer to define space in geometrical terms and the vacuum as a physical entity.

2.1.

Space

The simplest way of looking at space is as a coordinate system that serves to describe the relative positions of observable objects. The intuitively most obvious is a cartesian system on three orthogonal axes. However, this may not necessarily be the most convenient coordinate system. Although it works well in a laboratory environment, it is well known to be inappropriate in geographical context, which is simplified by using spherical trigonometry. By the same argument cartesian coordinates may not be the most convenient to map the relative positions of astronomical objects. To first approximation the planets and most of their moons have been assumed to move in a single plane, called the ecliptic, around the sun. Individual elliptic orbits of planets and moons have a simple description in terms of Kepler’s laws in a plane, but most of these planes are known to make non-zero angles with the ecliptic. Moving into interstellar, or even intergalactic, space the situation becomes more complicated when dealing with cosmological distances, times and velocities, which demands relativistic rather than Galilean kinematics. Special relativity is conveniently formulated in four-dimensional Minkowski space and general relativity requires non-Euclidean geometry, i.e. curved space. In general relativity the geometry of space is described by a curvature tensor, which is linearly related to a stress tensor that describes the distribution of matter in the cosmos.

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This reciprocal relationship shows that an empty universe has zero curvature and that curved space generates matter. The mechanism whereby curvature generates matter is visualized in the process of covering a curved surface with an inflexible sheet. The higher the curvature, the poorer the fit and the more the wrinkles in the cover that cannot be smoothed away. Such wrinkles in space are interpreted as matter and energy – the content of the stress tensor. 2.1.1.

Number

Figure 1. Spiral structure of a fossilized nautilus shell. Not only the topology of space-time, but also the physical content of the universe, resembles the natural number system in remarkable detail [2]. This explains the unreasonable effectiveness of mathematics as a scientific tool and the success of number theory to predict natural phenomena as a manifestation of cosmic symmetry [2, 3]. The physical world, as an image of the natural numbers, can never be known in more detail than the number system. Concepts such as infinity and singularity, poorly understood mathematically, therefore make no physical sense. On the other hand, the concept of high-dimensional space, readily manipulated mathematically, and, although physically difficult to visualize, has a legitimate place in scientific discourse. To deal with the ubiquitous, but bothersome, infinities of physics, the infinity concept of projective geometry can be used to define both the number system and the physical cosmos as closed. Realizing that any closed system is periodic, by definition, a wave structure of the vacuum and periodicity of matter are inferred. The implied cosmic symmetry is referred to as self-similarity. The chambered structure of a nautilus shell, shown in Figure 1, is one of the best-known examples of this symme-

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try type. All the chambers have the same shape and only differ in size, which increases regularly along a golden logarithmic spiral. The same pattern occurs in the arrangement of growing seed buds in a sunflower head and in the image of a spiral galaxy. Recognition of the same growth features [2, 4] in atomic nuclei, atoms, covalent molecules and the solar system, reveals self-similarity on a much wider scale. The number theory of self-similarity shows that all of these structures are based on the Fibonacci number sequence, which converges to the golden ratio.

2.2.

Vacuum

The fabric of space is a matter of conjecture. However, if tangible matter occurs on curving flat space, flat space is not void. A useful analogy is to picture the vacuum as a regular undulating expanse filled by waves of constant wavelength. When curved, interference of the primary waves produces persistent wave packets, earlier identified as wrinkles. In local space such a wave packet is conveniently described as the superposition of threedimensional spherical waves, converging to and diverging from a centre of mass. These waves are the retarded and advanced solutions of the general wave equation, which implies motion, either forward or backward in time. 2.2.1.

Wave Packets

A typical wave packet generated by such a superposition of waves is shown in Figure 2. The tangent curve follows the amplitude of the 1/r Coulomb potential, which reflects the actual charge density, except when r → 0. The secondary waves propagate with the group velocity vg of the system and the primary waves have phase velocity vφ , such that √ vg vφ = c2 , where c = 1/ ǫ0 µ0 , is the velocity of light in the vacuum. Such a wave packet [4]:   sin kr iωt cos kr (1) or Φ = Ae kr kr has been shown [5] to describe elementary waves, equivalent to the postulated elementary distortions of space, i.e. the elementary particles of atomic physics. Interpreted as an electron, the distance between nodal points represents λdB = h/me vg , the de Broglie wavelength of a free electron and λC = 2π/k = h/me c, the Compton wavelength. The amplitude of the standing wave is proportional to the electronic charge. Φ0 = A, in eqn.(1), represents a wave packet with charge proportional to 0 or ±A. Electrons and protons, despite their difference in mass have charges of ±e. The neutron is neutral. The field intensity ΦΦ∗ = A2 (sin kr/kr)2 = C/r2 defines the force between charges, in line with Coulomb’s law, except when r → 0. The breakdown of Coulomb’s law, which occurs naturally for charged wave structures is equivalent to the special renormalization postulate in quantum field theory. 2.2.2.

Electron and Atom

A particle image, as shown in Figure 3, is obtained by rotating the diagram of Figure 2 about two axes perpendicular to x. The charge density can either contract or expand,

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Figure 2. One-dimensional section through a spherical wave packet with components converging on and diverging from x0 . depending on environmental pressure. The minimum radius that can be reached on compression depends on the rest mass of the object. For an electron r0 = e2 /m0 c2 .

Figure 3. Wave structure of a free electron with de Broglie wavelength λdB = h/me vg . Given the inferred flexible structure of an electron as a continuous indivisible charge, the self-similarity of atoms and the solar system is not obvious. The pioneering work on the planetary model of atomic structure was firmly based on Kepler’s model of elliptic orbits. Two crucial parameters that define a Kepler ellipse are the semimajor axis and the eccentricity, which characterize the size and shape of an orbit. By Newton’s laws these respective parameters are related to the energy and angular momentum of the orbital motion. In retrospect Kepler’s laws are seen to embody the general conservation principles for energy and angular momentum in celestial mechanics. The efforts of Bohr and Sommerfeld to explain electronic motion in atoms by the same model were spectacularly successful, despite a few subtle, but fatal, defects. Whereas Kepler’s model is valid in a gravitational field, it needs modification in an electromagnetic field as an accelerated charge radiates energy and an accelerated point charge therefore cannot maintain a stable orbit. Conservation

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of angular momentum in a central electrostatic field should rather be interpreted as conservation of the spherical shape of a continuous charge. Polar deformation under an external influence is described by the three-dimensional surface harmonics, or eigenfunctions, of the circulation Laplacian, with discrete eigenvalues: L2 Ylml = l(l + 1)k 2 Ylml ,

Lz Ylml = ml kYlml .

This classical result acquires quantum-mechanical meaning by equating the arbitrary constant k with ~, the elementary unit of angular momentum [6].

Figure 4. Phase-locked cavity with perfectly reflecting walls, filled with radiation in the form of standing waves. The difference between atomic and planetary systems goes a long way towards understanding of cosmic self-similarity. Although electrons in an atom are spread in three dimensions and planets orbit the sun in an approximately two-dimensional plane, both arrangements depend parametrically on the golden ratio. In the same way sunflower seeds of varying size are closely packed in a plane, compared to the three-dimensional stacking of nucleons; both styles conditioned by the golden ratio. The only common factor in all cases is the general curvature of space. Evidently, the curvature of cosmic space must be a function of the golden ratio, from the sub-atomic to supergalactic scales. 2.2.3.

Mass

Not only the charge, but also the characteristic mass and spin of sub-atomic species are accounted for by their wave structure. Jennison and Drinkwater [7] demonstrated that microwave radiation trapped in a phase-locked cavity, as in Figure 4, generates an interaction pattern which is mathematically equivalent to a system with inertial mass. Disturbing the equilibrium by a pulse that moves a cavity wall at velocity δv for a period δt, which is matched to the wave propagation across the cavity, modifies the internal pressure by Doppler shifting of the waves and sets the entire system into motion with velocity 2δv. The radiation pressure on the walls is balanced by an electromagnetic field, which keeps the system in static equilibrium. In the real vacuum the analogue of the phase-locked cavity is a standing wave, filled with radiation of Compton wavelength, internal energy E, in equilibrium with the external radiation (wave) field. Simple calculation [7] shows that

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the inertial mass of the wave packet obeys Newton’s law, F = ma, on identifying E/c2 with the rest mass.

2.2.4.

Spin

The standing-wave description of an electron defines it as an integral part of the vacuum, not obviously free to move without impediment. Linear motion of an electron must then clearly lead to continuous drag and deformation of both electron and its immediate environment, culminating in rupture of the vacuum and creation of a turbulent state. An electron that rotates in the vacuum, although more symmetrical, winds up the connecting medium until it shears and develops a discontinuity along a cylindrical surface. The only motion that occurs without distortion of the spherical wave packet or mechanical entanglement of the environment is rotation around a point. Unlike axial rotation this mode is more like a continuous wobble that returns to the original situation after two complete revolutions. The three dimensions of space participate equally in the motion without the transfer of rotational energy from the spinning object to the connecting medium. The strain that builds up during the first part of the rotational cycle relaxes during the second part. Apart from half-frequency cyclic disturbance in the connecting medium, the electron is free to move through the vacuum without permanent entanglement. An object, which performs this type of spherical rotation, is described mathematically by a spinor, or a quantity that reverses sign on rotation through an odd multiple of 2π radians.

3.

Quantum Theory

Quantum theory started with the discovery of line spectra and Balmer’s observation that the spectral lines of atomic hydrogen obey a digital formula, later generalized to: ν = Rc



1 1 − 2 2 n1 n2



,

n2 > n1 = 1, 2, 3 . . .

(2)

The first sensible explanation of the formula was proposed in 1904 by Nagaoka who used the planet Saturn with its system of rings as the basis of an atomic model, with electrons at energy levels (rings) in simple numerical order, orbiting a heavy positively charged nucleus. Experimental confirmation of such an arrangement was found by Rutherford in 1910 and a dynamic model, based on Planck’s quantum condition, E = hν, was proposed in 1914 by Bohr. Where Nagaoka argued that electrodynamically stable orbits required a standing electron wave of length λ = 2πr/n at an average distance r from the nucleus, Bohr postulated quantum stability for an orbiting electron with angular momentum p = nh/2π ≡ n~. With electrostatic and mechanical forces in balance, (using electrostatic units, 4πǫ0 = 1): p2 e2 = , r2 mr

E =T +V =

e2 e2 e2 − =− , 2r r 2r

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the postulate leads to the Balmer formula 2π 2 me4 hν = ∆E = h2 En =

2π 2 me4 , n2 h2

 1 1 − , n21 n22 n2 h2 rn = 2 2 . 4π me 

(3) (4)

The rest is history. When de Broglie rediscovered the Nagaoka condition in 1924 by postulating that all matter has an associated wavelength of λ = h/mv, only the mathematical framework for defining a general wave formalism of electronic behaviour, was lacking.

3.1.

Wave Mechanics

The mechanical behaviour of a Newtonian particle is described correctly by three quantities – energy, momentum and angular momentum, which describe the motion as a function of either time, displacement or rotation. In wave formalism each of these parameters is specified as a periodic function [8] with respect to time (τ ), translation (λ) or rotation (ϕ): ω = 2π/τ, E = ~ω,

k = 2π/λ, p = ~k,

ml = 2π/ϕ, Lz = ~ml .

The carrier of the electromagnetic field is described by the differential wave equation: ∇2 Ψ = µǫ

1 ∂2Ψ ∂2Ψ = 2 2. 2 ∂t c ∂t

(5)

To remain consistent with the previous relationships the dynamic variables need to be specified as differential operators: E → −~i∂/∂t,

p → −~i∇,

Lz → ~i∂/∂ϕ,

which can be checked by direct substitution. To allow for the first-order temporal dependence of the energy, the equation for matter waves is restricted to processes which only depend on time through a factor exp(2πiνt), leading to the final form:  2  ~ ∂Ψ 2 ∇ + V Ψ = ±~i (6) 2m ∂t which formally resembles the classical Hamiltonian definition of total energy, as H =T +V =

p2 + V = E. 2m

(7)

By defining a density function ρ = ΨΨ∗ and a current density j=

~ (Ψ∗ ∇Ψ − ψ∇Ψ∗ ) 2mi

(8)

there follows a continuity equation as in classical hydrodynamics ∂ρ + divj = 0. ∂t

(9)

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A general expression for a one-electron wave function over all available states X ck ψk e2πiνt Ψ=

(10)

k

may be used to calculate the current density over two states k and l: j=

~e X ck cl (ψl ∇ψk − ψk ∇ψl ) e2πi(νk −νl )t . mi

(11)

k,l

If only a single eigenvibration is excited, the current disappears and the distribution of electron density remains constant. Otherwise an electron flows from one state to another in an exchange that involves a photon to keep the energy in balance. This flow of electricity can hardly be described as a quantum jump. More realistically the vibrations of the two affected states (emitter and acceptor) are seen to interact and generate a beat (wave packet) that moves to the state of lower energy. The virtual photon that links two equilibrium states turn into a real photon that carries the excess energy, either into or away from the system. In chemical applications Schr¨odinger’s equation is best known in its amplitude form, which is obtained by substituting Ψ = ψ exp(2πiνt), followed by elimination of the time parameter to give: 2m ∇2 ψ + 2 (E − V )ψ = 0. (12) ~ In spherical polar coordinates this equation, for the hydrogen problem, separates into independent radial and angular equations:   l(l + 1)  2m  d2 R 2 dR R = 0, + + (13) E − V (r) − dr2 r dr ~2 r2   1 ∂ ∂Y 1 ∂Y sin θ + + l(l + 1)Y = 0, (14) sin θ ∂θ ∂θ sin2 θ ∂ϕ2 with separation constant λ = l(l + 1), integer l. The angular part is further separable into: d2 Φ + m2l Φ = 0 (ml = −l . . . l), dϕ2     m2l dΘ 1 d sin θ + l(l + 1) − Θ = 0. sin θ dθ dθ sin2 θ

(15) (16)

An electron associated with a stationary proton (V = e2 /r) defines the only problem of some chemical significance for which the radial equation has been solved. Since the proton is here regarded as a point particle, the system does not represent a wave-mechanical model of a hydrogen atom, despite contrary claims in all chemistry texts. Like the Bohr model, it defines a set of quantized energy levels to match most spectroscopic measurements, apart from the Lamb shift, fairly well. The angular equations are valid for central-field problems and produce quantized values of the orbital angular momentum. These eigenvalues should not be confused with the angular momenta of an orbiting particle. They are, more appropriately, considered as symmetry

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parameters, such that ml = 0 defines a spherically symmetrical charge distribution. For given l there is always an odd number of 2l + 1 sub-levels with different quantum numbers P ml , which, for many-electron systems, can be chosen in such a way that l ml = 0, in all cases. The choice reflects the electrostatic property of the charge distribution to assume spherical symmetry. A hydrogen atom, by this model, has ml 6= 0 only for excited states, which spontaneously relax to the ml = 0 spherically symmetrical ground state. 3.1.1.

Electron Spin

Schr¨odinger’s equation appears incomplete in the sense of lacking an operator for spin, only because its eigenfunction solutions are traditionally considered complex variables. The wave function, interpreted as a column vector, operated on by square matrices, such that  abbreviated to



ei(ωt−kx) 0 0 e−i(ωt−kx)

φ1 e+ φ2 e−



φ1 φ2



=

φ1 ei(ωt−kx) φ2 e−i(ωt−kx)





.

 , represents a spinor that moves in the x-direction. By forming the

derivatives: ∂φ = iω ∂t

φ1 e+ φ2 e−



∂2φ = k2 ∂x2



φ1 e+ φ2 e−



, 

,

it follows that (in three dimensions): −i

∂φ ω = 2 ∇2 φ. ∂t k

This is Schr¨odinger’s equation, providing (~k)2 = 2m~ω, i.e. −i

~ 2 ∂φ = ∇ φ, ∂t 2m

as in (6):V = 0

which shows ~ω = E = p2 /2m, k = 2π/λ, p = h/λ. This result is interpreted [4] to show that a region of the continuum, which rotates in spherical mode, interacts with its environment by generating a wave-like disturbance at half the angular frequency of the core. The angular momentum on the surface of a unit sphere is L = mω. At λ = 2π, k = 1, the spin angular momentum follows as L = ~/2, with intrinsic magnetic moment µ = ~e/2mc.

3.2.

Bohmian Mechanics

The connection between wave mechanics and hydrodynamics, expressed by equations (7) and (8), was developed in more detail by Madelung, writing the time dependence of Ψ

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as an action function, Ψ = ψe2πiνt → ReiS/~, which seperates (5) into a coupled pair that resembles the field equations of hydrodynamics: ∂S (∇S)2 ~2 ∇2 R + − + V = 0, ∂t 2m 2mR  ∂R2 R2 ∇S = 0, +∇· ∂t m

(17) (18)

which describe the irrotational flow of a compressible fluid, assuming R2 to represent the density ρ(x) of a continuous fluid with stream velocity v = ∇S/m. It was shown that both density and flux vary periodically with the same periodicity as νik = (Ei − Ek )/h, that results from superposition of states i and k. This means that radiation is not due to quantum jumps, but rather happens by slow transition in a non-stationary state. An attractive feature of the hydrodynamic model is that it obviates the statistical interpretation of quantum theory, by eliminating the need of a point particle. It is worth noting that the assumption of a point electron derives from the observation that it responds as a unit to an electromagnetic signal, which must therefore propagate instantaneously through the interior of the electron, at variance with the theory of relativity. However, by now it is known from experiment that non-local (instantaneous) response is possible in quantum systems and the initial reservation against Madelung’s proposal and Lorentz’s definition of an electron as a flexible sphere should fall away. On reinterpretation it was pointed out by David Bohm that equation (17) differed from the classical Hamilton-Jacobi equation only in the term Vq = −

~ 2 ∇2 R . 2mR

(19)

The quantity Vq , called quantum potential vanishes for classical systems as h/m → 0. A gradual transition from classical to quantum behaviour is inferred to occur for systems of low mass, such as sub-atomic species. All dynamic properties of classical systems should therefore be defined equally well for quantum systems, although the relevant parameters are hidden [10]. 3.2.1.

Quantum Potential

As for the classical potential, the gradient of quantum potential energy defines a quan.. tum force. A quantum object therefore has an equation of motion, m x= −∇V − ∇Vq . For an object in uniform motion (constant potential) the quantum force must vanish, which requires Vq = 0 or a constant, −k say. Vq = 0 defines a classical particle; alternatively1 −(V + Vq ) = T , the kinetic energy of the system. Hence ~2 ∇2 R/2mR = −E, which rearranges into 2mE ∇2 R + 2 R = 0 ~ Schr¨odinger’s equation for a free particle. 1

It is a common misconception that Vq = T for a free electron – compare [11]. Stationary states do not occur for Vq = 0, but when Vq = −V .

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The quantum potential concept is vitally important for understanding the structure of an electron and of quantum systems in general. The fact that the amplitude function (R) appears in both the numerator and denominator of Vq implies that the effect of the wave field does not necessarily decay with distance and that remote features of the environment can affect the behaviour of a quantum object. The quantum potential for a many-body system: Vq =

n X i=1

~2 − 2mR



∇2i R mi



depends on the quantum state of the entire system. The potential energy between a pair of entities, Vq (xi , xj ) is not uniquely defined by the coordinates, but depends on the wave function of the entire system, Ψ. This condition defines a holistic system in that the whole is more than a sum of the parts. The instantaneous motion of one part depends on the coordinates of all other parts at the same time. That defines a non-local interaction of the type assumed to exist within an indivisible electron, and now inferred to occur in all quantum systems, including molecules. If the system is distorted locally, the entire system responds instantaneously. As the quantum potential is not a function of distance, the behaviour of a composite system depends non-locally on the configuration of all constituents, no matter how far apart. In a chemical context the properties, structure and rearrangement of molecules must depend intimately on the quantum potential. It is necessary to give up the notion that molecular rearrangement involves the breaking and making of bonds and rather consider it as a modification of the intramolecular electronic wave interference pattern. However, all systems are not correlated equally well. Whenever a wave function can be written as a product Ψ(r1 , r2 , t) = ΦA (r1 , t)ΦB (r2 , t) the quantum potential becomes the sum of two terms: Vq (r1 , r2 , t) = VqA (r1 , t) + VqB (r2 , t). The two sub-systems evidently behave largely independently. That is a good description of a molecular crystal, or liquid, with relatively weak interaction between molecular units. Systems like these are better described as partially holistic. The contentious issue of quantum-particle trajectories is put into perspective by the Bohmian model. One interpretation is that the quantum electron has an unspecified diffuse structure, which contracts into a classical point-like object when confined under external influences. The observed trajectory, as in a cloud chamber, may be considered to follow the centre of gravity. In a two-slit experiment an electron wave passes through both slits to recombine, with interference, but without rupture. The interference pattern disappears on closure of one slit or when the slits are too far apart, compared to the de Broglie wavelength. It now behaves exactly like a classical particle, when forced through a single slit2 [12]. 2

The de-Broglie – Bohm formulation of particle plus pilot wave is considered an unnecessary complication by this author. Instead, Ψ may be thought of as a state of vibration of empty space.

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3.2.2.

Stationary States

Writing the wave equation in two equivalent forms: Ψ(x, t) = Ψ0 e−iEt/~, Ψ(x, t) = R(x, t)eiS(x,t)/~, and noting that R(x, 0) = R0 (x); S(x, 0) = S0 (x); Ψ0 = R0 eiS0 /~, it follows that: S(x, t) = S0 (x) − Et,

(20)

R(x, t) = R0 .

The unexpected conclusion is that a real wave function, Ψ0 = ψ, implies S0 (x) = 0 and hence the momentum ∇S = p = 0 and E = V + Vq . Those states with ml = 0 all have real wave functions, which therefore means that such electrons have zero kinetic energy and are therefore at rest. The classical (electrostatic) and quantum forces on electrons in such stationary states are therefore balanced and so stabilize the position of the electron with respect to the nucleus. For the hydrogen atom in the ground state, R(r) = N e−r/a0 and hence, d2 R N = 2 e−r/a0 , 2 dr a0 such that, from (19), Vq = ~2 /2ma20 . In general Vq =

~2 , 2mr2

(21)

and the quantum force on the electron: Fq =

∂Vq ~2 =− 3 ∂r mr

whereas the electrostatic force F = e2 /r2 . These forces are in balance when ~2 e2 = ; mr3 r

r=

~2 = a0 , me2

the Bohr radius. This means that V = Vq at r = a0 /2, halfway between proton and electron. 3.2.3.

Orbital Angular Momentum

Orbital angular momentum is perhaps the most awkward concept to visualize as the property of a quantum-mechanical point electron, but is readily understood in hydrodynamic analogy. Like tidal motion, atomic orbital motion in a continuous spherical charge cloud consists of the propagation of a wave disturbance, without matter circulation, as first proposed by Nagaoka and described by the quantized spherical surface harmonics, Ylml = N Plml eiml ϕ , in terms of Legendre polynomials, P .

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In Bohmian formalism angular momentum is described by rotation of the phase function: S(x, t) = S0 (x) − Et

= ml ~ϕ − Et.

The wavefronts S=constant are planes parallel to and rotating about the z-axis, with angular velocity ∂ϕ/∂t = E/ml ~. Single-valuedness of Ψ = R exp(iS/~) requires that Ψ(S) = Ψ(S + 2πn~) = ψ(S + nh). This is interpreted to mean that n = |ml | wave crests occur during each cycle. Positive and negative values of ml represent anticlockwise and clockwise rotations respectively. This interpretation of orbital angular momentum has a formal resemblance to the semiclassical model of Bohr and Sommerfeld, but there is no physical rotation of charge. Two electrons with magnetic quantum numbers of ±ml have wave structures that rotate, in phase, in opposite directions, with resultant distortion of zero. Quenching of orbital angular momentum during chemical interaction between neighbouring atoms happens by the same principle. The wave pattern in the case where l 6= 0 and ml = 0 is to be interpreted as the three-dimensional analogue of the circular modes of a vibrating drumhead. There is no axial component to the disturbance. The wave motion is more like spherical vibration, compared to spherical rotation that causes electron spin and which can be oriented in the polar direction of a magnetic field.

4.

Chemical Change

In the same sense that biological activity is more than chemical change, chemical effects depend on a number of emergent properties unknown to physics. The concepts of chemical affinity, cohesion and structure were discovered experimentally and not anticipated from first principles. Although chemical events can therefore not be inferred from the laws of physics, the Bohmian interpretation of quantum mechanics provides an attractive framework for their understanding. The fundamental reason for this emergence is the chemical environment. The interaction between chemical species, partially characterized in isolation, is as hard to predict as the behaviour of an individual in a crowd. Not being acquainted with the concepts molecule, phase transition and free energy, there is no possibility of deriving the laws of chemical affinity, reactivity and composition from the quantum numbers that quantify the energy and angular momentum of electrons in isolated atoms. The problem is approached here by examining the possible modes of interaction between charges and the response of atoms to close confinement.

4.1.

Interaction Theory

Interaction at a distance is interpreted in modern theories as a field phenomenon. The electromagnetic field, described by Maxwell’s equations as waves, propagate through the vacuum, with a constant velocity that depends on the permittivity and permeability of free √ space, c = 1/ µ0 ǫ0 . The wave equation (4) has solutions Ψ(t) and Ψ(−t), known as retarded and advanced waves, respectively. The transmission of electromagnetic energy

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Jan C.A. Boeyens

between an emitter and a distant receptor is assumed to be negotiated by a pair of retarded and advanced waves. As a spherical wave signal from the emitter reaches an acceptor, it responds with an advanced return signal that reaches the emitter at the exact moment of first emission, to establish a standing wave, known as a photon. Further interaction depends on the potential energy difference between emitter and receptor. Transfer of excess energy occurs by relaxation of the standing wave, which is experimentally observed as photon emission. Alternatively the standing wave, known as a virtual photon, that exists between interacting sites, becomes balanced against external factors, at a distance that defines the electrostatic force of interaction between the charges as: F =

q1 q2 . 4πǫ0 r2

All chemical interactions are of this type [4]. In Bohmian formalism the theory predicts the stability of atomic matter as a function of the fine-structure constant. Sommerfeld [13] – (p.107) introduced the fine-structure constant as α = v1 /c = e2 /4πǫ0 c~ (= 2πe2 /ch, in esu), where v1 is the velocity of an electron in the first Bohr orbit. More generally, the parameter α′ = v/c for a freely moving electron with de Broglie wavength λdB = h/mv and Compton wavelength λC = h/mc is defined, more appropriately as α′ = λC /λdB . An electron in a hydrogenic stationary state has nλdB = 2πn2 a0 , hence: e2 . αn = n~c In the Bohmian interpretation an atomic stationary state occurs when the potential energy of the electron, at rest, is balanced by the quantum potential. The relativistic mass of an electron at the position of the nucleus, with respect to the rest mass mo in the ns state, would be

i.e.,

mo mo =√ m= p 2 2 1 − α2 1 − v /c α2 =

m2 − m2o 4π 2 e4 En = = , m2 n2 h2 c2 mc2

Hence En = ∆m′ c2 ,

where ∆m′ = m − m2o /m ≃ m − mo .

This is interpreted here to show that an electron in a stationary state has its mass reduced, with respect to the nucleus, by an amount ∆m′ , which reappears as the binding energy −En . The same argument explains nuclear binding energy as a mass defect. Transition of an electron with n > 1 to a lower unoccupied energy level by emission of a photon with energy hν and spin ~, is anticipated. However, in the 1s state with quantum number l = 0, there is no orbital angular momentum to transfer in promoting photon emission and the ground state remains stable. The calculation does not imply different velocities for the electron at different energy levels – only a quantized change in de Broglie wavelength. The mass-energy difference amounts to exchange of a (virtual) photon in the form of a standing wave between the charge centres.

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207

With the classical radius of the electron defined as r0 = e2 /mc2 it is noted that r0 me4 = = a0 m~2 c2



e2 ~c

2

= α2 ,

where a0 is the Bohr radius. This result follows from the two relationships: 2πe2 = 2πr0 , mc2 λC 2π~2 = = 2πa0 = λdB . α me2

αλC =

Now define λZ = 2πr0 . Whereas the wavelength λdB = λC /α represents a wavepacket with group velocity vg < c, the phase velocity vφ > c is associated with the Zitterbewegung of wavelength λZ = α · λC ; vg vφ = c2 [15]. This argument relates to two problematic parameters: α and the classical electron radius r0 which still awaits quantum-mechanical definition. The fine-structure constant appears firmly associated with the wave nature of an electron, seen as a standing wave that results from the superposition of diverging and converging spherical components. The internal wave structure of the electron is observed as high-frequency Zitterbewegung while the macroscopic effects in an electromagnetic field are fixed by the spread of the wavepacket, conveniently defined as a de Broglie wavelength. Trapped in the field of a proton the de Broglie wavelength is quantized to avoid self-destruction, such that e2 λC = αn = . λdB n~c For an effective charge separation of rn , the ratio αn may be considered the ratio of two energies:   2 e2 1 e = · n~c rn hν an electrostatic and a quantum-mechanical factor. The constant c = λ/τ = λν describes the virtual photon that occurs as a standing wave (nλ = 2πr) between the charge centres. The balance between the classical coulombic attraction and the quantum-mechanical repulsion (the quantum potential) defines the fine-structure constant with a value, fixed by the de Broglie wavelength of the virtual photon. In a strong field the size of an electronic wavepacket may be compressed below the Compton radius to an absolute minimum of λZ , which describes the minimum size to which an electron may be compressed, measuring r0 = λZ /2π, for an electron defined as an electric charge −e distributed over a sphere of radius r0 . The potential energy E = e2 /r0 corresponds to r0 = e2 /mo c2 , as measured classically.

4.2.

Environmental Effects

Schr¨odinger’s solution for the hydrogen electron serves as the starting point for the qualitative discussion of all chemical effects in quantum formalism. It is routinely forgotten that the simple hydrogen solution ignores all interactions that the electron would experience in

208

Jan C.A. Boeyens

a chemical environment. Even the use of hydrogen energy levels to rationalize the structure of the periodic table is of limited value. A useful approach to simulate environmental effects was pioneered by Sommerfeld on solving Schr¨odinger’s equation under modified boundary conditions. Non-zero environmental pressure was introduced by assuming that ψ(r) → 0 as r approaches some finite value rc , rather than infinity. All energy levels move to higher values with decreasing rc , until the ground level reaches the ionization limit at rc = r0 , the ionization radius. It is noted that on reaching the ionization limit by uniform compression the electron that becomes decoupled from the nucleus finds itself confined to a spherical cavity at zero potential and kinetic energy. However, the non-zero energy of a free electron in a hollow sphere, must therefore be interpreted as quantum potential energy. The Helmholtz equation for such an electron: p ∇2 + k 2 ψ = 0 , k = 2mE/~2 p has the radial solutions, R = 2kr/π · kl (kr). At the first zero of the spherical Bessel function 0 = sin(kr)/(kr), kr0 = π, and hence (compare 21) E0 =

h2 = Vq . 8mr02

(22)

The Fourier transform of 0 is the box function f (r) =

 √

2π/2r0 if |r| < r0 , 0 if |r| > r0 .

(23)

It follows that the decoupled (valence) electron of the hydrogen atom, compressed to r0 is uniformly spread across the ionization sphere.

4.3.

Emergent Properties

Chemical theory requires insight into more than atomic stability. One-electron quantum theory provides no guidance beyond the hydrogen atomic ground state and the structure of many-electron atoms must be inferred from the empirically known periodic table of the elements. However, the superficial correspondence between the calculated quantum states of hydrogen and the observed elemental periods strongly suggests a functional relationship between the two sets. To better appreciate the relationship it is noted that both sets can be generated by convergent sequences of Fibonacci or Lucas fractions as shown below. 4.3.1.

Periodicity

A modular pair of rational fractions



h1 k1

,

h2 k2



has the property:

h1 h2 k1 k2 = ±1.

Emergent Properties in Bohmian Chemistry

209

Such a pair is geometrically represented by two Ford circles with radii and y-coordinates of 1/2k 2 at x-coordinates of h/k. A series of rational fractions with all neighbouring terms in unimodular relationship is represented by a set of tangent Ford circles [4]. Examples of such modular series are the Farey sequences, Fn and the converging Fibonacci and Lucas fractions on the segment ( 12 35 32 ) of F5 :

Despite a number of uncertain half-lives, a reasonable estimate of 264 divides the stable (non-radioactive) nuclides into 11 periods of 24. Plotting the ratio of protons:neutrons (Z/N ) for all isotopes as a function of atomic number, the hem lines that separate the periods of 24, intersect a reference line, at Z = τ , in Z-coordinates which correspond to well-known ordinal numbers that define the periodic table of the elements [2, 3]. Remarkably, the same hem lines intersect a reference line at Z/N = 0.58 in atomic numbers that correspond to the closure of the calculated wave-mechanical energy levels for hydrogen. Noting that the radii of unimodular Ford circles are inversely proportional to the number (2k2 ) of atoms in elemental periods, we look for converging circles that match the two forms of periodicity. The primary circle

at x = 0 or 1, rF = 21 , is flanked by two tangent circles at x = 0.5 and 1.5 ≡ −0.5, 1 1 and x = ± 34 , rF = 32 . This rF = 81 ; further converging pairs are at x = ± 23 , rF = 18 arrangement mimics the periodic table:

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Jan C.A. Boeyens

What is probably the aesthetically most pleasing form of the periodic table is obtained by rearrangement in circular array, as shown in Figure 5 for the hundred naturally occurring cosmic elements. Interpreted in terms of electronic distribution it implies twelve 8-fold and three 2-fold energy levels, with all closed-shell elements grouped together. These are not hydrogen-like energy levels, but they agree with the valence levels, calculated for compressed atoms in Hartree-Fock-Slater approximation [14]. The hypothetical arrangement based on the hydrogen solution is recognized in the nested set of Ford circles at x = 1, predicting consecutive periods of 2n2 , n = 1, 2, . . . , arranged as follows:

If n is interpreted as Schr¨odinger’s principal quantum number, periods of the correct length (2n2 ) are predicted. Each of the periods consists of n subshells for subsidiary quantum numbers 0 ≤ l < n. The number of elements per subset equals 2(2l + 1), l ≤ ml ≤ l. This result provides the basis of Pauli’s exclusion postulate, which defines an emergent property, not of quantum-mechanical origin. The hypothetical and observed versions of the periodic table are in agreement for elements 1 to 18. The superficial agreement (e.g. for elements 28, 46 and 78; and 29–36) beyond that point is purely accidental. We conclude that the wave-mechanical hydrogen model fails to account for elemental periodicity mainly because it ignores all interactions apart from the central-field unitary electrostatic attraction. The common thesis of chemistry textbooks that Schr¨odinger’s equation, with due allowance for interelectronic effects, accounts for the periodic table, fails on two important counts. It predicts transition series of ten elements, compared to the observed eight. The guiding principle, known as the Aufbau procedure, is valid only for the alkaline-s and p blocks. Less than two-thirds of the nominal transition elements obey an Aufbau rule. The correct periodic system occurs in an environ-

Emergent Properties in Bohmian Chemistry

211

Figure 5. The Periodic Table of the elements in circular form. ment that requires the convergence of stable nuclear composition, Z/N to the golden ratio, τ , and subject to an emergent exclusion principle. Further new properties are expected to emerge in the analysis of chemical affinity, cohesion and conformation, at a higher hierarchical level.

4.3.2.

Electronegativity

Chemical affinity is the intuitive qualitative concept that guided experimental chemistry for centuries. The first quantitative measure of affinity was discovered by Lothar Meyer as the atomic volume of an element – his basis of periodicity. It served to differentiate between electropositive and electronegative elements, with a natural affinity between them. The concept was generalized by Pauling, Mulliken and others, by placing all elements on

212

Jan C.A. Boeyens

Figure 6. Electronegativity as quantum potential of the valence state a single empirical electronegativity scale. By demonstrating the equivalence of electronegativity and the quantum potential of the valence state [16] it was finally recognized as an emergent atomic property, readily reduced to fundamental quantum theory. Like hydrogen, an atom is said to be in its valence state when ionized by environmental pressure. The energy of the electron, decoupled from the nucleus but confined to the ionization sphere, is given by (22). Characteristic ionization radii, r0 , are obtained by numerical Hartree-Fock-Slater calculation [14] with boundary conditions modified as for H. Redefined on this basis, electronegativity, χ, is calculated as χ2 =

h2 , 8mr02

expressed in eV, such that χ relates to Pauling electronegativities on a linear scale and χ2 √ to the Mulliken scale. A plot of χ = E0 reveals the same periodicity as Figure 5 and as Lothar Meyer atomic volumes. The uniform electron density of the valence state, from (23): 1 ρ = ψ 2 (r), ψ(r) = (φ/V0 ) 2 exp {−(r/r0 )p } , p >> 1. (24) The scale factor, φ, which compensates for an inaccessible core, is proportional to r0 and varies inversely with the number of nodes as defined by an effective principal quantum number n. Hence, φ = cr0 /n. The wave function   p 1 exp {−(r/r0 )p } (25) ψ(r) = 3c/4πn r0 describes the interaction of an atom with its chemical environment.

Emergent Properties in Bohmian Chemistry 4.3.3.

213

Covalence

Chemical cohesion has for many years been the main topic of theoretical chemistry, conducted as an exercise in computational quantum physics, described by one practitioner [17] as if repeatedly ′′ ...validating Schr¨odinger’s equation!′′ . There is a curious conviction that the Born-Oppenheimer scheme enables molecular structure to be computed ab initio. An initially assumed structure is treated only as a device to kickstart the calculation. Once the electronic density has been obtained, the nuclear framework is computed theoretically, without assumption. It always comes out miraculously close to the assumed structure. To the uninitiated the procedure appears to be circular and unlikely to produce anything of physical significance beyond the assumed molecular structure. An obvious alternative is to model the electron exchange that constitutes atomic pairwise interactions, known as covalent bonds, before assembly into a three-dimensional structure is attempted. The computational details for this procedure, which requires atomic wave functions, have been documented as the well-known Heitler-London method. Appropriate wave functions (25) are obtained from empirically adjusted ionization radii that compensate for steric factors [4]. H–L calculations predict both dissociation energy and equilibrium interatomic distance for any first-order covalent interaction. High-order interaction results from the valence-level screening of the internuclear repulsion.

Figure 7. Covalent binding energy curve for homonuclear diatomics in dimensionless units. The same set of characteristic atomic radii (r) can be used to model covalent electron exchange by point-charge simulation, as a function of interatomic separation (d) only. It

214

Jan C.A. Boeyens

is found that with the ratio d/r and binding energy E, expressed in dimensionless units, all homonuclear diatomic interactions are described by a single interaction curve, shown in Figure 7. The curve turns where d/r = τ and E = −2τ , at the point where exactly two electron waves are concentrated in the interatomic region. The condition is seen to reflect the exclusion principle for fermions. Its relationship to the golden ratio defines the origin of the exclusion principle as the curvature of space-time. Without this emergent property there is no understanding of covalent interaction. 4.3.4.

Molecular Shape

The inability to derive molecular structures from fundamental quantum theory identifies molecular shape as another emergent property. Although it cannot be inferred from basic theory it is readily reduced to the conservation of orbital angular momentum. Conventional computational schemes, designed to minimize energy, with total neglect of orbital angular momentum, must, by definition converge to a spherically symmetrical arrangement. To prevent this from happening a potential field of lower symmetry is imposed by assuming a fixed nuclear framework. Instead of imposing an experimentally determined structure, conservation of orbital angular momentum provides a theoretically more satisfying algorithm to generate such a structure from first principles. Polarization of mutually approaching reactants resolve local angular-momentum vectors, just like an applied magnetic field. During the formation of a molecule, the alignment of reactants that minimizes angular momentum in the local polar direction is favoured. In many reaction systems there is sufficient symmetry for the orbital angular momentum in the polar direction to become quenched completely. Where the quenching in low-symmetry (chiral) systems is incomplete, the residual angular momentum will couple to the magnetic field of polarized light, causing optical activity. Should quenching be possible only for a specific angular alignment of neighbouring fragments, a rigid system, which resists torsional deformation, is obtained. So-called double bonds and aromatic systems are common examples. The empirical stereochemical rules, pioneered by Kekul´e, van’t Hoff and others, are consistent with the principles outlined here and these have been generalized into empirical computational schemes, collectively known as molecular mechanics. There is no more fundamental procedure to predict molecular structure.

References [1] H. Primas, Chemistry, Quantum Mechanics and Reductionism, 2nd ed., SpringerVerlag, Berlin, 1983. [2] J. C. A. Boeyens and D.C. Levendis, Number Theory and the Periodicity of Matter, Springer.com, 2008. [3] J. C. A. Boeyens, Periodicity of the stable isotopes, J. Radioanal. Nucl. Chem., 2003 (257) 33–41.

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[4] J. C. A. Boeyens, Chemistry from First Principles, Springer.com, 2008. [5] M. Wolff, Beyond the Point Particle – A Wave Structure for the Electron, Galilean Electrodynamics, 1995 (6) 83–91. [6] J. C. A. Boeyens, Angular Momentum in Chemistry, Z. Naturforsch. 2007 (62b) 373– 385. [7] R. C. Jennison and A. J. Drinkwater, An approach to the understanding of inertia from the physics of the experimental method, J. Phys. A, 1977 (10) 167–179. [8] J. C. A. Boeyens, Quantum theory of molecular conformation, C.R. Chimie, 8 (2005) 1527 – 1534. [9] E. Schr¨odinger, Collected Papers on Wave Mechanics, (Translated from the second German edition), 2nd ed., Chelsea, NY, 1978. [10] D. Bohm, A Suggested Interpretation of the Quantum Theory in Terms of ′′ Hidden′′ Variables, Phys. Rev., 1952 (85) 166–179, 180–193. [11] D. W. Belousek, Einstein’s 1927 Unpublished Hidden-Variable Theory: Its Background, Context and Significance, Stud. Hist. Phil. Mod. Phys., 1996 (27) 437–461. [12] P. R. Holland, The Quantum Theory of Motion, University Press, Cambridge, 1993. [13] A. Sommerfeld, Atombau und Spektrallinien, 4th ed., Vieweg, Braunschweig, 1924. [14] J. C. A. Boeyens, Ionization radii of compressed atoms, J. Chem. Soc. Faraday Trans., 1994 (90) 3377–3381. [15] J. C. A. Boeyens, New Theories for Chemistry, Elsevier, Amsterdam, 2005. [16] J. C. A. Boeyens, The Periodic Electronegativity Table, Z. Naturforsch., 2008 (63b) 199–209. [17] N. C. Handy, in: R. Broer, P.T.C. Aerts and P.S. Bagus, New Challenges in Computational Chemistry, University of Groningen, 1994.

In: Quantum Frontiers of Atoms and Molecules Editor: Mihai V. Putz, pp. 217-250

ISBN: 978-1-61668-158-6 © 2010 Nova Science Publishers, Inc.

Chapter 10

THE ALGEBRAIC CHEMISTRY OF MOLECULES AND REACTIONS Cynthia Kolb Whitney* Galilean Electrodynamics, 141 Rhinecliff Street, Arlington, MA 02476-7331, USA

Abstract A new line of research generally characterized as ‘Algebraic Chemistry’ is here applied to the problem of modeling energies involved in molecule formation and in chemical reactions. The approach is based on algebraic scaling laws that allow one to estimate energies of interest by evaluating simple algebraic expressions, without recourse to computer calculations based on detailed quantum mechanical formulations and phase-space integrations, such as found in traditional Quantum Chemistry. The simplicity of the algebraic approach means that it can address molecules and reactions involving more atoms than the ones that are presently convenient for traditional Quantum Chemistry. In fact, there is no complexity-related limit on the atom count amenable to molecule or reaction analysis with the algebraic method. Algebraic Chemistry is a simple tool always suitable for hand calculations. In the cases of many real molecules and reactions, data is available to test the algebraic approach, and thus build some confidence about it. This is important because the scaling laws used come from new, and not yet broadly known, theoretical extensions to the traditional quantum mechanics of atoms, and even to special relativity theory and classical electromagnetic theory. These extensions of traditional theory are briefly summarized in Appendices and detailed further in the References.

Keywords: Algebraic Chemistry, Quantum Chemistry, heat of molecule formation, heat of chemical reaction.

*

E-mail address: [email protected]

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Cynthia Kolb Whitney

1. Introduction The present work develops a technique for quantitative analysis of molecules and reactions of arbitrary complexity. The approach is called ‘Algebraic Chemistry’, because it is based on simple multiplicative scaling laws. The present work extends the development of Algebraic Chemistry [1-4] from the discussion of individual elements to the discussion of complete molecules and reactions. The objective is to make Algebraic Chemistry more ready for full adoption into Chemistry, and less likely to remain a disruptive stepchild within Physics. Much of the earlier work in the development of Algebraic Chemistry concerned ionization potentials of atoms. It showed that all the information necessary to specify ionization potentials of arbitrary order for all the elements is embodied in first-order ionization potentials for all the elements, and that, in fact, all the information necessary to specify the first-order ionization potentials for all the elements is embodied in the first-order ionization potential of Hydrogen, and that, in fact, this number is predictable from theory. Here the idea of scaling laws is used to extend the known information about ionization potentials of neutral atoms to estimate ionization potentials for ‘already-ionized’ atoms. This new information is not to be confused with ‘higher-order’ ionization potentials. The ‘higherorder’ ionization potentials of neutral atoms, apparently [1], describe multiple ionizations that occur simultaneously, whereas the ionization potentials of ‘already-ionized’ atoms describe single ionization events that occur sequentially. The ‘one-at-time’ physical process is gentler than the ‘all-at-once’ physical process, and is more typical of events that actually occur throughout most of normal Chemistry. Like the previously known information about the higher-order ionization potentials of neutral atoms, the new information about ionization potentials of already-ionized atoms is entirely derivable from first-order ionization potentials of all the elements. Section 2 shows exactly what the scaling laws have to be, given the algebraic model under consideration. The new information is then used in subsequent Sections to estimate energies involved in molecule formation and chemical reactions.

2. Scaling Laws The basis for having scaling laws at all lies in imagining atoms to consist of a nucleus and an electron ‘cluster’ sufficiently well defined that the atom as a whole is similar to a twobody system. Being imagined as similar to a two-body system, all atoms are then similar to Hydrogen, and scaling laws based on Hydrogen follow. This idea is developed in [1], and then applied and extended in [2-4], and is summarized for the present applications in Appendix 3. The key results are as follows: The magnitude of the potential energy for the one electron in the Hydrogen atom is:

e2 (re + rp ) = 3c 2 me2 25 mp

(1)

The Algebraic Chemistry of Molecules and Reactions

219

where e is electron charge, r is orbit radius, c is light speed, m is mass, and subscripts e and p distinguish electron from proton. Suppose we want to model the system potential energy, not for Hydrogen per se, but for its isotopes, Deuterium and/or Tritium. The proton mp and rp then need to be replaced with a more generic nuclear mass M and its orbit radius rM . The magnitude of potential energy for this more massive atom is:

e2 (re + rM ) = 3c 2 me2 25 M .

(2)

Next, suppose we want to deal with a neutral atom with nuclear charge number Z , as well as the generic nuclear mass M . Then we have Z electrons as well. For the more charged system, the magnitude of the potential energy becomes:

Z 2e2 (re + rM ) = Z 2 3c 2 me2 25 M .

(3)

This scaled-up expression represents the magnitude of the total potential energy of the system involving Z electrons. What is then comparable to the ionization potential for removing a single electron is:

(

)

(

Z e2 (re + rM ) = Z × 3c 2 me2 25 M ≡ (Z / M ) × 3c 2 me2 25

)

(4)

Thus we see that Z / M scaling that is predicted for measured ionization potentials. This is cancelled out by M / Z scaling to produce the IP ’s collected in Appendix 1 and used in following Sections. More generally, if the atom is in an ionized state, we have a distinct electron count Ze and proton count Z p . For the baseline nuclear-orbit part, we have for the total system:

)(

(

Z p Ze e2 (re + rM ) = Z p Ze M × 3c 2 me2 25

)

(5)

What is then generally comparable to the nuclear-orbit part of the ionization potential for removing a single electron? Appendix 3 shows that it is as if all factors of e changed to

Z p Ze e . What is then comparable to the ionization potential for removing a single electron is:

Z p Ze e2 (re + rM ) =

(Z Z

p e

)(

M × 3c 2 me2 25

)

(6)

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Cynthia Kolb Whitney

In the present work, all scaling laws are based on first-order ionization potentials of neutral elements. Removing the Z / M scaling by multiplying raw ionization data by the inverse, M / Z , produces IP ’s that are all comparable to the IP of Hydrogen. In this paper, the symbol IP always means ‘ M / Z -scaled ionization potential to compare with that of Hydrogen’. Let the M / Z -scaled first-order ionization potential for Hydrogen be represented by IP1,1 . The first subscript 1 means ‘first ionization potential’, and the second subscript 1 means ‘first element’; i.e. Hydrogen. The M / Z scaled first-order IP for the element Z is then represented by IP1,Z . In the earlier work [1], the problem addressed was modeling IP ’s of all orders higher than unity for all elements - the ‘all-at-once’ problem. The results showed that we could enlist information from one element to develop information about an ion of another element. That idea is exploited here to infer the first-order IP ’s of already-ionized atoms – the ‘one-at-atime’ problem. The IP1,Z for removing an electron separates into a baseline, nuclear-orbit part, IP1,1 , and a deviation, electron-cluster part, ΔIP1,Z = IP1,Z − IP1,1 . The baseline part is independent of Z , but the deviation part is a complicated function of Z . The present paper takes the deviation part as input data. But there exists a basis for future deeper analysis and modeling of the deviation part. The development of an Expanded SRT, detailed in [1] or [2], allows for superluminal speeds, which in turn allows for same-charge systems. Rings of multiple charges rotating at superluminal speeds are analyzed in [3], and such rings stacked like little magnets are used to model the electron populations in atoms. The deviation ΔIP1,Z can be positive or negative, depending on where the element is in the Periodic Table. The deviation generally tends to zero at mid period, between noble gasses. It is maximally positive at a noble gas, and maximally negative just after a noble gas. For example, [see Appendix 1] the deviation term for Helium is

ΔIP1,2 = IP1,2 − IP1,1 = 49.875 − 14.250 = 35.625 , whereas the deviation term for Lithium is

ΔIP1,3 = IP1,3 − IP1,1 =Ê12.469 − 14.250 = −1.781 . +

−

Let symbols like IP1,Z and IP1,Z represent first-order IP ’s for already-ionized atoms. The superscript + means positively charged due to previous electron removal, and multiple +’s would mean multiple electrons removed. Superscript − means negatively charged due to previous electron addition, and multiple − ’s would mean multiple electrons added. The

Z n Ze M scaling maps the IP1,1 baseline part of the IP1,Z for the neutral element into

The Algebraic Chemistry of Molecules and Reactions

221

+ − the baseline part of the IP1, Z or IP1, Z or whatever. The increment part is different; it depends only on the number of electrons involved in the charge cluster, so the scaling that

applies to it is just Ze / M , and what it applies to is the deviation term for the element whose Z matches the Ze needed. The IP1,1 and the ΔIP1,Z are the basic data used below to describe first-order IP ’s like

IP1,+Z and IP1,−Z for already-ionized atoms.

3. Example Atoms In Physics, it is traditional to begin any discussion about atoms with the simplest possible atom, Hydorgen 1 H . I am going to depart from this tradition: I am starting this discussion with Carbon 6 C . Hydrogen is just too simple; with Hydrogen too many important parts of the ionization problem disappear as invisible zero’s. Carbon has no such degeneracies, and yet it is still a lot like Hydrogen; after Hydrogen, it is the next ‘keystone’ element [see Appendix 4]. Like all keystone elements, it is just as willing to give as to take electrons, until the number reaches a ‘noble-gas’ number: 2 or 10 for Carbon (as compared to 0 or 2 for Hydrogen). For Carbon 6 C , IP1,6 = IP1,1 + ΔIP1,6 times Z / M = 6 / M 6 represents the work that must be supplied to take one electron off the neutral Carbon atom. That exercise produces + a positively charged ion 6 C . The transition also returns some heat, because after the ionization, the electron cluster is different: 5 electrons instead of 6, and all resting at a different energy. I call it ‘heat’, not ‘work’, because it is uncontrollable; Nature simple ‘does’ this. The heat returned is evidently (ΔIP1,6 × 6 − ΔIP1,5 × 5) / M 6 . Observe how the ‘5’ for 5 B information enters here.

+ + The ionization potential of the singly ionized 6 C , IP1,6 , times 6 / M 6 represents the work that must be supplied to remove another electron from the already singly ionized

(

Carbon atom. It is IP1,1 ×

)M

6 × 5 + ΔIP1,5 × 5

6 . Observe how the

6 × 5 due to the

+ previous ionization enters here. It means that IP1,6 is certainly not the same thing as the so-

called ‘second ionization potential’ of the neutral Carbon atom, IP2,6 . As before, heat is also returned as the electron cluster readjusts. This time, the amount of heat returned is

(ΔIP1,5 × 5 − ΔIP1,4 × 4) M6 . Observe how the 4 for 4 Be information enters here.

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Cynthia Kolb Whitney

Proceeding to the next step, the work for removing another electron from the now doubly ++ ++ ionized 6 C , IP1,6 , times 6 / M 6 is IP1,1 × 6 × 4 + ΔIP1,4 × 4 M 6 . The heat

(

(

)

) M6 . The now triply ionized 6 C+++ takes

returned this time is ΔIP1,4 × 4 − ΔIP1,3 × 3 +++ work IP1,6 × 6 / M 6 =

(IP

1,1 ×

)M

6 × 3 + ΔIP1,3 × 3

(

The M / Z -scaled heat returned is ΔIP1,3 × 3 − ΔIP1,2 × 2 negative number. This means that getting from 6 C environment.

And

the

(

now

quadruply

ionized

++++ × 6 / M 6 = IP1,1 × 6 × 2 + ΔIP1,2 × 2 IP1,6

+++

)M

6C

6 to remove another electron.

) M6 . This turns out to be a

to 6 C ++++

++++

cools the local

would

take

work

ΔIP1,2 is a large

6 . Because

number, this last ionization is not very likely to happen. That is, like the 2 He atom, the ++++ ion is quite stable. 6C With all this information, one can construct an energy tally for all the electron removal + ++ +++ , and scenarios that Carbon invites: 6 C → 6 C , 6 C → 6 C , 6 C → 6 C ++++ : 6C → 6C 6C → 6C

+

6C → 6C

++

6C → 6C

+++

takes IP1,1 × 6 / M 6 and (ΔIP1,6 × 6 − ΔIP1,5 × 5) / M 6 ;

( 6 × 5 ) M and (ΔIP takes IP (6 + 6 × 5 + 6 × 4 ) M

takes IP1,1 6 +

1,6 × 6 − ΔIP1,4 × 4) / M 6 ;

6

1,1

6

and (ΔIP1,6 × 6 − ΔIP1,3 × 3) / M 6 ; 6C → 6C

++++

(

takes IP1,1 6 +

6×5+ 6×4 + 6×3

)M

6

and (ΔIP1,6 × 6 − ΔIP1,2 × 2) / M 6 . Carbon also allows the addition of electrons, producing negatively charged ions. To −−−− quantify these, let use the pattern for electron removal, which means we start with 6 C . The

(IP

1,1 ×

work

to

remove

6 × 10 + ΔIP1,10 × 10

an

)M

electron 6 . Like

must

be

−−−− IP1,6 × 6 / M6 =

ΔIP1,2 , ΔIP1,10 is a large number, meaning

The Algebraic Chemistry of Molecules and Reactions that, like 10 Ne , 6 C

−−−−

is quite stable. The heat returned in this de-ionization is

(ΔIP1,10 × 10 − ΔIP1,9 × 9) M6 .

(IP

−−− × 6 / M6 = IP1,6

1,1 ×

The

another

−−

electron

1,1 ×

removal,

6 × 7 + ΔIP1,7 × 7

(

ion

and

)M

−−−

,

takes

work

6 to remove the next electron. The

(IP

1,1 ×

this −

6C

And

)M

6C

down,

(ΔIP1,9 × 9 − ΔIP1,8 × 8) M6 . Then the next ion

−− , takes work IP1,6 × 6 / M 6 =

(ΔIP1,8 × 8 − ΔIP1,7 × 7 ) M6 .

(IP

next

6 × 9 + ΔIP1,9 × 9

heat returned in this de-ionization is down, 6 C

223

6 × 8 + ΔIP1,8 × 8

de-ionization takes

work

)M

6 for

returns −

heat

IP1,6 × 6 / M 6 =

6 for the last electron removal, and this final de-

ionization then returns heat ΔIP1,7 × 7 − ΔIP1,6 × 6

) M6 .

The process of adding electrons to an atom is just opposite to the process of removing electrons. So the energy tally for all the electron addition scenarios is: 6C → 6C

−

6C → 6C

−−

takes − IP1,1 × takes − IP1,1

6 × 7 M 6 and −(ΔIP1,7 × 7 − ΔIP1,6 × 6) / M 6 ;

( 6×8 +

6×7

)M

6

and −(ΔIP1,8 × 8 − ΔIP1,6 × 6) / M 6 ; 6C → 6C

−−−

takes − IP1,1

( 6×9 +

6×8 + 6×7

)M

6

and −(ΔIP1,9 × 9 − ΔIP1,6 × 6) / M 6 ; 6C → 6C

−−−−

takes − IP1,1

( 6 × 10 +

6×9 + 6×8 + 6×7

)M

6

and −(ΔIP1,10 × 10 − ΔIP1,6 × 6) / M 6 . With Carbon now well in tow, we are ready to look back to Hydrogen. The transition + 1 H → H takes IP1,1 / M1 and (ΔIP1,1 − ΔIP1,0 ) / M1 , but the latter two IP data items are zero, so the pattern being followed isn’t well revealed by them. The transition 1 H → H

−

takes − IP1,1 1 × 2 M1 and (−ΔIP1,2 × 2 + ΔIP1,1 × 1) / M1 , but ΔIP1,1 is zero, and so doesn’t fully reveal the pattern. That is why it was better to violate tradition and start with Carbon.

224

Cynthia Kolb Whitney Since hydrocarbons are so important in the discipline of Chemistry, it will also be useful

to have at hand the corresponding results for Oxygen. Being so close to 10 Ne , 8 O is + probably characterized thoroughly enough by just four transitions: 8 O → 8 O , ++ − −− and 8 O → 8 O , 8 O → 8 O . 8O → 8O + 8 O → 8 O takes IP1,1 × 8 / M 8 and (ΔIP1,8 × 8 − ΔIP1,7 × 7) / M8 , 8O → 8O

++

8O → 8O

−

8O → 8O

−−

(

takes IP1,1 8 + 8 × 7

)M

8 and (ΔIP1,8 × 8 − ΔIP1,6 × 6) / M 8 ;

takes − IP1,1 × 8 × 9 M8 and −(ΔIP1,9 × 9 − ΔIP1,8 × 8) / M 8 , takes − IP1,1

( 8 × 10 +

8×9

)M

8 and

−(ΔIP1,10 × 10 − ΔIP1,8 × 8) / M8 .

4. Application to Analysis of Molecules The basic idea exploited in this work is that a molecule consists of ionized atoms, some positively ionized and some negatively ionized, exerting Coulomb attraction for each other. It is therefore possible to learn something about a molecule by determining what its constituent ions are, and modeling the energy requirements for creating those ions from the neutral atoms involved. The first part of that question was addressed in Ref. [1], which gave the following two mirror-image propositions: Proposition 1: Molecules that are relatively stable have total electron counts such that every atom present can be assigned an electron count equal to that of a noble gas, or else zero. Proposition 2: Molecules that are highly reactive have total electron counts such that not every atom present can be assigned an electron count equal to that of a noble gas, or else zero. Proposition 1 is very often fulfilled, and was illustrated by molecules from small ( NH 3 , NaOH )

to

larger

( CH 3CO 2C10 H17 ,

(CH 3CO2 )2 Pb ⋅ 3H 2O ,

(C17 H 35CO2 )2 Ca ). Proposition 2 was illustrated with some common atmospheric gasses ( O 2 , O3 , or NO ). So the Propositions are probably true, but we really need to know much more. We need not just a binary division into stable / reactive; we need numerical rankings within those categories. That would mean modeling the energy requirements for creating the ions involved. Furthermore, it is often true that a ‘stable’ molecule admits more than one possible set of noble-gas electron assignments, or that a ‘reactive’ molecule admits more than one possible set of not-quite noble-gas electron assignments. When multiple assignments are possible, which one is the one Nature picks? Again, we need numerical rankings.

The Algebraic Chemistry of Molecules and Reactions

225

The present work is aimed at developing such quantitative rankings, based on the algebraic modeling concepts developed in the previous Section. There are several preliminary remarks to be made: 1) In all of the following examples, the IP data used come from the algebraic model developed in [1] and summarized in Appendix 1 and Appendix 2. Occasional IP data points differ detectably from M / Z -corrected raw data. So be it. I wish to test the concept of algebraic modeling over all, and if errors that are manifest in molecule modeling actually arise from errors in atom modeling, they are nevertheless errors, and I want them to be seen; 2) The calculations reported carry several more digits than can be justified as ‘significant’. This level of numerical detail is provided only to make the calculations easier to follow and to reproduce, and not to imply an amazing level of precision; 3) Comparison to reported data is generally desired, but not always possible, for reasons discussed below. One problem is that the energies calculated here are just for the formation of the ions in a single molecule. Some of this energy will be consumed in actually forming the molecule; i.e. giving the ions enough energy to stay at some stand-off distance from each other. Some more energy will be consumed in getting to the ‘state of matter’ for which data are reported. That means many molecules, in bulk matter – a mole – endowed with collective attributes. Molar heats of formation are generally reported for ‘standard conditions’, i.e. temperature (typically 25o C), pressure (for gasses, one atmosphere), dilution (for solutions, infinite dilution), etc. A single molecule can hardly be said to possess such properties. Indeed, not even the ‘state’ of matter – solid, liquid, or gas – seems meaningful for a single molecule. So it is clear that the energy calculated here for forming the ions in a single molecule in isolation cannot be said to correspond directly to reported data on the heat of formation for a molecule as a constituent of bulk matter. The energies calculated here are generally are generally less than reported heats of formation. Why less? Because, for molecules that actually form spontaneously, thus releasing heat, the reported heats of formation are by convention negative. That means the energies for forming the ions in molecules should also come out negative, and indeed more negative than the heats of formation reported for bulk matter. “Less is more,” the saying goes! Another problem, all too pervasive throughout Chemistry, is the need for conversion among many different systems of units. That is why the edition of Lang’s Handbook that provided most of the data in this paper has some 42 pages devoted to specifying conversion factors. And even at that generous page count, it does not have the particular conversion factor needed here. Heats of formation are quoted there in ‘kilogram-calories per mole’