Multi-Component Molecular Orbital Theory

MULTI-COMPONENT MOLECULAR ORBITAL THEORY No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, 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 herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

MULTI-COMPONENT MOLECULAR ORBITAL THEORY

TARO UDAGAWA AND

MASANORI TACHIKAWA

Nova Science Publishers, Inc. New York

Copyright © 2009 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. 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. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA ISBN: 978-1-60741-912-9 (E-Book) Available upon request

Published by Nova Science Publishers, Inc.  New York

CONTENTS

Preface

vii

Chapter1

Introduction

1

Chapter 2

Multi-Component Molecular Orbital Methods

5

Chapter 3

Application of the Multi-Component Methods

21

Chapter 4

Conclusions

49

References

51

Index

57

PREFACE We report the multi-component first-principles methods which can take account of the quantum effect of light particles such as proton and positron, as well as electron. In particular, we introduce our multi-component molecular orbital (MC_MO) methods and multi-component “hybrid type” density functional theory (MC_DFT). Using these multi-component procedure, we can analyze many chemical phenomena, H/D isotope effect, positronic systems, and so on. We will show some examples of MC_MO and MC_DFT works.

Chapter1

INTRODUCTION The conventional molecular orbital (MO) methods which based on BornOppenheimer (BO) approximation[1] have been developed to describe the electronic motions. In fact, these conventional MO methods successfully describe the electronic structures and provide us useful information for some chemical phenomena. However, nuclear motion is known to be very important to elucidate the hydrogen-bonded systems and hydrogen/proton transfer reactions[2,3]. Furthermore, H/D isotope effects which are caused by difference of the mass between proton and deuteron induce many chemical phenomena. Geometrical parameters are changed by H/D isotope effect, and this geometric isotope effect (GIE) induces drastic change of phase transition temperature in ferroelectric materials. According to such background, the quantum mechanical description of nuclear motion is a problem of central interests in physics, chemistry, and interdisciplinary fields[4-7]. To obtain the nuclear motion or nuclear wave function within the framework of BO approximation, it is necessary to get a potential energy hypersurface. Many theoretical methods considering quantum nature on hydrogen nucleus are developed in the framework of BO approximation, such as path integral molecular dynamics method[7,8], surface hopping trajectory simulation[9,10], and grid-based BO analysis[11]. However, since the potential energy hypersurface corresponds to the eigenvalues with the electronic Hamiltonian for all possible nuclear configurations, this treatment is possible to only few-atoms containing system. On the other hand, nuclear motion is also described by extension of the MO method, in which both electrons and nuclei are simultaneously treated quantummechanically. Generalization of MO to a multi-component (MC) system can

2

Taro Udagawa and Masanori Tachikawa

include the non-adiabatic coupling between electronic and nuclear motions, as well as the quantum nature of the nuclei. The pioneering work of this theory was presented by Thomas[12-15] in 1969. He added the kinetic energy operators of nucleus to the electronic Hamiltonian and calculated HF, H2O, NH3, and CH4 molecules with fixed Slater-type basis functions. Monkhorst[16] derived the coupled-cluster formulation for MC systems. Recently, we have proposed “Multi-component MO (MC_MO)”, and successfully analyzed various chemical and physical phenomena, such as phase transition of ferroelectric materials[17], geometrical H/D isotope effect of C-H...O hydrogen bonds[18,19], kinetic H/D isotope effect in hydrogen transfer systems[20], and so on. Our MC_MO method can determine both nuclear and electronic wavefunctions simultaneously, and expresses the isotope effect including coupling effects between nuclei and electrons directly. In the conventional MO method, only the electronic state is described under the condition of fixed nuclei, i.e., the motion of electrons is evaluated in the field of fixed nuclear charges. It is noted that this electronic Hamiltonian expresses only the electronic state, and no operator terms for the nuclear kinetic energy are taken into account. As mentioned above, in order to describe the isotope effect of nuclei, one must solve the nuclear motion of the adiabatic potential under the BO approximation. Although the nonadiabatic effects are very small and the adiabatic approach suffices for most chemical systems, the coupling effect between electronic and nuclear motions and the nonadiabatic effect would be very important in the case of a system containing the hydrogen atom; therefore, the lightest nuclei should be treated as a quantum wave by appropriate method, such as our MC_MO. Several groups also studied by using this multi-component MO method. Nakai and coworkers[21-24] have also developed the nuclear orbital plus molecular orbital (NOMO) method, where translational and rotational motions are eliminated from corresponding Hamiltonian[24]. Hammes-Schiffer and coworkers[25-32] have developed the nuclear-electronic orbital (NEO) method and extended it to CI[25,31], MP2[30], multiconfigurational self-consistent-field (MCSCF)[25], and explicitly correlated Hartree-Fock (XCHF) method[32]. Shigeta and coworkers[33,34], and Gross[35] and coworkers have developed the multi-component density functional theory (MCDFT), and analyze the isotope effect of hydrogen molecule and its isotopomers. Adamowicz and coworkers[3646] have developed a explicit correlated Gaussians (ECG) for nonadiabatic calculations. They have calculated polarizability of several small molecules within ultra-precision with their ECG, and they also have presented the improved nonadiabatic ground-state energy upper bound for several small molecules.

Introduction

3

Nowadays, many methods of multi-component MO have been developed. Details of each theory and work would be introduced in next chapter.

Chapter 2

MULTI-COMPONENT MOLECULAR ORBITAL METHODS 2-1. PIONEERING WORKS BY I. L. THOMAS In 1969, the first study of multi-component systems in terms of molecular orbital method was reported by Thomas[12-15]. He added the nuclear kinetic term to conventional “nuclear fixed” Hamiltonian. Then Hamiltonian has a form

H =∑ a

Z Z Z 1 1 1 Δ a − ∑ Δ i − ∑∑ a + ∑ + ∑ a b 2m a i 2 a i rai i < j rij a
,

(1)

where the i and j indices refer to the electrons, the a and b refer to the nuclei,

Z

and a represents the nuclear charge. When a molecular problem is solved using the variational theorem, the wave function is often given the following form:

Ψtot = Φ elec Φ nuc ,

(2)

where

Φ elec =

∑ (− 1) P[φ (1)Λ φ (n )] n!

1

P

1

e

and

P

ne

e

(3)

6

Taro Udagawa and Masanori Tachikawa

Φ nuc = ∏ a

∑ (− 1) P[η (1)Λ η (n )] n ! 1

P

na

1

a

P

In Eqs. (3) and (4),

a

.

(4)

φi ’s and η a ’s are one-electron and one-nucleus

n

n

m

functions, e is the number of electrons, a is the number of a nuclei, and a is the mass of the a nucleus. He applied these schemes to calculate the protonic structure of small molecules with one-electron Slater functions.

2-2. MULTI-COMPONENT MOLECULAR ORBITAL METHODS A. Hartree-Fock level of multi-component molecular orbital method First, let us introduce our Hartree-Fock (HF) level of multi-component molecular orbital (MC_MO-HF) method[47-50]. The electronic Hamiltonian of an expressed as follows:

⎛ 1 2 M Zμ H e = ∑ ⎜ − ∇i − ∑ ⎜ 2 i =1 ⎝ μ =1 riμ Ne

N e electrons and M nuclei system is

⎞ N e 1 M Z μ Zν ⎟+∑ +∑ ⎟ i > j r μ >ν r ij μν ⎠ ,

(5)

where the i and j indices refer to the electrons, μ and ν to the nuclei and μ represents the nuclear charge. In a conventional molecular orbital (MO) calculation, the time-independent Schrödinger equation of the electronic Hamiltonian is solved approximately using the variational method with the fixed nuclei. Another way of saying, the motion of electrons is evaluated in the field of fixed nuclear charges[51,52]. This electronic Hamiltonian expresses only electronic states and includes no nuclear kinetic energy operator terms. To analyze the nuclear motion in the framework of Born-Oppenheimer (BO) approximation[1], we have to solve the electronic Hamiltonian for all possible nuclear configurations. Of course, this procedure is not practical and only applicable to few atom-containing systems.

Z

Multi-Component Molecular Orbital Methods

7

According to these backgrounds, we have extended a concept of MO to light particles such as proton, deuteron, positron, and muon, to analyze the nuclear motions. In order to obtain both the electronic and nuclear wavefunctions simultaneously, we use the total Hamiltonian which includes not only the electronic part but also the quantum-treated nuclear part: M Z ⎛ 1 2 M Z μ ⎞ Ne 1 N p ⎛ ⎟ + ∑ + ∑ ⎜ − 1 ∇ 2p + ∑ μ H tot = ∑ ⎜ − ∇ i − ∑ ⎜ 2 ⎟ i> j r ⎜ 2m i =1 ⎝ p =1 ⎝ μ =1 riμ ⎠ μ =1 r pμ ij p Np Z p Z q N e N p Z p M Z μ Zν +∑ − ∑∑ +∑ μ >ν rμν p > q r pq i p rip Ne

⎞ ⎟ ⎟ ⎠

,

(6)

where p and q indices refer to the quantum-treated nuclei. The total wavefunction could be expressed as the full-configuration interaction (CI) form: ⎛ Ψ = ∑ Φ L C L = Φ 0A Φ 0B Λ Φ 0M C0 + ⎜⎜ ∑∑ Λ L ⎝ LA LB

'

∑ LM

⎞ A ⎟ Φ L Φ BL Λ Φ ML C( L ,L ,Λ L ) A B M A B M ⎟ ⎠

(7) where the Φ s are symmetrized wavefunctions for bosons, or antisymmetrized wavefunctions for fermions, respectively. The superscript of Φ s refers to the type of particles, and the subscripts indicate the chosen configuration for each type of particle. The prime on parenthesis means the exclusion of

L A = LB = Λ LM = 0 . In the MC_MO-HF method, the total nuclear wavefunction

φ

Φp

is expressed

Φ

by the nuclear MO( p )s, as well as the total electronic wavefunction e . For simplicity, we here treat one kind of nuclear species as the quantum mechanical wave and other nuclei as the point charges. Of course, to treat lightest nucleus, such as protons, is better approximation. The zeroth-order wave function in Eq.

Ψ = Φe ⋅ Φ p

0 0 0 corresponds to the HF wavefunction. Thus the HF (7), wavefunction is expressed as simple product of the electronic and nuclear wavefunctions. The HF wavefunctions of electrons and nuclei are given by antisymmetric or symmetric products of the electronic and nuclear MOs,

8

Taro Udagawa and Masanori Tachikawa

Φ e0 = χ i χ j Λ χ k Φ

p 0

,

= χ p χq Λ χr

(8) ,

(9)

χ

χ

where i and p are the spin MOs of an electron and a nucleus. The energy of this system after integration of the spin coordinates is given as Ne

Ne

i

i, j

{ (

)

(

)}

Np

E = ∑ nie hiie + ∑ α ije φiφi φ jφ j + β ije φiφ j φiφ j + ∑ n pp h ppp Np

{ (

p

)

(

)}

(

Ne N p

p + ∑ α pq φ pφ p φqφq + β pqp φ pφq φ pφq − ∑∑ nie n pp φiφi φ pφ p p ,q

where

i

p

) ,

(10)

p φi and φ p are the spatial MOs of an electron and a nucleus. hiie and h pp φiφi φ jφ j φiφ j φiφ j

are one-electron and one-nucleus integrals,

(

Coulomb and exchange integrals of electrons, those of nuclei, and

(φ φ p

p

) and ( ) are the (φ φ φ φ ) are φφ ) and q q

p q

p q

(φ φ φ φ ) is the Coulomb integral between an electron and i i

p

p

p nie and n p are the occupation numbers of φi and φ p , N N the α and β are Coulomb and exchange coupling constants and e and p

a nucleus. The coefficients

are the number of electrons and nuclei, respectively. Then we can derive the HF equations for electrons and quantum-treated nuclei by the variational method as

f eφi = ε iφi , f φ p = ε pφ p

(11)

p

where

,

(12)

Multi-Component Molecular Orbital Methods Ne / 2

∑ (2 J

f = 2h + e

e

i

− Ki ) −

i

f

p

= 2h p +

∑ (2 J

Ne / 2

∑ 2J

p

p

Ne / 2

p

± K p )−

p

9

,

(13)

Ne / 2

∑ 2J

i

i

.

(14)

In Eqs. (13) and (14), J and K are the Coulomb and exchange operators,

φ

respectively. We can also see that the effective field of the electronic MO i is due to the motion of the nuclei and other electrons in Eq. (13). Similarly, that of

φ

nuclear MO p is due to the motion of the electrons and other nuclei. To solve these Fock equations, we use linear combination of Gauss type functions (LCGTF) approximation for both electronic and quantum nuclear MOs, which is follows

φi = ∑ Crie χ re r

φ p = ∑ Cν χν p

,

ν

where GTF

(15)

p

,

(16)

χ is

χ σ ( x, y, z ) = (x − X σ ) ( y − Y σ ) (z − Z σ ) l

{

(

× exp − α σ x − X σ

m

n

) + (y − Y ) + (z − Z ) }. 2

σ 2

σ 2

(17)

As mentioned above, electronic and nuclear MOs have three kinds of

(

e

parameters such as LCGTF coefficients C , C

(

e

and GTF centers R , R

(

e

p

)

p

p

) , GTF exponents (α

e

)

,α p ,

). In conventional MO calculations, only the LCGTF

are determined by the variational theorem with the other coefficients C , C parameters fixed. Using electronic and quantum nuclear basis functions,

φ

φ

electronic MO i and quantum nuclear MO p are obtained by solving the electronic and quantum nuclear Roothaan equations simultaneously. We have

10

Taro Udagawa and Masanori Tachikawa

implemented these schemes and gradient routine of energy with respect to the classical nuclear coordinate to GAUSSIAN03 program package[53] and our original program FVOPT[48,49,54]. The detail of the gradient routine will be shown in below. Then we encounter the problem of determination of nuclear GTF exponent. Since electronic GTF centers are settled on the nuclear point charge in the conventional BO calculations, we can restrict the electronic and nuclear GTF centers are identical. Of course this restriction is only approximation, however we think this approximation is allowed in many cases. In contrast to the case of nuclear GTF centers, there is no indicator for determination of nuclear GTF exponent, because the all nuclei are treated as point charge in the conventional BO calculations. In addition, there are no well known nuclear basis sets, while large amount of electronic basis sets are well known. Thus we will introduce the Fully variational molecular orbital procedure which can determine the not only LCGTF coefficients but also GTF exponents and centers variationally. B. Fully variational molecular orbital (FVMO) method Recently we have proposed the fully variational MO (FVMO) method[48,49,54], in which all parameters in the basis functions are optimized under the variational principle. We note here the advantages of FVMO method. First, the total energy is improved under the variational principle with the small number of basis functions due to the extension of variational space. The extension of variational space also provides the drastically improved values for the properties, such as dipole moments and polarizabilities. In addition, the basis sets of quantum light particles, such as positron, proton, muon, and other nucleus can be variationally determined. By optimizing the GTF centers[55-58] and exponents[59], Hellmann-Feynmann[60,61] and virial[62] theorems are completely satisfied in the FVMO method. It is noted that these two theorems are satisfied in the exact wavefunction, but not always obeyed in the conventional procedure.

N

e electrons and a Here, we consider a system which is constructed by proton for simplicity. Then HF energy of this system is given as Ne / 2

E HF = 2 ∑ hiie + i

∑ {2(φ φ φ φ ) − (φ φ

Ne / 2

i i

i, j

j

j

i

j

)}

Ne / 2

(

φiφ j + h ppp − 2 ∑ φiφi φ pφ p i

) .

(18)

Multi-Component Molecular Orbital Methods

11

{ } {χ }

Each molecular orbital is expanded in the basis set χ r or ν . Hereafter we denote the parameters, orbital exponents, and orbital centers of electronic and p

e

nuclear basis sets as a whole, by Ω . In order to optimize the HF energy of Eq. (18) with respect to these parameters, the analytical formulas of the HF energy derivatives are required. The first derivative of the HF energy of Eq. (32) with respect to parameter Ω is given as follows:

{(

Ne / 2 Ne / 2 ∂ E HF = 2 ∑ hiie (Ω ) + ∑ 2 φiφi φ jφ j ∂Ω i i, j Ne / 2

(

+ h ppp (Ω ) − 2 ∑ φiφi φ pφ p i

)( ) − (φ φ φ φ )( ) } Ω

)( ) − 2 ∑ S ( )ε Ne / 2

Ω

Ω

i

Ω ii

i

i

j

i

j

(Ω ) − S pp εp

,

(19)

where AO

S ij(Ω ) = ∑ Cμi Cνj μ ,ν

∂S μν ∂Ω

(20) (Ω )

h is the skeleton overlap derivative[63] integral with respect to parameter Ω , ii (Ω ) h pp

and

are the skeleton derivative of one-electron and one-proton integral,

(φ φ φ φ )( ) (φ φ φ φ )( ) Ω

i i

j

j

(φ φ φ φ )( ) Ω

and

i i

p

p

are that of Coulomb integrals, and

Ω

i

j

i

j

is that of exchange integral. In addition, the derivatives of GTF with respect to the GTF exponent α and x component of GTF center are given as

{

∂ χ = (x − X )l + 2 ( y − Y )m (z − Z )n + (x − X )l ( y − Y )m+ 2 (z − Z )n + (x − X )l ( y − Y )m (z − Z )n+ 2 ∂α

[ {

× exp − α (x − X ) + ( y − Y ) + ( z − Z ) 2

2

2

}],

{

∂ χ = 2α (x − X )l +1 ( y − Y )m (z − Z )n + l (x − X )l −1 ( y − Y )m (z − Z )n ∂X

[ {

× exp − α (x − X ) + ( y − Y ) + ( z − Z ) 2

2

2

}].

}

(21)

} (22)

12

Taro Udagawa and Masanori Tachikawa

The derivatives with respect to the y and z components of GTF center are also can be expressed in the similar way to that of x axis. By using FVMO method, we can determine the “unknown” optimum parameters such as GTF exponent and GTF center for nuclear GTFs variationally. C. Configuration interaction (CI) Møller-Plesset perturbation theory treatment of MC_MO method We have introduced that the total wavefunction of the multi-component system is given in the CI formalism, Eq. (7), and have derived the Hartree-Fock equation by treating only the first term of Eq. (7). The CI matrix element is calculated by modification of the GUGA technique for the multi-component system as M

H LL ' = ∑ Φ

HI Φ

I LI

M

I L 'I

I

∏δ K ≠I

M

LK , L 'K

+ ∑ Φ ILJ Φ ILI VIJ Φ IL 'I Φ LJ ' J I >J

M

∏δ

K ≠I ,J

LK , L ' K

,

(23)

where M is a number of kinds of quantum particles, the first term denotes the contribution of intra-correlation and the second term denotes that of intercorrelation. After diagonalization of Eq. (23), one obtains the CI coefficient and its energy expressed as IMO M ⎛ IMO M IMO JMO I (φiφ j φkφl )l ⎞⎟⎟ + ∑ ∑ ∑ ΓijklIJ (φiφ j φkφl )IJ ECI = ∑ ⎜⎜ ∑ γ ijI hijI + ∑ Γijkl i , j ,k ,l∈I I ⎝ i , j∈I ⎠ I > J i , j∈I k ,l∈J ,

where

γ ijI

and

I Γijkl

(24)

are the one- and two-body reduced density matrices (RDM)

in the MO base and

hijI

and

(φ φ i

φ k φl )

I

j

are the one- and two-particle MO

integrals of the I th and J th kinds of particle, and IMO and JMO are the number MOs of the I th and J th kinds of particle, respectively.

E The first derivative of CI with respect to the parameter Ω (e.g., GTF exponent and GTF center) is expressed as

Multi-Component Molecular Orbital Methods

(

M ⎛ IAO IAO ∂ I I (Ω ) I ECI = ∑ ⎜⎜ ∑ γ μν hμν φ μφν φ ρφσ + ∑ Γμνρσ ∂Ω I ⎝ μν ∈I μνρσ M

I > J μν ∈I

where

I γ μν

(AO) basis,

and

I Γμνρσ

∑ Γμνρσ (φμφν φρφσ ) ρσ

I Ω)

−

IAO

Wμν S μν( ∑ μν I

∈I

I Ω)

⎞ ⎟ ⎟ ⎠

IJ (Ω )

IAO JAO

+∑ ∑

)(

13

IJ

∈J

,

(25)

are the one- and two-particle RDM in the atomic orbital

I (Ω ) I (Ω ) S μν hμν

,

( ( φφ φφ ) , and

I Ω)

μ ν

ρ σ

are the skeleton derivatives[58] of

overlap, one-, and two-particle integrals of the I th kind particle. IAO is the numbers of AOs, and as

WμνI

is the energy-weighted density matrix which is given

IMO

WμνI = ∑ C μI i CνIj xijI ij∈I

.

(26)

Note that Eq. (25) also satisfies the multi-configuration self-consistent field (MCSCF) wavefunction as well as the full-CI wavefunction. Next, we introduce the second-order Møller-Plesset (MP2) perturbation theory of MC_MO method (MC_MO-MP2). Since MC_MO method treats light nuclei as quantum mechanically, we must consider three types of the MP2 (2 )

(2 )

E E energies, electron-electron ( (e−e ) ), electron-nucleus ( (e− p ) ), and nucleusE (2 )

nucleus ( ( p − p ) ). The explicit formulae of these schemes at closed shell system are given as

ij ab ijab ε i + ε j − ε a − ε b

E((e2−)e ) = ∑

E((e2−) p ) = ∑

iapp '

2 ip ap'

,

(27)

2

ε i + ε p − ε a − ε p'

,

(28)

14

Taro Udagawa and Masanori Tachikawa

E((p2−) p ) =

∑

pqp 'q '

pq p' q' (2 p' q' pq − p' q' qp ε p + ε q − ε p' − ε q'

) ,

(29)

where i, j and a, b indices refer to occupied and unoccupied electronic orbitals, and p, q and p ' , q ' are those for nuclear orbitals, respectively. We have implemented these schemes of MC_MO-CI and also MC_MO-MP2 to our original program FVOPT[48,49,54]. Of course, the FVMO procedure is available for these MC_MO-CI and MC_MO-MP2 methods. D. Multi-component “hybrid” density functional theory. Recently, we have developed the multi-component “hybrid” type of density functional theory (MC_MO-(HF+DFT))[64] by combining the multi-component technique and density functional theory (DFT). As mentioned above, our MC_MO has been already extended to Møller-Plesset perturbation and configuration interaction methods beyond the mean field approximation and several groups have also studied such non-BO treatment based on this MC_MO concept to evaluate the many-body effect. However, the disadvantages of enormous computational costs are still remained in these correlated multicomponent methods. According to this background, we focus on DFT[65,66] which provides quantitative results with reasonable computational costs. In particular, recent hybrid functionals[67-69] have provided a significant contribution for analyzing many chemical phenomena, such as molecular geometries, vibrational frequencies, and single-particle properties within chemical accuracy. Although DFT had several problems for van der Waals interaction energies, Tsuneda and Hirao[70-72] have already reproduced such weak interaction by modification of the exchange-correlation functional. As mentioned above, we again note that DFT will be more and more promising and powerful tool by the suitable density functionals. Several studies of MC_DFT have been already reported[33-35,73]. In 1982, the concept of MC_DFT was first presented by Capitani and Parr, et al.[73], who have established the existence theorem of Hohenberg-Kohn for multi-component system. Shigeta and coworkers[33,34] have shown the non-BO-DFT scheme based upon the real-space grid method, and applied to hydrogen molecule and its isotopomers. They have concluded that even though the electron-nucleus and

Multi-Component Molecular Orbital Methods

15

nucleus-nucleus correlation terms are completely neglected, isotope effects among these species can be discussed at least qualitatively. Kreibich and Gross[35] have also presented MC_DFT where electron-nucleus correlation term was approximation by the classical electrostatic interaction of the corresponding mean field charge distributions. To our knowledge, however, there are no reports of multi-component “hybrid” DFTs, despite a remarkable success of the conventional hybrid density functionals. First, let us introduce the concept of hybrid density functional theory briefly. Nowadays, several types of hybrid functionals are proposed. In particular, “Becke’s half and half” (BHandH) concept[67] is one of the most popular schemes, where the integrand of 1

λ E XC = ∫ U XC dλ 0

(30)

is approximated to

1 0 1 E XC ≅ U XC + U 1XC 2 2 ,

(31)

λ U XC is the potential energy of U0 exchange-correlation at intermediate coupling strength λ , XC is the exchange-

using a linear interpolation. In Eqs. (30) and (31),

correlation potential energy of the noninteracting reference system, and

U 1XC is

0 U XC is the exchange-correlation U0 potential energy of the noninteracting reference system, XC can be replaced by

that of the fully-interacting real system. Since

the Kohn-Sham (KS) exchange energy E X ,

E XC ≅

1 1 LSDA E X + U XC 2 2 .

(32)

At the practical point of view, Hartree-Fock orbital is used for the E X term in Eq. (32) instead of KS orbital. Becke has also developed the “Becke’s threeparameter hybrid functional” (B3)[68] by relaxing the linear

λ dependence of

16

Taro Udagawa and Masanori Tachikawa

the “half and half” theory. Today, B3LYP has become the most popular hybrid exchange-correlation functional. Next, we would extend our MC_MO scheme to multi-component hybrid density functinonal theory (MC_MO-(HF+DFT)) by combining the conventional DFT procedure. In our MC_MO-(HF+DFT) approach, the Kohn-Sham operators for electron and quantum nuclei are derived by adding the hybrid type exchangecorrelation potentials to Fock operators of MC_MO-HF scheme as Ne

Np

e

p

Np

Ne

p

e

( HF + DFT ) + DFT ) f e( HF + DFT ) = he + ∑ J e − ∑ J p + VXC + VC( HF ( e −e ) (e − p )

( HF + DFT )

fp

where

,

(33)

,

(34)

( HF + DFT ) ( HF + DFT ) = h p + ∑ J p − ∑ J e + V XC ( p − p ) + VC (e− p )

( HF + DFT ) V XC ( e −e )

( HF + DFT ) V XC = ( e −e )

is given as ( HF + DFT ) δE XC ( e −e ) δρ .

(35)

K

At this stage, the exchange term between two quantum nuclear particles p in Eq. (14) is ignored, and we take account of only electron-electron correlation by using conventional functionals because it accounts a major part of many-body

V ( HF + DFT )

V ( HF + DFT )

in Eqs. (33) effect. This corresponds to the neglect of XC ( p − p ) and C (e− p ) and (34), so the quantum nuclear KS operator of MC_MO-(HF+DFT) in Eq. (34) has a similar manner of the quantum nuclear Fock operator of MC_MO-HF in Eq. (14). Hereafter, thus we will denote the

( HF + DFT ) V XC (e −e )

in Eq. (33) as

N

( HF + DFT ) V XC .

e electrons and a For simplicity, let us consider the system containing proton. The energy of this system after integration of the spin coordinates in the framework of MC_MO-(HF+DFT) is given as follows:

Multi-Component Molecular Orbital Methods ( HF + DFT )

Etotal

Ne / 2

Ne / 2

i

ij

= 2 ∑ hii +

∑ 2J

( HF + DFT )

ij

+ E XC

17

Ne / 2

+ h pp − 2 ∑ J ip i

,

(36)

E ( HF + DFT )

the exchange-correlation energy contribution by hybrid type of where XC exchange-correlation functional. We have developed and implemented the two MC_MO-(HF+DFT) versions of BHandHLYP (MC_BHandHLYP) and B3LYP (MC_B3LYP) functionals by combining the MC_MO technique with conventional BHandHLYP and B3LYP hybrid density functionals, BHandHLYP E XC = 0.5E XHF + 0.5(E XSlater + ΔE XB 88 ) + ECLYP ,

(37)

B 3 LYP E XC = 0.2 E XHF + 0.8E XSlater + 0.72ΔE XB 88 + ECLYP + 0.81ΔECLYP .

(38)

HF

The notation E X

Slater

is Hartree-Fock exchange, E X

the Slater exchange

ΔE LYP

C the Becke’s gradient correction, and and the functional, ΔE local- and non-local correlation provided by the LYP expression, respectively. To optimize the molecular geometry, we have also implemented the first order energy derivative of Eq. (36) with respect to classical nuclear coordinate B 88 X

E

LYP C

“ R ”. The explicit formulae is as follows Ne / 2 Ne / 2 Ne / 2 ∂ ( HF + DFT ) (R ) Etotal = 2 ∑ hii( R ) + 2 ∑ J ij( R ) + h pp − 2 ∑ J ip( R ) ∂R i ij i Ne / 2 ∂ ( HF + DFT ) (R ) + E XC − 2 ∑ S ii( R )ε i − S pp εp ∂R i .

(39)

We have implemented these schemes to GAUSSIAN03 program package[53]. We mention that non-hybrid functionals (e.g., BLYP, BOP, etc.) are, of course, usable in the GAUSSIAN03 modified for our MC_MO-(HF+DFT) scheme. We will show several applications of our multi-component methods in the next chapter.

18

Taro Udagawa and Masanori Tachikawa

2-3. OTHER GROUP’S APPROACH Nakai and coworkers[21-24] have also developed the multi-component method which is named as nuclear orbital and molecular orbital (NOMO) methods. Original NOMO method is equivalent to our multi-component molecular orbital (MC_MO) method. They have extended NOMO method to configuration interaction single theory[21], many-body perturbation theory[23], and coupled-cluster method[23] in a similar way of extends the conventional HF method to these methods. Recently, they have developed the translation- and rotation-free (TRF) NOMO method by eliminating translational and rotational contributions from NOMO Hamiltonian. In the non-BO theory, Gauss type function (GTF) is frequently used as nuclear basis function. The GTF is proper to represent a vibrational motion, while it is not proper to represent translational- and rotational motion. They have proposed a scheme to eliminate the effect of the translational motion from the NOMO calculation. The Hamiltonian of the translational motion is the kinetic term of the center of mass which is given by

TT (x ) = −

1 2M

∑μ ∇(x μ )

2

−

1 M

∇(x μ ) ⋅ ∇(xν ) ∑ μ ν <

,

(40)

where M is the total mass of all particles. The translational-free (TF) Hamiltonian is constructed by subtracting these operators from total Hamiltonian. The calculated translation-free energy is used for the discussion of molecular vibrations, while previous non-BO works used the translation-contaminated energy for discussion. They have also developed the translational- and rotational-free (TRF) NOMO method by subtracting the rotational contribution from TF Hamiltonian. Rotational motion is difficult to separate from Hamiltonian because rotational motion and vibrational motion are coupled each other. Thus to separate rotational motion from Hamiltonian requires the iterative procedure. The Hamiltonian of the rotational motion is given by

TR =

Lα2 ∑ α 2Iα ,

x, y, z

(43)

Multi-Component Molecular Orbital Methods

19

α component of the total angular momentum operator, which is

L

where α is the expressed as

Lα = ∑ Lα , μ μ

.

(44)

Therefore, Eq. (43) is rewritten by

TR =

x, y,z

1 ⎛

⎞

⎜ ∑ Lα μ + 2∑ Lα μ Lα ν ⎟ ∑ ⎟ ⎜ 2 I α μ μ ν α

2

,

⎝

,

<

,

⎠.

By introducing the equilibrium coordinate

(45)

x 0μ = x μ0 , y μ0 , z μ0

(

) and expand

Δx μ

, Eq. (45) can

the denominator in Eq. (45) in a Taylor series with respect to be rewritten by

⎛ 0 2 2⎞ 2 ⎜ Lα , μ + L0α , μ ΔLα , μ + ΔLα ,μ L0α , μ − 0 ΔI α , μ L0α , μ ⎟ ⎜ ⎟ Iα ⎝ ⎠ x, y,z 2 1 − ∑ ∑ 0 L0α ,μ L0α ,ν − L0α ,μ ΔLα ,μ − ΔLα ,μ L0α ,ν − ΔLα ,μ L0α ,ν μ <ν α I α

x, y,z

TR = ∑ ∑ μ

α

1 2 I α0

2

− ΔIα ,μ L0α ,μ − −

(

(

2 ΔIα ,μ Lα0 ,μ Lα0 ,ν − ΔIα Lα0 ,μ L0α ,ν Iα0 x, y ,z

∑ ∑ μ ν μ λν λ α I ≠ , ≠ , <

2 02

α

)⎞⎟⎟ ⎠

( )

ΔIα ,μ Lα0 ,ν L0α ,λ + O Δx 2

.

(46)

By subtracting the respective part of rotational Hamiltonian from that of TF Hamiltonian, the TRF Hamiltonian is derived. They have applied TRF-NOMO method to series of H2 isotopomers and H2O molecule and confirmed that the energy was considerably improved by eliminating the translational and rotational contamination. Thus they suggested the TRF-NOMO method is required for high accuracy in the non-BO calculations. Hammes-Schiffer and coworkers[25-32] also have developed the non-BO theory, which is called as “Nuclear electronic orbital (NEO)” method. They have

20

Taro Udagawa and Masanori Tachikawa

implemented the NEO-multiconfigurational self-consistent-field (MCSCF) [25], NEO-MP2[30], NEO-nonorthogonal configuration interaction (NOCI)[31], and NEO-explicitly correlated Hartree-Fock (XCHF)[32] schemes and mainly analyzed the hydrogen/proton transfer systems. They have successful reproduced the delocalized nature of nuclear wavefunctions in hydrogen/proton transfer systems by using NEO-state averaged MCSCF and NEO-NOCI methods. In addition, they have presented the methodology for a vibrational analysis within the NEO framework. Adamowicz and coworkers[36-46] calculated several small molecules by nonadiabatic calculation with explicitly correlated basis set. They have proposed a new explicitly correlated basis set for performing very high accuracy nonadiabatic variational energy calculations on small “diatomic” systems. Their basis functions are of the form,

[

]

φk = r1m exp − r ' Ak r . k

(48)

These basis functions are correlated Gaussians with a premultiplying factor mk 1

r

, the m th power of the distance between the two heavy particles. The exponential factor is the simple correlated Gaussian, and provides full correlation mk

between all system particles. A premultiplying factor r1 eliminates or at least drastically reduces the linear dependence problems. They have reported the really high-level results which is the ground-state energy upper bound for H2, LiH, and LiD by nonadiabatic calculation with their explicit correlated basis sets. In addition, they have reported the calculated physical properties such as electron affinities and polarizabilities of these molecules within excellent accuracy.

Chapter 3

APPLICATION OF THE MULTI-COMPONENT METHODS In this chapter, we will present several applications of MC_MO method. The MC_MO methods have advantages compared with conventional MO methods. Since the nuclear kinetic terms contribution is positive quantity, the total energy obtained with MC_MO method is higher than that with conventional MO methods. However, this MC_MO ground-state energy is closer to that of real system than that of conventional MO methods. Since the difference of nuclear mass is directly reflected in MC_MO Hamiltonian through the nuclear kinetic term, we can easily analyze the isotope effect of given molecule which is difficult to analyze by using conventional MO methods. In addition, MC_MO method is also usable for positron (e+) systems. A positron is an antiparticle of an electron. Recently, many experimental and theoretical researchers have been interested in systems containing antiparticles[74-76]. When a positron injected into some species, pair annihilation between positron and electron occurs at the final step of reactions. In some species, however, a stable system containing a positron, which is called ‘positronic compounds’ is detected experimentally[77,78]. However, positronic and electronic structures of positronic compounds have yet been made clear neither from experiments nor theories. Some theoretical researchers have studied the possibility various methods[7984]. These studies showed that the positron orbital is more diffusive than the occupied electron orbitals because of the repulsive nature between the positron and the nuclei. It is necessary to add diffuse functions to express the positronic orbital adequately. Such positronic exponents of the diffuse functions, however,

22

Taro Udagawa and Masanori Tachikawa

were determined artificially. On the other hands, positronic exponents could be determined variationally by using our FVMO technique. We would present three applications of MC_MO method; (A) Application of the full-CI MC_MO method to the positron-molecule complexes, (B) An analysis of the isotope effects on hydrogen molecules by using full-CI MC_MO calculation, and (C) H/D isotope effect on porphine and porphycene molecules revealed by MC_DFT method.

3-1. APPLICATION OF THE FULL-CI MC_MO TO THE POSITRON-MOLECULE COMPLEXES WITH FVMO PROCEDURE The purpose of this research is to determine the best wavefunction of ground and excited states for positronic compounds, and to study the influence and convergence of the electron-positron correlation effect under the one-particle MO scheme with Gaussian type function (GTF). The GTF basis has been known to possess an advantageous feature in calculation of integrals, particularly for manyelectron systems. Table 1. Optimized energy and energy components of H- species with various basis functions. All units are in hartree. [6s ] Total energy (V/T) + 2 Kinetic components Te

[6s 1p ]

[6s 2p ]

[6s 3p ]

[6s 2p 1d ]

[6s 3p 1d ] [6s 3p 1d 1f ][6s 3p 2d 1f ]

-0.5140520 -0.5235361 -0.5257728 -0.5261440 -0.5261783 -0.5269229 -0.5267294 -0.5272624 8.52E-08 -6.65E-08 2.59E-08 2.00E-10 -1.16E-07 -7.86E-08 -1.61E-07 6.26E-09

0.5140520

0.5235361

0.5257728

0.5261440

0.5261783

0.5269229

0.5267294

0.5272624

Potential components Vee 0.3000551 0.3107114 Vne -1.3281590 -1.3577837 Most accurate energy is -0.527751 hartree [85]

0.3107560 -1.3623016

0.3106452 -1.3629332

0.3116162 -1.3639728

0.3110740 -1.3649199

0.3121190 -1.3655778

0.3111170 -1.3656418

We have chosen H- and [H-;e+] species as a pilot subject of this study. In the [H-;e+] species, we have calculated ground and low-lying positronic excited state, which we hereafter express as [H-;e+(1s)] and [H-;e+(2p)], respectively. We obtain the optimum full-CI wavefunctions for each state with state specific approach, since these states are spatially orthogonal to each other. In addition, we have also calculated a positron-molecular system, [OH-;e+] with the MC_MO-HF FVMO method. Since the optimization of all variational parameters is carried out in

Application of the Multi-Component Methods

23

multi-dimensional hyper energy surface, we might encounter the problem of multiple energy minima. Although we have found that this problem scarcely occurs and have met the difficulty of convergence, we have checked by using the different initial conditions for the GTF exponents, and have gotten the same result in most cases. 3-1-1. H- system In the H- system all configurations of single and double electronic excitations are taken into account. We have calculated these systems with [6s], [6s1p], [6s2p], [6s3p], [6s2p1d], [6s3p1d], [6s3p1d1f], and [6s3p2d1f] basis sets, in which all exponent values are optimized under full-CI treatment. It should be addressed that several standard basis sets, which have been determined by optimization of energy of atoms or tiny molecules, are not adequate for the electron- or positron-attached compounds. Table 1 summarizes the total energy, virial ratio, and energy components of kinetic ( T ) and potential ( V ) contributions for H- system with various optimum basis sets. The total potential energy ( V ) is divided into the electronic-electronic

V

V

( ee ) and nuclear(protonic)-electronic ( ne ) potential parts. The virial ratios are calculated to be very close to two within the accuracy of 10-6 for all cases, which confirm the reliability for the relation between energy components, T and V . As the number of basis function increases, the energy of full-CI rapidly converges toward the most accurate value, -0.5277510 hartree[85]. The difference between our [6s3p2d1f] best energy and the most accurate one is only 0.00049 hartree. The energy with the conventional primitive aug-cc-pVQZ [7s4p3d2f] is calculated as 0.5271692 hartree, which is slightly higher than our [6s3p2d1f] energy. The kinetic and potential energy components also rapidly converge in Table 1. 3-1-2. [H-;e+(1s)] For [H-;e+(1s)] species, which consists of two electrons and one positron, all configurations of single and double electronic excitations, single positronic excitations, single electronic-single positronic excitations, and double electronicsingle positronic excitations are taken into account in our calculation. The results of ground state of [H-;e+(1s)] species are tabulated in Table 2, in which the total kinetic energy ( T ) are divided into the electronic (

Te ) and

24

Taro Udagawa and Masanori Tachikawa

positronic (

Tp

) kinetic components, and the total potential energy ( V ) into the

electronic-electronic

V

(

Vee ), electronic-positronic ( Vep ), nuclear(protonic)Vnp

) potential parts, electronic ( ne ), and nuclear(protonic)-positronic ( - + respectively. The energies of [H ;e ] with [6s] GTFs under full-CI and MC_MOHF FVMO treatments are -0.6910097 and -0.666944[45] hartree, respectively, in which correlation energy is quite small due to the small basis function space. Adding 1p, 2p, and 3p GTFs to [6s] basis set as polarization functions and optimizing these exponent values, [H-;e+] energy lowers 0.041, 0.052, and 0.055 hartree, respectively, from the energy with [6s] GTFs. We have employed further calculation with [6s2p1d], [6s3p1d], [6s3p1d1f], and [6s3p2d1f] GTFs, in order to extend the space for explicit repulsion between two electrons and explicit attraction between electron and positron. We have also calculated with fixed basis set of the conventional primitive aug-cc-pVQZ [7s4p3d2f], which exponent values are not scaled for electron and positron basis sets, since diffuse functions are already included. The energy with the conventional primitive aug-cc-pVQZ [7s4p3d2f] is calculated as -0.7663716 hartree, which is 0.0028 hartree higher than our [6s3p2d1f] energy and 0.02 hartree higher than the Hylleraas CI one. It is noted that though the conventional basis set gives poor results for [H-;e+(1s)] system compared to H- system, our basis set gives improvement for positronic compound. Table 2. Optimized energy and energy components of [H-;e+(1s)] species with various basis functions. All units are in hartree. [6s 2p 1d ]

[6s 3p 1d ] [6s 3p 1d 1f ] [6s 3p 2d 1f ]

-0.6910097 4.49E-06

[6s ]

-0.7321757 5.46E-07

-0.7433360 1.48E-06

-0.7458065 1.50E-09

-0.7564082 2.64E-07

-0.7589646 -1.58E-06

0.6220029 0.0690099

0.6376979 0.0944781

0.6456695 0.0976675

0.6475726 0.0982338

0.6470537 0.1093546

0.6490069 0.1099566

0.6493367 0.1155044

0.6508235 0.1183408

0.4238378 -0.5972562

0.4180253 -0.7068657

0.4098066 -0.7223421

0.4076584 -0.7251449

0.4060305 -0.7618247

0.4040544 -0.7648399

0.4009470 -0.7816825

0.3981662 -0.7917504

Vne -1.5407249 -1.5372510 Vnp 0.3321208 0.3617398 Most accurate energy is -0.7891944 hartree [85]

-1.5373569 0.3632195

-1.5372220 0.3630956

-1.5239146 0.3668924

-1.5238464 0.3667037

-1.5151168 0.3661730

-1.5110704 0.3663229

Total energy (V/T) + 2

[6s 1p ]

[6s 2p ]

[6s 3p ]

-0.7648382 -0.7691673 3.79E-06 -4.0303E-06

Kinetic components Te Tp Potential components Vee Vep

In order to see a slow convergence of the energy, we have analyzed some energy components in Table 2. Showing the kinetic components, the positronic

Application of the Multi-Component Methods

25

kinetic energy with [6s3p2d1f] GTF is twice as large as that with [6s] GTF, that is, the convergence of positronic kinetic energy is slower than that of electronic one. Showing the potential components of [H-;e+], one-particle potentials,

V

V

Vne V

and np , converge much more rapidly than two-particle potentials, ep and ee , as the basis set size increases. Especially, the convergence of the electronic-

V

positronic component ep is slowest among them. Our result with a slow convergence of the basis set size indicates that an insufficient part of the electronic-positronic correlation energy has been obtained, even though with the optimized basis functions under the full-CI multi-component treatment. Much higher polarized orbitals may be required for the [H-;e+] case for more sufficient convergence under one-particle basis function scheme We analyze the CI coefficients of both H- and [H-;e+] systems with [6s3p2d1f] basis set. The coefficient of the Hartree-Fock configuration of H- and [H-;e+] systems are 0.977 and 0.905, respectively. The contribution of HartreeFock configuration in [H-;e+] is much smaller than that in H- species, which demonstrates the single determinant approximation for [H-;e+] is not more suitable than that for H- case. In the [H-;e+] system the coefficient of single-positron excitation configuration of [1s2:2s1], which refers to the configuration of electronic 1s2 and positronic 2s1, has a large contribution of -0.114. The coefficients of the configurations of single-electron and single-positron excitations to polarized orbitals, such as [1s12p1:2p1], have also large values as 0.140. These configurations are required for expression of the positronium contribution in oneparticle MO scheme, where positronium is a bound state of a positron and an electron. Table 3. Optimized energy and energy components of [H-;e+(2p)] species with various basis functions. All units are in hartree. Total energy (V/T) + 2 Kinetic components Te Tp Potential components Vee Vep Vne Vnp

[6s 1p ]

[6s 2p ]

[6s 3p ]

[6s 4p ]

-0.6231852 -1.60E-07

-0.6365852 1.57E-07

-0.6389969 -1.56E-07

-0.6395910 1.50E-09

[6s 2p 1d ] [6s 3p 1d ] [6s 3p 1d 1f ][6s 3p 2d 1f ] -0.6569636 3.04E-07

-0.6598826 6.06E-07

-0.6709070 -4.47E-07

-0.6763597 -3.55E-06

0.5439846 0.0792005

0.5550011 0.0815842

0.5570226 0.0819742

0.5577323 0.0818587

0.5572846 0.0996793

0.5593886 0.1004943

0.5627366 0.1081700

0.5647470 0.1116103

0.3508163 -0.3796974 -1.4086580 0.1911688

0.3399602 -0.4179832 -1.4021247 0.2069772

0.3375813 -0.4215960 -1.4020805 0.2081015

0.3369869 -0.4220047 -1.4022228 0.2080586

0.3241460 -0.4962372 -1.3663189 0.2244825

0.3221514 -0.5015595 -1.3658845 0.2255272

0.3067830 -0.5373851 -1.3387480 0.2275366

0.3025176 -0.5537491 -1.3319425 0.2304570

26

Taro Udagawa and Masanori Tachikawa 3-1-3. [H-;e+(2p)]

Kurtz and Jordan[80] calculated the positronic excited state of [H-;e+(2p)] with Hartree-Fock approximation as -0.588 hartree. We could not find any accurate calculation for [H-;e+(2p)] bound state system. Table 3 shows the results of [H-;e+(2p)] species with our full-CI multi-component FVMO scheme. Adding 2p, 3p, and 4p GTFs to [6s] basis set and optimizing these exponent values, [H;e+(2p)] energy lowers 0.013, 0.016, and 0.016 hartree, respectively, from 0.6231852 hartree with [6s1p] GTFs. The energy of the conventional primitive aug-cc-pVQZ [7s4p3d2f] GTFs is calculated as -0.6340616 hartree, which is 0.042 hartree higher than our energy with only [6s3p2d1f] GTFs. It should be addressed that the conventional basis set gives poor results for positronic-excited state [H-;e+(2p)] system, compared to H- and ground state [H-;e+(1s)] ones, and our basis set gives good improvement for positronic compound. Some energy components are also shown in Table 3. The convergences of two-particle potentials, and

Vnp

Vep

and

Vee are much slower than those of one-particle potentials, Vne

, as the basis set size increases. Especially, the convergence of the

electronic-positronic component

Vep

is slowest among them.

Table 4. Optimized basis functions for H- and [H-;e+] systems with [6s3p1d] GTFs. H－

[H－;e+(1s )]

Electron

Electron Exponent

[H－;e+(2p )] Positron

Population

Exponent

Positron

Electron

Exponent

Population

Exponent

Population

Exponent

1s 2s 3s 4s 5s 6s

59.617 8.9963 2.0681 0.52597 0.062847 0.014893

0.000 0.007 0.069 0.554 0.673 0.418

66.011 10.002 2.3585 0.73659 0.26847 0.0493777

0.000 0.007 0.061 0.235 0.795 0.517

8.9346 1.5904 0.51619 0.11092 0.0524718 0.0245624

Population

0.000 -0.003 -0.017 0.248 0.507 0.151

39.909 6.0215 1.3866 0.39899 0.12515 0.026618

0.001 0.014 0.124 0.425 0.745 0.300

3.7301 0.65793 0.19757 0.092797 0.049211 0.021165

Population

0.000 -0.001 -0.005 -0.082 0.238 0.044

1p 2p 3p

1.0060 0.27688 0.079031

0.002 0.012 0.012

0.67771 0.19922 0.0753036

0.007 0.041 0.061

0.47831 0.18726 0.0697295

-0.001 0.031 0.062

0.45503 0.11310 0.043333

0.008 0.048 0.140

0.16282 0.066130 0.024500

0.005 0.373 0.443

1d

0.29382

0.254

0.17232

0.278

0.15439

0.022

0.076506

0.195

0.067043

-0.015

3-1-4. Optimized basis functions Finally, we show the optimized exponents and populations of [6s3p1d] GTFs for H-, [H-;e+(1s)], and [H-;e+(2p)] systems in Table 4. The electronic exponents of s-type GTFs for the [H-;e+(1s)] system are a little larger than those for the H-,

Application of the Multi-Component Methods

27

while the positronic exponents are smaller than electronic ones. This means that the positronic MO spreads out in comparison with the electronic MOs due to the repulsion between a proton and a positron. The effect of the positron is to contract the inner electronic clouds slightly. According to the contraction of electronic

Vee ) of [H-;e+(1s)] is more repulsive V than that of H-, while the nuclear(protonic)-electronic potential ( ne ) of [H-

clouds, the electronic-electronic potential (

;e+(1s)] is more attractive than that of H-, as shown in Tables 1 and 2. Comparing the exponent values of [H-;e+(1s)] and [H-;e+(2p)] systems in Table 4, both electronic and positronic GTF exponents of [H-;e+(2p)] are smaller than those of [H-;e+(1s)]. Since the positronic orbital of excited [H-;e+(2p)] is more

V

spread than that of ground [H-;e+(1s)], the ne contribution of [H-;e+(2p)] becomes smaller than that of [H-;e+(1s)], as shown in Tables 2 and 3. According to the spread of the positronic orbital of [H-;e+(2p)], the electronic orbital of [H;e+(2p)] is also more spread than that of ground state of [H-;e+(1s)]. Then, the

Vee

contribution of [H-;e+(2p)] becomes smaller than that of [H-;e+(1s)], and the of [H-;e+(2p)] becomes more repulsive than that of [H-;e+(1s)].

Vne

3-1-5. Positron-molecular system [OH-;e+] We show the results of the [OH-;e+] system as the positron-molecular complex with MC_MO-HF FVMO method. In this calculation the distance between the hydrogen and oxygen nuclei of OH- is fixed at 1.822 bohrs from the measurements of Schulz et al.[86], and we placed them at (0.0, 0.0, -1.61955) and (0.0, 0.0, 0.20245) bohr in three-dimensional space. At first we optimized only

{α

e

,α p

}

{R , R } e

p

i i fixed orbital exponents i ' i under the condition of orbital centers on each nucleus (Table 5), and, second, the orbital centers and the orbital exponents of the electronic and positronic basis sets are optimized together (Table 6). In Table 5 the initial and optimized exponents of the basis sets for OH- and [OH-;e+] are shown. We have used the STO-3G primitive set with the orbital centers fixed on hydrogen and oxygen nuclei while the exponents of electronic and positronic basis sets are optimized. In the electronic results of OH- and [OH-

;e+], the exponents of hydrogen basis functions and oxygen

χ 1s , χ 2 s , χ 3 s , χ 4 s

28

Taro Udagawa and Masanori Tachikawa

and especially the

χp

z

increased, while oxygen

is caused by the fact that the

χp

z

χ3 p

and

x

χ3 p

y

decreased. This

are used for the bonding pair between hydrogen

χ

χ

and oxygen nuclei, while the 3 px and 3 p y for the lone pairs on oxygen atom. On the other hand, the exponents of the positronic basis set in [OH-;e+] become small and delocalize over the whole molecule. The optimum exponents of positronic basis set on hydrogen nucleus are much larger than those of [H-;e+][48]. Table 5. Optimum exponents of 6s3p (STO-3G primitive) basis set for OH and [OH-;e+] systems with orbital centers {Rie,Rip} fixed on each nucleus. -

OHType H χ1s χ2s χ3s

Initial

χ1s χ2s χ3s χ4s χ5s χ6s χ1pz χ1px, χ1py χ2pz χ2px, χ2py χ3pz χ3px, χ3py E HF (a.u.) Virial ratio

Final

3.42525 → 0.623914 → 0.168854 →

O 130.709 23.8089 6.44361 5.03315 1.16960 0.380389 5.03315 5.03315 1.16960 1.16960 0.380389 0.380389 -74.413977 2.0158189

-

[OH-;e+] Final (e-)

+

Initial (e and e )

7.31507 0.996781 0.197687

862.020 → 129.948 → 29.1523 → 7.78716 → 0.850538 → 0.202437 → 11.1070 → 6.46925 → 2.35725 → 1.29566 → 0.586815 → 0.246843 → → -75.177075 → 1.9999948

3.42525 → 0.623914 → 0.168854 → 130.709 23.8089 6.44361 5.03315 1.16960 0.380389 5.03315 5.03315 1.16960 1.16960 0.380389 0.380389 -74.426856 2.0101077

→ → → → → → → → → → → → → →

7.46497 1.03835 0.206552

Final (e+) 2.54078 0.300747 0.0765415

864.819 12.3492 130.366 1.230696 29.2455 0.428886 7.81170 0.0768184 0.873756 0.0369078 0.216166 0.0171953 9.88002 1.07887 6.62172 5.03315 2.07786 0.111166 1.33018 1.16960 0.505416 0.0327949 0.257252 0.380389 -75.361417 2.0052047

Next we show in Table 6 the results of optimization of both orbital centers and orbital exponents. The orbital centers are expressed as the displacements from original nuclear positions in bohr. Schematic illustrations of the initial and the optimum basis sets of [OH-;e+] system are also shown in Figure 1 in which only the 3s2p (

χ 4s , χ 5s , χ 6s, χ 2 p ,

and

χ3 p

) basis sets of oxygen atom and the 2s

χ 2 s and χ 3 s ) of hydrogen atom are expressed in arbitrary units. The electronic χ basis functions of the hydrogen atom of OH-, particularly hydrogen 3 s , move

(

Application of the Multi-Component Methods

29

toward the oxygen nucleus by the orbital center optimization, and the exponents change little compared with those of OH- (Table 5). These hydrogen basis functions are mainly used for expressing the σ orbital of this system. On the other

χ

χ

hand, the px and p y type of basis functions used to express the lone pairs. As expected from the difference of electronegativity between hydrogen and oxygen, all basis sets tend to move toward the oxygen atom. Thus we have predicted that the positronic orbital moves toward the oxygen nucleus, because of the attraction between the positron and the electrons gathering near the oxygen atom. In fact, Figure 1 shows that the positronic orbital not only becomes more diffusive but also moves toward the oxygen atom. Table 6. Optimum centers and exponents of 6s3p/3s (STO-3G primitive) basis set for OH- and [OH-;e+] systems. Type H χ1s χ2s χ3s O χ1s χ2s χ3s χ4s χ5s χ6s χ1pz χ1px, χ1py χ2pz χ2px, χ2py χ3pz χ3px, χ3py E HF (a.u.) Virial ratio

OHCenter Exponent -1.61955 +0.010799 8.08782 +0.118401 1.17179 +0.749584 0.237009 0.20245 -0.000001 867.952 -0.000057 130.834 +0.000029 29.3485 +0.000143 7.83933 -0.196227 0.922243 +0.214614 0.235636 -0.000838 17.1974 -0.001536 6.39218 -0.014867 3.74343 -0.020748 1.27854 -0.034186 1.00764 -0.131813 0.241988 -75.208073 2.0000000

Center (e-) -1.61955 +0.011881 +0.125454 +0.835971 0.20245 -0.000001 -0.000053 +0.000043 +0.000185 -0.184343 +0.255355 -0.000600 -0.001145 -0.014206 -0.017935 -0.029224 -0.088355

[OH-;e+] Exponent (e-) Center (e+) 8.20142 1.19261 0.247801

Exponent (e+)

-0.024118 +0.113473 +1.365186

4.54649 0.676053 0.170910

869.014 +0.093443 130.992 +0.098179 29.3839 +0.137154 7.84851 -0.334620 0.935227 +0.824741 0.243199 +1.495749 17.5995 -0.309754 6.57817 0.000000 3.83690 -0.216695 1.32076 0.000000 1.03775 -0.471549 0.254609 0.000000 -75.392926 1.9999987

3.64601 0.649922 0.177810 0.0490555 0.0230946 0.00862750 1.24373 5.03315 0.168798 1.16960 0.0538371 0.380389

In Table 7 total energies and positron affinities (PAs) of both OH- and [OH;e ] systems are shown. Our total energies with only a 6s3p basis set are within 0.2 a.u. deviation of those by Kao and Cade with the [5s5p2d/3s1p] basis set for the electrons and the [4s4p/2s] basis set for positron[87]. Of course, they calculated OH- and [OH-;e+] with the same basis sets. On the other hand we have optimized the parameters of the basis sets, that is, our PA includes the effect of electronic orbital relaxation explicitly. However, our PA is calculated by HF approximation, so it is less than the values of MP2 of QMC[88] computations. +

30

Taro Udagawa and Masanori Tachikawa

Figure 1. Schematic illustration of the initial and the optimum basis functions of the [OH-;e+] system. Only the 3s2p (

χ

χ 4s , χ 5s , χ 6s , χ 2 p ,

and

χ3 p

) basis sets of the oxygen

χ

atom and the 2s ( 2 s and 3 s ) basis sets of the hydrogen atom are shown in arbitrary units. (a) Basis functions for OH- are shown. (b) Electronic basis functions of the [OH-;e+] system are shown. The electronic basis functions of the hydrogen atom move toward the

χ

χp

y basis functions become more diffuse. (c) Positronic oxygen nucleus, while px and - + basis functions of the [OH ;e ] system are shown. The positronic orbital not only becomes more diffuse but also moves toward the oxygen atom.

Table 7. Optimum energies of OH- and [OH-;e+] systems with each basis set and method Method E HF

b

Basis set 6s3p (STO-3G primitive) 6s3p a Kao

OH- (a.u.) -74.413977 -75.208073 -75.41117 -75.61845

[OH-;e+] (a.u.) -74.426856 -75.392926 -75.58764 -75.81439

MP2 c QMC a [5s5p2d/3s1p] for electron and [4s4p/2s] for positron [87]. b (13s7p3d/6s3p) contracted to [6s5p3d/4s3d] for electron and positron [97]. c Ref. [88]

PA (eV) 0.350 5.030 4.80 5.332 5.57 ± 0.15

Application of the Multi-Component Methods

31

3-1-6. Comparison of optimum basis function and energies in Li-, [Li-;e+], LiH, and LiD systems. In Table 8 the initial and optimized exponents of a 6s basis set for the Li- and [Li ;e+] systems are shown. Since the initial basis set is the STO-3G primitive set, which is determined by the optimization of the energy of the Li atom, the optimum exponents of Li- significantly differ from the initial ones. The optimized -

χ

χ

χ

exponents of 1s , 2 s , and 3 s in Li- are four to six times larger than the initial values. These exponents are required to express the character of core electrons of

χ

χ

Li-. On the other hand, the exponents of 5 s and 6 s for valence electrons decrease. These results indicate that when an electron attaches to a system, the core electron orbitals shrink while valence electron orbitals become more diffuse. Using this optimum basis set, the HF energy is lowered by 0.113 a.u. from the initial value, and the virial ratio with the optimal basis set is very close to 2. On the other hand, in the [Li-;e+] system the exponents of

χe

χ 1es , χ 2es , and χ 3es

χe

increase, and exponents of 5 s and 6 s decrease. The optimum exponents of the electronic basis set of [Li-;e+] tend to be larger than those of Li-. In particular, the

χe

exponent of 6 s is remarkably large. In contrast, the exponents of the positronic basis set decrease due to the repulsion by Li nucleus. The energy improvement obtained with the optimum exponents is 0.112 a.u. and the virial ratio of [Li-;e+] is very close to 2. As shown above, the basis set of electronic and positronic MOs are determined by using the FVMO method. In the FVMO method it is possible to determine the basis sets of protonic, or nuclear, MOs directly, which are not proposed variationally yet. Table 9 shows the results of the FVMO calculation applied to [Li-;e+], LiH, and LiD systems. The positron or the proton is treated as the quantum wave, while the Li nucleus as the point charge. 6s basis functions are employed for electrons and 1s for the positively charged quantum particles. Since all the exponents and centers of basis functions are optimized, the virial ratios of these species are fairly close to 2. Figure 2 shows the schematic illustration of optimized basis functions for electron and positively charged particle in arbitrary units.

32

Taro Udagawa and Masanori Tachikawa Table 8. Optimum exponents of 6s (STO-3G primitive) basis set for Liand [Li-;e+] systems Li

Type χ1s χ2s χ3s χ4s χ5s χ6s E HF (a.u.) Virial ratio

Initial 16.1196 2.93620 0.794651 0.636290 0.147860 0.0480887 -7.301830 2.0142956

-

→ → → → → → → →

-

+

[Li ;e ] Final 100.566 15.1464 3.32431 0.842991 0.050394 0.0110185 -7.414790 2.0000000

-

+

Initial (e and e ) 16.1196 2.93620 0.794651 0.636290 0.147860 0.0480887 -7.404681 2.0107731

-

→ → → → → → → →

+

Final (e ) Final (e ) 102.79 3.23169 15.4705 0.424169 3.39979 0.104258 0.864439 0.0219853 0.0715922 0.0112615 0.0246763 0.00634184 -7.516885 1.9999999

All the optimized centers of the electronic and positronic basis functions are situated at the Li nucleus in the [Li-;e+] system, and the exponent of the positronic basis function is smaller than any other ones. The result of LiH molecule is significantly different from that of the [Li-;e+] system, i.e., the protonic orbital center shifts from Li nucleus toward the position of the point charge, having a large exponent of 15.97. It is noted that the center of the most diffuse s functions is separated from the Li nucleus by 2.64 bohrs. As illustrated in Figure 2(c), these functions seem to express the bonding pair between the Li and H atoms. On the other hand the orbital center of D+ in the LiD molecule is a little bit shorter than that of H+ in the LiH molecule, and close to the equilibrium distance calculated by conventional FVMO method. This cause is due to the anharmonicity. Since the kinetic energy of D+ is a smaller than that of H+, the total energy of LiD is lower than one of LiH molecule, and the orbital exponent is larger than that of H+. In the FVMO method this isotopic effect is clearly and directly shown. We have applied the full-CI and Hartree-Fock level of multi-component FVMO treatment to a system containing a positron. Although a conventional basis set gives good results in the H- system, it gives very poor results in the case of [H;e+] system, especially for the positronic excited state. Our full-CI multicomponent FVMO calculation gives the optimum GTF basis sets not only for electronic wavefunction but also for positronic wavefunction. It is noted that our full-CI multi-component FVMO calculation also gives good improvement for the positronic-excited states. Concerning to a molecular system, the results of the [Li;e+], LiH, and LiD systems seem to correspond to the change from a quantum wave to a classical particle.

Application of the Multi-Component Methods

33

Figure 2. Schematic illustration of the optimum basis functions for electron and positively charged particle. Only 3s (

χ 3es , χ 4es , and χ 6es ) electronic basis functions and

χp

1s ( 1s ) positively charged particle are shown in arbitrary units. (a) Basis functions of the Li- anion. (b) Basis functions of the [Li-;e+] system. (c) Basis functions of the LiH molecule.

Due to strong electron-positron correlation, the energy value of positronic compounds approaches rather slow to the exact value, as the basis set size increases. In order to correct the energy convergence, it seems to be more reliable to combine the present method with some other approach, such as R12[90] or Ten-no’s FROGG[91] techniques.

34

Taro Udagawa and Masanori Tachikawa Table 9. Optimum basis function and energies for Li-, [Li-;e+], LiH, and LiD systems

Positively particle E HF (a.u.) Virial ratio Dipole (a.u.) Electron χ1s χ2s χ3s χ4s χ5s χ6s Positively particle χ1s

Li-7.4147900 2.0000000 0.0000 Center Exponent 0.0000 100.566 0.0000 15.1464 0.0000 3.32431 0.0000 0.842991 0.0000 0.0530394 0.0000 0.0110185

[Li-;e+] e+ Quantum wave -7.5094065 2.0000000 0.0000 Center Exponent 0.0000 102.473 0.0000 15.4348 0.0000 3.39138 0.0000 0.862130 0.0000 0.0690911 0.0000 0.0229214 0.0000 0.0118690

LiH

LiD

LiH

H+ Quantum wave -7.9119399 2.0000000 -2.4680 Center Exponent 0.0000 96.6625 -0.0005 14.5586 -0.0003 3.19100 -0.0285 0.809183 3.1667 0.917424 2.6357 0.116406

D+ Quantum wave -7.9194841 2.0000000 -2.4698 Center Exponent 0.0000 96.5970 -0.0005 14.5488 -0.0004 3.18881 -0.0288 0.808629 3.1546 0.953425 2.6335 0.118731

Point Charge -7.9434074 2.0000000 -2.4724 Center Exponent 0.0000 96.3989 -0.0006 14.5191 -0.0005 3.18220 -0.0295 0.806931 3.1159 1.07607 2.6248 0.126232

3.2055

15.9704

3.1907

21.8376

R(Li-H)

3.1449

3-2. AN ANALYSIS OF THE ISOTOPE EFFECTS ON HYDROGEN MOLECULES BY USING FULL-CI MC_MO CALCULATION In this research, the configuration interaction (CI) treatment for the MC_MO method, beyond Hartree-Fock approximation, is applied to account for manybody effects on nuclear-nuclear and electronic-nuclear correlations, in addition to the electronic one. The CI MC_MO treatment has an advantage in that (nuclear) vibrational excited states can be obtained simultaneously with the electronic excited ones. We will demonstrate that the experimental values can be reproduced satisfactorily for properties such as the average internuclear distances, the dipole polarizabilities, and the (nuclear) vibrational excited energies for isotopic hydrogen molecules by this CI MC_MO treatment with optimized basis sets. Calculations have been employed for H2, D2, T2, and μ2 molecules with several optimized basis sets, in which all electrons and nuclei are treated quantum-mechanically. Since the center-of-mass motion is eliminated from the total Hamiltonian of Eq. (1), the translational motion of the whole molecule is excluded. On the other-hand, the rotational motion is not completely eliminated in this MC_MO scheme due to the coupling among the vibrational and rotational motions[34,35]. However, since all GTF centers are optimized along the vibrational z-axis starting from (0.0, 0.0, ±0.7 bohr) in the Cartesian coordinate, the low-lying excited states correspond to vibrational excited states, instead of electronic excited states in this MC_MO treatment.

Application of the Multi-Component Methods

35

The energy derivatives with respect to the GTF exponents have been evaluated in the scale of their natural logarithms. Individual exponent optimization is carried out for p and p GTFs, and a single exponent for d-type GTFs. To avoid the redundancy problem of the basis functions, the GTF center for electronic s-type GTFs on each side is assumed to have identical values through the optimization, and also for electronic p-type GTFs. Since all variational parameters are optimized in multi-dimensional energy hypersurface, possible problems of redundancy or multiple energy minima are checked by using different initial conditions for the GTF exponent; the same results are obtained in most cases. Table 10. Basis set convergence of H2 molecule with full-CI MC_MO calculation (0)

(V/T ) + 2 [Å ] -2.74E-09 0.7614 2.98E-09 0.7611 1.60E-08 0.7602 -1.75E-09 0.7606 2.32E-06 0.7529 -8.23E-07 0.7519 -8.36E-07 0.7511

Basis set E CI [hartree] [6s 1p : 1s 1p ] -1.130907 [6s 2p : 1s 1p ] -1.142004 -1.144425 [6s 3p : 1s 1p ] [6s 3p 1d : 1s 1p ] -1.145901 [6s 2p : 1s 1p 1d ] -1.145069 [6s 3p : 1s 1p 1d ] -1.147601 [6s 3p 1d : 1s 1p 1d ] -1.151602 a

0.7510 -1.164025 Results with explicitly correlated GTF [512 terms], Ref. [98]. b Calculated from Ref. [91]; the equilibrium distance is 0.7414Å .

b

a

3-2-1. Basis set dependence of H2 molecule To observe the basis set dependence of the full-CI MC_MO method, we show in Table 10 the results for H2 molecule with a number of basis sets. The basis set, e.g. [6s1p:1s1p], refers to the 6s1p and 1s1p GTFs for the electron and nucleus, respectively. Several basis sets are employed starting from [6s1p:1s1p] to [6s3p1d:1s1p1d], in which all values of GTF exponents and centers are optimized under the full-CI MC_MO treatment. The virial ratios are calculated to be very close to 2 within the accuracy of 10-6 for all cases. Table 1 also shows the internuclear distances calculated from the expectation value of the nuclear position. The internuclear distance

R

obtained by the present MC_MO

36

Taro Udagawa and Masanori Tachikawa

treatment corresponds to the average (instead of equilibrium) internuclear distance. The total energy and the internuclear distance calculated are improved as the number of basis functions increases. Although the energy of [6s3p1d:1s1p1d] basis set gains about 99% of that with 512 term explicitly correlated GTFs, the average internuclear distance agrees with the experimental one[91] to within 0.001 Å. Table 11. Optimized [6s3p1d:1s1p] GTFs with full-CI MC_MO calculation. H2 Centera Electron s

0.6401

ps

0.7009

pp

0.6936

d Nucleus s ps pp a b

Exponent

D2 Populationb

Centera

0.003 0.036 0.124 0.302 0.394 0.137 0.000 0.000 0.004 0.000 0.003 0.007 -0.010

0.6489

0.2942

11.3832 3.24974 1.18884 0.542615 0.224980 0.0901116 11.5644 3.09906 0.371958 7.37572 1.47148 0.356637 0.936967

0.7172 0.7151 0.7135

17.7347 19.5420 19.4177

0.984 0.010 0.007

Exponent

Centera

0.001 0.014 0.074 0.313 0.435 0.164 0.000 0.000 0.005 0.000 0.003 0.006 -0.015

0.6531

0.3069

17.1288 5.40195 1.94437 0.684541 0.254378 0.0966887 14.1006 3.43476 0.426480 8.57038 1.56962 0.371046 0.950705

0.7120 0.7108 0.7094

24.6613 26.8313 26.5987

0.987 0.008 0.005

0.6993

0.6910

μ2

T2 Populationb

Exponent

Populationb

Centera

0.001 0.013 0.073 0.314 0.436 0.165 0.000 0.000 0.006 0.000 0.002 0.006 -0.017

0.6352

0.3027

19.3371 5.79940 2.00581 0.695142 0.256918 0.0974195 15.8299 3.66544 0.435742 9.26954 1.61626 0.378079 0.965011

0.7097 0.7089 0.7077

29.8179 32.2271 31.9070

0.989 0.007 0.004

0.6985

0.6899

Exponent

Populationb

0.4997

6.20373 2.91566 1.22007 0.439167 0.187653 0.0790322 4.28817 1.20857 0.363194 3.81140 0.989354 0.277898 0.921183

0.003 0.023 0.139 0.317 0.361 0.108 0.000 0.002 0.001 0.002 0.008 0.010 0.025

0.7566 0.7462 0.7436

5.90683 6.86492 6.87529

0.965 0.019 0.017

0.7203

0.7148

Only the positive x coordinates are shown. Units are in bohr. Only the populations on one side of the hydrogen atom are shown.

3-2-2. Isotope effect of H2, D2, T2, and μ2 molecules Table 11 shows the optimized [6s3p1d:1s1p] GTF centers, exponents, and populations for H2, D2, T2, and m2 molecules. Since the nuclear kinetic effect is directly taken into account by this MC_MO method, the H/D/T/μ isotope effect appears in the nuclear basis functions. The nuclear s-type GTF exponent of triton (29.8) is greater than those of deuteron (24.7), proton (17.7), and muon (5.91); that is, the nuclear wavefunction of triton is the most localized among them. As the nuclear mass increases form μ to T, the values of electronic GTF exponents also increase. This demonstrates that the electronic structure of T2 molecule is more localized than those of D2, H2, and μ2 molecules. Thus the effect of relaxation on the electronic structure is clearly observed in the MC_MO procedure.

Application of the Multi-Component Methods

37

Table 12. Energies, internuclear distances, and polarizabilities with fullCI MC_MO calculation Basis set

E CI [hartree]

(0)

[Å ]

α zz [au]

H2 [6s 3p 1d : 1s 1p ] [6s 3p 1d : 1s 1p 1d ]

-1.145901 -1.151602 -1.164025 a

0.7606 0.7511 0.7510 b

6.82 6.75 6.78 c

[6s 3p 1d : 1s 1p ] [6s 3p 1d : 1s 1p 1d ]

-1.154273 -1.157740 -1.167169 a

0.7555 0.7481 0.7483 b

6.69 6.65 6.66 c

[6s 3p 1d : 1s 1p ] [6s 3p 1d : 1s 1p 1d ]

-1.157910 -1.160202 a -1.168536

0.7530 0.7472 b 0.7469

6.64 6.61 c 6.61

0.8012 0.7705

7.74 7.64

D2

T2

μ2

[6s 3p 1d : 1s 1p ] -1.089001 -1.109949 [6s 3p 1d : 1s 1p 1d ] a Results with explicitly correlated GTF [512 terms], Ref. [98]. b Ref. [91]. c Ref. [92].

Table 12 shows the total energies, internuclear distances, and dipole polarizabilities of these molecules with optimized [6s3p1d:1s1p] and [6s3p1d:1s1p1d] basis sets. The calculated average internuclear distances with the latter set are 0.7511, 0.7481, and 0.7472 Å for H2, D2, and T2 molecules, which coincide with the experimental average internuclear distances, 0.7510, 0.7483,

R

corresponds and 0.7469 Å[91], respectively. Since this internuclear distance to the average internuclear distance as shown above, we observe a significant difference among these isotopic species due to the anharmonicity of the potential as

RT2 < RD2 < RH 2 < Rμ2

.

(49)

Thus the H/D/T/μ isotope effect on the internuclear distance of hydrogen molecule is adequately reproduced by the MC_MO method. Polarizability is calculated with the finite-field approximation as

α zz

∂ 2 E (ε ) . = ∂ε z ∂ε z z →0

(50)

38

Taro Udagawa and Masanori Tachikawa A fully variational MC_MO calculation is carried out under the finite electric

field of ε z = 0.001 a.u. The optimization of GTF centers and the exponents under the electric field significantly improves the dipole polarizabilities, 6.75, 6.65, and 6.61 a.u. for H2, D2, and T2 molecules, which coincide with the calculation by Wolniewics and coworkers[92], 6.78, 6.66, and 6.61 a.u. The non-adiabatic effect on the internuclear distance and polarizability of μ2 molecule must be greater than those of other isotopic hydrogen species due to the light mass of the muon. This reflects tha fact that the basis set convergence of μ2 molecule is the slowest among the isotopic hydrogen analogs, since the nonadiabatic effect is improved by the contribution of electron-muon correlation. Table 13. Nuclear excitation energies and averaged bond lengths of vibrational excited states with full-CI MC_MO calculation H2 Basis set [6s 3p : 1s 1p ] [6s 3p 1d : 1s 1p ] [6s 3p : 3s 3p ] [6s 3p 1d : 3s 3p ] a

Exptl. Ref. [91].

-1

D2 (1)

DE [cm ] [Å ] 6524 0.7599 6700 0.7605

-1

T2 (1)

DE [cm ] [Å ] 5086 0.7543 5089 0.7561

-1

(1)

DE [cm ] [Å ] 4384 0.7518 4386 0.7536

4194 4182

0.7730 0.7737

3017 3006

0.7633 0.7641

2487 2477

0.7591 0.7599

4161

0.7705

2994

0.7619

2465

0.7579

a

The excitation energies and the expectation values of the nuclear position at

R (1)

, are shown in Table 13. The results with the the first-excited state, [6s3p:1s1p] and [6s3p1d:1s1p] basis sets, determined by optimization of the ground-state energy, are far from the experimental values. To improve the excitation energy, 2s2p GTFs are added for the nuclear basis set. The exponent values of the additional 2s2p GTFs are determined by optimization with the state average approach, where the sum of the first and second roots of density matrices is used in Eq. (25). By recalculation with new [6s3p1d:3s3p] basis set, the excitation energies are found to be 4182, 3006, and 2447 cm-1 for H2, D2, and T2 molecules, respectively, which agree with the experimental ‘vibrational frequency’ values, 4161, 2994, and 2465 cm-1[91], to within 21 cm-1. This result indicates that the first excited state corresponds to the (nuclear) vibrational excited states, since all GTF centers are optimized along the vibrational z-axis only. The internuclear distance of (nuclear) vibrational excited state for isotopic hydrogen molecule with [6s3p1d:3s3p] basis set also coincide with the experimental values to within 0.003 Å. To obtain higher (nuclear) vibrational excited states, basis

Application of the Multi-Component Methods

39

functions with higher quantum numbers are required, e.g., d-type functional for first excited state, and f-type functions for the second one.

3-3. AN ANALYSIS OF THE H/D ISOTOPE EFFECT ON PORPHINE AND PORPHYCENE MOLECULES REVEALED BY THE MC_DFT METHOD Theoretical studies on porphyrin derivatives have been reported in large amounts[93-95]. Almlöf and coworkers[93] clearly showed that D2h “highsymmetric” geometry of porphine molecule is not able to be theoretically reproduced without including electron correlation. In fact, post-HF methods of Møller-Plesset perturbation and local density functional calculations accurately reproduce “high-symmetric” geometries of metal porphyrin[94], while the Hartree-Fock level of calculation provides “low-symmetric” (CS) geometry regardless of the basis sets used. Many-body effect, thus, plays a very important role for the equilibrium geometry of porphine molecule, as well as porphycene molecule[95]. In this study, we would reveal the effect of quantum nature of two innerprotons or deuterons for porphine and porphycene molecules by using our multicomponent “hybrid” density functional theory (MC_(HF+DFT)). We have calculated porphine and porphycene molecules using our MC_(HF+DFT). These molecules are known to posses D2h and C2h point group for porphine and porphycene, respectively, and these “high-symmetric geometries cannot be reproduced without electron-electron correlation[93]. We use the 631G** electronic basis function for central two hydrogen atoms and 6-31G for other atoms. In the framework of multi-component procedure, several nuclei can be treated as quantum waves, as well as electrons under the field of nuclear point charges. In this calculation, only central two hydrogen nuclei are treated as quantum wave. We have employed the single s-type Gaussian type function

{

(GTF), exp − α (r − R )

2

} for each protonic and deuteronic basis function, and

optimized two variational parameters ( α and R ), simultaneously. While nuclear wavefunction should be expanded by suitable numbers of basis function as well as electronic case, we have already demonstrated that the H/D isotope effect could be analyzed at least qualitatively even employed only single s-type GTF, as nuclear basis function[18,19]. All of the geometric optimization has been carried out with planar constraint. The center of electronic GTFs were fixed on each

40

Taro Udagawa and Masanori Tachikawa

nucleus or quantum nuclear GTF. All calculations were carried out at the MC_BHandHLYP and MC_B3LYP level, using the modified version of GAUSSIAN 03 program[53]. 3-3-1. Optimized geometries of HH-molecules Tables 14 and 15 list the optimized geometric parameters of porphine and porphycene molecules by using MC_HF, MC_BHandHLYP, and MC_B3LYP methods, respectively. The notations used in Tables 14 and 15 are depicted in Figures 3 and 4, respectively. The geometries of porphine and porphycene molecules optimized with MC_BHandHLYP and MC_B3LYP methods are D2h and C2h “high-symmetries”, while those by MC_MO-HF are C2v and CS “lowsymmetries”. These optimized “high-symmetric” geometries are clearly demonstrated that electron-electron correlation is actually taken into account by our MC_MO-(HF+DFT) method. We note here that the hydrogen-bonded distances in porphycene molecule (1.532Å with MC_BHandHLYP and 1.497Å with MC_B3LYP) are much shorter than those of porphine molecule (2.284Å with MC_BHandHLYP and 2.296Å with MC_B3LYP). Table 14. Optimized geometric parameters [Å] for HH-porphine with MC_HF, MC_BHandHLYP, MC_B3LYP

MC_HF MC_BHandHLYP MC_B3LYP upper left lower right RN-H 1.015 1.017 1.027 1.038 RH…N 2.301 2.321 2.284 2.296 RN…N 2.914 2.946 2.913 2.934 R1 1.373 1.362 1.369 1.382 R2 1.373 1.362 1.369 1.382 R3 1.461 1.400 1.431 1.440 R4 1.341 1.389 1.365 1.377 R5 1.461 1.400 1.431 1.440 a Only geometric parameters on pyrrole ring and hydrogen bond are shown for simplicity. In the MC_(HF+DFT) methods, the geometric parameters of upper left and lower right pyrrole rings are the same. Table 16 shows the optimized α value, which indicates a distribution of protonic wavefunction. The optimized α values of each central hydrogen-bonded

Application of the Multi-Component Methods

41

proton with MC_HF are different each other, because of the “low-symmetric” backbone structure as shown in Tables 14 and 15. On the other hand, in the case of MC_MO-(HF+DFT) (MC_BHandHLYP and MC_B3LYP), no different α values are obtained between each two central inner-protons in both molecules, reflecting the geometric symmetry. Table 15. Optimized geometric parameters [Å] for HH-porphycene with MC_HF, MC_BHandHLYP, and MC_B3LYP MC_HF MC_BHandHLYP MC_B3LYP upper left lower right RN-H 1.056 1.052 1.096 1.127 RH…N 1.648 1.660 1.532 1.497 RN…N 2.628 2.638 2.567 2.567 R1 1.371 1.354 1.366 1.380 R2 1.354 1.359 1.359 1.372 R4 1.455 1.403 1.430 1.441 R5 1.348 1.388 1.369 1.380 R6 1.457 1.407 1.432 1.443 a Only geometric parameters on pyrrole ring and hydrogen bond are shown for simplicity. In the MC_(HF+DFT) methods, the geometric parameters of upper left and lower right pyrrole rings are the same.

R4 R5

R3 R1

R2 RN-H

RN…N RH…N

Figure 3. Optimized structure of porphine molecule.

The α values in porphycene molecule are found to be smaller than those in porphine molecule. The exponent α values, as well as the hydrogen-bonded

42

Taro Udagawa and Masanori Tachikawa

distance, indicate that the interaction of intramolecular hydrogen-bond in porphycene molecule is stronger than that in porphine molecule.

R5

R4

R3

R6 R7

R1

R2 RN-H

RN…N

R8

RH…N R9 R14

R13

R10 R11

R12

Figure 4. Optimized structure of porphycene molecule.

Table 16. Optimized exponent α value for inner-hydrogens in porphine and porphycene molecules

Porphine

α1 α2

MC_HF H D 23.38 34.55 23.42 34.61

Porphycene

α1 α2

21.88 21.61

32.71 32.36

MC_BHandHLYP H D 23.23 34.36 23.23 34.36

MC_B3LYP H D 23.18 34.28 23.17 34.28

21.08 21.07

20.91 20.91

31.63 31.63

31.40 31.40

3-3-2. Geometric isotope effect on DD-molecules We have calculated porphine (HH-porphine), porphycene (HH-porphycene), and its isotopomers replaced two inner-hydrogens to deuteriums (denoted as DDporphine and DD-porphycene). As shown in Table 16, the optimized α values of each proton (deuteron) by MC_BHandHLYP and MC_B3LYP methods have same values, which reflect the geometric symmetry in HH- (DD-) molecules. The α values of deuteron are larger than that of proton, which means deuteron is more localized than proton.

Application of the Multi-Component Methods

43

Tables 17 and 18 list the representative optimized geometric parameters of HH- and DD- molecules, and the notation of these parameters are depicted in Figures 3 and 4, respectively. As shown in Table 17. Optimized geometric parametersa [Å] for H-porphine and Dporphine molecules with MC_HF, MC_BHandHLYP, and MC_B3LYP MC_HF MC_BHandHLYP MC_B3LYP H-Complex D-Complex H-Complex D-Complex H-Complex D-Complex RN-H 1.015 1.008 1.027 1.020 1.038 1.031 RH…N 2.301 2.306 2.284 2.290 2.296 2.302 RN…N 2.914 2.915 2.913 2.914 2.934 2.934 R1 1.382 1.383 R2 1.382 1.383 R3 1.431 1.430 R4 1.431 1.430 a Only different parameters by replacing a hydrogen to deuterium are shown in table.

Table 18. Optimized geometric parametersa [Å] for H-porphycene and Dporphycene molecules with MC_HF, MC_BHandHLYP, and MC_B3LYP MC_HF MC_BHandHLYP MC_B3LYP H-Complex D-Complex H-Complex D-Complex H-Complex D-Complex RN-H 1.052 1.038 1.096 1.077 1.127 1.103 RH…N 1.660 1.691 1.532 1.569 1.497 1.538 RN…N 2.638 2.651 2.567 2.581 2.567 2.581 R1 1.366 1.365 R2 1.372 1.373 R3 1.418 1.419 R4 1.403 1.402 1.430 1.429 1.441 1.440 R5 1.388 1.389 R6 1.407 1.406 1.443 1.442 R7 1.428 1.429 1.408 1.407 R8 1.358 1.357 R9 1.434 1.435 R10 1.451 1.452 1.448 1.449 1.455 1.457 R11 1.360 1.359 R12 1.440 1.441 1.443 1.444 R13 1.372 1.373 1.354 1.353 R14 1.328 1.327 1.364 1.363 a Only different parameters by replacing a hydrogen to deuterium are shown in table.

Tables 17 and 18, central N-H lengths (RN-H) in HH-molecules are longer than those in DD-molecules (RN-D), due to the direct inclusion of the quantum effect of

44

Taro Udagawa and Masanori Tachikawa

proton and deuteron. Thus, our method can directly take account of the anharmonicity of the potential which induces the geometrical change between HH and DD species. Such geometrical change is called as primary geometric isotope effect (GIE). Next, we have clearly observed that the hydrogen-bonded H…N distance (RH…N) and the N…N distance (RN…N) in HH-molecules are shorter than the DDmolecules’ counterparts. Such secondary GIE is hard to explain from the anharmonicity of only one dimensional potential curve. Thus, we deal with this problem from a different point of view using the idea of the hydrogen-bonded crystals, which is known as the Ubbelohde effect[2]. This effect represents the lattice expansion of crystals as a result of the deuterium-substitution in hydrogenbonds, which is explained based on the assumption that proton/deuteron mediates the attractive interaction of two nitrogen atoms. Chemically, the resonance is more likely to occur and two nitrogen atoms are more attractive by proton rather than by deuteron, because of the delocalized character of proton. This idea can be used for the secondary GIE of porphine and porphycene molecules[96], since we have seen that proton is indeed more delocalized than deuteron. In addition, we have slightly observed the geometric change of the “backbone” parts (R1 to R14 in the Table 15 and 16) between HH- and DDspecies. Such tertiary GIE is induced by the primary GIE, secondary GIE, and originally from the difference of the quantum nature between the proton and deuteron. In porphine molecule, the tertiary GIE appears in only pyrrole ring region, while that appears over a whole structure in the case of porphycene. As mentioned in the previous section, the strength of the hydrogen-bonded interaction in porphycene molecule is greater than that in porphine molecule. Thus, the structural difference between porphine and porphycene makes tertiary GIE be more prominent in porphycene molecule than that in porphine molecule. 3-3-3. Symmetric breaking down in HD-molecules To investigate the effect of quantum nature of inner-hydrogens in detail, we have also calculated another isotopomers of porphine and porphycene, which is replaced an inner-hydrogen by a deuterium (denoted as HD-porphine and HDporphycene). Figures 5 and 6 list the optimized geometric parameters by using MC_MO-(HF+DFT) method in HD-porphine and HD-porphycene, respectively. The difference of the quantum nature between inner-hydrogen and innerdeuterium induces the longer RN-H bond length than RN-D length, and then we

Application of the Multi-Component Methods

45

clearly demonstrate that the “high-symmetric” geometry of porphycene and porphine is broken down. Such symmetric breaking down is more prominent in porphycene molecule than that in porphine molecule, because the strength of hydrogen bond interaction of porphycene molecule is greater than the porphine molecule as mentioned above. Due to the anharmonicity of the potential, of course, RN-H is longer than RN-D in both HD-molecules. Furthermore, in the case of HD-porphycene, symmetric breaking down has appeared not only around the hydrogen bond but also in pyrrole rings, while that has appeared only on RN-H and RN-D bond lengths in the HD-porphine molecule. Consequently, HD-porphine and HD-porphycene molecular geometries have C2v and CS symmetries, respectively.

1.431

1.028 1.038

1.431

2.288 H 2.300 2.914 2.913 2.934 2.933 2.284 2.296 D 1.020 1.031

1.430

1.430

Optimized parameters of MC_BHandHLYP and MC_B3LYP

Figure 5. Optimized geometric parameters for HD-porphine molecule with MC_BHandHLYP and MC_B3LYP. Only the structurally-asymmetrical parameters are shown. 1.430 1.444 1.441 1.453 1.397 1.443 1.365 1.407

1.448 1.456

1.353 1.368

1.359

1.363 1.379

1.096 1.124 1.562 1.530 2.576 2.571 H 2.572 1.540 D 2.576 1.506 1.078 1.105 1.354 1.364 1.366 1.369 1.380

1.360

1.449 1.457

1.408 1.442

1.398 1.443 1.429 1.452 1.440

Optimized parameters of MC_BHandHLYP, and MC_B3LYP

Figure 6. Optimized geometric parameters for HD-porphycene molecule with MC_BHandHLYP and MC_B3LYP. Only the structurally-asymmetrical parameters are shown.

46

Taro Udagawa and Masanori Tachikawa

Table 19 lists the optimized exponent α values for inner-proton and innerdeuteron in HD-porphine and HD-porphycene. The α values of HD-molecules are not drastically changed from corresponding values of HH- and DD-molecules, where α values for porphycene molecule are smaller than those in porphine molecule for both HH- and DD-species. Table 19. Optimized exponent α value for inner-hydrogen and innerdeuterium in HD-porphine and HD-porphycene molecules

Porphine

α(H) α(D)

MC_BHandHLYP 23.22 34.36

MC_B3LYP 23.15 34.28

Porphycene

α(H) α(D)

21.10 31.55

20.96 31.32

Table 20 lists the calculated dipole moment of HH-, DD-, and HD-porphine and porphycene molecules. We have obtained the interesting results of dipole moment for the HD-molecules. Although D2h and C2h structures of H- and Dmolecules have zero dipole moments, small polarization appears in HDmolecules. Since the dipole moment values of HD-molecules clearly reflect the symmetric breaking down of molecular structures, dipole moment of HDporphycene is larger than that of HD-porphine. Table 20. Calculated dipole moment [Debye] on porphine and porphycene molecules and its isotopomers with MC_(HF+DFT) method

Porphine Porphycene

H 0.000 0.000

MC_BHandHLYP D HD 0.000 0.015 0.000 0.032

H 0.000 0.000

MC_B3LYP D 0.000 0.000

HD 0.015 0.045

To treat the nuclear quantum effect with electron correlation efficiently, we have proposed a new scheme of multi-component “hybrid” DFT (MC_MO(HF+DFT)) method, which is combined a concept of multi-component HartreeFock (MC_MO-HF) with density functional theory (DFT) to evaluate the manybody effect. We have clearly demonstrated that our MC_MO-(HF+DFT) method can take account of electron-electron correlation by combining with DFT, since the experimentally observed “high-symmetric” structures are actually reproduced for porphine and porphycene molecules.

Application of the Multi-Component Methods

47

We revealed that the molecular symmetries were broken down by replacing only an inner-hydrogen atom to a deuterium in porphine and porphycene molecules. In addition, these molecules have dipole moment values accompanying this replacing. These facts clearly demonstrate that the quantum nature of light particles, such as proton and deuteron, influences the molecular geometry and electronic structure. Because of the computational advantages of DFT, our method is expected to be a powerful tool for analyzing variety phenomena induced by the nuclear quantum effect in large scale molecules such as biomacromolecule. In this study, since we use the conventional exchange-correlation functional LYP, we can evaluate only electron-electron correlation. It is known that nucleus-nucleus correlation is very small and can be negligible, while electron-nucleus correlation should not be completely ignored since the electron-nucleus correlation must explicitly couple electron-electron correlation with each other. Such effective functional for electron-nucleus correlation has, unfortunately, not been known yet. Next stage of our research is the construction of new correlation functionals for electron-nucleus correlation.

Chapter 4

CONCLUSIONS We have developed the multi-component molecular orbital (MC_MO) method which can take account of the quantum nature of light particles such as proton, deuteron, positron, and so on. Nowadays, several research groups are studying related work of MC_MO method. Our MC_MO method has already extended to Møller-Plesset perturbation theory, configuration interaction method, and density functional theory and successfully analyzing many systems and phenomena, such as positron systems, hydrogen-bonded systems, and the H/D isotope effects. We have also developed the fully variational molecular orbital (FVMO) method which can variationally optimize energy with respect to not only nuclear coordinates but also exponent values and centers of basis functions. We can obtain optimum basis functions for not only electrons but also nuclei and other light particles such as positron by FVMO procedure, which has not determined variationally yet. We believe that our MC_MO method has a tremendous amount of potential, because our MC_MO method is a newborn method, yet. Since the original idea of the MC_MO method is not far apart from conventional MO method, our MC_MO method will be more and more promising and powerful tool as the conventional MO method does.

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INDEX

A accuracy, 14, 19, 20, 23, 35 adiabatic, 2, 38 angular momentum, 19 anharmonicity, 32, 37, 44, 45 anion, 33 annihilation, 21 antiparticle, 21 ASI, 54 atoms, 1, 23, 32, 39, 44

B basis set, 10, 11, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 55 binding, 55 binding energy, 55 bonding, 28, 32 bonds, 44 BOP, 17 bosons, 7

C CH4, 2 charged particle, 31, 33 chemical, vii, 1, 2, 14 chemistry, 1 classical, 10, 15, 17, 32 clouds, 27 components, 12, 22, 23, 24, 25, 26 compounds, 21, 22, 23, 33 configuration, 7, 13, 14, 18, 20, 25, 34, 49 Congress, iv construction, 47 contamination, 19 convergence, 22, 23, 24, 26, 33, 35, 38 correlation, 12, 14, 15, 16, 17, 20, 22, 24, 25, 33, 38, 39, 40, 46, 47 correlation function, 14, 16, 17, 47 correlations, 34 costs, 14 Coulomb, 8, 9, 11 coupling, 2, 8, 15, 34 coupling constants, 8 crystals, 44 CS, 39, 40, 45

58

Index

D DD, 42, 43, 44, 46 Debye, 46 density, vii, 2, 12, 13, 14, 15, 16, 17, 38, 39, 46, 49 density functional theory, vii, 2, 14, 15, 39, 46, 49 density matrices, 12, 38 derivatives, 11, 12, 13, 35, 39 deuteron, 1, 7, 36, 42, 44, 46, 47, 49 deviation, 29 DFT, vii, 14, 16, 17, 22, 39, 40, 41, 44, 46, 47 dipole, 10, 34, 37, 38, 46, 47 dipole moment, 10, 46, 47 dipole moments, 10, 46 distribution, 40

E ECG, 2 eigenvalues, 1 electric field, 38 electron, vii, 6, 8, 11, 13, 14, 16, 20, 21, 22, 23, 24, 25, 31, 33, 35, 38, 39, 40, 46, 47, 55 electronegativity, 29 electronic, iv, 1, 2, 6, 7, 9, 10, 11, 14, 19, 21, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 47, 53 electronic structure, 1, 21, 36, 47, 53 electronic structure theory, 53 electrons, 1, 2, 5, 6, 7, 8, 9, 10, 16, 23, 24, 29, 31, 34, 39, 49 electrostatic, iv, 15 energy, 1, 2, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 31, 33, 35, 36, 38, 49, 55 equilibrium, 19, 32, 36, 39 excitation, 25, 38 exclusion, 7 expert, iv exponential, 20

F fermions, 7 Feynman, 53 free energy, 18

G Gaussian, 20, 22, 39, 40, 53 groups, 2, 14, 49

H H2, 19, 20, 34, 35, 36, 37, 38 Hamiltonian, 1, 2, 5, 6, 7, 18, 19, 21, 34 hands, 22 Hartree-Fock approximation, 26, 34 HD, 44, 45, 46 heavy particle, 20 high-level, 20 hybrid, vii, 14, 15, 16, 17, 39, 46 hydrogen, 1, 2, 14, 20, 22, 27, 28, 30, 34, 37, 38, 39, 40, 41, 44, 46, 47, 49, 55 hydrogen atoms, 39 hydrogen bonds, 2

I inclusion, 43 indices, 5, 6, 7, 14 injury, iv integration, 8, 16 interaction, 7, 12, 14, 15, 18, 20, 34, 42, 44, 45, 49 interdisciplinary, 1 isotope, vii, 1, 2, 15, 21, 22, 34, 36, 37, 39, 42, 44, 49

J Jordan, 26, 54

Index

K kinetic energy, 2, 6, 23, 25, 32

L lattice, 44 linear, 9, 15, 20 linear dependence, 20 London, 53, 54 lying, 22, 34

59

O one dimension, 44 operator, 2, 6, 16, 19 optimization, 22, 23, 28, 31, 35, 38, 39 oxygen, 27, 28, 30, 55

P

M magnetic, iv matrix, 12, 13 mechanical, iv, 1, 7 molecular dynamics, 1 molecular structure, 46 molecules, 2, 6, 20, 22, 23, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47 Møller, 12, 13, 14, 39, 49 motion, 1, 2, 6, 9, 18, 34 multi-component systems, 5 muon, 7, 10, 36, 38

N NATO, 54 natural, 35 neglect, 16 New York, iii, iv, 53, 54 nitrogen, 44 nuclear, 1, 2, 5, 6, 7, 9, 10, 11, 12, 14, 16, 17, 18, 20, 21, 23, 24, 27, 28, 31, 34, 35, 36, 38, 39, 46, 47, 49 nuclear charge, 2, 5, 6 nuclei, 1, 2, 5, 6, 7, 8, 9, 10, 13, 16, 21, 27, 34, 39, 49 nucleus, 1, 2, 6, 7, 8, 10, 13, 14, 27, 28, 29, 30, 31, 32, 35, 40, 47

PA, 29, 53, 55 parameter, 11, 12, 15 particles, vii, 7, 10, 12, 16, 18, 20, 31, 47, 49 perturbation, 12, 13, 14, 18, 39, 49 perturbation theory, 12, 13, 18, 49 physical properties, 20 physics, 1 planar, 39 polarizability, 2, 38 polarization, 24, 46, 55 polarized, 25 poor, 24, 26, 32 positron, vii, 7, 10, 21, 22, 23, 24, 25, 27, 29, 31, 32, 33, 49 potential energy, 1, 15, 23, 24 power, 20 preparation, iv program, 10, 14, 17, 40 property, iv protons, 7, 39, 41 p-type, 35 pyrrole, 44, 45

Q quantum, vii, 1, 2, 7, 8, 9, 10, 12, 13, 16, 31, 32, 34, 39, 43, 44, 46, 47, 49, 53 quantum chemistry, 53

R redundancy, 35 reference system, 15 relaxation, 29, 36

60

Index

reliability, 23 research, 22, 34, 47, 49 researchers, 21 rings, 45

S Schrödinger equation, 6 series, 19 services, iv simulation, 1 Singapore, 54 skeleton, 11, 13 spatial, 8 species, 7, 15, 21, 22, 23, 24, 25, 26, 31, 37, 38, 44, 46 spin, 8, 16 STO, 27, 28, 29, 31, 32 strength, 15, 44, 45 substitution, 44 symmetry, 41, 42 systems, vii, 1, 2, 6, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 34, 49

TF, 18, 19 theoretical, 1, 21 theory, 2, 3, 14, 16, 18, 19 three-dimensional, 27 three-dimensional space, 27 time, 6 total energy, 10, 21, 23, 32, 36 trajectory, 1 transfer, 1, 2, 20 transition, 1, 2 transition temperature, 1 translation, 18 translational, 2, 18, 19, 34 TRF, 18, 19

V valence, 31 values, 10, 23, 24, 25, 26, 27, 29, 31, 34, 35, 36, 38, 40, 41, 42, 46, 47, 49 van der Waals, 14 vibrational, 14, 18, 20, 34, 38

W T weak interaction, 14 Taylor series, 19