Hyperspherical Harmonics and Generalized Sturmians

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Progress in Theoretical Chemistry and Physics 92/80( 

Editor-in- Chief:

- 0DUXDQL (Laboratoire de Chimie Physique, Paris, France) 6 :LOVRQ (Rutherford Appleton Laboratory Oxfordshire, United Kingdom)

Editorial Board: ' $YQLU (Hewbrew University of Jerusalem, Israel) - &LRVORZVNL (Florida State University, Tallahassee, FL, U.S.A.) 5 'DXGHO (European Academy of Sciences, Paris, France) (.8 *URVV (Universität Würburg Am Hubland, Würzburg, Germany) :) 9DQ *XQVWHUHQ (ETH-Zentrum, Zurich, Switzerland) . +LUDR (University of Tokyo, Japan) , +XEDF (Komensky University, Bratislava, Slovakia 03 /HY\ (Tulane University, New Orleans, LA, U.S.A.) :1 /LSVFRPE (Harvard University, Cambridge, MA, U.S.A.) */0DOOL(Simon Fraser University, Burnaby, BC, Canada) -0DQ](Universität Würzburg, Germany) 50F:HHQ\(Università di Pisa, Italy) 3* 0H]H\ (University of Saskatchewan, Saskatoon, SK, Canada) ,3ULJRJLQH(Université Libre de Bruxelles, Belgium) 3 3\\NN| (University of Helsinki, Finland) - 5\FKOHZVNL (Polish Academy of Sciences, Poznan, Poland) '5 6DODKXE (Université de Montreal, Quebec, Canada) 6' 6FKZDUW] (Yeshiva University, Bronx, NY, U.S.A.) <* 6PH\HUV (Institute de Estructura de la Materia, Madrid, Spain) 6 6XKDL (Cancer Research Center, Heidellberg, Germany) 2 7DSLD (Uppsala University, Sweden) 35 7D\ORU (University of California, La Jolla, CA, U.S.A.) 5* :RROOH\ (Nottingham Trent University United Kingdom)

Hyperspherical Harmonics and Generalized Sturmians by John Avery HC ø UVWHG Institute, University RI Copenhagen. Copenhagen, Denmark

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3 construct the wave functions of more complicated systems  for example manyelectron atoms or molecules. However, it was soon realized that unless the continuum is included, a set of hydrogenlike orbitals is not complete. To remedy this defect, Shull and Löwdin [273] introduced sets of radial functions which could be expressed in terms of Laguerre polynomials multiplied by exponential factors. The sets were constructed in such a way as to be complete, i.e. any radial function obeying the appropriate boundary conditions could be expanded in terms of the Shull-Löwdin basis sets. Later Rotenberg [256, 257] gave the name “Sturmian” to basis sets of this type in order to emphasize their connection with Sturm-Liouville theory. There is a large and rapidly-growing literature on Sturmian basis functions; and selections from this literature are cited in the bibliography. In 1968, Goscinski [138] completed a study of the properties of Sturrnian basis sets, formulating the problem in such a way as t o make generalization of the concept very easy. In the present text, we shall follow Goscinski’s easily generalizable definition of Sturmians. Generalized Sturmian basis sets can be understood from the following simple argument: Suppose that we are able to find a set of solutions to the wave equation

Here

is the generalized Laplacian operator and [ = ^x x xG` are the mass-weighted Cartesian coordinates of the particles in a system. If we are discussing a single particle, then d = 3, while if we are discussing an NSDUWLFOH system, d = N. We let ßv be a set of constants chosen in such a way that the functions all correspond to the same value of energy, ( regardless of their quantum numbers. Then it is easy t o show, using the Hermiticity of the operator

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Acknowledgements I am extremely grateful to the Carlsberg Foundation for travel grants which allowed me to make several visits to the laboratory of Prof. Dudley Herschbach at Harvard University. I would also like to thank the Danish Science Research Council which made possible two periods of research at the laboratory of Prof. Vincenzo Aquilanti at the University of Perugia, Italy, and also a short visit to the laboratory of Prof. Jamie Fernández Rico in Madrid. Prof. Fernández Rico and Dr. Rafael López have generously allowed me to use their computer program for evaluating many-center two-electron integrals over Slater-type orbitals. It has been an extremely great pleasure for me to receive encouragement from two of the most important pioneers of momentum-space quantum theory  Prof. Roy McWeeny and Prof. Carl Wulfman. I am also grateful to Professors Mario Raimondi, Robert Nesbet, Luis Tel, Jens Bang and Jan Vaagen for encouragement and for stimulating discussions. Finally, I would like to thank my co-workers, Dr. Frank Antonsen, Dr. Wensheng Bian, Prof. John Loeser, Cand. Scient. Tom Børsen Hansen, Stud. Scient. Rune Shim, Dr. Cecilia Coletti, and Assist. Prof. Stephan Sauer for their important contributions.

Chapter 1

MANY-PARTICLE STURMIANS Suppose that we are able to find a set of functions øv[ which are solutions to the many-electron equation: (1.1) In equation (1.1),

is the generalized Laplacian operator (1.2)

while [ represents the set of coordinate vectors for all the electrons in the system: [ = {;1, [2, ..., ;1} (1.3) and [^xJ, yJ, zJ` j = 1,2, ..., N (1.4) The potential, V [ , is chosen to be the Coulomb attraction potential due to the nuclei in the system. The simplest case is that of an Nelectron atom or ion, where

(1.5) 7



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9 and so on. In equation (1.6), µ represents the set of quantum numbers ^n, O, m, s`. The functions satisfy the relationships: (1.11) (1.12) and (1.13) We now introduce the subsidiary conditions (1.14) and (1.15) Provided that the subsidiary conditions are satisfied, the product shown in equation (1.6) will be a solution to (1.1), since

(1.16) It can easily be seen that an antisymmetrized function of the form

(1.17)

will also satisfy (1.1), since the antisyrnmetrized function can be expressed as a sum of terms, each of which has the form shown in equation



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11 Combining (1.21) and (1.22), and making use of the orthonormality of the spin functions and the spherical harmonics, we obtain the potentialweighted orthonormality relations: (1.23) where v stands for the set of quantum numbers {µ, µ', µ'', ...}.

The secular equation Having constructed a set of many-electron Sturmian basis functions by the method just described, we would like t o use this basis set t o solve  the Schrodinger equation for an NHOHFWURQ atom. Using atomic units,  we can write the Schrodinger equation in the form: (1.24) where (1.25) Here V  [ is the nuclear attraction potential shown in equation (1.5), while (1.26) is the interelectron repulsion potential. Expanding the wave function as series of generalized Sturmian basis functions, and making use of equation (1.1), we obtain:

(1.27) Multiplying (1.27) from the left by a conjugate basis function, integrating over the coordinates, and making use of the potential-weighted orthonormality relation, (1.23), we obtain: (1.28)



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excited states. The accuracy of both the ground-state and excitedstates increases when the number of configurations included in the basis set is increased. The evaluation of off-diagonal interconfigurational matrix elements requires the use of the generalized Slater-Condon rules, which are discussed in an appendix.

The many-center problem Let us end this chapter by looking briefly a t the many-center problem, although we shall postpone a detailed treatment of this problem until a later chapter. In the case of molecules, we need to solve the one-electron wave equation (1.55) where Y[ is the many-center nuclear attraction potential: (1.56) The weighting factor bµ kµ is introduced into (1.55) in order to give the one-electron orbitals flexibility, so that they may be used to build up a solution to the many-electron equation, (1.1). Equation (1.55) is the many-center analog of (1.11), while the many-center analog of (1.6) is (1.57)

In order that this product should be a solution to (1.1), with (1.58) we require that the subsidiary conditions (1.59) and (1.60)

19 be satisfied. Then

(1.61) Just as in the case of atoms, we of course use Slater determinant basis functions of the form in order that the Pauli principle shall be fulfilled. To solve (1.55), we can expand the orbital in terms of hydrogenlike basis functions located on the various atomic centers ;D (1.62) where

stands for the set of indices {D, Q, OP, V} (1.63)

Substituting the expansion (1.62) into the wave equation (1.55) we obtain (1.64) Next we make use of equation (1.11), which allows us t o rewrite (1.64) in the form: (1.65) where we have also multiplied from the left by a complex conjugate function from the basis set and integrated over the electron’s coordinates. The integrals needed for the solution of the many-center oneelectron secular equation can most easily be calculated in momentum space, making use of the properties of hyperspherical harmonics; and therefore we shall discuss these topics in the following chapters.



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Exercises 1. Use Table 1.2 to show that the one-electron hydrogenlike Sturmian x1, , obeys equations (1.11)-(1.13). 2. Show that if k1 = and k 2 = then (x j )obey the orthogonality relation

3. Show that if k 1 = k 2 = kµ , then and the potential-weighted orthonormality relation

x 1 ,0 ,0

and

obey

4. Use equations (1.7) and (1.8) to evaluate the direct-space hydrogenlike orbitals χ1s, χ2s, χ2pj, χ3s and χ3pj.

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CHAPTER 2.

MOMENTUM-SPACE WAVE FUNCTIONS

where jlSU is a spherical Bessel function. The integrals needed for the evaluation of the radial transform are of the form: (2.5) T h e most simple of these integrals is (2.6) which can be evaluated by elementary integration, yielding (2.7) By differentiating J with respect to kµ we obtain (2.8)

(2.9) and so on. The integrals corresponding t o other values of l can then be generated using the recursion relations [30, 146]: (2.10) and (2.11) In general the integrals Jl+,l have the form: (2.12) The first few integrals of this type are shown in Table 2.1; and using them we can derive the Fourier-transformed hydrogenlike orbitals shown in Table 2.2.

25

)RFN¶V WUHDWPHQW RI K\GURJHQOLNH DWRPV V. Fock [124, 125] introduced the following projection, which maps 3-dimensional S-space onto the surface of a 4-dimensional hypersphere:

(2.13)

He was then able to show (using a method which we will discuss in a later section) that the Fourier-transformed hydrogenlike orbitals are given by a universal factor, (2.14)

which is independent of' the quantum numbers, niultiplicd by a 4dimensional hyperspherical harmonic whose argument is the unit vector defined by equation (2.13): 

The first few 4-dimensional hyperspherical harmonics are shown in Table 2.3 as functions of the unit vector X u , X   u , u   These functions ai e a generalization of the familiar 3-dimensional spherical harmonics; and ZH will discuss their properties in detail in Chapter 3. The 4-dimensional hyperspherical harmonics are given by > 38] (2.16) where



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Table 2.2 Hydrogenlike atomic orbitals and their Fourier transforms W N—[, W N —U

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CHAPTER 2. MOMENTUM-SPACE WAVE FUNCTIONS

31 Table 2.4 Alternative 4-dimensional hyperspherical harmonics

32

CHAPTER 2. MOMENTUM-SPACE WAVE FUNCTIONS

([HUFLVHV 1. Calculate the integral in equation (2.6) and show that it yields the result shown (2.7). 2. Starting with J calculate the integrals J and J by differentiating with respect t o kµ , as shown in equations (2.8) and (2.9). 3. Starting with J, use the recursion relation of equation (2.11) t o generate J, J and J. 4. Use the integrals JVO, in Table 2.1 to evaluate the Fourier transform of' the direct-space hydrogenlike orbitals and Show that the transforms correspond to the solutions of Fock, equation (2.15).

&KDSWHU  +<3(563+(5,&$/ +$5021,&6 +DUPRQLF SRO\QRPLDOV In the previous chapter, we saw that when 3-dimensional momentum space is mapped onto the surface of a 4-dimensional hypersphere by Fock’s projection, the Fourier-transformed hydrogenlike atomic orbitals are represented by 4-dimensional hyperspherical harmonics, multiplied by a universal factor, 0 ( S )  which is independent of the quantum num bers. We mentioned also that our aim will be to use the momentumspace hydrogenlike functions as basis sets for constructing atomic and molecular orbitals, as well as crystal orbitals. Thus, hyperspherical harmonics play a n important role in momentum-space quantum chemistry; and for this reason, we shall devote the present chapter t o discussing some of their properties. T h e simplest approach t o hyperspherical harmonics is through the theory of harmonic polynomials [30, 38, 296, 300]: If we consider a G-dimensional space, with Cartesian coordinates [1 , [ 2 , ..., [ G , then a KDUPRQLF polynomial is, by definition, a homogeneous polynomial which satisfies the generalized Laplace equation:

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superposition of monomial terms of the form (3.2) where (3.3) Thus, for example, + [ 1 [ 2 is a homogeneous polynomial of order 2, but it is not harmonic; whereas is a harmonic polynomial of order 2. Harmonic polynomials in a G-dimensional space are related t o hyperspherical harmonics by –

(3.4) where U is the K\SHUUDGLXV defined by (3.5) Thus, for example, apart from a normalization factor, (3.6) is a hyperspherical harmonic with principal quantum number = 2 in our G -dimensional space. Any homogeneous polynomial IQ can be decomposed into a series of harmonic polynomials multiplied by powers of the hyperradius:

(3.7)

In order to carry out this decomposition, we first notice that if we differentiate the monomial PQ of equation (3.2) with respect to one of the coordinates, we obtain: (3.8)

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40

CHAPTER 3. HYPERSPHERICAL HARMONICS

On the left-hand side of equation (3.37), we express this integral in terms of the hyperradius and hyperangles, while on the right-hand side, we express it in terms of the individual Cartesian coordinates. The integral over the hyperradius can be evaluated in terms of the gamma function: (3.38) and similarly, for each of the integrals on the right, we have (3.39) Combining equations (3.37)-(3.39),we obtain an expression for the total solid angle: (3.40) For example, when d = 3, this expression yields the familiar result: (3.41) while when d = 4 it gives (3.42) It is interesting to consider the integral of the product of two harmonic polynomials and taken over the generalized solid angle. If A A’, i.e. if the orders of the two harmonic polynomials are different, then (as we saw in the previous section) they are eigenfunctions of the grand angular momentum operator belonging to different eigenvalues. In that case, it follows from the Hermiticity of the operator with respect to integration over that

(3.43) For the case where A’ = 0, (3.43) reduces to (3.44)

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44

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Gegenbauer polynomials Each of the familiar theorems which one can derive for 3-dimensional spherical harmonics has a G-dimensional generalization. In the theory of hyperspherical harmonics, Legendre polynomials are replaced by Gegenbauer polynomials, which are defined by the generating function:

(3.65) w here

(3.66) (3.67) and

(3.68) The polynomials defined by this generating function are given by [ 153]:

(3.69) Equation (3.65) is the G-dimensional generalization of the familiar expansion of the Green’s function of the Laplacian operator in terms of Legendre polynomials:

(3.70) and in fact, the function x –x is the Green’s function of the generalized Laplacian operator, (3.27). When G = 3 and = 1/2, the

45 Gegenbauer polynomials reduce to Legendre polynomials. Table 3.2 shows the first few Gegenbauer polynomials for the case where G and are general, and in the special case where G 4 and 1. The familiar 3-dimensional spherical harmonics obey a sum rule involving Legendre polynomials: (3.71) Similarly, one can show that hyperspherical harmonics obey a sum rule involving Gegenbauer polynomials: (3.72) The sum rule for hyperspherical harmonics can be established in the following way: If we let [ [ [B, and note that A then from (3.34) we have –

(3.73) We now replace the Green’s function of the generalized Laplacian operator by its expansion in terms of Gegenbauer polynomials, (3.65). Equation (3.73) then becomes: (3.74) For the expansion in equation (3.74) to hold for all values o f t h e coordinates, each term must vanish separately, and thus (3.75) The radial operator in A, acting on

gives (3.76)

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46 and therefore

(3.77) In other words, x’) is an eigenfunction of the grand angular momentum operator corresponding t o the eigenvalue, + G – 2). However, we know that the Hilbert space of all such functions is spanned by the hyperspherical harmonics belonging t o this eigenvalue. Therefore . X’) as a linear combination of this we must be able to express set of hyperspherical harmonics: (3.78) From the invariance of that

+

X’)

under coordinate rotations, it follows (3.79)

where

is some constant. Thus,

(3.80)

If is a function of the hyperangles in a G-dimensional space, then it follows from (3.71) that

(3.81) onto the part of Hilbert space spanned will be a projection of by eigenfunctions of corresponding to the eigenvalue + G – 2). Of course, if we perform the projection on a function which is entirely within this subspace, the function will be unchanged; and this condition can be used to show that the constant is given by: (3.82) where ,(0) is the total generalized solid angle.

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48

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was developed by a number of Russian authors [176, 177, 184, 202, 230, 277, 282, 296, 297] and by Prof. V. Aquilanti and his co-workers at the University of Perugia [12, 13, 70] . We said above that an orthonormal set of 4-dimensional hyperspherical harmonics can be constructed according t o the chain of subgroups 62 (4) 62 (3) 62 (2) or according t o the alternative chain, 62 (4) 62 (2) x 62 (2). These two alternative ways of constructing an orthonormal set can be symbolized by the two WUHHV shown in Figure 3.1.

Figure 3.1

The meaning of these two alternative trees is as follows: If we wish to construct a set of hyperspherical harmonics according t o the first tree (the VWDQGDUG WUHH  we begin with the harmonic polynomials (3.90)

49 in the 2-dimensional space whose Cartesian coordinates are [ 1 and [ 2  Then (3.91) will be a set of homogeneous polynomials of order O in the 3-dimensional space whose coordinates are [1 , [ 2  and [ 3 . Equation (3.23) tells us that if we act on IO ([1 [ 2, [ 3 ) with the operator

(3.92) we will project out a set of harmonic polynomials KOP([1, [ 2, [ 3). (If we divide these harmonic polynomials by and normalize them, we will have obtained the familiar 3-dimensional spherical harmonics
+ +

(3.93) and we can once more use equation (3.23) to project out the harmonic polynomials of highest order contained in it, (x1, . . . ,x4) The standard tree can be used to construct an orthonormal set of hyperspherical harmonics in a space of arbitrarily high dimension: Suppose that we have already constructed an harmonic polynomial KO([ 1, ... , in a ( G – 1)-dimensional space. Then (3.94) will be a homogeneous polynomial of order in the G-dimensional space obtained by adding the final coordinate. Equation (3.23) tells us that if we act on this homogeneous polynomial with the operator (3.95)

50

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we will project out from it an harmonic polynomial which will also be of order The action of on is easy to calculate because (3.96) from which it follows that (3.97) Combining (3.94) (3.95) and (3.97), we have

(3.98) Looking carefully at the series in equation (3.98), we can see that it can be expressed in terms of a Gegenbauer polynomial, since (from equation (3.69))

(3.99) Comparing this with the series in equation (3.98), we can see that we ought t o let Q = – O and G = G + 2 O . With these substitutions, (3.99) becomes:

(3.100) Combining equations (3.98) and (3.100), we obtain the relationship (3.101)

( G – 2)/2, and where is a constant. If KO ([ 1, ..., is where already normalized, then the harmonic polynomial ( [ ) will also be

51 properly normalized if the constant

is chosen to be (3.102)

rather than the constant which results from projection. Equation (3.101) allows us t o generate an orthonormal set of hyperspherical harmonics in a space of any dimension; and indeed this is the method which was used to construct the standard 4-dimensional hyperspherical harmonics shown in equation (2.16). However, in physical applications, the standard hyperspherical harmonics are not necessarily the most convenient ones to use. Therefore we shall now turn our attention to non-standard trees, an example of which is the second tree shown in Figure 2.1. To make the argument as general as possible, let us consider a general fork joining two subspaces whose dimensions are respectively G1 and G2, where G1 +G2 = G Suppose that we have constructed the harmonic polynomials hO1 (x1) and KO2 (x2) in the two subspaces. We would like to use them as building blocks for constructing an harmonic polynomial of order in the G-dimensional space whose position vector is (3.103) If we let

(3.104) then (3.105) will be a homogeneous polynomial of order in the coordinates [ 1 , ..., [G , provided that and are even and + O 1 + O2 = If we act on (x) with the projection operator shown in equation (3.95), we will obtain an harmonic polynomial of order The simplest case is that for which = = 0 and O +O = In this simple case, the homogeneous polynomial shown in equation (3.105) is already harmonic because

+

(3.106)

52

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where

(3.107) The next simplest case is that for which so that

= 2,

= 0, and O1+ O2 +2 = (3.108)

Making use of equation (3.12), we have: (3.109) and (3.110) so that

(3.111) This can be rewritten in a more symmetrical form:

(3.112) from which we can see that no new linearly independent harmonic polynomial would have been generated if we had let = 0 and = 2. The reader might now be wondering why we do not also try a homogeneous polynomial of the form (3.113) since we could certainly project out from this an harmonic polynomial of order Our reason for not doing so is that we are considering the

 general fork S O ( d ) 62(G1) [ 62G  The indices O and O label irreducible representations of SO G  and 62 G  respectively, and this property would in general be lost if we multiplied by , but it is retained when we multiply by The powers of and must be even in order t h a t we should stay within the domain of polynomials in [  [ G  unless G 1 or G 1. To illustrate our discussion of the general fork, let us consider the alternative 4-dimensional hyperspherical harmonics constructed by means of the second tree shown in Figure 2.1. When ; 0, the only possible harmonic polynomial is just a constant, and this fits with the predicted degeneracy,  1)2 12 1. When ; 1, there are two linearly independent harmonic polynomials with O 1 and 0, namely 0 and 1, namely [  L[  This and two with also checks with the predicted degeneracy  1) 2 2 2 4. When ; 2, there are two harmonic polynomials with 2 and O 0: [  L[   two with O1 0 and O 2: [ four withO  1 and O2 1: ([ L[  and one with l 0, O 0 and 2: This makes a total of 9 linearly independent harmonic 2 , which checks with the predicted degenerpolynomials of order 2 2 acy,  1) 3 9. We can find the corresponding hyperspherical harmonics by dividing by r  and normalizing, as illustrated in Table 4.1. As we shall see, the alternative hyperspherical harmonics shown in this table correspond in Fock’s mapping to the Fourier transforms of the orbitals which result when the hydrogenlike Schrödinger equation is separated in paraboloidal coordinates, and the quantum numbers la beling the hyperspherical harmonics in Table 4.1 are those which are often used t o label the direct-space paraboloidal orbitals. –

–

One can show that for a general fork between subspaces of dimension d and G  , a n harmonic polynomial of order can be written in the form: (3.114)

where

and where

is a Jacobi polynomial. When

54

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(3.115) For example, when

= 3, O1 = 1 and O2

= 0, equation (3.115) becomes:

(3.116) which checks with equation (3.112). For more details concerning these relationships, the interested reader is referred t o the Russian literature and t o the papers of Aquilanti and his coworkers.

55 Table 3.1 Decomposition of monomials into harmonic polynomials The indices i j k are assumed to be all unequal. The terms in brackets are harmonic.



CHAPTER 3. HYPERSPHERICAL HARMONICS Table 3.2 Gegenbauer polynomials

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With the help of (6.44) and (5.74), we can also obtain the terms which are off-diagonal in the indices a1 and a:

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as a function of R = s/kµ can be generated by substituting these values of K² and K into equation (6.32). Such a curve, with 3 basis functions on each center, is shown in Figure 6.1, compared with the nearly exact, results of Koga and Matsuhashi [190-192]. Comparison shows that equation (6.32) yields a more accurate result with 3 basis functions on each center than could be obtained from (6.30) with 15 basis functions on each center. The numerical values of the ground-state energies for obtained using equation (6.32) with 3 basis functions on each center are shown in column a of Table 6.1, compared with the extremely high-precision results of Koga and Matsuhashi (column b ) and the best available direct-space results (column c).



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Exercise 4.3 From equation (4.38) and the associated Laguerre polynomials in Table 4.2, calculate the parabolic hydrogenlike orbitals shown in Table 4.3. Express these functions as linear combinations of

Solution Let t

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= r–z



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Exercise 8.1 Show that for j = 1/2, l = 0, and M = 1/2, the 4-component solution to the hydrogenlike Dirac equation can be written in the form:

What is the form of the solution corresponding t o j = 1/2, l = 0, and M = –1/2?

Solution From (8.8) we have:

while from (8.9) with l = j – l = 1,

161 Therefore (8.5) yields

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Exercise 8.2 Letting bµ = 1, find the values of k , nr , to (8.1) in the n = 1 and n = 2 shells.

N , and

for the solutions

Solution From equations (8.12)-(8.15) and (8.19) we obtain:

Exercise 8.3 The energies calculated in Exercise 8.2 include the electron rest energy mc² and are expressed in units of mc². Subtract the rest energy from the calculated values, and express the results in Hartrees.



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168 56. Bransden, B.H.; Noble, C.J.; and Hewitt, R.N., On the reduction of momentum space scattering equations to Fredholm form, J Phys B: At, Mol Opt Phys, Vol 26 (16), p 2487-99, 1993. 57. Brion, C.E., Looking at orbitals in the laboratory: the experimental investigation of molecular wave functions and binding energies by electron momentum spectroscopy, Int J Quantum Chem, Vol 29 (5), p 1397-428, 1986. 58. Brink, D.M. and Satchler, G.R., Angular Momentum Oxford University Press, 1968. 59. Calais, J-L.; Defranceschi, M.; Fripiat, J.G. and Delhalle, J., Momentum space functions for polymers, J Phys: Condens Matter, Vol 4 (26), p 5675-91, 1992. 60. Calais, Jean-Louis, Fukutome classes in momentum space, Theor Chim Acta, Vol 86 (1-2), p 137-47, 1993. 61. Calais, Jean-Louis, Orthogonalizationin momentum space, Int J Quantum Chem, Vol 35 (6), p 735-43, 1989. 62. Calais, Jean-Louis, Pathology of the Hartree-Fock method in configuration and momentum space, J Chim Phys Phys-Chim Biol, Vol 84 (5), p 601-6, 1987. 63. Chen, Joseph Cheng Yih and Ishihara, Takeshi, Hydrogenic- and Sturrnian-functionexpansions in three-body atomic problems, Phys Rev, Vol 186 (1), p 25-38, 1969. 64. Chiu, T.W., Non-relativistic bound-state problems in momentum space, J Phys A: Math Gen, Vol 19 (13), p 2537-47, 1986. 65. Cinal, Marek, Energy functionals in momentum space: exchange energy, quantum corrections, and the Kohn-Sham scheme, Phys Rev A, Vol 48 (3), p 1893-902, 1993. 66. Clark, Charles W. and Taylor, K.T., The quadratic Zeeman effect in hydrogen Rydberg series: application of Sturmian functions, J Phys B, Vol 15 (8), p 1175-93, 1982.

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171 87. Defranceschi, M.; Suard, M.; and Berthier, G., Epitome of theoretical chemistry in momentum space, Folia Chim Theor Lat, Vol 18 (2), p 65-82, 1990. 88. Defranceschi, M., Theoretical investigations of the momentum densities for molecular hydrogen, Chem Phys, Vol 115 (3), p 34958, 1987. 89. Defranceschi, Mireille and Delhalle-Joseph., Numerical solution of the Hartree-Fock equations for quasi-one- dimensional systems: prototypical calculations on the (hydrogen atom) x chain, Phys Rev B: Condens Matter, Vol 34 (8, Pt. 2), p 5862-73, 1986. 90. Defranceschi, Mireille and Delhalle-Joseph, Momentum space calculations on the helium atom, Eur J Phys, Vol 11 (3), p 172-8, 1990. 91. Delande, D. and Gay, J.C., The hydrogen atom in a magnetic field. Spectrum from the Coulomb dynamical group approach, J Phys B: At Mol Phys, Vol 19 (6), p L173-L178, 1986. 92. Delhalle, Joseph and Defranceschi and Mireille, Toward fully numerical evaluation of momentum space Hartree-Fock wave functions. Numerical experiments on the helium atom, Int J Quantum Chem, Quantum Chem Symp, Vol 21, p 425-33, 1987. 93. Delhalle, Joseph; Fripiat, Joseph G.; and Defranceschi, Mireille, Improving the one-electron, states of ab initio GTO calculations in momentum space. Tests on two-electron systems: hydride, helium, and lithium,(1+), Bull Soc Chim Bclg, Vol 99 (3), p 135-45, 1990. 94. Delhalle, Joseph and Harris, Frank E., Fourier-representation method for electronic structure of chainlike systems: restricted Hartree-Fock equations and applications to the atomic hydrogen (H)x chain in a basis of Gaussian, functions, Phys Rev B: Condens Matter, Vol 31 (10), p 6755-65, 1985. 95. Deloff, A. and Law, J., Sturmian expansion, method for bound state problems, Phys Rev C, Vol 21 (5), p 2048-53, 1980.

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Index Amos, A.T., 135 Angular integrals, 135 Angular integration, 41, 42 Angular integration theorem, 85 Antisymmetry, 11, 112 Antonsen, Frank, 5, 7 Aquilanti, Vincenzo, 2, 3, 5, 7, 50, 71 Automatic optimization of basis set, 81 Bang, Jens, 5, 7 Basis potential, 10 Basis set truncation, 101 Bian, Wensheng, 7 Chain of subgroups, 45, 50, 66, 67 Clebsch-Gordan coefficients, 12 2, 142 Clementi, Enrico, 18, 133 Coletti, Cecilia, 7 Combined Coulomb attraction and repulsion, 82 Completeness relations, 70 Correlation, 6 Coulomb integrals, 18, 138 Coulson, Charles, 2 Duchon, C., 3 Decomposition of a homogeneous polynomial, 36

193

Decomposition of monomials, 57 Degeneracy of hydrogenlike orbitals, 2 Degeneracy of hyperspherical harmonics, 49 Diatomic molecules, 3, 104 Dirac equation, 121, 124 Dirac-Coulomb equation, 128 Dumont-Lepage, M.C., 3 Duncanson, W.E., 2 Dynamical symmetry, 2 Effective nuclear charges, 123 Exchange integrals, 18, 138 Excited states, 6, 18, 22 Expansion of the kernel, 77 Exponential decay, 81 ' Fernandez Rico, Jamie, 6, 7 Fock, V., 2, 77 Fock’s projection, 2, 27, 35, 63 Fock's treatment of hydrogenlike atoms, 27, 63 Forks, 53 Fourier convolution theorem, 68, 97 Fourier transform, d-dimensional, 61 Fourier transformed hydrogenlike orbitals, 2, 25, 30, 35 Fourier transforms, 97, 102

194 Gamma function, 41, 138 Gazeau, J.P., 3, 5 Gegenbauer polynomials, 46, 52, 58, 65, 82, 86 Generalized Laplace equation, 35 Generalized Laplacian operator, 4, 9, 37 Generalized Slater-Condon rules, 135 Generalized Sturmian basis functions, 2, 10 Generalized solid angle, 41, 64 Generating function, 46 Goscinski, Osvaldo, 4 Grand angular momentum, 2,40 Green’s function, 98 Green’s function of the generalized Laplacian, 46 Hall, G., 135 Hansen, Tom Børsen, 7 Harmonic polynomials, 35, 45, 85 Harmonic polynomials, orthogonality, 42 Hartree-Fock approximation, 18, 133 Hermiticity, 12 Herschbach, Dudley R., 5, 7 Homogeneous polynomials, 35, 142 Hydrogen molecule, 115 Hydrogen molecule ion, 106 Hydrogenlike Sturmian basis functions, 102 Hydrogenlike Sturmians in d dimensions, 75 Hydrogenlike atomic orbitals, 10,

INDEX

23, 30 Hydrogenlike orbitals in parabolic coordinates, 74 Hyperangles, 42 Hyperangular integration theorems, 43, 85 Hypergeometric function, 16, 138 Hyperradius, 36, 41 Hypersphere, 63 Hyperspherical harmonics, 2, 21, 27, 31, 35, 45, 72, 77 Incomplete gamma function, 138 Integral equation, 61, 100 Inter-configurational matrix elements, 135 Interelectron repulsion, 114 Interelectron repulsion matrix, 15, 138 Interelectron repulsion potential, 13 Irreducible representations, 55 Isoelectronic series, 17, 133 Iteration of one-electron equation, 100 Iteration of the many-electron wave equation, 97 Jacobi polynomials, 55 Jeziorski, B., 3 Johansen, W.R., 133 Judd, Brian, 2 Kernel of the momentum-space wave equation, 77 Kinetic energy term, vanishing of, 6, 98 Koga, Toshikatsu, 2, 3, 81, 101, 105, 106, 108 Löwdin, Per-Olov, 4, 68, 131, 135

INDEX López, Rafael 6, 7 Laguerre polynomials, 4 Laguerre polynomials, associated, 67, 74 Laplace equation, generalized, 35 Laplacian operator, generalized, 37 Legendre polynomials, 15, 46 Lie groups, 2 Loeser, John, 7 Many-center Coulomb potential, 2 Many-center one-electron equation, iteration, 100 Many-center problem, 20, 77 Many-center problem in direct space, 102 Many-electron Sturmians, 6, 111 Many-electron problems, 2, 9 Many-particle Sturmians, 9 Many-particle wave equation, 97 Maquet, Alfred, 5 Mass-weighted coordinates, 62 Matsuhashi, T., 3, 81, 101, 105, 106, 108 McWeeny, Roy, 2, 7 Method of trees, 3, 49 Molecular Sturmians, 6, 111 Momentum-space Schrödinger equation, 2 Momentum-space hydrogenlike orbitals, 28, 35 Momentum-space orthonormality of Sturmians, 67, 88 Momentum-space quantum theory, 2 Momentum-space wave equation,

195 61, 78 Momentum-space wave functions, 25 Monkhorst, Hendrik J., 3 Monomial, 36 Normalization, 5, 113, 116 Novosadov, B.K., 3 Nuclear attraction, 114 Nuclear attraction integrals, 89, 104, 107 Nuclear attraction potential, 10, 13 Nuclear physics, 5 Orthogonality of harmonic polynomials, 42 Orthonormality of Sturmians in Momentum space, 67 Orthonormality of hyperspherical harmonics, 45, 65, 84 Orthonormality relation, 12 Orthonormality, potential-weighted, 69, 124 Parabolic coordinates, 3, 55, 66, 72, 74 Pauli principle, 12 Pauli spin matrices, 121 Pauling, Linus, 2 Plane waves, Sturmian expansion of, 71, 88, 98 Plante, D.R., 133 Podolski, B., 2 Potential-weighted orthonormality relation, 5, 12, 14, 69, 104, 124, 128 Relativistic effects, 121 Relativistic energy, 121, 125, 133 Relativistic hydrogenlike Sturmi-

196 ans, 121 Relativistic many-electron Sturmians, 127 Relativistic radial functions, 123 Rotenberg, Manuel, 4 SCF method, 6 Sapirstein, J., 133 Sauer, Stephan, 6, 7 Schrödinger equation, 5 Second- iterated equation, 3, 81, 101 Secular equation, 5, 13, 14, 80 Secular equation, ith order 99, 102 Secular equation, accurate 102, 104, 113 Secular equation, many-electron 114 Secular equation, many-electron, relativistic 128 Secular equation, relativistic 125 Self-adjointness, 128 Self-consistent-field approximation, 2 Shells, filling of, 18 Shibuya, Tai-ichi, 2, 77, 84 Shibuya-Wulfman integrals, 79, 83, 90, 94 Shibuya-Wulfman integrals and translations, 88 Shim, Rune, 7 Shull, H., 4, 68 Slater determinants, 21, 135 Slater exponents, 6 Slater-Condon rules, generalized, 135 Sobolev space, 70

INDEX Solid angle, generalized, 41, 64 Spherical Bessel functions, 26,44 Spherical spinors, 122, 124 Standard tree, 50 Stark effect, 3 Sturm-Liouville theory, 4 Sturmian basis functions, 3 Sturmian expansion of plane waves, 71, 88, 98 Sturmian overlap integrals, 118 Sturmian secular equation, 14, 81 Subgroups, chain of, 45, 49, 66, 67 Subsidiary conditions, 10, 11, 14, 17, 20, 112, 114, 115 Sum rule for hyperspherical harmonics, 47, 65 Total generalized solid angle, 48 Translation group, 89 Trees, method of, 49 Truncated basis set, 101 Unit hypersphere, 2 Unitary transformation, 135 Vaagen, Jan, 5, 7 Weighting factor, 20, 112 Weniger, E., 70 Wigner coefficients, 84 Wulfman, Carl E., 2, 7, 77, 84 Zeeman effect, 3