Iteration of the Schrödinger equation starting from Hartree–Fock wave functions

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY. VOL. 42.87-101 (1992)

Iteration of the Schrodinger Equation Starting from

Hartree-Fock Wave Functions JOHN AVERY AND FRANK ANTONSEN

Department of Physical Chemistry, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark

Abstract

Methods are discussed for iterating the many-particle Schrodinger equation starting from Hartree-Fock wave functions. It is shown that when the Hartree-Fock wave function is expressed as a linear combination of the appropriate basis functions in hyperspherical coordinates the hyperangular integrals needed for iteration can be evaluated analytically. It is also shown that the first-iterated solutions contain terms corresponding to angular correlation.

Introduction

Let us consider an N-particle system whose Hamiltonian is given by N r .. 1

H = - ~ - V j ’ + V ( x ) ; j = l 2mj

then the wave function of the system will satisfy the Schrodinger equation

( H - = 0, (2)

where

x = (XbX2,. . . , X N ) (3)

is a vector in a space with dimension d = 3N. If we introduce a d-dimensional wave vector

k = (ki, kz,. . , b), (4)

then the Fourier transform of the wave function can be written as

@(k) = (2?r)-”’ I dx exp(ik . x)+(x),

where

8 1992 John Wiley & Sons, Inc. CCC 0020-7608/92/010087-15$04.00

88 AVERY AND ANTONSEN

Conversely, the wave function can be expressed in terms of its Fourier transform:

$(XI = ( 2 ~ ) - ~ / ' J dk exp(-ik * x)$'(k) , (7)

where

I dk = I d3k, I d3k2.. . I d3k,.

Substituting (7) into (2), multiplying on the left by exp(-ik' x), and integrating over dx, we obtain the many-particle reciprocal space Schrodinger equation:

(ki + Kf2)$'(k') = - 2 ( 2 ~ ) - ~ " I dkV'(k' - k)$'(k), (9)

where

k i I -2E

and

V'(k' - k) I ( 2 ~ ) - ~ " dx exp{i(k' - k) * x}V(x). (12)

In Eq. (ll), mi represents the mass of the jth particle expressed in units of the electron mass. If $(O'(x) and $(')'(k) represent the zeroth-order wave function and its Fourier transform, while $(')(x) and $("'(k) represent the first-order functions, then the reciprocal space Schrodinger equation can be iterated by writing

(ki + Kf2)$('"(k) = - 2 ( 2 ~ ) - ~ " I dkV'(k' - k)$'"'(k). (13)

This iterative approach to the solution of the reciprocal space Schrodinger equation was tried as early as 1941 [l-61 by Coulson, Duncanson, and McWeeny, but the integrals proved to be difficult and the method was not developed further at that time. In the present paper, we will try to show that, with the help of hyperspherical coordinates, the integrals needed can be evaluated without difficulty and that it therefore might be interesting to take a fresh look at the iterative method of Coulson, Duncanson, and McWeeny. We will also try to show that Hartree-Fock solutions can be used as a starting point for iteration.

The Green's Function

Equation (13) can be put into a more convenient form by means of a general- ization of the Fourier convolution theorem [7], from which it follows that

dk'Y'(k - k')$(')'(k') = dx exp(ik . ~)V(X)$(~)(X). (14)

ITERATION OF THE SCHRODINGER EQUATION 89

Thus, (13) can be rewritten in the form

(K’ + ki)$(”‘(k) = - 2 ( 2 ~ ) - ~ ” I dx exp(ik * x)V(x)$(”‘(x). (15)

Dividing (15) by K’ + k ; and taking the Fourier transform of both sides, we obtain:

which can be rewritten in the form:

$(‘)(x) = - dx’G(x - x ‘ )V (~ ’ )$ (~ ) (x ’ ) , (17)

where

exp(ik * (x - x’)) K’ + k i

G(x - x’) = 2 ( 2 ~ ) - ~ I dk

If mass-weighted coordinates are used, then mi can be set equal to 1 in Eqs. (1) and (11). In that case,

K’ = k 2 k * k (19)

and the Green’s function shown in Eq. (18) can be expanded in terms of Gegen- bauer polynomials. One can show [7] that a many-dimensional plane wave can be expanded in the form

m

exp(ik . x) = 2 iA(d + 2A - 2) (d - 4)!!j,d(kr)C,“(uk * u ) , (20) A=O

where C,” is a Gegenbauer polynomial with a = ?hd - 1 and where m

(21) - r(a)2a-’Jo+A(kr) - ( -l)‘(kr)’‘+*

j,d(kr) c f = O (2t)!!(d + 2t + 2A - 2)!! (d - 4)!!(kr)”

is a “hyperspherical Bessel function.” Equation (20) is the d-dimensional gener- alization of the familiar expansion

m

exp(ik * x) = x i ’ (21 + l) j l(kr)fi(uk - u) (22) 1=0

and reduces to (22) if we let d = 3 and A = 1. In Eqs. (20) and (22), uk and u are unit vectors pointing, respectively, in the directions of k and x. The Gegenbauer polynomials that appear in &. (20) can be shown to obey the orthogonality rela- tions [7]


where

and where

I(0) = dR.

The generalized solid-angle element dR that appears in (23) and (25) is defined by the relation

dx = rd- ’drdR, (26) while the hyperradius, r, is defined by

Expanding the two plane waves of Eq. (18) independently by means of (20), let- ting K’ = kZ, and making use of the orthormality relation (23), we obtain [7]

O D . 1

C(X - x‘) = 2 - C ~ ( U * u’)CA(r,r’), A=O K A

where

and where JA+,(kr) is a Bessel function of order A + a. The integral over k can be evaluated exactly [8], yielding

where

(r,r’) r < r‘ ( r ’ , r ) r > r’ ( r < , r > ) =

In Eq. (30), I and K are modified Bessel functions [9]. From the definition of the Gegenbauer polynomials,

it follows that the Green’s function can be written in an alternative form: m

G(x - x’) = C (U - u’yGn(r7r’), n=n

(33)


where

Iteration in the n-Representation

In order to express Hartree-Fock wave functions in terms of hyperspherical coordinates, we begin by noticing that

where

t E 3a - 2 . (36)

By means of (39, we can express a Hartree-Fock wave function as a linear combination of hyperspherical basis functions of the form

N 4a,m,,,(x) = e-8" n rpx2y2+lz;+*

a=l

or, alternatively, in terms of basis functions of the form

(37)

where d

j=l

and

(39)

The nj's in Eqs. (37)-(40) are positive integers or zero. The expression (u * u')", which appears in Eq. (33), can be expanded in terms of the functions x.(n) CEq. (4011:


where

(42) 1 1 r r u = -(x1,yl,zl,xZ,yz,. . . , ZN) -(xl,xzr. . . , X d )

and where the prime on the summation in Eq. (41) means that the sum is re- stricted to values of the nj’s for which nl + n2 + ... + nd = n. For a system of particles interacting through Coulomb forces, the potential can be written in the form

where Z(n) is independent of the hyperradius, r. Combining Eqs. (26), (33), (38), and (43), we can see that

where

Rn(r‘) = Jomdr rd-’R(r)Gn(r, r’) .

If we substitute (41) into the hyperangular integral in (44), we obtain

(45)

where

Thus, if the zeroth-order wave function has the form

@O’(x) = R(r)W(R),

where

w(n) = C a !io’xn(n) 3

n

then the first-order wave function will have the form

(49)

with


and where the first-order and zeroth-order coefficients are related by

Let us now turn to the problem of evaluating the integrals, Z,,, defined by Eq. (47). One can show [7] that

where

n = nl + n2 + ... + nd

v, = n, + n1+1 f nr+2

t z 3 a - 2

d = 3N

and where

Z(n) = I dRx,(R) = Z(0)ho

If all the nj’s are even, ho is given by

(54)

while if any of the nj’s are uneven, ho vanishes. In addition to integrals of the form J(n) [Eq. (53)], we need to evaluate integrals of the form

where


If we transform to the relative coordinates

x + = 2-”2(x. + X b )

2+ = 2-1/72. + 2 6 )

x - = 2-”2(x, - X b )

y- = 2-’”(y, - yb)

2- = 2--”2(2, - Zb),

y+ = ~- ‘” (Yu + y6)

(59) then

and

K(n) = 2-1’2~J(n’)UII~,. , II‘

where Urn,,, is the transformation matrix needed to express x.(R) as a linear combination of functions of the same form in the rotated coordinate system of Eq. (59). A method for generating the transformation matrix U,,,,. (based on a modification of Pascal’s triangle) is discussed in Appendix C of Ref. 7.

Iteration in the A-Representation

In the previous section, we discussed methods for iterating Eq. (17) when the Green’s function is represented by the expansion shown in Eq. (33). Let us now consider iteration using the expansion shown in Eq. (28).

The Gegenbauer polynomials, C?(u * u’), are the d-dimensional generaliza- tion of the Legendre polynomials, and, just as the familiar 3-dimensional spherical harmonics obey a sum rule involving Legendre polynomials,

(62) 21 + 1 2 Xm(R)Y&(R’) = 7 Pl(u . u’) ,

m

the hyperspherical harmonics, YA,(R), obey a sum rule involving Gegenbauer polynomials:

where KA is given by Eq. (24). From (62), it follows that if F(R) is an angular function for d = 3, then

O,[F(R)] = 2 Yh(R) I dR’Y&(fl’)F(R’) = y I dR’B(u * u’)F(R‘) m

(64)


is the component corresponding to 1 in the decomposition of F(R) into eigenfunctions of angular momentum. Similarly, if F(R) is an angular function in a d-dimensional space, then

1 O,[F(R)] = I: Y,(R) 1 dR’Y?JR’)F(R’) = 1 dR’CI(u . u’)F(R’) (65)

Ir.

is the component corresponding to A in the decomposition of W(R) into eigenfunctions of the generalized angular momentum operator, A’:

of which the hyperspherical harmonics and the Gegenbauer polynomials are eigenfunct ions:

A’Y,,(n) = A(A + d - 2)Y~,(n)

A’C,“(u * u‘) = A(A + d - 2)Cf(u * u’). (67)

Substituting (28) into (17), we can see that if

+‘O)(x) = R(r)W(R) ,

then

where

and

(71)

From Eq. (65), it follows that if On is an operator that projects out from any function the component corresponding to the generalized angular momentum quantum number A, then

1 K A

wA(n’) = - 1 dR cp(u * u’)w(n)z(fk).

wA(n) = oA[w(fl)z(n)l* (72) One can show [7] that for even values of A,

and

%

where

AVERY AND ANTONSEN

iA(d + 2A - 2)r A =

For example,

(75)

and so on. Only the even projections are nonzero because Z(n) has even parity under the inversion operation, x + -x, and the parity of eigenfunctions of A' is given by (-l)A. By means of Eqs. (73) and (74), Z ( n ) can be written in the form

(78) 1 1

Z ( n ) = ho + 7 h 2 + ph4 + .. . ,

where hA = rAOAIZ(n)] is a homogeneous polynomial obeying

We now let A represent the generalized Laplacian operator:

From (79) and (80), and from the fact that rWAhA is a pure function of the hyper- angles, it follows that

In other words, hr is an harmonic polynomial. As a simple example of the resolu- tion of Z ( n ) into harmonic polynomials, we can consider the case of a neutral N-electron atom, where

ITERATION OF THE S C H R ~ I N G E R EQUATION 97

Then, from (73) and (74), we obtain the leading terms:

If W ( n ) is similarly resolved into harmonic polynomials, so that

1 1 r r2 w(n) = wo + -w1 + -w2 + ...,

where AwA = 0, then

O,@"n)Z(n)] = Oh

1 r

+ ~ ( ~ o h 2 + ~ 2 h o ) + ...

and

and so on. Having expressed Z(n) and W ( n ) in terms of the harmonic pdy- nomiab, hA, we can perform the projections by successive applications of the generalized Laplacian operator with appropriate coefficients, as discussed in Ap- pendix A. Notice that since both wA and hr, are harmonic, only the cross-terms survive when we apply A to their product:

and, similarly,


For example, if

X . w(n) = A , r

then

1 ah2 d + 2 ax,

01[~jZ(n)] = xjho + --

xjhz 1 ah2 1 ah4 o,[xjz(n)] = 2 - -- +

r d + 2 ax, r2(d + 6)z

~ j h 4 1 ah4 + o,[xjz(n)] = - - r 4 rz(d + 6)z r4(d + 1O)z

and so on, the even projections being zero. Thus, through the theory of harmonic polynomials and hyperspherical harmonics, the angular integration problem im- plied by Eq. (71) can be converted into a problem of differentiation!

Discussion

In the previous sections, we have discussed two alternative methods for iterating the Schriidinger equation, starting from Hartree-Fock wave functions. In both cases, we saw that if the Hartree-Fock solutions are expressed as a linear combination of the appropriate hyperspherical basis functions the hyperangular integrals needed for iteration can be evaluated analytically. Thus, when hyperspherical coordinates are used, the iterative method is not blocked by intractable integrals, although whether or not it can compete with conventional methods for treating correlation remains to be seen. We can see immediately, however, that the iterated solutions contain correlation effects. For example, the Hartree-Fock solutions for helium S-states are independent of the angle between x1 and XZ; but from Eqs. (84) and (87), we can see the iterated solutions contain a term propor- tional to x1 * xz, corresponding to angular correlation.

Appendix A Harmonic Projectien

Let xl, xz, . . . , x d be the Cartesian coordinates of a d-dimensional space, and let d

fn * llxi., j=l

where the nj’s are positive integers or zero and

Then,

af. ax,

z x j - = nfn.


From (3) it follows that if A is the generalized Laplacian operator

and if r is the hyperradius

then

Let h, be a homogeneous polynomial of order a satisfying

Ah, = 0 . (A.7)

Such a homogeneous polynomial is said to be harmonic. We would like to re- solve f. into a series of harmonic polynomials of the form

(A.8)

Since h, is a linear combination of terms of the form shown in Eq. (l), it follows from (6) that

(A.9)

f. = h, + r2h,-2 + r4h,-4 + . . a .

A(r’h,) = p(@ + d + 2 a - 2)r’-2ha.

Applying A repeatedly to both sides of (A.8), we obtain

Af, = 2(d + 2n - 4)h,-2 + 4(d + 2n - 6)r2h,-4 + -.- AY. = 8(d + 2n - 6)(d + 2n - 8)h,-4 + .-. (A.lO)

and, in general,

(d + 2n - 2k - 2)!! rwr-2uhn-wr. (A.ll) [.El (2k)!!

= zu (2k - 2v)!! (d + 2n - 2k - 2v - 2)!!

For example, when n is even and Y = n/2, we obtain

n!!(d + n - 2)!! (d - 2)!! Ani2fn = ho

or

(d - 2)!! ho = An!?..

n!(d + n - 2)!!

(A.12)

(A.13)


Equations (10) or (11) constitute a set of simultaneous equations that can be solved for ha. For the first few values of n, we obtain

f2 =

ho =

hz =

f3 =

hi =

h3 =

f4 =

ho =

h2 =

h4 =

and, in general [7],

h2 + r2ho

1 g A f 2

r2 f 2 - A f 2

h3 + r2hl

1 2(d + 2)

f 3 - 2(d + 2)

A f 3

r2 A f 3

h4 + r2h2 + r4ho

A'f4 1

8d(d + 2)

A 7 4 1 r2

2(d + 4) Af4 - 4d(d + 4)

r 2 r4 f4 - 2(d + 4) Af4 + 8(d + 2)(d + 4) A2f4 7 (A.14)

(d + 2n - 41, - 2)!! (2u)!!(d + 2n - 21, - 2)!! hn-2v = .

[fl'2-u1 (-l)"(d + 2n - 41, - 2k - 4)!! Zk k + v r A f f l . (A.15) z0 (2k)!!(d + 2n - 41, - 4)!!

The harmonic polynomials, h A, are eigenfunctions of the generalized angular momentum operator, A2:

Thus, if OA is a projection operator corresponding to the A-th eigenfunction of

OA[ fn] = +hA. (A.17)

Since A2 is Hermitian with respect to integration over the generalized solid an-

A2,

gle, a, I d a hrhr, = 0 if A # A ' . (A.18)


But ho is a constant and, therefore, (18) implies that

/ t i a h r = 0 i f A # O . (A.19)

Thus, if n is even,

/ dRf, = / dR (h, + r2hn-2 + + r”h0) = Z(O)r‘ho, (A.20)

where Z(0) E JdR. From (1) and (13), it follows that if all the nj’s are even

(d - 2)!! ho = n (nj - l)!!, (d + n - 2)!! j = l

(A.21)

while if any of the nj’s are uneven, ho vanishes. Equations (A.20) and (A.21) provide us with a powerful theorem for evaluating angular integrals in a d-dimensional space.

Bibliography

[l] C. A. Coulson, Proc. Camb. Philos. SOC. 37, 55, 74 (1941). [2] C. A. Coulson and W. E. Duncanson, Proc. Camb. Philos. SOC. 37,67 (1941). [3] W. E. Duncanson and C. A. Coulson, Proc. Camb. Philos. SOC. 37, 406, (1941). [4] W. E. Duncanson, Proc. Camb. Philos. SOC. 37,37 (1941). [5] R. McWeeny, Proc. Phys. SOC. (Lond.) A62,519 (1949). [6] T. Koga, J. Chem. Phys. 83,2328 (1985). [7] J. Avery, Hyperspherical Harmonics; Applications in Quantum Theory (Kluwer Academic,

[8] J. Avery, Chem. Phys. Lett. 138, 520 (1987). [9] I. S. Gradshteyn and I. M. Ryzhik, “Tables of Integrals, Series and Products” (Academic

Press, 1965, page 679, formula 6.541.1).

Dordrecht, Netherlands, 1989).

Received June 6, 1990 Accepted for publication November 29, 1990

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Iteration of the Schrödinger equation starting from Hartree–Fock wave functions