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Transformation of the helium atom Schrödinger equation into integral form
Chemical Physics 228 Ž1998. 61–72
Transformation of the helium atom Schrodinger equation into ¨ integral form Hendrik F. Hameka Department of Chemistry, UniÕersity of PennsylÕania, Philadelphia, PA 19104, USA Received 18 December 1996
Abstract We define the Green’s function for two identical particles moving in a Coulomb field. We use this Green’s function as a basis for deriving an integral equation of the helium atom Schrodinger equation by treating the electron repulsion term as a ¨ perturbation. The two-particle Green’s function is further simplified by means of a contour integration. An additional transformation does not lead to further simplification of the result. q 1998 Elsevier Science B.V.
1. Introduction It is possible to transform a differential equation with boundary conditions into an integral equation. The transformation requires the introduction of a Green’s function, which may be derived from the eigenvalues and eigenfunctions of the differential equation. In general the two approaches are equivalent, that is the required efforts to solve either the differential equation or the integral equation are of comparable magnitude. However, in quantum mechanical problems where approximate solutions are derived by means of perturbation theory the integral approach may be more convenient than the differential approach. This is the case in situations where the Green’s function may be reduced to a more simple form. For instance, the Green’s function technique has been quite effective in dealing with scattering problems since the free-particle Green function has a very simple from. It is possible to use the Green function approach also for the study of bound systems but the applications to atomic and molecular structure have met with limited success only. In the present paper we wish to show how the introduction and evaluation of a Green’s function may be used to transform the helium atom Schrodringer equation into an integral equation which may be solved by means of ¨ iterative procedures. We first present the relation between perturbation theory and the Green’s function in order to define the latter function. We consider a Hamiltonian H Ž x . of a set of variables that we denote symbolically by x and we assume that H Ž x . may be taken as the sum of an unperturbed Hamiltonian H0 Ž x . and a perturbation lV Ž x ., H Ž x . s H0 Ž x . q lV Ž x . . Ž 1. We denote the eigenvalues of H0 Ž x . by En and f nŽ x . and the eigenvalues and eigenfunctions of H Ž x . by e n and cnŽ x ., H 0 Ž x . f n Ž x . s En f n Ž x . , Ž 2a. H Ž x . c n Ž x . s e n cn Ž x . . Ž 2b . 0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 1 - 0 1 0 4 Ž 9 7 . 0 0 2 8 9 - 9
H.F. Hamekar Chemical Physics 228 (1998) 61–72
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We define the Green’s function GŽ x, xX ; e . and the modified Green’s function G Ž1. Ž x, xX ; e 1 . of H0 Ž x . by G Ž x , x X ; e . s y Ý Ž x . Ž En y e .
y1
f n Ž x . f n) Ž xX . ,
Ž 3a .
n
G Ž1. Ž x , xX ; e 1 . s
Ý Ž En y e 1 . f n) xX . .
Ž 3b .
n/1
It is then easily shown that the eigenvalue e 1 and eigenfunction c 1Ž x . of H Ž x . are determined by the following equations
c 1 Ž x . s f 1 Ž x . y lHG Ž1. Ž x , xX ; e 1 . V Ž xX . c 1 Ž xX . d xX ,
Ž 4a .
e 1 s E1 q lHf 1) Ž x . V Ž x . c 1 Ž x . d x .
Ž 4b .
It is assumed here that the eigenvalue e 1 is nondegenerate. Eqs. Ž4a. and Ž4b. have general validity but they are of practical use only if the perturbation lV is small so that the eigenvalue spectra of H0 Ž x . and H Ž x . are similar and the differences between e 1 and E1 and between f 1Ž x . and c 1Ž x . are small. The above approach may be used for perturbation calculations on the hydrogen atom since the infinite expansion Ž3. may be reduced to a relatively simple form. It was shown as early as 1933 by Meixner w1x that in the case of the one-dimensional Kepler problem the infinite sum over the bound states and the integral over the continuum states could be reduced to a single term only by means of a clever contour integration in the complex plane. Meixner w1x also derived the three-dimensional hydrogen atom Green’s function in terms of parabolic coordinates. More recently Hostler w2x derived the hydrogen atom Green’s function for the nonrelativistic three-dimensional Schrodinger equation in closed form by means of a partial-wave expansion. Hostler obtained ¨ the Green’s function as a sum of products of Whittaker functions of different arguments. At almost the same time Schwinger w3x derived an expression for the momentum space Green’s function of the hydrogen atom. We derived the three-dimensional hydrogen atom Green’s function as an expansion in terms of spherical harmonics w4,5x by means of procedures similar to Meixner’s treatment of the one-dimensional Kepler problem w1x. It is possible to obtain exact analytical solutions for a variety of perturbation effects on the hydrogen atom by making use of the hydrogen atom Green’s function so that the derivation of the Green’s function has led to some useful applications. On the other hand, the extension of this approach to larger atomic and molecular systems has met with little success up to now. The first step in extending the Green’s function approach to more complex systems is to tackle the next largest atom, which is the helium atom. We therefore report our efforts to derive Green’s function of the helium atom, that is to say the Green’s function of the helium atom without the electron repulsion term. If we are successful in this attempt then we may treat the electron repulsion term as a perturbation so that the helium atom Schrodinger equation is transformed into an integral equation. The efficiency of this approach depends on the ¨ extent that we can simplify the Green’s function. Many years ago we presented a derivation of the helium atom Green’s function w6x and we reported a relatively simple expression. Unfortunately, we reported subsequently that this derivation and the simple result were incorrect w7x, at the time we failed to report the correct derivation. In the present paper we report the correct derivation and result of the helium atom Green’s function. We also report an interesting transformation of the Green’s function which is derived by a contour integration in the complex plane. Alternative approaches to the solution of the two-electron atomic Schrodinger equation were reviewed by ¨ Bethe and Salpeter w8x and by Hylleraas w9x. The latter review emphasizes the work by Kinoshita w10,11x and by Pekeris w12–14x. Our work on the helium atom is to some extent an expansion of the previous work on the hydrogen atom. We therefore review the hydrogen atom derivation first, this also presents an opportunity to define the various functions and relations that are used in the subsequent sections.
H.F. Hamekar Chemical Physics 228 (1998) 61–72
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2. The hydrogen atom Green’s function In deriving the hydrogen atom Green’s function we follow the customary procedure in multiplying the Hamiltonian by a factor minus two. The discrete eigenvalues are then given by En s 1rn2
Ž 5.
and the corresponding radial eigenfunctions f n,l Ž r . are f n ,l Ž r . s Ž 2rn2 . Ž n q l . !r Ž n y l y 1 . !
1r2
Ž 2 l q 1. !
y1
Ž 2 rrn .
l
=exp Ž yrrn . 1 F1 Ž yn q l q 1, 2 l q 2; 2 rrn . .
Ž 6.
The continuum state radial eigenfunctions g l Ž r, s . are g l Ž r , s . s Ž 2rp .
1r2
G Ž l q 1 q irs .
=G Ž l q 1 y irs . x
1r2
Ž 2 l q 1. !
y1
l
s Ž 2 s r . exp Ž pr2 s .
= exp Ž yi s r . 1 F1 Ž l q 1 q irs , 2 l q 2; 2i s r . .
Ž 7.
The corresponding energies are E Ž s . s ys 2 .
Ž 8.
The function 1 F1Ž a, c; x . is the well-known confluent hypergeometric function w15x. In our approach we expand the hydrogen atom Green’s function G in terms of spherical harmonics `
Gs
Ý Ž 2 l q 1. r4p
Pl Ž cos u X . G l Ž r , r X ; d . ,
Ž 9.
1s0
where u X is the angle between the vectors r and rX and the functions G l are defined as Gl Ž r , r X ; d . s
`
H0 g Ž r , s . g l
) l
Ž rX , s . w s 2 q d 2 x
y1
ds
`
q
Ý
f n , l Ž r . f n , l Ž r X . w yny2 q d 2 x
y1
sIqS .
Ž 10 .
n s lq1
We assume here that d is a positive real number different from any 1rn since this is the case for our subsequent helium atom application and since it is not practical to consider all alternatives. In order to evaluate the integral I we first assume that r G r X so that the asymptotic behavior of the integrand is determined by r and we write the function 1 F1Ž l q 1 q irs , 2 l q 2; 2i s r . as a sum of two Meixner functions F1 and F2 w1x 1 F1
Ž a, c; x . s Ž 1r2. F1 Ž a, c; x . q Ž 1r2. F2 Ž a, c; x . .
Ž 11 .
For large values of x the function F1 behaves asymptotically as xya and the function F2 as x ay c e x . They obey the relations w1x F1 Ž c y a, c; xe p i . e x s F2 Ž a, c; xe 2 p i . ey2 p i Ž c y a . ,
Ž 12a.
F2 Ž c y a, c; xe p i . e x s F1 Ž a, c; x . ey2 p i .
Ž 12b.
The function 1 F1Ž a, c; x . obeys Kummer’ relation w15x 1 F1
Ž a, c; x . s1 F1 Ž c y a, c; yx . e x .
Ž 13 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
64
The integrand of I has simple poles at s s "irn due to the G functions and at s s "i d . The poles and their residues are: nyly1
s s irn ,
residues Ž irn2 . Ž y1 .
s s yirn ,
residues Ž yirn2 . Ž y1 . n s l q 1, l q 2, . . . residues Ž 1r2i d . , residues Ž y1r2i d . .
s s ird , s s yird ,
Ž n y l y 1. !
nyly1
y1
Ž n y l y 1. !
, y1
,
Ž 14 .
All single poles are located on the imaginary axis In order to evaluate I we substitute Eq. Ž11. and we integrate the part containing F2 along the path C 1 of Fig. 1 and the part containing F1 along the path C 2 of Fig. 1. The contributions from the circles will tend to zero if we let their radii R tend to infinity. The integral I is then the sum of contributions from the straight parts of the paths along the imaginary axis and of the contributions from the small half-circles around the singularities. The
Fig. 1. The integration paths C 1 and C 2 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
65
straight part contributions are l
I1 s Ž 2rp . Ž 4 rr X . Ž 2 l q 1 . !
y2
Ž 1r2.
`
=
½H
G Ž l q 1 q 1ry . G Ž l q 1 y 1ry . Ž i y .
2 lq2
exp Ž pri y . yy 2 q d 2
y1
0
= exp Ž yr y yr X . F2 Ž l q 1 q 1ry, 2 l q 2; y2 yr . 1 F1 Ž l q 1 y 1ry, 2 l q 2; 2 yr X . Ž i d y . `
H0 G Ž l q 1 y 1ry . G Ž l q 1 q 1ry . Ž yi y .
q
2 lq2
exp Ž ypri y . yy 2 q d 2
y1
exp Ž yyr q yr X .
=F1 Ž l q 1 y 1ry, 2 l q 2; 2 yr . 1 F1 Ž l q 1 q 1ry, 2 l q 2; y2 yr X . Ž yi d y . s 0 .
5
Ž 15 .
It is easily derived from Eqs. Ž11., Ž12a. and Ž12b. that the two integrals cancel. The sum of the contributions from the singularities s s "i n is given by l
I2 s Ž yp ir2 . Ý Ž 2rp . Ž 4 rr X . Ž 2 l q 1 . !
y2
n
=G Ž l q 1 q n . Ž irn2 . Ž y1 .
nyly1
y1
2 lq2
exp Ž p nri . Ž n y l y 1 . ! Ž irn . 2 y2 y1 = exp Ž rrn . F2 Ž l q 1 q n, 2 l q 2; y2 rrn . w d y n x =exp Ž yr Xrn . 1 F1 Ž l q 1 y n, 2 l q 2; 2 r Xrn . q Ž p ir2 . y2 l nyly1 = Ý Ž 2rp . Ž 4 rr X . Ž 2 l q 1 . ! G Ž l q 1 q n . Ž yirn2 . Ž y1 . n
= Ž n y l y 1. !
y1
Ž yirn .
2 lq2
exp Ž yp nri . exp Ž yrrn . y1
=F1 Ž l q 1 y n, 2 l q 2; 2 rrn . w d 2 y ny2 x exp Ž r Xrn . 1 F1 Ž l q 1 q n, 2 l q 2; y2 r Xrn . . We note that w1x F1 Ž l q 1 q n, 2 l q 2; y2 rrn . s 0 if n s l q 1, l q 2, etc. and we substitute Eqs. Ž12a. and Ž12b.. It is then found that I2 may be reduced to
Ž 16 . Ž 17 .
`
I2 s y
f n , l Ž r . f n , l Ž r X . w yny2 q d 2 x
Ý
y1
.
Ž 18 .
nslq1
The contributions from the remaining singularities are lq1 2 l
l
y2
2 Ž rr X . d 2 l q 1 Ž 2 l q 1 . ! exp Ž ypird . e d r X =F2 Ž l q 1 q 1rd , 2 l q 2; y2 d r . e d r 1 F1 Ž l q 1 q 1rd , 2 l q 2; y2 d r X . . We may summarize the above results as follows I3 s Ž y1 .
`
H0 g Ž r , s . g l
) l
Ž rX , s . w s 2 q d 2 x
y1
Ž 19 .
ds
`
s Gl Ž r , r X ; d . y
Ý
f n , l Ž r . f n), l Ž r X . w d 2 y ny2 x
y1
Ž 20 .
nslq1
with lq1
l
lq1
l
y2
G l Ž r , r X ; d . s Ž y1 . Ž 4 rr X . d 2 lq1 Ž 2 l q 1 . ! exp Ž p ird . G Ž l q 1 q 1rd . G Ž l q 1 y 1rd . =exp Ž yd r . F1 Ž l q 1 y 1rd , 2 l q 2; 2 d r . exp Ž yd r X . 1 F1 Ž l q 1 y 1rd , 2 l q 2; 2 d r X . y2
s Ž y1 . Ž 4 rr X . d 2 l q 1 Ž 2 l q 1 . ! exp Ž ypird . G Ž l q 1 q 1rd . G Ž l q 1 y 1rd . =exp Ž d r . F2 Ž l q 1 q 1rd , 2 l q 2; y2 d r . exp Ž d r X . 1 F1 Ž l q 1 q 1rd , 2 l q 2; y2 d r X . l
y1
s Ž 4 rr X . d 2 lq1 Ž 2 l q 1 . ! G Ž l q 1 y 1rd . exp Ž yd r . U Ž l q 1 y 1rd , 2 l q 2; 2 d r . = exp Ž yd r X . 1 F1 Ž l q 1 y 1rd , 2 l q 2; 2 d r X . . Ž 21 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
66
In the latter expression we have substituted the function UŽ a, b; x . defined in Slater’s book w15x, we assume in all three expressions that r G r X . We use these results in deriving the helium atom Green’s function.
3. The helium atom Green’s function In defining the helium atom Green’s function we again multiply its Hamiltonian by a factor Žy2.. We have then H Ž x . s H0 Ž x . q lV Ž x . ,
Ž 22a.
H0 Ž x . s D1 q Ž 2rr1 . q D2 q Ž 2rr 2 . ,
Ž 22b.
with
lV Ž x . s lrr12 . The definition of the Green’s function of the Hamiltonian H0 Ž x . is to some extent similar to the case of the hydrogen atom but it is more complex since we now have two particles to consider instead of one. We again expand the Green’s function in terms of spherical harmonics `
G Ž x , xX ; d . s
`
Ý Ý Ž 2 l q 1. Ž 2 lX q 1. r16p 2 X
ls0 l s0
=Pl Ž cos u 1X . PlX Ž cos u 2X . G l , lX Ž r 1 , r 1X , r 2 , r 2X ; d . .
Ž 23 .
The two-particle radial Green’s functions G l,lX are given by G l , lX Ž r 1 , r 1X ; r 2 , r 2X ; d . s
`
`
H0 H0 g Ž r , s . g l
1
1
= s 12 q s 22 q d 2
) l
Ž r1X , s 1 . g Xl Ž r 2 , s 2 . g Xl ) Ž r 2X , s 2 .
y1
`
`
d s1 d s 2 q
g l Ž r1 , s 1 .
H0 Ý X
X
n sl q1
=g l) Ž r 1X , s 1 . f nX , lX Ž r 2 . f nX , lX Ž r 2X . s 12 y Ž nX . `
q
Ý nslq1
`
H0
= yŽ n.
y2
y2
qd 2
y1
d s1
f n , l Ž r 1 . f n , l Ž r 1X . g lX Ž r 2 , s 2 . g l)X Ž r 2X , s 2 . q s 22 q d 2
y1
d s2 q
`
`
Ý
Ý
X
X
f n , l Ž r 1 . f n , l Ž r 1X . f nX , lX Ž r 2 .
nslq1 n sl q1
=f nX ,lX Ž r 2X . y Ž n .
y2
y Ž nX .
y2
qd 2
y1
.
Ž 24 .
We limit ourselves to values of d that are relevant to practical applications, consequently we will assume that d is a positive real number, larger than unity. We may then introduce the abbreviations
dn s d 2 y Ž n .
y2 1r2
dnX s d 2 y Ž nX .
,
y2 1r2
Ž 25a. ,
Ž 25b.
where we take dn and dnX as the positive roots so that they are real positive numbers also. We simplify the expansion Ž24. of the two-particle function by repeated substitution of Eq. Ž20.. We first
H.F. Hamekar Chemical Physics 228 (1998) 61–72
67
apply this relation to one-half of the second and third terms of Eq. Ž24.. Since both dn and dnX are always real positive numbers we have G l , lX Ž r 1 , r 1X ; r 2 , r 2X ; d . `
s
`
H0 H0 g Ž r , s . g l
1 q 2
1 q 2
Ž r1X , s 1 . g lX Ž r 2 , s 2 . g l)X Ž r 2X , s 2 . s 12 q s 22 q d 2
g l Ž r 1 , s 1 . g l) Ž r 1X , s 1 . f nX , lX Ž r 2 . f nX , lX Ž r 2X . s 12 q sn2
H0
f nX , l Ž r 1 . f n , l Ž r 1X . g lX Ž r 2 , s 2 . g l)X Ž r 2X , s 2 . s 22 q Ž dnX .
X
X
n sl q1 ` `
Ý nslq1 `
q 2
) l
H0 Ý
1 1
1
X
y1
d s1 d s 2
`
`
q 2
1
y1
2
d s1
d s2
G l Ž r 1 , r 1X ; dnX . f nX , lX Ž r 2 . f nX , lX Ž r 2X .
Ý X
n sl q1 `
f n , l Ž r 1 . f n , l Ž r 1X . G lX Ž r 2 , r 2X , dn . .
Ý
Ž 26 .
nslq1
Clearly, in using Eqs. Ž20. and Ž21. we have assumed that r 1 G r 1X and r 2 G r 2X . Next, we split the first term of Eq. Ž26. in two and we apply Eq. Ž20. to the s 1 integration in the first half and to the s 2 integration in the second half. The result is 1 ` G l , lX Ž r 1 , r 1X ; r 2 , r 2X ; d . s G Ž r , r X ; u . g X Ž r , s . g )X Ž r 2X , s 2 . d s 2 2 0 l 1 1 2 l 2 2 l 1 ` q g Ž r , s . g ) Ž rX , s . G X Ž r , rX ; u . d s1 2 0 l 1 1 l 1 1 l 2 2 1 1 ` q Ý f Ž r . f Ž rX . G X Ž r , rX ; d . 2 nslq1 n , l 1 n , l 1 l 2 2 n
H
H
`
1 q 2
X
Ý X
G l Ž r 1 , r 1X ; dnX . f nX , lX Ž r 2 . f nX , lX Ž r 2X . .
Ž 27 .
n sl q1
Here G l is defined by Eq. Ž21. in three alternative forms and the continuum wave functions g l are defined in Eq. Ž7.. The variables u1 and u 2 are functions of d and of s 1 or s 2 , u1 s s 12 q d 2
1r2
u 2 s s 22 q d 2
1r2
,
Ž 28a.
.
Ž 28b.
They are real on the integration paths and it follows from our derivations that we must take the positive root in each case. The complexity of expression Ž27. is comparable to the hydrogen atom Green’s function before its simplification by means of contour integration. In the next section we explore possible further simplification of Eq. Ž27. by means of integration in the complex plane.
4. Transformations of the helium atom Green’s function We derived Eq. Ž27. for the helium atom Green’s function by making use of the results for the hydrogen atom which were in turn derived from a contour integration in the complex plane along the paths described in Fig. 1. We wish to investigate whether further simplification of Eq. Ž27. is possible by suitable contour
H.F. Hamekar Chemical Physics 228 (1998) 61–72
68
integration related to the second integration. Since the two integrals in Eq. Ž27. are identical we consider the first one only. By substituting Eqs. Ž7. and Ž21. we find that the integral is given by Jl , lX s Ž y1 .
lq1
Ž 2rp . Ž 2 l q 1 . !
y2
Ž 2 lX q 1 . !
y2
l
Ž 4 r1 r1X . Ž 4 r 2 r 2X .
l
X
`
H0 u
2 lq1
X
s 2 l q2
=exp Ž ypiru . exp Ž prs . G Ž l q 1 q 1ru . G Ž l q 1 y 1ru . G Ž lX q 1 q irs . G Ž lX q 1 y irs . =exp Ž ur1 . F2 Ž l q 1 q 1ru, 2 l q 2; y2 ur 1 . exp Ž ur1X . 1 F1 Ž l q 1 q 1ru, 2 l q 2; y2 ur 1X . =exp Ž yi s r 2 . 1 F1 Ž lX q 1 q irs , 2 l q 2; 2i s r 2 . = exp Ž i s r 2X . 1 F1 Ž lX q 1 y irs , 2 l q 2; y2i s r 2X . d s ,
Ž 29 .
with us w s 2qd 2 x
1r2
.
Ž 30 .
Along the integration path the variable s is real positive and we must also take the variable u as real and positive, in other words we must take the positive root of Eq. Ž30. along the integration path. The integrand of Eq. Ž29. is therefore well defined along the integration path. In order to continue the integrand of Eq. Ž29. beyond the integration path along the real positive axis we ought to realize that u is a double-valued function of s . Its Riemann surface consists of two plane sheets with branch points at s s "i d . The line segment along the imaginary axis between yi d and i d is the cut that should not be crossed while selecting integration paths. It may be helpful to discuss the behavior of s and u along the real and imaginary axes. First we move s along the positive real axis from infinity to the origin. The function u is also real and positive. If we then move s along the positive imaginary axis u will be positive real for the line segment between 0 and i d . We now move s in a small half-circle on the right-hand side of the imaginary axis around the point i d . Since us Ž sqis . Ž syis .
1r2
Ž 31 .
the phase of u changes by 908 if the phase of s changes by 1808 so that u is now given by u s iw s 2 y d 2 x
1r2
.
Ž 32 .
If we move s along the negative imaginary axis then u is real and positive until we encounter the point yi d . If we move s around yi d along a small half-circle on the right-hand side of the imaginary axis the phase of u changes by y908 so that u is now given by u s yi w s 2 y d 2 x
1r2
.
Ž 33 .
The integrand of Eq. Ž29. has two infinite sets of simple poles. Both sets are related to the G functions. The first set are at the points
s s "irnX ,
n s lX q 1, lX q 2, . . . , etc. ,
Ž 34 .
and the residues are given by Eq. Ž14.. The second set of simple poles are given by u s 1rn , n s l q 1, l q 2, . . . , etc.
Ž 35.
We are restricted to the sheet where the real values of u are all positive but each point of Eq. Ž35. corresponds to two simple poles s s "irdn of the variable s . These poles and their residues are
s s irdn , s s yirdn ,
residues Ž yirdn P n3 . Ž y1 . 3
residues Ž irdn P n . Ž y1 .
nyly1
nyly1
Ž n y l y 1. !
Ž n y l y 1. !
y1
y1
.
,
Ž 36 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
69
In order to evaluate Eq. Ž29. we assume that r 2 ) r 2X and we substitute 1 F1
Ž l q 1 q 1rs , 2 l q 2; 2i s r 2 . 1
s 2
F1 Ž l q 1 q 1rs , 2 l q 2; 2i s r 2 . q
1 2
F2 Ž l q 1 q irs , 2 l q 2; 2i s r 2 .
Ž 37 .
so that Jl,lX is the sum of two integrals, I Žcontaining F2 . and II Žcontaining F1 .. We integrate I along the path CX1 of Fig. 2 and II along the path CX2 of Fig. 2. In both cases we assume that all singularities are located along the line segment between the points "i d on the imaginary axis. The integration paths contain little half-circles around each singularity on the right-hand side of the imaginary axis and also little half-circles around the two branch points "i d . The contributions from the large quarter-circles with radius R will become zero when R sends to infinity so that the integrals along the positive real axis become equal to the integrals along the imaginary axes. Both integrals I and II are the sum of four contributions: ŽA. Contributions from straight line segments between 0 and i d ŽB. Contribution from path from i d to infinity
X
X
Fig. 2. The integration paths C 1 and C 2 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
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ŽC. Sum of the residues s s "irn ŽD. Sum of the residues u s 1rn We consider these various contributions separately. The contributions A of both parts I and II cancel. In part I we substitute s s i t and in part II we substitute s s yi t so that we have us w d 2yt2 x
ssit ,
I,
2 1r2
us w d 2yt x
s s yit ,
II ,
d 2 lq1
Hu u
1r2
,
0FtFd
,
0FtFd
X
Žit .
2 l q2
exp Ž ypiru . exp Ž ypirt . G Ž l q 1 q 1ru . G Ž l q 1 y 1ru .
= G Ž lX q 1 q irt . G Ž lX q 1 y irt . exp Ž ur1 . F2 Ž l q 1 q 1ru, 2 l q 2; y2 ur 1 . exp Ž ur1X . = 1 F1 Ž l q 1 q 1ru, 2 l q 2; y2 ur 1X . exp Ž tr 2 . F2 Ž l q 1 q 1rt , 2 lX q 2; y2 tr 2 . exp Ž ytr 2X . = 1 F1 Ž lX q 1 y 1rt , 2 l q 2; 2 tr 2X . Ž i d t . q
d 2 lq1
H0 u
X
Ž yit .
2 l q2
exp Ž ypiru . exp Ž p irt .
=G Ž l q 1 q 1ru . G Ž l q 1 y 1ru . G Ž lX q 1 y lrt . G Ž lX q 1 q lrt . exp Ž ur1 . =F2 Ž l q 1 q 1ru, 2 l q 2; y2 ur1 . exp Ž ur 1X . 1 F1 Ž l q 1 q 1ru, 2 l q 2; y2 ur1X . exp Ž ytr 2 . = F1 Ž l q 1 y 1rt , 2 lX q 2; 2 tr 2 . exp Ž tr 2X . 1 F1 Ž l q 1 q 1rt , 2 l q 2; y2 tr 2X . Ž yi d t . s 0
Ž 38 .
because of Eqs. Ž12a., Ž12b. and Ž13.. In order to evaluate the contributions B to I and II we substitute: 1r2
I,
ssit ,
u s iw t 2 y d 2 x
II ,
s s yit ,
u s yi w t 2 y d 2 x
s in ,
1r2
n w t2yd 2 x
1r2
s yi n
according to Eqs. Ž32. and Ž33.. The sum of the two contributions B becomes `
Hd
Ž in .
2 lq1
X
Žit .
2 l q2
exp Ž yprn . exp Ž ypirt . G Ž l q 1 y irn . G Ž l q 1 q irn . G Ž lX q 1 q 1rt .
=G Ž lX q 1 y 1rt . exp Ž i n r 1 . F2 Ž l q 1 y irn , 2 l q 2; y2i n r 1 . =exp Ž i n r 1X . 1 F1 Ž l q 1 y irn , 2 l q 2; y2i n r 1X . exp Ž tr 2 . F2 Ž lX q 1 q 1rt , 2 lX q 2; y2 tr 2 . =exp Ž ytr 2X . 1 F1 Ž lX q 1 y 1rt , 2 lX q 2; 2 tr 2X . Ž i d t . y2
s Ž 2rp . Ž 2 l q 1 . !
Ž 4 r1 r1X .
l
`
Hd n
2 lq1
=exp Ž yprn . G Ž l q 1 y irn . G Ž l q 1 q irn . exp Ž i n r 1 . F2 Ž l q 1 y irn , 2 l q 2; y2i n r 1 . =exp Ž i n r 1X . 1 F1 Ž l q 1 y irn , 2 l q 2; y2i n r 1X . tGlX Ž r 2 , r 2X ; t . d t `
s Ž 2rp . =
X 1
Hd G Ž r , r ; i n . G l
1
l
X
Ž r 2 , r 2X ; t . d t .
Ž 39 .
The contribution due to the singularities at s s "i n in may be derived in the same way as for the hydrogen atom in Eq. Ž16. since the Green’s function part of the integral is not involved in these singularities. The result is `
Sl ,lX Ž C . s y
X
Ý X
n sl q1
G l Ž r 1 , r 1X ; dnX . f nX , lX Ž r 2 . f nX , lX Ž r 2X . .
Ž 40 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
71
In a similar fashion it may be derived that the contribution from the singularities s s "i dn or u s Ž1rn. with n s l q 1, l q 2, . . . , etc., is given by Sl , lX Ž D . s q Ý q f n , l Ž r 1 . f n , l Ž r 1X . G lX Ž r 2 , r 2X ; dn . .
Ž 41 .
n
The above results may be summarized as follows `
HGl Ž r 1 , r 1X ; u . g lX Ž r 2 , s . g l)X Ž r 2 , s . d s q
X
Ý
G l Ž r 1 , r 1X ; dnX . f nX , lX Ž r 2 . f nX , lX Ž r 2X .
n slq1 `
`
s Ž 2rpi .
X 1
Hd G Ž r , r ; in . G l
1
X l Ž r2 , r2 ; t . t d t q
f n , l Ž r 1 . f n , l Ž r 1X . G lX Ž r 2 , r 2X ; dn .
Ý
X
Ž 42 .
nqlq1
and the helium atom Green’s function Ž27. has been transformed to G l , lX Ž r 1 , r 1X ; r 2 , r 2X ; d . s Ž 1rpi .
` l
1
l
X
Ž r 2 , r 2X ; t . t d t q Ž 1rpi . `
`
=
X 1
Hd G Ž r , r ; i n . G X 1
Hd G Ž r , r ; t . G l
1
l
X
Ž r 2 , r 2X ; i n . t d t q Ž 1r2.
Ý
f n , l Ž r 1 . f n , l Ž r 1X .
nqlq1 `
=G lX Ž r 2 , r 1X ; dn . q Ž 1r2 .
X
Ý X
G l Ž r 1 , r 1X ; dnX . f nX , lX Ž r 2 . f nX , lX Ž r 2X . .
Ž 43 .
n sl q1
5. Discussion The procedure outlined above presents an alternative approach to the derivation of the eigenvalues and eigenfunctions of the helium atom. we have limited our discussion to the nondegenerate ground state but it is in principle possible to extend it to excited states as long as we take the changes in distribution of the singularities into account. The relative convenience of the proposed procedure depends for a great deal on the complexity of the Green’s function. We were able to reduce the original form Ž24. of the Green’s function to the more convenient forms of Eq. Ž27. or Eq. Ž43. but we were disappointed to find that our transformations did not lead to even more convenient forms. We anticipated that the contour integration described in Section 4 would lead to elimination of the infinite sum over the discrete states. Our assumption turned out to be correct but we also found that a second infinite set of singularities produced another infinite sum over discrete states which is equally cumbersome as the other one. As a matter of fact, the new pair of infinite sets is identical to the old pair. Nevertheless, the transformation described in Section 4 produced an interesting result even if it did not lead to significant further simplifications of the Green’s function. The key question that we should address is whether it is feasible to derive the exact helium atom wave function from the integral equation Ž4.. We cannot supply the answer to that question at this time but we plan to pursue the matter further. Recent developments in hardware and software have produced significant progress in iterative procedures for dealing with mathematical problems and the present computer technology should be better suited to deal with integral equations than with differential equations w16,17x. At the least we hope to have outlined a new approach to solving the helium atom Schrodinger equation and we hope that further exploration ¨ of this approach leads to useful results.
H.F. Hamekar Chemical Physics 228 (1998) 61–72
72
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x
J. Meixner, Math. Z. 36 Ž1933. 677–707. L. Hostler, J. Math. Phys. 5 Ž1964. 591–611. J. Schwinger, J. Math. Phys. 5 Ž1964. 1606–1608. H.F. Hameka, J. Chem. Phys. 47 Ž1967. 2728–2735. H.F. Hameka, J. Chem. Phys. 48 Ž1968. 4810. H.F. Hameka, J. Chem. Phys 49 Ž1968. 2002–2008. H.F. Hameka, J. Chem. Phys. 50 Ž1969. 1495–1496. H.A. Bethe, E.E. Salpeter, Handbuch der Physik, vol. 35 ŽS. Flugge, Ed.., Springer, Berlin, 1957, pp. 82–86. ¨ E.A. Hylleraas, Advances in Quantum Chemistry, vol. 1, ŽP. Lowdin, Ed.., Academic Press, New York, 1964, pp. 1–33. ¨ T. Kinoshita, Phys. Rev. 105 Ž1957. 1490. T. Kinoshita, Phys. Rev. 115 Ž1959. 336. C.L. Pekeris, Phys. Rev. 112 Ž1958. 1649. C.L. Pekeris, Phys. Rev. 115 Ž1959. 1216. C.L. Pekeris, Phys. Rev. 126 Ž143. Ž1962. 1470. L.J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, Cambridge, 1960. H. Exton, Multiple Hypergeometric Functions and Applications, Ellis Horwood, Chichester, 1976. H. Exton, Handbook of Hypergeometric Integrals, Ellis Horwood, Chichester, 1978.
Transformation of the helium atom Schrodinger equation into ¨ integral form Hendrik F. Hameka Department of Chemistry, UniÕersity of PennsylÕania, Philadelphia, PA 19104, USA Received 18 December 1996
Abstract We define the Green’s function for two identical particles moving in a Coulomb field. We use this Green’s function as a basis for deriving an integral equation of the helium atom Schrodinger equation by treating the electron repulsion term as a ¨ perturbation. The two-particle Green’s function is further simplified by means of a contour integration. An additional transformation does not lead to further simplification of the result. q 1998 Elsevier Science B.V.
1. Introduction It is possible to transform a differential equation with boundary conditions into an integral equation. The transformation requires the introduction of a Green’s function, which may be derived from the eigenvalues and eigenfunctions of the differential equation. In general the two approaches are equivalent, that is the required efforts to solve either the differential equation or the integral equation are of comparable magnitude. However, in quantum mechanical problems where approximate solutions are derived by means of perturbation theory the integral approach may be more convenient than the differential approach. This is the case in situations where the Green’s function may be reduced to a more simple form. For instance, the Green’s function technique has been quite effective in dealing with scattering problems since the free-particle Green function has a very simple from. It is possible to use the Green function approach also for the study of bound systems but the applications to atomic and molecular structure have met with limited success only. In the present paper we wish to show how the introduction and evaluation of a Green’s function may be used to transform the helium atom Schrodringer equation into an integral equation which may be solved by means of ¨ iterative procedures. We first present the relation between perturbation theory and the Green’s function in order to define the latter function. We consider a Hamiltonian H Ž x . of a set of variables that we denote symbolically by x and we assume that H Ž x . may be taken as the sum of an unperturbed Hamiltonian H0 Ž x . and a perturbation lV Ž x ., H Ž x . s H0 Ž x . q lV Ž x . . Ž 1. We denote the eigenvalues of H0 Ž x . by En and f nŽ x . and the eigenvalues and eigenfunctions of H Ž x . by e n and cnŽ x ., H 0 Ž x . f n Ž x . s En f n Ž x . , Ž 2a. H Ž x . c n Ž x . s e n cn Ž x . . Ž 2b . 0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 1 - 0 1 0 4 Ž 9 7 . 0 0 2 8 9 - 9
H.F. Hamekar Chemical Physics 228 (1998) 61–72
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We define the Green’s function GŽ x, xX ; e . and the modified Green’s function G Ž1. Ž x, xX ; e 1 . of H0 Ž x . by G Ž x , x X ; e . s y Ý Ž x . Ž En y e .
y1
f n Ž x . f n) Ž xX . ,
Ž 3a .
n
G Ž1. Ž x , xX ; e 1 . s
Ý Ž En y e 1 . f n) xX . .
Ž 3b .
n/1
It is then easily shown that the eigenvalue e 1 and eigenfunction c 1Ž x . of H Ž x . are determined by the following equations
c 1 Ž x . s f 1 Ž x . y lHG Ž1. Ž x , xX ; e 1 . V Ž xX . c 1 Ž xX . d xX ,
Ž 4a .
e 1 s E1 q lHf 1) Ž x . V Ž x . c 1 Ž x . d x .
Ž 4b .
It is assumed here that the eigenvalue e 1 is nondegenerate. Eqs. Ž4a. and Ž4b. have general validity but they are of practical use only if the perturbation lV is small so that the eigenvalue spectra of H0 Ž x . and H Ž x . are similar and the differences between e 1 and E1 and between f 1Ž x . and c 1Ž x . are small. The above approach may be used for perturbation calculations on the hydrogen atom since the infinite expansion Ž3. may be reduced to a relatively simple form. It was shown as early as 1933 by Meixner w1x that in the case of the one-dimensional Kepler problem the infinite sum over the bound states and the integral over the continuum states could be reduced to a single term only by means of a clever contour integration in the complex plane. Meixner w1x also derived the three-dimensional hydrogen atom Green’s function in terms of parabolic coordinates. More recently Hostler w2x derived the hydrogen atom Green’s function for the nonrelativistic three-dimensional Schrodinger equation in closed form by means of a partial-wave expansion. Hostler obtained ¨ the Green’s function as a sum of products of Whittaker functions of different arguments. At almost the same time Schwinger w3x derived an expression for the momentum space Green’s function of the hydrogen atom. We derived the three-dimensional hydrogen atom Green’s function as an expansion in terms of spherical harmonics w4,5x by means of procedures similar to Meixner’s treatment of the one-dimensional Kepler problem w1x. It is possible to obtain exact analytical solutions for a variety of perturbation effects on the hydrogen atom by making use of the hydrogen atom Green’s function so that the derivation of the Green’s function has led to some useful applications. On the other hand, the extension of this approach to larger atomic and molecular systems has met with little success up to now. The first step in extending the Green’s function approach to more complex systems is to tackle the next largest atom, which is the helium atom. We therefore report our efforts to derive Green’s function of the helium atom, that is to say the Green’s function of the helium atom without the electron repulsion term. If we are successful in this attempt then we may treat the electron repulsion term as a perturbation so that the helium atom Schrodinger equation is transformed into an integral equation. The efficiency of this approach depends on the ¨ extent that we can simplify the Green’s function. Many years ago we presented a derivation of the helium atom Green’s function w6x and we reported a relatively simple expression. Unfortunately, we reported subsequently that this derivation and the simple result were incorrect w7x, at the time we failed to report the correct derivation. In the present paper we report the correct derivation and result of the helium atom Green’s function. We also report an interesting transformation of the Green’s function which is derived by a contour integration in the complex plane. Alternative approaches to the solution of the two-electron atomic Schrodinger equation were reviewed by ¨ Bethe and Salpeter w8x and by Hylleraas w9x. The latter review emphasizes the work by Kinoshita w10,11x and by Pekeris w12–14x. Our work on the helium atom is to some extent an expansion of the previous work on the hydrogen atom. We therefore review the hydrogen atom derivation first, this also presents an opportunity to define the various functions and relations that are used in the subsequent sections.
H.F. Hamekar Chemical Physics 228 (1998) 61–72
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2. The hydrogen atom Green’s function In deriving the hydrogen atom Green’s function we follow the customary procedure in multiplying the Hamiltonian by a factor minus two. The discrete eigenvalues are then given by En s 1rn2
Ž 5.
and the corresponding radial eigenfunctions f n,l Ž r . are f n ,l Ž r . s Ž 2rn2 . Ž n q l . !r Ž n y l y 1 . !
1r2
Ž 2 l q 1. !
y1
Ž 2 rrn .
l
=exp Ž yrrn . 1 F1 Ž yn q l q 1, 2 l q 2; 2 rrn . .
Ž 6.
The continuum state radial eigenfunctions g l Ž r, s . are g l Ž r , s . s Ž 2rp .
1r2
G Ž l q 1 q irs .
=G Ž l q 1 y irs . x
1r2
Ž 2 l q 1. !
y1
l
s Ž 2 s r . exp Ž pr2 s .
= exp Ž yi s r . 1 F1 Ž l q 1 q irs , 2 l q 2; 2i s r . .
Ž 7.
The corresponding energies are E Ž s . s ys 2 .
Ž 8.
The function 1 F1Ž a, c; x . is the well-known confluent hypergeometric function w15x. In our approach we expand the hydrogen atom Green’s function G in terms of spherical harmonics `
Gs
Ý Ž 2 l q 1. r4p
Pl Ž cos u X . G l Ž r , r X ; d . ,
Ž 9.
1s0
where u X is the angle between the vectors r and rX and the functions G l are defined as Gl Ž r , r X ; d . s
`
H0 g Ž r , s . g l
) l
Ž rX , s . w s 2 q d 2 x
y1
ds
`
q
Ý
f n , l Ž r . f n , l Ž r X . w yny2 q d 2 x
y1
sIqS .
Ž 10 .
n s lq1
We assume here that d is a positive real number different from any 1rn since this is the case for our subsequent helium atom application and since it is not practical to consider all alternatives. In order to evaluate the integral I we first assume that r G r X so that the asymptotic behavior of the integrand is determined by r and we write the function 1 F1Ž l q 1 q irs , 2 l q 2; 2i s r . as a sum of two Meixner functions F1 and F2 w1x 1 F1
Ž a, c; x . s Ž 1r2. F1 Ž a, c; x . q Ž 1r2. F2 Ž a, c; x . .
Ž 11 .
For large values of x the function F1 behaves asymptotically as xya and the function F2 as x ay c e x . They obey the relations w1x F1 Ž c y a, c; xe p i . e x s F2 Ž a, c; xe 2 p i . ey2 p i Ž c y a . ,
Ž 12a.
F2 Ž c y a, c; xe p i . e x s F1 Ž a, c; x . ey2 p i .
Ž 12b.
The function 1 F1Ž a, c; x . obeys Kummer’ relation w15x 1 F1
Ž a, c; x . s1 F1 Ž c y a, c; yx . e x .
Ž 13 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
64
The integrand of I has simple poles at s s "irn due to the G functions and at s s "i d . The poles and their residues are: nyly1
s s irn ,
residues Ž irn2 . Ž y1 .
s s yirn ,
residues Ž yirn2 . Ž y1 . n s l q 1, l q 2, . . . residues Ž 1r2i d . , residues Ž y1r2i d . .
s s ird , s s yird ,
Ž n y l y 1. !
nyly1
y1
Ž n y l y 1. !
, y1
,
Ž 14 .
All single poles are located on the imaginary axis In order to evaluate I we substitute Eq. Ž11. and we integrate the part containing F2 along the path C 1 of Fig. 1 and the part containing F1 along the path C 2 of Fig. 1. The contributions from the circles will tend to zero if we let their radii R tend to infinity. The integral I is then the sum of contributions from the straight parts of the paths along the imaginary axis and of the contributions from the small half-circles around the singularities. The
Fig. 1. The integration paths C 1 and C 2 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
65
straight part contributions are l
I1 s Ž 2rp . Ž 4 rr X . Ž 2 l q 1 . !
y2
Ž 1r2.
`
=
½H
G Ž l q 1 q 1ry . G Ž l q 1 y 1ry . Ž i y .
2 lq2
exp Ž pri y . yy 2 q d 2
y1
0
= exp Ž yr y yr X . F2 Ž l q 1 q 1ry, 2 l q 2; y2 yr . 1 F1 Ž l q 1 y 1ry, 2 l q 2; 2 yr X . Ž i d y . `
H0 G Ž l q 1 y 1ry . G Ž l q 1 q 1ry . Ž yi y .
q
2 lq2
exp Ž ypri y . yy 2 q d 2
y1
exp Ž yyr q yr X .
=F1 Ž l q 1 y 1ry, 2 l q 2; 2 yr . 1 F1 Ž l q 1 q 1ry, 2 l q 2; y2 yr X . Ž yi d y . s 0 .
5
Ž 15 .
It is easily derived from Eqs. Ž11., Ž12a. and Ž12b. that the two integrals cancel. The sum of the contributions from the singularities s s "i n is given by l
I2 s Ž yp ir2 . Ý Ž 2rp . Ž 4 rr X . Ž 2 l q 1 . !
y2
n
=G Ž l q 1 q n . Ž irn2 . Ž y1 .
nyly1
y1
2 lq2
exp Ž p nri . Ž n y l y 1 . ! Ž irn . 2 y2 y1 = exp Ž rrn . F2 Ž l q 1 q n, 2 l q 2; y2 rrn . w d y n x =exp Ž yr Xrn . 1 F1 Ž l q 1 y n, 2 l q 2; 2 r Xrn . q Ž p ir2 . y2 l nyly1 = Ý Ž 2rp . Ž 4 rr X . Ž 2 l q 1 . ! G Ž l q 1 q n . Ž yirn2 . Ž y1 . n
= Ž n y l y 1. !
y1
Ž yirn .
2 lq2
exp Ž yp nri . exp Ž yrrn . y1
=F1 Ž l q 1 y n, 2 l q 2; 2 rrn . w d 2 y ny2 x exp Ž r Xrn . 1 F1 Ž l q 1 q n, 2 l q 2; y2 r Xrn . . We note that w1x F1 Ž l q 1 q n, 2 l q 2; y2 rrn . s 0 if n s l q 1, l q 2, etc. and we substitute Eqs. Ž12a. and Ž12b.. It is then found that I2 may be reduced to
Ž 16 . Ž 17 .
`
I2 s y
f n , l Ž r . f n , l Ž r X . w yny2 q d 2 x
Ý
y1
.
Ž 18 .
nslq1
The contributions from the remaining singularities are lq1 2 l
l
y2
2 Ž rr X . d 2 l q 1 Ž 2 l q 1 . ! exp Ž ypird . e d r X =F2 Ž l q 1 q 1rd , 2 l q 2; y2 d r . e d r 1 F1 Ž l q 1 q 1rd , 2 l q 2; y2 d r X . . We may summarize the above results as follows I3 s Ž y1 .
`
H0 g Ž r , s . g l
) l
Ž rX , s . w s 2 q d 2 x
y1
Ž 19 .
ds
`
s Gl Ž r , r X ; d . y
Ý
f n , l Ž r . f n), l Ž r X . w d 2 y ny2 x
y1
Ž 20 .
nslq1
with lq1
l
lq1
l
y2
G l Ž r , r X ; d . s Ž y1 . Ž 4 rr X . d 2 lq1 Ž 2 l q 1 . ! exp Ž p ird . G Ž l q 1 q 1rd . G Ž l q 1 y 1rd . =exp Ž yd r . F1 Ž l q 1 y 1rd , 2 l q 2; 2 d r . exp Ž yd r X . 1 F1 Ž l q 1 y 1rd , 2 l q 2; 2 d r X . y2
s Ž y1 . Ž 4 rr X . d 2 l q 1 Ž 2 l q 1 . ! exp Ž ypird . G Ž l q 1 q 1rd . G Ž l q 1 y 1rd . =exp Ž d r . F2 Ž l q 1 q 1rd , 2 l q 2; y2 d r . exp Ž d r X . 1 F1 Ž l q 1 q 1rd , 2 l q 2; y2 d r X . l
y1
s Ž 4 rr X . d 2 lq1 Ž 2 l q 1 . ! G Ž l q 1 y 1rd . exp Ž yd r . U Ž l q 1 y 1rd , 2 l q 2; 2 d r . = exp Ž yd r X . 1 F1 Ž l q 1 y 1rd , 2 l q 2; 2 d r X . . Ž 21 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
66
In the latter expression we have substituted the function UŽ a, b; x . defined in Slater’s book w15x, we assume in all three expressions that r G r X . We use these results in deriving the helium atom Green’s function.
3. The helium atom Green’s function In defining the helium atom Green’s function we again multiply its Hamiltonian by a factor Žy2.. We have then H Ž x . s H0 Ž x . q lV Ž x . ,
Ž 22a.
H0 Ž x . s D1 q Ž 2rr1 . q D2 q Ž 2rr 2 . ,
Ž 22b.
with
lV Ž x . s lrr12 . The definition of the Green’s function of the Hamiltonian H0 Ž x . is to some extent similar to the case of the hydrogen atom but it is more complex since we now have two particles to consider instead of one. We again expand the Green’s function in terms of spherical harmonics `
G Ž x , xX ; d . s
`
Ý Ý Ž 2 l q 1. Ž 2 lX q 1. r16p 2 X
ls0 l s0
=Pl Ž cos u 1X . PlX Ž cos u 2X . G l , lX Ž r 1 , r 1X , r 2 , r 2X ; d . .
Ž 23 .
The two-particle radial Green’s functions G l,lX are given by G l , lX Ž r 1 , r 1X ; r 2 , r 2X ; d . s
`
`
H0 H0 g Ž r , s . g l
1
1
= s 12 q s 22 q d 2
) l
Ž r1X , s 1 . g Xl Ž r 2 , s 2 . g Xl ) Ž r 2X , s 2 .
y1
`
`
d s1 d s 2 q
g l Ž r1 , s 1 .
H0 Ý X
X
n sl q1
=g l) Ž r 1X , s 1 . f nX , lX Ž r 2 . f nX , lX Ž r 2X . s 12 y Ž nX . `
q
Ý nslq1
`
H0
= yŽ n.
y2
y2
qd 2
y1
d s1
f n , l Ž r 1 . f n , l Ž r 1X . g lX Ž r 2 , s 2 . g l)X Ž r 2X , s 2 . q s 22 q d 2
y1
d s2 q
`
`
Ý
Ý
X
X
f n , l Ž r 1 . f n , l Ž r 1X . f nX , lX Ž r 2 .
nslq1 n sl q1
=f nX ,lX Ž r 2X . y Ž n .
y2
y Ž nX .
y2
qd 2
y1
.
Ž 24 .
We limit ourselves to values of d that are relevant to practical applications, consequently we will assume that d is a positive real number, larger than unity. We may then introduce the abbreviations
dn s d 2 y Ž n .
y2 1r2
dnX s d 2 y Ž nX .
,
y2 1r2
Ž 25a. ,
Ž 25b.
where we take dn and dnX as the positive roots so that they are real positive numbers also. We simplify the expansion Ž24. of the two-particle function by repeated substitution of Eq. Ž20.. We first
H.F. Hamekar Chemical Physics 228 (1998) 61–72
67
apply this relation to one-half of the second and third terms of Eq. Ž24.. Since both dn and dnX are always real positive numbers we have G l , lX Ž r 1 , r 1X ; r 2 , r 2X ; d . `
s
`
H0 H0 g Ž r , s . g l
1 q 2
1 q 2
Ž r1X , s 1 . g lX Ž r 2 , s 2 . g l)X Ž r 2X , s 2 . s 12 q s 22 q d 2
g l Ž r 1 , s 1 . g l) Ž r 1X , s 1 . f nX , lX Ž r 2 . f nX , lX Ž r 2X . s 12 q sn2
H0
f nX , l Ž r 1 . f n , l Ž r 1X . g lX Ž r 2 , s 2 . g l)X Ž r 2X , s 2 . s 22 q Ž dnX .
X
X
n sl q1 ` `
Ý nslq1 `
q 2
) l
H0 Ý
1 1
1
X
y1
d s1 d s 2
`
`
q 2
1
y1
2
d s1
d s2
G l Ž r 1 , r 1X ; dnX . f nX , lX Ž r 2 . f nX , lX Ž r 2X .
Ý X
n sl q1 `
f n , l Ž r 1 . f n , l Ž r 1X . G lX Ž r 2 , r 2X , dn . .
Ý
Ž 26 .
nslq1
Clearly, in using Eqs. Ž20. and Ž21. we have assumed that r 1 G r 1X and r 2 G r 2X . Next, we split the first term of Eq. Ž26. in two and we apply Eq. Ž20. to the s 1 integration in the first half and to the s 2 integration in the second half. The result is 1 ` G l , lX Ž r 1 , r 1X ; r 2 , r 2X ; d . s G Ž r , r X ; u . g X Ž r , s . g )X Ž r 2X , s 2 . d s 2 2 0 l 1 1 2 l 2 2 l 1 ` q g Ž r , s . g ) Ž rX , s . G X Ž r , rX ; u . d s1 2 0 l 1 1 l 1 1 l 2 2 1 1 ` q Ý f Ž r . f Ž rX . G X Ž r , rX ; d . 2 nslq1 n , l 1 n , l 1 l 2 2 n
H
H
`
1 q 2
X
Ý X
G l Ž r 1 , r 1X ; dnX . f nX , lX Ž r 2 . f nX , lX Ž r 2X . .
Ž 27 .
n sl q1
Here G l is defined by Eq. Ž21. in three alternative forms and the continuum wave functions g l are defined in Eq. Ž7.. The variables u1 and u 2 are functions of d and of s 1 or s 2 , u1 s s 12 q d 2
1r2
u 2 s s 22 q d 2
1r2
,
Ž 28a.
.
Ž 28b.
They are real on the integration paths and it follows from our derivations that we must take the positive root in each case. The complexity of expression Ž27. is comparable to the hydrogen atom Green’s function before its simplification by means of contour integration. In the next section we explore possible further simplification of Eq. Ž27. by means of integration in the complex plane.
4. Transformations of the helium atom Green’s function We derived Eq. Ž27. for the helium atom Green’s function by making use of the results for the hydrogen atom which were in turn derived from a contour integration in the complex plane along the paths described in Fig. 1. We wish to investigate whether further simplification of Eq. Ž27. is possible by suitable contour
H.F. Hamekar Chemical Physics 228 (1998) 61–72
68
integration related to the second integration. Since the two integrals in Eq. Ž27. are identical we consider the first one only. By substituting Eqs. Ž7. and Ž21. we find that the integral is given by Jl , lX s Ž y1 .
lq1
Ž 2rp . Ž 2 l q 1 . !
y2
Ž 2 lX q 1 . !
y2
l
Ž 4 r1 r1X . Ž 4 r 2 r 2X .
l
X
`
H0 u
2 lq1
X
s 2 l q2
=exp Ž ypiru . exp Ž prs . G Ž l q 1 q 1ru . G Ž l q 1 y 1ru . G Ž lX q 1 q irs . G Ž lX q 1 y irs . =exp Ž ur1 . F2 Ž l q 1 q 1ru, 2 l q 2; y2 ur 1 . exp Ž ur1X . 1 F1 Ž l q 1 q 1ru, 2 l q 2; y2 ur 1X . =exp Ž yi s r 2 . 1 F1 Ž lX q 1 q irs , 2 l q 2; 2i s r 2 . = exp Ž i s r 2X . 1 F1 Ž lX q 1 y irs , 2 l q 2; y2i s r 2X . d s ,
Ž 29 .
with us w s 2qd 2 x
1r2
.
Ž 30 .
Along the integration path the variable s is real positive and we must also take the variable u as real and positive, in other words we must take the positive root of Eq. Ž30. along the integration path. The integrand of Eq. Ž29. is therefore well defined along the integration path. In order to continue the integrand of Eq. Ž29. beyond the integration path along the real positive axis we ought to realize that u is a double-valued function of s . Its Riemann surface consists of two plane sheets with branch points at s s "i d . The line segment along the imaginary axis between yi d and i d is the cut that should not be crossed while selecting integration paths. It may be helpful to discuss the behavior of s and u along the real and imaginary axes. First we move s along the positive real axis from infinity to the origin. The function u is also real and positive. If we then move s along the positive imaginary axis u will be positive real for the line segment between 0 and i d . We now move s in a small half-circle on the right-hand side of the imaginary axis around the point i d . Since us Ž sqis . Ž syis .
1r2
Ž 31 .
the phase of u changes by 908 if the phase of s changes by 1808 so that u is now given by u s iw s 2 y d 2 x
1r2
.
Ž 32 .
If we move s along the negative imaginary axis then u is real and positive until we encounter the point yi d . If we move s around yi d along a small half-circle on the right-hand side of the imaginary axis the phase of u changes by y908 so that u is now given by u s yi w s 2 y d 2 x
1r2
.
Ž 33 .
The integrand of Eq. Ž29. has two infinite sets of simple poles. Both sets are related to the G functions. The first set are at the points
s s "irnX ,
n s lX q 1, lX q 2, . . . , etc. ,
Ž 34 .
and the residues are given by Eq. Ž14.. The second set of simple poles are given by u s 1rn , n s l q 1, l q 2, . . . , etc.
Ž 35.
We are restricted to the sheet where the real values of u are all positive but each point of Eq. Ž35. corresponds to two simple poles s s "irdn of the variable s . These poles and their residues are
s s irdn , s s yirdn ,
residues Ž yirdn P n3 . Ž y1 . 3
residues Ž irdn P n . Ž y1 .
nyly1
nyly1
Ž n y l y 1. !
Ž n y l y 1. !
y1
y1
.
,
Ž 36 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
69
In order to evaluate Eq. Ž29. we assume that r 2 ) r 2X and we substitute 1 F1
Ž l q 1 q 1rs , 2 l q 2; 2i s r 2 . 1
s 2
F1 Ž l q 1 q 1rs , 2 l q 2; 2i s r 2 . q
1 2
F2 Ž l q 1 q irs , 2 l q 2; 2i s r 2 .
Ž 37 .
so that Jl,lX is the sum of two integrals, I Žcontaining F2 . and II Žcontaining F1 .. We integrate I along the path CX1 of Fig. 2 and II along the path CX2 of Fig. 2. In both cases we assume that all singularities are located along the line segment between the points "i d on the imaginary axis. The integration paths contain little half-circles around each singularity on the right-hand side of the imaginary axis and also little half-circles around the two branch points "i d . The contributions from the large quarter-circles with radius R will become zero when R sends to infinity so that the integrals along the positive real axis become equal to the integrals along the imaginary axes. Both integrals I and II are the sum of four contributions: ŽA. Contributions from straight line segments between 0 and i d ŽB. Contribution from path from i d to infinity
X
X
Fig. 2. The integration paths C 1 and C 2 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
70
ŽC. Sum of the residues s s "irn ŽD. Sum of the residues u s 1rn We consider these various contributions separately. The contributions A of both parts I and II cancel. In part I we substitute s s i t and in part II we substitute s s yi t so that we have us w d 2yt2 x
ssit ,
I,
2 1r2
us w d 2yt x
s s yit ,
II ,
d 2 lq1
Hu u
1r2
,
0FtFd
,
0FtFd
X
Žit .
2 l q2
exp Ž ypiru . exp Ž ypirt . G Ž l q 1 q 1ru . G Ž l q 1 y 1ru .
= G Ž lX q 1 q irt . G Ž lX q 1 y irt . exp Ž ur1 . F2 Ž l q 1 q 1ru, 2 l q 2; y2 ur 1 . exp Ž ur1X . = 1 F1 Ž l q 1 q 1ru, 2 l q 2; y2 ur 1X . exp Ž tr 2 . F2 Ž l q 1 q 1rt , 2 lX q 2; y2 tr 2 . exp Ž ytr 2X . = 1 F1 Ž lX q 1 y 1rt , 2 l q 2; 2 tr 2X . Ž i d t . q
d 2 lq1
H0 u
X
Ž yit .
2 l q2
exp Ž ypiru . exp Ž p irt .
=G Ž l q 1 q 1ru . G Ž l q 1 y 1ru . G Ž lX q 1 y lrt . G Ž lX q 1 q lrt . exp Ž ur1 . =F2 Ž l q 1 q 1ru, 2 l q 2; y2 ur1 . exp Ž ur 1X . 1 F1 Ž l q 1 q 1ru, 2 l q 2; y2 ur1X . exp Ž ytr 2 . = F1 Ž l q 1 y 1rt , 2 lX q 2; 2 tr 2 . exp Ž tr 2X . 1 F1 Ž l q 1 q 1rt , 2 l q 2; y2 tr 2X . Ž yi d t . s 0
Ž 38 .
because of Eqs. Ž12a., Ž12b. and Ž13.. In order to evaluate the contributions B to I and II we substitute: 1r2
I,
ssit ,
u s iw t 2 y d 2 x
II ,
s s yit ,
u s yi w t 2 y d 2 x
s in ,
1r2
n w t2yd 2 x
1r2
s yi n
according to Eqs. Ž32. and Ž33.. The sum of the two contributions B becomes `
Hd
Ž in .
2 lq1
X
Žit .
2 l q2
exp Ž yprn . exp Ž ypirt . G Ž l q 1 y irn . G Ž l q 1 q irn . G Ž lX q 1 q 1rt .
=G Ž lX q 1 y 1rt . exp Ž i n r 1 . F2 Ž l q 1 y irn , 2 l q 2; y2i n r 1 . =exp Ž i n r 1X . 1 F1 Ž l q 1 y irn , 2 l q 2; y2i n r 1X . exp Ž tr 2 . F2 Ž lX q 1 q 1rt , 2 lX q 2; y2 tr 2 . =exp Ž ytr 2X . 1 F1 Ž lX q 1 y 1rt , 2 lX q 2; 2 tr 2X . Ž i d t . y2
s Ž 2rp . Ž 2 l q 1 . !
Ž 4 r1 r1X .
l
`
Hd n
2 lq1
=exp Ž yprn . G Ž l q 1 y irn . G Ž l q 1 q irn . exp Ž i n r 1 . F2 Ž l q 1 y irn , 2 l q 2; y2i n r 1 . =exp Ž i n r 1X . 1 F1 Ž l q 1 y irn , 2 l q 2; y2i n r 1X . tGlX Ž r 2 , r 2X ; t . d t `
s Ž 2rp . =
X 1
Hd G Ž r , r ; i n . G l
1
l
X
Ž r 2 , r 2X ; t . d t .
Ž 39 .
The contribution due to the singularities at s s "i n in may be derived in the same way as for the hydrogen atom in Eq. Ž16. since the Green’s function part of the integral is not involved in these singularities. The result is `
Sl ,lX Ž C . s y
X
Ý X
n sl q1
G l Ž r 1 , r 1X ; dnX . f nX , lX Ž r 2 . f nX , lX Ž r 2X . .
Ž 40 .
H.F. Hamekar Chemical Physics 228 (1998) 61–72
71
In a similar fashion it may be derived that the contribution from the singularities s s "i dn or u s Ž1rn. with n s l q 1, l q 2, . . . , etc., is given by Sl , lX Ž D . s q Ý q f n , l Ž r 1 . f n , l Ž r 1X . G lX Ž r 2 , r 2X ; dn . .
Ž 41 .
n
The above results may be summarized as follows `
HGl Ž r 1 , r 1X ; u . g lX Ž r 2 , s . g l)X Ž r 2 , s . d s q
X
Ý
G l Ž r 1 , r 1X ; dnX . f nX , lX Ž r 2 . f nX , lX Ž r 2X .
n slq1 `
`
s Ž 2rpi .
X 1
Hd G Ž r , r ; in . G l
1
X l Ž r2 , r2 ; t . t d t q
f n , l Ž r 1 . f n , l Ž r 1X . G lX Ž r 2 , r 2X ; dn .
Ý
X
Ž 42 .
nqlq1
and the helium atom Green’s function Ž27. has been transformed to G l , lX Ž r 1 , r 1X ; r 2 , r 2X ; d . s Ž 1rpi .
` l
1
l
X
Ž r 2 , r 2X ; t . t d t q Ž 1rpi . `
`
=
X 1
Hd G Ž r , r ; i n . G X 1
Hd G Ž r , r ; t . G l
1
l
X
Ž r 2 , r 2X ; i n . t d t q Ž 1r2.
Ý
f n , l Ž r 1 . f n , l Ž r 1X .
nqlq1 `
=G lX Ž r 2 , r 1X ; dn . q Ž 1r2 .
X
Ý X
G l Ž r 1 , r 1X ; dnX . f nX , lX Ž r 2 . f nX , lX Ž r 2X . .
Ž 43 .
n sl q1
5. Discussion The procedure outlined above presents an alternative approach to the derivation of the eigenvalues and eigenfunctions of the helium atom. we have limited our discussion to the nondegenerate ground state but it is in principle possible to extend it to excited states as long as we take the changes in distribution of the singularities into account. The relative convenience of the proposed procedure depends for a great deal on the complexity of the Green’s function. We were able to reduce the original form Ž24. of the Green’s function to the more convenient forms of Eq. Ž27. or Eq. Ž43. but we were disappointed to find that our transformations did not lead to even more convenient forms. We anticipated that the contour integration described in Section 4 would lead to elimination of the infinite sum over the discrete states. Our assumption turned out to be correct but we also found that a second infinite set of singularities produced another infinite sum over discrete states which is equally cumbersome as the other one. As a matter of fact, the new pair of infinite sets is identical to the old pair. Nevertheless, the transformation described in Section 4 produced an interesting result even if it did not lead to significant further simplifications of the Green’s function. The key question that we should address is whether it is feasible to derive the exact helium atom wave function from the integral equation Ž4.. We cannot supply the answer to that question at this time but we plan to pursue the matter further. Recent developments in hardware and software have produced significant progress in iterative procedures for dealing with mathematical problems and the present computer technology should be better suited to deal with integral equations than with differential equations w16,17x. At the least we hope to have outlined a new approach to solving the helium atom Schrodinger equation and we hope that further exploration ¨ of this approach leads to useful results.
H.F. Hamekar Chemical Physics 228 (1998) 61–72
72
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x
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