Fourier transform approach to potential harmonics

— —< <

Fourier Transform Approach toPotential Harmonics

JOHN AVERYU AND WENSHENG BIAN†

H. C. Orsted Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark; e-mail for J.A.:[email protected]

JOHN LOESERDepartment of Chemistry, Oregon State University

FRANK ANTONSENNiels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark

Received December 4, 1995; revised manuscript received January 15, 1996; accepted February 8, 1996

ABSTRACT

Because of the high degeneracy of hyperspherical harmonics, a method is needed forselecting the most important ones for inclusion in hyperangular basis sets. Such a methodwas developed by M. Fabre de la Ripelle, who showed that the most importantharmonics are l-projections of the product of the potential and a zeroth-order wavefunction; and he gave these the name, ‘‘potential harmonics.’’ In the present study wedevelop Fourier-transform-based methods for generating potential harmonics and forevaluating matrix elements between them. These methods are illustrated by a smallcalculation on three-body Coulomb systems with a variety of mass ratios. Q 1997 JohnWiley & Sons, Inc.

Introduction

f mass-weighted coordinates are used, theI Schrodinger equation of a system of N parti-¨cles can be written in the form

1w Ž .x Ž . Ž . Ž .y D q V x c x s Ec x , 12

* To whom correspondence should be addressed.† Permanent address: Institute of Theoretical Chemistry,

Shandong University, 250100 Jinan, People’s Republic of China.

where

d 2Ž .D s , d s 3N 2Ý 2 x jjs1

is the generalized Laplacian operator. In recentyears, a number or problems where correlation isimportant have been treated by avoiding single-particle approximations, and solving the many-particle Schrodinger equation directly in a space of¨dimensions d s 3N using hyperspherical coordi-

( )International Journal of Quantum Chemistry, Vol. 63, 5]14 1997Q 1997 John Wiley & Sons, Inc. CCC 0020-7608 / 97 / 010005-10

AVERY ET AL.

nates, and basis functions of the form:

Ž . Ž . Ž .F s R r Y u . 3nlm n l lm

Ž .In Eq. 3 ,

d2 2 Ž .r ' x 4Ý j

js1

w xis the hyperradius 1]16 ,

x 1Ž . Ž .u ' s x , x , . . . , x 51 2 dr r

Ž .is a d-dimensional unit vector, and Y u is alm

w xhyperspherical harmonic 17]19 satisfying

2 Ž .L Y s l l q d y 2 Y , l s 0, 1, 2, . . .lm lm

2d 2 Ž .L ' y x y x , 6Ý i jž / x xj ii)j

L2 being the generalized angular momentum oper-ator. The hyperspherical harmonics satisfy an or-thonormality relation of the form

U Ž .X X X XdV Y Y s d d , 7H l m lm l l m m

where dV is the generalized solid-angle elementdefined by

dy1 Ž .dx s dx dx . . . dx s r dr dV . 81 2 d

Ž . Ž .For d s 3, Eqs. 8 and 9 reduce to the familiarrelationships

2 Ž .L Y s l l q 1 Y , l s 0, 1, 2, . . .lm lm

U Ž .X X X XdV Y Y s d d 9H l m lm ll m m

Although we have used a notation which empha-sizes the similarity between hyperspherical har-monics and the familiar spherical harmonics of athree-dimensional space, a difference should benoted: m is not a single index but stands for a setof indices which are the eigenvalues of a completeset of operators which commute with each otherand with L2.

All the familiar theorems for spherical harmon-ics have d-dimensional generalizations, and it maybe worthwhile to mention a few of them here. Likethe familiar spherical harmonics, hyperspherical

harmonics obey a sum rule:

2l q d y 2X XU aŽ . Ž . Ž . Ž .Y u Y u s C u ? u , 10Ý lm lm lŽ . Ž .d y 2 I 0m

Ž .where I 0 is the total solid angle in the d-dimen-sional space:

2p d r2

Ž . Ž .I 0 ' dV s 11H Ž .G dr2

aŽ X.and where C u ? u is a Gegenbauer polynomiall

with a ' dr2 y 1:

w xlr21Xa Ž .C u ? u s Ýl Ž .G a ts0

t ly2 tXŽ . Ž .Ž .y1 G a q l y t 2u ? uŽ .= . 12Ž .t ! l y 2 t !

Ž .When d s 3, the Gegenbauer polynomials in 12reduces to a Legendre polynomial, and the sum

Ž .rule in 10 reduces to the familiar sum rule:

2 l q 1X XU Ž . Ž . Ž . Ž .Y u Y u s P u ? u . 13Ý lm lm l4pm

Ž . Ž .From Eq. 10 it follows that if f u is an arbitraryfunction of the hyperangles in a d-dimensionalspace, then

w Ž .x Ž . X X Ž X . Ž X .O f u s Y u dV Y u f uÝ Hl lm lmm

2l q d y 2X X Xa Ž . Ž .s dV C u ? u f uH lŽ . Ž .d y 2 I 0

Ž .14

Ž .is the component of f u which is an eigenfunctionof the generalized angular momentum operator L2

Ž .corresponding to the eigenvalue l l q d y 2 . Justas a plane wave in three-dimensional space can beexpanded in terms of Legendre polynomials andspherical Bessel functions, so a d-dimensionalplane wave can be expanded in terms of Gegen-bauer polynomials and what might be called ‘‘hy-perspherical Bessel functions’’:

eik ?x s eiŽk1 x1q. . .qk d x d .

`l d aŽ . Ž . Ž . Ž .s d y 4 !! i d q 2l y 2 j kr C u uÝ l l k

ls0

k 1Ž . Ž .u ' s k , k , . . . , k , 15k 1 2 dk k

VOL. 63, NO. 16

FOURIER TRANSFORM APPROACH TO POTENTIAL HARMONICS

where

Ž . ay1 Ž .G a 2 J kraqld Ž .j kr ' al Ž . Ž .d y 4 !! krt 2 tql` Ž . Ž .y1 kr

Ž .s . 16Ý Ž . Ž .2 t !! d q 2 t q 2l y 2 !!ts0

Ž .When d s 3, Eq. 15 reduces to the familiar rela-tionship

ìk ?x l Ž . Ž . Ž . Ž .e s i 2 l q 1 j kr P u ? u . 17Ý l l k

ls0

Ž . Ž .From Eqs. 14 and 15 it follows that a function ofŽ . Ž .the form R r f u has the d-dimensional Fourier

transform:

1t ik ?xw Ž . Ž .x Ž . Ž .R r f u s dx e R r f uHdr2Ž .2p`

t Ž . w Ž .x Ž .s R k O f u , 18Ý l l kls0

w Ž .x Ž .where O f u is defined by Eq. 14 and wherel k

Ž .Ž . lÌ 0 d y 2 !!i

t dy1 dŽ . Ž . Ž .R k s dr r j kr R r .Hl ldr2Ž . 02pŽ .19

Potential Harmonics

In spite of the attractive features of hyperspheri-cal harmonics, there is a serious difficulty whichmust be overcome if they are to be used as ahyperangular basis set. The difficulty is that highvalues of l are needed for accuracy; but the de-generacy of the hyperspherical harmonics is given

w xby 17 :

Ž .Ž .d q 2l y 2 d q l y 3 !Ž .v s , 20Ž .l! d y 2 !

and for values of l, this degeneracy becomes verylarge. In order that the hyperangular part of thecalculation should not grow to an unmanageablesize, one needs a method for choosing the mostimportant hyperspherical harmonics for inclusionin our basis set and for rejecting those which willnot contribute importantly. For this reason, M.

w xFabre de la Ripelle 20, 21 and others introduced aŽset of hyperspherical harmonics ‘‘potential har-

.monics’’ which are generated by taking l-projec-

tions of the potential multiplied by an appropriatezeroth-order wave function. We see by the follow-ing argument that these harmonics are the oneswhich contribute most importantly to the wavefunction.

Let us suppose that our zeroth-order wave func-Ž0Ž .tion is c x . Then, if we take the l-projection of

Ž . Ž0.Ž . w Ž .xV x c x as in Eq. 14 , we obtain:

w Ž . Ž0. Ž .x Ž . Ž . Ž .O V x c x s F r Y u . 21Ýl l p l pp

In other words, if we apply the projection operator

< :² < Ž .O s Y Y 22Ýl lm lmm

Ž . Ž0.Ž .to the function V x c x , we will obtain a set ofŽ .hyperspherical harmonics, Y u , multiplied byl p

functions of the hyperradius. In this way we canassociate with each zeroth-order wave function asmall set of hyperspherical harmonics.

In order to see that it is the potential harmonics,Ž . Ž .Y u , defined by Eq. 21 , which contribute mostl p

importantly to the wave function at high values ofl, we can notice that

U Ž . Ž . Ž0. Ž .dV Y u V x c xH lm

Ù Ž0.Ž . w Ž . Ž .xYs dV Y u O V x c xÝH lm l

Yl s0

U Ž . w Ž . Ž0. Ž .xs dV Y u O V x c xH lm l

Ž . U Ž . Ž . Ž .s F r dV Y u Y u . 23Ý Hl p lm l pp

Thus, if we first construct the set of potentialŽ . Ž .harmonics, Y u by means of Eq. 21 , and thenl p

construct the remaining hyperspherical harmonicsŽ .so that they are orthogonal to the set Y u , thel p

only harmonics which will have a nonzero angularŽ . Ž0.Ž .matrix element with the function V x c x will

be the set of potential harmonics.One can also show that the potential harmonics

are those which appear in the first iterated solutionof the integral form of the many-particle Schrod-¨

Ž0.Ž .inger equation when c x is used as a startingpoint. The many-particle Schrodinger equation of a¨system in a bound state can be written as an

w xintegral equation 17 :

Ž . X Ž X . Ž X . Ž X . Ž .c x s y dx G x y x V x c x , 24H

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 7

AVERY ET AL.

where

` 2l q d y 2X X XaŽ . Ž . Ž .G x y x s C u ? u G r , r ,Ý l lŽ . Ž .d y 2 I 0ls0

Ž .25

and where

2XŽ . Ž . Ž . Ž .G r , r s I k r K k r . 26al lqa 0 - lqa 0 )XŽ .r r

Ž .In Eq. 26 , I and K are modified Bessel functions,

dŽ .a ' y 1 27

2

and

r if r - rX

Ž .r ' . 28X X- ½ r if r - r

Ž .The parameter k in Eq. 26 is related to the0energy by

2 Ž .k s y2 E, 290

where E is the binding energy of the system.Ž .Equation 24 can be iterated, starting with some

zeroth-order wave function:

Ž1. Ž . X Ž X . Ž X . Ž0. Ž X . Ž .c x s y dx G x y x V x c x . 30HŽ .Then, from Eq. 14 , it follows that the first-iterated

solution will be given by

Ž1. Ž . Ž . Ž . Ž .c x s R r Y u , 31Ý l p l pl p

where

`X X dy1 X XŽ . Ž . Ž . Ž .R r s y dr r G r , r F r 32Hl p l l p

0

Ž .and where F is defined by Eq. 21 .l p

Fourier Transform Representations ofPotential Harmonics

The application of Fourier transforms to poten-tial harmonics can perhaps best be illustrated withthe simple case where the zeroth-order wave func-tion of the system is independent of the hyperan-gles, i.e., if

Ž0. Ž . Ž . Ž .c x s R r , 33

then the potential harmonics associated with c Ž0.

Žcan be found by taking the l-projections harmonic. Ž .projections of V x :

w Ž . Ž0. Ž .x Ž . w Ž .x Ž .O V x c x s R r O V x . 34l l

Suppose that the potential for the system has theform:

N NXŽ . Ž . Ž .V x s g r 35Ý Ý ab

a)b as1

when expressed in terms of the non-mass-weightedŽ . Ž X .primed coordinates. Here g r is some functionab

X < X X <of the interparticle distance, r ' r y r . As aab a bfirst step in calculating its harmonic projection, we

Ž X .can express g r as a three-dimensional Fourier12transform:

1 X XX 3 t ik ?Žr yr .1 2Ž . Ž .g r s d k g k eH12 3r2Ž .2p

`1 X X2 t ik ?Žr yr .1 2Ž .y dk k g k dV e ,H H k3r2Ž . 02pŽ .36

where rX , rX , and k are three-dimensional vectors,1 2and where

p2p Ž .dV s df sin u du . 37H H Hk k k k0 0

If we now define the d-dimensional unit vectorsw x17 :

1Žw ' g k , g k , g k , yg k , yg k ,12 1 1 1 2 2 3 2 1 2 2g k12

.yg k , 0, . . . , 0 ,2 3

1Ž . Ž .u ' x , x , x , x , x , . . . , x , 381 2 3 4 5 dr

where, for example,

1g ' , a s 1, 2a m' a

1r21 1Ž .g ' q . 3912 ž /m m1 2

VOL. 63, NO. 18


Ž .Then, using Eq. 15 , we can write:

ik ?Žr 1X yr 2

X . ig 12 k r w12 u Ž .e s e s d y 4 !!`

l d aŽ . Ž . Ž . Ž .= i 2l q d y 2 j g kr C u ? w . 40Ý l 12 l 12ls0

Ž . Ž .If we substitute 40 into 36 , we obtain

`Ž .d y 4 !!X lŽ . Ž .g r s i 2l q d y 2Ý12 3r2Ž .2p ls0

`2 t d aŽ . Ž . Ž . Ž .= dk k g k j g kr dV C u ? w . 41H Hl 12 k l 12

0

Ž .Equation 41 can be written in the form

`X 12Ž . Ž . Ž . Ž .g r s A g r U u , 42Ý12 l 12 l

ls0

where

112 aŽ . Ž . Ž .U u ' dV C u ? w 43Hl k l 124p

and

2lŽ . Ž .Ž .A r ' i 2l q d y 2 d y 4 !!(l p

`2 t dŽ . Ž . Ž .= dk k g k j kr . 44H l

0

Ž .The angular integral in Eq. 43 can be performedwithout difficulty: Using the definition of the

Ž .Gegenbauer polynomials, Eq. 12 , together withthe fact that

1 ly2 tŽ .dV 2u ? wH k 124p

ly2 tXp1 2 r12 ly2 tŽ .s du sin u cos uH k k kž /2 g r 012

ly2 tX¡ 1 2 r12l s even~s ž /l y 2 t q 1 g r12¢

0 l s oddŽ .45

we obtain for even l

tlr2 Ž . Ž .1 y1 G l q a y t12 Ž .U u s Ýl Ž . Ž .G a t ! l y 2 t q 1 !ts0

ly2 tX2 r12 Ž .= . 46ž /g r12

12 Ž . Ž . Ž .The function U u shown in Eqs. 43 and 46 isl

an eigenfunction of generalized angular momen-Ž .tum, since, from the sum rule, 10 ,

Ž .Ž .I 0 d y 2U2 a 2Ž . Ž . Ž .L C u ? w s L Y u Y wÝl 12 lm lm 12Ž .2l q d y 2 m

Ž . a Ž . Ž .s l l q d y 2 C u ? w 47l 12

Ž . Ž X .Thus, Eq. 42 represents a resolution of g r into122 Ž . Ž .eigenfunctions of L . Combining 42 and 45 , we

can write:

w Ž X .x Ž .O g r s A g r ,0 ab 0 12

X 2d rabXw Ž .x Ž . Ž .O g r s a A g r y 1 , 482 ab 2 12 ž /3 g r12

and so on, all odd projections being zero. In theparticular case where all the particles in the systemhave the same mass, we obtain only one potentialharmonic for each value of l:

w Ž .x Ž . Ž . Ž .O V x s A g r U u , 49l l l

where

N NabŽ . Ž .U u s U uÝ Ýl l

b)a as1

ly2 tXlr2 N N 2 rab Ž .s b 50Ý Ý Ýl, t ž /g rts0 b)a as1

and

tŽ . Ž .y1 G l q a y tŽ .b ' , 51l, t Ž . Ž .G a t ! l y 2 t q 1 !

while for odd values of l the projections vanish.Apart from a normalization factor, the functions

Ž . Ž .U u shown in Eq. 50 are the potential harmonicsl

associated with the zeroth-order ground state inŽ .equation 34 . The potential harmonics depend on

the angular form of the zeroth-order wave func-tion; but they are independent of the form of the

Ž X .interparticle interaction potential, g r .ab


AVERY ET AL.

Normalization

Ž .From Eq. 10 , and from the orthonormality ofŽ .the hyperspherical harmonics, 7 , it follows that

1Xa aŽ . Ž .XdV C u ? w C u ? wH l ab l cdŽ .I 0

Ž .Xd d y 2l l Xa Ž . Ž .s C w ? w . 52l ab cdŽ .2l q d y 2

This relationship provides us with a convenientmethod for normalizing the potential harmonics,since, if we define the unit vector w in such aabway that

Ž X X .ig kr u ? w s ik ? r y rab ab a b

Ž .w ? w s 1 53ab ab

w Ž .xthe generalization of Eq. 38 , we have

1ab aŽ . Ž . Ž .U V s dV C u ? w . 54Hl k l ab4p

Then

2Ž .4p 2ab< Ž . <dV U VH lŽ .I 01

a Ž .Xs dV dV dV C u ? wH H Hk k l abŽ .I 0

Xa Ž .X=C u ? wl ab

d y 2Xa Ž .Xs dV dV C w ? wH Hk k l ab ab2l q d y 2

d y 2s Ž . Ž .2l q d y 2 G a

tw xlr2 Ž . Ž .y1 G l q a y t= Ý Ž .t ! l y 2 t !ts0

ly2 tXŽ . Ž .X= dV dV 2w ? w . 55H Hk k ab ab

But

1X X Ž .w ? w s k ? k s cos u 56Xab ab kk k

so that

2Ž .4ply2 tXŽ . Ž .XdV dV w ? w s . 57H Hk k ab ab l y 2 t q 1

Therefore

1 2ab< Ž . <dV U VH lŽ .I 0w xlr2Ž .d y 2 ly2 tŽ . Ž .s b 2 , 58Ý l, tŽ .2l q d y 2 ts0

Ž .where b is defined by Eq. 51 . It is easy to showl, tin a similar way that when b / c,

1ab acŽ . Ž .dV U V U VH l lŽ .I 0

ly2 tw x 2lr2d y 2 2ga Ž .s b , 59Ý l, t ž /2l q d y 2 g gab acts0

while when all four indices are different,

1ab cdŽ . Ž .dV U V U VH l lŽ .I 0

lr2Ž .Ž . Ž .d y 2 y1 G lr2 q aŽ .s . 60Ž . Ž .Ž .2l q d y 2 G a lr2 !

Ž . Ž .Equations 58 and 59 can be simplified by notic-ing that

wŽ . xlq1 r21ay1Ž .C j s Ýlq1 Ž .G a y 1 ts0

t lq1y2 tŽ . Ž .Ž .y1 G l q a y t 2jŽ .= . 61Ž .t ! l y 2 t q 1 !

Ž . Ž . Ž .Comparing Eqs. 61 , 58 , and 51 , and makingw xuse of the fact that 17

Ž .l q d y 4 !ay1Ž . Ž .C 1 s , 62lq1 Ž . Ž .l q 1 ! d y 5 !

Ž .we can rewrite 58 in the form:

1 2ab< Ž . <dV U VH lŽ .I 0Ž .Ž .d y 2 l q d y 4 !

Ž .s , 63Ž .Ž . Ž .2l q d y 2 l q 1 ! d y 4 !

VOL. 63, NO. 110


Ž .while 59 becomes

1ab acŽ . Ž .dV U V U VH l lŽ .I 0

d y 2ay1Ž . Ž .s C j , 64lq1Ž .Ž .2l q d y 2 d y 4 j

where

2g m ma b c Ž .j s s 65(Ž .Ž .g g m q m m q mab ac a b a c

Hyperangular Matrix Elements

w xIt has been shown by Avery and Ørmen 22w xand by Michels 23 that

1 n n n1 2 3Ž . Ž . Ž .dV w ? u w ? u w ? uH 1 2 3Ž .I 0Ž .d y 2 !!

s Ž .n q n q n q d y 2 !!1 2 3

n !n !n !1 2 3= Ý Ž . Ž . Ž .2n !! 2n !! 2n !!11 22 33� 4n i j

n n n12 13 23Ž . Ž . Ž .w ? w w ? w w ? w1 2 1 3 2 3= ,

n !n !n !12 13 23

Ž .66

Ž .where the sum over the n in Eq. 66 is restrictedi jby the requirement

n s 2n q n q n ,1 11 12 13

n s 2n q n q n ,2 22 12 23

Ž .n s 2n q n q n , 673 33 13 23

the n ’s being positive integers or zero. Equationsi jŽ . Ž . Ž .66 and 67 can be used in conjunction with 43to evaluate hyperangular matrix elements involv-

ing the potential harmonics. If we let

1X XŽ .w ? u s k ? x y x ,1 a bg krab

1X XŽ .w ? u s k ? x y x ,2 c dg krcd

1X XŽ . Ž .w ? u s k ? x y x , 683 e fg kre f

and

j¡ i j Xk ? k i / jX X~ Ž .w ? w s , 69k ki j ¢1 i s j

then, in a manner similar to the derivation of Eq.Ž .45 , we obtain, when all the n ’s are even:j

1 n1Ž .X YdV dV dV w ? uH H Hk k k 13Ž .4pn nX Y2 3Ž . Ž .= w ? u w ? u2 3

1s Ž .Ž .Ž .n q 1 n q 1 n q 11 2 3

nn nX X X 31 2r r rab cd e f Ž .= , 70ž / ž / ž /g r g r g rab cd e f

while if any of the n ’s are odd the integral van-jŽishes. Combining these relations, we obtain for n ,1

.n , and n all even :2 3

nn nX X X 31 21 r r rab cd e fdVH ž / ž / ž /Ž .I 0 g r g r g rab cd e f

3Ž .d y 2 !!Ž .s n q 1 !Ł jŽ .n q n q n q d y 2 !! js11 2 3

n12 n13 n 23 � 4j j j J nŽ .12 13 23 i j= ,Ý Ž . Ž . Ž .n !n !n ! 2n !! 2n !! 2n !!12 13 23 11 22 33� 4n i j

Ž .71

where

n !n ! 1¡ 12 13 ÝŽ . Ž . Ž . Ž .n q n q 1 !! m! n y m !! n y m !! n q m q 112 13 12 13 23m~� 4 Ž .J n s . 72Ž .i jn , n , and n all even or all odd12 13 23¢0 otherwise


AVERY ET AL.

In the ‘‘all-even’’ case, m s 0, 2, 4, . . . , m, while inthe ‘‘all-odd’’ case, m s 1, 3, 5, . . . , m, m being the

Ž .smaller of n or n . In Eq. 71 the sum is re-12 23Ž .stricted by the condition shown in Eq. 67 .

An Illustrative Example

To illustrate the discussion presented above, wecan consider a simple example—a system consist-ing of three particles interacting through Coulombforces. If the charges and masses of the threeparticles are q and m , a s 1, 2, 3 in units of ana aelectron’s charge and mass, then the Hamiltonianof the system, in atomic units, is

1Ž . Ž .H s y D q V x , 73

2

where

q q q q q q1 2 1 3 2 3Ž . Ž .V x s q q 74X X Xr r r12 13 23

and where

9 2 3 2 2 2 1 D ' ' q q .Ý Ý2 X 2 X 2 X 2ž /m x x y zajjs1 as1 a a a

Ž .75

Ž .The unprimed coordinates are mass weighted.For systems interacting through Coulomb forces,

Ž .the integral in Eq. 44 can be evaluated exactlyw x w x17 , 24 , and we obtain

1lŽ . Ž . Ž .i d q 2l y 2 G a G lr2 q 2Ž . Ž .A r s . 76l 1Ž .p rG lr2 q a q 2

The l-projections of the three-particle Coulombpotential are:

q q1 2 12w Ž .x Ž . Ž .O V x s A r U xl l lg12

q q q q1 3 2 313 23Ž . Ž . Ž .q U u q U u , 77l lg g13 23

so that, apart from a normalizing constant, thepotential harmonics are:

q q q q1 2 1 312 13Ž . Ž . Ž .Y u ; U u q U ul, p l lg g12 13

q q2 3 23Ž . Ž .q U u , 78lg23

12 Ž .where g and U u are defined, respectively, by12 l

Ž . Ž .Eqs. 39 and 46 , with analogous definitions forg , etc. We can see that for Coulomb systems,13there is only one potential harmonic for each value

Ž .of l, since g can be factored out from A g r .ab l abWhen m s m s 1, m s 4m , q s q s y11 2 3 p 1 2

and q s 2, the three-particle Coulomb system cor-3responds to a helium atom; while when m s m1 2s m , m s 1, q s q s y1, and q s 1, the sys-p 3 1 2 3tem corresponds to a charge-reversed Hq mole-2cule. We can go between these two cases in acontinuous way by changing the masses and bychanging q . This smooth atom-to-molecule transi-3

w xtion, first studied by Burden 25 , is interestingŽand we have chosen it for our calculational exam-

.ple because in the atomic limit, the motions ofparticles 1 and 2 are approximately independentand uncorrelated. Correlation effects are present inhelium, but they are relatively small. However, ifm and m are gradually increased, the kinetic1 2energy of these two particles is less and less ableto liberate them from the effects of the potential,and their motion becomes gradually more andmore highly correlated. In the molecular limit, thedistance between particles 1 and 2 is quite sharplydefined, and the correlation between these twoparticles is very great. Thus the independent parti-cle approximation breaks down gradually as onegoes smoothly from atom to molecule; and as thishappens, the Born]Oppenheimer approximationbecomes progressively more valid. As Berry and

whis co-workers at Chicago have emphasized 26]x29 , our usual way of looking at atoms is only an

approximation: Electron correlation effects giveatoms a moleculelike structure. Conversely, ourusual picture of molecular structure is only anapproximation: By giving molecules large amountsof vibrational energy one can cause them to ‘‘melt,’’so that the nuclear wave functions become delocal-ized.

We have made a calculation of the ground stateand the first few 1S excited states for the atom ªmolecule transition described above using a verysmall hyperangular basis set, consisting of onlyseven potential harmonics. Since Coulomb systemshave only one potential harmonic for each value ofl, and since 1S states involve only even l values,the potential harmonics included in the basis setcorresponded to l s 0, 2, 4, . . . , 12. Hyperangularmatrix elements were evaluated by means of Eqs.Ž . Ž .66 ] 72 . The hyperradial part of the wave func-tion was treated by the method of generalizedLaguerre functions, which is discussed in refer-

VOL. 63, NO. 112


TABLE Iq = q = y1, q = 2, m = m = 1. Effect on1 2 3 1 2energies of increasing m from 1 to 4m .3 p

1 1 1m 1 S 2 S 3 S3

1 y1.41865 y0.88280 y0.652572 y1.88794 y1.18118 y0.894413 y2.12305 y1.32281 y0.995694 y2.26809 y1.42414 y1.087545 y2.36542 y1.49088 y1.14971

10 y2.58983 y1.64178 y1.2720720 y2.72182 y1.73607 y1.345314m y2.86945 y1.84663 y1.45084p

w x w xences 30 and 31 . The hyperradial basis set con-sisted of six generalized Laguerre functions.

Table I and Figure 1 show the effect of increas-ing m from m s 1 to m s 4m , where m sp3 3 3 p1836.15 is the proton mass. When all three particles

FIGURE 1. Effect of increasing m , starting with a3three-body Coulomb system whose charges correspondto those in the helium atom, but where the masses of all

( )three particles are equal to the electron mass Table I .As the kinetic energy of the third particle becomes lessimportant, the system decreases in size and becomesmore tightly bound. The results of our small-basiscalculation with potential harmonics are indicated bysolid lines, while the dashed line indicates theground-state energy of the system in the

( )pseudo-classical limit e calculated by means of`

dimensional scaling.

TABLE IIq = q = y1, q = 2, m = 1. Effect on1 2 3 3energies of increasing m = m from 1 to 10.1 2

1 1 1m 1 S 2 S 3 S3

1 y1.41865 y0.88280 y0.652572 y1.88514 y1.16890 y0.839714 y2.23142 y1.41378 y1.00403

10 y2.45636 y1.66396 y1.21009

have a mass equal to the electron mass, the aver-age interparticle distances are large, and the sys-tem’s binding energy is correspondingly small; butas m becomes larger, the system decreases in size3and its binding energy increases. In the atomic

Ž .limit He , our small basis set yielded a bindingenergy of y2.86945 atomic units for the groundstate.

Table II and Figure 2 show the effect of startingwith equal masses and increasing m s m from 11 2to 10. Again, the average interparticle distancesdecrease and the system’s binding energy in-creases. With our very small hyperangular basis

FIGURE 2. Effect on energies of increasing m = m1 2( )Table II . We again start with a system of three equalmasses, with heliumlike charges. As the masses ofparticles 1 and 2 are increased, the motion of these twoparticles becomes highly correlated, and our small-basispotential harmonic calculation underestimates theground-state binding energy.


AVERY ET AL.

FIGURE 3. Final step in the atom ª moleculetransition. As q is decreased, the system expands in3size and becomes less tightly bound. As in the otherfigures, the dashed line indicates e .`

set, we were unable to obtain accurate energies forlarger values of m s m , since the l-convergence1 2becomes slow as one approaches the highly corre-lated molecular limit. In fact, the ground-state en-ergy curve shown in Figure 2 differs appreciablyfrom the dotted curve, corresponding to the e , the`

w Žpseudo-classical limit. The pseudo-classical D s. w x` results were obtained by one of us 32 using

w x .dimensional scaling techniques 19 . By contrast,there is good qualitative agreement between theground-state curve in Figure 1 and the pseudo-classical limit, e , shown in Figures 1]3.`

We conclude from our small illustrative calcula-tion that the potential harmonic method is apromising way of treating correlation and non-Born]Oppenheimer effects in Coulomb systems;that only moderately high values of l aremaxneeded for the atomic case; but that much highervalues of l are needed to treat molecules.max

References

Ž .1. J. Macek, J. Phys. B 1, 831 1968 .Ž .2. J. Macek and K. A. Jerjian, Phys. Rev. A 33, 233 1986 .

Ž .3. U. Fano, Rep. Prog. Phys. 46, 97 1983 .

4. U. Fano and A. R. P. Rao, Atomic Collisions and SpectraŽ .Academic Press: New York, 1986 .

5. M. I. Haftel and V. B. Mandelzweig, Phys. Lett. A 120, 232Ž .1987 .

6. B. R. Judd, Angular Momentum Theory for Diatomic MoleculesŽ .Academic Press, New York 1975 .

7. A. Kupperman and P. G. Hypes, J. Chem. Phys. 84, 5962Ž .1986 .

8. M. E. Kellman and D. R. Herrick, Phys. Rev. A 22, 1536,Ž .1980 .

Ž .9. D. R. Herrick, Adv. Chem. Phys. 52, 1 1983 .

Ž .10. H. Klar, J. Phys. A 18, 1561 1985 .

Ž .11. H. Klar and M. Klar, J. Phys. B 13, 1057 1980 .

Ž .12. D. L. Knirk, J. Chem. Phys. 60, 1 1974 .

Ž .13. T. Koga and T. Matsuhashi, J. Chem. Phys. 89, 983 1988 .

Ž .14. C. D. Lin, Phys. Rev. A 23, 1585 1981 .¨15. J. Linderberg and Y. Ohrn, Int. J. Quant. Chem. 27, 273

Ž .1985 .

16. J. Avery, D. Z. Goodson, and D. R. Herschbach, Int. J.Ž .Quantum Chem. 39, 657 1991 .

17. J. Avery, Hyperspherical Harmonics; Applications in QuantumŽ .Theory Kluwer Academic, Dordrecht, Netherlands, 1989 .

18. N. K. Vilenken, Special Functions and the Theory of GroupŽRepresentations American Mathematical Society, Provi-

.dence, RI, 1968 .

19. D. R. Herschbach, J. Avery, and O. Goscinski, Eds., Dimen-Žsional Scaling in Chemical Physics Kluwer Academic, Dor-

.drecht, Netherlands, 1993 .

20. M. Fabre de la Ripelle, in Proceedings of the InternationalŽ .School on Nuclear Theoretical Physics Predeal, 1969 , A. Cor-

Ž .ciovi, Ed., Inst. Atom. Phys., Bucharest, 1969 ; Rev. Roum.Ž .Phys. 14, 1215 1969 .

Ž .21. M. Fabre de la Ripelle, Ann. Phys. 147, 281 1983 .

22. J. Avery and P.-J. Ørmen, Int. J. Quantum Chem. 18, 953Ž .1980 .

Ž .23. M. A. J. Michels, Int. J. Quantum Chem. 20, 951 1982 .

24. I. S. Gradshteyn and I. M. Ryshik, Tables of Integrals, SeriesŽ .and Products Academic Press, New York, 1965 .

Ž .25. F. R. Burden, J. Phys. B: At. Mol. Phys. 16, 2289 1983 .

Ž .26. R. S. Berry and J. L. Krause, Adv. Chem. Phys. 70, 35 1988 .

27. R. S. Berry, in Understanding Molecular Properties, J. Avery,ŽJ. P. Dahl, and A. E. Hansen, Eds. Reidel, Dordrecht,

.Netherlands, 1987 .

Ž .28. G. S. Ezra and R. S. Berry, Phys. Rev. A 28, 1974 1983 .

29. R. S. Berry, in Structure and Dynamics of Atoms and Molecules:Conceptual Trends, J. L. Calais and E. S. Kryachko, Eds.,

.Kluwer Academic, Dordrecht, Netherlands, 1995 .

Ž .30. W. Bian and C. Deng, Int. J. Quantum Chem. 50, 395 1994 ;Ž . Ž .51, 285 1994 ; 54, 273 1995 .

Ž .31. W. Bian and C. Deng, Theor. Chim. Acta 92, 135 1995 .

32. J. Avery, in Structure and Dynamics of Atoms and Molecules:Conceptual Trends, J. L. Calais and E. S. Kryachko, Eds.Ž .Kluwer Academic, Dordrecht, Netherlands, 1995 .

VOL. 63, NO. 114

Documents

Fourier transform approach to potential harmonics