Methods for the calculation of spherically averaged Compton profiles with GTOs

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Methods for the Calculation ofSpherically Averaged Compton Profileswith GTOs

ALEJANDRO SAENZ, TANJA ASTHALTER, WOLF WEYRICHFakultat fur Chemie, Universitat Konstanz, Fach M 722, D-78457 Konstanz, Germany¨ ¨ ¨

Received 19 November 1996; revised 8 April 1997; accepted 17 April 1997

ABSTRACT: For the first time, an analytical and efficient algorithm for the evaluationŽ . ² Ž .:of spherically averaged reciprocal form factors B s s B s using Gauss-type basisV s

functions is presented. The spherically averaged Compton profile is available by Fouriertransformation of the reciprocal form factor. The algorithm has been successfullyimplemented in connection with the quantum chemistry codes GAMESS and CRYSTAL92,which perform Hartree]Fock calculations for molecules and solids. In addition, ananalytical algorithm for the direct evaluation of spherically averaged Compton profiles

² n: Ž .and the moments p n G y1 via the momentum density is proposed for Gauss-typebasis functions. Q 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 213]223, 1997

Key words: Compton profile; reciprocal form factor; spherical average; autocorrelationfunction

Introduction

powerful tool for assessing the validity ofA different quantum-chemical models ofmolecules and solids is provided by comparison toexperimentally observable quantities that are re-lated as closely as possible to wave functions inposition or momentum space, such as coherentform factors and Compton profiles.

Correspondence to: W. Weyrich.Contract grant sponsor: Fonds der Chemischen Industrie.Contract grant sponsor: European Union.Contract grant number: HCM-CHR-CT93-0155.

Coherent form factors—and by Fourier transfor-mation the electronic position density—are acces-sible from elastic scattering of photons or electronsat a target which may be solid, liquid, or gaseousw x1]4 . Complementary information can be gainedfrom high-energy inelastic scattering experimentsŽ w x.Compton scattering 5]6 , which, within the

w xframe of the sudden-impulse approximation 7]9 ,lead to the projection of the electronic momentumdensity onto the scattering vector, the so-calledCompton profile.

Since position-space and momentum-spacewave functions are related to each other by aFourier]Dirac transformation, they contain thesame information about the electronic structure of

( )International Journal of Quantum Chemistry, Vol. 65, 213]223 1997Q 1997 John Wiley & Sons, Inc. CCC 0020-7608 / 97 / 030213-11

SAENZ, ASTHALTER, AND WEYRICH

matter. On the other hand, according to thew xHohenberg]Kohn theorem 10 , the position-

space density alone determines all ground-stateone-electron properties. Diffraction experimentsshould therefore in principle be sufficient for thepurpose of testing quantum-chemical models. Inpractice, however, three important points must betaken into account.

First, the deviation of any approximate wavefunction from the correct one will show up differ-ently in position and momentum space owing tothe reciprocity of both spaces.

Second, precise position-density data of valenceelectrons, whose behavior is especially interestingwith respect to chemical bonding, are difficultto obtain because of extinction problems and thelimited accessibility and accuracy of high-k reflec-tions; moreover, an appropriate model is neces-sary. On the other hand, inelastic-scattering experi-ments are very sensitive to the behavior of valenceelectrons, possess a very good signal-to-noise ratio,and depend only moderately on the structuralperfection of the sample. However, their resultslack the illustrative quality of position space whendiscussed in momentum space. An alternative ap-

w xproach has been suggested 11]14 where use ismade of the wave function autocorrelation proper-ties of the Fourier transform of the Compton pro-

w xfile, a quantity that was first introduced in 15and is now usually called reciprocal form factor.

w xThird, as it has been shown by Henderson 16 ,the statement of a one-to-one relationship betweenposition-space density and all ground-state prop-erties of the Hohenberg]Kohn theorem also holds

Žfor the momentum-space density and the recipro-.cal form factor because of the equivalence of the

position and the momentum representation. Thus,the experimental momentum-space density is ofequal importance as the position-space density.

For all three reasons it is advisable to follow acombined approach using experimental data fromboth position and momentum space and compar-ing them with the respective theoretically obtainedquantities.

In the majority of cases, the substance of interestis only available in gaseous, liquid, or powderform, and inelastic-scattering experiments yieldonly spherically averaged Compton profiles. Thecorresponding theoretical Compton profiles are ac-cessible via two different ways.

The first one involves the calculation of thespherically averaged momentum density, which

can be expressed analytically in the case of Slater-type as well as of spherical and Cartesian Gauss-

w xtype basis functions 17 . The spherically averagedCompton profile is then obtained by a semi-infinite integration over the momentum coordi-nate.

The second way involves the evaluation of thespherically averaged reciprocal form factor fromthe wave function in position space. The spheri-cally averaged Compton profile is then availableby one-dimensional Fourier transformation. Theseeming detour of the second way becomes partic-ularly advantageous when considering the resultsof inelastic-scattering experiments in terms of the

Ž .reciprocal form factor B s rather than the Comp-Ž . Ž .ton profile J q . On the one hand, B s establishes

a link to well-known chemical concepts in positionw xspace 11]14 ; on the other hand, multiple scatter-

ing effects that may not have been completelyeliminated from the experimental data do not showup in the region of large s values, which is domi-nated by ‘‘chemically interesting’’ valence electroncontributions. Therefore, the second way with cal-culating reciprocal form factors directly in positionspace is an attractive alternative.

Up to now, the path via momentum space isusually adopted for spherically averaged dataŽeven though it requires a semi-infinite, but one-dimensional integration that is usually performed

.numerically . The alternative via the reciprocalform factor requires either numerical spherical av-eraging of a sufficiently large set of directionalŽ .B s over the unit sphere—which is very time-

consuming, as will be shown later—or the evalua-tion of spherically averaged autocorrelation inte-grals between the basis functions, but such analgorithm did not yet exist. In this work we willpresent such an algorithm for Gauss-type basisfunctions that is efficient and completely analyti-cal. The algorithm has been successfully imple-mented in connection with the quantum chemistry

w x w xcodes CRYSTAL92 18]19 and GAMESS 20 .After a brief outline of the basic relations in the

next section, a closed expression is presented forthe spherically averaged Compton profile and the

² n:moments p with n G y1 for Gaussian basisfunctions in the third section. The new algorithmfor calculating the spherically averaged reciprocal

Ž .form factor and from that the Compton profilewill be derived in the fourth section. Finally, theperformance of the new algorithm is demonstratedin the last section.

VOL. 65, NO. 3214

SPHERICALLY AVERAGED COMPTON PROFILES

Basic Relations

Atomic and molecular Compton profiles can beobtained from position-space LCAO wave func-tions as provided by standard quantum chemistryprograms. The relation between all one-particlequantities involved in the calculation of Comptonprofiles is visualized in Figure 1.

For the calculation of directional Compton pro-Ž .files DCPs , two well-established techniques

w x17]21 are in use, which will be called momen-tum-space method and position-space method inthis study. The momentum-space method pro-ceeds via the momentum-space orbitals

1yi p?r˜ Ž . Ž . Ž .c p s c r e dr 1HHHj j3r2Ž .2p

Ž .using atomic units with " s 1 , the momentumdensity

˜ 2Ž . < Ž . < Ž .p p s n c p 2Ý j jj

with orbital occupation numbers n , and the defi-jnition of the DCP

p ? qŽ . Ž . Ž .J q s p p d y q dp, 3HHH ž /q

where the last step reduces to a two-dimensionalintegration over momentum components perpen-dicular to the scattering vector q.

For both Slater- and Gauss-type basis functions,Ž . Ž .the steps given in Eqs. 1 to 3 can be carried out

w xanalytically 17 .w xThe position-space method 21 yields the one-

dimensional Fourier transform of the ComptonŽ .profile the reciprocal form factor , which may be

decomposed into autocorrelation functions of allorbitals c ,j

Ž . Ž .B s s n B sÝ j jj

Ž . U Ž . Ž .s n c r c r q s dr, 4Ý HHHj j jj

followed by

q`1qi q sŽ . Ž .J q s B s e dsH2p y`

`1Ž . Ž . 5 Ž .s B s cos qs ds, q s. 5H

p 0

Thus the calculation of the momentum density isnot necessary. Instead, one has multiple sums overoverlap integrals of the position-space basis func-

FIGURE 1. Scheme for the calculation of directional and spherically averaged Compton profiles from orbital< < 2contributions, where j is the orbital index. stands for square modulus, FT for Fourier or Fourier ]Dirac transformation,

( )HT for Hankel transformation, AC for autocorrelation, Proj. for the projection according to Eq. 3 , Int. for the integration( )according to Eq. 7 , and Diff. for its reverse operation.

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 215


Ž .tions with the variable argument s s s ? qrq .These integrals can be solved analytically as well.

ŽBoth methods momentum-space and position-.space have been implemented for directional

Compton profiles from LCAO wave functions ofw xsolids 21]24 as yielded by the CRYSTAL92 pro-

w xgram 18]19 .In the momentum-space method the spherically

Ž .averaged Compton profile J q is calculated fromthe spherically averaged momentum density

1Ž . ² Ž .: Ž . Ž .p p s p p s p p dV 6V HH pp 4p V p

by the relation

`Ž . Ž . Ž .J q s 2p p p p dp. 7H

< <q

To the authors’ knowledge the integration in Eq.Ž .7 is up to now performed only numerically.However, as it will be shown in the followingsection, a closed expression exists for the integralsoccurring for Gaussian basis functions.

Ž .In the position-space method, J q is obtainedby a Fourier transformation of the spherically av-

Ž .eraged reciprocal form factor B s . However, theuse of the position-space method for the calcula-

Ž .tion of J q has so far been hampered by the needto calculate the integral

Ž . ² Ž .:B s s B s Vj j s

1UŽ . Ž .s c r c r q s dr dV .HH HHH j j s4p V s

Ž .8

In the fourth section we present a method to solvethis integral in the case of Gaussian basis func-tions.

Thus it is now possible to obtain sphericallyaveraged Compton profiles and reciprocal formfactors of molecular or periodic systems via twoalternative ways analytically, which enables one,for example, to gain computational speed and toassess the numerical accuracy of all methods morereliably.

Momentum-Space Method

Since the evaluation of the spherically averagedŽ .momentum density p p from position-space wave

w xfunctions is described in detail in 17 for a num-

ber of different types of basis functions includingGaussians, this problem will not be addressed inthis article. It is only worthwhile to emphasize thatin contrast to the three-dimensional momentumdensity, only finite sums over spherical harmonics

w xoccur 25 , as can be seen from comparison of Eq.Ž . w x Ž . w x126 in 17 with Eq. C.17b in 26 .

For spherical and Cartesian Gauss-type orbitalsŽ . Ž . Ž . w Ž .xGTOs the evaluation of J q from p p Eq. 7involves integrals that are a special case of thetype

` 2llqnq2 yh pŽ . Ž .I h , ll , n , l, R ; q s p e j pR dp ,H lq

Ž .9

w Ž . w xx Ž .with n s y1 cf. Eq. 203 in 17 . In Eq. 9 ,Ž .j pR is a spherical Bessel function, the parameterl

Ž . Ž .h s a q b r 4ab with a and b as the Gaussianexponents of basis function 1 and 2, respectively,and R is the distance between the centers of thesebasis functions. If spherical Gaussians are consid-ered, one has ll s ll q ll with ll and ll as the1 2 1 2quantum numbers of angular momentum of thetwo basis functions, respectively. For CartesianGaussians, on the other hand, one has ll s l q m1 1q n q l q m q n , where l , m , and n are the1 2 2 2 1 1 1respective exponents of x y A , y y A , and z yx y

wA of the basis function 1 centered at A cf. Eq.zŽ .x25 , and l , m , and n are the corresponding2 2 2quantities of basis function 2.

w x Ž .In 17 no solution for I h, ll , n s y1, l, R; q isgiven for the general case q / 0, and to the au-thors’ knowledge this integral has so far beensolved only numerically. However, as shown in

Ž .the following, I h, ll , n G y1, l, R; q can be ob-tained by a recursion relation based on the errorfunction.

wExpressing the spherical Bessel functions as cf.Ž . w xxEq. 10.1.10 in 27

lyky1 ykŽ . Ž . Ž .j pR s c p sin pR q d p cos pR ,Ýl k k

ks0

Ž .10

and using the Euler relations

1i pR yi pRŽ . Ž .sin pR s e y e and

2 i1

i pR yi pRŽ . Ž . Ž .cos pR s e q e , 112

VOL. 65, NO. 3216


Ž .the evaluation of I h, ll , n , l, R; q can be reducedto the evaluation of

` 2" n yh p " i pRŽ . Ž .K h , n , R ; q s p e dp. 12Hq

< < ŽUsing ll s ll q ll and ll y ll F l F ll q ll or1 2 1 2 1 2the corresponding relation that holds for Cartesian

.Gaussians , the upper limit of the possible valuesof n is given by n s ll q n q 2 y k y 1 s llmax minq n q 1, whereas the lower limit is given by nmins ll qn q2 yk y 1 s ll y l q n q 1 s n q1.max maxThus, in the general case we have n q 1 F n F ll qn q 1, which reduces to 0 F n F ll for the Comp-

Ž .ton profile n s y1 . Since n is a positive integerin the case of n G y1, the relation

` 2" n yh p " i pRŽ . Ž .K h , n , R ; q s p e dp 13Hq

n` 2yn yh p " i pRŽ .s "i e dpHn R q

nyn "Ž . Ž . Ž .s "i K h , 0, R ; q 14n R

"Ž . "Žcan be used to obtain K h, n, R; q , if K h, n s.0, R; q is known analytically. By the substitution

Ž .'z r h s p . iRr 2h , the integration yields"

"Ž .K h , 0, R ; q2R1 p iR

y Ž .'s e erfc h q . 154h( ž /2 h '2 h

1 p R2yh q " i qR 's e e w " q h qi ,( ž /2 h '2 h

Ž .16

Ž .where erfc z is the complementary error functionŽ . yz 2 Ž . w xand w z s e erfc yiz , as defined in 27 , Eqs.

Ž . Ž .7.1.2 and 7.1.3 , respectively. Therefore, one has

"Ž .K h , n , R ; qn1 p 2yn yh q "Ž . Ž . Ž .s "i e W h , R ; q 17n(2 h R

with

R" " i qRŽ . Ž .'W h , R ; q s e w " q h qi . 18ž /'2 h

"Ž .The first derivative of W h, R; q is obtainedŽ . w Ž . w xxusing the properties of w z Eq. 7.1.20 in 27 ,

"Ž . W h , R ; q

R

R i" " i qRŽ . Ž .s y W h , R ; q " e . 19

2h 'hp

This relation can be used to derive the recursionrelation

nq2 "Ž . W h , R ; qnq2 R

"nq1 Ž . W h , R ; q R"Ž .s y W h , R ; qnq1 2h R

i" i qR Ž ." e 20'hp

Rnq1yk ynq1 ž /2hn q 1s Ý nq1ykž /k Rks0

=k "Ž . W h , R ; q

k Rnq1Ž " i qR .i e

Ž ." 21nq1 R'hp

nq1 "Ž .R W h , R ; qs y nq12h R

n "Ž .n q 1 W h , R ; qy n2h R

nq1Ž ."qn " i qR Ž .. i e . 22'hp

This demonstrates that in the case of Gaussianbasis functions, a closed expression exists for thespherically averaged Compton profile, if the mo-mentum-space method is used. However, we havenot yet implemented these formulas in computercode and therefore not checked the numerical sta-bility of the recursion. For certain values of q, R,and h it may be useful in numerical applications

Ž .to express the complementary error function erfc zŽ .occurring in Eq. 15 in terms of the error function



Ž . Ž . werf z as 1 y erf z and to use the property Eq.Ž . w xx7.1.19 in 27

nq1 Ž .d erf z 2 2n yzŽ . Ž . Ž .s y1 H z e , 23nnq1 'dz p

Ž .where H z are the Hermite polynomials.nIt is interesting to note that setting q s 0 in

Ž .I h, ll , n , l, R; q also provides a closed solution² n:for the moments p of the electron momentum

w xdensity, as defined in 5 , for the case n G y1. Therelation of some of these moments to physical

w xquantities is discussed in 28 .

Position-Space Method

Ž .Expressing the orbitals c r as a linear combina-Ž .tion of contracted Gaussians, the integrals to be

calculated are given by

1² Ž .: Ž .b s s f a , A; rV HH HHH1, 2 l , m , ns 1 1 14p V s

Ž . Ž .= f b , B; r q s dV dV , 24l , m , n r s2 2 2

where

Ž .f a , A; rl , m , n1 1 1

m n 2l 1 11 ya ŽryA.Ž . Ž . Ž .s x y A y y A z y A ex y z

Ž .25

and

Ž .f b , B; r q sl , m , n2 2 2

ml 22Ž . Ž .s x q s y B y q s y Bx x y y

=n 22 yb ŽrqsyB.Ž . Ž .z q s y B e 26z z

m n X 2lX X X2 22 yb ŽryB .Ž . Ž . Ž .s x y B y y B z y B ex y z

Ž X . Ž .s f b , B ; r 27l , m , n2 2 2

are Cartesian Gaussians centered at A and B, re-Ž .spectively. As is evident from Eq. 27 ,

Ž .f b , B; r q s can be expressed as anotherl , m , n2 2 2

Gaussian centered at BX s B y s.The derivation of the algorithm splits into two

parts. In the first step the problem is reduced tothe angular integration over a Cartesian Gaussianby performing the integration over dV . In thersecond step the solution for the angular integrationis presented.

Integration over dVr

Ž .The integration over dV given in Eq. 24 canrbe easily performed, since it is the overlap integralbetween two Cartesian Gaussians, whose solution

w xis well-known 29, 30 . The integral over dV canrbe written as a product of integrals over x, y,and z,

Ž .I s f a , A; rHHHr l , m , n1 1 1

Ž . Ž .= f b , B; r q s dV s I I I , 28l , m , n r x y z2 2 2

wwhere using the Gaussian-product theorem andŽ .xg s abr a q b

q` 2l lX1 2ya Ž xyA .xŽ . Ž .I s x y A e x y BHx x xy`

= yb Ž xyBXx .

2 Ž .e dx 29

s eyg Ž A xyB Xx .

2

l l1 2 l l l yl1 2 2 2l yl X1 1= PA PBÝ Ý xxž / ž /l l1 2l s0 l s01 2^ ` _l q l s even1 2

Ž .p l q l y 1 !!1 2 Ž .= . 30l ql q1 Žl ql .r2( 1 2 1 22Ž .a q b

XIn this equation we have PA s P y A and PBx x x xX Ž X. Ž .s P y B where P s aA q b B r a q b is thex x

center of the new Gaussian. Therefore, one finds

a A q bBX bx x XŽ .PA s y A s B y Ax x x xa q b a q b

bŽ .s ys q B y Ax x xa q b

bŽ . Ž .s y s y C 31x xa q b

and

a A q bBXax xX X XŽ .PB s y B s y B y Ax x x xa q b a q b

aŽ .s y ys q B y Ax x xa q b

aŽ . Ž .s s y C 32x xa q b

VOL. 65, NO. 3218


with C s B y A. Inserting these results into Eq.Ž .30 yields

l l1 2pŽ .I s f l , l , l , l , a , bÝ Ýx 1 2 1 2(a q b l s0 l s01 2^ ` _

l q l s even1 2

l ql yl yl 21 2 1 2 yg Ž s yC .x xŽ . Ž .= s y C e 33x x

with

Ž .f l , l , l , l , a , b1 2 1 2

Ž .l q l y 1 !!l l 1 21 2s Žl ql .r21 2ž / ž /l l 21 2

=

l yl1 1l yl2 2 Ž .a ybŽ .. 34Ž .l ql y l ql r21 2 1 2Ž .a q b

The final result of the integration over dV in-rŽ .serted into Eq. 24 is therefore

² Ž .:b s V1, 2 s

3r2ps ž /a q b

l l1 2

Ž .= f l , l , l , l , a , bÝ Ý 1 2 1 2l s0 l s01 2^ ` _

l q l s even1 2

m m1 2

Ž .= f m , m , m , m , a , bÝ Ý 1 2 1 2m s0 m s01 2^ ` _

m q m s even1 2

n n1 2

Ž .= f n , n , n , n , a , bÝ Ý 1 2 1 2n s0 n s01 2^ ` _

n q n s even1 2

² Ž .: Ž .= f g , C; s . 35Vl , m , n s

Ž .In Eq. 35 , we have defined l s l q l y l y l ,1 2 1 2m s m q m y m y m , n s n q n y n y n ,1 2 1 2 1 2 1 2and

² Ž .:f g , C; s Vl , m , n s

1Ž . Ž .s f g , C; s dV . 36HH l , m , n s4p

Obviously, the procedure of spherical averaging isnow reduced to an integration of the CartesianGaussian

Ž .f g , C; sl , m , n

m n 2l yg ŽsyC.Ž . Ž . Ž . Ž .s s y C s y C s y C e 37x x y y z z

Ž .over the angular variables according to Eq. 36 .This integration will be subject of the followingsection.

Integration over dVs

SPECIAL CASE OF AN S-TYPE GAUSSIAN

The integration over the angular variables issimplified, if the special case of an s-type GaussianŽ .l s m s n s 0 is considered. In this case one has

1 2yg ŽsyC.² Ž .:f g , C; s s e dVV HH0, 0, 0 ss 4p

eyg Ž s2qC 2 .2g s?Cs e dV .HH s4p

Ž .38

Using 2g sC s yisD, i.e., D s 2 ig C, and perform-ing the usual plane-wave expansion into spherical

w Ž . w xxwaves Eq. B.102 in 31 ,

1 2yg ŽsyC.e dVHH s4p` L

2 2 Lyg Ž s qC . Ž . Ž .s e yi j sDÝ Ý LLs0 MsyL

UM MŽ . Ž . Ž .= Y V Y V dV 39HHL D L s s

` L2 2 Lyg Ž s qC . Ž . Ž .s e yi j sDÝ Ý L

Ls0 MsyL

M Ž . Ž .= Y V d 40L D L , 0

yg Ž s2qC 2 . Ž . Ž .s e j 2 ig sC 410

Ž w x. M Ž .is obtained cf. 32 . In these equations Y V areLŽ .spherical harmonics and j x are spherical BesselL

functions.

GENERAL CASE I: STEP-OPERATOR METHOD

If at least one of the two basis functions in Eq.Ž .24 is not of s-type, the spherical averaging willrequire the integration over the angular variables



Ž .of Gaussians f g , C; s with l, m, andror nl, m , nunequal to zero. Since the integral for the specialcase l s m s n s 0 can be calculated analyticallyw Ž .xEq. 41 , a standard approach to the solution ofthe integrals with larger values of l, m, and n isgiven by the well-known step-operator method.This method is based on the property

² Ž .:f g , C; s Vlq1, m , n s

² Ž .:1 f g , C; s Vl , m , n ss2g Cx

² Ž .: Ž .ql f g , C; s 42Vly1, m , n s

and similar relations with respect to m and n. Allpossible combinations of l, m, and n occurring forGauss-type basis functions up to d symmetry havebeen derived and implemented in this way. In this

Ž .derivation the relation j s 2 ig s

Ž . j j C ll 2 Ž .s j C j j Cx ly2Ž .Ž . C 2l q 1 2l y 1x

1Ž .q j j ClŽ .Ž .2l y 1 2l q 3

l q 1Ž . Ž .y j j C , 43lq2Ž .Ž .2l q 1 2l q 3

which has been obtained using the chain rule andw Ž . Ž . w xxthe relations Eqs. 10.1.20 and 10.1.19 in 27

Ž . j z 1l w Ž . Ž . Ž .xs l j z y l q 1 j z ,ly1 lq1 z 2l q 1Ž .44

and

Ž .j z 1l w Ž . Ž .x Ž .s j z q j z , 45ly1 lq1z 2l q 1

has proved to be very useful, because it preventsC or C from appearing in any denominator whenxfurther derivatives are required. As it is evident

Ž .from Eq. 43 , only spherical Bessel functions ofeven order will occur. They may be evaluated inthe usual way with the aid of recursion formulas.Numerical precision is gained by use of a Horner-

Ž .2type scheme in 1r j C and by inclusion of thew Ž 2 2 .xfactor exp yg s q C into the recursion rela-

Ž w x.tion cf. a comparable scheme given in 33 .

GENERAL CASE II: EXPRESSION GIVEN INCLOSED FORM

As an alternative to the step-operator methodwhere the integrals are obtained by successivelyforming derivatives, it is also possible to obtain aclosed-form expression. A similar procedure is de-

w xrived in 34 . Using the relations

ll l l lyl lŽ . Ž . Ž .s y C s y1 s C 46Ýx x x xž /l

ls0

and

lqmqn LX

Xl m n lqmqn l , m , n M Ž .X X Xs s s s s c Y V ,Ý Ýx y z L , M L sX X XLs0 M syL

Ž .47

the equation

² Ž .:f g , C; s Vl , m , n s

l m nlqmqnŽ .s y1Ý Ý Ý

ls0 ms0 ns0

l m n l m n L= C C C sx y zž /mž /ž / nl

L LX

lyl , mym , nynX X= cÝ Ý L , M

X X XLs0 M syL

1 X 2M yg ŽsyC.Ž . Ž .X= Y V e dV 48HH L s s4p

w Ž .xwith L s l q m q n y l q m q n is obtained.The coefficients

X Ul , m , n yŽ lqmqn. l m n M Ž .X X Xc s s s s s Y V dVHHL , M x y z L s sV s

Ž .49

1r2X X XŽ < <.2 L q 1 L y M !X XM q < M <s i X XŽ < <.4p L q M !

=2p Xm l yi M w sŽ . Ž .sinw cosw e dwH s s s

0

p Xnlqmq1 < M <Ž . Ž . Ž .X= sinq cosq P cosq dwH s s L s s0

Ž .50

w M XŽ . xXP z being the associated Legendre functionLŽ .occurring in Eq. 48 relate simply the one-center

Cartesian Gaussians with the spherical ones.

VOL. 65, NO. 3220


Ž w x2 .Expanding the term exp yg s y C in theŽ .same way as in Eq. 39 ,

² Ž .:f g , C; s Vl , m , n s

l m n2 2 lqmqnyg Ž s qC . Ž .s e y1Ý Ý Ý

ls0 ms0 ns0

l m n l m n L= C C C sx y zž /mž /ž / nl

=L LX

lyl , mym , nynX XcÝ Ý L , M

X X XLs0 M syL

` LL MŽ . Ž . Ž .= yi j sD Y VÝ Ý L L D

Ls0 MsyL

X UM MŽ . Ž . Ž .X= Y V Y V dV 51HH L s L s s

is obtained. The integration over V can now besperformed using the orthogonality relations of thespherical harmonics. Recognizing also V s V ,D Cthe result can be written as

² Ž .:f g , C; s Vl , m , n s

l m n2 2 lqmqnyg Ž s qC . Ž .s e y1Ý Ý Ý

ls0 ms0 ns0

=l m n l m n LC C C sx y zž /mž /ž / nl

L Llyl , mym , nyn= cÝ Ý L , M

Ls0 MsyL

L MŽ . Ž . Ž . Ž .= yi j 2 ig sC Y V . 52L L C

Ž . Ž .Inserting Eq. 52 into Eq. 35 yields the final² Ž .:analytical expression for b s between twoV1, 2 s

Ž .Cartesian Gaussians. The equivalence of Eq. 52 tothe results obtained by the step-operator methodgiven in the previous section has been explicitelychecked for the implemented combinations of l, m,and n.

Spherically Averaged Compton Profile

The calculation of the spherically averagedŽ .Compton profile J q via the position-space

method requires the Fourier transformation asŽ . Ž .given in Eq. 5 . A direct evaluation of J q from

the basis functions thus requires the calculation of

integrals of the type

q` 2L yg s i qsŽ .s e j 2 ig sC e dsH Ly`

` 2L yg s Ž . Ž . Ž .s 2 s e j 2 ig sC cos qs ds, 53H L0

which may either be performed numerically basedon efficient fast-Fourier-transformation algorithmsor by use of the error function.

However, in order to obtain both quantities,Ž . Ž .J q and B s , it is not efficient to calculate both of

Ž .them at the level of basis functions, i.e., b s1, 2Ž . Ž . Ž .from Eq. 35 and J q from Eq. 53 or from the1, 2

momentum-space method outlined above. Instead,it is much more efficient to evaluate the quantityŽ . Ž . Ž .B s from b s and to access J q by a fast-Four-1, 2

Ž .ier transformation position-space method or toŽ . Ž . Ž .evaluate J q from J q and to access B s by a1, 2

Žfast-Fourier transformation momentum-space.method .

Computer Program

The algorithm presented in the preceding sec-tion is part of the program BRG, which evaluatesdirectional and spherically averaged reciprocalform factors and Compton profiles by theposition-space method using wave functions fromthe quantum chemistry codes GAMESS and CRYS-TAL92. For reasons of computational efficiency, thedensity-matrix representation is used instead ofthe wave function representation.

Table I shows the performance of the new algo-rithm presented in this study, compared with thedirect numerical averaging procedure mentionedin the introduction. All calculations have beencarried out at the Computer Center of the Univer-sitat Konstanz on a CRAY J916r6-1024 running¨UNICOS 9.0.2.2. Four systems were chosen as

Ž‘‘typical’’ representatives: solid a-NaOH ortho-. Ž . Žrhombic and TiO tetragonal , a NaOH slab or-2

.thorhombic and a cluster of oxalic acid dihydrate,Ž .COOH ? 2H O, having the geometry of the2 2Ž .monoclinic solid. The wave functions for all theexamples presented in this work have been ob-tained with CRYSTAL92.

In the NaOH calculations the extended basisw xsets for Na and O given in 35 were used, whereas

for the H atom a p polarization function with theGaussian exponent 0.2 was added to the STO-3G



TABLE IPerformance of the spherical-averaging algorithm presented in this work in comparison to a

a( )numerical averaging procedure see text .

( )System s Mesh Size No. of B s Abs. Dev. Rel. Dev. t tCPU CPU

˚ ˚( ) ( ) ( )A A Points s tnew

NaOH 0.0 1 39.9984668 0.0 0.0 2.5 0.7( )solid New algorithm: 39.9984668 0.0 0.0 3.4 1.0

y4 y 51.0 0.20 57 5.3208267 4.0 ? 10 7.5 ? 10 160 53y4 y 50.15 100 5.3210117 2.1 ? 10 3.9 ? 10 277 92y4 y 50.10 196 5.3211223 1.0 ? 10 1.9 ? 10 543 181y5 y 60.05 716 5.3211970 2.5 ? 10 4.7 ? 10 1981 660

New algorithm: 5.3212220 0.0 0.0 3 1

TiO 0.0 1 75.9999789 0.0 0.0 19 12( )solid New algorithm: 75.9999789 0.0 0.0 19 1

y6 y 71.0 0.05 381 8.7135548 1.9 ? 10 2.2 ? 10 7046 371New algorithm: 8.7135529 0.0 0.0 19 1

y6 y 62.0 0.05 1378 y0.6624830 4.7 ? 10 7.0 ? 10 24586 1294New algorithm: y0.6624783 0.0 0.0 19 1

NaOH 0.0 1 39.9984668 0.0 0.0 2.5 0.9( )slab New algorithm: 39.9984668 0.0 0.0 2.9 1.0

1.0 0.05 716 5.3201272 3.8 ? 10 y5 7.1 ? 10y6 1690 563New algorithm: 5.3200892 0.0 0.0 3 1



( )COOH 0.0 1 65.9999833 0.0 0.0 0.9 0.82?2H O New algorithm: 65.9999833 0.0 0.0 1.1 1.02( )cluster

y5 y 71.0 0.05 1385 12.4081754 1.1 ? 10 8.9 ? 10 1097 1097New algorithm: 12.4081864 0.0 0.0 1 1

a ( ) < ( ) ( ) < <( ( ) ( )) ( ) <The absolute and relative deviations columns 6 and 7 are defined as B s y B s and B s y B s / B s ,num new num new newrespectively. t stands for the required CPU time and t / t for its ratio relative to the CPU time used by the new algorithm.CPU CPU new

basis. For oxalic acid dihydrate, a 6-21G basis setplus a p polarization function centered on the Hatom and having a Gaussian exponent 0.25 wasused. These basis sets thus contain only s and pfunctions. In order to investigate the dependenceon the maximal quantum number of the basis setalso results for solid TiO are given in Table I. The2

w xcorresponding extended basis set 36 includes alltypes of basis functions for which our algorithm

Ž .has so far been implemented s, p, and d .The numerical averaging was carried out over

Ž .values of B s calculated at grid points uniformlydistributed over a sphere of radius s. For a given

Ž .mesh size column 3 in Table I , the number ofŽ .points column 4 thus increases quadratically with

the value of s. It should be emphasized that the

number of sampling points can be reduced accord-ing to the symmetry of the system considered. Inparticular, for oxalic acid dihydrate one quarter of

Žthe unit sphere has to be covered 0 F q F 908,s. Ž0 F w F 1808 , for NaOH one eighth 0 F q Fs s

. Ž908, 0 F w F 908 and for TiO one sixteenth 0 Fs 2.q F 908, 0 F w F 458 .s s

From Table I it is evident that the new algo-rithm leads to a considerable gain of computa-tional speed, even if an accuracy of four digits is

w xconsidered as sufficient 37 . The maximum quan-tum number of the basis set does not play asignificant role.

The computational costs for one directional orŽ .for one spherically averaged B s value are very

similar, if the latter is calculated with the aid of

VOL. 65, NO. 3222


the new algorithm. This can especially be seenfrom the entries in Table I corresponding to s s

˚0 A. A comparison of the number of points perŽ .unit sphere column 4 and the CPU time ratio

Ž .t rt column 9 reveals that the speed gain ofCPU newthe new algorithm mainly depends on the numberof sampling points needed in the numerical aver-

Ž .aging to achieve convergence and on the smallspeed difference between the directional and thespherical averaging algorithm.

From the data for solid NaOH an importantconclusion about the convergence behavior of thenumerical averaging algorithm may be drawn:Since it is essentially identical to a two-dimen-sional integration according to the trapezoidal rule,a quadratic convergence behavior toward the truevalue is expected. Columns 6 and 7 of Table Ishow that the deviation from the value obtainedwith the new algorithm indeed decreases by afactor of 4 if the grid width is reduced by a factorof 2, which may be taken as an indication that thevalue obtained by our new algorithm is correct.

Ž .The program code is available for personal usefrom the authors on request.

ACKNOWLEDGMENTS

The authors wish to thank Dr. H. Schmider andDr. M. Weisser for valuable discussions. The con-tinuous financial support by the Fonds derChemischen Industrie is gratefully acknowledged.

Ž .Furthermore, one of us T. A. acknowledges thehospitality of the Theoretical Chemistry Group,University of Turin, Italy, where parts of the pre-sent work were carried out with the support of theEuropean Union under contract No. HCM-CHR-CT93-0155.

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Documents

Methods for the calculation of spherically averaged Compton profiles with GTOs