Transferable integrals in a deformation density approach to crystal orbital calculations. IV. Evaluation of angular integrals by a vector-pairing method

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XVIII, 953-971 (1980)

Transferable Integrals in a Deformation Density Approach to Crystal Orbital Calculations. IV. Evaluation of Angular Integrals by a

Vector-Pairing Method

JOHN AVERY AND PER-JOHAN ORMEN Department of Ph ysrcal Chemistry, H. C. 0rsted Institute, University of Copenhagen, Universitetsparken

5, D K - 2 1 0 0 Copenhagen 0, Denmark

Abstracts

A general method for performing angular integrations is presented. The method depends on the fact that the integral must be invariant under rotations of the coordinate system, and it also makes use of combinatorial analysis. In most cases the method presented is computationally much faster than alternative methods of angular integration using Condon-Shortley coefficients. Applications to charge density analysis and Fourier transforms are discussed, and a general formula for the action of angular momentum projection operators on functions of the Cartesian coordinates is derived. A general angular integration formula for an m-dimensional space is also given.

On presente une mtthode gtntrale pour des integrations angulaires. Cette mtthode est basee sur I’invariance de l’inttgrale sous des rotations du systeme de coordonnees; elle utilise aussi I’analyse combinatoire. Dans la plupart des cas a traiter cette mtthode est beaucoup plus rapide en temps d’ordinateur que les methodes alternatives pour l’integration angulaire qui utilisent les coefficients de Condon-Shortley. On discute des applications a I’analyse des densites de charge et aux transformtes de Fourier et on obtient une formule gknerale pour I’action des projecteurs du moment cinetique sur des fonctions de coordonnCes Cartesiennes. On donne aussi une formule gCnCrale pour I’inttgration angulaire dans un espace a m dimensions.

Eine allgemeine Methode fur Winkelintegration wird vorgelegt. Diese Methode ist auf der Tatsache gegrundet, dass das Integral unter Drehungen des Koordinatensystems invariant bleiben muss. Sie verwendet auch Kornbinatorikanalyse. In den meisten Fallen ist diese Methode mit Rucksicht auf die Rechenzeit vie1 schneller als alternative Methoden fur Winkelintegration, die die Condon-Shortley-Koeffizienten benutzen. Anwendungen auf Ladungsdichten und Fouriertransfor- men werden diskutiert und eine allgemeine Formel fur die Wirkung der Drehimpulsprojektoren auf Funktionen von Kartesischen Koordinaten wird abgeleitet. Eine allgemeine Formel fur Winkelin- tegration in einem m-dimensionalen Raum wird auch gegeben.

Introduction

Parts 1-111 of this series of papers [l-31 discussed a method for crystal orbital calculations making use of the concept of deformation density [4-61, i.e., the change in electron density due to chemical bonding. In this method, matrix elements of the deformation potential were converted into reciprocal lattice sums of the form

@ 1980 John Wiley & Sons, Inc. 0020-7608/80/0018-0953$01.90

954 AVERY AND 0RMEN

while matrix elements of the neutral-atom Coulomb potential V0 were expressed in terms of transferable integrals (moments of the neutral-atom potentials). In order to relate the matrix elements of V0 to these moments, it was necessary to expand the atomic orbitals xa and Xb, located respectively at the points x = a and x = b, about a third point x = c. When the atomic orbitals xa and X b correspond to high values of angular momentum, complicated angular integrals have to be evaluated, both in order to relate the matrix elements of Vo to the moments of the neutral atom potentials, and also in order to evaluate the generalized scattering factors [2-211

Xab(K)= J d3x eiK."Xaxb,

which appear in Eq. (1). We therefore tried to develop improved methods for evaluating angular integrals. We found it most convenient to express the angular dependence of the atomic orbitals xa and Xb in terms of Cartesian coordinates, and to use tensor methods in evaluating the integrals. We hope that our methods for evaluating angular integrals will be useful, not only in the context of the present series of papers, but also in other quantum chemical applications.

Evaluation of Angular Integrals by a Vector-Pairing Method

Let us consider the angular integral

dR (A * u)(B . u), (3)

where A and B are arbitrary vectors of unit length,

u = x/lxj = (sin 0 cos 4, sin 0 sin 4, cos 0), (4)

and

dR = sin 0 d0 d4.

The integral I2 must be a scalar, i.e., it must be invariant under rotations of the coordinate system. Furthermore, it must contain A and B on equal footing and only in first power. From this it follows that the integral I2 must be proportional to (A 3 B), the scalar product of the two vectors A and B, since this is the only scalar which can be constructed from the two vectors in this way:

(6)

The constant of proportionality c2 can be found by considering the particular case where B = A. In that case, the t axis of our coordinate system can be chosen in the direction of A, and we can write

12 = 62(A * B).

I 2 = - dRcos20=- /2Td4/:sin 0 cos2 0d0 =;. (7) 47T 47r 0

CRYSTAL ORBITAL CALCULATIONS. IV 955

This determines the constant of proportionality, and we can finally write

1 2 = - dR (A . u)(B * U) = +(A * B). 4rr 'I

The integral

1 3 =- dS1 ( A . u)(B . u)(C . U) 47r ' I (9)

vanishes because the argument has odd parity, and in general, integrals involving odd powers of u will vanish because of the odd parity of the integrand.

Next let us consider the integral

1 4 =- dR (A . u)(B . u)(C . u)(D * u), (10) 47r ' I where A, B, C, and D are arbitrary vectors of unit length. Like 12, the integral I4 must be a scalar, invariant under rotations of the coordinate system. Also, since the integrand is symmetric with respect to interchanges of the various vectors, the integral too must be symmetric with respect to such interchanges. It follows that

14 = &[(A. B)(C . D) + (A . C)(B . D) + ( A . D)(B . C)], (11)

where t4 is a constant of proportionality. We cannot construct any other scalar with the necessary symmetry properties from the four vectors A, B, C, and D. The constant t4 is determined by considering the particular case where B = C = D = A. Then, choosing the z axis of our coordinate system in the direction of A, we have

I4 = - /dClcos4B=$. 4rr

Therefore, we can write the final result

I4 = - I dCl (A . u)(B. u)(C . u)(D . u) 47r

= &[(A * B)(C * D) + (A * C)(B . D)+ (A . D)(B . C)]. (13)

Comparing (13) with (12), we can see that & is the factor that is needed to give the correct answer in the special case where all four unit vectors are pointing in the same direction. There are three terms in the numerator of (13), each of which reduces to 1 in the special case where all the unit vectors are identical, and a factor of 3 in the numerator cancels with a factor of 3 in the denominator.

956 AVERY AND 0 R M E N

By an argument similar to that used in deriving (13) it follows that

16 =- dfl (A * u)(B . u)(C . u)(D * u)(E . u)(F . U) 4T ' I 1

7!! =-[(A. B)(C. D)(E. F) +(A * B)(C. E)(D * F)

+(A * B)(C * F)(D E) + (A * C)(B. D)(E * F)

+(A . C)(B . E)(D . F) + ( A . C)(B * F)(D * E)

+(A . D)(B . C)(E. F) + ( A . D)(B. E)(C. F)

+(A . D)(B . F)(C E) + ( A . E)(B * C)(D . F)

+(A * E)(B * D)(C * F) + ( A . E)(B * F)(C * D)

+(A. F)(B . C)(D . E) + (A * F)(B . D)(C . E)

+(A. F)(B * E)(C. D)], (14)

where 7!! = 7 . 5 * 3 . 1. The 5!! = 5 . 3 . 1 possible pairings of the six unit vectors of Eq. (14) are shown in Figure 1. The factor 1/7!! which appears in Eq. (14) is derived by considering the case where all six unit vectors are identical. In that case,

16 = - I d f l c o s 6 B = 4 . 4T

There are 5!! = 15 terms in the numerator of (14), each of which reduces to 1 in the special case where all six vectors are identical. Therefore, in order that (14) should reduce to (15) in this special case, we need the factor 1/7!! =f(1/5!!).

Figure 1


In general, when N is even, we have

N

47r 'I j = l IN =- dR (Aj * U)

=[1/(N+ l)!!]{(Ai . A2)(A3 * A4) . . . (AN-1 * AN)

+. . . [all ( N - l)!! possible pairings of A1, A2 * * . AN]}. (16)

Now let us consider the integral

1 I,,,,, = - 47r

dR (A . u)"'(B . u ) ~ ~ , ('7)

where A and B are arbitrary unit vectors, and where n1 and n2 are either both even integers or both odd integers. For example, let us consider the integral

I2,2 = - dR (A * u)'(B * u ) ~ . 47r 'I

Comparing (18) with (13), and letting C = A, D = B, we can see that

1 I2,2=- dR ( A . u)'(B. ~ ) '= - [1+2(A * B)'].

4T 'I 5 ! !

Similarly, comparing

13.3 = - ' 1 dR (A * u ) ~ ( B u ) ~ , 47r

with (14), and letting D = C = A, F = E = B, we obtain

(2') 1

dR (A . u ) ~ ( B * u ) ~ = -[9(A . B) + 6(A . B)3]. 13.3 := - 47r 'I 7!!

In general, we can show that if n1 and n2 are both even, then

1 I,,,,, = - 47r

dR (A . u)"'(B * u)n2

- (n1- l)!!(n2- l)!! (n1+ n2 + l)! ! -

958 AVERY AND 0RMEN

while if n l and nz are both odd, then

I,,,,, = -!- I d o (A . u)"'(B . u ) " ~ 4%-

If (-l)!! = 1, then Eq. (22) is seen to be valid for n1 = 0 and n2 = 0. This definition is maintained for the remainder of the paper. The line of reasoning by which Eq. (22) is derived is as follows: The factor l / (n l+ nz+ l)!! is the same as the factor 1/(N + l)!! in Eq. (16), since n1 + nz = N is the total number of vectors to be paired. Let us first consider the terms that are formed by pairing A's with other A's and B's with other B's, with no (A . B) pairs at all. There are (nl - l)!!(nz- l)!! such terms, and all of them are equal to 1, since A and B are unit vectors. For example, with %=z and n2 = 2, there is only (2- 1)!!(2- l)!! = 1 such term symbolized by AABB. With nl= 2 and n2 = 4, there are (2- 1)!!(4- l)!! = 3 such terms, symbolized by

AABBBB, AABBAB, and AAB-. We cannot have any terms involving only a single (A . B) scalar product, because then an odd number of A's would have to form pairs exclusively with other A's, which is impossible. Therefore the next category to be considered is the category of terms proportional 'to ( A . B)*. The number of these terms is found to be (nl - l)!!(nz- l)!!(nlnz/2!) by the following argument: Consider the first (A . B) scalar product formed. Since there are n1 A's and nz B's, there are nln2 ways to form this pair. In forming the second (A B) scalar product, we have n l - 1 remaining A's to draw on, and nz- 1 remaining B's; so there are (nl - l ) (nz- 1) ways to form thesecond (A B) pair. Therefore the total number of ways in which the two ( A . B) scalar products can be formed is n l (n l - l)nz(nz- 1)/2!, where we have divided by 2! because the two pairs are identical. We still have n l - 2 A's which have to be paired with other A's, and nz - 2 B's which have to be paired with other B's; and the number of ways in which this last operation can be done is (nl -3)!!(n2-3)!!. This gives the total number of terms proportional to ( A . B)' as

n - r - --

For example, when n l = 2 and n2 = 2, there are

2 . 2 (2- 1)!!(2- l ) ! ! y = 2

L

terms of this form, symbolized by


while when n1 = 2 and n2 = 4, there are

2 . 4 2

( 2 - 1)!!(4-1)!!-= 12

such terms, symbolized by the pairings shown in Figure 2. Continuing in this way, we derive the series of Eq. (22) . Equation (23) is derived by similar arguments. Note that the series (22) and (23) terminate after a finite number of terms because

Figure 2

beyond a certain point, all of the terms will contain a factor equal to zero in the numerator. For example, when nl = 2 and n2 = 4, we obtain

1 12.4 = - 4.n

dfl (A * u)*(B 1 u ) ~

( A . B)4+ . a )

(2-1)!!(4-l)!! 2 . 4 2 . 0 . 4 - 2 -_ - (1 + y ( A * B)2+ (2 + 4 + l)!! 4!

(24) 1

7!! = -[:3 + 12(A . B)2].

The two relations (22) and (23) are very easy to check, because we know from the addition theorem for spherical harmonics that

and

960 AVERY A N D 0RMEN

where the PI'S are Legendre polynomials. From this it follows that

-!- dR P((A.0 u)Pl,(B u) 47r

Pi(A * B). 611, 21+1

-- -

We can decompose the integral of Eq. (27) into terms of the form given in Eqs. (22) and (23), and in this way we can check the two relations. For example, when 1 = 1 and1'=3, then

d o P1(A + u)P3(B . u) = - dR (A . u):[5(B * u ) ~ -3(B . u)] 47r 47r 'I

5 3 = 211.3 -211.1

5 (1)!!(3)!! 3 (l)!!(l)!! 2 (1+3+1)!! 2 ( l + l + l ) ! !

- _ - ( A . B)-- (A B) = 0,

which checks with Eq. (27). Let us next consider integrals of the form

1 47r

= - dR (A u)"'(B u)"'(C u ) ~ ~ , I n l , n z , n g

where A, B, and C are arbitrary unit vectors, and nl , n2, and n3 are integers. By arguments similar to those used in deriving Eqs. (22) and (23), we can show that when n1, n2, and 123 are all even, then

1 =- 4.77

- (nl - l)!! ( n 2 - l)!! ( n 3 - l)!!

dR (A - u)"'(B * u)"'(C . I n l , n z , n 3

- (nl + n 2 + n 3 + l)!!

n2n3 * B)'+-(B. C)3

2!

+-(A n 1 n 3 C)2+nln2ns(A. B)(B * C)(A C) 2!

CRYSTAL ORBITAL CALCULATIONS. IV 96 1

while when n1 and n2 are odd, with n3 even, we obtain

1 477

= - / dn (A * u)"'(B . u)"'(C u ) ~ ~ I n l r n 2 , " 3

(A * B)(A. C)' c ) 2 + (n1- l h 3

2!

Equations (22) , (23) , (30), and (31) can be written in a more condensed form. If we look at Eq. (22) , we can see that it can be written in the form

1 I , , , , , = / dR (A . u)"*(B . u)",

since

n j ! ! (ni -A)!!

nj(nj-2)(nj-4) . . . (nj-A + 2 ) =

Taking nl ! ! and nz! ! outside. the summation, and noting that

(ni - l)!! nj!!,= nj! ,

we obtain Eq. (22) in the condensed form

1 4rr

I, , , , , = - / dR (A * u)"'(B . u),,

(33)

(34)


where n1 and n2 are both positive even integers, and where m, the maximum value of A , is given by

Similarly, looking at Eq. (23) we can see that the series can be written in the form

I,,,,, =- dR (A . u)"'(B . u)"'

(37) (A . B)" (nl - l)!! (122 - l)!!

417 'I n l ! ! n 2 ! ! c - - - - -

( n l + n 2 + 1 ) ! ! A = 1 , 3 , 5 , ... A ! ( n l - A ) ! ! ( n 2 - A ) ! ! '

Taking ( n l - l ) ! ! and (n2 - l)!! outside the summation, and making use of Eq. (34), we obtain Eq. (23) in the condensed form

1 I,,,,, = dR (A * u)"'(B * u),,

(38)

where n 1 and n2 are both positive odd integers, dnd where m, the maximum value of A, is given by (36). Thus, in their condensed form, the even-even integral, Eqs. (22) and (35), and the odd-odd integral, Eqs. (23) and (38), look alike except that in the even-even case, the A sum runs over even integers, while in the odd-odd case it runs over odd integers.

By a similar line of reasoning, both Eqs. (30) and (31) can be written in the condensed form

- n l ! nz! m (A * B)" c - (n l + n 2 + l)!! h z 1 . 3 . 5 , ... A ! ( n l -A) ! ! (nz -A)!! '

dR ( A . u)"'(B . u)"'(C . u)"' I n , , n z , n 3 ='J 417

- n l ! n2! n3! - ( n l +n2 + n 3 + l)!!

A ~ Z , A ~ ~ . A , ~ A 1 2 ! A 2 3 ! A 1 3 ! ( n l - A I Z - A 1 3 ) ! ! ( n 2 - A12 - A23)!!(n3 - A 1 3 - A 2 3 ) ! ! '

( A . B)"'"(B . C)""(A - C)"I3 x c

(39) where each Aji runs from zero up to a maximum value which is either ni or ni, which ever is smallest, subject to the following restrictions:

A12fA135nlr ( _ 1 ) A l 2 + A 1 3 = (-1P,

A 2 3 + A 1 2 1 n 2 , (-1)A23+A12 = (-l)nz, (40) A 1 3 + A 2 3 5 n 3 , (-1)A13+h23=

If the criteria of Eqs. (40) are not fulfilled by the Aii's, then the corresponding term is not included in the sum.


Equations (38) and (39) can be rewritten in a more general form: Let n stand for the set of positive integers n1,n2, . . . , nN, while Al , A2, . . . , AN form a set of unit vectors. Then

N N

= [ (jil ni+ l)!!] C Cj (Aj . Ai)*li, j=1 i = j

where N

i = j + l

j-1

Cj = n j ! / ( ni- 1 hii- i = l i = j + l

and where the sum 1, denotes a sum over all values of hii which are positive integers or zero and which fulfill the criteria

i N

i = l j = j 1 hij+ hji=nj, j = 1 , 2 , . . . , N . (43)

It can easily be verified that when N = 2, Eq. (41) reduces to Eq. (38), while if N = 3, it reduces to Eq. (39). A Fortran IV computer program implementing Eqs. (41)-(43) for values of N up to N = 5 is available from authors on request. We have tested this program in many particular cases and found that the results check with alternative methods of integration.

Angular Momentum Projection Operators Acting on Functions of the Cartesian Coordinates x, y, and z

In a previous paper [22], we discussed the properties of angular momentum projection operators acting on tensor functions. The angular momentum projection operators were defined in the following way: Letf(O,c$) be any function of the angles 8 and q5, and let O1 be the projection operator corresponding to the angular momentum quantum number 1. Then

O,[f(e , 4)1= i yixe, 4 ) I d o ’ ylmte‘, mw, 4’). (44) m=-1

In other words, 0, is an operator which projects out that component of a function which belongs to the l‘th irreducible representation of the rotation group.

Using the addition theorem for spherical harmonics, [Eq. (25 ) ] , we can rewrite (44) in the form

(45) 21+1 o,[f(e, 441 = I don’ . w ( e ’ , 41

where Pl is a Legendre polynomial, u is defined by Eq. (4), and u’ is defined by an equation analogous to Eq. (4) but with primes everywhere.

964 AVERY AND ORMEN

The results of the previous section can be used to obtain a general expression for the action of O1 on a function of the Cartesian coordinates x, y, and z : Let Al, AZ, A3, and u be unit vectors, and let A1, A2, and A3 be mutually perpendicular, but not necessarily perpendicular to u. Then

and from Eqs. (41)-(43),

1 4lr In1,n2,ns,n4 =- [ dfl' (A1 * U'),'(A~ * u')"'(A3 * U')"'(U * u')"'

with the sum 1, being taken over values of A i j satisfying (43) with the additional requirement, A1,2 = Al,3 = A2,3 = 0. If the three mutually perpendicular unit vectors Al, AZ, and A3 are taken in the direction of the axis of our Cartesian coordinate system, then Eq. (47) becomes

The next step is to express the Legendre polynomial in the form

1 ( 1 ) p/(U * U') = 1 K , (U * U')fl,

f l = O

where the constant coefficients ~ l f ' are given by

(49)

(-l)(1-fl)'2(l + n ) ! l + n even,

2"(1+ n ) / 2 ] ! [ ( 1 - n ) / 2 ] ! n ! , (50) 0 I + n odd.


Combining Eqs. (45), (48), and (49), we obtain the general relation

O1 (X “l y n2t ”’)

where the sum 1, includes values of Aii such that

hi4=ni ,n , -2 ,n i -4 , . . . , etc., j = 1 , 2 , 3 , ( 5 2 )

and such that the double factorials in the denominator of Eq. (51) never have negative arguments. Table I shows a number of relations generated from Eq. (51).

TABLE I. Relations derived from Eq. (51).

O*(x) = x

0 2 i X Y 1 = xy

O d X Y 1 = 0 0 2 ( x 2 ) = x 2 - Sr’

00(x ’ ) = f r 2

0 3 ( x y z ) = xyz

O l ( X Y Z ) = 0 O3(X2y) = x2y -+yr2 0 l ( x 2 y ) =$yr2

0 3 ( x 3 ) = x 3 - f 3x r2

01(x3 ) = j x r 2

0 4 ( x 2 y z ) =x2y.r -+yzr2

0 2 i x 2 y z ) =+yzr2

0 , ( x 2 y z ) = 0 0 4 ( x 3 y ) = x 3 y -$xyr2

0 2 ( x 3 y ) = +xyr’ o O ( x 3 y ) = o

0 4 i x 2 y 2 ) = x 2 y 2 -+ (x ’+ y2 i r2 +&r4 0 2 ( x 2 y 2 ) =+(x ’+ y2)r2+&r‘

O o ( x 2 y 2 ) = &r4 04(x4) =x4-$x2r2+&r4

0 2 ( x 4 ) =$x2r2-$r4

00(x4) = 4r4

0 5 ( x ~ ) = x ~ - ~ x ~ r ~ + & x r ~

0 3 ( x s ) =+?x3r2-$xr4

01(x5) =7xr 3 4

0 5 ( x 4 y ) =x4y -$x2y r2+&yr4

0 3 ( x 4 y ) =$x2yr2-&yr4

0 1 ( x 4 y ) =&yr4

0 5 i x 3 y 2 ) = x 3 y 2 - 4xy2r2 - 6.t 3 r 2 + &xr4

0 3 ( x 3 y 2 ) = $xy2r2 + 4x3r2 - &xr4

O ~ ( X ~ ~ Z ) = X ~ ~ Z -$xyzr

0 3 ( x 3 y z ) = jxyzr’

0 1 ( x 3 y z ) = 0 0 5 ( x ~ y ~ z ) = x 2 y 2 z -6 (x2+y2) t r2+&zr4 0 3 ( x 2 y 2 z ) =$(x2+y2)zr2-&zr4

~ ~ ( x ~ y ~ z ) =&zr4

0 1 ( x 3 y 2 ) = &xr4

Other relationships can be found from those shown in Table I by permuting x , y and z. For example, if we know that O z ( x y ) = xy, it follows that Oz(yt) = y t and OZ(X2) = XZ.


A particularly simple special case of Eq. (51) is the case where 1 = 0. In that case, all the hij's must be zero, and the sum reduces to a single term

1 41r

OO(xn1yn2zn3) =- I d a xn1yn2zn3

= - I)!! ( n z - I)!! ( n s - 1)!! r " 1 + ~ 2 + " 3 if nl , n2, and n3

(n1+nz+n3+1)!! ' are all even otherwise (53)

Applications

Suppose that a function of the space coordinates can be written in the form

F ( X ) = A ( r ) f ( e , 4) . (54)

(In other words, the function can be written as a radial part multiplied by an angular part.) Then the Fourier transform of the function will be given by

where j l is a spherical Bessel function and PI is a Legendre polynomial. Comparing this with Eq. (43, we can see that the Fourier transform of A(r ) f (B , 4 ) can be written in the form

where

and where 0 1 is an angular momentum projection operator. As a simple example of the way in which Eqs. (Sl) , (56), and (57) can be used

to evaluate the generalized scattering factors of Eq. ( 2 ) , let us consider the case where the two atomic orbitals xa and X b are on the same center. Writing the angular parts of the two orbitals in terms of Cartesian coordinates, we have


Then, using Eqs. ( 2 ) and (56), we can write the generalized scattering factor corresponding to the two orbitals as

where

and where Eq. (51) or Table I may be used to find the effect of the projection operator 0, on powers of the Cartesian coordinates in K space. 0”s effect in K space is, of course, the same as its effect in x space. Thus, for example,

O , ( K x ) = K x , 02(KxKy)=KxK, , , 0 2 ( K t ) = K t - $ K 2 .

As a second example of how the methods discussed above may be applied, let us consider the angular integral

where A, B, and C are unit vectors. An integral of this form occurs in the evaluation of the matrix element (VO)ab when ,ya is an s orbital, while X b is a p orbital [as shown in Part I of this series, Eqs. (38)-(40)]. The integral M of Eq. (61) is always zero, except when I / - - / ’ \ = 1, so we can confine our attention to integrals of the form

dR (A * u)P/(B . u)P/+I(C. u). (62)

Using Eq. (39), we can easily evaluate Ml. For the first few values of I, the results are

M -1 0 -3A 9 C,

M 1 -1 - 5(A . C)(B . C) -&(A . B),

M z = & ( A . C ) ( B . C)’-&(A. B) (B . C)-&(A * C),

M3 = & ( A . C)(B * C)3-&(A * B)(B. C)’-&(A. C)(B * C)+&(A * B),

etc. (63 ) As we showed in Part I of this series of papers [I], the integral of Eq. (62) can also be evaluated by means of Condon-Shortley coefficients and the addition theorem for spherical harmonics. However, the method illustrated in Eq. (63) is computationally much faster.

968 AVERY AND 0RMEN

Spaces of Higher Dimensionality

It is interesting to note that the methods discussed above can be applied with only very minor modifications in spaces of higher dimensionality. As an example, let us consider angular integrals in a four-dimensional space. The four- dimensional element of solid angle is given by

dR4 = sin2 x sin 8 d8 d4 dx. (64)

The extra angle x which appears in Eq. (65) runs from 0 to n, while the angles 8 and 4 have their usual meaning and range. By elementary integration, we obtain

dR4 = 2T2

and

Now suppose that we would like to evaluate the integral

1 J z = T [ 2 n dR4(A.u) (B .u) ,

where A and B are four-dimensional unit vectors and where

ui = x i / r , j = 1 ,2 ,3 ,4 ,

. 2 112 r = (x: + x i +x: + x 4 )

As before, we know that the integral must be invariant under rotations of the coordinate system, and therefore

J2 = &(A . B), (70)

since A * B is the only invariant which can be formed from A and B. The constant & can be determined by considering the special case where B = A. In that case, if we take A in the direction of x4 axis and make use of the fact that x4 = r cos x, then using Eq. (67), we obtain

1 2rr J2 = 7 dR4 (cos x)' = $,

so that, in general,

J 2 = 7 do4 (A * u)(B . u)=:A * B. 2T ' I


Similarly, if A, B, C, and D are four-dimensional unit vectors, then

dCl4 ( A . u)(B u)(C * u)(D * U)

= <[(A. B)(C * D) + ( A . C)(B . D) + (A - D)(B * C)], (73)

since the quantity in square brackets is the only invariant with the proper permutational symmetry which can be formed from A, B, C, and D. With the help of Eq. (67) we then find that in the special case where A = B = C = D,

4 2 . (3)!! 1 J 4 = 7 ~ S ~ ~ ( C O S X ) =--- - 89 277 ' I 6!! (74)

so that

5 4 = 7 dS14 (A * u)(B * u)(C * u)(D . U)

(75)

277 'I = &[(A * B)(C . D) +(A * C)(B . D) + (A * D)(B . C)],

and so on. The four-dimensional angular integrals which we have been discussing have an

interesting application: As was shown by Fock [23], the Fourier transforms of the hydrogenlike orbitals can be expressed in terms of four-dimensional spherical harmonics. Recently, Monkhorst and Jeziorski [24], extending an earlier treat- ment by Shibuya and Wulfman [25], have used Fock's result to convert the one-electron many-center problem of quantum chemistry into the problem of performing four-dimensional angular integrations. We hope that the methods presented above may useful in this context.

In the examples given above, it can be seen that the arguments based on the rotational invariance and permutational symmetry of the angular integrals, as well as the combinatorial arguments, are all independent of the dimension of the space. However, in going to a space of higher dimensionality, a minor modification is necessary to take into account the dependence of

on the dimensionality of the space. In an m-dimensional space, the general angular integral formula analogous to (4 1) becomes

where the coefficients Cj and the allowed values of A j i in the sum are given respectively by Eqs. (42) and (43), and where

N

n = C ni. i = l

970 AVERY A N D 0 R M E N

In an m-dimensional space, where the Cartesian coordinates are related to the polar coordinates by

x1 = r sin el sin e2 . . . sin sin em-2 sin 4, x 2 = r sin el sin e2 . . sin em-3 sin Om-* cos 4, x3 = r sin el sin e2. . . sin em-3 cos em-2, x4 = r sin el sin e2 . cos em-3, (79)

x,-~ = r sin el cos e2, x, = r cos 01,

the element of solid angle is given by

dR, = (sin 01)m-2(sin e2)m-3 + . . sin gmP2 del do2 . dern-2 dd. (80)

The angles el , . . . , em-2 run from 0 to T, while the angle 4 runs from 0 to 2 ~ . Letting x , / r = cos el and integrating, we obtain

(2T)"'*(n - I)!! ( m + n - 2)!! '

2(2rr)'m-1"2(n - I)!! I (m+n-2)!! '

m = even,

(81) m = odd,

sm," = 1 dam (cos el)" =

where n is even, so that the integral does not vanish. This result, combined with Eqs. (77), (78), (42), and (43) gives us the general angular integral formula for an rn-dimensional space. In particular, the equation analogous to Eq. (53) for an m-dimensional space becomes

Nmrn if nl , n2, n 3 , . . . , n, n (ni - l)!!, - ( m + n -2)!! i = 1 are all even, -

L 0, otherwise,

where n = C,:, nj and where

if m is even,

, if m is odd.

Discussion

The results reported in this paper are part of a long-term project. We are building up a program system for calculating crystal orbitals using transferable integrals and perhaps also making use of measured deformation densities (if

CRYSTAL ORBITAL CALCULATIONS. IV 97 1

sufficiently accurate crystallographic data become available). We would like to establish collaborative contacts with other groups working in the fields of charge density analysis or crystal orbital calculations, and we would be happy to make our tabulations and program components available to them. Although the methods of angular integration reported in this paper were derived in connection with our deformation density project, we hope that they will be useful in other contexts as well.

Acknowledgments

The authors are grateful to Professor Hendrik J. Monkhorst for helpful conversations and for a preprint of one of his papers. They would also like to thank the Northern European University Computing Center at Lundtofte, Denmark, for access to their facilities.

Bibliography

[I] J. Avery, Int. J. Quantum Chem. (in press). [2] J. Avery and E. Berg, Int. J. Quantum Chem. (in press). 131 J. Avery, Int. J. Quantum Chem. (in press). [4] P. Coppens and E. D. Stephens, Aduances in Quantum Chemistry (Academic, New York, 1976). [S] E. N. Maslen, Acta Crystallogr. Sec. B 24, 1172 (1968). [6] P. Becker, Phys. Scripta 15, 119 (1977). [7] F. E. Harris and H. H. Michels, Adv. Chem. Phys. 13,205 (1967). [8] J. S. Avery and K. J. Watson, Acta Crystallogr. Sec. A 33, 679 (1977). [9] J. S. Avery, Acta Crystallogr. Sec. A 34, 582 (1978).

[ 101 J. S. Avery and P.-J. armen, Acta Crystallogr. Sec. A 35, 849 (1979). [ l l ] J. S. Avery, Theor. Chim. Acta 39, 281 (1975). [12] G. S. Changler and M. A. Spackman, Acta Crystallogr. Sec. A 34, 341 (1978). [13] A. Graovac, H. J. Monkhorst, and T. Zivkovik, Int. J. Quantum Chem. 7,233 (1973). [ 141 F. E. Harris, Computation Methods in Quantum Chemistry, Preprint, Quantum Chemistry

[lS] F. E. Harris and H. H. Michels, Adv. Chem. Phys. 13, 205 (1967). [16] H. J. Monkhorst and F. E. Harris, Int. J. Quantum Chem. 6,601 (1972). [17] R. F. Stewart, J. Chem. Phys. 51, 4569 (1969). [18] J. Bentley and R. F. Stewart, J. Comput. Phys. 11, 127 (1973). [19] R. F. Stewart, Chem. Phys. Lett. 26, 121 (1974). [20] R. F. Stewart and J. Bentley, J. Chem. Phys. 63, 3786 (1975). [21] R. F. Stewart, Chem. Phys. Lett. 49, 281 (1977). [22] J. Avery and M. Cook, Theor. Chim. Acta 35, 99 (1974). [23] V. Fock, Z . Phys. 98, 145 (1935). [24] H. J. Monkhorst and B. Jeziorski (preprint). [25] T. Shibuya and C . E. Wulfman, Proc. R. SOC. London Ser. A 286,376 (1965). [26] J. R. Jasperse, Phys. Rev. A 2, 2232 (1970). [27] B. R. Judd, Aizgular Momentum Theory for Diatomic Molecules (Academic, New York, 1975).

Group, Uppsala University, Uppsala, Sweden (1973).

Received September 7, 1979 Accepted for publication December 5 , 1979

Documents

Transferable integrals in a deformation density approach to crystal orbital calculations. IV. Evaluation of angular integrals by a vector-pairing method