Transferable integrals in a deformation density approach to crystal calculations. I. Crystal harmonics and their properties

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XXI, 515-542 (1982)

Transferable Integrals in a Deformation Density Approach to Crystal Calculations.

I. Crystal Harmonics and Their Properties JOHN AVERY, PER-JOHAN @RMEN, AND IBHA CHATTERJEE*

Department of Physical Chemistry, H. C. 0rsted Institute, University of Copenhagen, Universitetsparken 5, DK-2100 K%benhavn 0, Denmark

Abstract

The term “crystal harmonic” is introduced to denote a symmetrized plane wave in the special case where the wave vector is a reciprocal lattice vector. Crystal harmonics, thus defined, have the translational symmetry of the lattice, and they also have the transformation properties of the irreducible representations of the crystal’s point group. An expansion is derived expressing crystal harmonics in terms of spherical Bessel functions and in terms of the functions Tll,c (eigenfunctions of L2 which are also basis functions for IRS of the crystal’s point group). A sum rule for the functions Tll,c is derived. Methods are given for expanding periodic functions of special symmetry in terms of crystal harmonics. Methods are also presented for calculating matrix elements of the potential in a crystal using crystal harmonics as a basis and for transforming to a STO basis. It is shown that the invariant component of the product of two crystal harmonics can be expressed as a sum of a few invariant crystal harmonics, and expressions for the coefficients in the sum are derived. Orthogonality with respect to summation over networks of points and normalization are also discussed. The properties mentioned above are illustrated in detail in the case of cubic crystals with point group Oh.

Introduction

This paper is part of a series [l-51 discussing methods for calculating crystal orbitals based on the concept of deformation density [6-lo]. In the deformation density approach, the charge density of a crystal is divided into two parts-a large part which can be represented as a superposition of a neutral-atom densities, and a small correction due to the flow of charge caused by chemical bonding. Correspondingly, the potential is divided into a large part Vo arising from the neutral atom density, and a small correction AV arising from the deformation density. Matrix elements of VO are evaluated by means of transferable integrals, while matrix elements of A V are converted into reciprocal lattice sums.

When matrix elements are converted into reciprocal lattice sums, one is faced with the problem of evaluating generalized scattering factors. A generalized scattering factor is the Fourier transform of the product of two basis functions. If the basis functions are Gaussians, the evaluation of the Fourier transform of the product is always easy, but if Slater-type orbitals are used as a basis, the two-center case presents difficulties. Nevertheless, a number of methods are available for the evaluation of generalized scattering factors, even when Slater- type orbitals are used [ll-171. When a crystal possesses a high degree of

* Present address: Saha Institute of Nuclear Physics, Calcutta, 700 009 India.

@ 1982 John Wiley & Sons, Inc. CCC 0020-7608/82/030515-28 $02.80

516 AVERY, DRMEN, AND CHATTERJEE

symmetry, a method based on group theory may be the most convenient. As mentioned in a previous paper in this series [S], the two basis functions can be expanded in terms of symmetrized functions which might be called “crystal harmonics.” These functions are symmetrized plane waves whose wave vectors are reciprocal lattice vectors. The invariant component of the product of two crystal harmonics can be expressed as a sum of crystal harmonics belonging to the invariant representation of the crystal’s point group, the coefficients in the sum being derived from group theory. Therefore if we expand our basis functions in terms of crystal harmonics, the invariant component of the product of two basis functions can be expressed as a sum of invariant crystal harmonics, and its Fourier transform can easily be evaluated.

In the present paper, we shall discuss in detail the properties of crystal harmonics, and we shall also discuss in detail methods for expressing basis functions in terms of them. Although this study was undertaken as part of our deformation density project, we hope that the results will also be useful in other contexts.

Definition of “Crystal Harmonics”

Let R1, R2, . . . , R, be the elements of the point group of a crystal. Then

will be a projection operator corresponding to the p th basis function of the y irreducible representation of the group. In Eq. (l), g is the order of the group, D;,,(Rj) is the y irreducible representation of R , and d , is the dimension of the representation.

If we let P; act on a plane wave, we will obtain a symmetrized sinusoidal function [18]

where

q.=R. I A- (3)

This “symmetrized plane wave” will transform under the elements of the point group like the p th basis function of the yth irreducible representation of the group. We now specialize to the case of a plane wave whose wave vector K is a reciprocal lattice vector

K = 2 ~ ( h b l + k b 2 + I b s ) , h, k, 1=0,*1, *2 , . . . . (4)

In this special case, we will obtain a function

CRYSTAL HARMONICS A N D THEIR PROPERTIES 517

which not only has the symmetry of a particular irreducible representation of the crystal point group, but also has the translational symmetry of the crystal lattice. We can call such a function a “crystal harmonic” in order to distinguish it from a more general symmetrized plane wave with arbitrary q. In Eq. (4), h, k, and 1 are integers: bl, b2, and b3 are basis vectors of the reciprocal lattice; and

K, R jK. (6)

(For example, if Ri is the inversion operator, the Ki = -K.)

Crystal Harmonics for Cubic Crystals

In order to illustrate the properties of crystal harmonics, let us consider in detail the case of a cubic crystal with point group Oh. In that case, the group has 48 elements. If we introduce the notation

where C.C. denotes complex conjugate and where

Table I gives the integers ni which appear in Eq. (8) for the various irreducible representations of oh. In Eq. (8), the plus (+) sign is used for the gerade representations, while the minus ( - ) sign is used for the ungerade representations. Thus the gerade representations are real, while the ungerade representations are imaginary.

From Eq. (8) and Table I, we can construct contracted expressions for the crystal harmonics. Introducing the notation

5 18

I

I

I

-

____

AVERY, ORMEN, AND CHATTERJEE

m o

o m

0 0

0 0

o m

m o

0 0

0 0

o m I

o m I

m o

m o

m m I I

m m

m m I

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

m m

x > N N

0 0

CRYSTAL HARMONICS AND THEIR PROPERTIES 519

and making use of various trigonometric identities, we obtain

P>gIh,k,l)=-$cos -hx sin -ky sin -12 r s in -1y sin (7 >[ (7 ) (7 ) (7 ) (%k.>3,

P$gIh,k,l)=-:cos

Pk I h, k, 1) = - sin (F k y) [ cos ($ hx) cos ($ 12) * (h & l ) ] , 2

In the last six expressions, the upper sign corresponds to j = 1, while the lower sign corresponds to j = 2. Some of these functions are shown in Figures 1-2.

Taylor Series Expansions of the Oh Crystal Harmonics About x = 0

If we expand the expressions of Eqs. (11) about the point x = 0, we obtain the leading terms

P"'glh, k, 1) = 1 +. *

p"iUlh, k, ~ ) = ~ ( ~ ) 9 ~ ( h k 3 1 ' - h k " i ' + h 5 k 1 3 - h 3 k 1 5 + h 3 k S l - h ' k 3 1 ) 6 d 3!5!

x ( ~ ~ ~ ~ ~ - ~ ~ ~ ~ ~ + ~ ~ ~ ~ ~ - ~ ~ ~ ~ ~ + ~ ~ ~ ~ ~ - X 5 y 3 r ) + . . . ,

520 AVERY, ORMEN, AND CHATTERJEE

Fig. 1. Invariant crystal harmonic Pa1pl0,2, 2) in the plane 2 = 0. The origin of the coordinate system is at the lower left-hand corner.

x { x 4 ( y " z ~ ) + y 4 ( 2 ~ - x ~ ) + 2 ~ ( x 2 - y z ) } + ~ . . ,

-i (7' ~1 6 d P;"Ih, k, l ) = - - - ( 2 h 2 - k 2 - 1 2 ) ~ y ~ ( 2 ~ 2 - y 2 - ~ 2 ) + * * * ,

* * ,

521

Fig. 2. Successive sections through P"*elO, 2,3). The planes of the sections are Z = 0, Z = 0.05d, Z = O.ld, and Z = 0.15d. Note the nodal planes corresponding

to a function with angular momentum 1 = 6.

522 AVERY, 0RMEN. AND CHAnERJEE

3 - i 21r P?"lh, k, l )=-( - ) 4 d h ( k 2 - 1 2 ) x ( y 2 - z 2 ) + . . . ,

3 -i 21r P9.Ih, k, l )=-(-) 4 d k ( h 2 - l 2 ) y ( X 2 - z 2 ) + . - ,

P?-Ih, k, l )=-( - ) l ( h 2 - k 2 ) z ( X 2 - y 2 ) + . . . . 3 -i 21r

4 d

Looking at the leading terms in these Taylor series expansions, we can see that in the near neighbourhood of x = 0 the alg projection of Ih, k, 1) behaves


Fig. 2. (Continued from previous page.)

like an s orbital, the three components of P k ( h , k, I ) behave like the three real p orbitals, the two components of P$lh, k, I), together with three components of P:"lh, k, I) behave like d orbitals, and so on. We can understand these relationships in the following way: The angular momentum eigenfunctions are basis functions for irreducible representations of the full rotation group 0 ( 3 ) , of which Oh is a subgroup. Representations of oh based on angular momentum eigenfunctions will therefore in general be reducible. We can use the group- theoretical formula for determining how many times an irreducible representation is contained in a reducible one, and if we do this we find that

D U = O ) = Dalg,

D(I=1) - - D".

o"=*' = D'. + D'Z.,

D"=3'-D"Zu - +DflU +DfZ"

,

3

5 24 AVERY, ORMEN, AND CHATTERJEE

Fig. 2. (Continued from previous page.)

etc.

(An unavoidable difficulty of notation is caused by the fact that it is conventional to use 1 to denote the third Miller index, and at the same time it is conventional to use 1 as the orbital angular momentum quantum number. We hope, however, that it will always be clear from the context which of these meanings 1 has.)

In Eqs. (13) we can note, for example, that the lowest angular momentum quantum number for which al, occurs in the decomposition is 1 = 9. This corresponds to the fact that the leading term in the Taylor series expansion of P"l~lh , k, I) is an angular momentum eigenfunction with 1 = 9.

From Eqs. (12) we can see that each of the octahedral angular momentum eigenfunctions (Table 11) can be associated uniquely with a particular type of

CRYSTAL HARMONICS AND THEIR PROPERTIES

TABLE 11. 9[,*(0, c p ) .

525

YU

a19

5 u l

5 u 2

t l u 3

egl

eg 2

t 2 r J 1

292 t

t2g3

a2u

5 u l

5 " 2

t l u 3

t 2 u l

t2u2

t2u3

crystal harmonic. For example, p x orbitals are associated with P?ulh, k, 1), p y orbitals are associated with P:'.lh, k, Z), and p z orbitals are associated with P$-lh, k, 1). We shall see below that a p x orbital (for example) can be built up from a superposition of crystal harmonics of the type P:'.lh, k, 1). The reverse

526 AVERY, 0RMEN. AND CHA'lTERJEE

type of association however does not work. For instance, we could not say that P"'zlh, k, I) is associated uniquely with s orbitals or that P"'zlh, k, 1) can be represented by a superposition of s orbitals. In fact, if we carry the Taylor series expansion of P"'.lh, k, 1) about x = 0 to higher terms, we find that it contains components corresponding to higher 1 values.

Looking at Eqs. (12), we might ask why it is that the leading term in the Taylor series expansion of Pzlh, k, 1) can always be expressed as the product of a factor which depends only on h, k, and 1 and an exactly similar factor with h, k, and 1 replaced by x, y, and z . The reason for this is as follows: Suppose that H is a one-electron Hamiltonian operator with eigenfunctions

H4i(x) = Ej4j(x). (14)

Then the set of eigenfunctions cbl(x), d2(x), . . . , q5co(x) will be complete and orthonormal. The corresponding set of Fourier transforms will be 4 ; (x), 4h(k), . . . , & , ( k ) , where

,

The set of functions C#J~(X), #2(x), . . . obeys the completeness relation

p$T(x')=S(x-x') . i

If we multiply (15) by ~ T ( x ' ) , sum over j and make use of (16), we obtain

This relationship will of course also hold if k = K is a reciprocal lattice vector and if x' = x, and therefore we can write

Ih, k, 1) = e iK.x = [ ( 2 ~ ) ~ ] ' / ~ 1 47 (x)C#Jf(K). (18) i

Equation (18) holds for the eigenfunctions of every one-particle Hamiltonian H, and it thus gives us an infinite number of expansions of eiK'%.

Let us now specialize to the case, where H is spherically symmetric about x = 0. In that case, the eigenfunctions of H can be separated into a radial part multiplied by a spherical harmonic

4nlm(x) =Rni(r)Yim(@, (PI (19)

(20)

and from Eq. (15), we obtain [19]

4 film (K) = R ,TI(K) Yirn (0, (PK 1, where K, OK, and ( P ~ are the polar coordinates of K,


and where j l is a spherical Bessel function. Substituting Eqs. (19) and (20) into (18), and making use of the addition theorem for spherical harmonics

where PI is a Legendre polynomial, we obtain

We now consider the case where H is changed by adding an infinitesimal perturbation with octahedral symmetry. This perturbation is too small to have any effect on the radial function R,r(r) or on its transform R S ( K ) : but no matter how small the perturbation is, it will be enough to hybridize the 21 + 1 degenerate eigenfunctions corresponding to a particular angular momentum quantum number 1. Therefore, instead of Eq. (23) we will now have

where

and

In Eqs. (25) and (26), the functions are octahedral angular momentum eigenfunctions, such as the ones listed in Table 11. They are superpositions of the 21 + 1 spherical harmonics belonging to a particular value of 1, and are therefore eigenfunctions of L2 but not of L,. Each of these functions is also a basis function of an irreducible representation of the point group Oh. The index 6 labels the different mutually orthogonal functions of this type belonging to a particular value of 1. A more complete tabulation of these functions has been given by Watanabe [20]. Inserting Eqs. (25) and (26) into Eq. (24), we have

Finally, combining Eqs. (27) and (23) we obtain the sum rule

An analogous formula holds for every point group, since we have not appealed to any special property of O h . In other words, for every point group there is a

528 AVERY, 0 R M E N . AND CHATTERJEE

sum rule analogous to the sum rule for spherical harmonics, the role of the spherical harmonics being played by eigenfunctions of L2 which are at the same time basis functions for the irreducible representation of the group. Apart from a normalizing factor, the functions can be generated by letting the projection operator PL act on the spherical harmonics Ylm. However, in order for the sum rule to hold, the functions 91c must be normalized in such a way that

J

Using Eq. (28) we can finally write down a complete expansion of the crystal harmonics in the form

m

P L eiK.* = 4~ C i ? , ~ ) C 9d4K, c p K ) 9 % e , cp ) , (30) i=o e = Y 3 &

with the sum Ce3,,, fi including only those functions 9 l e which transform like the g t h basis function of the 7th irreducible representation of the group. An expansion of the form shown in Eq. (30) holds for every point group; and the wave vector can be arbitrary, i.e., not necessarily a reciprocal lattice vector.

Fourier Series Representations Based on Crystal Harmonics

Let us consider a function of the form

where the sum Cx is taken over all the direct lattice vectors X of a crystal. Here

X = mlal + m2a2+ m3a3, mi = O,* 1,*2, . . . , (32)

where the vectors ai are basis vectors of the direct lattice, and hence related to the bi’s of Eq. (4) by

(33) a . . b! = 8 ._, I I 11

From Eq. (31) we can see that F(x) will have the periodicity of the direct lattice, and therefore we can represent it by a Fourier series of the form

K

where (F)K are constant coefficients. We can also write F(x) in the form

where

1 Ff(k) = [(2.rr)311’/2 j d 3 x eik.% Cf(x-X) = C e’k’Xf‘(k)

X X

(34)

(35)

CRYSTAL HARMONICS AND THEIR PROPERTIES

and

Substituting Eq. (36) into (35) and making use of the identity

where v is the volume of the unit cell, we obtain

[(2.rr)31i/2 d 3 X e- lk'xf ' (k)S(k+K) F J F(x) =

Comparing Eqs. (39) and (34), we can see that the constant coefficients in the Fourier series (31) are given by

where z1 is the volume of the unit cell and f ' is defined by Eq. (37). For example, suppose that f(x) is an atomic orbital of the form

f ( x ) = Rtdr)%*(O, cp). (41) Then if F(x) is a periodic superposition off's as defined by Eq. (31), the Fourier series representation of F(x) will be given by Eq. (34) with the coefficients

where RTi(K) is defined by Eq. (21). [Note that the parity of the functions ??',* is given by (-l)'.]

As a second example, let us consider the case where f(x) represents the spherically averaged electronic charge density of a neutral atom located at the point x = 0. Suppose that the function representing the charge density has the form

(Coefficients C,,, mi and l, for carbon, nitrogen, and oxygen are given in Table 111, and coefficients for other neutral atoms up to 2 = 36 can be obtained by writing to the authors.) Then F(x) will be given by Eq. (34) with the Fourier coefficients

530 AVERY, PRMEN, AND CHATTERJEE

TABLE 111. Coefficients mi, Cj, and l, for analytic representation of the neutral-atom charge densities and potentials. The charge density is given by Eq. (43) with r measured in atomic units. The neutral-atom density in a crystal can be calculated by means of Eqs. (43)-(50) and subtracted from the crystallographically measured density to obtain the deformation density and deformation potential. The coefficients were calculated from Clementi's Hartree-Fock atomic wave functions

[21]. Coefficients for other elements up to Z = 36 can be obtained from the authors.

carbon nitrogen oxygen

m . c .

0 3 . 0 6 5 6 0 3 3 . 3 4 3 6 1 1 . 3 3 4 2 2 0 . 2 5 9 0

0 9 1 . 0 9 4 6 1 - 0 . 5 0 6 1 1 - 0 . 0 1 5 6 1 1 9 . 1 9 0 6 2 1 . 1 8 9 7 2 5 .1920 1 - 6 . 6 9 1 4

2 0 . 1 0 3 5 1 - 3 . 9 8 2 0 2 - 7 . 8 4 2 6 1 - 0 . 1 2 8 1

2 - 0 . 1 4 7 1 2 2 . 9 6 4 5 2 1 . 3 6 6 0 2 3 . 2 4 9 8 2 1 . 4 9 7 2 2 0 . 1 1 1 2 2 0 . 2 3 7 8 2 0 . 8 9 0 9 2 0 . 4 1 0 2 2 0 . 0 6 0 3 2 0 . 1 3 0 3

2 0 . 0 1 0 3

1

1 - 0 . 8 2 0 7

2 0 . 6 5 1 9

2 - 4 . 3 0 1 1

2 O . O O I . O

5 , 1 6 . 5 7 2 5 1 4 . 6 9 8 7 1 3 . 5 4 5 7 1 2 . 6 8 7 5 1 1 . 8 7 6 0 1 0 . 8 2 4 9 1 0 . 7 8 8 2 1 0 . 3 1 7 3

9 . 6 7 1 9 8 .9310 8 . 5 1 6 9

7 .7646 7 . 2 9 9 1 6 . 9 1 4 4 6 . 8 4 9 2 6 . 4 4 3 5 5 . 7 6 1 4

5 . 1 7 9 4 5 . 1 7 4 5

8 . 0 0 2 2

5 . 2 9 0 5

4 . 0 9 1 7 4 . 0 0 8 1 3 . 6 2 3 6 3 .5426 3 . 0 0 3 9 2 . 6 4 1 7 2 . 5 3 3 0 2 . 3 7 6 2 2 . 0 6 2 1 1 . 9 1 0 7

m . C .

0 3 .5196 0 4 6 . 5 9 6 3 1 3 . 4 0 9 3 2 1 . 0 5 1 1 1 - 1 . 3 2 1 3 0 1 5 5 . 8 3 6 4 1 - 0 . 5 9 9 5 1 - 0 , 1 6 3 7 1 3 1 . 8 0 1 8 2 5 . 1 1 1 1 2 1 4 . 0 4 3 2 1 - 1 5 . 7 0 7 6 2 2 . 1 4 8 0 1 - 7 . 2 0 2 5 2 0 . 5 3 5 4 2 -20 .5689 1 - 1 . 9 2 6 7 2 - 9 . 5 4 8 0 2 - 2 . 4 9 2 8 2 7 . 6 3 6 0 2 6 . 2 1 3 3 2 7 . 0 9 1 6 2 5 . 2 2 2 4 2 1 . 8 5 0 2 2 1 . 3 0 1 8 2 1 . 6 4 6 5 2 1 . 0 9 7 3 2 0 . 8 5 9 1 2 0 . 1 1 2 0 2 0 . 5 4 7 1 2 0 . 0 6 8 1

5 . 7

2 1 . 6 7 7 7 1 7 . 2 9 8 3 1 5 . 8 7 2 6 1 4 . 2 0 3 5 1 3 . 9 9 4 8 1 2 . 9 1 8 9 1 2 . 7 5 5 0 1 2 . 3 0 8 7 1 1 . 4 9 3 2 1 0 . 1 1 2 9 1 0 . 0 6 7 5

9 . 6 1 5 4 8 . 8 1 4 1 8 . 3 7 5 6 8 . 2 9 5 4 8 . 1 8 9 7 7 . 9 2 9 3 6 . 9 4 9 9 6 . 5 0 3 6 6 . 3 1 1 9 6 . 0 2 2 3 5 . 0 7 2 1 4 .7235 4 . 6 2 5 8 4 . 2 0 4 8 3 . 8 3 2 2 3 . 4 2 4 7 3 . 3 8 6 0 2 .9397 2 .9060 2 . 3 8 1 3

m . c .

0 2 . 2 5 2 7 0 4 8 . 3 5 4 8 1 8 . 0 6 6 1 1 - 2 . 2 2 4 2

2 2 . 9 3 1 8 0 2 6 1 . 2 0 8 0 1 - 0 . 3 5 3 1 1 1 0 4 . 3 4 1 4 1 - 3 4 . 2 9 6 8 2 5 2 . 9 6 3 5 2 1 6 . 3 2 5 9 2 -57 .6489 1 - 1 8 . 7 4 1 2 2 5 . 5 5 5 0 1 - 5 . 9 2 0 8 2 0 . 5 2 2 0 2 1 6 . 2 8 9 8 2 - 3 7 . 2 5 0 1 2 - 1 1 . 2 7 9 3 2 2 2 . 7 2 7 5 2 2 1 . 2 3 4 8 2 6 . 4 1 6 7 2 1 5 . 4 6 6 6 2 6 .9336 2 1 . 4 5 3 5 2 4 .1884 2 2 . 6 3 1 3 2 0 . 6 3 2 6 2 0 . 4 9 4 5 2 0 . 0 2 3 2

1 - 1 . 1 0 7 3

'1 2 6 . 6 4 8 5 2 0 . 9 4 0 2 1 9 . 2 6 8 7 1 7 . 6 0 7 4 1 5 . 8 8 6 9 1 5 . 8 1 3 9 1 5 . 2 3 1 9 1 5 . 0 8 2 4

1 1 . 8 9 9 1 1 1 . 8 8 8 9 1 1 . 3 4 4 8 1 0 . 2 2 7 6 1 0 . 1 7 8 6

9 . 7 0 2 9 9 . 3 7 4 1 9 . 0 6 0 5 8 . 5 6 6 3 8 . 5 0 7 1

6 . 8 7 5 7 6 . 8 4 5 8 6 . 0 4 1 3 5 . 2 3 3 8 5 . 1 2 5 3 4 . 5 9 1 4 4 . 3 2 0 8 3 .5919 3 . 5 1 6 3 2 .9495 2 . 3 0 7 1

1 3 . 5 6 0 4

7 .7026

where

with

abx a(a + l ) b ( b + 1)x2 F(alblclx)= 1+-+ , . . .

C c ( c + 1)2!

so that


If we replace f(x) in Eq. (31) byf(x-6) so that

F(x)=Cf(x-X-S), (48) X

then Eq. (40) must be replaced by -iK.S (F)K= {[(2~)’]’’~/l~}f‘(-K) e ,

as can be seen from the fact that iK.S r [fb - Wl‘ = e f (K).

Having discussed the methods for obtaining Fourier coefficients for the representation of functions with the periodicity of a crystal lattice, let us turn to the question of how to obtain series representations based on crystal harmonics. Suppose that the function F(x) transforms under the operations of the crystal’s point group in such a way that

PLF(x) = F(x). (51)

In other words, suppose that F(x) behaves like the p th basis function of the 7th irreducible representation. Then acting with PL on both sides of Eq. (34), we obtain

P;F(X) = F(X) = 1 (F)KPE e iK’x, (52) K

or, written in a slightly different notation, with (F)K = F h k l ,

The sum C h , k , l still runs over all the values of the Miller indices (i.e., over all reciprocal lattice points), but many terms in the sum will involve the same crystal harmonic, and we can greatly reduce the number of terms by lumping the repetitious ones together.

Thus instead of Eq. (53), we write

where the sum 1’ now includes only one point in each symmetry-related set of reciprocal lattice points, and where Wki) is an appropriate weighting factor. The sum 1’ also skips reciprocal lattice points for which

Pzlh, k, I ) = O .

To illustrate the construction of a series representation of the form shown in Eq. (54), let us consider the case where F(x) is a hydrogenlike p x orbital

532 AVERY, 0RMEN. AND CHAnERJEE

with

and 3 1 , t l u l (e, cp) = ( 3 / 4 ~ Y x / r .

Then from Eqs. (42), (21), and (45), we obtain

2i [ (2T)3]7/2 h d4 ((2.r/d)'(h2 + k 2 + 1') + (1/2)2}" Fhkr =

In this example, the first few terms in the series (53) are

F ( X ) = ~ : ' ~ ~ ~ o , o , o + ~ l , o , o l ~ , 0, o ) + ~ ~ , ~ , ~ l o , L O ) +Fo,o,iIO, 0, 1 ) + ~ - i , o , o I - ~ , 0, O)+Fo,-i,oIO, -1, 0)

+F0,0,-110, 0 , -1)+* * * ) . (59)

Omitting the terms which are identically zero, and making use of the fact that [from Eq. ( 1 l ) ]

[Note that because of Eq. (42), the Fourier coefficients Fhkl will always transform in the same way as PElh, k, l ) .]

The weighting factors for various categories of reciprocal lattice points and for various irreducible representations of O h can be calculated in a systematic way from Eq. ( l l ) , and the results are shown in Table IV. Using weighting factors taken from Table IV, we can easily construct series representations of periodic functions based on crystal harmonics. For example, in the case of the periodically repeated p x orbital discussed above, the series can be carried to higher terms with the help of the weighting factors listed in Table IV. In the case of a simple cubic lattice it becomes


TABLE IV. Wjlyk).

I

A

0

-5. 0 -

A

-5. 0

0 -

@f 0 0

4- 1 !

0 1 o j o j 0 w- 0 0 0

0 0 0 1 0

0 1 0

O 1 O

534 AVERY, PIRMEN, AND CHATTERJEE

In the case of a face centered cubic lattice, the series includes only those reciprocal lattice points whose Miller indices are either all odd or else all even, and in that case, the series for the periodicly repeated p x orbital becomes

F(x) =PP(8Fi,i,iIl, 1, 1)+2F2,0,012, 0,O)

+8Fz,o,zI2, 0, 2)+8F3,1,113, 1, 1)+16Fi,3,1\17 3, 1) (64)

+ 8Fz,z,zI2, 2, 2)+2F4,0,014, 0, O)+* * *I, with F h k l given by Eq. (58) and

P:'-lh, k , 1 )

- - sin ($ hx) [ cos ($ ky) cos ($12) + cos ($ l y ) cos ($ k z ) ] . 2

(65) The contour plot of the series representation of a periodicly repeated p x orbital is shown in Figure 3 for a simple cubic lattice, and in Figure 4 for a face centered

Fig. 3. Contour diagram for the series of Eq. (63)-a periodically repeated p x orbital built up from crystal harmonics of the form P?.lh, k, l ) . The series is

truncated after eight terms, and 5 = 15 /d .


Fig. 4. Series of Eq. (64). It is the same as the series of Eq. (63) except that it includes only those terms where the Miller indices are either all odd or else all even. The result is that the p x orbital appears on the lattice points of a face-centered-

cubic lattice instead of a simple cubic lattice.

cubic lattice. In both figures, the series is based on crystal harmonics and is truncated after the eighth term, but eight terms are sufficient to give a good representation of the function.

Coupling Coefficients for Crystal Harmonics

Suppose that we wish to construct a matrix representation of the potential V(x) in a crystal, using as a basis MPW functions of the form (19)

4 4 . a ( ~ ) = (eiqX/4%va(x). (66) Here N is the number of unit cells in the crystal and v,(x) is a periodic function of the form

536 AVERY, 0RMEN. AND CHATTERJEE

In Eq. (67), xa is an atomic orbital. The quasimomentum q is assumed to obey Born-von Karman boundary conditions imposed at the faces of the crystal. Then, since V will not mix different q values, the matrix will have the form

where

In the previous section we saw how to express periodically repeated atomic orbitals like 7, in terms of crystal harmonics. Thus we can calculate Vaca by first finding a representation of V based on crystal harmonics, and then transforming to our atomic orbital basis. The representation of V based on crystal harmonics will be block diagonal in the symmetry indices ( y p ) and we thus need only consider the diagonal blocks

1 J d3x (P; eiK"")*V(x)(P; eiK'x). (70) V(Ylr.) _ _ K',K -

Since

and since

Jd3x (p;', p'-)*

1 P;: = 1, Y'lr'

we can rewrite Eq. (70) in the form


here Kj is defined by Eq. (6), and where we have also made use of Eq. (5). In Eq. (73), V' is the Fourier transform of the potential [Eq. (37)]. Equation (73) gives us a practical method for evaluating V 2 2 and hence also for evaluating

A similar argument can be used to derive an expression for the invariant V a , , a *

component of the product

iK'.x)* (p ; iK.x 1. (p;

The product of two functions with different symmetry under a group cannot have a component which is invariant under the group. (This is, of course, the reason why the invariant potential V cannot mix two functions of different symmetrv ) Therefore, using Eq. (71), we can write

Then from Eqs. ( 5 ) and (9), we have

From Eq. (75) we can construct a multiplication table expressing the invariant component of the product of two crystal harmonics of the same symmetry as a sum of crystal harmonics belonging to the invariant representation.

In the case of cubic crystals with point group Oh, Eq. (75) becomes

P"'.[(P;lh', k', l'))*(PLIh, k, l ) ) ]

=;i18Pa1x[(nilh, k, l)+n,lk, 1, h)+n,l-k, 1, -h)+* *

x n&h, -1, -k) ) ( l -h ' , -k', -l')*lh', k', l))]. (76)

In Eq. (76), the intervening terms which have been left out are exactly the same as those occuring in Eq. (8). As in Eq. (8), the integers n j are those given in Table I. The plus (+) sign is to be used for gerade representations, while the minus ( - ) sign is appropriate for ungerade representations.

From Eq. (76) we can derive simple multiplication rules for special categories of crystal harmonics. For example, one finds that

Pnlx[(P;Ih', O,O))*(P;Ih, h, h ) ) ] = P a 1 ~ [ ~ l h - h ' , h, h)+blh +h ' , h, h )] ~yj-+ Ul, 2 2 7 (77)

538 AVERY, 0RMEN. AND CHATTERJEE

P"'g[(PzIh', O,O))*(PzIh, h, O)]=P"'g[~lh - h', h, O)+blh, h, h')+clh +h' , h, O)]

P"1"[(PLIh', O,O))*(P1:Ih, 0, o)]=P"'qaIh -h' , O,O)+blh, h', O)+clh +h' , 0,O)l

(PzIh', O,O))*(P:lh, k, k ) ) ] = P " ' g [ ~ I h -h ' , , k)+blh, k -h' , k) +clh, k +h' , k)+dlh +h', k, k ) ]

(For the irreducible representations not listed, one or the other of the functions in the product vanishes identically.) Note that in Eqs. (77)-(80), we have

while

These relationships follow from Eq. (71) and from the fact that the product of two functions with different symmetry cannot have an invariant component.


Thus we have

P"'z[((h', k', Z'))*(lh, k, Z))] = P"'zlh - h', k - k', 1 - 1')

= P"'g[(Pl Ih', k', Z'))*(P: Ih, k, Z)], Y.,

from which Eqs. (81) and (82) follow.

Normalization of the O h Crystal Harmonics

If we set h = h' in Eq. (79), we will have

P"lz[(P;lh, O,O))*(P:Ih, 0, O))]=P"ls[alO, 0, O)+blh, h, O)+c12h, 0, O)].

Integrating both sides of Eq. (84) over the volume of the crystal gives (84)

j d 3 x IPzlh, 0, 0)l' = Nva, (85)

where N is the number of unit cells in the crystal and v is the volume of a unit cell. Since Eq. (79) was derived from Eq. (76), we have, in effect, used Eq. (76) to find the normalization of the crystal harmonic P:Jh, 0,O). We can see from Eq. (79) that

Nu

Nv Id3x IP:'ulh, 0, 0)' = -, 2

or, directly from Eq. (76),

Nu 48 [d3X IPLIh, 0, 0>1' =- (n l+ nlof rill * nl2+ 1213 + n14* n19f %4), (87)

where the ni's are the integers given in Table I and where the + and - signs should be used, respectively, for the gerade and ungerade representations. Continuing in this way to make use of Eq. (76), we find that

Nu 48

Nu 48

Id3x IP:IO, h, O)l'= - (nl* n ~ o + n11 f n ~ z + 1215 + n16f nzof n 2 3 ) ,

j d 3 X IPLlO, 0, h>I2 = - (n1 fn1of rill +n12+ n17fn1ef n21 f nzz),

540 AVERY, 0 R M E N , AND C H A m E R J E E

[In Eqs. (88), h, k, and 1 are assumed to be not equal to zero and not equal to each other.] It follows, of course, that whenever the right-hand side of one of the relationships in Eqs. (88) vanishes for a particular representation, the corresponding crystal harmonic is identically zero in that representation, as can be verified from Table I and Eq. (11).

Orthogonality

A final property of crystal harmonics should be mentioned. They are orthogonal to one another not only with respect to integration over space coordinates but also with respect to summation over certain networks of points.

Let us discuss first orthogonality with respect to integration. The integral of the product of two crystal harmonics of different symmetries under the point

CRYSTAL HARMONICS A N D THEIR PROPERTIES 541

group of course vanishes. In addition, if IKI # IK’I then

since, when IKI # IK’I, Ki = RjK can never be equal to K;, = Rj,K’.

points follows from the fact that The orthogonality property with respect to summation over networks of

1 ifu=O,&M,*2M , . . . ,

0 otherwise,

where M and v are integers. Let x, be a set of points defined by the relation

x, = (l/M)(T1al + ~~a~ + na3) , T I , ~ 2 , 7 7 3 = 1,2 , . . . , M, (91)

where al, a2, and a3 are basis vectors for the direct lattice of the crystal. Then from Eqs. (4) and (33), we have

7 q = l r z = l r3=l

Looking at the first factor in Eq. (92) and comparing it with Eq. (90), we can see that if I h - h’l < M, then

1 i f h = h ‘ ,

0 otherwise, 1 M i(27i/M)(h-h’)‘r1 =

- - C e M 7 , = i (93)

with similar relations for the other two terms in the product. Thus Eq. (92) can be written in the form

provided that

Ih - h’l <M, Ik - k’l <M, I I - I f ) <M. (95)

Making use of Eqs. (3, (9), and (94), we obtain

542 AVERY, 0RMEN, AND CHA'ITERJEE

Comparing Eq. (96) with (89), we can see that if M is large enough so that

(Kj - K;*)*x, < 2 ~ , j , j ' = 1, . . . , g, (97)

wh,ere Ki = RjK and where x, is defined by Eq. (91), then

Therefore, subject to the condition on M, whenever the integral vanishes, the sum over the array of points will vanish exactly also.

Acknowledgments

We are grateful to Lektor Helge Johansen for the use of his contour plotting program. We would also like to thank the Northern European University Com- puting Center at Lundtofte, Denmark, for the use of their facilities.

Bibliography

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[7] E. N. Maslen, Acta Crystallogr. B 24, 1172 (1968). [8] P. Becker, Phys. Scrip. 15, 119 (1977). [9] F. L. Hirshfield, Acta Crystallogr. A 31, S224 (1975).

1977), Vol. 10.

[lo] Electron and Magnetization Densities in Molecules and Crystals, P. Becker, Ed. (Plenum, New

[ l l ] A. Graovak, H. J. Monkhorst, and T. Zivkovik, Int. J. Quantum Chem. 7, 233 (1973). [12] J. Avery and K. J. Watson, Acta Crystallogr. A 34, 582 (1978). [13] J. Avery, Acta Crystallogr. A 34, 582 (1978). [14] R. F. Stewart, J. Chem. Phys. 51,4569 (1969). [15] G. S. Chandler and M. A. Spackman, Acta Crystallogr. A 34, 342 (1978). [16] H. J. Monkhorst and F. E. Harris, Int. J. Quantum Chem. 6, 601 (1972). [17] J. Avery and P. -J. 0rmen, Acta Crystallogr. A 35, 849 (1979). [18] J. C. Slater, Rev. Mod. Phys. 37,68 (1965). [19] F. E. Harris, in Electronic Structure of Polymers and Molecular Crystals, J. M. Andrk and J.

[20] H. Watanabe, Operator Methods in Ligand Field Theory (Prentice-Hall, Englewood Cliffs, N.J.,

[Zl] E. Clementi, Tables of Atomic Functions, IBM J. Res. Dev. Suppl. 9, 2 (1965). [22] J. Avery, Theor. Chim. Acta 39, 281 (1975).

Received January 8, 1981 Accepted for publication May 25, 1981 Corrected proofs received December 1, 198 1

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Documents

Transferable integrals in a deformation density approach to crystal calculations. I. Crystal harmonics and their properties