Overlap integrals with g-orbitals

Computers and Chemistry 24 (2000) 685–691

Overlap integrals with g-orbitals

Christian Opitz *Chemisch-Geowissenschaftliche Fakultat, Friedrich-Schiller-Uni6ersitat Jena, Institut fur Physikalische Chemie,

Am Steiger 3/Haus 3, D-07743 Jena, Germany

Received 9 December 1999; accepted 14 February 2000

Abstract

Formulae are derived for the overlap integrals between s-, p-, d-, f- or g-orbitals at one centre and g-orbitals atthe other one for any position of the two centres in the three dimensional space. © 2000 Elsevier Science Ltd. Allrights reserved.

Keywords: Overlap integrals; g-Orbitals

www.elsevier.com/locate/compchem

1. Introduction

For the quantum chemical investigation of matter inthe mesoscopic region between ‘normal’ molecules andthe infinite solid the so-called giant atom concept (Saitoand Ohnishi, 1987a,b) may be very useful. Clustersformed by electron gas like atoms are considered asnew units, the giant atoms, in the frame of that con-cept. It is based on the jellium model with its assump-tion of a uniform spherical shaped charge distributionin the inner of such giant atoms. A more detaileddescription of the giant atom concept and the deriva-tion of a periodic system of giant atoms is given in(Muller and Opitz, 1999).

The eigenfunctions

Cn,l,m(r, q, 8)

=!Rn,l(r)Yl,m(q, 8)=Rn,l(r)Ul,m(q)Fm(8) (05r5R)

0 (r\R)

(1)

05q5p, 05852p

of the model Schrodinger equation are obtained on ananalytical way by the technique of separation of vari-

ables and are called giant atom orbitals. The functions

Yl,m(q, 8)

=1

2ll !'2l+1

2(l+m)!(l−m)!

1sinm q

dl−m

d(cos q)l−m (cos2 q−1)l

×1

2pe im8 (2)

are spherical harmonics. The radial dependence leads toBessel functions of half-integer order, not important indetail here. R is the giant atom radius, n, l and m arequantum numbers.

n=1, 2, 3, … (3)

l=0, 1, 2, … (4)

m=0, 91, 91, 92, …, 9 l. (5)

The eigenvalues Enl of the Schrodinger equation are(2l+1)-fold degenerated. 2l %+1 eigenfunctions Cn%,l%,m

belong to En%,l%, with fixed n % and l %. Sorting the valuesEnl in ascending order, and using the symbolss, p, d, f, g, … for l=0, 1, 2, 3, 4, … yields the sequence

1s1p1d2s1f 2p1g2d… (6)

In contrast to ‘normal’ atomic orbitals, here f- andespecially g-orbitals are found in the forepart. This factmust be noted in the development of quantum chemicalcalculation methods.

* Tel.: +49-3641-948334; fax: +49-3641-948002.E-mail address: [email protected] (C. Opitz).

0097-8485/00/$ - see front matter © 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0097 -8485 (00 )00066 -8

C. Opitz / Computers & Chemistry 24 (2000) 685–691686

Table 1Orthonormalized real linear combinations of spherical harmonics for l=4 (and l=2)

q hl,q(8)·hl,q

(q) Spherical representation Cartesian representation

gz 4 0 (35z4−30z2r2+3r4)/r41

2p

32

16(35 cos4 q−30 cos2 q+3)

1gxz 3 xz (7z2−3r2)/r41

p

310

8(7 cos4 q−3) cos q sin q cos 8

gyz 3 −1 yz (7z2−3r2)/r4

(7 cos4 q−3) cos q sin q sin 81

p

310

8

g(x 2−y 2)z 2 2 (x2−y2) (7z2−r2)/r4

sin2 q (7 cos4 q−1) cos 2 81

p

35

8

−2gxyz 2 2xy(7z2−r2)/r41

p

35

8sin2 q (7 cos4 q−1) sin 2 8

gxz(x 2−3y 2) 3 xz (x2−3y2)/r41

p

370

8sin3 q cos q cos 38

−3gyz(3x 2−y 2) yz(3x2−y2)/r4

sin3 q cos q sin 381

p

370

8

4 (x4−6x2y2+y4)/r4gx 4−6x 2y 2+y 41

p

335

16sin4 q cos 48

−4gxy(x 2−y 2) 4xy (x2−y2)/r41

p

335

16sin4 q sin 48

0 (3z2−r2)/r2dz 21

2p

10

43 cos2 q−1

1dxz xz/r21

p

15

2cos q sin q cos 8

dyz −1 yz/r2

cos q sin q sin 81

p

15

2

2dx 2−y 2 (x2−y2)/r21

p

15

4sin2 q cos 28

dxy −2 2xy/r2

sin2 q sin 281

p

15

4

The following linear combinations of complex spheri-cal harmonics

1

2[−Yl,m+Yl,−m ] and

1

i2[−Yl,m−Yl,−m ]

m=1, 3 (7)

and

1

2[Yl,m+Yl,−m ] and

1

i2[Yl,m−Yl,−m ]

m=2, 4 (8)

can be taken to form real giant atom orbitals. The realorbitals are completely equivalent (except for the spa-tial orientation). They are the products of the sameradial function by the quantities given in Eqs. (7) and(8).

For g-orbitals (l=4) the orthonormalized real linearcombinations in spherical polar and Cartesian represen-

tations (with r2=x2+y2+z2) are listed in Table 1. Toget an illustrative example of overlap integrals betweeng- and other orbitals, the well-known functions forl=2 (d-orbitals) are added. The new quantity q is usedto label the linear combinations. q starts also with − land runs up to + l. hl,q

(8) and hl,q(q) are normalization

constants.

2. Problem

Overlap integrals

S=S(CA, CB)=& & &

CACBdt (9)

occur frequently in quantum chemical calculationmethods. The orbitals CA CnA,lA,qA

and CB CnB,lB,qB

are located at points A and B, respectively. D is thedistance between A and B.

C. Opitz / Computers & Chemistry 24 (2000) 685–691 687

Table 2Terms z

A

eq,k(A)

zB

eq,k(B)

7eq,k(7)

and coefficients bq,k in the basic integrals for S(nAd, nBg)

zA

eq,k(A)

zB

eq,k(B)

7eq,k(7)

1 2 3 4k 5 6 7 8zA

2 zB4 zA

2 zB472 zA

2 74 zAzB372 zAzB74 zB

472q zB274 76

16 −48 6bq,k 0 (s) −8 24 −34 −31 (p)

2 (d) 6 −1

2.1. Basic o6erlap integrals

In a first step, only one overlap integral is considered.The point A (atom A, giant atom A) lies in the originof the local Cartesian coordinate system and B atz=D. The allocation of one orbital to A and the otherone to B is just a question of definition. For thecalculation of the integral, a useful coordinate systemhas to be chosen in dependence on the form of inte-grand and integration volume. Since the functions Fq

are orthonormalized the integral (Eq. (9)) becomes

S=S(CA(x, y, z),CB(x, y, z−D))

=dqa,qb

&&RARBUAUBdf. (10)

For qA=qB=q=0, 91, 92, 93, 94, … theseintegrals are called s, p, d, 8, g, …-integrals, respec-tively. They are basic o6erlap integrals in the calculationof more general overlap integrals and shall be denotedby the symbols Ss, Sp, Sd, S8, Sg, ….

In the case of giant atom orbitals, the integrationvolume is the intersection of two spheres. Circularcylindrical coordinates

7A=7B=7, 8A=8B=8, zA=z,

zB=z−D (11)

are used. The definition of centres A and B is given bynABnB or nA=nB and RA\RB. The area integral (Eq.(10)) is integrated numerically (Opitz, 1999) usingGaussian quadrate with 20 abscissas and weights foreach dimension. Values are taken from Krylov andS& ulgina (1966).

Henceforth the basic integrals are assumed to begiven. Notice the x- and y-directions of the local coor-dinates are involved implicitly in the integrals.

As an example, the overlap S(nAd, nBg) betweend-orbitals at A and g-orbitals at B is considered. Afterthe integration over 8, the three basic integrals

Ss(nAd, nBg)=Sdz 2, gz 4=

3

325

&&RARB

16zA2 zB

4−48zA2 zB

272+6zA2 74−8zB

472+24zB274−376

rA2 rB

4df

(12)

Sp(nAd, nBg)=Sdxz, gxz 3=Sdyz, gyz 3

=1516

6&&

RARB

4zAzB372−3zAzB74

rA2 rB

4 df

(13)

Sd(nAd, nBg)=Sdw 2−y 2,g (x 2−y 2)z 2=Sdxy, gxyz 2

=1532

3&&

RARB

6zB274−76

rA2 rB

4 df (14)

are got with

72=x2+y2 (15)

and

r i2=x2+y2+z i

2, i=A,B. (16)

In the opposite case, if the d-orbitals are located at Band the g-orbitals at A, the subscripts A and B are tobe exchanged in Eqs. (12)–(14). Therefore, the ex-change of the centres is incorporated in the basic inte-grals by the uniform centre definition. Further basicintegrals appear if f- or g-orbitals interact with g-orbitals.

The integrands consist, besides the product RARB

and the denominator rnAA , rnB

B only of a sum of terms

bq,kzAeq,k

(A)zB

eq,k(B)

7eq,k(7)

(17)

Table 2 shows these terms for our example S(nAd,nBg) following from Eqs. (12)–(14).

2.2. General o6erlap integrals

Generally more than two non-collinear centres aregiven in a global coordinate system j, h, z. For eachpair A and B of centres overlap integrals are calculatedusing a uniform centre definition. Fig. 1 shows such asituation qualitatively. The dependence between theglobal and each local coordinate system is given by

ji=l1x+m1y+61zi

hi=l2x+m2y+62zi

zi=l3x+m3y+63zi

}i=A, B. (18)

where 61, 62, 63 are the direction cosines of the line AB(z-axis), ll, l2, l3 and m1, m2, m3, the direction cosines ofthe x- and y-axis, respectively.


If the coordinate axes are rotated arbitrarily, anarbitrary spherical harmonic Yl,q% becomes a linear com-bination of the spherical harmonic Yl,q of the newcoordinates, all having the same value l. Therefore,general overlap integrals may be written as linear com-bination of basic integrals

S=S(CA(jA, hA, zA), CB(jB, hB, zB))= %min(lA ,lB )

q=0

cqSq.

(19)

The problem is to calculate the coefficients cq as afunction of the direction cosines of AB

cq=cq (61, 62, 63) (20)

3. Calculation of the coefficients cq

These coefficients cq are known for overlap integralsbetween s-, p-, d-orbitals and part of the sources ofrelated programs. It is partly the same for f-orbitals(Smith and Clack, 1975; Opitz, 1999). In the followingthe coefficients shall be calculated for g-orbitals.

To this end, x and y in the left side of Eq. (19) aresubstituted by

x=7 cos 8 y=7 sin 8, (21)

then Eq. (19) is integrated over 8 and the result isordered after the terms zA

eq,k(A)

zBeq,k

(B)7eq,k

(7)of the basic inte-

grals with new coefficients aq,k. This is readily carriedout using computer algebra; Maple V in our case.

Comparing the new coefficients aq,k with bq,k weobtain cq. But often we have more then one equationfor the same cq, so the most convenient may be used inthe aspect mentioned below.

In our example we get the following equations

a0,k=c0h2,0(q,A)h4,0

(q,B)b0,k, k=1, 2, 3, 6

a1,k=c1h2,1(q,A)h4,1

(q,B)b1,k, k=4, 5

a2,k=c0h2,0(q,A)h4,0

(q,B)b0,k+c2h2,2(q,A)h4,2

(q,B)b2,k, k=7, 8(22)

Generally cq is a function of all nine direction cosines

cq=cq(l1, l2, l3, m1, m2, m3, n1, n2, n3). (23)

Now the job is to substitute the six quantities l1, l2, l3,and m1, m1, m3, by 61 62, 63 using the six fundamentalrelations between the nine direction cosines. No specialtransformation (j, h, z)� (x, y, z)1 is necessary sinceeach local system is incorporated in the basis integrals.This substitution process is more difficult than integra-tion over 8, but here Maple V is also a powerful tool.All equations for cq with fixed q are equivalent, butdiffer in their convenience to handle them by the com-puter algebra.

In Table 3, the coefficients c0, c1, c2 for our exampleS(nAd, nBg) are listed in dependence on n1, n2, n3 onlyusing the identities

n1 X, n2 Y, n3 Z. (24)

These and the remaining coefficients cq=cq(n1, n2, n3)for integrals between s-, p-, f-, and g-orbitals on onecentre and g-orbitals on the other, one may be obtainedas Fortran source text on the ftp server

ftp. uni-jena.de

at the university of Jena.

login name: ftp

password: e-mail address

directory: /pub/forschung/cco

file: coeff – g – orb

Additionally tables of all coefficients are held inthe form of tex-Files for LATEX on the same place.Further commented records of Maple V sessionsfor the examples dxy−gxyz 2, fxz 2−gxz(x 2−3y 2), andgxz(x 2−3y 2)−gxy(x 2−y 2) are held in the subdirectory

MapleV – examp

as LATEX, ASCII, and PostScript files DG, FG, andGG denoted by the suffixes tex, txt, and ps, respec-tively. Notice that Maple V is an interpreter not acompiler. That means in repeated (interactive!) runs,the mathematical content of any formula will be repro-duced in each case, but the form of the formula can bechanged (another sequence of factors for instance).

Fig. 1. General overlap between two orbitals in the globalcoordinate system.

1 Special transformations are that in Smith and Clack(1975) or the following symmetric, l1=1−n1

2/(1+n3), m1=l2= −n1n2/(1+n3), m2=1−n2

2/(1+n3), l3= −n1, m3= −n2,obtained by the shifting of A in the origin and rotating thez-axis in the z-axis through an rotation axis perpendicular toboth the z- and the z-axis.


Table 3The coefficients c0, c1, and c2 for the general overlap integral

c0gp c1p c2

Between dz2 and g orbitalsg

z 4 0 116

(3Z2−1)(35Z4−30Z2+3)30

4Z2(1−Z2)(7Z2−3)

158

(Z2−1)2(7Z2−1)

gxz 3 +1 10

8XZ(3Z2−1)(7Z2−3)

64

XZ(7Z2−4)2(Z2−1)3

4XZ(−28Z4+27Z2−3)

−1gyz 3

108

YZ(3Z2−1)(7Z2−3)3

4YZ(−28Z4+27Z2−3)

64

YZ(7Z2−4)(Z2−1)

+2g(x 2−y 2)z 2

58

(X2−Y2)(3Z2−1)(7Z2−1)6

2Z2(X2−Y2)(4−7Z2)

34

(X2−Y2)(7Z4−6Z2+1)

−2gxyz 2

32

XY(7Z4−6Z2+1)6XYZ2(4−7Z2)54

XY(3Z2−1) (7Z2−1)

+3gxz(x 2−3y 2)

708

XZ(X2−3Y2) (3Z2−1)21

4XZ(X2−3Y2)(1−4Z2)

424

XZ3(X2−3Y2)

−3g(yz(3x 2−y 2)

708

YZ(3X2−Y2) (3Z2−1)42

4YZ3(3X2−Y2)

214

YZ(3X2−Y2)(1−4Z2)

gx 4−6x 2y 2+y 4 +4 35

16(X4−6X2Y2+Y4) (3Z2−1)

218

(X4−6X2Y2+Y4) (Z2+1)−42

4Z2(X4−6X2Y2+Y4)

−4gxy(x 2−y 2)

354

XY(X2−Y2) (3Z2−1) −42XY Z2(Y2−X2)21

2XY(X2−Y2) (Z2+1)

Between dxz and g orbitals

0gz 4

38

XZ(35Z4−30Z2+3) −10

4XZ(2Z2−1)(7Z2−3)

54

XZ(Z2−1) (7Z2−1)

+1gxz 3

304

X2Z2(7Z2−3)24

(Z2−1){(14Z2−1)X2−7Z2+1}14{X2(−56Z4+33Z2−3)+Z2(7Z2−3)}

−1gyz 3

304

XY Z2(7Z2−3)24

XY(Z2−1)(14Z2−1)−14

XY(56Z4−33Z2+3)

+2g(x 2−y 2)z 2

154

XZ(X2−Y2) (7Z2−1)2

4XZ{(X2−Y2)(9−28Z2)+7Z2−1}

12XZ{(X2−Y2−1)(7Z2−6)−1}

gxyz 2 −2 15

2X2YZ(7Z2−1)

12YZ{(2X2−1)(7Z2−6)−1}

24

YZ{2X2(9−28Z2)+7Z2−1}

+3gxz(x 2−3y 2)

2104

X2Z2(X2−3Y2)7

4{X2(1−8Z2)(X2−3Y2)+3Z2(X2−Y2)}

144

{X2(X2−3Y2)(2Z2−1)−(3Z2−1)(X2−Y2)}

g(3x 2−y 2)yz

−3 2104

XYZ2(3X2−Y2)14

4XY{(3X2−Y2)(2Z2−1)−2(3Z2−1)}

74

XY{(1−8Z2)(3X2−Y2)+6Z2}

gx 4−6x 2y 2+y 4 +4 1058

XZ(X4−6X2Y2+Y4)14

4XZ{X2−3Y2−2(X4−6X2Y2+Y4)}

74

XZ{X4−6X2Y2+Y4−2(X2−3Y2)}

−4gxy(x 2−y 2)

1052

X2YZ(X2−Y2)72

YZ{2X2(X2−Y2)−(3X2−Y2)}14

4YZ{3X2−Y2−8X2(X2−Y2)}


Table 3 (Continiued)

gp c0 c1 c2p

Between dyz and g orbitals

0gz 4

38

YZ(35Z4−30Z2+3) −10

4YZ(2Z2−1)(7Z2−3)

54

YZ(Z2−1)(7Z2−1)

+1gxz 3

304

XYZ2(7Z2−3)24

XY(Z2−1)(14Z2−1)−14XY(56Z4−33Z2+3)

gyz 3 −1 30

4Y2 Z2(7Z2−3)

14{Y2(−56Z4+33Z2−3)+Z2(7Z2−3)}

24

(Z2−1){(14Z2−1)Y2−7Z2+1}

+2g(x 2−y 2)z 2

154

YZ(X2−Y2) (7Z2−1)2

4YZ{(X2−Y2)(9−28Z2)−7Z2+1}

12YZ{(X2−Y2+1)(7Z2−6)+1}

−2gxyz 2

152

XY2Z(7Z2−1)2

4YZ{2Y2(9−28Z2)+7Z2−1}

12XZ{(2Y2−1)(7Z2−6)−1}

gxz(x 2−3y 2)

+3 2104

XYZ2(X2−3Y2)14

4XY{(X2−3Y2)(2Z2−1)+2(3Z2−1)}

74

XY{(1−8Z2)(X2−3Y2)−6Z2}

−3gyz(3x 2−y 2)

144

{Y2(3X2−Y2)(2Z2−1)−(3Z2−1)(X2−Y2)}210

4Y2Z2(3X2−Y2)

74

Y2{Y2(1−8Z2)(3X2−Y2)

+3Z2(X2−Y2)}

gx 4−6x 2y 2+y 4 +4 105

8YZ(X4−6X2Y2+Y4)

74

YZ{X4−6X2Y2+Y4+2(3X2−Y2)}−14

4YZ{3X2−Y2+2(X4−6X2Y2+Y4)}

−4gxy(x 2−y 2)

1052

XY2Z(X2−Y2)14

4XZ{X2−3Y2−8Y2(X2−Y2)}

72

XZ{2Y2(X2−Y2)−(X2−3Y2)}

Between dx 2−y 2 and g orbitals

0gz 4

316

(X2−Y2)(35Z4−30Z2+3)10

4Z2(X2−Y2)(3−7Z2)

54

(X2−Y2)(Z2+1)(7Z2−1)

gxz 3 +1 30

8XZ(X2−Y2)(7Z2−3)

24

XZ{(X2−Y2)(7Z2+3)−7Z2+1}14XZ{(X2−Y2)(6−28Z2)+7Z2−3}

−1gyz 3

308

YZ(X2−Y2)(7Z2−3)14YZ{(X2−Y2)(6−28Z2)−7Z2+3}

24

YZ{(X2−Y2)(7Z2+3)+7Z2−1}

+2g(x 2−y 2)z 2

158

(X2−Y2)2(7Z2−1)14{(X2−Y2)2(7Z2+)+Z2(14Z2−10)}2

4{(X2−Y2)2(1−14Z2)

+(7Z2−1)(1−Z2)}

gxyz 2 −2 15

4XY(X2−Y2)(7Z2−1)

22

XY(X2−Y2)(1−14Z2)12XY(X2−Y2)(7Z2+1)

gxz(x 2−3y 2)

+3 2108

XZ(X2−Y2)(X2−3Y2)14

4XZ{(X2−Y2)(X2−3Y2)+3Z2−1}−

74

XZ{4(X2−Y2)(X2−3Y2)+3(Z2−1)}

gyz(3x 2−y 2)

−3 2108

YZ(X2−Y2) (3X2−Y2) −7

4YZ{4(X2−Y2)(3X2−Y2)+3(Z2−1)}

gx 4−6x 2y 2+y 4 +4

−14

4(X2−Y2)(X4−6X2Y2+Y4+Z2−1)

10516

(X2−Y2)78

(X2−Y2)(X4−6X2Y2+Y4+4Z2)

(X4−6X2Y2+Y4)

gxy(x 2−y 2)

−4 1054

XY(X2−Y2)2 72

XY{(X2−Y2)+2Z2}−14

2XY{2(X2−Y2)2+Z2−1}


Table 3 (Continiued)

gp c0p c1 c2

Between dxy and g orbitals

gz 4 0 3

8XY(35Z4−30Z2+3)

102

XY Z2(3−7Z2)54

XY(Z2+1)(7Z2−1)

+1gxz 3

304

X2YZ (7Z2−3)14YZ{4X2(3−14Z2)+7Z2−3}

24

YZ{2X2(7Z2+3)−7Z2+1}

gyz 3 −1 30

4XY2Z(7Z2−3)

14XZ{4Y2(3−14Z2)+7Z2−3}

24

XZ{2Y2(7Z2+3)−7Z2+1}

+2g(x 2−y 2)z 2

154

XY(X2−Y2)(7Z2−1)2

2XY(X2−Y2)(1−14Z2)

12XY(X2−Y2)(7Z2+1)

−2gxyz 2

152

X2Y2(7Z2−1) 12{2X2Y2(7Z2+1)+Z2(7Z2−5)}

24

{(4X2Y2)(1−14Z2)+(1−Z2)(7Z2−1)}

gxz(x 2−3y 2)

+3 2104

X2YZ(X2−3Y2) −7

4YZ{8X2(X2−3Y2)−3(Z2−1)}

144

YZ{2X2(X2−3Y2)−3Z2+1}

−3gyz(3x 2−y 2)

2104

XY2Z(3X2−Y2) −7

4XZ{8Y2(3X2−Y2)+3(Z2−1)}

144

XZ{2Y2(3X2−Y2)+3Z2−1}

+4gx 4−6x 2y 2+y 4

1058

XY(X4−6X2Y2+Y4) −14

2XY(X4−6X2Y2+Y4−Z2+1)

74

XY(X4−6X2Y2+Y4−4Z2)

gxy(x 2−y 2)

−4 1052

X2Y2(X2−Y2)72

(X2−Y2)(2X2Y2+Z2)14

4(X2−Y2)(1−Z2−8X2Y2)

References

Krylov, V.A., S& ulgina, L.T., 1966. Spravocnaja kniga pocislennomu integrirovaniju. Nauka, Moskva, p. 160 (Rus-sian).

Muller, H., Opitz, Ch., 1999. Ein Periodisches System furMetall-Cluster. Westdeutscher Verlag, Opladen/Wiesbaden.

Opitz, Ch., 1999. Bull. Soc. Sci. Lett. Lodz. XLIX, 23.Saito, S., Ohnishi, S., 1987. Proceedings of the International

Symposium of Physical Chemical Small Clusters. Plenum,New York, p. 115.

Saito, S., Ohnishi, S., 1987. Phys. Rev. Lett. 59, 190.Smith, W., Clack, D.W, 1975. Rev. Roum. Chim. 20,

1243.

.

Overlap integrals with g-orbitals

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