INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, NO. 5, 307-3 16 (1 971 )

Dipole Moments and Orbital Energies from ARCANA: A Semi-Empirical Molecular

Orbital Calculation Program JOYCE H. CORRINGTON

Xavier University, New Orleans, Louisiana 70125

AND

H. S. ALDRICH, C. W. McCURDY AND L. C. CUSACHS Tulane University, New Orleans, Louisiana 701 18

Abstract

Approximations employed in calculating two-center nuclear attraction and electron repulsion integrals are discussed. A procedure for economical reduction of the number of integrals to be evaluated which satisfies the requirement of rotational invariance is dis- cussed. This calculation technique is incorporated into a semiempirical molecular orbital calculation program and results in a realistic account of the anisotropy of atoms in molecules of low symmetry. Good agreement is found between electric dipole moments and molecular orbital energies determined experimentally and calculated employing ARCANA.

1. Background

ARCANA is a semi-empirical molecular orbital calculation program including effects due to the electrostatic potential of neighbor atoms in a charge self-con- sistent calculation. The semi-empirical approach was motivated by the difficulty of evaluating the many electron interaction integrals arising in ab initio calculations and by a desire to obtain results in better agreement with experiment than can be expected from economical calculations from first principles.

ARCANA, which is written in FORTRAN IV for an IBM 7044 with a 32K core memory, will handle I20 valence orbitals; thus, it has been possible to study organic molecules of the size of histamine, serotonin and norepinephrine [ 11 and inorganic systems such as water polymers [2]. Each valence atomic orbital is completely defined by atomic, rather than molecular, data :

(I) A single Slater type orbital, employed only for the calculation of overlap integrals [3].

(2) A characteristic energy parameter, the reciprocal radius, 1/R, = (iI 1 / ~ 1;).

(3) A valence state ionization potential for the orbital doubly occupied in the neutral free atom. The function for overlap is determined by matching moments

307 @ 1971 by John Wiley & Sons, Inc.

308 JOYCE H. CORRINGTON AND COWORKERS

(il r li) and (il r2 1;) to the best available atomic orbitals, numerical solutions to the atomic SCF equations, from which the reciprocal radii are also obtained. In the absence of valence state data from atomic spectra, SCF atomic orbital energies may be substituted for the ionization potentials.

Since the uniqueness of ARCANA lies in the neighbor atom potential incor- porated into the calculation of the one-electron Hamilton matrix elements, this paper will discuss the calculation assumptions and the results as shown in calculated dipole moments and orbital energies for a range of molecules.

2. Neighbor Atom Potential

In a local coordinate system with orbitals h and i belonging to a set of type u on atom A (with common radial function and I quantum number but different n) and with orbitalsj and k belonging to a different set B on atom B, it is possible to decompose a charge distribution into a spherically symmetric term and gener- alized higher multipole terms.

(iil = Go,,(Mal + G,,,( Tal + higher multipoles

($1 = Go,,(MBI + G2,j( TBl + higher multipoles

where Ma and M, are spherically symmetric charge distributions determined by the radial function of the type u orbitals and the type /-I orbitals. T, and TB are analogous quadrupoles. Gk,i and Gk,, are constants defining the orientation of orbitals i a n d j in space (Go,, = Go,, = 1). Employing this notation, the two- center nuclear attraction integral can be similarly expressed as :

(;I - l/r, Ii) = - (Ma Ib) - G2.j( T, Ib) + etc

where (Ma Ib) is the nuclear attraction integral for the spherical charge distribution and a unit charge at B. (T, Ib) is the corresponding integral for the quadrupole charge distribution, and the two-center electron repulsion integral as :

(iil 1/rij I j j> = (Ma1 I M P ) + GZ,i(Tal I M B ) + G*.i(TBl I M a )

+ G2.iGz,j(TaI IT') + etc

where (M.1 IMP) is the electron repulsion integral for the two spherical charge distributions. ( TaI IMP) is the electron repulsion integral for the quadrupole- charge interaction, etc. Other integrals of the form (hi1 I j j) can be eliminated by orthogonalizing the atomic orbital basis [4,5], and terms of the form (hi1 ljk) lead only to quadrupole-quadrupole and higher terms which we shall choose to ignore.

The approximations used in ARCANA for the two-center nuclear attraction and electron repulsion integrals have been discussed in previous papers [6, 71 and the results will be repeated here only briefly for reference. If a type a radial

DIPOLE MOMENTS AND ORBITAL ENERGIES 309

charge distribution is characterized by a reciprocal mean radius :

then the two-center nuclear attraction integral for a unit charge at distance R from A can be estimated by approximating the spherical charge-charge integral by

(Ma I b ) = Fo(x)/Ra where

Fo(x) = (1 - exp (-2x)(1 + x ) ) / x x = R/Ra

and the quadrupole-charge integral by

( 'a I 6) = F2(x) /'a where

Fk(x) = 1.875( 1 - exp ( - 0 . 2 7 5 ~ ) ~ ~ + ' / ~ ~ + '

k = 2 for the quadrupole; k = 4 for the hexadecapole, etc. Expressing the two- center electron repulsion integral as

(j j l l/yji I$ = ( j l (il l/yji 1;) l j ) where the operator for thej-th orbital is the average value of the charge distri- bution i divided by the distance between, suggests that it is possible to approximate this operator by the two-center nuclear attraction integral:

(;I 1/yij Ii) N (;I l/r, 1;)

Thus, employing the approximation used above for the two-center nuclear attrac- tion integral, the electron repulsion becomes :

(j j l lbji 1;;) N (jl Fo(x)/R, + c2,iF2(x)/Ra + * * * I j ) Employing the approximation a second time results in the desired multipole- multipole expansion for electron repulsion where the charge-charge integral (averaged to eliminate any slight asymmetry in the approximations) is approxi- mated by:

where S{(MaI I M P ) + ( M p I IMa)} = i (Fo(y) /Ra + FO(z)/Rp)

Y = RplRaFo(R/Rp)

z = Ra/RpFo(R/Ra)

The quadrupole-charge integrals are approximated by :

( TaI IMb) = F2(y)/Ra

(TpI IMa) = F2(z)/Rb

310 JOYCE H. CORRINGTON AND COWORKERS

3. Orthogonalization and Rotational Invariance

Orthogonalization of the atomic orbital basis produces a remarkable reduction in the number of electron interaction integrals which need be evaluated. If the atomic orbital basis is transformed to the symmetrically orthogonalized orbitals of Lowdin [ 5 ] , Parr’s analysis [4] of electron repulsion in the Lowdin basis suggests that three-center and four-center integrals can be eliminated. A similar treatment for one-electron integrals has been formulated by Cusachs and Trus [8 ] , and ap- plied successfully to the calculation of oscillator strengths and dipole moments for large molecules of low symmetry containing heavy atoms. Thus, only integrals of the form (iil Ijj) evaluated in the local coordinate system seem to be required.

However, while orthogonalization indicates how a large number of electro- static integrals can be discarded, it introduces difficulties in the form of problems of rotational invariance. A typical example is provided by Pople and coworkers [9], in the coulomb repulsion integral of the form (xyl Iss), where (xyl describes a charge distribution resulting from the product of a p o and a p , orbital on one atom and Iss) describes the square of an s orbital on a different atom. A 45 degree rotation about the t axis converts this integral into the form ((yyl Iss) - (xxl lss))/2 indicating that the p5 and py orbitals are anisotropic. Thus, if electrostatic inte- grals of the form (xyl Iss), where x andy are on the same atom, are to be discarded, then rotational invariance can be achieved only by the drastic simplification of making all electron interaction integrals the same for all orbitals of a type on the same atom, that is by spherically averaging the integrals. It is thus desirable to evaluate integrals of the form (xyl Iss) and generalized multipole expansion leads to a simple way of achieving this.

When the integrals for electron interaction, for example between sets of p orbitals on two centers, are developed in multipole form, they lead to charge- charge, charge-quadrupole, and quadrupole-quadrupole contributions. The charge-charge terms are spherically symmetric, like the integrals retained by Pople and coworkers. The quadrupole-charge terms produce terms of the form (xyl Iss) upon rotation from the local coordinate system to some absolute coordi- nate system, but since the charge part is unchanged by rotation, only a twofold sum need be evaluated. However, the quadrupole-quadrupole terms evaluated from integrals of the form (iij l j j} are not rotationally invariant since there are additional quadrupole-quadrupole terms which come from integrals of the form (hi1 ljk) which we choose not to evaluate. Thus, it is necessary to eliminate all terms which do not contain at least one charge factor, i.e. the higher multipole- higher multipole terms, in order to maintain rotational invariance.

4. Rotation of Coordinates

If orbitals h and i belonging to set GC on atom A in the local coordinate system are employed to expand orbitals p and q belonging to a set on A in some absolute

DIPOLE MOMENTS AND ORBITAL ENERGIES 31 1

coordinate system, and orbitals j and k of set 8 on atom B are employed to expand orbitals r and s belonging to a set on B in the absolute coordinate system, then:

Y ) P = D P B h v h f D V . i v i

v q = DO.hvh + D g . i v i

Y . = Dr.jYj + D r , k ~ / c

Ys = D s J v j f Ds, l cvk

and in general

(@!?I I T S ) = 2 Dv,hDp. .hDr. iDs. j ljk) hiik

where the D’s are the expansion coefficients. By employing orthogonalization to eliminate all integrals where i # h and

j # k and discarding all terms which do not contain at least one charge factor the evaluation of (pql I T S ) reduces to:

Computationally, for groups of p type orbitals on A and B, this reduces thirty-six possible distinct integrals of the form (Pq) I T S ) to a t most nine of the form (!PI ~ Y Y )

and three each of the form (pql IT^) and (@@I I T S ) which consist of sums of products of transformation coefficients and G coefficients multiplied by a charge-multipole integral general for the two sets. While sets of) orbitals have been employed as an example, this method is general and applicable to d and f type orbitals also. For example, seven hundred and eighty-four integrals resulting from interactions between two groups off orbitals reduce to forty-nine diagonal terms and forty-two possible distinct off-diagonal contributions made up of charge-multipole terms.

5. Calculation of Hamiltonian Elements

By expanding the orbital charge distributions in multipoles, calculating all charge-multipole factors for two-center electrostatic integrals for orthogonalized orbitals expressed in local coordinates, and rotating to a selected absolute coordi- nate system, we end up with diagonal Hamiltonian elements of the form:

- H p p = Aa + B ~ Q A +(; &(PI l/r, IP) - L:Pr(PPI try)) r

and off-diagonal elements of two forms:

-Hpr = P p r ( 2 - l svr l ) (Hw + Hrr)

312 JOYCE H. CORRINCTON AND COWORKERS

TABLE I. Atomic data used in ARCANA molecular orbital calculations.

Overlap STO A , A , Ra Atom Orbital n z non-re1 re1 (a.u.)

C 2s 2 1.57 19.54 1.12 C 2P 1 0.88 9.69 1.28 Si 3s 3 1.58 14.79 1.65 Si 3P 2 0.91 7.58 2.22 Ge 4s 3 1.58 15.15 1.72 Ge 4P 2 0.87 7.33 2.26 Sn 5s 3 1.36 13.04 2.04 Sn 5P 3 1.06 6.76 2.61 Pb 6s 3 1.29 12.48 16.66 2.17 Pb 6P 3 1.01 6.52 7.64 2.72

0 0 S S Se Se Te Te

2s 2P 3s 3P 4s 4P 5s 5P

2.19 1.22 2.03 1.21 1.87 1.09 1.57 1.30

32.00 15.30 2 1.29 11.38 22.85 10.68 19.14 9.54

0.79 0.90 1.28 1.54 1.44 1.78

21.41 1.74 10.22 2.12

B 2s 2 1.26 13.46 1.40 B 2P 1 0.68 8.43 1.65 A1 3s 2 0.96 10.70 1.97 A1 3P 2 0.73 5.71 2.64 Ga 4s 2 1.41 11.55 1.94 Ga 4P 2 0.73 5.67 2.69 In 5s 3 1.67 22.35 1.64 In 5P 3 1.40 10.97 1.96 F 2s 2 2.50 39.00 0.70 F 2P 1 1.38 18.20 0.79

where p and r belong to different sets.

wherep and q belong to the same set.

A, = Spectroscopic atomic orbital energy for u type orbitals on atom A

3, = Change in orbital energy with change in net electronic charge C, = Change in orbital energy with change in effective nuclear charge Q-, = Net charge of atom A calculated employing molecular coefficients

2, = Nuclear charge (minus "core" charge) for atom B

(doubly occupied values preferable) [ 101

expressed in Lowdin orthogonalized atomic basis

DIPOLE MOMENTS AND ORBITAL ENERGIES 313

P , = Electron population of orbital r calculated from Lowdin coefficients S,, = Overlap integral calculated employing single STO’S optimized for over-

lap calculation [3].

While B, and C, can often be determined from spectroscopic data, an approxi- mation consistent with our integral approximations and invariant orbitals is :

C, = F,(O)/R, = 1/R, Hartrees = 27.2/Ra ev

B, = Fo(l)/Ra = 0.73/Ra Hartrees = 19.%/R, ev

Experience and spectroscopic data indicate that for real orbitals these values are consistently high, and for all calculations reported herein B, and C, were estimated to be two-thirds of the above. Table I tabulates aII atomic data employed to characterize orbitals for calculation purposes.

6. Dipoles Calculated from ARCANA

Hoeft and coworkers [I 13 have reported electric dipole moments of the diatomic Group IV/VI compounds for the ground vibrational state. The equi- librium distances and experimental dipole moments are tabulated in Tables I1 and 111. Since diatomic molecular dipole moments are sensitive indicators of

TABLE 11. Equilibrium distances for diatomic group IV/VI compounds, A. (Values in parentheses are estimated.)

C Si Ge Sn Pb

0 1.128 1.51 1.625 1.835 1.922 S 1.535 1.929 2.012 2.209 2.287 Se (1.67) 2.058 2.135 2.326 2.402 Te (1.87) (2.27) 2.340 2.523 2.595

TABLE 111. Electric dipole moments for diatomic Group IV/VI compounds, D.a (Values in parentheses are estimated.)

C Si Ge Sn Pb

0 -0.112 3.098 3.282 4.32 4.64 S - 1.985 1.73 2.00 3.18 3.59 Se (-2.6) 1.1 1.648 2.82 3.29 Te (-3.5) (0.6) 1.06 2.19 2.73

a See Bibliography [l 11.

atomic anisotropy, this work prompted us to attempt to reproduce this molecular series through calculations employing ARCANA to generate molecular wave functions and energies and a diatomic dipole calculation program exact for the

3 14 JOYCE H. CORRINGTON AND COWORKERS

wave functions employed. The resulting dipole moments are tabulated in Table IV.

TABLE IV. Calculated dipole moments for diatomic Group IV/VI compounds, D. (Values in parentheses include relativistic corrections.)

C Si Ge Sn Pb ~~ ~~ ~~ ~ ~

0 0.042 2.807 4.144 3.536 6.57 S - 1.648 1.238 2.058 3.2 17 4.304

(2.894) Se -2.505 0.392 1.266 2.697 3.628

(0.656) (1.611) (3.045) (2.142) Te -3.467 -0.669 0.092 1.564 2.387

(+0.235) (0.919) (2.553) (1.3 12)

In general, good agreement is found between the experimental and caIcuIated values. For the heavier atoms (Se, Te and Pb) relativistic corrections to the calculated Hartree-Fock atomic orbital energies are significant. Since such relativistic orbital energies were not available to us, the Herman-Skillman [ 121 relativistic corrections were added to the Froese [13] orbital energies, resulting in significantly different calculated dipoles.

The electric dipole moments of Group I11 fluorides were also reported by Hoeft and coworkers [ l l ] and their reported experimental and ARCANA calculated values are reported in Table V. Again, good agreement is found.

TABLE V. Equilibrium distances and dipole moments for group I11 flourides.

Distance, Molecule A Exp. dipolea Calc. dipoleb Calc. dipoleC

BF 1.262 (-1.0) -1.733 - 1.429 AIF 1.654 +1.53 +0.953 +0.849 GaF 1.774 2.45 2.1 19 2.385 InF 1.985 3.41 2.956 3.166

a See Bibliography [ll]. Atomic orbital data from Table I, A values for Group 111 elements from SCF-AO data,

Atomic orbital A from same SCF for all atoms. for F from VSIP.

7. Orbital Energies Calculated from ARCANA

While, of course, orbital energies were calculated for all of the Group IV/VI diatomic compounds and fir the Group IIIfluorides for which dipole moments were calculated, few experimental or SCF calculated values are available for comparison.

DIPOLE MOMENTS AND ORBITAL ENERGIES 315

McLean and Yoshimine [ 141 have reported SCF calculated orbital energies for CO, BF, and AIF, and Al-Joboury and coworkers [ 15J have reported experimental molecular beam values for CO. These values are compared in Table VI with

TABLE VI. Highest occupied molecular orbital energies.

-Orbital energy, eV Molecule Orbital Exp." SCFb ARCANA

co d

x d

U

BF U

7r

U

d

AlF U

x d

d

13.98 15.07 16.58 17.39 19.67 21.86

41.36

11.00 20.24 23.20 46.16

8.83 17.76 19.42 42.93

13.06 15.14 19.35 32.36

11.20 16.53 18.01 37.04

9.33 13.68 14.71 35.40

See Bibliography [15]. See Bibliography [14].

values calculated employing ARCANA and allowing for the anisotropy of atomic orbitals belonging to the same set. The ARCANA values are in good agreement both with regard to absolute value and with regard to the splitting of G and T orbitals.

In polyatomic molecules the effects of orbital anisotropy are less conspicuous, but it is still possible to compare ionization potentials with experiment as a test

TABLE VII . Ionization potenjials of CF,.

ARCANA ESCAa CNDOa

16.58 It, 16.1 20.5 3t, 17.35 3t, 17.4 22.5 It, 18.69 le 18.5 23.6 le 21.03 2t, 22.2 28.5 2t, 22.95 2a, 25.1 29.8 2a, 38.41 It, 40.3 48.5 It, 43.87 la, 43.8 55.0 lal

a ESCA experimental values and CNDO calculations by K. Siegbahn and coworkers, ESCA Applied to Free Molecules, North-Holland, Amsterdam, 1970. The authors cite un- published ab initio calculations as indicating the same level ordering as ARCANA. The experimental data does not distinguish between the two level orderings.

316 JOYCE H. CORRINGTON AND COWORKERS

of ARCANA’s ability to reproduce the pattern of valence molecular orbital energies without the use of parameters determined from any molecular experi- ment or calculation. In Table VII the predictions of ARCANA for CF, are compared with experiment and with competitive CNDO calculations. Very similar quality results are reported elsewhere [16] for SF, and SO2.

8. Discussion By expanding orbital charge distributions in terms of generalized multipoles

and calculating a11 charge-multipole factors for two-center electrostatic integrals, it is possible of allow for the anisotropy of atomic orbitals belonging to the same n, 1 set while still maintaining rotational invariance. This leads to realistic approxi- mations of the actual electronic structure of molecules containing heavy atoms and reasonable calculated molecular properties, such as dipole moments and or- bital energies. By avoiding the use of parameters derived from molecular prop- erties or calculations this method is capable of making bona fide predictions of at least a limited number of properties of molecules.

ARCANA employs a calculation procedure quite comparable in computational effort to CNDO, but systematically avoids the Hartree-Fock equations. I t is an approach from another direction toward a one-electron approximation for mole- cules in the sense discussed for atoms and solids by Slater and coworkers [17] whose discussion section clearly and concisely describes the attitude and purpose of this work.

Bibliography [l] J. H. Corrington, M. Coleman and V. B. Haarstad, to be published. [2] H. S. Aldrich, L. P. Gary, H. J. Lader, L. C. Cusachs and J. H. Corrington, Proc. Intern.

[3] L. C. Cusachs and J. H. Corrington, in Sigma Molecular Orbital Theory, Ed. 0. Sinanoglu and

[4] R. G. Parr, Quunturn Theory of Molecular Electronic Structure, (Benjamin, New York, 1963). [5] P. 0. Lowdin, J. Chem. Phys. 18, 365 (1950). [6] J. H. Corrington and L. C. Cusachs, Int. J. Quantum Chem. 3S, 207 (1969). [7] J. H. Corrington and L. C. Cusachs, Spectry Letters, 1,67 (1968). [8] L. C. Cusachs and B. L. Trus, J. Chem. Phys. 46, 1532 (1967). [9] J. A. Pople, D. P. Santry and G. A. Segal, J. Chem. Phys. 43, S129 (1965).

Conf. Polywater, 1, (1970).

K. B. Wilber (Yale University Press, 1970), p. 256.

[lo] L. C. Cusachs and J. W. Reynolds, J. Chem. Phys. 43, S160 (1965); L. C. Cusachs, J. W. Reynolds and D. Barnard, J. Chem. Phys. 44, 835 (1966); L. C. Cusachs and J. R. Linn, Jr., J. Chem. Phys. 46,2919 (1967).

[ l l ] J. Hoeft, F. J. Lovas, E. Tiemanns and T. Torring, J. Chem. Phys. 53,2736 (1970). [ 121 F. Herman and S. Skillman, Atomic Structure Calculations (Prentice-Hall, 1960). [13] C. Froese, J. Chem. Phys. 45, 1417 (1966) and supplement. [ 141 A. D. McLean and M. Yoshimine, IBM J. Res. Develop (1967) and supplement. [15] M. T. Al-Joboury, D. P. May and D. W. Turner, J. Chem. SOC. 5141 (1963). [I61 L. C. Cusachs and D. J. Miller, in Trends in Sulfur ReJearch, Ed. T. Wiewiorowski and

[17] J. C. Slater, J. B. Mann, T. M. Wilson and J. H. Wood, Phys. Rev. 184,672 (1969). D. J. Miller, American Chemical Society, in press.