INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XXII, 415420 (1982)

Ab initio Molecular Calculations with Pseudopotentials: Calculations of

Double-Zeta Quality on Ethylene, Acetylene, and Water

R. GASPAR, JR.

R. GASPAR Department of Bioph ysics, Medical University of Debrecen, H-4012 Debrecen, Hungary

Institute of Theoretical Physics, Kossuth Lajos University, H-4010 Debrecen, Hungary

Abstract

Calculations with pseudopotentials of double-zeta quality have been performed on ethylene, acetylene, and water molecules. A description of the carbon-carbon double and triple bonds is presented in the framework of the pseudo-FSGO method. A possible model of the oxygen lone pairs has been established and its functioning has been proved by calculations on the water molecule.

Introduction

The calculation of properties of molecules by ab initio methods usually demands large computing resources. The introduction of pseudopotentials into the molecular calculations may reduce the numerical problem to the level of all-valence electron SCF calculations. The floating spherical Gaussian orbitals (FSGO) method of Frost has been modified by the incorporation of pseudopoten- tials of double-zeta quality [l]. The resulting pseudo-FsGo method has been used for the quantum mechanical description of small molecules as well as molecular fragments [2]. Moreover, the same method has been used for the construction of larger hydrocarbons as well as other molecules from pseudopoten- tial molecular fragments [3].

The pseudo-FsGo method has been specially designed to account for chemical bonds and lone pairs. In the present paper calculations on ethylene and acetylene molecules are presented and the capability of the pseudo-FsGo method to describe the double and triple carbon-carbon bonds in these molecules is investi- gated. Moreover, a description of the oxygen lone pairs is established in the framework of the pseudo-FsGo method and calculations on the water molecule are discussed.

Method

In the pseudo-FsGo method effect of the core orbitals of the original FSGO

method are replaced by pseudopotentials. The N, valence electrons in a molecule

@ 1982 John Wiley & Sons, Inc. CCC 0020-7608/82/080415-06$01.60

416 GASPAR JR. AND GASPAR

containing n atoms can be described by the valence pseudo-Hamiltonian

N o 1 ZPs = ( -IA(i)+ f V f ( i ) ) + 1 -, i = l k = l . i>j rij

where V f is the pseudopotential belonging to the core of atom k. For a given atom, we have

Z r i

Vf = --+C A! exp (-arr2)Pi,

where Z is the net charge of the ion core consisting of the core electrons and the nucleus of the atom; Al and ai are the pseudopotential parameters; and r, 1, and Pl have their usual meanings. Double-zeta parameters of Vf are presented in Table I for the carbon and oxygen atoms. A detailed procedure of double-zeta pseudopotential parameter determination has been published earlier.*

TABLE I. Double-zeta parameters of the pseudopotential V,(r) = A! exp (-a& for 1 = 0 and I = 1.

- E l e m e n t Z A. d o A1 4 C 4 31.6899 8,0076 -15,2716 20.6316

0 6 54.9425 13.3183 -17,0904 37.8607

Molecular orbitals are constructed as the linear combinations of normalized FSGOS of the following form:

(3) -1 -2 3/4 Gi(r )=(27r p i ) exp[-pf2(r-Ri)2],

where pi is the radius of the orbital and Ri is the vector pointing to the center of the FSGO.

The total valence energy Eva, of the molecule can be determined as

where z k and Zm are the charges of the atomic cores, and Rkm is the internuclear distance between atoms k and m. During a molecular calculation the nuclear geometry of the molecule as well as the linear coefficients and the Ri and pi parameters of the FSGOS in the molecule are varied until the valence energy Eval of the molecule is minimized.

* See Ref. 1 ; for the values of parameters, see Ref. 4.

CALCULATIONS ON PSEUDOPOTENTIALS 417

Results and Discussion

In Tables 11, 111, and IV results of pseudo-FsGo calculations, utilizing the double-zeta parameters of Table I, are presented for ethylene, acetylene, and water molecules. Parameters describing the floating spherical Gaussian orbitals, calculated and experimental geometrical parameters, as well as valence only total energies are presented in Tables 11-IV belonging to the minimum energy configuration of the ground state of the molecules. Results gained by the original FSGO procedure are included in Tables 11-IV for comparison. The partitioning of the valence only total energy of the molecules into its components as kinetic TE, electronic Coulombic repulsion V E ~ , point core-electron attraction VEN, point core-point core repulsion VNN, and pseudo-core-electron interaction VpE energies is also given.

TABLE 11. Parameters of the ethylene molecule."

Gauss ian pa ramete rs

O r b i t a l

C-H

c = c

C-H

c=c

UCH

Radius D i s t a n c e f rom t h e C atom R

P Comments

1.83080 1,65910 R ,f = f r e e l y v a r i e d

1.32755 10.2 ? = f r e e l y v a r i e d R = f i x e d

I n t e r n u c l e a r d i s t a n c e s and bond a n g l e s

FSGOC E x p e r i m e n t a l b C a l c u l a t e d

2.33939 2.081 2.02

2.02064 2.554 2.47

106.79' 118.7' 11025°

E n e r g i e s E x p e r i m e n t a l d

C a l c u l a t e d FSGO'

T o t a l valenceml2. 01747 - -13.82650 energy Tot a 1 energye-76. 82967 -65.835 -78,63860

C o n t r i b u t i o n s t o t h e v a l e n c e o n l y t o t a l ene rgy

I<ine t i c 9.04893

E l e c t r o n r e p u l s i o n 24.63724

P o i n t c o r e - e l e c t r o n a t t r a c t i o n -67.55409 Core-core r e p u l s l o n 20.13843

Ps e u do -co r e - e l e c t r on r epu Is i o n 1.71202

a All energies and lengths are given in a.u. see Ref. 5. See Ref. 6. See Ref. 7. Estimated value of valence only total energy using experimental ionization

"Total energy of the molecule in the pseudo-FsGo approximation has been potentials and bond energies.

calculated according to the procedure outlined in Ref. 3.

418 GASPAR JR. AND GASPAR

TABLE 111. Parameters of the acetylene molecule."

Gauss ian pa ramete rs

O r b i t a l

c - ! i

c-c

c -c

C-H

c:c

Radivs D i s t a n c e f rom t h e Comments C atom R

.? 1.55469 1.05095 R , j = f r e e l y v a r i e d

1.47723 0.69792 R ,f = f r e e l y va r l e d

1.70023 20.1 $ = f r e e l y v a r i e d R= f i x e d

I n t e r n u c l e a r d i s t a n c e s b

C a l c u l a t e d FSGOC E x p e r l m e n t a l

1.92865 2.039 2.00

2.34539 2.295 2.28

E n e r g i e s d

C a l c u l a t e d FSGOC E x p e r i m e n t a l

T o t a l v a l e n c e -11.36729 - -13,82650 energy

T o t a l ene rgye -76.17949 -64.678 -78.63870

C o n t r i b u t i o n s t o t h e v a l e n c e o n l y t o t a l ene rgy I < i n e t i c 7,44380

E l e c t r o n r e p u l s i o n 18.67943

P o i n t c o r e - e l e c t r o n a t t r a c t i o n -52.74647 Core-core repulsion 13,00288

Pseudo -co r e -e l e c t ron r e pu 1 s i on 2.2 5307

a All energies and lengths are given in a.u. see Ref. 5 . See Ref. 6. See Ref. 7. Estimated value of valence only total energy using experimental ioniz-

Total energy of the molecule in the pseudo-FsGO approximation has ation potentials and bond energies.

been calculated according to the procedure outlined in Ref. 3.

For the description of the electronic configuration of the molecules simple orbital basis sets have been used. The ethylene molecule has been constructed from six primitive Gaussians, four describing the C-H bonds and two describing the C=C double bond. The latter two have been positioned at the center of the molecule 0.2 a.u. above and below the plane of the molecule. As Table I1 shows the calculated energy and geometrical parameters are in fair agreement with the experimental and FSGO results in spite of the very simple orbital representation employed.

The acetylene molecule has been assembled from 12 primitive orbitals. Two of them account for the C-H bonds and the others for the C=C triple bond in the molecule. To represent the sigma character of the C=C triple bond primitive orbitals have been positioned along the carbon-carbon axis of the

CALCULATIONS ON PSEUDOPOTENTIALS 419

TABLE IV. Parameters of the water molecule."

Gauss i a n pa r a me t e r s

Radius D i s t a n c e f rom t h e Comments 0 atom R P

OH 1.28959 0.67918 R , f = f r e e l y v a r i e d

LPll 1.90549 +O. 5 R=f i x e d f = f r e e l y v a r i e d

R = f i x e d

R= f i x e d

p = f r e e l y v a r i e d

? = f r e e l y v a r i e d

LP12 1.33029 -0.5

t o . 1 LP2 ; LP22 1.2 1323

I n t e r n u c l e a r d i s t a n c e s , bond a n g l e s and d i p o l e

moments

b C a l c u l a t e d FSGOC Exper imen t a 1

OH

HOH

D i p o l e moment

1.70472 1.666 1 .81

103.91' 88.4' 104.45'

0.75220 0.7278ge

Ene r g i e s C a l c u l a t e d FSGOC E x p e r i m e n t a l d

- -17.26717 T o t a l v a l e n c e -15.02760 energy

T o t a l ene rgy -74.20419 -64.288 -76.44380

C o n t r i b u t i o n s t o t h e v a l e n c e o n l y t o t a l ene rgy

I<ine t 1 c 10.31821

E l e c t r o n r e p u l s i o n 17,33671

P o i n t c o r e - e l e c t r o n a t t r a c t i o n -52.48472

Core-core r e p u l s i o n 7 .4116g

Pseudo -co r e - e l e c t ron i n t e r ac t i o n -2.39051

a All energies and lengths are given in a.u. see Ref. 5. See Ref. 6. See Ref. 8. Estimated value of valence only total energy using experimental ionization

Handbook of Chemistry and Physics, 55th ed., R. C. Weast, Ed. (Chemical

Total energy of the molecule in the pseudo-FSGO approximation has been

potentials and bond energies.

Ru!ber Co., Cleveland, OH, 1974-1975).

calculated according to the procedure outlined in Ref. 3.

molecule at equal distances from the carbon atoms. The remaining eight primitive orbitals have been used to construct the two T bonds of the molecule in planes at right angles to each other. The distance of these orbitals from the carbon- carbon axis was chosen to be 0.1 a.u. and their position along the C-C axis coincided with the position of the C atoms. The results of the pseudo-FsGo calculations on this molecule are in good agreement with the experimental and FSGO ones as can be seen from Table 111.

420 GASPAR JR. AND GASPAR

Six primitive Gaussians have been used to describe the valence shell of the water molecule in the framework of the pseudo-FsGo model. Two primitive orbitals have been positioned between the oxygen and the two hydrogens to account for the OH sigma bonds in the molecule. The description of the lone pairs of the oxygen atom required four basis orbitals. Two of them were placed 0.1 a.u. above and below the plane of the H 2 0 molecule at the position of the oxygen atom. The remaining two orbitals were situated along the symmetry axis of the molecule k0.5 a.u. from the oxygen atom. The pseudo-FSGO results on the H 2 0 molecule are displayed in Table IV. The good agreement of the experimental and calculated energy and geometrical parameters as well as dipole moments suggests that the simple lone pair description employed in the present paper is satisfactory for further applications. It may be seen from Table IV that the pseudo-FsGo results on the H 2 0 molecule are more reasonable than those gained by the original FSGO procedure.

The results presented in this paper have increased the applicability of the pseudo-FsGo method. It is worth mentioning that highly restricted basis sets have been employed in the present calculations resulting in good energy predic- tions combined with reasonable results for the geometrical parameters. Finally we wish to emphasize that calculations with the pseudo-FsGo model are ab initio, for no adjustment of the pseudopotential parameters happens at the molecular level of the calculations, and all matrix elements are explicitly calculated. The pseudopotential parameters are determined at the atomic level according to procedures described elsewhere [4].

Bibliography

[l] R. Giispar and R. Gaspar, Jr., Acta Phys. Acad. Sci. Hung. 45, 27 (1978). [2] R. Gaspar and R. Gaspar, Jr., Int. J. Quantum Chem. 15, 567 (1979). [3] R. Gaspar, Jr. and R. GBspar, Int. J. Quantum Chem. 15,567 (1979); 16, 57 (1979). [4] R. Gaspar and I. KO&, Acta Phys. Acad. Sci. Hung. 45, 123 (1978). 51 H. Shull and G. G. Hall, Nature (London) 184, 1559 (1959). [6] L. E. Sutton, Tables of Interatomic Distances and Configurations in Molecules and Ions, Spec.

[7] A. A. Frost and R. A. Rouse, J. Am. Chem. SOC. 90,1965 (1968). [8] A. A. Frost, J. Phys. Chem. 72, 1289 (1968).

Received July 15, 1981 Accepted for publication February 26, 1982

Publ. No. 11 (Chemical Society, London, 1958).