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Universal Gaussian basis set for relativistic calculations on atoms and molecules
Volume 20 1, number I ,2,3,4
CHEMICALPHYSICSLETTERS
1 January 1993
Universal Gaussian basis set for relativistic calculations on atoms and molecules G.L. Malli, A.B.F. Da Silva ’ ofChemistry, Simon Fraser University, Burnaby, Britisch Columbia, Canada VSA IS6
Department
and Yasuyuki Ishikawa Department of Chemistry and the Chemical Physics Program, University ofPuerto Rico, P.O. Box 23346, U.P.R. Station, San Juan 00931, Puerto Rico
Received I September 1992
A universal Gaussian basis set is developed for ab initio relativistic Dirac-Fock calculations on atoms and molecules. The results of the matrix Dirac-Fock-Coulomb and Dirac-Fock-Breit self-consistent field calculations using the universal Gaussian basis set are presented for the atoms, He (Z=2) through Xe (Z= 54). The total Dirac-Fock-Coulomb energies calculated with the relativistic universal Gaussian basis set are in excellent agreement with the corresponding energies obtained by the numerical finite difference Dirac-Fock method. The computed Breit interaction energiesare convergentto at leastfive figuresfor all the systems studied.
1. Introduction The use of the Gaussian basis set in relativistic quantum chemistry was proposed more than a decade ago [ 1] _ Gaussian basis sets have a number of advantages in relativistic Dirac-Fock calculations [ 2,3]. Although Gaussian basis sets may be at a disadvantage with respect to Slater basis sets in nonrelativistic calculations because of their improper behavior near the origin in the point nucleus approximation, Slater basis sets are no longer advantageous for systems involving heavy atoms where the use of a finite nucleus is mandatory [ 2-41. Moreover, within the finite nucleus of uniform proton charge distribution, the solutions of the radial Dirac equation can be represented by Gaussians [ 2-41, and there is an obvious advantage to using this model in atomic and molecular calculations. In previous stud’ Permanent address: Departamento de Quimica e Fisica Molecular, Instituto de Fisica e Qulmica de Sfo Paulo, C.P. 369, 13560SHoCarlos, SP, Brazil.
ies [ 561, matrix Dirac-Fock-Coulomb (DFC) and Dirac-Fock-Breit (DFB) self-consistent field (SCF) calculations were performed using various basis sets of Gaussians. In these studies, we have shown that finite basis sets of Gaussians can yield results comparable in accuracy with the corresponding results obtained by using the numerical finite difference Dirac-Fock approach [ 5,6]. Since the computational cost of using a finite basis set in atomic and molecular calculations increases approximately as N4 (where N is the number of basis functions used), various attempts have been made to economize the cost of computations as much as possible by adopting various strategies. One such approach was the introduction of a universal basis set of Slater functions [7,8]. The major advantage in designing a universal basis set lies in the fact that the use of such a single set of basis functions can lead to significant computational savings in molecular calculations due to the transferability of all the one- and two-electron integrals over the universal basis set from system to system (apart from multiplicative
0009-2614/93/%06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
37
Volume 201, number 1,2,3,4
scale factors due to the change in nuclear charges). A universal Gaussian basis set (UGBS) was reported for the atoms He through Xe (Z= 54) for use in nonrelativistic calculations [ 91. Recently, we reported DFC and DFB SCF calculations for the atoms, He through Xe, by using the UGBS [ lo], Reasonably good results were obtained, but only for the lighter atoms, He through Ca. In order to investigate the feasibility of devising a UGBS for relativistic atomic and molecular structure calculations, we have generated a new set of UGBS by employing the generator coordinate Hartree-Fock (GCHF) method [ 9,111. In this Letter, the results of the matrix DFC and DFB SCF calculations obtained by using the relativistic UGBS are presented for a number of,atoms, He through Xe (Z=54).
2. Theoretical In matrix DFB calculations for closed-shell systems, the SCF equation for symmetry type K takes the form [ 121, (1)
,
F,C,=S,C,E,
where following the notation used by Quinley et al. [ 13 1, the overlap matrix is given in a block diagonal form:
(2) The superscripts LL and SS indicate which of the large (L) or small (S) component basis sets have been employed. The Fock matrix can be written as F,=f,+q,+b,,
(3)
where the one-electron part f, is fx=
ViL
[ cn= K
v= K
cw+ -2c2p
k
(4)
1.
The two-electron part gK, which consists of the matrices of two-electron Coulomb and exchange interactions, is given by
- K,LX JZS _K:S
38
1 January I993
CHEMICAL PHYSICS LETTERS
1 ’
(5)
Breit-interaction leads The frequency-independent to the term b, in the matrix SCF equation, viz.
1
BLL BLs bK= ,;, B:s [ I( K
(6)
The one- and two-electron matrix elements in eqs. (4)-( 6) were expressed in terms of Gaussian basis functions in our previous work [ 61. The exponents of the universal GTFs used in our work are generated by using the GCHF method [ 9,111. The GCHF method is characterized by the representation of the one-electron functions as an integral transforms, viz. $i(l,(~)J;((~)da,
i=l,...,n,
(7)
where the $,,J and (Yare the generator functions, the weight functions, and the generator coordinate, respectively. The minimization of the energy value built with the one-electron functions, vi, in eq. (7) leads to the Griffm-Wheeler-HF (GWHF) equations, viz. I
]F(a, P) -eiS(a, i= 1, .... n .
P)
IA(P) V=O > (8)
The GWHF equations are integrated numerically by using the integral discretization (ID) technique [ 9,111 that preserves the integral character of the GCHF method. The ID technique is implemented with a relabelling of the generator coordinate space ((Yspace), to the a space by the following relation: Q+,
A>l.
An equally spaced N-point mesh {sl,} is selected so that one can obtain an adequate numerical integration range for the s, p and d symmetries of several atoms. The integration range is then characterized by a stating point, coin, an increment, M! and N (number of points). The scaling parameter, A, is the same (A=6.0) for all the calculations. The exponents of the universal Gaussian basis set are generated by the following discretization parameters. Symmetry: s, p and d; G,,,,,= -0.79; A.0=0.13; N= 32. As one can see from the discretization parameters, we have obtained a unique set of 32 exponents common for all three symmetries (s, p and d). These 32
exponents generated with the discretization parameters represent a single set of Gaussian exponents that can be used in atomic and molecular calculations involving the atoms He (Z=2) through Xe (Z= 54).
3. Results and discussion The DFC and DFC SCF calculations on the raregas atoms: He, Ne, Ar, Kr and Ze, and the alkalineearth atoms: Be, Mg, Ca and Sr, are performed using the UGBS as outlined above. The DFC SCF calculations are performed by omitting the Breit term, b,, from the DFB matrix. The finite nuclear model of uniform proton charge distribution and the kinetic balance constraints [2,3,13-l 51 are used in all the calculations. The atomic masses used for the He, Ne, Ar, Kr, and Xe atoms are: 4.0026, 20.183, 39.948, 83.80, and 13 1.30, respectively. For Be, Mg, Ca and Sr, the atomic masses used are 9.0 122, 24.3 12, 40.08, and 87.62, respectively. The speed of light used in our calculations is 137.0370 au.
Table I Total Dirac-Fock-Coulomb Atom
I January 1993
CHEMICAL PHYSICS LETTERS
Volume 201, number 1,2,3,4
(&,-),
UGBS size
Dirac-Fock-Breit
Tables 1 and 2 display the results of the calculated DFC (ED&, DFB (EDFB), and Breit interaction (I$,) energies for the rare-gas and the alkaline earth atoms, respectively. The EB denotes the variational Breit interaction energy computed as the difference between the EDFBand EDFC_The’variational Breit interaction energy, E,, is the level shift in the total SCF energy due to the inclusion of the low-frequency Breit interaction term in the SCF process. In the last column of tables 1 and 2, we tabulate the DFC energies calculated for the various atomic systems by using the Oxford finite difference numerical DiracFock program [ 161 #I The UGBS size displayed in tables 1 and 2 represents the number of Gaussian exponents for each s, p and d symmetry necessary to attain a DFC energy value with an accuracy of a few micro- or millihartrees as compared to the corresponding numerical finite difference results. The addition of higher ” OxfordMCDF program (ref. at Cambridge University.
(E DFB),and variational Breit
EDFC=’
(EB)energies of the rare-gas atoms (in hanree) E,”
EDFEn)
[ I6 1) modified by N.C. Pyper
Finite difference b,
(-&FC) He Ne Ar Kr Xe
18s 32~29~ 32~29~ 32s29p16d 32s29p18d
-2.861813335 - 128.6919310 -528.6837530 -2788.860655 - 7446.892977
-2.861749571 - 128.6752902 -528.5514300 -2787.434761 -7441.123179
a) This work (using the universal Gaussian basis set). b1 E DcFcomputed by the numerical finite-difference Dirac-Fock program (ref.
Table 2 Total Dirac-Fock-Coulomb hartree) Atom
UGBS size
(E,,),
Dirac-Fock-Breit
E ma’
(I&,),
0.000063763 0.0 166408 0.1323230 1.425894 5.769798
[ I6]! and footnote I ).
and variational
E rx=Ba
-2.86181335 - 128.691938 - 528.683840 -2788.86168 -7446.90018
Breit (I$,) energies of the alkaline-earth atoms (in
EBa’
Finite difference b, (&FC)
Be Mg Ca Sr
25s 32~29~ 32~29~ 32s29p16d
- 14.57589219 - 199.9350665 -679.7101239 -3178.080060
-14.57518897 - 199.9032396 -679.5191300 -3176.361450
‘) This work (using the universal Gaussian basis set). b, EoFc computed by the numerical finite-difference Dirac-Fock program (ref.
0.00070322 0.0318269 0.1909939 1.718560
-14.5758919 - 199.935083 -679.710276 -3178.01833
[ 16 1,and footnote 1). 39
Volume 20 I, number
I ,2,3,4
CHEMICAL PHYSICS LETTERS
exponents beyond the size given for each atom in tables 1 and 2 do not lead to any substantial improvement in the DFC and DFB energies. The results show that the total DFC energies computed with the UGBS are in very good agreement with the corresponding results obtained by the finite difference Dirac-Fock calculations for all the atoms except Xe. The UGBS is capable of reproducing the total DFC energies to within one millihartee (mhartree) or better as compared to the corresponding finite difference values. For the He and Be atoms, the agreement is within a few microhartrees ( lOA hartree). For heavier atoms, the results differ from the corresponding finite difference results by up to one mhartree. The results obtained by using the UGBS are in closer agreement with the corresponding numerical limits than those obtained previously by using the well-tempered Gaussian basis sets [ 5 1. For the heaviest Xe atom, the total energy lies seven mhartrees above the numerical limit and two mhartrees above the total DFC energy obtained by using the well-tempered Gaussian basis set [ 51.
5. Conclusions The relativistic universal Gaussian basis set developed in our work yields total DFC and DFB energies accurate to within a few mhartrees or better as compared to the corresponding numerical finite difference results for all the systems studied. The computed variational Breit interaction energies are convergent to at least live figures. The results obtained in our study clearly indicate that it is feasible to develop a universal Gaussian basis set for accurate atomic and molecular relativistic Dirac-Fock calculations.
Acknowledgement
This work has been supported by the Natural Sci-
40
1 January I993
ences and Engineering Research Council (NSERC) of Canada to GLM (Grant No. A3598), and by the CNPq (Brazilian Agency) to ABFDaS and GLM, and by the National Science Foundation to YI (Grant PHY-9008627). All the calculations reported in this work were performed on an IBM RISC/6000 workstation at Simon Fraser University. The authors thank Dr. Steve Kloster of the Academic Computing Services at Simon Fraser University and Dr. Chan Kyung Kim for their help with the calculation on the IBM RISC/6000. We gratefully acknowledge Kirsten Masse for wordprocessing the manuscript cheerfully and diligently.
References [ I] G.L. Malli, Chem. Phys. Letters 68 (1979) 529. [2] Y. Ishikawa, R. Baretty and R.C. Binning Jr., Chem. Phys. Letters I21 (1985) 130. [3] Y. Ishikawa and H.M. Quiney, Intern. J. Quantum Chem. Symp. 21 (1987) 523. [4] F.G. Werner and J.A. Wheeler, Phys. Rev. 109 (1958) 126. [5] Y. Ishikawa, Chem. Phys. Letters 166 (1990) 321; Y. Ishikawa, H.M. Quiney and G.L. Malli, Phys. Rev. A 34 (1991) 8270. [6] Y. Ishikawa, G.L. Malli and A.J. Stacey, Chem. Phys. Letters 188 (1992) 145. [7] D.M. Silver and W.C. Nieuwpoort, Chem. Phys. Letters 57 (1978) 421; D.M. Silver, S. Wilson and W.C. Nieuwpoort, Intern. J. Quantum Chem. 14 (1978) 635. [S] S. Wilson, Advan. Chem. Phys. 67 (1967) 439. [9]A.B.F.DaSilva,H.F.M.DaCostaandM.Trsic,Mol.Phys. 68 (1989) 433. [IO] G.L. Malli, A.B.F. Da Silva and Y. Ishikawa, Can. J. Chem. 70 (1992), in press. [I I ] J.R. Mohallem, R.M. Dreizler and M. Trsic, Intern. J. Quantum Chem. Symp. 20 ( 1986) 45. [ 121 Y.K. Kim, Phys. Rev. 154 (1967) 17. [ 131 H.M. Quiney, LP. Grant and S. Wilson, J. Phyr B 20 (1987) 1413. [ 141 R.E. Stanton and S. Havriliak, J. Chem. Phys. 81 (1984) 1910. [ 151K. Dyall, I.P. Grant and S. Wilson, J. Phys. B 17 (1984) 1201. [ 161 I.P. Grant, B.J. McKenzie, P.H. Norrington, D.F. Mayers and NC. Pyper, Computer Phys. Commun. 2 1(1980)207.
CHEMICALPHYSICSLETTERS
1 January 1993
Universal Gaussian basis set for relativistic calculations on atoms and molecules G.L. Malli, A.B.F. Da Silva ’ ofChemistry, Simon Fraser University, Burnaby, Britisch Columbia, Canada VSA IS6
Department
and Yasuyuki Ishikawa Department of Chemistry and the Chemical Physics Program, University ofPuerto Rico, P.O. Box 23346, U.P.R. Station, San Juan 00931, Puerto Rico
Received I September 1992
A universal Gaussian basis set is developed for ab initio relativistic Dirac-Fock calculations on atoms and molecules. The results of the matrix Dirac-Fock-Coulomb and Dirac-Fock-Breit self-consistent field calculations using the universal Gaussian basis set are presented for the atoms, He (Z=2) through Xe (Z= 54). The total Dirac-Fock-Coulomb energies calculated with the relativistic universal Gaussian basis set are in excellent agreement with the corresponding energies obtained by the numerical finite difference Dirac-Fock method. The computed Breit interaction energiesare convergentto at leastfive figuresfor all the systems studied.
1. Introduction The use of the Gaussian basis set in relativistic quantum chemistry was proposed more than a decade ago [ 1] _ Gaussian basis sets have a number of advantages in relativistic Dirac-Fock calculations [ 2,3]. Although Gaussian basis sets may be at a disadvantage with respect to Slater basis sets in nonrelativistic calculations because of their improper behavior near the origin in the point nucleus approximation, Slater basis sets are no longer advantageous for systems involving heavy atoms where the use of a finite nucleus is mandatory [ 2-41. Moreover, within the finite nucleus of uniform proton charge distribution, the solutions of the radial Dirac equation can be represented by Gaussians [ 2-41, and there is an obvious advantage to using this model in atomic and molecular calculations. In previous stud’ Permanent address: Departamento de Quimica e Fisica Molecular, Instituto de Fisica e Qulmica de Sfo Paulo, C.P. 369, 13560SHoCarlos, SP, Brazil.
ies [ 561, matrix Dirac-Fock-Coulomb (DFC) and Dirac-Fock-Breit (DFB) self-consistent field (SCF) calculations were performed using various basis sets of Gaussians. In these studies, we have shown that finite basis sets of Gaussians can yield results comparable in accuracy with the corresponding results obtained by using the numerical finite difference Dirac-Fock approach [ 5,6]. Since the computational cost of using a finite basis set in atomic and molecular calculations increases approximately as N4 (where N is the number of basis functions used), various attempts have been made to economize the cost of computations as much as possible by adopting various strategies. One such approach was the introduction of a universal basis set of Slater functions [7,8]. The major advantage in designing a universal basis set lies in the fact that the use of such a single set of basis functions can lead to significant computational savings in molecular calculations due to the transferability of all the one- and two-electron integrals over the universal basis set from system to system (apart from multiplicative
0009-2614/93/%06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
37
Volume 201, number 1,2,3,4
scale factors due to the change in nuclear charges). A universal Gaussian basis set (UGBS) was reported for the atoms He through Xe (Z= 54) for use in nonrelativistic calculations [ 91. Recently, we reported DFC and DFB SCF calculations for the atoms, He through Xe, by using the UGBS [ lo], Reasonably good results were obtained, but only for the lighter atoms, He through Ca. In order to investigate the feasibility of devising a UGBS for relativistic atomic and molecular structure calculations, we have generated a new set of UGBS by employing the generator coordinate Hartree-Fock (GCHF) method [ 9,111. In this Letter, the results of the matrix DFC and DFB SCF calculations obtained by using the relativistic UGBS are presented for a number of,atoms, He through Xe (Z=54).
2. Theoretical In matrix DFB calculations for closed-shell systems, the SCF equation for symmetry type K takes the form [ 121, (1)
,
F,C,=S,C,E,
where following the notation used by Quinley et al. [ 13 1, the overlap matrix is given in a block diagonal form:
(2) The superscripts LL and SS indicate which of the large (L) or small (S) component basis sets have been employed. The Fock matrix can be written as F,=f,+q,+b,,
(3)
where the one-electron part f, is fx=
ViL
[ cn= K
v= K
cw+ -2c2p
k
(4)
1.
The two-electron part gK, which consists of the matrices of two-electron Coulomb and exchange interactions, is given by
- K,LX JZS _K:S
38
1 January I993
CHEMICAL PHYSICS LETTERS
1 ’
(5)
Breit-interaction leads The frequency-independent to the term b, in the matrix SCF equation, viz.
1
BLL BLs bK= ,;, B:s [ I( K
(6)
The one- and two-electron matrix elements in eqs. (4)-( 6) were expressed in terms of Gaussian basis functions in our previous work [ 61. The exponents of the universal GTFs used in our work are generated by using the GCHF method [ 9,111. The GCHF method is characterized by the representation of the one-electron functions as an integral transforms, viz. $i(l,(~)J;((~)da,
i=l,...,n,
(7)
where the $,,J and (Yare the generator functions, the weight functions, and the generator coordinate, respectively. The minimization of the energy value built with the one-electron functions, vi, in eq. (7) leads to the Griffm-Wheeler-HF (GWHF) equations, viz. I
]F(a, P) -eiS(a, i= 1, .... n .
P)
IA(P) V=O > (8)
The GWHF equations are integrated numerically by using the integral discretization (ID) technique [ 9,111 that preserves the integral character of the GCHF method. The ID technique is implemented with a relabelling of the generator coordinate space ((Yspace), to the a space by the following relation: Q+,
A>l.
An equally spaced N-point mesh {sl,} is selected so that one can obtain an adequate numerical integration range for the s, p and d symmetries of several atoms. The integration range is then characterized by a stating point, coin, an increment, M! and N (number of points). The scaling parameter, A, is the same (A=6.0) for all the calculations. The exponents of the universal Gaussian basis set are generated by the following discretization parameters. Symmetry: s, p and d; G,,,,,= -0.79; A.0=0.13; N= 32. As one can see from the discretization parameters, we have obtained a unique set of 32 exponents common for all three symmetries (s, p and d). These 32
exponents generated with the discretization parameters represent a single set of Gaussian exponents that can be used in atomic and molecular calculations involving the atoms He (Z=2) through Xe (Z= 54).
3. Results and discussion The DFC and DFC SCF calculations on the raregas atoms: He, Ne, Ar, Kr and Ze, and the alkalineearth atoms: Be, Mg, Ca and Sr, are performed using the UGBS as outlined above. The DFC SCF calculations are performed by omitting the Breit term, b,, from the DFB matrix. The finite nuclear model of uniform proton charge distribution and the kinetic balance constraints [2,3,13-l 51 are used in all the calculations. The atomic masses used for the He, Ne, Ar, Kr, and Xe atoms are: 4.0026, 20.183, 39.948, 83.80, and 13 1.30, respectively. For Be, Mg, Ca and Sr, the atomic masses used are 9.0 122, 24.3 12, 40.08, and 87.62, respectively. The speed of light used in our calculations is 137.0370 au.
Table I Total Dirac-Fock-Coulomb Atom
I January 1993
CHEMICAL PHYSICS LETTERS
Volume 201, number 1,2,3,4
(&,-),
UGBS size
Dirac-Fock-Breit
Tables 1 and 2 display the results of the calculated DFC (ED&, DFB (EDFB), and Breit interaction (I$,) energies for the rare-gas and the alkaline earth atoms, respectively. The EB denotes the variational Breit interaction energy computed as the difference between the EDFBand EDFC_The’variational Breit interaction energy, E,, is the level shift in the total SCF energy due to the inclusion of the low-frequency Breit interaction term in the SCF process. In the last column of tables 1 and 2, we tabulate the DFC energies calculated for the various atomic systems by using the Oxford finite difference numerical DiracFock program [ 161 #I The UGBS size displayed in tables 1 and 2 represents the number of Gaussian exponents for each s, p and d symmetry necessary to attain a DFC energy value with an accuracy of a few micro- or millihartrees as compared to the corresponding numerical finite difference results. The addition of higher ” OxfordMCDF program (ref. at Cambridge University.
(E DFB),and variational Breit
EDFC=’
(EB)energies of the rare-gas atoms (in hanree) E,”
EDFEn)
[ I6 1) modified by N.C. Pyper
Finite difference b,
(-&FC) He Ne Ar Kr Xe
18s 32~29~ 32~29~ 32s29p16d 32s29p18d
-2.861813335 - 128.6919310 -528.6837530 -2788.860655 - 7446.892977
-2.861749571 - 128.6752902 -528.5514300 -2787.434761 -7441.123179
a) This work (using the universal Gaussian basis set). b1 E DcFcomputed by the numerical finite-difference Dirac-Fock program (ref.
Table 2 Total Dirac-Fock-Coulomb hartree) Atom
UGBS size
(E,,),
Dirac-Fock-Breit
E ma’
(I&,),
0.000063763 0.0 166408 0.1323230 1.425894 5.769798
[ I6]! and footnote I ).
and variational
E rx=Ba
-2.86181335 - 128.691938 - 528.683840 -2788.86168 -7446.90018
Breit (I$,) energies of the alkaline-earth atoms (in
EBa’
Finite difference b, (&FC)
Be Mg Ca Sr
25s 32~29~ 32~29~ 32s29p16d
- 14.57589219 - 199.9350665 -679.7101239 -3178.080060
-14.57518897 - 199.9032396 -679.5191300 -3176.361450
‘) This work (using the universal Gaussian basis set). b, EoFc computed by the numerical finite-difference Dirac-Fock program (ref.
0.00070322 0.0318269 0.1909939 1.718560
-14.5758919 - 199.935083 -679.710276 -3178.01833
[ 16 1,and footnote 1). 39
Volume 20 I, number
I ,2,3,4
CHEMICAL PHYSICS LETTERS
exponents beyond the size given for each atom in tables 1 and 2 do not lead to any substantial improvement in the DFC and DFB energies. The results show that the total DFC energies computed with the UGBS are in very good agreement with the corresponding results obtained by the finite difference Dirac-Fock calculations for all the atoms except Xe. The UGBS is capable of reproducing the total DFC energies to within one millihartee (mhartree) or better as compared to the corresponding finite difference values. For the He and Be atoms, the agreement is within a few microhartrees ( lOA hartree). For heavier atoms, the results differ from the corresponding finite difference results by up to one mhartree. The results obtained by using the UGBS are in closer agreement with the corresponding numerical limits than those obtained previously by using the well-tempered Gaussian basis sets [ 5 1. For the heaviest Xe atom, the total energy lies seven mhartrees above the numerical limit and two mhartrees above the total DFC energy obtained by using the well-tempered Gaussian basis set [ 51.
5. Conclusions The relativistic universal Gaussian basis set developed in our work yields total DFC and DFB energies accurate to within a few mhartrees or better as compared to the corresponding numerical finite difference results for all the systems studied. The computed variational Breit interaction energies are convergent to at least live figures. The results obtained in our study clearly indicate that it is feasible to develop a universal Gaussian basis set for accurate atomic and molecular relativistic Dirac-Fock calculations.
Acknowledgement
This work has been supported by the Natural Sci-
40
1 January I993
ences and Engineering Research Council (NSERC) of Canada to GLM (Grant No. A3598), and by the CNPq (Brazilian Agency) to ABFDaS and GLM, and by the National Science Foundation to YI (Grant PHY-9008627). All the calculations reported in this work were performed on an IBM RISC/6000 workstation at Simon Fraser University. The authors thank Dr. Steve Kloster of the Academic Computing Services at Simon Fraser University and Dr. Chan Kyung Kim for their help with the calculation on the IBM RISC/6000. We gratefully acknowledge Kirsten Masse for wordprocessing the manuscript cheerfully and diligently.
References [ I] G.L. Malli, Chem. Phys. Letters 68 (1979) 529. [2] Y. Ishikawa, R. Baretty and R.C. Binning Jr., Chem. Phys. Letters I21 (1985) 130. [3] Y. Ishikawa and H.M. Quiney, Intern. J. Quantum Chem. Symp. 21 (1987) 523. [4] F.G. Werner and J.A. Wheeler, Phys. Rev. 109 (1958) 126. [5] Y. Ishikawa, Chem. Phys. Letters 166 (1990) 321; Y. Ishikawa, H.M. Quiney and G.L. Malli, Phys. Rev. A 34 (1991) 8270. [6] Y. Ishikawa, G.L. Malli and A.J. Stacey, Chem. Phys. Letters 188 (1992) 145. [7] D.M. Silver and W.C. Nieuwpoort, Chem. Phys. Letters 57 (1978) 421; D.M. Silver, S. Wilson and W.C. Nieuwpoort, Intern. J. Quantum Chem. 14 (1978) 635. [S] S. Wilson, Advan. Chem. Phys. 67 (1967) 439. [9]A.B.F.DaSilva,H.F.M.DaCostaandM.Trsic,Mol.Phys. 68 (1989) 433. [IO] G.L. Malli, A.B.F. Da Silva and Y. Ishikawa, Can. J. Chem. 70 (1992), in press. [I I ] J.R. Mohallem, R.M. Dreizler and M. Trsic, Intern. J. Quantum Chem. Symp. 20 ( 1986) 45. [ 121 Y.K. Kim, Phys. Rev. 154 (1967) 17. [ 131 H.M. Quiney, LP. Grant and S. Wilson, J. Phyr B 20 (1987) 1413. [ 141 R.E. Stanton and S. Havriliak, J. Chem. Phys. 81 (1984) 1910. [ 151K. Dyall, I.P. Grant and S. Wilson, J. Phys. B 17 (1984) 1201. [ 161 I.P. Grant, B.J. McKenzie, P.H. Norrington, D.F. Mayers and NC. Pyper, Computer Phys. Commun. 2 1(1980)207.