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Comparison of slater and contracted gaussian basis sets in SCF and CI calculations on H2O
Volwne
7. number
CHEMICAL
3
COMPARISON
OF
SLATER
IN
SCF
R. P. HOSTEW, Battelle
Memorial
PHYSICS
AND
AND
CI
L November
LETTERS
CONTRACTED
GAUSSIAN
CALCULATIONS
ON
BASIS
505 K0zg Avenue,
Columbus,
SETS
H20
R. R. GILMAN *, T. H. DUNNING Jr., A. PIPANO Institute,
L970
Ohio 43202,
**
USA
and I. SHAV-ITT Battelle Memorial Insfitute, 505 King Avenue, Columbus, Ohio 13201, US% and Department of Citemist~, The Ohio State University, Columbu, Ohio 43210, Received
22
US.4
September 1970
series of SCF and CI calculations for the electronic ground state of H,O have been carried out with different 14-function basis sets; one a Clementi-type double zeta ST0 l%sis and the other a contracted GTO set. The results obtained with the two bases are compared and analyzed in terms of inner and outer shell correktion and of the contributions of different IeveIs of excitation. A
two
A considerable amount of experience is accumulating concerning self-consistent field and configuration interaction calculations of moleculzr wave functions with both Slater-type and contracted gaussian basis sets. It seems, however, that there is little published information on a comparison of the two tmes of basis functions in equivalent calculations, i.e., calculations in which the number of independent basis functions is the same, and in which the same procedures (except for the integral calculation, of course) are used. In the present note we report one such comparison involving double-zeta SCF and CI calculations on the ground state of the water molecule near the experimental equilibrium geometry. The Slater basis used‘in the calculations is described in table 1. It is composed of the oxygen atom double-zeta basis of Clementi [l], with exponents rounded off to two decimal places, and two hydrogen 1s orbitals chosen on the basis of previous calculations on similar systems, without further optimization. Guidotti and Salvetti [2], also using Clementi’s double-zeta set for the oxygen basis but different values for the hydrogen Is exponents, obtained an SCF energy * Present address: Graduate School Computing Center, University of Colorado, Botider, Colorado
80304, USA. ** Present address: Department
The Johns IiopkIns University, of Chemistry. Baltimore, Maryland
21218, USA.
--
Atom 0 0 0 0 0 0 H H
Table L ST0 basis set for Et20 __.--_-Exponent -Type a) --__^ -- -__ IS
Is 2s 2s
$ 1s
__---.-
a) p-type includes all three components 2pl).
LO.LL 7.06 2.62 1.63 3.6s L.65 L-5
1.2
(2p,,
2pJ,~
0.0002 hartree lower than that reported here. Scarzafava [3] optimized the double-zeta basis for the water molecuZe with a (Is, 2s) pair for hydrogen instead of (as, ls’), and obtained an SCF energy 0.002 hartree Iower than in the present calculation. It is felt, however, thst the method of selection used here for the Slater basis, particularly with regard to the use of atom-optimized basis functions for oxygen, is quite comparable to the way the gaussian basis was chosen. It is doubtfui that a complete re325
.
CHEMICAL PHYSICS LETTERS
Volume 7, number 3
--___
---~ Calculation
; 3 4 5
a)
-__-__.-_._________ No. of configurations
-------
SCF(1-k 2 excitations) CI CI (l+ 2 minus inner shell) CI (l+ 2+3 minus inner shell) CI (l+ 2+3+4 minus inner Shell)
The geometry
units.
-------.--_
- -
3611 224 1558
E(ST0)
E(GT0)
-76.005 -76.144588 253 ~76.129 203 -76.131105
-76.148161 -76,069 256 -76.135406 -76.136481
O.QO5376
-76.142 247
0.007 262
6779 -76.134985 _....__.-___ ._--.--_-_____.____.______
is RGI, --_ :‘: 1.811’1 boil:, LHOH = 104.45 O, taken from ref. [12]. All energies
optimization of both bases would alter the conclusions obtained in this study to a very significant degree. The gaussian basis set used is the ‘optimally contracted’ [4s 2p/2s] basis of Dunning [4]. The oxygen functions of this basis were contracted from the atom-optimized (9s 5p) set of Huzinaga [5]; the hydrogen functions were scaled by a factor of 1.2 (exponents multiplied by 1.44) relative to those tabulated by Dunning 141. Both the Slater and gaussian bases thus have 14 independent functions consisting of two core s-functions, two valence s-functions and two p-functions on the oxygen, and two s-functions on each of the hydrogen atoms. The calculations performed with each basis included closed-shell restricted SCF and a number of CI calculations in which the canonical occupied and virlual SCF orbitals were used. Single- and double-excitation CI was carried out in full, while in another series of Cl calculations all excitations from the o-xygen inner shell orbital (lal) were omitted in order to compare the effectiveness of the two bases for representing inner shell and valence shell correlation 163. In this second series of calculations the contributions of all tri- and tetra-excited configurations were also examined. The energy results are summarized in table 2. The computation times (on the CDC 6400 computer) for the oneand two-electron integrals over atomic orbitals were 569 seconds for the ST0 basis (using a
-
_____
Calculation No. 2 3 4 5
0.003 003 0.004 573 0.006 203
are in hartree atomic
a)
Excitation levels
No. of non-zero matrix elements
Time for code tape
Time for H matrix
Time for eigenvalue
1+2 1+ 2 minus Inner shell l+ 2+ 3 minus inner shell lc2+3+4 minus inner sheiI
17 782 8093 146 543 1204 730
51 23 487 5177
36 17 307 2 870
11 5 149 1215
------
a) AII times are in seconds of central processor
326
E(STO)-E(GT0)
program written by Stevens [7]) and 49 seconds for the GTO set (using a modification, written by Basch [8], of the Polyatom system [9]). The time needed for the SCF iterations in both cases was about 30 seconds, while the transformation of the A0 integrals to the basis of the SCF orbitals, in preparation for the CI calculation, took 23 seconds in each case. The times for the other stages of the CI calculations are summarized in table 3. (The ‘code tape’ referred to in table 3 contains a compact code from which the formulas for the matrix elements of the hamiltonian are very easily determined in the next step [lo, 111; it need not be recomputed for calculations which differ only in values of numerical parameters.) The energy results are analyzed in tables 4 and 5. In table 4 the inner and outer shell contributions to the computed correlation energy (for single and double excitations only) are compared for the two basis sets; in this comparison it was convenient to include the mixed excitation contributions (one inner shell electron and one from the outer shell) with the inner shell values. The gaussian basis did somewhat more poorly for the inner shell correlation energy than the ST0 basis, but this was compensated by a larger contribution to the outer shell correlation. In fact, the gaussians gave nearly as much improvement’over the ST0 basis in the valence shell as the STO’s did oLer the gaussians in the inner shell, making the total correlation energy
Table 3 Ccmputarion times for various CI calculations
-.-.-
1970
Table 2 Energies for the ground state of H20 by various methods a) Type
NO.
1 November
_-
time on the CDC 6400 computer.
Volume ‘7. number 3
CHEMICAL PHYSICS LETTERS
1 Xoveml& 1970
Table 4 Analysis of correlation contributions of single and double excitation configurations in terms of inner and outer -----
..- -.__-._
Contribution ___
._ ---.
shells
.._______ ___.,_ Obtained as
__.- _ _...______..__._._
inner shell outer shell
‘,;; ; 1;;
total
(2) -. (1)
a)
4E(STO) ._.___._..___-___.._ -0.015385 -0.123 950 ~-0.139 335
bE(ST0) - L’Z(GTO) 4 E(GT0) .._... -______..___..._..._____ .._. _. ____.__ -0.012 755 -0.126 150
-0.002 63O +0.002200
-0.138 905
-0.000 431
’
a) &lived excitations (one electron from the inner shell, the other from the outer shell) are incIu<.ed in the inner shell contributions in &is table. All quantities are in hartree atomic units.
b) Compare table 2.
computed (through all doubles) nearly the same for both bases. In the CI minus inner shell calculations (table 5), the tri-excited configurations contributed more with the ST0 basis, though this contribution was small in both bases (I-lf‘% of the calculated correlation energy). The tetraexcited configurations were considerably more important than the triples in both bases (3-40/o of the total correlation energy), and contributed appreciably more in the gaussian basis than in the ST0 set, resulting in a total excess correlation energy of 0.003 hartree for the GTO basis relative to the STWs. It is expected that this difference would be substantially reduced lf inner shell excitations had been included. Because the energy differences are small and the optimum choice of basis functions (orbltal exponents, contraction coefficients, etc.) not fully investigated, !zhe relative merits of the two sets, aside from economic considerations
(for the integral evaluations), are still debatable. It is clear, however, that properly chosen ST0 and contracted GTO basis sets with comparable numbers of independent basis functions are about equally good for both SCF and CK calculations,
at least by the tot21 energy
criterion.
They may, of course, differ more slgnlficantly in some other calculated molecular properties. In fact, for the basis sets used Sere, based on the SCF and the CI calculations, the gaussian set appears to give a better description of the valence shell, which is the region of primary interest in the determination of potential surfaces, equilibrium geometries, excitation energies, etc., by ab initio calculations. We wish to thank Dr. R. M. Stevens and Dr. Harold Basch for the use of their integral pro-
grams.
Table 5 Contributions of various levels of excitation to the calculated “CI minus inner shells” energies a) ____-__.____-__--___ _~-----_____--_- _____ Obtained as
Excitation level
cti~$%%YnoO:
1+2 3 4 ~--
total
(3) - (1) (4) - (3) (6) - (4) ____
__..__!?L1.(?L____.
b)
A E(ST0)
4 E(GT0)
-0.123 950 -0.001902 -0.003 880
-0.126 150 -0.001075 -0.005 766
_.._-o-‘.Y?L-.-_
4 E(STO)- ti(GTO)
~~1~~~1_._____._-+0:003
+o.ooz 200 -0.000 827 +o.ooL 886 -.--25g ---.-_-_
a) All quantities in hartree atomic units. b) Compare table 2.
REFERENCES VI E. Clementi, J. Chem. Phys. 40 (1964) 1944. 121C. Guidotti and 0. Salvettii Theoret. Chim. Acta 10 (1968) 454. 131 E. A. Scarzafava, Thesis, Indiana University (1969).
[4] T. H. Dunning, J. Chem. Phys., to be pubkhed. 151 S.Huzinza. J. Chem. Phvs. 42 (19651 1293. i6i A. Pipang, k. R. Gilman a&d I. ShHvitt~ Chem. Phys. Letters 5 (1970) 285. [7] R. M. Stevens, Quantum Chemistry Program Exchange, Indiana University (1970) program 161.
327
Volume
i,
number
3
CHEMICAL
[8] H. Basch. privnte communication. [9] I. G. Csizmadia, M. C. Harrison, J. W.Moskowitz. S. Seung, B. T. Sutcliffe and M. P. Barnett. Quanturn Chemistry Program Exch‘ange, Indiana University (1964) program 47.
328
PHYSICS [lo]
LETTERS
1 November
C.M. Reeves, Thesis, Cambridge University (1957). [llj A. Pipano and 1. Shavitt, to be published. [12] S. Aung, R. M. Pitzer and S. I. Ghan, J. Chem. Phys. 49 (1968) 2071.
1970
7. number
CHEMICAL
3
COMPARISON
OF
SLATER
IN
SCF
R. P. HOSTEW, Battelle
Memorial
PHYSICS
AND
AND
CI
L November
LETTERS
CONTRACTED
GAUSSIAN
CALCULATIONS
ON
BASIS
505 K0zg Avenue,
Columbus,
SETS
H20
R. R. GILMAN *, T. H. DUNNING Jr., A. PIPANO Institute,
L970
Ohio 43202,
**
USA
and I. SHAV-ITT Battelle Memorial Insfitute, 505 King Avenue, Columbus, Ohio 13201, US% and Department of Citemist~, The Ohio State University, Columbu, Ohio 43210, Received
22
US.4
September 1970
series of SCF and CI calculations for the electronic ground state of H,O have been carried out with different 14-function basis sets; one a Clementi-type double zeta ST0 l%sis and the other a contracted GTO set. The results obtained with the two bases are compared and analyzed in terms of inner and outer shell correktion and of the contributions of different IeveIs of excitation. A
two
A considerable amount of experience is accumulating concerning self-consistent field and configuration interaction calculations of moleculzr wave functions with both Slater-type and contracted gaussian basis sets. It seems, however, that there is little published information on a comparison of the two tmes of basis functions in equivalent calculations, i.e., calculations in which the number of independent basis functions is the same, and in which the same procedures (except for the integral calculation, of course) are used. In the present note we report one such comparison involving double-zeta SCF and CI calculations on the ground state of the water molecule near the experimental equilibrium geometry. The Slater basis used‘in the calculations is described in table 1. It is composed of the oxygen atom double-zeta basis of Clementi [l], with exponents rounded off to two decimal places, and two hydrogen 1s orbitals chosen on the basis of previous calculations on similar systems, without further optimization. Guidotti and Salvetti [2], also using Clementi’s double-zeta set for the oxygen basis but different values for the hydrogen Is exponents, obtained an SCF energy * Present address: Graduate School Computing Center, University of Colorado, Botider, Colorado
80304, USA. ** Present address: Department
The Johns IiopkIns University, of Chemistry. Baltimore, Maryland
21218, USA.
--
Atom 0 0 0 0 0 0 H H
Table L ST0 basis set for Et20 __.--_-Exponent -Type a) --__^ -- -__ IS
Is 2s 2s
$ 1s
__---.-
a) p-type includes all three components 2pl).
LO.LL 7.06 2.62 1.63 3.6s L.65 L-5
1.2
(2p,,
2pJ,~
0.0002 hartree lower than that reported here. Scarzafava [3] optimized the double-zeta basis for the water molecuZe with a (Is, 2s) pair for hydrogen instead of (as, ls’), and obtained an SCF energy 0.002 hartree Iower than in the present calculation. It is felt, however, thst the method of selection used here for the Slater basis, particularly with regard to the use of atom-optimized basis functions for oxygen, is quite comparable to the way the gaussian basis was chosen. It is doubtfui that a complete re325
.
CHEMICAL PHYSICS LETTERS
Volume 7, number 3
--___
---~ Calculation
; 3 4 5
a)
-__-__.-_._________ No. of configurations
-------
SCF(1-k 2 excitations) CI CI (l+ 2 minus inner shell) CI (l+ 2+3 minus inner shell) CI (l+ 2+3+4 minus inner Shell)
The geometry
units.
-------.--_
- -
3611 224 1558
E(ST0)
E(GT0)
-76.005 -76.144588 253 ~76.129 203 -76.131105
-76.148161 -76,069 256 -76.135406 -76.136481
O.QO5376
-76.142 247
0.007 262
6779 -76.134985 _....__.-___ ._--.--_-_____.____.______
is RGI, --_ :‘: 1.811’1 boil:, LHOH = 104.45 O, taken from ref. [12]. All energies
optimization of both bases would alter the conclusions obtained in this study to a very significant degree. The gaussian basis set used is the ‘optimally contracted’ [4s 2p/2s] basis of Dunning [4]. The oxygen functions of this basis were contracted from the atom-optimized (9s 5p) set of Huzinaga [5]; the hydrogen functions were scaled by a factor of 1.2 (exponents multiplied by 1.44) relative to those tabulated by Dunning 141. Both the Slater and gaussian bases thus have 14 independent functions consisting of two core s-functions, two valence s-functions and two p-functions on the oxygen, and two s-functions on each of the hydrogen atoms. The calculations performed with each basis included closed-shell restricted SCF and a number of CI calculations in which the canonical occupied and virlual SCF orbitals were used. Single- and double-excitation CI was carried out in full, while in another series of Cl calculations all excitations from the o-xygen inner shell orbital (lal) were omitted in order to compare the effectiveness of the two bases for representing inner shell and valence shell correlation 163. In this second series of calculations the contributions of all tri- and tetra-excited configurations were also examined. The energy results are summarized in table 2. The computation times (on the CDC 6400 computer) for the oneand two-electron integrals over atomic orbitals were 569 seconds for the ST0 basis (using a
-
_____
Calculation No. 2 3 4 5
0.003 003 0.004 573 0.006 203
are in hartree atomic
a)
Excitation levels
No. of non-zero matrix elements
Time for code tape
Time for H matrix
Time for eigenvalue
1+2 1+ 2 minus Inner shell l+ 2+ 3 minus inner shell lc2+3+4 minus inner sheiI
17 782 8093 146 543 1204 730
51 23 487 5177
36 17 307 2 870
11 5 149 1215
------
a) AII times are in seconds of central processor
326
E(STO)-E(GT0)
program written by Stevens [7]) and 49 seconds for the GTO set (using a modification, written by Basch [8], of the Polyatom system [9]). The time needed for the SCF iterations in both cases was about 30 seconds, while the transformation of the A0 integrals to the basis of the SCF orbitals, in preparation for the CI calculation, took 23 seconds in each case. The times for the other stages of the CI calculations are summarized in table 3. (The ‘code tape’ referred to in table 3 contains a compact code from which the formulas for the matrix elements of the hamiltonian are very easily determined in the next step [lo, 111; it need not be recomputed for calculations which differ only in values of numerical parameters.) The energy results are analyzed in tables 4 and 5. In table 4 the inner and outer shell contributions to the computed correlation energy (for single and double excitations only) are compared for the two basis sets; in this comparison it was convenient to include the mixed excitation contributions (one inner shell electron and one from the outer shell) with the inner shell values. The gaussian basis did somewhat more poorly for the inner shell correlation energy than the ST0 basis, but this was compensated by a larger contribution to the outer shell correlation. In fact, the gaussians gave nearly as much improvement’over the ST0 basis in the valence shell as the STO’s did oLer the gaussians in the inner shell, making the total correlation energy
Table 3 Ccmputarion times for various CI calculations
-.-.-
1970
Table 2 Energies for the ground state of H20 by various methods a) Type
NO.
1 November
_-
time on the CDC 6400 computer.
Volume ‘7. number 3
CHEMICAL PHYSICS LETTERS
1 Xoveml& 1970
Table 4 Analysis of correlation contributions of single and double excitation configurations in terms of inner and outer -----
..- -.__-._
Contribution ___
._ ---.
shells
.._______ ___.,_ Obtained as
__.- _ _...______..__._._
inner shell outer shell
‘,;; ; 1;;
total
(2) -. (1)
a)
4E(STO) ._.___._..___-___.._ -0.015385 -0.123 950 ~-0.139 335
bE(ST0) - L’Z(GTO) 4 E(GT0) .._... -______..___..._..._____ .._. _. ____.__ -0.012 755 -0.126 150
-0.002 63O +0.002200
-0.138 905
-0.000 431
’
a) &lived excitations (one electron from the inner shell, the other from the outer shell) are incIu<.ed in the inner shell contributions in &is table. All quantities are in hartree atomic units.
b) Compare table 2.
computed (through all doubles) nearly the same for both bases. In the CI minus inner shell calculations (table 5), the tri-excited configurations contributed more with the ST0 basis, though this contribution was small in both bases (I-lf‘% of the calculated correlation energy). The tetraexcited configurations were considerably more important than the triples in both bases (3-40/o of the total correlation energy), and contributed appreciably more in the gaussian basis than in the ST0 set, resulting in a total excess correlation energy of 0.003 hartree for the GTO basis relative to the STWs. It is expected that this difference would be substantially reduced lf inner shell excitations had been included. Because the energy differences are small and the optimum choice of basis functions (orbltal exponents, contraction coefficients, etc.) not fully investigated, !zhe relative merits of the two sets, aside from economic considerations
(for the integral evaluations), are still debatable. It is clear, however, that properly chosen ST0 and contracted GTO basis sets with comparable numbers of independent basis functions are about equally good for both SCF and CK calculations,
at least by the tot21 energy
criterion.
They may, of course, differ more slgnlficantly in some other calculated molecular properties. In fact, for the basis sets used Sere, based on the SCF and the CI calculations, the gaussian set appears to give a better description of the valence shell, which is the region of primary interest in the determination of potential surfaces, equilibrium geometries, excitation energies, etc., by ab initio calculations. We wish to thank Dr. R. M. Stevens and Dr. Harold Basch for the use of their integral pro-
grams.
Table 5 Contributions of various levels of excitation to the calculated “CI minus inner shells” energies a) ____-__.____-__--___ _~-----_____--_- _____ Obtained as
Excitation level
cti~$%%YnoO:
1+2 3 4 ~--
total
(3) - (1) (4) - (3) (6) - (4) ____
__..__!?L1.(?L____.
b)
A E(ST0)
4 E(GT0)
-0.123 950 -0.001902 -0.003 880
-0.126 150 -0.001075 -0.005 766
_.._-o-‘.Y?L-.-_
4 E(STO)- ti(GTO)
~~1~~~1_._____._-+0:003
+o.ooz 200 -0.000 827 +o.ooL 886 -.--25g ---.-_-_
a) All quantities in hartree atomic units. b) Compare table 2.
REFERENCES VI E. Clementi, J. Chem. Phys. 40 (1964) 1944. 121C. Guidotti and 0. Salvettii Theoret. Chim. Acta 10 (1968) 454. 131 E. A. Scarzafava, Thesis, Indiana University (1969).
[4] T. H. Dunning, J. Chem. Phys., to be pubkhed. 151 S.Huzinza. J. Chem. Phvs. 42 (19651 1293. i6i A. Pipang, k. R. Gilman a&d I. ShHvitt~ Chem. Phys. Letters 5 (1970) 285. [7] R. M. Stevens, Quantum Chemistry Program Exchange, Indiana University (1970) program 161.
327
Volume
i,
number
3
CHEMICAL
[8] H. Basch. privnte communication. [9] I. G. Csizmadia, M. C. Harrison, J. W.Moskowitz. S. Seung, B. T. Sutcliffe and M. P. Barnett. Quanturn Chemistry Program Exch‘ange, Indiana University (1964) program 47.
328
PHYSICS [lo]
LETTERS
1 November
C.M. Reeves, Thesis, Cambridge University (1957). [llj A. Pipano and 1. Shavitt, to be published. [12] S. Aung, R. M. Pitzer and S. I. Ghan, J. Chem. Phys. 49 (1968) 2071.
1970