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Gaussian expansions of orbital products for the evaluation of two-electron integrals
Volume 6, number 6
CHEMICALPHYSICSLETTERS
GAUSSIAN THE
EXPANSIONS
EVALUATION
OF
OF
ORBITAL
TWO-ELECTRON
L5 March 1971
PRODUCTS
FOR
INTEGRALS
P. D. DACRE and M. ELDER Deparhteni
of Citemistq,
The University.
Received 23 December
Sheffield S3 7HF. UK 1970
A method is described for evaluating multicenter integrals over contrscted gaussim-trye orbit& by the Use of gaussian expansion Of orbital products. The expansions are determined by the method of nonlinear least swares with constraints. There ia no restriction tiponthe symmetry of the orbital product and the method is applicable to all products arising from s , p and d-type orbit&. Results are given to indicate the accuracy of the method.
Multicenter electron repulsion integrals for Slater-type orbitals (STO’s) are very time-consuming to evaluate so that most ab initio molecular calculations have been carried out using linear combinations of gaussian-type orbitals (CGTO’S). However, for large molecules even the use oi CGTO’s leads to heavy demands on computing time. One possible device for obtaining integrals more rapidly is to use gaussian expansions for the orbital products which occur in such integrals. Such a method has been described by Monkhorst and Harris [l] for the evaluation of integrals for s-type STO’s. They used constrained least-squares expansions in s-type gausSian functions to represent the orbital products. Once the expansions are determined the required integrals may be rapidly evaluated in terms of the gaussian integrals arising from the expansions. We have extended this method to the computation of gaussian expansions for a general twocenter orbital product arising from CGTO’s of s, p or d symmetry, and present here some preliminary results. A set of computer programs make it’straightforward to compute for a given molecule. a) A good starting approximation to the desired expansion for each of the symrpetry-unique orbital products. b) A least-squares fit for each orbital product, subject td monopole, dipole and quadrupoIe moments being constrained to the exkct values for the ml CGTO expansion. : c) The complete set _of unique two-electron ’ ‘lintegrals arising from these symmetry-unique orb+ products; : ./-. -. : ._
Step (a) is important for the foIlowing reason. A Cartesian CGTO with centre R has the form
C cixzymZn
exp (Qi: r - R)2] )
i
where the ci are coefficients, ai the exponents, r a radius vector, x, y, t coordinates with origin R, and I, )n and n the cartesian exponents, are integers. Because the product of two gausSian functions is again a gaussian function, with centre on the line joining the two centres, the orbital product QO formed from two CGTO’s is itself a sum of gaussian functions. The carteSian exponents of each term assume integral values between zero and the sum of the corresponding exponents in- the parent CGTO’s. If a0 is to be represented as a short expansion Cl of gaussian functions then the Cartesian exponents of each term must be specified, in addition to the values of the adjusfabfe parameters (the coefficients, gaussian exponents and the position of the centres along the line joining the two CGTO’s centres). We have used the functions which occur in the full expansion of a0 as a basis for the determination of the trial 0. The function with the greatest overlap with fi, is chosen as the first term of Q. The remaining functions are then Schmidt orthogonalised with respect to the chosen functions. Successive functions are chosen from_-the remaining orthogonalised functions by repetition of the procedure until either enough fuidions are obtained, or the mean square deviation. frqm a.0 is sufficiently small. When a satisfactory t-ial expansion for 510 has ” been obtained, the-parameters are refined by 625
Volume 8, number 6
CHEMICALPHYSICSLETTERS-
minimising
Table1 Expansion
i 15March1971
lengths andmean-square deviations
OrbitaIproducta)
N
nl
9- 4 9 4 1s 4 27 10 27 5 18 3 36 6 36 6 72 8
s3,sl 89, 83 px3, sl PX3, PXl PX5. PXl pr6,.sl PY5,pxl d&Fe), sl d*z(Fe),pyl
El
"z
.1(-Z) b)
6
.2(-2) .5(-Z) .Z(-2) .2(-Z) .1(-l) .5(-2) .5(-2) .3(-Z)
1: 16 10 4 13 12 17
y2 .1(-4\_ ‘.l(-5) .l(-3) .l(-4) .7(-3) .2(-2) .2(-2) .4(-3) .31-3)
3 Nis the number of terms in the exact expansion of the product. ni, Ei are the number of terms and mean square deviation respectively. for expansion set i. b) In this and subsequent tables 0.1 x ldr is written ae A(n).
by conventional non-linear least-squares methods [2]. The final parameters are then further modified to satisfy the constraints by a nonlinear least squares with non-linear constraints procedure. We found, as in [l], that the constraints greatly improve the accuracy of the integrals obtained. To test the procedure.we computed a selection of orbital products for ferrocene (Fe.0, 0, 0; Cl 2.297, 9.0, 3.096 (au); C2 -Cl6 generated by successive application of the S : e(Z) symmetry operation. The basis CGTO’s were 3GT0 leastsquares fits to Slater 1s (exponent 5 = 1.0);
Table 2 Two-‘and three-centre integrals (in hartrees) computed from the orbital product expansions of table1 Exact value
Integral a) 93 pr3 Pd PX5 PX5 PY5 PY9 PY8 dxz(Fe)
Sl Sl PXl PXl P%l
s3 a3 PX3 PX3 PX3 PX3 s3 I=3 s5
DXl
sl al Z';' -pxl sl Sl
.84174(-l) .39705(-l) .41634(-Z) -.41898(-2) .10479(-2) -.13665(-2) -.15127(-3) .14399(-4) -.71515(-4)
Error @)
set !. .84363(-l) .39642(-l) .4X35(-2) -.41461(-2) .10535(-Z) -.13451(-2) -.15344(-3) .14421(-4) -.72093(-4)
0.23 -0.17 -0.34 3.03 0.54 1.54 -0.14 0.15 -0.82
Set 2
Error (%)
.84172(-l) .39693(-l) .41639(-2) -.41768(-2) .10436(-Z) -.13584(-2) -.15136(-3) .14397(-4) -.71431!-4)
0.00 -0.03 0.01 0.32 -0.43 0.62 -0.06 -0.01 0.11
a) CIb c d means{a(l)b(l)~~~c(2)d(2)drldT2
Table 3 Four-csntre integrals (in hartrees) computed from tineorbitaI produot expa&iona of table 1 Integral s5
B?
w3 -pr3 w3 px6. px8 pr8 PY5. PY5
&z(Fe)
&z(Fe)
d=+e)
Exact value s3
s7 sl pxl s4 Ix1 s7 sl px6. $3 ~6 px5 prl -. a9 pxl pr8 :al, -97. :- al ~8 ,’ pyl : 84
Sl 85 S8
s5 92 pxl prl. s3 w6 ;6
: a2
.45491(-l) .12134f-1). --.13301(-2) -.13514(-2) .28751(-3) .51661(-4) .19646(-3) -+8590(-2) .80332(-4) .37984(-2) “-.13948(-3)' -.80384(-i) :
Error fi)
Set 1 .45747(-l)
0.57
.12075(-l) -.13337(-2) -.13866{-2) .28987<-3) ‘.51629<:4) .19653{-3) -.18601+2) ..8033'7:-4) .382?1;;-2)
-.J3321+3)
-.819~~~-4),
Set2 .45511(-i)
_.,I:_
:
--.
‘.. _’
0.04
.12127(-l) -0.49 -0.05 -0.23 -.13302(-2) 0.00 -.13531(-2) 2.66 -0.13 0.84 .28735(-3) -0.05 0.60 .51660(-4) 0.00 0.03 .19650(-3) '_ 8-x" 0.09 -.18576(-2) .: -0.00 .80322(-4), -0.01 ,.0.60 _ .3&002(-Z) 0.05 ‘- 0.91 :.:.13853(-3) 0.68 -.60661('4) -0.35 :_-1.93
. . _-. :
.
._
., &;
Error fi)
_.
,.:
Volume 8, number 6
CHEMICAL PHYSICS LETTERS
Table 4 SCF results for CH4 comparing the orbItal product expansion method with conventional caicuiations Orbital expansion Product expansion (8.. . 1%GTO) (S-GTO) (L-STO)
Orbital energy
1Al
-11.341
-11.339
2Al lT2
-0.961
-0.962
-0.574
-0.575
3Al
0.426
0.434
n’2
0.503
0.503
Total energy Populations
-40.0567
-40.0601
H
0.870
0.869
C(ls)
1.996
1.996
CGa)
1.344
1.345
CG%)
1.060
1.061
-40.0606
2p (5 = 2.0); 3d (5 = 3.0) [3]. The nomenclature for example px3 is the 2pn orbital on C3. The tables show the results for two sets of expansions for representative orbital products. It is apparent from table 1 that the expansions converge more slowly with increasing complexity of the orbital product. As the separation between orbital centres increases the number of terms required for a given mean-square deviation decreases (compare px3, pxl and pa5, pxl). The ‘exact’ integral values in tables 2 end 3 are those calculated by the full expansion using contracted gaussians. Gur results show that even with the poorer expansion set it is possible consistently to obtain reliable values of integrals over a wide range of character and magnitude. The results obtained with the second set are very encouraging and are probably accurate enough for most molecular calculations. For example we have observed [5] that a calculation on benzene in which all integrals of magnitude less than 0.01 hartrees are given a random error lying between -2% and +2%, gives a total energy within 0.01 hartrees of the value obtained with the correct integrals, and one-electron orbital energies to 0.002 hartrees. We anticipate that this method of integral evalut&ion will have its greatest application in ‘ab initio’ calcuIations on large molecular systems. As the system becomes larger the average sepa.ration between atomic- centres increases and hence the average =pansion length will decrease: Moreover because an ST0 can’usualiy be adequ-
used in the tables is self-explanatory,
L5 March
1971
ately represented by an expansion of between 6 and 8 gaussian functions [4] the procedure is also applicable to the calculation of integrals for STO’s as it is especially suited to calculations using large
CGTO’s
.for a basis set.
Thus it is
probable that integrals computed in this manner would be sufficiently accurate for use in a mixed basis calculation [6,7]. As a further test we have used the orbital product expansion method to repeat the methane calculation of KIessinger and McWeeny [a]. The same parameters were used (carbon; S(k) = 5.7, [(Bs) = ((2~) = 1.625; hydrogen; (;&a) = 1.0; C-H = 2.067419 au) and each basis orbitaL was represented by en 8-GTO expansion taken from Huzinaga [4]. The 12 unique orbita] products were represented by moment-constrained GTO expansions of between 8 and I5 terms, and used to compute the two-electron integrals. Gneelectron terms were computed accurately over the 8-GTO orbital expansions. The results, listed in table 4, indicate that an expansion length equivalent to using 3-4 GTO’s per orbital yields closely comparable energies to the comparison 8-GTO calculation, and to the singleSlater orbital result. A complete account of the method will be given elsewhere. We are at present appLying the technique to some representative LCAO MO calculations on polyatomic systems with 50-100 electrons. We thank Professor R. McWeeny and Professor R-Mason for helpful discussions and encouragement. We are grateful to the Science Research Council for financial support of this project.
REFERENCES
[l] J. Monkhorst and F. E. Harris,
Chem.Phy~. titters
3 (1969) 537. [Z] W. C. Hamilton, Statistfcs in physfcai science (Ronald Press, New York, L964). [3] R. F. Stewart,
J. Chem. Phys. 50 (l969) 2485.
141K.Odhato, II. T&eta and S.Hxzfnaga, .I. Phya. Sot. Japan 21 (1966) 2306. [5] P. D. Dacrc and M. ELder, unpubIf&cd. [6] D. 3. Cook, P.C.HoHfs and R.McWeeny, Mol. Phye. 13 (19671 553. [‘I] D. H. Cook, P. D. Dacre, .I. L. Dcdda a.ndM. Elder, Theoret. C&II. Acta, tu he pubiiahed. [S] MKIcssinger and R. McWeeny; J.Chem. Phya. 42 (1965) 3343. -_
627
CHEMICALPHYSICSLETTERS
GAUSSIAN THE
EXPANSIONS
EVALUATION
OF
OF
ORBITAL
TWO-ELECTRON
L5 March 1971
PRODUCTS
FOR
INTEGRALS
P. D. DACRE and M. ELDER Deparhteni
of Citemistq,
The University.
Received 23 December
Sheffield S3 7HF. UK 1970
A method is described for evaluating multicenter integrals over contrscted gaussim-trye orbit& by the Use of gaussian expansion Of orbital products. The expansions are determined by the method of nonlinear least swares with constraints. There ia no restriction tiponthe symmetry of the orbital product and the method is applicable to all products arising from s , p and d-type orbit&. Results are given to indicate the accuracy of the method.
Multicenter electron repulsion integrals for Slater-type orbitals (STO’s) are very time-consuming to evaluate so that most ab initio molecular calculations have been carried out using linear combinations of gaussian-type orbitals (CGTO’S). However, for large molecules even the use oi CGTO’s leads to heavy demands on computing time. One possible device for obtaining integrals more rapidly is to use gaussian expansions for the orbital products which occur in such integrals. Such a method has been described by Monkhorst and Harris [l] for the evaluation of integrals for s-type STO’s. They used constrained least-squares expansions in s-type gausSian functions to represent the orbital products. Once the expansions are determined the required integrals may be rapidly evaluated in terms of the gaussian integrals arising from the expansions. We have extended this method to the computation of gaussian expansions for a general twocenter orbital product arising from CGTO’s of s, p or d symmetry, and present here some preliminary results. A set of computer programs make it’straightforward to compute for a given molecule. a) A good starting approximation to the desired expansion for each of the symrpetry-unique orbital products. b) A least-squares fit for each orbital product, subject td monopole, dipole and quadrupoIe moments being constrained to the exkct values for the ml CGTO expansion. : c) The complete set _of unique two-electron ’ ‘lintegrals arising from these symmetry-unique orb+ products; : ./-. -. : ._
Step (a) is important for the foIlowing reason. A Cartesian CGTO with centre R has the form
C cixzymZn
exp (Qi: r - R)2] )
i
where the ci are coefficients, ai the exponents, r a radius vector, x, y, t coordinates with origin R, and I, )n and n the cartesian exponents, are integers. Because the product of two gausSian functions is again a gaussian function, with centre on the line joining the two centres, the orbital product QO formed from two CGTO’s is itself a sum of gaussian functions. The carteSian exponents of each term assume integral values between zero and the sum of the corresponding exponents in- the parent CGTO’s. If a0 is to be represented as a short expansion Cl of gaussian functions then the Cartesian exponents of each term must be specified, in addition to the values of the adjusfabfe parameters (the coefficients, gaussian exponents and the position of the centres along the line joining the two CGTO’s centres). We have used the functions which occur in the full expansion of a0 as a basis for the determination of the trial 0. The function with the greatest overlap with fi, is chosen as the first term of Q. The remaining functions are then Schmidt orthogonalised with respect to the chosen functions. Successive functions are chosen from_-the remaining orthogonalised functions by repetition of the procedure until either enough fuidions are obtained, or the mean square deviation. frqm a.0 is sufficiently small. When a satisfactory t-ial expansion for 510 has ” been obtained, the-parameters are refined by 625
Volume 8, number 6
CHEMICALPHYSICSLETTERS-
minimising
Table1 Expansion
i 15March1971
lengths andmean-square deviations
OrbitaIproducta)
N
nl
9- 4 9 4 1s 4 27 10 27 5 18 3 36 6 36 6 72 8
s3,sl 89, 83 px3, sl PX3, PXl PX5. PXl pr6,.sl PY5,pxl d&Fe), sl d*z(Fe),pyl
El
"z
.1(-Z) b)
6
.2(-2) .5(-Z) .Z(-2) .2(-Z) .1(-l) .5(-2) .5(-2) .3(-Z)
1: 16 10 4 13 12 17
y2 .1(-4\_ ‘.l(-5) .l(-3) .l(-4) .7(-3) .2(-2) .2(-2) .4(-3) .31-3)
3 Nis the number of terms in the exact expansion of the product. ni, Ei are the number of terms and mean square deviation respectively. for expansion set i. b) In this and subsequent tables 0.1 x ldr is written ae A(n).
by conventional non-linear least-squares methods [2]. The final parameters are then further modified to satisfy the constraints by a nonlinear least squares with non-linear constraints procedure. We found, as in [l], that the constraints greatly improve the accuracy of the integrals obtained. To test the procedure.we computed a selection of orbital products for ferrocene (Fe.0, 0, 0; Cl 2.297, 9.0, 3.096 (au); C2 -Cl6 generated by successive application of the S : e(Z) symmetry operation. The basis CGTO’s were 3GT0 leastsquares fits to Slater 1s (exponent 5 = 1.0);
Table 2 Two-‘and three-centre integrals (in hartrees) computed from the orbital product expansions of table1 Exact value
Integral a) 93 pr3 Pd PX5 PX5 PY5 PY9 PY8 dxz(Fe)
Sl Sl PXl PXl P%l
s3 a3 PX3 PX3 PX3 PX3 s3 I=3 s5
DXl
sl al Z';' -pxl sl Sl
.84174(-l) .39705(-l) .41634(-Z) -.41898(-2) .10479(-2) -.13665(-2) -.15127(-3) .14399(-4) -.71515(-4)
Error @)
set !. .84363(-l) .39642(-l) .4X35(-2) -.41461(-2) .10535(-Z) -.13451(-2) -.15344(-3) .14421(-4) -.72093(-4)
0.23 -0.17 -0.34 3.03 0.54 1.54 -0.14 0.15 -0.82
Set 2
Error (%)
.84172(-l) .39693(-l) .41639(-2) -.41768(-2) .10436(-Z) -.13584(-2) -.15136(-3) .14397(-4) -.71431!-4)
0.00 -0.03 0.01 0.32 -0.43 0.62 -0.06 -0.01 0.11
a) CIb c d means{a(l)b(l)~~~c(2)d(2)drldT2
Table 3 Four-csntre integrals (in hartrees) computed from tineorbitaI produot expa&iona of table 1 Integral s5
B?
w3 -pr3 w3 px6. px8 pr8 PY5. PY5
&z(Fe)
&z(Fe)
d=+e)
Exact value s3
s7 sl pxl s4 Ix1 s7 sl px6. $3 ~6 px5 prl -. a9 pxl pr8 :al, -97. :- al ~8 ,’ pyl : 84
Sl 85 S8
s5 92 pxl prl. s3 w6 ;6
: a2
.45491(-l) .12134f-1). --.13301(-2) -.13514(-2) .28751(-3) .51661(-4) .19646(-3) -+8590(-2) .80332(-4) .37984(-2) “-.13948(-3)' -.80384(-i) :
Error fi)
Set 1 .45747(-l)
0.57
.12075(-l) -.13337(-2) -.13866{-2) .28987<-3) ‘.51629<:4) .19653{-3) -.18601+2) ..8033'7:-4) .382?1;;-2)
-.J3321+3)
-.819~~~-4),
Set2 .45511(-i)
_.,I:_
:
--.
‘.. _’
0.04
.12127(-l) -0.49 -0.05 -0.23 -.13302(-2) 0.00 -.13531(-2) 2.66 -0.13 0.84 .28735(-3) -0.05 0.60 .51660(-4) 0.00 0.03 .19650(-3) '_ 8-x" 0.09 -.18576(-2) .: -0.00 .80322(-4), -0.01 ,.0.60 _ .3&002(-Z) 0.05 ‘- 0.91 :.:.13853(-3) 0.68 -.60661('4) -0.35 :_-1.93
. . _-. :
.
._
., &;
Error fi)
_.
,.:
Volume 8, number 6
CHEMICAL PHYSICS LETTERS
Table 4 SCF results for CH4 comparing the orbItal product expansion method with conventional caicuiations Orbital expansion Product expansion (8.. . 1%GTO) (S-GTO) (L-STO)
Orbital energy
1Al
-11.341
-11.339
2Al lT2
-0.961
-0.962
-0.574
-0.575
3Al
0.426
0.434
n’2
0.503
0.503
Total energy Populations
-40.0567
-40.0601
H
0.870
0.869
C(ls)
1.996
1.996
CGa)
1.344
1.345
CG%)
1.060
1.061
-40.0606
2p (5 = 2.0); 3d (5 = 3.0) [3]. The nomenclature for example px3 is the 2pn orbital on C3. The tables show the results for two sets of expansions for representative orbital products. It is apparent from table 1 that the expansions converge more slowly with increasing complexity of the orbital product. As the separation between orbital centres increases the number of terms required for a given mean-square deviation decreases (compare px3, pxl and pa5, pxl). The ‘exact’ integral values in tables 2 end 3 are those calculated by the full expansion using contracted gaussians. Gur results show that even with the poorer expansion set it is possible consistently to obtain reliable values of integrals over a wide range of character and magnitude. The results obtained with the second set are very encouraging and are probably accurate enough for most molecular calculations. For example we have observed [5] that a calculation on benzene in which all integrals of magnitude less than 0.01 hartrees are given a random error lying between -2% and +2%, gives a total energy within 0.01 hartrees of the value obtained with the correct integrals, and one-electron orbital energies to 0.002 hartrees. We anticipate that this method of integral evalut&ion will have its greatest application in ‘ab initio’ calcuIations on large molecular systems. As the system becomes larger the average sepa.ration between atomic- centres increases and hence the average =pansion length will decrease: Moreover because an ST0 can’usualiy be adequ-
used in the tables is self-explanatory,
L5 March
1971
ately represented by an expansion of between 6 and 8 gaussian functions [4] the procedure is also applicable to the calculation of integrals for STO’s as it is especially suited to calculations using large
CGTO’s
.for a basis set.
Thus it is
probable that integrals computed in this manner would be sufficiently accurate for use in a mixed basis calculation [6,7]. As a further test we have used the orbital product expansion method to repeat the methane calculation of KIessinger and McWeeny [a]. The same parameters were used (carbon; S(k) = 5.7, [(Bs) = ((2~) = 1.625; hydrogen; (;&a) = 1.0; C-H = 2.067419 au) and each basis orbitaL was represented by en 8-GTO expansion taken from Huzinaga [4]. The 12 unique orbita] products were represented by moment-constrained GTO expansions of between 8 and I5 terms, and used to compute the two-electron integrals. Gneelectron terms were computed accurately over the 8-GTO orbital expansions. The results, listed in table 4, indicate that an expansion length equivalent to using 3-4 GTO’s per orbital yields closely comparable energies to the comparison 8-GTO calculation, and to the singleSlater orbital result. A complete account of the method will be given elsewhere. We are at present appLying the technique to some representative LCAO MO calculations on polyatomic systems with 50-100 electrons. We thank Professor R. McWeeny and Professor R-Mason for helpful discussions and encouragement. We are grateful to the Science Research Council for financial support of this project.
REFERENCES
[l] J. Monkhorst and F. E. Harris,
Chem.Phy~. titters
3 (1969) 537. [Z] W. C. Hamilton, Statistfcs in physfcai science (Ronald Press, New York, L964). [3] R. F. Stewart,
J. Chem. Phys. 50 (l969) 2485.
141K.Odhato, II. T&eta and S.Hxzfnaga, .I. Phya. Sot. Japan 21 (1966) 2306. [5] P. D. Dacrc and M. ELder, unpubIf&cd. [6] D. 3. Cook, P.C.HoHfs and R.McWeeny, Mol. Phye. 13 (19671 553. [‘I] D. H. Cook, P. D. Dacre, .I. L. Dcdda a.ndM. Elder, Theoret. C&II. Acta, tu he pubiiahed. [S] MKIcssinger and R. McWeeny; J.Chem. Phya. 42 (1965) 3343. -_
627