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Kinetic energy, potential energy, and virial ratio from calculations using systematic sequences of even-tempered basis sets
Volume 81, number 3
CHEMICAL PHYSICS LETTERS
1 August 1981
KINETIC ENERGY, POTENTIAL ENERGY, AND VIRIAL RATIO FROM CALCULATIONS USING SYSTEMATIC SEQUENCES OF EVEN-TEMPERED BASIS SETS Stephen WILSON * Department of Theoretical Chemistry, University of Oxford, Oxford OX1 3TG, UK Received 10 April 1981; in final form 25 April 1981 The use of systematic sequences of even-tempered basis sets in calculations of the kinetic energy, potential energy and vffial ratio within the self-consistent-field molecular-orbital approach is discussed. The components of the total energy are shown to converge smoothly with increasing size of basis set. Systematic sequences of universal even-tempered basis sets are also considered.
The components of the total energy of an atomic or molecular system are known [1] to be more sensitive to the quality of the basis set used to parameterize the orbitals employed in the construction of the wavefunction than the total energy itself. For example, in a study of the nitrogen molecule using the self-consistent-field molecular-orbital method, Cade et al. [2] examined the dependence of the total energy, E, the kinetic energy, T, the potential energy, V, and the orbital energies, % on the quality of the basis set and demonstrated that although the total energy and the orbital energies are not very sensitive to the basis set, the kinetic energy and the potential energy can behave erratically as the basis set is extended. It is for this reason that the virial ratio, R = -v/2r,
(1)
is often regarded as a more sensitive test of the quality of a basis set than the total energy since in the Hartree-Fock limit R = 1. The basis set truncation error is often the most significant source of error in ab initio computations. The use of systematic sequences of even-tempered basis sets [ 3 - 5 ] and universal basis sets [ 6 - 8 ] has been advocated since by extending basis sets in an orderly fashion extrapolation procedures can be employed to obtain accurate estimates of the basis set limit. This approach has been demonstrated in calcu:t: Science Research Council Advanced Fellow.
lations of total energies both within the self-consistentfield molecular-orbital method and in calculations which take account of electron correlation effects. In this letter, we examine the behaviour of the components of the total energy and the virial ratio in self-consistent-field calculations when a systematic sequence of even-tempered basis sets is employed. The use of a universal systematic sequence of basis sets is also considered. The orbital exponents in an even-tempered basis set are given by the geometric relation ~'k = °~3k,
k = 1, 2 ..... N.
(2)
In order that such basis sets approach a complete set as the value of N is increased, the values of a and 13 must depend on N. Empirical forms for a and/3 have been suggested by Schmidt and Ruedenberg [3] which are equivalent to the recursions Olu= [(fiN-- 1 ) / ( f i N - 1 - 1)]a0~N-1 ,
a>O,
(3)
and l n ~ N : [ N / ( N - 1)]bln/3N_l ,
--1 < b < 0 .
(4)
These recursions enable the number of functions in an even-tempered basis set to be increased, or decreased, smoothly. It is instructive to examine the eigenvalues of the metric matrix corresponding to basis sets generated in this fashion. In fig. I a, the eigenvalues of the metric matrix for the sequence of basis sets given in ref. [3] for the beryllium atom are
0 0 0 9 - 2 6 1 4 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
467
Volume 81, number 3
1 August 1981
CHEMICAL PHYSICS LETTERS
plotted as a function of N. For comparison the eigenvalues of the metric matrices corresponding to the basis sets obtained by using a fixed value of a and/3 are shown in fig. lb. In fig. l a the logarithm of the eigenvalues of the metric matrices have an approximately linear dependence on the number of basis functions, N, and the spread of the eigenvalues becomes greater as N is increased. On the other hand, when a and ~ are fixed, it can be seen in fig. l b that the spread of the eigenvalues of the metric matrix tends to a definite limit as N goes to infinity.
Table 1 Total energy, kinetic energy, and potential energy for the beryllium atom from self-consistent-field calculations using the regularised even-tempered basis sets of Schmidt and Ruedenberg a) Set
E
T
V
6s 8s 10s 12s 14s 16s 18s 20s
-14,53490 -14.56644 -14.57173 -14.57271 -14.57295 -14.57300 -14.57302 -14.57302
14.5147 14.5715 14.5707 14.5728 14.5730 14.5730 14.5730 14.5730
-29.0496 -29.1380 -29.1425 -29.1455 -29.1459 -29.1460 -29.1460 -29.1460
a) All energies are in hartree atomic units. Using the regularised, even-tempered basis sets given by Schmidt and Ruedenberg [3], the kinetic energy, T, and tile potential energy, V, of the beryllium atom were calculated within the self-consistent-field approximation. The results of these calculations are given in table 1 together with values of the total energy which have been reported previously [3]. Similar calculations were performed for the neon atom using the regularised, energy-balanced systematic sequence of basis sets given in ref. [3]. The results are displayed in table 2. In order to examine the convergence of the energy components and the virial ratio in more detail, we define the quantity x(X, iV) = lnIX[N] -
X[N'] I,
(S)
Table 2 Total energy, kinetic energy, and potential energy for the neon atom from self-consistent-field calculations using the regularised even-tempered basis sets of Schmidt and Ruedenberg a)
Fig. 1. Plots of In ei against size of basis set, where ei is an eigenvalue of the metric matrix, for (a) the regularised eventempered basis sets of gaussian-type orbitals given in ref. [ 3] for the beryllium atom and (b) even-tempered gaussian-type orbitals with fixed a and/3 (a = 1/9, t3 = 9/4). 468
Set
E
T
V
6~3p 8~4p 10~5p 12~6p 14~7p 16~8p 18~9p
-128.07977 -128.43917 -128.51933 -128,53936 -128.54474 -128.54630 -128,54682
128.0147 128.4535 128.5075 128.5410 128.5456 128.5469 128.5472
-256.0944 -256.8926 -257.0268 -257.0803 -257.0933 -257.0932 -257.0940
a) All energies are in hartree atomic units.
Volume 81, number 3
CHEMICAL PHYSICS LETTERS
where X = E, T, V or R. N again denotes the size of the basis set. x(X, iV), X = E, T, 11, R, is plotted as a function of N for the beryllium atom in fig. 2a. A similar plot for the neon atom is given in fig. 2b. For all of the quantities displayed in figs. 2a and 2b, convergence with increasing size of basis set is observed. ×(E, iV) and x(R, N') behave somewhat more smoothly as a function of N than do x(T, N) and x(V, N). For both Be and Ne, ×(R, N) is considerably smaller than x(E, N), x(T, N), and x(V, N); the virial ratio appears to be less sensitive to the basis set than either the total energy or its components. We also tested the dependence of X on the accuracy
BASIS SET
"-i
--2
1 August 1981
to which the self-consistent-field process was converged. We repeated the calculations on the Be atom using the [6s] and the [20s] basis sets and continued the self-consistent-field iterations until no off-diagonal element of the Fock matrix was greater than 10 - 8 instead of 10 - 5 which was used in all other calculations. If 8X = In IX - X ' I where X and X ' are the values of E, T, V or R obtained using the higher and lower accuracy, then for the [6s] basis set: 8 T = - 1 1 . 8 , 8 V= - 1 1 . 8 , 8E = - 2 3 . 0 , 8R = - 1 5 . 1 ; and for the [20s] basis set: S T = - 1 1 . 8 , 6 V = - 1 1 . 8 , BE= - 2 2 . 3 , 8R = - 1 5 . 2 . For the calculations reported in this letter, the basis sets have been extended until X and 6
Ill .-3
-4 --5
--6
I-6 "~
"-7 "~
-7 x
T~E
--8
--9
-
i
-10 R
' -11
-12
:-13
• -13
(a) Fig. 2. Convergence of the total energy, kinetic energy, potential energy and the virial ratio obtained from self-consistent-field calculations using the regularised even-tempered basis sets of ref. [ 3 ] for (a) the beryllium atom and (b) the neon atom. 469
-14
(b)
-17
-14
×
z
(c)
3e
Fig. 3. Convergence of (a) the kinetic energy, (b) the potential energy, and (c) the virial ratio for Be-like ions from self-consistent-field calculations using the universal systematic sequence of even-tempered basis sets given in ref. [8 ].
(a)
-16
-15
-13
-.LZ
-14
-13
-12
-11
-10
-9
H
Volume 81, number 3
CHEMICALPHYSICS LETTERS
are of comparable magnitude. It should be noted that whereas for a given basis set the magnitude of bE is about twice that of 6 T or 6 V, x(E, N) is of the same order of magnitude as x(T, N) and x(V, N). The dependence of the components of the total energy and the virial ratio on N has also been examined in self-consistent-field calculations employing a universal systematic sequence of even-tempered basis sets. Calculations have been performed for the beryllium-like ions B+, C 2+, N 3+, O 4+, F 5+ and Ne 6+. Calculations of the total energy of these ions, including correlation effects, have been reported previously [8]. The variation of x(T, N) with N for the beryllium atom and the beryllium-like ions is shown in fig. 3a. x(V, N) is plotted in fig. 3b and x(R, At) in fig. 3c. The variation of x(E, N) with N has been illustrated in ref. [8]. It can be seen by comparing figs. 2 and 3 that the convergence of the components of the energy and the virial ratio obtained for the beryllium-like ions is much the same as that observed for the beryllium atom and the neon atom. By using a universal systematic sequence of even-tempered basis sets we have obtained fairly smooth convergence of the components of the energy and the virial ratio with basis set size even though these basis sets were not specifically designed for the particular system being treated. To summarise, we have shown in this letter that by using a systematic sequence of even-tempered basis sets fairly smooth convergence of quantities which are more sensitive to the quality of the basis set - the kinetic and potential energy, and the virial ratio - can be obtained. Universal basis sets are also useful in such calculations even though they have not been designed for a particular system. It is envisaged that systematic
1 August 1981
sequences of even-tempered basis sets and universal basis sets will prove valuable in calculations of molecular properties (cf. McCullough's recent discussion of the accuracy of calculated quadmpole moments [9] ). In particular, polarisabilities and nuclear electric shielding factors offer a severe test because of their sensitivity to the basis set. Polarisabilities are determined largely by the valence shell of a molecule and thus they are sensitive to the accuracy of the wavefunction far from the nuclei. On the other hand, nuclear shielding factors are sensitive to the accuracy Of the wavefunction in the vicinity of the nucleus. Polarisabilities and nuclear shielding factors provide complementary measures of the adequacy of a basis set. This work was supported by S.R.C. Research Grant GR/B/4738.6.
References [1] P. ~rsky and M. Urban, Ab initio calculations. Methods and applications in chemistry (Springer, Berlin, 1980) p. 15. [2] P.E. Cade, K.D. Sales and A.C. Wahl, J. Chem. Phys. 44 (1966) 1973. [3] M.W. Schmidt and K. Ruedenberg, J. Chem. Phys. 71 (1979) 3951. [4] D.F. Feller and K. Ruedenberg, Theoret. Chim. Acta 52 (1979) 231. [5] S. Wilson, Theoret. Chim. Acta 57 (1980) 53. [6] D.M. Silver, S. Wilson and W.C. Nieuwpoort, Intern. J. Quantum Chem. 14 (1978) 635. [7] S. Wilson and D.M. Silver, J. Chem. Phys. 72 (1980) 2159. [8] S. Wilson, Theoret. Chirn. Acta 58 (1980) 31. [9] E.A. McCuUoughJr., Mol. Phys., to be published.
471
CHEMICAL PHYSICS LETTERS
1 August 1981
KINETIC ENERGY, POTENTIAL ENERGY, AND VIRIAL RATIO FROM CALCULATIONS USING SYSTEMATIC SEQUENCES OF EVEN-TEMPERED BASIS SETS Stephen WILSON * Department of Theoretical Chemistry, University of Oxford, Oxford OX1 3TG, UK Received 10 April 1981; in final form 25 April 1981 The use of systematic sequences of even-tempered basis sets in calculations of the kinetic energy, potential energy and vffial ratio within the self-consistent-field molecular-orbital approach is discussed. The components of the total energy are shown to converge smoothly with increasing size of basis set. Systematic sequences of universal even-tempered basis sets are also considered.
The components of the total energy of an atomic or molecular system are known [1] to be more sensitive to the quality of the basis set used to parameterize the orbitals employed in the construction of the wavefunction than the total energy itself. For example, in a study of the nitrogen molecule using the self-consistent-field molecular-orbital method, Cade et al. [2] examined the dependence of the total energy, E, the kinetic energy, T, the potential energy, V, and the orbital energies, % on the quality of the basis set and demonstrated that although the total energy and the orbital energies are not very sensitive to the basis set, the kinetic energy and the potential energy can behave erratically as the basis set is extended. It is for this reason that the virial ratio, R = -v/2r,
(1)
is often regarded as a more sensitive test of the quality of a basis set than the total energy since in the Hartree-Fock limit R = 1. The basis set truncation error is often the most significant source of error in ab initio computations. The use of systematic sequences of even-tempered basis sets [ 3 - 5 ] and universal basis sets [ 6 - 8 ] has been advocated since by extending basis sets in an orderly fashion extrapolation procedures can be employed to obtain accurate estimates of the basis set limit. This approach has been demonstrated in calcu:t: Science Research Council Advanced Fellow.
lations of total energies both within the self-consistentfield molecular-orbital method and in calculations which take account of electron correlation effects. In this letter, we examine the behaviour of the components of the total energy and the virial ratio in self-consistent-field calculations when a systematic sequence of even-tempered basis sets is employed. The use of a universal systematic sequence of basis sets is also considered. The orbital exponents in an even-tempered basis set are given by the geometric relation ~'k = °~3k,
k = 1, 2 ..... N.
(2)
In order that such basis sets approach a complete set as the value of N is increased, the values of a and 13 must depend on N. Empirical forms for a and/3 have been suggested by Schmidt and Ruedenberg [3] which are equivalent to the recursions Olu= [(fiN-- 1 ) / ( f i N - 1 - 1)]a0~N-1 ,
a>O,
(3)
and l n ~ N : [ N / ( N - 1)]bln/3N_l ,
--1 < b < 0 .
(4)
These recursions enable the number of functions in an even-tempered basis set to be increased, or decreased, smoothly. It is instructive to examine the eigenvalues of the metric matrix corresponding to basis sets generated in this fashion. In fig. I a, the eigenvalues of the metric matrix for the sequence of basis sets given in ref. [3] for the beryllium atom are
0 0 0 9 - 2 6 1 4 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
467
Volume 81, number 3
1 August 1981
CHEMICAL PHYSICS LETTERS
plotted as a function of N. For comparison the eigenvalues of the metric matrices corresponding to the basis sets obtained by using a fixed value of a and/3 are shown in fig. lb. In fig. l a the logarithm of the eigenvalues of the metric matrices have an approximately linear dependence on the number of basis functions, N, and the spread of the eigenvalues becomes greater as N is increased. On the other hand, when a and ~ are fixed, it can be seen in fig. l b that the spread of the eigenvalues of the metric matrix tends to a definite limit as N goes to infinity.
Table 1 Total energy, kinetic energy, and potential energy for the beryllium atom from self-consistent-field calculations using the regularised even-tempered basis sets of Schmidt and Ruedenberg a) Set
E
T
V
6s 8s 10s 12s 14s 16s 18s 20s
-14,53490 -14.56644 -14.57173 -14.57271 -14.57295 -14.57300 -14.57302 -14.57302
14.5147 14.5715 14.5707 14.5728 14.5730 14.5730 14.5730 14.5730
-29.0496 -29.1380 -29.1425 -29.1455 -29.1459 -29.1460 -29.1460 -29.1460
a) All energies are in hartree atomic units. Using the regularised, even-tempered basis sets given by Schmidt and Ruedenberg [3], the kinetic energy, T, and tile potential energy, V, of the beryllium atom were calculated within the self-consistent-field approximation. The results of these calculations are given in table 1 together with values of the total energy which have been reported previously [3]. Similar calculations were performed for the neon atom using the regularised, energy-balanced systematic sequence of basis sets given in ref. [3]. The results are displayed in table 2. In order to examine the convergence of the energy components and the virial ratio in more detail, we define the quantity x(X, iV) = lnIX[N] -
X[N'] I,
(S)
Table 2 Total energy, kinetic energy, and potential energy for the neon atom from self-consistent-field calculations using the regularised even-tempered basis sets of Schmidt and Ruedenberg a)
Fig. 1. Plots of In ei against size of basis set, where ei is an eigenvalue of the metric matrix, for (a) the regularised eventempered basis sets of gaussian-type orbitals given in ref. [ 3] for the beryllium atom and (b) even-tempered gaussian-type orbitals with fixed a and/3 (a = 1/9, t3 = 9/4). 468
Set
E
T
V
6~3p 8~4p 10~5p 12~6p 14~7p 16~8p 18~9p
-128.07977 -128.43917 -128.51933 -128,53936 -128.54474 -128.54630 -128,54682
128.0147 128.4535 128.5075 128.5410 128.5456 128.5469 128.5472
-256.0944 -256.8926 -257.0268 -257.0803 -257.0933 -257.0932 -257.0940
a) All energies are in hartree atomic units.
Volume 81, number 3
CHEMICAL PHYSICS LETTERS
where X = E, T, V or R. N again denotes the size of the basis set. x(X, iV), X = E, T, 11, R, is plotted as a function of N for the beryllium atom in fig. 2a. A similar plot for the neon atom is given in fig. 2b. For all of the quantities displayed in figs. 2a and 2b, convergence with increasing size of basis set is observed. ×(E, iV) and x(R, N') behave somewhat more smoothly as a function of N than do x(T, N) and x(V, N). For both Be and Ne, ×(R, N) is considerably smaller than x(E, N), x(T, N), and x(V, N); the virial ratio appears to be less sensitive to the basis set than either the total energy or its components. We also tested the dependence of X on the accuracy
BASIS SET
"-i
--2
1 August 1981
to which the self-consistent-field process was converged. We repeated the calculations on the Be atom using the [6s] and the [20s] basis sets and continued the self-consistent-field iterations until no off-diagonal element of the Fock matrix was greater than 10 - 8 instead of 10 - 5 which was used in all other calculations. If 8X = In IX - X ' I where X and X ' are the values of E, T, V or R obtained using the higher and lower accuracy, then for the [6s] basis set: 8 T = - 1 1 . 8 , 8 V= - 1 1 . 8 , 8E = - 2 3 . 0 , 8R = - 1 5 . 1 ; and for the [20s] basis set: S T = - 1 1 . 8 , 6 V = - 1 1 . 8 , BE= - 2 2 . 3 , 8R = - 1 5 . 2 . For the calculations reported in this letter, the basis sets have been extended until X and 6
Ill .-3
-4 --5
--6
I-6 "~
"-7 "~
-7 x
T~E
--8
--9
-
i
-10 R
' -11
-12
:-13
• -13
(a) Fig. 2. Convergence of the total energy, kinetic energy, potential energy and the virial ratio obtained from self-consistent-field calculations using the regularised even-tempered basis sets of ref. [ 3 ] for (a) the beryllium atom and (b) the neon atom. 469
-14
(b)
-17
-14
×
z
(c)
3e
Fig. 3. Convergence of (a) the kinetic energy, (b) the potential energy, and (c) the virial ratio for Be-like ions from self-consistent-field calculations using the universal systematic sequence of even-tempered basis sets given in ref. [8 ].
(a)
-16
-15
-13
-.LZ
-14
-13
-12
-11
-10
-9
H
Volume 81, number 3
CHEMICALPHYSICS LETTERS
are of comparable magnitude. It should be noted that whereas for a given basis set the magnitude of bE is about twice that of 6 T or 6 V, x(E, N) is of the same order of magnitude as x(T, N) and x(V, N). The dependence of the components of the total energy and the virial ratio on N has also been examined in self-consistent-field calculations employing a universal systematic sequence of even-tempered basis sets. Calculations have been performed for the beryllium-like ions B+, C 2+, N 3+, O 4+, F 5+ and Ne 6+. Calculations of the total energy of these ions, including correlation effects, have been reported previously [8]. The variation of x(T, N) with N for the beryllium atom and the beryllium-like ions is shown in fig. 3a. x(V, N) is plotted in fig. 3b and x(R, At) in fig. 3c. The variation of x(E, N) with N has been illustrated in ref. [8]. It can be seen by comparing figs. 2 and 3 that the convergence of the components of the energy and the virial ratio obtained for the beryllium-like ions is much the same as that observed for the beryllium atom and the neon atom. By using a universal systematic sequence of even-tempered basis sets we have obtained fairly smooth convergence of the components of the energy and the virial ratio with basis set size even though these basis sets were not specifically designed for the particular system being treated. To summarise, we have shown in this letter that by using a systematic sequence of even-tempered basis sets fairly smooth convergence of quantities which are more sensitive to the quality of the basis set - the kinetic and potential energy, and the virial ratio - can be obtained. Universal basis sets are also useful in such calculations even though they have not been designed for a particular system. It is envisaged that systematic
1 August 1981
sequences of even-tempered basis sets and universal basis sets will prove valuable in calculations of molecular properties (cf. McCullough's recent discussion of the accuracy of calculated quadmpole moments [9] ). In particular, polarisabilities and nuclear electric shielding factors offer a severe test because of their sensitivity to the basis set. Polarisabilities are determined largely by the valence shell of a molecule and thus they are sensitive to the accuracy of the wavefunction far from the nuclei. On the other hand, nuclear shielding factors are sensitive to the accuracy Of the wavefunction in the vicinity of the nucleus. Polarisabilities and nuclear shielding factors provide complementary measures of the adequacy of a basis set. This work was supported by S.R.C. Research Grant GR/B/4738.6.
References [1] P. ~rsky and M. Urban, Ab initio calculations. Methods and applications in chemistry (Springer, Berlin, 1980) p. 15. [2] P.E. Cade, K.D. Sales and A.C. Wahl, J. Chem. Phys. 44 (1966) 1973. [3] M.W. Schmidt and K. Ruedenberg, J. Chem. Phys. 71 (1979) 3951. [4] D.F. Feller and K. Ruedenberg, Theoret. Chim. Acta 52 (1979) 231. [5] S. Wilson, Theoret. Chim. Acta 57 (1980) 53. [6] D.M. Silver, S. Wilson and W.C. Nieuwpoort, Intern. J. Quantum Chem. 14 (1978) 635. [7] S. Wilson and D.M. Silver, J. Chem. Phys. 72 (1980) 2159. [8] S. Wilson, Theoret. Chirn. Acta 58 (1980) 31. [9] E.A. McCuUoughJr., Mol. Phys., to be published.
471