- Email: [email protected]
Spin and charge densities from the Hiller—Sucher—Feinberg identity: double-perturbation calculations for BeH
Volume 175, number I,2
30 November1990
CHEMICAL PHYSICS LETTERS
Spin and charge densities from the Hiller-Sucher-Feinberg identity: double-perturbation calculations for BeH Kazuhiro Ishida Department of Chemistry, Faculty ofScience, Science University of Tokyo, Shinjuku-ku, Tokyo 162, Japan
Received 4 June 1990
Meller-Plesset (MP) and Epstein-Neabet (EN) types of double-perturbation theory are used for the calculation of the spin and charge densities at the nuclei of the groundstateof BeH with the Hiller-Sucher-Feinberg (HSF) identity. Within single- and double-electron excitations for the configuration state functions, it is not sufficient to take the second-order contribution with respect to the electron correlation into account; indeed, for a good spin density it is necessary to take the fourth-order contribution into account, in each MP (SD-MP) or EN (SD-EN). The fourth-order MP (or EN) valueof 0.309 (or 0.306) bohr-3 for the HSF spin density at the Be nucleus is in excellent agreement with the experimental value of 0.317 bohr-‘. For the HSF spin density at H, the fourth-order MP (or EN) value of 0.0421 (or 0.0437 ) bohr-’ is also in excellent agreement with the experimental value of 0.0434 bohre3. It can be shown that the HSF identity is an excellent tool for calculating spin and charge densities accurately, not only in configuration-interaction theory but also in double-perturbation theory.
1. Introduction
the EN type, H,, is the shifted one given by
In paper I [ 11, configuration-interaction (CLSD) calculations with the use of the HSF identity [2,3] for the spin and charge densities at the nuclei of BeH molecule were reported. A very good spin density (compared with experiment) is obtained with only a moderate-sized CI-SD calculation with the CT0 basis set when using the HSF identity. In contrast with this, for the usual spin-density calculations, one can never obtain a good value even when one uses the ST0 basis set in any U-SD level of calculation, for example, as shown by Schaefer, Klemm, and Harris [ 41. In this paper, we present the double-perturbation calculations of the HSF spin and charge densities for comparison with the previous CI-SD calculations. Both Mlzrller-Plesset (MP) and Epstein-Nesbet (EN) types of double-perturbation theory are used in this paper, i.e. the Hamiltonian is given by
Ho= c I~K)(~KI(~o+~)I~K)(~KI,
H=H,+V+W, where H,, is the zeroth-order Hamiltonian, which is the Hartree-Fock Hamiltonian for the MP type. For
in which OK is the Kth configuration state function (CSF). The CSF is constructed with the Yamanouchi-Kotani spin eigenstate [ 5 ] together with spacesymmetry considerations. The perturbation V is the usual electron correlation in both MP and EN. The perturbation W is the HSF Hamiltonian given by W= & C gi I
(for the HSF charge density) ,
or W= & c 2&g, I
(for the HSF spin density) ,
where
in which T,~= I r,-FI, r, is the coordinate of the ith electron, S,, is the z component of the spin angular momentum operator of the ith electron, LfF is the
0009-2614/90/$ 03.50 0 1990 - Elsevier Science Publishers B.V. ( Nonh-Holland)
133
Volume 175, number 1,2
CHEMICAL PHYSICS LETTERS
square of the angular momentum operator of the ith electron around any given point F, and & is the nuclear charge of the Cth nucleus. The actual calculation in the present paper is an algebraic approximation [ 61 of double-perturbation theory in each of the types MP and EN using the GTO basis set and within single- and double-electron excitations for the CSFs. A more detailed description of the present method can be found elsewhere [ 7 1. The contribution from the nth-order perturbation with respect to V and the first-order with respect to W is called the nth-order (MPn or ENn) contribution to the HSF spin/charge density and denoted as Wcn,‘) in this paper. The molecular integrals over GTOs which are necessary for the present calculations have been derived elsewhere [ 7-9 1.
2. Results and discussion Table 1 shows the calculated values of the HSF spin and charge densities from the fourth-order MP within single- and double-electron excitations for the CSFs (SD-MP4), compared with the experimental value by Knight et al. [ lo]. The basis sets used are denoted by (12s4p/4slp), (12s6p/4s2p), and (16s6p/ 6slp), where (12s4p/4slp) and (16s6p/6slp) are the same basis sets as in paper I and (12s6p/4s2p) is constructed as follows: The regularized even-tempered (RET) ( 12s) GTOs by Schmidt and Ruedenberg [ 111 centered at Be are augmented by the RET (6~) GTOs, in which a,(Be)=a&c,(Be)2 (k=l to 6), where o!, and ps are the same factors as in the above ( 12s) and I$,( Be) = 1.O.The RET (4s) scaled by Q(H) = 1.2 centered at H are augmented by the RET (2p), in which (Y~(H)=Q~&(H)~ (k= 1,2), where a! and j? are the same factors as in the above Table 1 The HSF spin and charge densities at the SD-MP4 level for
(4s) and 1;,(H) = 1.6. These scaling factors c,(H), c,(H) and CD(Be) are roughly optimized at the Hartree-Fock level. The (16s6p/6slp) basis set reaches near the Hartree-Fock limit by 1.2 mhartree, as shown in paper I. The basis-set truncation error for the present basis sets can be almost neglected for the HSF spin and charge densities as described in paper I. The contribution in each order W (no’)to the HSF spin density at Be, say @(Be), is shown in table 2. Table 3 shows W(“,‘) for p”(H). For p’(Be), the value of each contribution in each order is almost independent of the basis set used, and the perturbation series tend to converge, as shown in table 2, leading to a good value of 0.309 bohrW3 (2.5% in relative error to the experimental value of 0.3 17). For p’(H), although each contribution is not so dependent on the basis set used, the perturbation does not converge at second order (i.e. W (2~’ ) r W ( ‘J ) ), but does gradually tend to convergence at the fourth orW(4.1) der (i.e. W VJ)> pfA3,‘)> ), as seen in table 3, finally leading to a good value of 0.0421 (3.0% in relative error to the experimental 0.0434), as shown in table 1. Table 4 shows the contribution in each order Wcn*‘) to the HSF charge density at Be, say pHSF(Be). Table 5 shows WcnJ) forpHSF(H). There is no experimental value for the charge density at the nucleus for BeH. Therefore, one cannot discuss whether the calculated value is accurate or not. However, it appears that the perturbation converges even at the second order for bothpHSF(Be) andpHSF(H), as seen in tables 4 and 5. Each value of Wcn,‘) depends only slightly on the basis set used. The Hartree-Fock value W (‘,I) is extremely dominant in both pHSF(Be) and PHSF(H). In conclusion, the MP2 level is not sufficient for p=(H), and, at least, the SD-MP4 level is necessary.
BeH(in atomic units) ‘)
Basisset
Number of CSFs
p’(Be)
P’(I+
pHsF(Be)
P=(H)
(12s4p/4slp) ( 12s6p/4s2p) (16s6p/6slp) experiment b,
2907 4704 5885
0.309 0.309 0.309 0.317
0.0465 0.0421 0.0421 0.0434
35.005 35.02 1 35.055
0.428 0.427 0.432
a) R&,=2.538 bohr.
134
30 November 1990
w Ref.
[IO].
Volume 175, number I,2
CHEMICAL PHYSICS LETTERS
30 November 1990
Table 2 The basis-set dependence of the contribution W (“J ) to p’( Be) in each order at the SD-MP4 level for BeH (in atomic units) Basis set
BWU’
W’1.l’
W(2.i)
BW.l)
W(%I)
(12s4p/4slp) ( 12s6p/4s2p) (16s6p/6slp)
0.2893 0.2888 0.2896
0.0300 0.0301 0.0303
-0.0017 -0.0012 -0.0019
- 0.0052 -0.0050 -0.0053
-0.0039 -0.0036 -0.0038
Table 3 The basis-set dependence of the contribution W r”.‘) top’(H)
in each order at the SD-MP4 level for BeH (in atomic units)
Basis set
w (0.1)
WCLl,
W’2,l)
WC3.l)
(12s4p/4stp) ( IZs6p/4sZp) (16s6p/6sip)
0.0288 0.0263 0.0259
0.0043 0.0033 0.0032
0.0067 0.0063 0.0065
0.0044 0.004
WW.1)
0.0022 0.0020 0.002 1
1
0.0043
Table 4 The basis-set dependence of the contribution W (“J) topnsF(Be) in each order at the SD-MP4 level for BeH (in atomic units)
34.9796 34.9744 35.0006
(12s4p/4slp) (12s6p/4s2p) (16s6p/6slp)
0.0435 0.0477 0.0486
-0.0118 0.0043 0.0118
0.0001 - 0.0007 - 0.0009
-0.0064 -0.0048 -0.0048
Table 5 The basis-set dependence of the contribution W w’) topnsF(H) in each order at the SD-MP4 level for BeH (in atomic units) Basis set
WuJJ,
W(1.1,
~‘2.”
WC’J,
W(4,”
(12s4p/4slp) ( IZs6p/4sZp) (16s6p/6slp)
0.4197 0.4204 0.4267
0.0097 0.0106 0.0103
- 0.0002 -0.0019 -0.0028
-0.0009 -0.0016 -0.0016
-0.0003 - 0.0004 - 0.0003
Table 6 The HSF spin and charge densities at the CI-SD level for BeH (in atomic units) Basis set
Number of CSFs
Energy
@(Be)
P’(H)
pHsF(Be)
P”*~W)
( IZs4pj4sl p) ‘) (12s6p/4s2p) b, ( 16s6p/6sl p) *r
2907 4704 5885
- 15.22052 - 15.22969 - 15.23042
0.305 0.307 0.306
0.0482 0.0432 0.0436
35.007 35.022 35.056
0.428 0.427 0.432
a) Paper I, ref. [ 11. ‘) This work.
On the other hand, the MP2 level appears to be sufGent for p’(Be), pHSF(Be) and pHSF(H). Table 6 showsthevaluesofpHSF(Be),pHSF(H),pf(Be),and p’(H) of the Cl-SD level calculated in paper I for comparison with the present SD-MP4 values. The CI-SD value for the (12s6p/4s2p) basis set is added
in this paper. Each SD-MP4 value is almost the same as the corresponding CLSD value, as seen in tables 1 and 6. For the Epstein-Nesbet case, the situation is very similar to the above Moller-Plesset case. Table 7 shows the calculated values of the HSF spin and 135
Volume 175, number 1,2
CHEMICAL PHYSICS LETTERS
charge densities at the SD-EN4 level with the same basis sets as in the SD-MP4 case. Each of tables 811 shows each contribution W(‘,‘) for p”(Be), ~‘(H),pHSF(Be),orpHSF(H), respectively. The SDEN4 value ofp*(Be) is 0.306 bohre3, which is in excellent agreement with the experimental value of
30 November 1990
0.3 17, in relative error by 3.5%. For p’(H), the SDEN4 value of 0.0437 is also in excellent agreement with the experimental value of 0.0434, in relative error by 0.7%. These SD-EN4 values for the HSF spin and charge densities are rather closer to the corresponding CLSD values than the SD-MP4 values, as
Table 7 The HSF spin and charge densities at the SD-EN4 level for BeH (in atomic units) Basis set
p’(Be)
P’(H)
pHSF(Be)
P-(H
(12s4p/4slp) ( 12s6p/4s2p) (16s6p/6slp)
0.305 0.306 0.306
0.0482 0.0437 0.0437
35.007 35.021 35.056
0.428 0.427 0.432
1
Table 8 The basis& dependence of the contribution W (“J) top”(Be) in each order at the SD-EN4 level for BeH (in atomic units)
(12s4p/4slp) ( 12s6p/4s2p) (l@P/6SlP)
0.2893 0.2888 0.2896
0.0325 0.0309 0.0326
Table 9 The basis-set dependence of the contribution W t”,‘) top’(H)
-0.0103 - 0.0070 - 0.0099
-0.0039 - 0.0047 -0.0041
-0.0027 -0.0020 -0.0025
in each order at the SD-EN4 level for BeH (in atomic units)
Basis set
lJ/ CO,l)
W(V)
WV,‘)
WV>11
w
(12s4p/4slp) (12s6p/4sZp) (16s6p/6slp)
0.0288 0.0263 0.0259
0.0067 0.0060 0.0055
0.0088 0.0077 0.0083
0.0033 0.0033 0.0034
0.0006 0.0004 0.0006
(4.1)
Table 10 The basis-set dependence of the contribution W W) topnSF(Be) in each order at the SD-EN4 level for BeH (in atomic units) Basis set
W’0.1’
W’L.1’
w (2.I)
w 0.1I
W’4.l’
(12s4p/4slp) (12s6p/4s2p) (16s6p/6slp)
34.9796 34.9744 35.0006
0.0466 0.0504 0.05 15
-0.0121 0.0098 0.0159
- 0.0099 -0.0122 -0.0121
0.0024 -0.0010 -0.0001
Table 11 The basis-set dependence of the contribution W (“J) topHSF(H) in each order at the SD-EN4 level for BeH (in atomic units) Basis set (12s4p/4slp) ( 12s6p/4sZp) (16s6p/6slp)
136
0.4197 0.4204 0.4267
0.0116 0.0125 0.0120
-0.0032 -0.0048 -0.0064
-0.0007 -0.0017 -0.0009
0.0005 0.0006 0.0008
Volume 175, number 1,2
CHEMICAL PHYSICS LETTERS
seen in tables 1, 6, and 7. Since the SD-EN4 results are very similar to those of the SD-MP4, both MP and EN perturbation series seem to be converged at the fourth order within, at least, two significant figures for both p’( Be) and p’( I-I). Further, one can consider that any possible error arising from the present algebraic approximation is negligible within the above accuracy. Thus, the HSF identity is an excellent tool for calculating the spin and charge densities at any given point accurately, not only in contiguration-interaction theory (as in paper I) but also in double-perturbation theory.
Acknowledgement All computations were carried out on HITAC M680H/S820 computers at the Computer Centre of the University of Tokyo. The library program F2/ TC/Emorl [ 12 ] (coded by Dr. N. Kosugi) was used for diagonalizing the CI matrix.
30 November 1990
References [ I] K. Ishida, Chem. Phys. Letters 158( 1989) 217. [ 21 J. Hiller, J. Sucher and G. Feinberg, Phys. Rev. A I8 ( 1978) 2399.
[ 31 J. Sucher and R.J. Drachman, Phys. Rev. A 20 (1979) 424. [ 4] H.F. Schaefer III, R.A. Klemm and F.E. Harris, Phys. Rev. 181 (1969) 137.
[ 51T. Yamanouchi, Proc. Phys. Math. Sot. Japan 18 (1936) 623; 19 (1937) 436; 20 (1938) 547; M. Kotani, A. Amemiya, E. Ishiguro and T. Kimura, Tables of molecular integrals (Maruzen, Tokyo, 1955). [ 61 S. Wilson, Electron correlation in molecules (Oxford Univ. Press, Oxford, 1984) ch. 6. [7] K. Ishida, Intern. .I. Quantum Chem. 36 (1989) 213. [8] K. Ishida, Intern. J. Quantum Chem. 28 (1985) 349; 34 (1988) 89. [ 91 K. Ishida, Intern. J. Quantum Chem. 30 (1986) 543; 34 (1988) 195. [lo] L.B. Knight Jr., J.M. Brom Jr. and W. Weltner Jr., J. Chem. Phys. 56 (1972) 1152. [ 111 M.W. Schmidt and K. Ruedenberg, J. Chem. Phys, 7 1 (1979) 3951. [ 121 N. Kosugi, J. Comput. Phys. 55 ( 1984) 426.
137
30 November1990
CHEMICAL PHYSICS LETTERS
Spin and charge densities from the Hiller-Sucher-Feinberg identity: double-perturbation calculations for BeH Kazuhiro Ishida Department of Chemistry, Faculty ofScience, Science University of Tokyo, Shinjuku-ku, Tokyo 162, Japan
Received 4 June 1990
Meller-Plesset (MP) and Epstein-Neabet (EN) types of double-perturbation theory are used for the calculation of the spin and charge densities at the nuclei of the groundstateof BeH with the Hiller-Sucher-Feinberg (HSF) identity. Within single- and double-electron excitations for the configuration state functions, it is not sufficient to take the second-order contribution with respect to the electron correlation into account; indeed, for a good spin density it is necessary to take the fourth-order contribution into account, in each MP (SD-MP) or EN (SD-EN). The fourth-order MP (or EN) valueof 0.309 (or 0.306) bohr-3 for the HSF spin density at the Be nucleus is in excellent agreement with the experimental value of 0.317 bohr-‘. For the HSF spin density at H, the fourth-order MP (or EN) value of 0.0421 (or 0.0437 ) bohr-’ is also in excellent agreement with the experimental value of 0.0434 bohre3. It can be shown that the HSF identity is an excellent tool for calculating spin and charge densities accurately, not only in configuration-interaction theory but also in double-perturbation theory.
1. Introduction
the EN type, H,, is the shifted one given by
In paper I [ 11, configuration-interaction (CLSD) calculations with the use of the HSF identity [2,3] for the spin and charge densities at the nuclei of BeH molecule were reported. A very good spin density (compared with experiment) is obtained with only a moderate-sized CI-SD calculation with the CT0 basis set when using the HSF identity. In contrast with this, for the usual spin-density calculations, one can never obtain a good value even when one uses the ST0 basis set in any U-SD level of calculation, for example, as shown by Schaefer, Klemm, and Harris [ 41. In this paper, we present the double-perturbation calculations of the HSF spin and charge densities for comparison with the previous CI-SD calculations. Both Mlzrller-Plesset (MP) and Epstein-Nesbet (EN) types of double-perturbation theory are used in this paper, i.e. the Hamiltonian is given by
Ho= c I~K)(~KI(~o+~)I~K)(~KI,
H=H,+V+W, where H,, is the zeroth-order Hamiltonian, which is the Hartree-Fock Hamiltonian for the MP type. For
in which OK is the Kth configuration state function (CSF). The CSF is constructed with the Yamanouchi-Kotani spin eigenstate [ 5 ] together with spacesymmetry considerations. The perturbation V is the usual electron correlation in both MP and EN. The perturbation W is the HSF Hamiltonian given by W= & C gi I
(for the HSF charge density) ,
or W= & c 2&g, I
(for the HSF spin density) ,
where
in which T,~= I r,-FI, r, is the coordinate of the ith electron, S,, is the z component of the spin angular momentum operator of the ith electron, LfF is the
0009-2614/90/$ 03.50 0 1990 - Elsevier Science Publishers B.V. ( Nonh-Holland)
133
Volume 175, number 1,2
CHEMICAL PHYSICS LETTERS
square of the angular momentum operator of the ith electron around any given point F, and & is the nuclear charge of the Cth nucleus. The actual calculation in the present paper is an algebraic approximation [ 61 of double-perturbation theory in each of the types MP and EN using the GTO basis set and within single- and double-electron excitations for the CSFs. A more detailed description of the present method can be found elsewhere [ 7 1. The contribution from the nth-order perturbation with respect to V and the first-order with respect to W is called the nth-order (MPn or ENn) contribution to the HSF spin/charge density and denoted as Wcn,‘) in this paper. The molecular integrals over GTOs which are necessary for the present calculations have been derived elsewhere [ 7-9 1.
2. Results and discussion Table 1 shows the calculated values of the HSF spin and charge densities from the fourth-order MP within single- and double-electron excitations for the CSFs (SD-MP4), compared with the experimental value by Knight et al. [ lo]. The basis sets used are denoted by (12s4p/4slp), (12s6p/4s2p), and (16s6p/ 6slp), where (12s4p/4slp) and (16s6p/6slp) are the same basis sets as in paper I and (12s6p/4s2p) is constructed as follows: The regularized even-tempered (RET) ( 12s) GTOs by Schmidt and Ruedenberg [ 111 centered at Be are augmented by the RET (6~) GTOs, in which a,(Be)=a&c,(Be)2 (k=l to 6), where o!, and ps are the same factors as in the above ( 12s) and I$,( Be) = 1.O.The RET (4s) scaled by Q(H) = 1.2 centered at H are augmented by the RET (2p), in which (Y~(H)=Q~&(H)~ (k= 1,2), where a! and j? are the same factors as in the above Table 1 The HSF spin and charge densities at the SD-MP4 level for
(4s) and 1;,(H) = 1.6. These scaling factors c,(H), c,(H) and CD(Be) are roughly optimized at the Hartree-Fock level. The (16s6p/6slp) basis set reaches near the Hartree-Fock limit by 1.2 mhartree, as shown in paper I. The basis-set truncation error for the present basis sets can be almost neglected for the HSF spin and charge densities as described in paper I. The contribution in each order W (no’)to the HSF spin density at Be, say @(Be), is shown in table 2. Table 3 shows W(“,‘) for p”(H). For p’(Be), the value of each contribution in each order is almost independent of the basis set used, and the perturbation series tend to converge, as shown in table 2, leading to a good value of 0.309 bohrW3 (2.5% in relative error to the experimental value of 0.3 17). For p’(H), although each contribution is not so dependent on the basis set used, the perturbation does not converge at second order (i.e. W (2~’ ) r W ( ‘J ) ), but does gradually tend to convergence at the fourth orW(4.1) der (i.e. W VJ)> pfA3,‘)> ), as seen in table 3, finally leading to a good value of 0.0421 (3.0% in relative error to the experimental 0.0434), as shown in table 1. Table 4 shows the contribution in each order Wcn*‘) to the HSF charge density at Be, say pHSF(Be). Table 5 shows WcnJ) forpHSF(H). There is no experimental value for the charge density at the nucleus for BeH. Therefore, one cannot discuss whether the calculated value is accurate or not. However, it appears that the perturbation converges even at the second order for bothpHSF(Be) andpHSF(H), as seen in tables 4 and 5. Each value of Wcn,‘) depends only slightly on the basis set used. The Hartree-Fock value W (‘,I) is extremely dominant in both pHSF(Be) and PHSF(H). In conclusion, the MP2 level is not sufficient for p=(H), and, at least, the SD-MP4 level is necessary.
BeH(in atomic units) ‘)
Basisset
Number of CSFs
p’(Be)
P’(I+
pHsF(Be)
P=(H)
(12s4p/4slp) ( 12s6p/4s2p) (16s6p/6slp) experiment b,
2907 4704 5885
0.309 0.309 0.309 0.317
0.0465 0.0421 0.0421 0.0434
35.005 35.02 1 35.055
0.428 0.427 0.432
a) R&,=2.538 bohr.
134
30 November 1990
w Ref.
[IO].
Volume 175, number I,2
CHEMICAL PHYSICS LETTERS
30 November 1990
Table 2 The basis-set dependence of the contribution W (“J ) to p’( Be) in each order at the SD-MP4 level for BeH (in atomic units) Basis set
BWU’
W’1.l’
W(2.i)
BW.l)
W(%I)
(12s4p/4slp) ( 12s6p/4s2p) (16s6p/6slp)
0.2893 0.2888 0.2896
0.0300 0.0301 0.0303
-0.0017 -0.0012 -0.0019
- 0.0052 -0.0050 -0.0053
-0.0039 -0.0036 -0.0038
Table 3 The basis-set dependence of the contribution W r”.‘) top’(H)
in each order at the SD-MP4 level for BeH (in atomic units)
Basis set
w (0.1)
WCLl,
W’2,l)
WC3.l)
(12s4p/4stp) ( IZs6p/4sZp) (16s6p/6sip)
0.0288 0.0263 0.0259
0.0043 0.0033 0.0032
0.0067 0.0063 0.0065
0.0044 0.004
WW.1)
0.0022 0.0020 0.002 1
1
0.0043
Table 4 The basis-set dependence of the contribution W (“J) topnsF(Be) in each order at the SD-MP4 level for BeH (in atomic units)
34.9796 34.9744 35.0006
(12s4p/4slp) (12s6p/4s2p) (16s6p/6slp)
0.0435 0.0477 0.0486
-0.0118 0.0043 0.0118
0.0001 - 0.0007 - 0.0009
-0.0064 -0.0048 -0.0048
Table 5 The basis-set dependence of the contribution W w’) topnsF(H) in each order at the SD-MP4 level for BeH (in atomic units) Basis set
WuJJ,
W(1.1,
~‘2.”
WC’J,
W(4,”
(12s4p/4slp) ( IZs6p/4sZp) (16s6p/6slp)
0.4197 0.4204 0.4267
0.0097 0.0106 0.0103
- 0.0002 -0.0019 -0.0028
-0.0009 -0.0016 -0.0016
-0.0003 - 0.0004 - 0.0003
Table 6 The HSF spin and charge densities at the CI-SD level for BeH (in atomic units) Basis set
Number of CSFs
Energy
@(Be)
P’(H)
pHsF(Be)
P”*~W)
( IZs4pj4sl p) ‘) (12s6p/4s2p) b, ( 16s6p/6sl p) *r
2907 4704 5885
- 15.22052 - 15.22969 - 15.23042
0.305 0.307 0.306
0.0482 0.0432 0.0436
35.007 35.022 35.056
0.428 0.427 0.432
a) Paper I, ref. [ 11. ‘) This work.
On the other hand, the MP2 level appears to be sufGent for p’(Be), pHSF(Be) and pHSF(H). Table 6 showsthevaluesofpHSF(Be),pHSF(H),pf(Be),and p’(H) of the Cl-SD level calculated in paper I for comparison with the present SD-MP4 values. The CI-SD value for the (12s6p/4s2p) basis set is added
in this paper. Each SD-MP4 value is almost the same as the corresponding CLSD value, as seen in tables 1 and 6. For the Epstein-Nesbet case, the situation is very similar to the above Moller-Plesset case. Table 7 shows the calculated values of the HSF spin and 135
Volume 175, number 1,2
CHEMICAL PHYSICS LETTERS
charge densities at the SD-EN4 level with the same basis sets as in the SD-MP4 case. Each of tables 811 shows each contribution W(‘,‘) for p”(Be), ~‘(H),pHSF(Be),orpHSF(H), respectively. The SDEN4 value ofp*(Be) is 0.306 bohre3, which is in excellent agreement with the experimental value of
30 November 1990
0.3 17, in relative error by 3.5%. For p’(H), the SDEN4 value of 0.0437 is also in excellent agreement with the experimental value of 0.0434, in relative error by 0.7%. These SD-EN4 values for the HSF spin and charge densities are rather closer to the corresponding CLSD values than the SD-MP4 values, as
Table 7 The HSF spin and charge densities at the SD-EN4 level for BeH (in atomic units) Basis set
p’(Be)
P’(H)
pHSF(Be)
P-(H
(12s4p/4slp) ( 12s6p/4s2p) (16s6p/6slp)
0.305 0.306 0.306
0.0482 0.0437 0.0437
35.007 35.021 35.056
0.428 0.427 0.432
1
Table 8 The basis& dependence of the contribution W (“J) top”(Be) in each order at the SD-EN4 level for BeH (in atomic units)
(12s4p/4slp) ( 12s6p/4s2p) (l@P/6SlP)
0.2893 0.2888 0.2896
0.0325 0.0309 0.0326
Table 9 The basis-set dependence of the contribution W t”,‘) top’(H)
-0.0103 - 0.0070 - 0.0099
-0.0039 - 0.0047 -0.0041
-0.0027 -0.0020 -0.0025
in each order at the SD-EN4 level for BeH (in atomic units)
Basis set
lJ/ CO,l)
W(V)
WV,‘)
WV>11
w
(12s4p/4slp) (12s6p/4sZp) (16s6p/6slp)
0.0288 0.0263 0.0259
0.0067 0.0060 0.0055
0.0088 0.0077 0.0083
0.0033 0.0033 0.0034
0.0006 0.0004 0.0006
(4.1)
Table 10 The basis-set dependence of the contribution W W) topnSF(Be) in each order at the SD-EN4 level for BeH (in atomic units) Basis set
W’0.1’
W’L.1’
w (2.I)
w 0.1I
W’4.l’
(12s4p/4slp) (12s6p/4s2p) (16s6p/6slp)
34.9796 34.9744 35.0006
0.0466 0.0504 0.05 15
-0.0121 0.0098 0.0159
- 0.0099 -0.0122 -0.0121
0.0024 -0.0010 -0.0001
Table 11 The basis-set dependence of the contribution W (“J) topHSF(H) in each order at the SD-EN4 level for BeH (in atomic units) Basis set (12s4p/4slp) ( 12s6p/4sZp) (16s6p/6slp)
136
0.4197 0.4204 0.4267
0.0116 0.0125 0.0120
-0.0032 -0.0048 -0.0064
-0.0007 -0.0017 -0.0009
0.0005 0.0006 0.0008
Volume 175, number 1,2
CHEMICAL PHYSICS LETTERS
seen in tables 1, 6, and 7. Since the SD-EN4 results are very similar to those of the SD-MP4, both MP and EN perturbation series seem to be converged at the fourth order within, at least, two significant figures for both p’( Be) and p’( I-I). Further, one can consider that any possible error arising from the present algebraic approximation is negligible within the above accuracy. Thus, the HSF identity is an excellent tool for calculating the spin and charge densities at any given point accurately, not only in contiguration-interaction theory (as in paper I) but also in double-perturbation theory.
Acknowledgement All computations were carried out on HITAC M680H/S820 computers at the Computer Centre of the University of Tokyo. The library program F2/ TC/Emorl [ 12 ] (coded by Dr. N. Kosugi) was used for diagonalizing the CI matrix.
30 November 1990
References [ I] K. Ishida, Chem. Phys. Letters 158( 1989) 217. [ 21 J. Hiller, J. Sucher and G. Feinberg, Phys. Rev. A I8 ( 1978) 2399.
[ 31 J. Sucher and R.J. Drachman, Phys. Rev. A 20 (1979) 424. [ 4] H.F. Schaefer III, R.A. Klemm and F.E. Harris, Phys. Rev. 181 (1969) 137.
[ 51T. Yamanouchi, Proc. Phys. Math. Sot. Japan 18 (1936) 623; 19 (1937) 436; 20 (1938) 547; M. Kotani, A. Amemiya, E. Ishiguro and T. Kimura, Tables of molecular integrals (Maruzen, Tokyo, 1955). [ 61 S. Wilson, Electron correlation in molecules (Oxford Univ. Press, Oxford, 1984) ch. 6. [7] K. Ishida, Intern. .I. Quantum Chem. 36 (1989) 213. [8] K. Ishida, Intern. J. Quantum Chem. 28 (1985) 349; 34 (1988) 89. [ 91 K. Ishida, Intern. J. Quantum Chem. 30 (1986) 543; 34 (1988) 195. [lo] L.B. Knight Jr., J.M. Brom Jr. and W. Weltner Jr., J. Chem. Phys. 56 (1972) 1152. [ 111 M.W. Schmidt and K. Ruedenberg, J. Chem. Phys, 7 1 (1979) 3951. [ 121 N. Kosugi, J. Comput. Phys. 55 ( 1984) 426.
137