Hartree-Fock CISD correlation energy

Table 8.12 The Hartree-Fock eneigy hf and the CISD correlation energy faso — hf for the ground state of the carbon atom calculated using correlation-consistent basis sets. All energies are given in atomic units. The Hartree-Fock basis-set limit is —37.688619 Eh and the estimated CISD correlation-energy basis-set limit is —99.7 0.4 mEh...

Fig. 8.16. Errors (in mEh) relative to the basis-set limit in the Hartree-Fock energy (dotted line) and in the valence CISD correlation energy (full line) for the ground state of the carbon atom calculated using correlation-consistent basis sets cc-pVXZ. On the left, we have plotted the errors on a linear scale as a function of the cardinal number X on the right, we have plotted the errors on a logarithmic scale.

A later study also focused on various means of computing the correlation contribution to the interaction energy in the HE dimer and reached very similar conclusions. All of the correlated methods (MP2, MP4, CCSD(T) and CISD) based on the Hartree-Fock reference configuration gave essentially the same binding energy. The results deteriorate when multireference methods are used. 

Presently, the widely used post-Hartree-Fock approaches to the correlation problem in molecular electronic structure calculations are basically of two kinds, namely, those of variational and those of perturbative nature. The former are typified by various configuration interaction (Cl) or shell-model methods, and employ the linear Ansatz for the wave function in the spirit of Ritz variation principle (c/, e.g. Ref. [21]). However, since the dimension of the Cl problem rapidly increases with increasing size of the system and size of the atomic orbital (AO) basis set employed (see, e.g. the so-called Paldus-Weyl dimension formula [22,23]), one has to rely in actual applications on truncated Cl expansions (referred to as a limited Cl), despite the fact that these expansions are slowly convergent, even when based on the optimal natural orbitals (NOs). Unfortunately, such limited Cl expansions (usually truncated at the doubly excited level relative to the IPM reference, resulting in the CISD method) are unable to properly describe the so-called dynamic correlation, which requires that higher than doubly excited configurations be taken into account. Moreover, the energies obtained with the limited Cl method are not size-extensive. 

Table 11 The CISD monomer correlation energy (in mEh) for one to eight noninteracting water molecules, calculated in the cc-pVDZ basis with and without the Davidson correction applied. The weight of the Hartree-Fock configuration in the CISD wave function IVhf is also listed. The calculations have been carried out for the C2 water molecule with R = / ref and R = IR ct where Rra = 1.84345ao- The HOH bond angle is fixed at 110.565°. At Rnt and 2R,et, the Hartree-Fock energies are —76.0240 and —75.5877 Eh, respectively, and the t CI correlation energies are —217.8 and —364.0 mEh, respectively...

In order to overcome the shortcommings of standard post-Hartree-Fock approaches in their handling of the dynamic and nondynamic correlations, we investigate the possibility of mutual enhancement between variational and perturbative approaches, as represented by various Cl and CC methods, respectively. This is achieved either via the amplitude-corrections to the one- and two-body CCSD cluster amplitudes based on some external source, in particular a modest size MR CISD wave function, in the so-called reduced multireference (RMR) CCSD method, or via the energy-corrections to the standard CCSD based on the same MR CISD wave function. The latter corrections are based on the asymmetric energy formula and may be interpreted either as the MR CISD corrections to CCSD or RMR CCSD, or as the CCSD corrections to MR CISD. This reciprocity is pointed out and a new perturbative correction within the MR CISD is also formulated. The earlier results are briefly summarized and compared with those introduced here for the first time using the exactly solvable double-zeta model of the HF and N2 molecules. 

To illustrate the performance of the cc-pVXZ sets, we have, for cardinal numbers X < 5, calculated the Hartree-Fock and CISD energies of the ground state of the carbon atom see Table 8.12. The correlation-eneigy contributions compare quite favourably with those of the ANOs in Table 8.10, keeping in mind that, for small cardinal numbers, the ANOs are constructed from much larger primitive sets. Thus, from the point of view of energy contributions at least, the use of primitive rather than contracted correlating orbitals spears to be fully justified. Also,... 

Table 8.15 The Hartree-Fock energy and the CISD valence correlation energy for the ground state of the carbon atom and for the 5 ground state of the carbon anion calculated using the cc-pVXZ and aug-cc-pVXZ basis sets (atomic units). The Hartree-Fock basis-set limits for the neutral and anionic atoms are —37.688619 and —37.708844 Eh, respectively. The estimated CISD basis-set limits are —99.7 0.4 mEh and —120.7 0.4 mEh, respectively...

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