Valence correlation energies

The correlation energy is expected to have an inverse power dependence once the basis set reaches a sufficient (large) size. Extrapolating the correlation contribution for n = 3-5(6) with a function of the type A + B n + I) yields the cc-pVooZ values in Table 11.8. The extrapolated CCSD(T) energy is —76.376 a.u., yielding a valence correlation energy of —0.308 a.u. 

Table 1.6 CCSD valence correlation energies calculated with the cc-pVXZ basis sets and using the extrapolation formula Eq. (5.14) are compared with the R12 values (mEh). The last column contains the mean absolute deviations from the R12 energies. All calculations have been carried out at the optimized all-electron CCSD(T)/cc-pCVQZ geometries [25].

As expected from our previous discussion of the CCSD valence correlation energies, the convergence towards the experimental energies is slow, with mean absolute errors of 511, 163, 61, 28, and 17 kJ/mol as... 

In the calculations presented so far, all electrons have been correlated. However, chemical reactions involve mainly the valence electrons, leaving the core electrons nearly unaffected. It is therefore tempting to correlate only the valence electrons and to let the core orbitals remain doubly occupied. In this way, we avoid the calculation of the nearly constant core-correlation energy, concentrating on the valence correlation energy. The freezing of the core electrons simplifies the calculations as there are fewer electrons to correlate and since it enables us to use the cc-pVXZ basis sets rather than the larger cc-pCVXZ sets. 

Nevertheless, core-correlation contributions to AEs are often sizeable, with contributions of about 10 kJ/mol for some of the molecules considered here (CH4, C2H2, and C2H4). For an accuracy of 10 kJ/mol or better, it is therefore necessary to make an estimate of core correlation [9, 56]. It is, however, not necessary to calculate the core correlation at the same level of theory as the valence correlation energy. We may, for example, estimate the core-correlation energy by extrapolating the difference between all-electron and valence-electron CCSD(T) calculations in the cc-pCVDZ and cc-pCVTZ basis sets. The core-correlation energies obtained in this way reproduce the CCSD(T)/cc-pCV(Q5)Z core-correlation contributions to the AEs well, with mean absolute and maximum deviations of only 0.4 kJ/mol and 1.4 kJ/mol, respectively. By contrast, the calculation of the valence contribution to the AEs by cc-pCV(DT)Z extrapolation leads to errors as large as 30 kJ/mol. 

Inner-shell correlation is a substantial part of the absolute correlation energy even for late first-row systems for second-row systems, it in fact rivals the absolute valence correlation energy in importance. However, its relative contribution to molecular TAEs is fairly small in benzene, for instance, it amounts to less than 0.7 % of the TAE. Even so, at 7 kcal/mol, its contribution is important by any reasonable thermochemical standard. By the same token, a 1 % relative error in a 7 kcal/mol contribution is tolerable even by benchmark thermochemistry standards, while the same relative error in a 300 kcal/mol contribution would be unacceptable even by the chemical accuracy standards. 

We have then found that the values of the intra- and inter-orbital energy contributions can be empirically determined in such a way that, for about 50 molecules with Ec ranging from 26 to about 650 kcal/mol, the total ab-initio valence correlation energies are reproduced within near-chemical accuracy of the theoretically calculated values. For one set of molecules, the ab-initio values had been obtained by valence CCSD(T) calculations in a cc-pVTZ basis (57) for another set, they were obtained by extrapolation to the complete basis 68),... 

The seven group correlation contributions El, Eb, Ell. Elb. .. were then determined by a least-mean-squares fit of Ec to the ab initio valence correlation energies of the afore-mentioned sets of molecules. They were found to have the following values 38) ... 

Table 2. Numbers of orbitals and orbital-pairs and comparison of valence correlation energies (mb) estimated by q. (3.3) with CCSD(T) values from reference (57) for selected systems. ...

A close estimate of the dynamic correlation energy was obtained by a simple formula in terms of pair populations and correlation contributions within and between localized molecular orbitals. The orbital and orbital-pair correlation strengths rapidly decrease with the distance between the orbitals in a pair. For instance, the total valence correlation energy of diamond per carbon atom, estimated as 164 mh, is the result of 84 mh from intra-orbital contributions, 74.5 mh from inter-orbital closest neighbors contributions, and 6.1 mh from interorbital vicinal contributions. The rapid decay of the orbital correlation contributions with the distance between the localized orbitals explains the near-... 

Figures 2 and 3 plot the valence correlation energies of Ne and FH obtained by various combinations of the CC or CC-R12 methods (using Ten-no s Slater-type correlation function) and basis sets (see ref. 35 for details). These figures are the stunning illustration of the extremely rapid convergence of correlation energies...

Figure 2 The valence correlation energies of Ne obtained by the CC or CC-R12 methods and the aug-cc-pVXZ basis sets (abbreviated as aXZ) [35].

Table 7.2 Valence correlation energies (—Econ, hiEa) from standard and R12 CCSD calculations and from extrapolation using Eq. (7.57) for seven closed-sheU singlet molecules...

For example, the first d-function provides a laree energy lowering, but the contribution 65% of the total (valence) correlation energy, the cc-pVTV 85%, cc-pVQZ 93%,... 

Table 11.8. The extrapolated CCSD(T) energy is —76.376 a.u., yielding a valence correlation energy of —0.308 a.u.

Besides the reduction of frozen-core errors when going from large-core to medium-core or small-core potentials also the valence correlation energies obtained in pseudopotential calculations become more accurate since the radial nodal structure is partially restored [97,98]. Clearly the accuracy of small-core potentials is traded against the low computational cost of the large-core po-... 

The performance of energy-consistent quasirelativistic 7-valence electron PPs for all halogen elements has been investigated in a study of the monohydrides and homonuclear dimers [242]. Special attention was also paid to the accuracy of valence correlation energies obtained with pseudo valence orbitals [97,98]. Some of the results for the halogen dimers is presented in Tables 13... 

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