About Correlation Energy

The post self-consistent era was mainly dedicated to the implementation of the so nominated correlation energy in the computation (Putz Chiriac, 2008). 

Firstly, it was noticed that a single Slater determinant (on which base the current HF analysis was exposed) can never account for a complete description of the many-electronic interaction. That is, the correlation 

The next step was sustained by the assumption that the correlation energy can be seen as the perturbation of the self-consistent-field energy, which is associated with a wave function derived for a single electronic configuration. At this point the basic methods of approximation used in quantum chemistry, namely the perturbation and variational, can be considered. 

In the case that perturbation method is employed, assuming the unperturbed wave function and energy as the HF solutions the exact eigen-functions and eigen-values can be written as expanded series 

On the other side, the linear variational method can be practiced within the configuration interaction (Cl) approach of the many-electronic wave-function  

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The direct term Ej resembles the classical Coulomb interaction of the electron density, while the exchange term Ex has no parallel with any classical term. It can be regarded as the energy due to the indistinguishability of the electrons. Notice that up to this point it has not been necessary to talk about correlation energy it is included in both the Ej and the Ex terms. 

In order to gather more information about this problem, it was deemed worthwhile to follow the energetics of the alkylation reaction of water by methyl-, ethyl-, and fluo-roethyldiazonium ions. The main goal of these calculations was to establish whether transition-state calculations can provide information about hard versus soft electrophilic character of these species.12 Computations at Hartree-Fock and MP2 level were performed using the 6-31G basis set. It was found that both at the Hartree-Fock level and when correlation energy affects were included, the ethyl and fluoroethyl species do not show the presence of a transition state, while the methyl species show a small transition state. It was concluded that transition state computations cannot shed light on the characters of these species. 

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. 

The correlation energy can in principle be resolved as a sum of contributions from tT> ii> ti correlations. Such a resolution even in the uniform density limit, is not really needed for the construction of density functional approximations, and no assumption about the spin resolution has been made in any of the functionals from our research group (which are all correct by construction in the uniform density limit). 

For the total correlation energy (r Q, so much is known about the r, — 0 and r, — 00 no limits that accurate values for all r, and can be found by interpolation [58], without ever using the Monte Carlo or other data. For the spin resolution e r, Q/e (r, Q, however, so little is known about these limits that we must and do rely on the Monte Carlo data. The spin resolution of Eq. (24) has recently been generalized to all [59]. 

Usually, SD-CI wavefiinctions recover about 90-95% of the correlation energy, inclusion of triply and quadruply substituted configurations is adequate for many purposes, and inclusion of quintuply and sextuply excited configurations is sufficient for describing most chemical reactions. Since the explosion of the number of configurations typically starts with the quadruple excitations, our first priority is the predictive deletion of deadwood from this group. 

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),... 

Polarography and ESR data provide important information about the energies and electron distribution of the excited states of annelated benzenes. " By incorporating rehybridization effects into the Hiickel model of electron densities, a correlation between ring strain, experimental spin densities, and redox potentials is obtained for a series of naphthalenes and naphthoquinones. These studies provide further support for ring-strain induced rehybridization. 

A second observation about the LYP functional is that it predicts no correlation energy for a fully spin-polarized system of electrons. Yet, in the uniform-density limit, the correlation energy at full spin-polarization is about half that of the unpolarized system [3, 4, 57]. Even in the Ne atom, the parallel-spin contribution accounts for about 24% of the total correlation energy (Sect. 3.4). 

From the last column of the table, we see that the ratio of the parallel-spin to the total correlation energy is remarkably independent of the size of the basis set. Contrary to expectation, the parallel-spin correlation contribution appears to be about as difficult to account for within a finite basis-set approach as the antiparallel-spin correlation. Our investigation does not provide a careful study of the basis-set saturation behavior in MP2 calculations, such as is given in Refs. [74,72,75,33]. However, our results show that, with small- and moderate-sized basis sets which are sufficiently flexible for most purposes and computationally tractable in calculations on larger systems, there is no evidence that the parallel-spin correlation contribution converges more rapidly than the antiparallel-spin contribution. A plausible explanation for this effect is that, for small interelectronic separations, the wavefunction becomes a function of the separation, which is difficult to represent in a finite basis-set approach for either spin channel. The cusp condition of Eq. (19) is a noticeable manifestation of this dependence, but does not imply that the antiparallel-spin channel is more difficult to describe with a moderate-sized basis set than the parallel channel. In fact, in the parallel correlation hole, there is a higher-order cusp condition, relating the second and third derivatives with respect to u [76]. 

It is also clear from Table 5 that the absolute basis-set truncation error in Ne is about three times bigger for the antiparallel-spin correlation energy than for parallel. Thus the proposed spin-analysis hybrid of Ref. [35] may yet have some (limited) utility. 

The discussion of the hierarchy of functionals in Section 10.2 highlighted the observation that while the exchange energy of a collection of electrons can, in principle, be evaluated exactly, the correlation energy cannot. As a result, it is reasonable to be particularly concerned about the reliability of DFT... 

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