HF Correlation

When calculating barriers for bowl-to-bowl inversions, MM methods generally give values that are too low, while semiempirical methods produce results which are either too high (AMI and PM3) or too low (MNDO). Ab initio methods, on the other hand, give satisfactory results if post-HF correlation effects are included. 

It follows from the relation (79) that both contributions to the HF correlation energy AT + AE = E " are of comparable magnitude and partially cancel out, leading to... 

We see that the HF correlation energy Eg " is a very small quantity indeed - four orders of magnitude smaller than the exchange energy. When compared with the known results for the KS correlation energy Eg s E or the QM one [Eq. (27)], it happens to be two orders of magnitude smaller. 

Functional Taylor series expansion of the functional minimized in Eq. (87), in powers of noK ") = [nGs( ) - gs( )] has been employed first, and Eq. (88) used in the last step. So E " is close to KS correlation energy functional taken for the GS density of HF approximation, corrected by the (much smaller) HF correlation energy, and a small remainder of the second order in the density difference. The last quantity gives an estimate to the large parentheses term of Eq. (28) in [12]. 

Thus Eq. (92) gives an approximation to this functional in terms of KS and HF correlation energy functionals. Rewriting Eq. (27) with the help of (93) and (64) we obtain... 

Finally, in careful comparative studies of the molecular electron densities generated by HF, correlated ab initio, and pure, self-interaction corrected, and hybrid DFT calculations, Cremer et al have made a very interesting observation [72, 73]. They found that the pure DFT generated densities differed from those obtained with accurate ab initio methods in a particular way, and that both hybrid, and self-interaction corrected DFT methods, yielded densities closer to the correct ones. Based on this observation, they suggested that mixing in of exact exchange in hybrid functionals serves as a proxy for the self-interaction correction. 

The energy barrier for the trans-cis isomerization is significantly lower than that for the trans-sin (48.6 versus 71.6 kcal/mol). Again the better agreement occurs with the HFS computations. Both HF correlated methods give barriers that are too large. 

There is also dramatic improvement when correlation is included (i.e., everything in Table 2 besides HF). Correlation is important whenever the number of electron pairs changes, as here. As discussed above, the HF and B3LYP results are less sensitive to the basis set than the post-HF methods (compare the aug-cc-pVTZ and aug-cc-pVQZ results). The strong dependence on both basis set and electron correlation makes EA(F) a difficult quantity to calculate. The most rigorously calculated value yet published is 328.32 kJmol [100], only 0.16 kJmol above the experimental benchmark. 

Hd correlates with Ha and Hb He correlates with He and Hf Hf correlates with He... 

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