Equation correlation effects

C. Valdemoro, M. P. de Lara-Castells, E. Perez-Romero, and L. M. Tel, The first order contracted density equations correlation effects. Adv. Quantum Chem. 31, 37 (1999). 

Kohn W and Sham L J 1965 Self-consistent equations including exchange and correlation effects Phys. Rev A 140 1133-8... 

Kohn W and L J Sham 1965. Self-consistent Equations Including Exchange and Correlation Effects. Physical Review A140 1133-1138. 

Self-Consistent Equations Including Exchange and Correlation Effects W. Kohn and L. J. Sham Physical Review 140 (1965) All33... 

The projection operator formalism also gives interesting aspects on the correlation problem. Previously one mainly used the secular equation (Eq. III.21) for investigating the symmetry properties of the solutions, and one was often satisfied with those approximate wave functions which were the simplest linear combinations of the basic functions having the correct symmetry. In our opinion, this problem is now better solved by means of the projection operators, and the use of the secular equations can be reserved for handling actual correlation effects. This implies also that, in place of the ordinary Slater determinants (Eq. III.17), we will essentially consider the projections of these functions as our basis. 

The Hammett equation is not the only LFER." ° Some, like the Hammett equation, correlate structural changes in reactants, but the Grunwald-Winstein relationship (see p. 452) correlates changes in solvent and the Brpnsted relation (see p. 337) relates acidity to catalysis. The Taft equation is a structure-reactivity equation that correlates only field effects. ... 

Note that the Kohn-Sham Hamiltonian hKS [Eq. (4.1)] is a local operator, uniquely determined by electron density15. This is the main difference with respect to the Hartree-Fock equations which contain a nonlocal operator, namely the exchange part of the potential operator. In addition, the KS equations incorporate the correlation effects through Vxc whereas they are lacking in the Hartree-Fock SCF scheme. Nevertheless, though the latter model cannot be considered a special case of the KS equations, there are some similarities between the Hartree-Fock and the Kohn-Sham methods, as both lead to a set of one-electron equations allowing to describe an n-electron system. 

For simple power law rate equations the effectiveness can be expressed in terms of the Thiele modulus, Eq 7.28. In those cases restriction is to irreversible, isothermal reactions without volume change. Other cases can be solved, but then the Thiele modulus alone is not sufficient for a correlation. 

Equation (96) shows that the effective KS potential may be simply obtained by adding to the standard KS potential of the isolated solute, an electrostatic correction which turns out to be the RE potential Or, and the exchange- correlation correction 8vxc. It is worth mentioning here, that Eq (96) is formally equivalent to the effective Fock operator correction bfteffi defined in the context of the self consistent reaction field (SCRF) theory [2,3,14] within the HF theory, the exchange contribution is exactly self-contained in Or, whereas correlation effects are completely neglected. As a result, within the HF theory 8v = Or, as expected. 

Another of our present aims is to continue investigating the theoretical properties of the 2- and 3-body terms. Thus, there are many exact relations linking the different 3-body terms arising from each of the 36 options of equation (9) as well as their holes counterparts [16]. These relations are interesting by themselves, because they widen our understanding of the problem and may be helpful for improving our approximations. Another important theoretical question is, as mentioned above, to investigate the reason why one Ai corrects the VCP error which in some cases is only due to the 3-body contribution but which may also involve other kind of correlation effects. 

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