Correlation energy equations estimating

GombAs, P., Acta Phys. Hung. 4, 187, Erweiterung der Hartree-Fockschen Gleichungen durch die Korrelation der Elektronen/ Extension of the Hartree-Fock equations through the correlation of the electrons. The correlation energy of the alkali metals is estimated with a statistical method. 

We have recently proposed (3) that the correlation energy can be estimated from equations of the form... 

Equation (85) has been confirmed very recently for the first row atoms (Gladney and Allen later obtained similar empirical values), dementi evaluated the correlation energies of these atoms and their ions empirically using his H.F. results and estimates of relativistic effects. The additivity observed is within the empirical uncertainty of the data. Figures 1 and 2 are based on his data and show the correlation energy increments in nitrogen and neon ions and atoms as more and more electrons are added. 

The only unknown quantity in eqn (5) is the exchange-correlation energy <, which is therefore defined by this equation. This quantity plays a crucial role in DF theory. It is, after a KS calculation, the only quantity for which a reliable estimate is needed to obtain a good total energy. 

Let y be the fraction of the basis-set correlation energy obtained by a CI-SD calculation on a molecule with n electrons. Sasaki showed that an approximate equation satisfied by y is y — 1 = pyn — p, where is a quantity whose value is typically 0.015 to 0.03 for molecules consisting of first-row atoms. For such molecules, use this equation to estimate the percent of the basis-set correlation energy obtained by CI-SD calculations for n = 20,50,100, and 200. 

In order to maintain the wave function antisymmetry, the diffusion QMC is normally used within the fixed node approximation, i.e. the nodes are fixed by the initial trial wave function. Unfortunately, the location of nodes for the exact wave function is far from trivial to determine, although simple approximations such as HF can give quite reasonable estimates. The fixed node diffusion QMC thus determines the best wave function with the nodal structure of the initial trial wave function. If the trial wave function has the correct nodal structure, the QMC will provide the exact solution to the Schrodinger equation, including the electron correlation energy. It should be noted that the region near the nuclei contributes most to the statistical error in QMC methods, and in many apphcations the core electrons are therefore replaced by a pseudopotential. 

Bond dissociation energies have also been obtained (Table 3.3) from calorimetric measurements of the heat of protonation of neutral complexes with CF3SO3H in dichlo-roethane. These heats correlate linearly with differences in pK (even differences in pK in another solvent, CH3CN) and thus can be used along with measured electrochemical potentials to estimate bond dissociation energies (Equation 3.127). 

VLE data are correlated by any one of thirteen equations representing the excess Gibbs energy in the liquid phase. These equations contain from two to five adjustable binary parameters these are estimated by a nonlinear regression method based on the maximum-likelihood principle (Anderson et al., 1978). 

More recent research provides reversible oxidation-reduction potential data (17). These allow the derivation of better stmcture-activity relationships in both photographic sensitization and other systems where electron-transfer sensitizers are important (see Dyes, sensitizing). Data for an extensive series of cyanine dyes are pubflshed, as obtained by second harmonic a-c voltammetry (17). A recent "quantitative stmcture-activity relationship" (QSAR) (34) shows that Brooker deviations for the heterocycHc nuclei (discussed above) can provide estimates of the oxidation potentials within 0.05 V. An oxidation potential plus a dye s absorption energy provide reduction potential estimates. Different regression equations were used for dyes with one-, three-, five-methine carbons in the chromophore. Also noted in Ref. 34 are previous correlations relating Brooker deviations for many heterocycHc nuclei to the piC (for protonation/decolorization) for carbocyanine dyes the piC is thus inversely related to oxidation potential values. 

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