Parameterizations, correlation energy

The Parameterized Configuration Interaction (PCI-X) method simply takes the correlation energy and scales it by a constant factor X (typical value 1.2), i.e. it is assumed that the given combination of method and basis set recovers a constant fraction of the correlation energy. 

In Eq. [47], epc ( ) and exc (n) are the exchange-correlation energy densities for the nonpolarized (paramagnetic) and fully polarized (ferromagnetic) homogeneous electron gas. The form of both exc(n) and exc(n) has been conveniently parameterized by von Barth and Hedin. Other interpolations have also been proposed24,33 for eKC(n, J ). The results for the homogeneous electron gas can be used to construct an LSDA... 

An alternative approach to the calculation of accurate thermochemical data is to scale the computed correlation energy with multiplicative parameters determined by fitting to the experimental data. Pioneering methods using such an approach include the scaling all correlation (SAC) method of Gordon and Truhlar [32], the parameterized correlation (PCI-X) method of Siegbahn et al. [33], and the multi-coefficient correlation methods (MCCM) of Truhlar et al. [34-36]. Such methods can be used... 

The quality of a variational quantum Monte Carlo calculation is determined by the choice of the many-body wavefunction. The many-body wavefunction we use is of the parameterized Slater-Jastrow type which has been shown to yield accurate results both for the homogeneous electron gas and for solid silicon (14) (In the case of silicon, for example, 85% of the fixed-node diffusion Monte Carlo correlation energy is recovered). At a given coupling A, 4>A is written as... 

The methods have been very successful, but they do suffer drawbacks. The lack of parameters for many elements seriously limits the types of problems to which the methods can be applied and their accuracy for certain problems is not very good (for example, both MNDO and AM 1 do not well describe water-water interactions). There are also questions about the theoretical foundations of the models. The parameterization is performed using experimental data at a temperature of 298K and implicitly includes vibrational and correlation information about the state of the system. Therefore, the parameterization is used, in part, to compensate for quantities that the HF method cannot, by itself, account for. But what happens if vibrational or correlation energy calculations are performed With these caveats and if one can be certain of their accuracy in given circumstances, the methods are very useful as calculations can be performed with them much more quickly than ab initio QM calculations. Even so, they are probably still too computationally intensive to treat complete condensed phase systems in a routine manner. 

The first generation is the local density approximation (LDA). This estimation involves the Dirac functional for exchange, which is nothing else than the functional proposed by Dirac [15] in 1927 for the so-called Thomas-Fermi-Dirac model of the atoms. For the correlation energy, some parameterizations have been proposed, and the formula can be considered as the limit of what can be obtained at this level of approximation [16-18], The Xa approximation falls into this category, since a known proportion of the exchange energy approximates the correlation. 

The analytical form for the correlation energy of a uniform electron gas, which is purely dynamical correlation, has been derived in the high and low density limits. For intermediate densities, the correlation energy has been determined to a high precision by quantum Monte Carlo methods (Section 4.16). In order to use these results in DFT calculations, it is desirable to have a suitable analytic interpolation formula, and such formulas have been constructed by Vosko, Wilk and Nusair (VWN) and by Perdew and Wang (PW), and are considered to be accurate fits. The VWN parameterization is given in eq. (6.36), where a slightly different spin-polarization function has been used. 

However, extended parameterization of the local correlation energy may be unfolded since considering the fit with an LSDA p and p ) ana-l>4ical expression by Vosko, Wilk andNusair (VWN) (Vosko et al., 1980),... 

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