Hybrid methods correction

The Monte Carlo approach, although much slower than the Hybrid method, makes it possible to address very large systems quite efficiently. It should be noted that the Monte Carlo approach gives a correct estimation of thermodynamic properties even though the number of production steps is a tiny fraction of the total number of possible ionization states. 

A more complex set of functionals utilizes the electron density and its gradient. These are called gradient-corrected methods. There are also hybrid methods that combine functionals from other methods with pieces of a Hartree-Fock calculation, usually the exchange integrals. 

Gradient corrected methods usually perform much better than LSDA. For the G2-1 data set (see Section 5.5), omitting electron affinities, the mean absolute deviations shown in Table 6.1 are obtained. The improvement achieved by adding gradient terms is impressive, and hybrid methods (like B3PW91) perform almost as well as the elaborate G2 model for these test cases. For a somewhat larger set of reference data, called the G2-2 set, the data shown in Table 6.2 are obtained. 

Muradoglu, M., S. B. Pope, and D. A. Caughey (2001). The hybrid method for the PDF equations of turbulent reactive flows Consistency conditions and correction algorithms. 

Table 7 shows the results from DFT and selected MO calculations. The geometries obtained by the local and nonlocal DFT methods are very similar to the ones calculated by MO-based methods. The LDA calculations, however, underestimate the activation energy by 10-13 kcal/mol. The use of nonlocal gradient corrections improves the performance of the calculations, although the activation energy is still underestimated by 3-5 kcal/mol. Unfortunately, no calculations using hybrid methods have been published yet. 

Iq this study the B3LYP hybrid method proposed by Becke [31] and included in the Gaussian 98 [30] package was used. This method includes a mixture of Hartree-Fock and DFT exchange terms associated with the gradient corrected correlation functional of Lee et al [32], In all the calculations the metal cluster geometry was kept frozen. 

Use of this damping term within the D3 correction -D3(BJ) is the method now promoted by Grimme. Mean deviations for a benchmark data set that involve weak interactions, chemical reactions, and conformations are reduced by at least 1 kcal mol with the inclusion of the D3 corrections for a range of functionals. Even the performance of the double hybrid methods can be improved by about... 

Kozuch and Martin have proposed combining the double hybrid method with the SCS-MP2 treatment along with a dispersion correction. This so-called DSD-DFT method (dispersion corrected, spin-component-scaled double hybrid) has the exchange-correlation term... 

To summarize, abundances are not necessarily better determined from model fitting. However, model fitting, if done with a sufficient number of constraints, provides ionization correction factors relevant for the object under study that should be more accurate than simple formulae derived from grids of photoionization models. This could be called a hybrid method to derive abundances. Such a method was for example used by Aller Czyzak (1983) and Aller Keyes (1987) to derive the abundances in a large sample of Galactic planetary nebulae, and is still being used by Aller and his coworkers. It must however be kept in mind that if photoionization models do not reproduce the temperature sensitive line ratios, this actually points to a problem that has to be solved before one can claim to have obtained reliable abundances. 

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