Semiempirical Functionals

Semiempirical functionals intend to reproduce accurate properties using as many semiempirical parameters as needed. The concept of these functionals may be based on the force fields of molecular mechanics, for example, CHARmm (Brooks et al. 1983) and Amber (Pearlman et al. 1995), which have been established to determine the structures of biomacromolecules such as folded proteins. In most cases, the development of such force fields focuses only on constructing potentials to yield highly accurate molecular structures. Likewise, semiempirical functionals have been developed to provide highly accurate properties. Note, however, that 

The B97 semiempirical functional (Becke 1997) is the first semiempirical functional and has become the basis for conventional semiempirical functionals. 

As easily presumed from the concept, the above semiempirical functionals are often superior to other types of functionals in terms of the ability to reproduce chemical properties and reactions. Nevertheless, it has recently been reported that 

Finally, it should be emphasized that conventional functionals have their own advantages and disadvantages and have been used to trade off these characteristics, depending on the calculated systems. It is, therefore, too optimistic to consider that functionals steadily approach the universal functional, as if climbing Jacob s ladder (Fig. 5.2) year by year, and consequently that the latest, state-of-the-art functionals are superior to the conventional ones. 

Antony, J., Grimme, S. Phys. Chem. Chem. Phys. 8, 5287-5293 (2006) 

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As we will see presently, several suggestions for the explicit dependence of the integrand f on the densities and their gradients exist, including semiempirical functionals which contain parameters that are calibrated against reference values rather than being derived from first principles. In practice, EXcA is usually split into its exchange and correlation contributions... 

These approaches may include (1) purely empirical methods that try to simulate conformations by using classical molecular mechanics and adjustable parameters, still employed in very large molecular systems (2) potential energy determination with empirical and semiempirical functions consisting... 

Wauchope and Getzen [52] employ a semiempirical function including the molar heat of fusion to fit Xw =f(T) for PAHs. May et al. [53] employ an empirical, cubic temperature function for PAHs. A quadratic function has been derived by Yaws et al. [29] for alkanes (C5-C17) and cycloalkanes (C5-C15) ... 

Before beginning our discussion of wave function-based electronic structure theory, we note that an alternative, rigorous approach to electronic structure is provided by DFT (this volume, chapter by Ayers and Yang). DFT is based on the premise that all information about the electronic system can be extracted from the electron density, rather than from the electronic wave function. The attraction of DFT is that the electron density is a much simpler entity than the wave function, depending on just three spatial coordinates rather than on the An spatial and spin coordinates of n electrons. However, a difficulty of DFT is that no accurate, non-empirical method has yet been devised to extract the necessary information from the electron density. Current DFT calculations are therefore, to a large extent, based on semiempirical functionals [12], in which a set of parameters is fitted to experimental data. Nevertheless, the fitted parameters are universal in the sense that they are not atom-dependent or molecule-dependent. Also, the accuracy achieved in this manner is often high, surpassed only by the most elaborate wave function methods [13]. 

This transformation from the dipole to the monopole approximation in the calculations of the intermolecular forces has very significant consequences and represents an essential step in the improvement of the calculations. More recently further refinements have still been introduced in the calculations, prominent among which are the replacement of molecular polarizability by bond polarizabilities and the introduction of a supplementary short-range repulsion term generally in the form of a semiempirical function of the type proposed by Kitaygorodskii and used more extensively by Faidni and Simonetta. Details of these refinements should be looked for in the original papers. 

Semiempirical functionals Functionals that are developed to reproduce accurate properties with many semiempirical parameters. 

This functional contains 7 semiempirical parameters (a through / and a) for each exchange, parallel-spin or opposite-spin pair correlation functional, and consequently has 21 semiempirical parameters in total (for the parameter values, see van Voorhis and Scuseria 1998). Therefore, this functional is also classified as a semiempirical functional (see Sect. 5.6) and is actually taken as the first meta-GGA semiempirical functional using Za- This functional is also characteristic in associating the correlation functional with the exchange one through the same hg. 

The long-range correction has also been applied to a semiempirical functional (see Sect. 5.6). The first long-range corrected semiempirical functional is the ft)B97X functional (Chai and Head-Gordon 2008a) in the form. 

Finally, the semiempirical dispersion-corrected functionals are a modification of conventional semiempirical functionals (see Seet. 5.6) for dispersion effects. DFT-D functionals such as the BLYP-D, B3LYP-D, and B97-D functionals (Antony and Grimme 2006) and several Mx-series funetionals sueh as the M05-2x and M06-2x functionals (Zhao and Truhlar 2008) are ineluded in these semiempirical dispersion-corrected functionals. In the DFT-D functionals, there are three versions, DFT-Dl, DFT-D2, and DFT-D3, based on the level of dispersion corrections. For a deep understanding of the dispersion corrections, it is interesting to examine the highest level DFT-D3 functional (Grimme et al. 2010),... 

The semiempirical functionals are fitted to selected data from experiment or from the ab-initio calculations. The higher the rung of the functional the larger is the number of parameters (functionals with as many as 21 fit parameters are popular in chemistry). Is DFT ab-initio or semiempirical As was suggested in [357] it can fall in between as a nonempirical theory when the functionals are constructed without empirical fitting. 

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