CHAPTER 4 Density Functional Theory
4.2 Functionals
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correct the imperfections, in view of the fact that the underlying reasons for the limitations in the theory are a long way from being understood. This is true when the reason for failure is not the selection of the functional, integration grid, or basis set.335
DFT methods account reasonably well for hydrogen bonding, which is mainly electrostatic. There are, however, indications that DFT methods less accurately predict relative energies and inadequately portray transition structures. Nevertheless, it must be remembered that for DFT methods the number of systems for which it has been calibrated is still somewhat undersized.342 DFT methods differ according to the choice of the functional form of the exchange-correlation energy. The most preferable DFT method was found to be the gradient functional method of Becke.346 Theory gives rather little guidance about how such functionals should be chosen and therefore many different potentials have been put forward. A variety of functionals has been defined in computational chemistry history and they are characterised by the means by which they deal with the exchange and correlation components.331
approximation, the acronym SVWN is used. The LSDA method does, unfortunately, have limitations. Although its results are mostly superior to that of the Hartree-Fock method, it has a tendency to overestimate outcomes.335 This LSDA approximation, unfortunately, has a tendency to underestimate the exchange energy by about 10%, which in turn causes errors larger than the whole correlation energy, which is itself overestimated (regularly by a factor close to 2). Consequently, the bond strengths will also be overestimated.342 LSDA techniques may offer outcomes with comparable accuracy to those acquired from Hartree-Fock methods, in spite of the simplicity of their fundamental assumptions.331,342
4.2.2 Gradient Methods
In the early eighties, the first successful expansions to the purely local approximations were developed.335 To improve on the LSDA approach, a non-uniform electron gas needs to be considered. One way to do this is to make the correlation and exchange energies dependent on electron density and derivatives of the density; thus gradient- corrected functionals will entail electron spin densities as well as their gradients.331,342 The name given to these processes is Gradient Corrected or Generalised Gradient Approximation (GGA) methods (also sometimes called non-local functionals in the literature, which is somewhat misleading). 342
In 1986, Perdew and Wang (PW86)348 suggested altering the existing LSDA exchange expression. And in 1988 Becke349 proposed a gradient-corrected correlation functional (B or B88) which became rather popular, followed by another widely-used functional (not a correction) by Lee, Yang and Parr (LYP) in the same year.350 (These two forms were also combined to make the B-LYP method.) There is one empirical parameter in the LYP functional and it differs from other GGA functionals because it includes a few local components.335 Another functional, with a correction to the LSDA energy, was proposed by Perdew and Wang in 1991,331,342 PW91.351 It should, however, be noted that quite a few of these proposed functionals defied fundamental restrictions. P86 and PW91, for example, predict correlation energies for one-electron systems and for others, the exchange energy may be unsuccessful in cancelling the Coulomb self-repulsion.342
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4.2.3 Hybrid Functionals
Generally, the exchange contributions are considerably larger in absolute numbers than the analogous correlation effects. Therefore, a prerequisite for acquiring useful outcomes from DFT is a precise expression for the exchange functional specifically.335 A precise link can be made between the exchange-correlation energy and the corresponding potential connecting the non-interacting reference and actual system using the Hamiltonian and the definition of the exchange-correlation energy.342 The resulting equation involves integration over a parameter that allows for the electron-electron interaction. This equation is known as the Adiabatic Connection Formula (ACF).342
The Half-and-Half method may be described by writing the exchange energy as a blend of LSDA, a gradient correction term and exact exchange. Hybrid methods frequently refer to models that include exact exchange. The Adiabatic Connection Model (ACM) and Becke 3 parameter functional (B3)352 are therefore examples of hybrid models.
These functionals operate well and thus the Half-and-Half model is hardly ever used.344 Several hybrid functionals define the exchange functionals as a linear combination of Hartree-Fock, local, and gradient-corrected exchange terms. This exchange functional is then joined with a local and/or gradient-corrected correlation functional.331 Becke’s three- parameter formulation, B3 (B3LYP),352 is the best known of these hybrid functionals and these Becke-style hybrid functionals have been found to be superior to the conventional functionals defined thus far.331
From the time of their manifestation in the early nineties these hybrid functionals have experienced unparalleled success.335 In particular, the B3LYP functional has become a well known and extremely useful functional, due to its surprisingly good performance in many chemical applications.335 It was suggested by Stevens et al. in 1994353 and has an unsigned error of only slightly above 2 kcal/mol (with respect to the G2 data base). It is related to that originally suggested by Becke (using B88 and PW91); however, the PW91 correlation functional has been exchanged for the LYP functional.335
In 1996 Becke made further progress by dropping the number of parameters to one where the amount of exact exchange was empirically ascertained. This resulted in the B1B95 functional.354 Becke presented a new type of exchange-correlation functional, founded on an intricate fitting procedure, in the closing stages of his string of papers on
density functional thermochemistry. It was aptly labelled B97.355 Together, Schmider and Becke, then reparameterised this functional (with respect to the extended G2 set)356 and the ensuing B98 functional preserves the good absolute average and low maximum errors (11.9 kcal/mol and 9.1 kcal/mol, respectively). Additional development was also carried out in the same year on the original B97 functional by Hamprecht et al.; they called their result B97-1.335 The year 1998 also saw van Voorhis and Scuseria357 offer a new exchange-correlation functional known as VSXC; it was dependent on the non- interacting kinetic energy density, as well as on ρ and its gradient, Δρ.335
Development and research into the discovery of new and improved functionals continues. Some of the current functionals do have the ability to yield energy related results approaching alleged ―chemical accuracy‖. This means that the results are within less than 2 kcal/mol of experimentally determined energies, which is very high accuracy.
There are many literature records showing high accuracy obtained by means of modern functionals, and more are being published each year.335 Theory does not provide much assistance in choosing functionals, thus numerous different potentials have been suggested. To determine the best performing functional involves a comparison with experiments of high-level wave mechanics calculations or with functionals known to have been successful for similar compounds in the past.