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In the crudest approximation (taking V"6 to be linear in l), the integral is given as the average of the values at the two end-points.

Exc« i ((Vxhc°le(0)|^o) + |Vxhcole(1)|^i)) (6.50)

In the l = 0 limit, the electrons are non-interacting and there is consequently no correlation energy, only exchange energy. Furthermore, since the exact wave function in this case is a single Slater determinant composed of KS orbitals, the exchange energy is exactly that given by Hartree-Fock theory (eq. (3.33)). If the KS orbitals were identical to the HF orbitals, the exchange energy would be precisely the energy calculated by HF wave mechanics methods. The last term in eq. (6.50) is still unknown. Approximating it by the LSDA result defines the Half-and-Half (H + H) method.58

Since the GGA methods give a substantial improvement over LDA, a generalized version of the H+H method may be defined by writing the exchange energy as a combination of LSDA, exact exchange and a gradient correction term. The correlation energy may similarly be taken as the LSDA formula plus a gradient correction term. Models that include exact exchange are often denoted hybrid methods, the Adiabatic Connection Model (ACM) and Becke 3 parameter functional (B3) methods are examples of such hybrid models, with the popular B3LYP method defined by eq. (6.52).59 An alternative version uses the PW91 correlation functional and has the acronym B3PW91, and an O3LYP combination has also been used.

ExBc3LYP = (1 - a)ExLSDA + aExexact + bAExB88 + (1 - c )EcLSDA + cELYP (6.52)

The a, b and c parameters are determined by fitting to experimental data and depend on the chosen forms for EGGA and EGGA, with typical values being a ~ 0.2, b ~ 0.7 and c ~ 0.8. Subsequent versions denoted B97 and B98 employed ten fitting parameters,60 but the improvements were rather marginal relative to the three parameters version.

The t-HCTH functional has been augmented with exact exchange to produce the acronym t-HCTH-hybrid.61 The PBE functional has also been improved by addition of exact exchange to give the PBE0 functional (also denoted PBE1PBE in the literature),62 where the mixing coefficient for the exact exchange is argued to have a value of 0.25 from perturbation arguments.63 Similarly, the third-rung TPSS functional has been augmented with ~10% exact exchange to give the TPSSh method.64

Inclusion of exact HF exchange is often found to improve the calculated results, although the optimum fraction to include depends on the specific property of interest. The improvement of new functionals by inclusion of a suitable fraction of exact exchange is now a standard feature. At least part of the improvement may arise from reducing the self-interaction error, since HF theory is completely self-interaction-free.

6.5.5 Generalized random phase methods

At the fifth level of the Jacob's ladder classification, the full information of the KS orbitals is employed, i.e. not only the occupied but also the virtual orbitals are included. The formalism here becomes similar to those used in the random phase approximation (Section 10.9), but very little work has appeared on such methods. Inclusion of the virtual orbitals is expected to significantly improve on, for example, dispersion (such as van der Waals) interactions, which is a significant problem for almost all current functionals.

One approach that can be considered as falling into this category is the class of Optimized Effective Potential (OEP) methods.65 The central idea is that the energy as a functional of the density is unknown (or at least the exchange-correlation part is), but the energy as a function of the orbitals is well known from wave function theory to a given order in the correlation, as defined for example by a perturbation expansion. Since the density is given by the sum of the square of the orbitals, this implicitly defines the energy as a function of the density. By requiring that the density derived from a Kohn-Sham calculation using a single-determinant wave function exactly matches the density derived from a (correlated) wave function, this implicitly defines the exchange-correlation potential.

The reference wave functions have so far been based on an MBPT type expansion (Section 4.8).The OEP1 method is defined by terminating the reference density at first order in the perturbation series. Since correlation only enters the perturbation expansion at order two, this yields the exchange-only potential. Terminating the expansion at second order defines the OEP2 method and corresponds to constructing a KS determinant that yields the (generalized) MP2 density. From the condition that the MP2-like density matrix matches that from the KS determinant, one may derive a set of coupled equations at the orbital level that provides the exchange-correlation potential correct to second order in the correlation. The OEP2 method is computationally equivalent to an iterative MP2 calculation, i.e. such calculations are computationally more expensive than standard DFT methods. Furthermore the OEP2 method has basis set requirements similar to other correlated wave function methods and thus cannot benefit from the faster basis set convergence of other DFT methods. Not surprisingly, OEP2 provides results of roughly MP2 quality although, in favourable cases, the performance may approach those from coupled cluster calculations. It does have the desirable feature that it can describe for example dispersion interactions, which are problematic with almost all traditional functionals.

Whether one should consider the OEP method as a density or wave functional theory is an open question, as it clearly tries to combine the best of both worlds. It has the advantage of being able to systematically improve the results towards the exact limit, but inherits also the wave function disadvantages of a slow convergence with respect to basis set size.

6.5.6 Functionals overview

The introduction of GGA and hybrid functionals during the early 1990s yielded a major improvement in terms of accuracy for chemical applications, and resulted in the Nobel prize being awarded to W. Kohn and J. A. Pople in 1998. Progress since this initial exciting developments has been slower, and the (in)famous B3LYP functional59 proposed in 1993 still represents one of the most successful in terms of overall performance. Unfortunately, neither the addition of more fitting parameters, the addition of more variables in the functionals, nor imposing more fundamental restrictions for the functional form have (yet) provided models with a significantly better overall per-formance.66 Although the performance for a given property can be improved by tailoring the functional form or parameters, such measures often result in the deterioration of the results for other properties.

It should be noted that the implicit cancellation of the long-range part of the exchange and correlation energies implies that the two functional parts should be at the same level of the ladder, and preferably developed in an integrated fashion. A popular topic in the literature is to search for a magic combination of exchange and correlation functionals, perhaps with a few adjustable scaling parameters and a choice of basis set, in order to reproduce a selected set of experimental data. This is not a theoretically justified procedure and should be considered merely as data fitting without much physical relevance. Nevertheless, such a procedure can of course be taken as an "experimental" fitting function that can be useful for predicting specific properties for a series of compounds.

Table 6.1 shows an overview of commonly used functionals given by their acronym, and placed in the Jacob's ladder classification. One may furthermore differentiate the functionals based on their use (or lack) of experimental data for assigning values to the parameters in the functional forms. The non-empirical ones such as the PW86,

Table 6.1 Perdew classification of exchange-correlation functionals

Level

Name

Variables

Examples

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