Exact-exchange Methods

In Section 2 we discussed the theorem of Hohenberg and Kohn that states that once the electron density of the ground state is known any other groimd-state property, including the total electronic energy E, can be calculated somehow . Schematically, this procedure was depicted in Eq. (17). The approach of Kohn and Sham introduced an intermediate step, i.e. fi om the electron density one calculates first some effective potential and fi om that one obtains E, cf Eq. (18). The Kohn-Sham approach leads actually also to a set of Kohn-Sham orbitals 

E is the total exchange-correlation energy that may be split into an exchange and a correlation part. 

In the preceding parts of this report we have discussed studies based on approximations to both and e. In many of these approximations, each part by itself (i.e. exchange and correlation separately) are not optimally approximated but their sum benefits from a favomable error cancellation so that E is more accurate than E and E separately. This is, e.g., the case for (most of) the LDAs and GGAs. 

One may, however, replace the approximate treatment of the exchange energy through the exact expression. 

The only difference from the Hartree-Fock expression, Eq. (12), is that in Eq. (81) the Kohn-Sham orbitals tpi and not the Hartree-Fock orbitals (p, enter. 

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Here, we shall briefly discuss three recent studies using exact-exchange methods. In all of them, smaller atomic and molecular systems were examined... 

In a later paper, Casida et used this formalism to calculate the excitation energies of some smaller molecules (N2, CO, CH2, and C2H4). In Table 12 we have collected their results for N2 and in Table 13 those for CO. Those for N2 can be compared directly with those of Table 11 obtained with an exact-exchange method. The results of both tables show that the time-dependent density-functional methods give results that are almost as accurate as those of the sophisticated correlation methods (like coupled-cluster, configuration-interaction, multiple-configuration, or polarization-propagator methods) and considerably... 

Another, maybe even more important, field is that of excitations and ionizations. Here, the facts that the conventional density-functional methods are ground-state methods and that the most common approximate density functionals suffer from a non-physical self-interaction of the electrons and from a wrong asymptotic behaviour (Section 9) lead to problems when attempting to study excitations and ionizations. The exact-exchange methods remove some of the problems of the approximate density functionals (Section 10) and were found to... 

Orbital-dependent exact-exchange methods in density functional theory... 

To avoid numerical problems in gaussian basis set due to the unbalance description of the basis set and the vanishing small eigenvalue of the density response matrix, the approximated KLI and LHF (or CEDA or ELP) effective exact-exchange methods can be used. These methods are still computationally much more elaborated than conventional local, semilocal or hybrid functionals, because the calculation of the Slater potential as well as the (self-consistent) correction term is required. A straightforward construction of the Slater potential... 

Pisani C and R Dovesi 1980. Exact-Exchange Hartree-Fock Calculations for Periodic Systems. I. Illustration of the Method. International Journal of Quantum Chemistry XVII 501-516. 

Models which include exact exchange are often called hybrid methods, the names Adiabatic Connection Model (ACM) and Becke 3 parameter functional (B3) are examples of such hybrid models defined by eq. (6.35). The <, d and parameters are determined by fitting to experimental data and depend on the form chosen for typical values are a 0.2, d 0.7 and c 0.8. Owing to the substantially better performance of such parameterized functionals the Half-and-Half model is rarely used anymore. The B3 procedure has been generalized to include more filling parameters, however, the improvement is rather small. 

The calculated dipole moment is remarkably insensitive to the size of the basis set. Note that the SVWN value in this case is substantially better than BLYP and BPW91, i.e. this is a case where the theoretically poorer method provides better results than the more advanced gradient methods. Inclusion of exact exchange again improves the performance, and provides results very close to the experimental value, even with quite small basis sets. 

It is not possible to give exact design methods for plate heat exchangers. They are proprietary designs, and will normally be specified in consultation with the manufacturers. Information on the performance of the various patterns of plate used is not generally... 

This method is usually thought as an approach allowing one to find the exact exchange potential. It may be considered [17] as an approximation to the exact GS problem, similar to the HF approximation namely, the solution of the optimized potential (OP) approximation - the energy Egg and the wave function Gs - stems from the following minimization problem... 

The difficulty of this problem can be appreciated by noticing that in order to solve the Kohn-Sham equations exactly, one must have the exact exchange-correlation potential which, moreover, must be obtained from the exact exchange-correlation functional c[p( )] given by Eq. (160). As this functional is not known, the attempts to obtain a direct solution to the Kohn-Sham equations have had to rely on the use of approximate exchange-correlation functionals. This approximate direct method, however, does not satisfy the requirement of functional iV-representability,... 

Development of methods related to DFT that can treat this situation accurately is an active area of research where considerable progress is being made. Two representative examples of this kind of work are P. Rinke, A. Qteish, J. Neugebauer, and M. Scheffler, Exciting Prospects for Solids Exact Exchange Based Functional Meet Quasiparticle Energy Calculations, Phys. Stat. Sol. 245 (2008), 929, and J. Uddin, J. E. Peralta, and G. E. Scuseria, Density Functional Theory Study of Bulk Platinum Monoxide, Phys. Rev. B, 71 (2005), 155112. 

The classes of functionals shown in Fig. 10.2 do not exhaust the kinds of functionals that can be constructed. Any functional that incorporates exact exchange is called a hybrid functional. The hypcr-GGA functionals listed above can therefore also be referred to as hybrid-GGA methods. Hybrid-meta-GGA functionals also exist—these combine exact exchange with meta-GGA functionals this group of functionals is not shown in Fig. 10.2. 

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