Explicit Density Functionals

Another similar approach applies an explicit density-functional theory treatment to the solute molecules, while representing the contribution of the solvent molecules as an effective potential [105]. 

Atomic units will be used throughout. The explicit density functionals representing the different contributions to the energy from the different terms of the hamiltonian are found performing expectation values taking Slater determinants of local plane waves as in the standard Fermi gas model. Those representing the first relativistic corrections are calculated in the Appendix. 

In view of these results there is little doubt that the density dependence of explicit density functionals for the x-only energy has to be modified in the relativistic regime. 

As is well known, the LDA overestimates the exact Ec of neutral atoms by roughly a factor of 2. This error increases to a factor of 5 or more for highly charged ions. Moreover, while the PW91-GGA is rather close to the exact Ec for neutral helium, the error increases to a factor of 2 for Fm98+. Obviously, these explicit density functionals do not scale properly with Z. The same is true for the orbital-dependent Colie-... 

If we separate out the Hartree energy from this, which can easily be done since it can be written as an explicit density functional, we can define the exchange-correlation hole, nxc(x,x ), around an electron as... 

The presently available explicit approximations for the relativistic xc-energy functional are presented in Section 4. Both implicit functionals (as the exact exchange) and explicit density functionals (as the RLDA and RGGA) are discussed (on the basis of the information on the RHEG in Appendix C and that on the relativistic gradient expansion in Appendix E). Section 4 also contains a number of illustrative results obtained with the various functionals. However, no attempt is made to review the wealth of RDFT applications in quantum chemistry (see e.g.[74-88]) and condensed matter theory (see e.g.[89-l(X)]) as well as the substantial literature on nonrelativistic xc-functionals (see e.g.[l]). In this respect the reader is referred to the original literature. The review is concluded by a brief summary in Section 5. 

As a fully nonlocal alternative to these explicit density functionals orbital-dependent (implicit) density functionals have been suggested. In addition to the exact exchange [51,52] some approximate correlation functionals are available, both empirical [166] and first-principles forms [57,59,60], As is already clear from Section 3.4 this concept can also be used in the relativistic situation. The status of relativistic implicit functionals [54] will be reviewed in Section 4.1. In particular, the various ingredients of the exact exchange will be analyzed. Subsequently the results obtained with the exact exchange will then serve as reference data for the analysis of the RLDA and RGGA. 

The corresponding definition of the nonadditive component of a property Blp], defined by an explicit density functional, is ... 

For some computational techniques in quantum chemistry a simple zero-th order approximation of the electron density of any atom of the system can be useful as the starting point of an iterative procedure. A very simple description of the electron density and binding energy of any atom or ion allows a rapid evaluation of very complex stmctures. This is the spirit of the orbital-free, explicit density functional approaches, usually based on the Thomas-Fermi-Dirac model and its extensions [1]. 

One important property that has been emphasized a number of times now is the — 1/r decay of the exact exchange potential of finite systems. In view of the long-range character of the underlying self-interaction integral, one might ask whether it is possible to reproduce this behavior by some explicit density functional In Sect. 2.1.2 it has been demonstrated that the LDA potential decays exponentially. In the case of the GGA, the second explicit density functional of interest,... 

The high accuracy attained by complex orbital functionals implemented via the OEP, and the fact that it is easier to devise orbital functionals than explicit density functionals, makes the OEP concept attractive, but the computational cost of solving the OEP integral equation is a major drawback. However, this computational cost is significantly reduced by the KLI approximation and other recently proposed simplifications. " In the context of the EXX method (i.e. using the Fock exchange term as an orbital functional) the OEP is a viable way to proceed. For more complex orbital functionals, additional simplifications may be necessary. " ... 

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