Orbital-Dependent Exchange-Correlation Functional

Hock, A. and Engel, E. (1998). Pseudopotentials from orbital-dependent exchange-correlation functionals, Phys. Rev. A 58, 3578-3581. 

The Pair Density. Orbital-dependent Exchange-correlation Functionals... 

In the next subsection we consider two orbital-dependent exchange-correlation functionals used for soMds with the strong electron correlation. These functionals are used mainly with non-LCAO (PW, LMTO) basis and from this point of view are not quantum-chemical approaches. Therefore, we discuss them only briefly. 

Semi-empirical Orbital-Dependent Exchange-Correlation Functionals... 

This chapter is devoted to orbital-dependent exchange-correlation (xc) functionals, a concept that has attracted more and more attention during the last ten years. After a few preliminary remarks, which clarify the scope of this review and introduce the basic notation, some motivation will be given why such implicit density functionals are of definite interest, in spite of the fact that one has to cope with additional complications (compared to the standard xc-functionals). The basic idea of orbital-dependent xc-functionals is then illustrated by the simplest and, at the same time, most important functional of this type, the exact exchange of density functional theory (DFT for a review see e.g. [1], or the chapter by J. Perdew and S. Kurth in this volume). 

Eds., NATO ASI Series B, Plenum, New York, 1995, pp. 191-216. Recent Developments in Kohn-Sham Theory for Orbital Dependent Exchange-Correlation Energy Functionals. 

The dependence of the DFT results on the basis set used to expand the Kohn-Sham orbitals is illustrated in Table 4.3, which collects equilibrium geometry properties of water dimer obtained with the same exchange-correlation functional (B88/P86) but with different basis sets. 

To perform excited-state calculations, one has to approximate the exchange-correlation potential. Local self-interaction-free approximate exchange-correlation potentials have been proposed for this purpose [73]. We can try to construct these functionals as orbital-dependent functionals. There are different exchange-correlation functionals for the different excited states, and we suppose that the difference between the excited-state functionals can be adequately modeled through the occupation numbers (i.e., the electron configuration). Both the OPM and the KLI methods have been generalized for degenerate excited states [37,40]. 

Ec = E c - Ex have been employed. On the one hand, LDA and GGA type correlation functionals have been used [14], However, the success of the LDA (and, to a lesser extent, also the GGA) partially depends on an error cancellation between the exchange and correlation contributions, which is lost as soon as the exact Ex is used. On the other hand, the semiempirical orbital-dependent Colle-Salvetti functional [22] has been investigated [15]. Although the corresponding atomic correlation energies compare well [15] with the exact data extracted from experiment [23], the Colle-Salvetti correlation potential deviates substantially from the exact t)c = 8Ecl5n [24] in the case of closed subshell atoms [25]. 

Z is the nuclear charge, R-r is the distance between the nucleus and the electron, P is the density matrix (equation 16) and (qv Zo) are two-electron integrals (equation 17). f is an exchange/correlation functional, which depends on the electron density and perhaps as well the gradient of the density. Minimizing E with respect to the unknown orbital coefficients yields a set of matrix equations, the Kohn-Sham equations , analogous to the Roothaan-Hall equations (equation 11). 

Exchange-correlation functionals, which depend explicitly on Kohn-Sham orbitals such as meta-GGAs, hybrid- and hyper functionals discussed before fall also in this cathegory. 

In view of the fact that density functional theory and the approximations to the exchange-correlation functional is the field of intensive research, we expect further progress and intend to keep in pace with these developments. An example of these developments are our recent attempts to approximate the exchange-correlation component of the KSCED effective potential in a hybrid way53. The orbital-dependent SAOP potential157 was used to approximate in Eq. 52, whereas the... 

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