Orbital-dependent functionals

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]. 

A fully relativistic extension of the scheme put forward in [12] has been introduced in [19], including the transverse electron-electron interaction (Breit +. .. ) and vacuum corrections. Restricting the discussion to the no-pair approximation [28] for simplicity, we here compare this perturbative approach to orbital-dependent Exc to the relativistic variant of the adiabatic connection formalism [29], demonstrating that the latter allows for a direct extraction of an RPA-like orbital-dependent functional for Exc- In addition, we provide some first numerical results for atomic Ec. 

Finally we mention some basic relations which are essential in the discussion of explicitly orbital dependent functionals. Examples of such functionals are the Kohn-Sham kinetic energy and the exchange energy which are dependent on the density due to the fact that the Kohn-Sham orbitals are uniquely determined by the density. The functional dependence of the Kohn-Sham orbitals on the density is not explicitly known. However one can still obtain the functional derivative of orbital dependent functionals as a solution to an integral equation. Suppose we have an explicit orbital dependent approximation for in terms of the Kohn-Sham orbitals then... 

The importance of orbital dependent functionals for a correct representation of the atomic shell structure, the correct properties of v for heteronuclear molecules, and the related particle number dependent properties will be discussed in Sect. 5.5. 

Other types of functionals include hybrid functionals [140], incorporating a fraction of Hartree-Fock exchange, orbital dependent functionals, kinetic energy functionals and so on [120]. 

As we have discussed already, the kinetic energy, Tsj can be readily rep>-resented as a functional written in terms of single-particle orbitals (1.37), whereas expressing it in terms of the density directly is much harder. Such functionals are termed orbital functionals. Another important example of an orbital-dependent functional is the exchange energy (1.39). The Meta-GGAs and hybrid functionals mentioned above are also orbital functionals. 

These equations can be solved through minimization, using an algorithm for orbital-dependent functionals [46,47]. 

There is one further orbital-dependent functional which can be mentioned at this point. In the nonrelativistic context it has been realized rather early [184] that, as a matter of principle, the self-interaction corrected (SIC) LDA of Perdew and Zunger [143] represents an implicit functional for which the 0PM should be used. A relativistic version of the Perdew-Zunger SIC has been proposed by Rieger and Vogl [185] as well as Severin et al. [186,187,46]. This functional, however, has not yet found widespread use, neither within the conventional,... 

Eqs. (20), (24), and (25), known as the Kohn-Sham equations, are formally exact and contain only one unknown term, EUp]- It is Exc that is approximated in Kohn-Sham DPT, not the conventional exchange-correlation energy Exact treatment of the kinetic energy as an orbital-dependent functional is cmcial to the practicality of this scheme because TTp] and Tsip] are notoriously difficult to approximate as explicit... 

In the general case of orbital-dependent functionals, minimization with respect to orbitals is only an approximation to the true Kohn-Sham scheme [281-285] (see also Ref. [58] concerning the gauge invariance problem with conventional r-dependent functionals). 

In the class of orbital-dependent functionals, the LDA-l-U method [24] also deserves special mention for its importance in solid-state calculations on open-shell transition metal compounds. The LDAh-U Hamiltonian contains an on-site term U, whose features reproduce in a parametric form the orbital-dependence of the HE theory. 

Figure 27 Excitation energies for the He atom obtained with different approaches. KS marks the orbital eigenvalues for the exact Kohn-Sham potential, ALDA the results with the adiabatic local-density approximation, and for the TDOEP approaches different orbital-dependent functionals have been used. Both the ALDA and the TDOEP results have been obtained using the time-dependent density-functional theory...

The next step to go beyond GGA (and meta-GGA °) functionals, whilst staying within the original KS framework, is to consider funetionals which depend explicitly on the Kohn-Sham orbitals. The formalism of these orbital-dependent functionals is the central part of this work and it will... 

The main advantage of the orbital-dependent functionals is that the exchange-interaction can be treated exactly because its representation in term of KS orbitals is equivalent to the well-known Flartree-Fock expression. Thus only the correlation part of the XC funetional remains to be approximated. In Exact-Exchange (EXX) methods the energy-funetional is free from the self-interaction error (SIE), whieh is of the main eurrent... 

For simplicity throughout this work we consider a spin-unpolarized closed-shell system where each (non degenerate) oceupied orbital is filled by n = 2 electrons with opposite spins. We can thus explicitly represent the non-interacting kinetic energy as an orbital-dependent functional ... 

In this section the basic formalism for orbital-dependent XC-functionals is derived. The orbital-dependent KS potential can be derived by applying the chain-rule of functional derivatives (subsection 3.3), which requires the use of Green s functions (subsection 3.1) and of the density response (subsection 3.2). An equivalent approach is the Optimized Effective Potential (OEP) method (subsection 3.4). The main properties of the exact OEP exchange-correlation potential are discussed in subsection 3.5. In subsection 3.6 well-established approximations to the Green s function are presented, while in subsection 3.7 alternative derivations of orbital-dependent functional are discussed. 

As an example of occupied orbital-dependent functional we recall the generalized exchange in Eq. (42) and the SIC functional. ... 

Orbital-dependent functional represent the next step towards more accurate DFT methods. KS orbitals are ultra-non-local functionals of the density and thus represent an advantageous alternative to the gradient expansion to include non-local contributions in the XC functionals. In addition KS orbitals are readily available in the self-consistent procedure without additional cost... 

In this appendix we evaluate the functional derivative of the non interacting kinetic energy functional which is simple occupied-orbital dependent functional and thus can be treated with the chain-rule formalism developed... 

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