Optimized effective potential , for

In this section a number of rigorous statements on the optimized effective potential for finite systems will be derived. For this purpose, the exchange-only potential and the correlation potential have to be treated separately within the OEP scheme. The exact exchange potential of DFT is defined as... 

Hamel S, Casida ME, Salahub DR (2002b) Exchange-only optimized effective potential for molecules from resolution-of-the-identity techniques Comparison with the local density approximation, with and without asymptotic correction, J Chem Phys, 116 8276-8291... 

We emphasize that the calculation of excitation energies from Eqs. (362) and (363) involves only known ground-state quantities, i.e., the ordinary static Kohn-Sham orbitals and the corresponding Kohn-Sham eigenvalues. Thus the scheme described here requires only one selfconsistent Kohn-Sham calculation, whereas the so-called Ajcf procedure involves linear combinations of two or more selfconsistent total energies [209]. So far, the best results are obtained with the optimized effective potential for in the KLI x-only approximation. Further improvement is expected from the inclusion of correlation terms [6,225] in the OPM. 

Kiimmel, S., Perdew, J. (2003). Simple iterative construction of the optimized effective potential for orbital functionals, including exact exchange. Phys. Rev. Lett. 90, 043004. 

So an optimized effective potential can be found for the given excited state. The Knieger-Li-Iaffate (KLI) approximation to the optimized effective potential can also be derived [37]. 

On the other hand, development of the xc kernel beyond ALDA does not necessarily bring better results. To illustrate this point, the singlet-singlet excitation energies 0)ks calculated in [24] for the He atom with the combination (accurate vxc)/ALDA are compared in Table 2 with (Ok calculated in [46] with the accurate vxc and with the spatially non-local kernel /ffI(TD0F P)(ri, r2), which is a part of the exchange-only kernel of the time-de-pendent optimized effective potential method (OEP) [47]. They are also compared with the TDDFRT zero-order estimate 0)kso, the difference (10) of the orbital energies obtained with the accurate vxc. 

The calculation of Table 2 uses the x-only optimized effective potential (OPM) for in the approximation of Krieger, Li and lafrate (KLI) [224] and... 

The optimized effective potential was used for and the ALDA for the xc kernels. The corresponding Kohn-Sham orbital-energy differences coo are shown in the last column (All values in rydbergs)... 

Since hybrid functionals, Meta-GGAs, SIC, the Fock term and all other orbital functionals depend on the density only implicitly, via the orbitals i[n, it is not possible to directly calculate the functional derivative vxc = 5Exc/5n. Instead one must use indirect approaches to minimize E[n and obtain vxc. In the case of the kinetic-energy functional Ts[ 0j[rr] ] this indirect approach is simply the Kohn-Sham scheme, described in Sec. 4. In the case of orbital expressions for Exc the corresponding indirect scheme is known as the optimized effective potential (OEP) [120] or, equivalently, the optimized-potential model (OPM) [121]. The minimization of the orbital functional with respect to the density is achieved by repeated application of the chain rule for functional derivatives,... 

Note that this correction has the problem that the Kohn-Sham equation is not invariant for the unitary transformation of occupied orbitals, even after the correction, differently from the Hartree-Fock equation. In the Hartree-Fock equation, the variations of the Coulomb self-interaction energy and its potential for the unitary transformations of occupied orbitals cancel out with those of the exchange self-interaction, while these are not compensated, even after the correction in the Kohn-Sham equation. Therefore, the effect of the self-interaction correction depends on the difference in occupied orbitals before and after the unitary transformation. For removing this difference, it is usual to localize the orbitals before the self-interaction correction (Johnson et al. 1994). Note, however, that there are various types of orbital localization methods, and the effect of the selfinteraction correction inevitably depends on them. Combining with the optimized effective potential (OEP) method (see Sect. 7.5) may be one of the most efficient ways to solve this problem. This combination enables us to consistently obtain localized potentials with no self-interaction error. 

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. 

In this chapter, we discuss some new developments in TDDFT beyond the linear response regime for accurate and efficient nonperturbative treatment of multiphoton dynamics and very-high-order nonlinear optical processes of atomic and molecular systems in intense and superintense laser fields. In Section 2, we briefly describe the time-dependent optimized effective potential (OEP) method and its simplified version, i.e., the time-dependent Krieger-Li-Iafrate (KLI) approximation, along with self-interaction correction (SIC). In Section 3, we present the TDDFT approaches and the time-dependent generalized pseudospectral (TDGPS) methods for the accurate treatment of multiphoton processes in diatomic and triatomic molecules. In Section 4, we describe the Floquet formulation of TDDFT. This is followed by a conclusion in Section 5. Atomic units will be used throughout this chapter. 

All of these approximations are density functionals, because the Kohn-Sham orbitals are implicit functionals of the density. Finding the exchange-correlation potential for nmgs (3) (5) requires the construction of the optimized effective potential [119], which is now practical even for fully three-dimensional densities [120]. For many purposes a non-selfconsistent implementation of nmgs (3)-(5) using GGA orbitals will suffice. 

Prom such an action functional, one seeks to determine the local Kohn-Sham potential through a series of chain rules for functional derivatives. The procedure is called the optimized effective potential (OEP) or the optimized potential method (OPM) for historical reasons [15,16]. The derivation of the time-dependent version of the OEP equations is very similar to the ground-state case. Due to space limitations we will not present the derivation in this chapter. The interested reader is advised to consult the original paper [13], one of the more recent publications [17,18], or the chapter by E. Engel contained in this volume. The final form of the OEP equation that determines the EXX potential is... 

Ensemble-Hartree-Fock Scheme for Excited States. The Optimized Effective Potential Method. 

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