The optimized effective potential OEP

For this limited Cl problem, it is convenient to diagonalize the one-electron density matrix, defining a set of natural orbitals that vary at each step of an iterative procedure. The one-electron density operator is defined by 

Natural orbitals / and occupation numbers nl are determined by diagonahzing the density matrix dl. The reference state T is constructed from N orbitals with the largest occupation numbers. The energy functional is of the form 

Orbital Euler-Lagrange equations are determined by functional derivatives 

Orbital construction and diagonalization of the Cl matrix are alternated until the calculations converge. 

The computational effort of solving orbital Euler-Lagrange (OEL) equations is significantly reduced if the generally nonlocal exchange-correlation potential vxc can be replaced or approximated by a local potential vxc(r). A variationally defined optimal local potential is determined using the optimized effective potential (OEP) method [380, 398]. This method can be applied to any theory in which the model 

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Rigorous properties of the optimized effective potential (OEP) are derived. We present a detailed analysis of the asymptotic form of the OEP, going beyond the leading term. Furthermore, the asymptotic properties of the approximate OEP scheme of Krieger, Li and Iafrate [Phys. Lett. A 146, 256 (1990)] are analysed, showing that the leading asymptotic behavior is preserved by this approximation. 

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,... 

Kohn and Sham took E c as a functional of the density p. An alternative procedure, the optimized effective potential (OEP) method, takes as a functional of the occupied KS orbitals, in the hope that this will make it easier to develop accurate functionals. The OEP method leads to equations that are hard to solve. Kreiger, Li and lafrate (KLI) developed an accurate approximation to the OEP equations, thereby making them easier to deal with, and the KLI method has given good results in DF calculations on atoms [J. B. Krieger, Y. li, and G. I. lafrate in E. K U. Gross and R. M. Dreizler (eds.). Density Functional Theory, Plenum, 1995, pp. 191-216]. 

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. 

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... 

Before leaving our tutorial on ground-state formalism, we mention the optimized effective potential (OEP) method in which the XC functional is written as a functional of the KS orbitals (which in turn are functionals of... 

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... 

As it will be shown in Section 3.4, Eq. (89) is also called as Optimized Effective Potential (OEP) equation. Exchange-correlation energy functional that depends on all orbitals and eigenvalues are, e.g., the ones based on the second-order many-body perturbation-theory (MBPT) and the ones which uses virtual orbitals to model the static-correlation. ... 

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. 

The method is sometimes also termed optimized effective potential (OEP). 

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. 

For any orbital-functional model, an optimal effective (local) potential (OEP) can be constructed following a well-defined variational formalism [24,25]. If a Frechet derivative existed for the exchange-correlation energy E,lc for ground states, it would be obtained in an OEP calculation, while the minimum energy and corresponding reference state would coincide with OFT results. Thus numerically accurate OEP calculations test the locality hypothesis. 

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