A detailed analysis of functional approximations within the KS scheme of DFT requires knowledge of their local, multiplicative xc potential based on the functional derivative in Eq. (2.18). However, while some DFAs are constructed using only the electron density itself (so-called density-dependent functionals), it becomes beneficial under certain aspects to design functionals by directly using KS orbitals (see Sec. 3.2 and 3.3 for an introduction to such DFAs). These functionals are termed orbital-dependent or simply orbital functionals, indicating that their xc energy is an explicit func-tional of the KS orbitals and only an implicit funcfunc-tional of the density. Thus, the question arises how to evaluatevxcσ[{nσ}](r) =δExc[{ϕiσ[{nσ}]}]/δnσ(r)in practice? An answer is provided by the optimized effective potential (OEP) scheme, which I briefly outline in the following.
The OEP formalism has its roots in early attempts to construct a local potential to the integral of Eq. (2.21) [SH53, TS76, SGP82]. The derivation of an expression for vxcσ(r) of any orbital functional is based on Eq. (2.18) with the chain-rule argument (see, e.g., Refs. [GL94, GG95, GKG97, FNM03, KK08])
vxcσ(r) =
∑
µ,ν=↑,↓ Nµ
∑
iZ Z δExc[{ϕjτ}] δϕiµ(r0)
δϕiµ(r0) δvKSν (r00)
δvKSν (r00)
δnσ(r) d3r0d3r00+c.c.. (2.52) Explicit evaluation of this expression leads to
Nσ
i=1
∑
ψiσ∗(r)ϕiσ(r) +c.c.=0, (2.53) which is one possible representation of the OEP equation [KLI92b]. Here, theψiσ∗ (r) are termed orbital shifts. They represent the first-order change in the KS orbitalϕiσ(r)if the KS potential is
2.6 The Optimized Effective Potential
replaced by the orbital-specific potentialuiσ(r), which is defined as [KP03a, KK08]
uiσ(r) = 1 ϕiσ∗ (r)
δExc[{ϕjτ}]
δϕiσ(r) . (2.54)
The orbital shifts can be obtained via
(hˆKSσ −εiσ)ψiσ∗(r) =−[vxcσ(r)−uiσ(r)−(v¯xciσ−u¯iσ)]ϕiσ∗(r). (2.55) Here, ˆhKSσ represents the KS Hamiltonian of Eq. (2.11), while the quantities ¯vxciσ and ¯uiσ denote the orbital-averaged potentials
¯ vxciσ =
Z ϕiσ∗ (r0)vxciσ(r0)ϕiσ(r0)d3r0 and (2.56)
¯ uiσ =
Z ϕiσ∗ (r0)uiσ(r0)ϕiσ(r0)d3r0. (2.57) Based on these expressions, the OEP equation can be reformulated as
vxcσ(r) = 1 2nσ(r)
Nσ
i=1
∑
|ϕiσ(r)|2[uiσ(r) + (v¯xciσ−u¯iσ)]−∇[ψiσ∗(r)∇ϕiσ(r)] +c.c.. (2.58) In this representation, an important property of the OEP equation becomes evident. Since the xc potential appears both on the left- and the right-hand side of Eq. (2.58) (via its orbital average), the OEP defines an integral equation forvxcσ(r)that has to be solved self-consistently.
In principle, both representations of the OEP are identical. The formulation in Eq. (2.53) is of special relevance for an efficient iterative construction of the xc potential [KP03b, KP03a, KKP04, MKHM06]. The alternative OEP expression in Eq. (2.58) readily sets the stage for an important approximation first suggested by Krieger, Li and Iafrate (KLI) [KLI90, KLI92b, LKI93, IK13]. It is obtained by neglecting the last term on the right-hand side of Eq. (2.58), i.e.,
vKLIxcσ(r) = 1 2nσ(r)
Nσ i=1
∑
|ϕiσ(r)|2
uiσ(r) + v¯KLIxciσ−u¯iσ +c.c., (2.59) and allows for a solution with drastically reduced numerical effort in contrast to the full OEP equation [KLI90, GG97, KK08].
A special feature of the OEP/KLI scheme are nonvanishing asymptotic constants, which were first discussed in the context of pure EXX in Refs. [DG02, KP03a]. These are related to the condition ¯vxcNσσ =u¯Nσσ, which is typically enforced within the OEP/KLI formalism in order to ensure thatvxcσ(r)of an orbital-dependent functional approaches zero asymptotically [KKGG98].
If evaluated along a nodal plane of the ho state, in contrast, the xc potential asymptotically ap-proaches the constant
Cσ =v¯xcMσσ−u¯Mσσ, (2.60)
withMσ denoting the highest lying KS state that does not vanish along the nodal plane of the ho orbital in this particular spin channel [DG02, KP03a]. This is a remarkable finding, since it means that the local xc potential of orbital-dependent functionals approaches different asymptotic limits in different spatial directions [KK08] (see Sec. 4.6 for an illustration).
Lastly, it is important to mention that a feasible alternative to the OEP exists outside the KS framework in DFT. Termed generalized Kohn-Sham (GKS) scheme [SGV+96], it is based on the
idea of mapping the interacting system of electrons into an auxiliary system that partially interacts, but is still describable by a single Slater determinant. Consequently, singe-particle equations in the spirit of Eq. (2.11) can be derived for this system, with orbitals that correctly reproduce the electron density. However, the difference to the KS realization is that the potential ceases to be strictly local and becomes a nonlocal and orbital-specific operator (see Ref. [SGV+96, KK08, KK10] for a detailed derivation and discussion regarding differences between KS and GKS).
3 Approximate Exchange-Correlation Functionals
Due to its central role in DFT, numerous approximations toExc were developed over the years. In this chapter, I provide a brief introduction to important DFAs with a focus on hybrid functionals, which are of special relevance for this thesis. Based on explicit results, I illustrate the fundamental parameter problem of global hybrids and outline the concept of EXX with compatible correlation.
In this context, I introduce local hybrid functionals as a more flexible extension to the global hybrid approach and discuss established local hybrid constructions. For more detailed reviews on functional approximations I refer the interested reader to, e.g., Refs. [PS01, KK08, Bec14].
3.1 Local and Semilocal Functionals
The LDA (and its spin-polarized formulation LSDA [vBH72]) is the most basic functional approx-imation. Already introduced in Ref. [HK64], it is as old as DFT itself and relies on a simple and efficient principle. The LSDA uses the parametrization of the exact xc energy density of the homogeneous electron gas (cf. Sec. 2.4) evaluated with the spin densities nσ(r) of the, not necessarily homogeneous,Nelectron system according to
eLSDAxc [{nσ(r)}](r) =ehomxc ( n0σ )
n0
σ→nσ(r). (3.1)
The xc energy density and potential atrare determined entirely by the density at this particular point in space. Thus, the LSDA and other DFAs that use onlynσ(r)are labeled local approximations.
Intriguingly, the LSDA performs qualitatively well not only for systems with slowly varying densities such as solid states, but also in other cases [PS01, KK08] (see Sec. 3.5 for a reasoning based on the properties of the LSDA xc hole). However, in general the results of LSDA calculations are not of sufficient precision, as, for instance, binding energies [Bec92a, Bec92b] are drastically over- and bond lengths underestimated.
The generalized gradient approximations (GGAs) were introduced in the 1980s to remedy some of the shortcomings of the LSDA [LM83, PY86, Per86a, Per86b]. GGAs include the gradients of the spin densities, i.e., eGGAxc (r) =eGGAxc [{nσ(r)},{∇nσ(r)}] (r), and are thus labeled semilocal functionals. The construction of GGAs in general does not follow from a direct gradient expansion of the xc energy of the homogeneous electron gas [LP80, SGP82]. In fact, such an expansion is known to perform poorly in comparison to the LSDA [Per85].
Instead, GGAs are constructed either to meet known constraints on the xc energy or by in-troducing parameters that are determined empirically in order to optimize the functionals per-formance. The most prominent example of the former category is the GGA of Perdew-Burke-Ernzerhof (PBE) [PBE96, PBE97], which was designed to reproduce the limits of slowly and rapidly varying densities. The latter type of GGA is represented by, e.g., the BLYP functional,
which consists of the B88 exchange functional [Bec88b, EB09] in combination with the correlation functional of Lee, Yang and Parr (LYP) [LYP88].
A related, yet more elaborate class of functionals are the so-called meta-GGAs. Meta-GGAs make use of higher-order derivatives of the density, e.g, the Laplacian∇2nσ(r), and employ the KS kinetic energy density
τ(r) =
∑
σ τσ(r) =1 2
∑
σ Nσ
∑
i |∇ϕiσ(r)|2. (3.2)Here,τσ(r)is directly related to Eq. (2.14), as its integral reproduces the noninteracting kinetic energy Tni =∑σRτσ(r) d3r. The KS kinetic energy density introduces an explicit dependence on the set of occupied KS orbitals{ϕiσ(r)}to the meta-GGA xc energy expression. While being of semilocal nature, meta-GGAs therefore belong to the class of orbital-dependent functionals in contrast to the LSDA and GGAs, which are strictly density dependent.
Due to their components, meta-GGAs allow for a more flexible construction and offer the possibility to fulfill more exact constraints [SRP15]. Such a construction is demonstrated, e.g., in Ref. [PKZB99] for the meta-GGA of Perdew, Kurth, Zupan and Blaha (PKZB), where the functional was additionally optimized by the inclusion of empirical parameters. Based on that ex-pression, the meta-GGA of Tao, Perdew, Staroverov and Scuseria (TPSS) was developed [TPSS03, PTSS04]. In general, the quality of DFT calculations for, e.g., atomization energies increases when upgrading from the LSDA to GGAs and meta-GGAs [SSTP03]. Yet, it is important to state that more evolved functional constructions do not necessarily lead to universal improvement in the functional’s performance, as, for instance, the PKZB and TPSS meta-GGAs predict molecular geometries and lattice constants of solids with less accuracy when compared to GGAs [AES00].
While semilocal functionals certainly allow for efficient and, for some applications, sufficiently accurate calculations, it is worthwhile to mention that they typically suffer from common draw-backs such as, for instance, the incorrect asymptotic decay of their local xc potential, the ab-sence of a derivative discontinuity, and electronic self-interaction. These shortcomings strongly affect the prediction of IPs and fundamental gaps using KS eigenvalues [KSRAB12]. Recently, an intriguing construction of a GGA addressing the issues of the potential asymptotics and the derivative discontinuity was introduced [AK13], yielding promising results for finite systems and solid states [COM14, VSNL+15]. Yet, the question of self-interaction remains difficult to resolve in the context of semilocal functionals, as discussed in the next section.