2.5.1 Approximations for the exchange-correlation energy functional The practical usability and reliability of DFT and TDDFT strongly depends on the quality of the used xc density functionals. This aspect has not been settled so far. I give an introduction into the most important and for this thesis most relevant approximations to Exc in the following.
Local and semilocal functionals: The earliest and still one of the most wide-spread approximations is the local density approximation (LDA) [HK64] and its extension to spin-dependent cases, the local spin-density approximation (LSDA) [vBH72]. The rationale behind LDA is to use the functionals of the exchange and correlation energies (homx (nhom) and homc (nhom)) of the homogeneous electron gas and replace the homogeneous electron density nhom by the local densityn(r) according to
homxc [n(r)] =h
homx (n0) +homc (n0)i
n0=n(r). (2.35)
The exchange part of the homogeneous electron gas xc energy has an analytical expression.
The correlation contribution is only known from highly accurate quantum Monte-Carlo simu-lations [CA80] and needs to be parametrized for application in DFT, e.g., the parametrization of Perdew and Wang [PW92]. The LDA xc energy reads
ExcLDA[n] = Z
homxc (n(r))n(r) d3r. (2.36)
2.5. EXCHANGE-CORRELATION FUNCTIONALS 13 Semilocal functionals are the rst class of beyond-LDA functionals. Historically, the rst step beyond purely local functionals was to include also gradients of the density into the xc functional. However, consistent improvements were obtained only when the so-called generalized gradient approximations (GGAs) [LM83, PY86, Per86] were introduced. One of the most popular GGAs, the non-empirical GGA of Perdew, Burke, and Ernzerhof (PBE) [PBE96], is based on exact constraints, e.g., to the xc hole. Other functionals use free parameters and t those to data sets from reference calculations or experimental ndings.
For instance, the semiempirical BLYP functional combines Becke88 exchange [Bec88] with the correlation functional of Lee, Yang, and Parr (LYP) [LYP88]. A second class of semilocal functionals are the so-called meta-GGAs that may include also higher-order derivatives of the density, the kinetic energy density, and gradients of the latter. Note that meta-GGAs may already fall into the next class of functionals, the so-called orbital functionals, because they may include explicit orbital dependence although they are semilocal in nature.
Orbital functionals: Orbital-dependent functionals comprise explicit dependence of the orbitals of the KS system beyond semilocal contributions [KK08]. They are still implicit density functionals because the orbitals themselves are implicit functionals of the density.
Prominent representatives of this class of functionals are the self-interaction correction (SIC) of Perdew and Zunger [PZ81] and the exact exchange (EXX) functional
Ex[{ϕjτ}] =−1 2
X
σ=↑,↓
Nσ
X
i,j=1
Z Z ϕ∗iσ(r)ϕ∗jσ(r0)ϕjσ(r)ϕiσ(r0)
|r−r0| d3rd3r0, (2.37) the Fock exchange integral known from Hartree-Fock (HF) theory computed with KS or-bitals. The latter includes exact exchange only and nding a compatible correlation func-tional is known to be dicult. In the SIC approach, exchange and correlation are based on the underlying xc functional approximation on top of which the self-interaction correction is performed (see Pub3 and Chap. 3 for a discussion and more details).
The price one has to pay when using orbital functionals are diculties when computing the xc potential via the functional derivative ofExc with respect to the density, because one does not know the explicit density dependence of the orbitals. A solution to this problem, the optimized eective potential (OEP) method [SH53, TS76, GKG97, KK08], yields a mul-tiplicative xc potential in the KS sense. This method is introduced in the context of SIC in Chap. 3. As an alternative to the OEP, one may leave the grounds of KS theory and rely on the generalized KS (GKS) approach [SGV+96]. In the GKS scheme, the constraint of strictly non-interacting reference systems is relaxed and interacting reference systems that use a sin-gle Slater determinant are allowed. The potential in the GKS approach is no longer a local but an orbital-specic one. Typically, functionals implemented via the GKS method involve at least a fraction of EXX, thus most GKS potentials include a fraction of the nonlocal Fock potential [KK10]. Details about the GKS approach and dierences to the theoretical framework of the KS scheme are discussed in Refs. [SGV+96, KK10, BLS10].
Hybrid functionals: In the hybrid functional idea, basically a fraction of the EXX functional is mixed with some semilocal (sl) density functional. For instance, a one-parameter hybrid can be written as
Exchyb =aEx+ (1−a)Exsl+Ecsl (2.38) with the mixing parameter a. In this case, one mixes the semilocal exchange Exsl with exact exchange and takes the full correlation Ecslof the semilocal functional. Typically, the
mixing parameter of such hybrid functionals is chosen empirically, for instance by tting the functional to a test set of atomic and molecular properties. Probably the most prominent hybrid functional is B3LYP [Bec93, SDCF94], a three-parameter hybrid functional based on a weighted mixture of LDA exchange and correlation, LYP correlation, Becke88 exchange, and EXX. The three parameters are obtained empirically from tting to a set of atomic properties. Other approaches emphasize the density dependence of such mixing parameters and suggest approaches to compute mixing parameters from the density alone [MVO+11].
Recently, the range-separated hybrid functional idea became increasingly popular. It rests upon a range-separation scheme [Sav95] of the electron-electron interaction into a short-range and a long-range part. In those two parts, the electron-electron interaction is treated with dierent functional approximations. Each of those approximations is supposed to play a specic role: Typically semilocal functionals are used in the short-range part, whereas EXX is supposed to dominate the long-range contribution. The transition between short and long range is determined by a partitioning scheme and a range-separation parameter γ [Sav95, VS06, LB07, BLS10, KSSB11]. The inverse of this parameter 1/γcan be interpreted to be a characteristic length scale that distinguishes between short and long rang. The choice ofγ is the key element of the performance of such range-separation ideas. First approaches of this kind were based on empirical range-separation parameters [YTH04, VS06, LB07, CHG07], but only recently parameter tuning to some additional theoretical constraints [SKB09a, SKB09b, BLS10, KSSB11] has been employed. Tuned range-separated hybrid functionals, as long as the underlying xc functionals are non-empirical, do not rely on empirical input data.
Hybrid functionals involve an explicit dependence on the orbitals. Thus, the diculties with orbital functionals already discussed in the previous section apply again. Usually, hybrid functionals are implemented via the GKS scheme [SGV+96, KSSB11]. An implementation within the KS framework of DFT can be performed based on the OEP method.
2.5.2 Exchange-correlation functionals in TDDFT
The xc action functional and the corresponding TD xc potential in TDDFT are very complex quantities, presumably even more complex than their static counterparts. Yet, the validity of results from TDDFT calculations strongly depends on the quality of the description of xc eects. Therefore, although the action functional formalism provides a solid starting point for developing functional approximations, nding reliable TD xc potentials for practical calculations can be tedious, in particular, for dicult applications like charge transfer.
Today, most applications of TDDFT rely on the so-called adiabatic approximation [GDP96, EBF07]. The rationale behind this approach is that in cases where the external potential varies slowly enough in time, the time evolution of the system looses its dependence on the past, and can be well described by the instantaneous density. Thus, the adiabatic approximation amounts to using well-known functionals and the corresponding potentials from GS DFT as TD xc potentials according to
vxc,σadia[n](r, t) =vxc,σ[nt](r) = δExc[nt]
δnt,σ(r), (2.39)
where the time variabletis considered as a parameter of the densitynt. For linear response calculations the adiabatic approximations may also be applied to the xc kernel. The adia-batic xc kernel is completely local in time, thus transforming it into Fourier space yields a frequency-independent xc kernel [MUN+06].
2.6. NUMERICAL REALIZATION 15