GS xc approximation. Many applications of TDDFT rely on the adiabatic local density approximation (ALDA, TDLDA), which is the simplest extension of GS DFT to the TD case. Surprisingly, ALDA works quite well far beyond its obvious range of validity, namely slowly varying densities both in space and in time [MUN+06]. Other applications use GGAs, hybrid functionals, or orbitals functionals as introduced in Sec. 2.5.1. In any case, one should be careful in choosing the right functional for each application depending on the system and observable one is interested in.
Care should be taken when using the term adiabatic and considering its implications on the history dependence of such approaches. Explicit density functionals implemented via Eq.
(2.39) neglect all memory dependence of the system evolution of timest0 ≤t. When it comes to explicitly orbital-dependent functionals, however, the TD KS orbitals in general depend on the entire history of the density n(r, t0) for t0 ≤t, thus recover in a natural way part of the memory dependence that is not in the TD density [GDP96, MBW02, MMN+12]. In this case, the adiabatic approximation should better be termed orbital-adiabatic approximation in contrast to density-adiabatic approximation.
2.6 Numerical realization
Most DFT and RT TDDFT investigations presented throughout this thesis are based on the Bayreuth version [MK07, Mun07, Mun09] of the PARSEC program package [KMT+06].
PARSEC is a real-space electronic-structure code that uses norm-conserving pseudopoten-tials of Troullier-Martins [TM91] type and a high-order nite dierence schemes for numeri-cally representing the Laplacian operator [CTS94, KMT+06]. The GS version is designed for solving the KS equations by numerical diagonalization of the KS Hamiltonian. The Bayreuth version includes solution of the OEP equation [KK08, Kör09] via the construction scheme of Refs. [KP03a, KP03b] and approximations to the OEP, as for instance the Krieger-Li-Iafrate (KLI) approximation [KLI92] and the Slater approximation [Sla51].
The practical realization of the RT propagation idea is based on stepwise numerical propagation with time steps∆tusing the propagatorU(t+∆t, t)[CAO+06, Mun07, Mun09].
The PARSEC real-time TDDFT implementation uses a Taylor series up to fourth order to numerically expand U(t+ ∆t, t) combined with the exponential midpoint rule [MK07, Mun07, Mun09]. In this scheme, the potential needs to be determined twice per time step.
Other propagation techniques are explained in Ref. [CMR04]. To avoid spurious reection of density that moves to the boundary of the numerical grid, RT PARSEC oers absorbing boundaries [RSA+06, Mun07, Mun09]. Last but not least, the Bayreuth version of TD PARSEC [MK07, Mun07] includes RT propagation of orbital functionals implemented via the time-dependent KLI approximation.
For the feasibility of most RT calculations presented in this thesis, numerical optimiza-tion of the original PARSEC version and the implementaoptimiza-tion of new algorithms were needed to reach acceptable computation times. One of the most time-consuming steps during time propagation is the evaluation of the Hartree potential via solution of Poisson's equation.
Numerical eciency of the Poisson solver is especially important when orbital functionals as, e.g., the EXX or the SIC are used. Therefore, I implemented a multigrid solver as an alternative to the existing conjugate gradient solver. Details about the numerical
realiza-tion are described in Appendix B. Further code optimizarealiza-tions are compiled in Appendix E. It includes an extrapolation scheme to avoid explicit determination of the potential for the midpoint Hamiltonian, some evaluation tools and extra features for the propagation, an adaptation scheme of the diagonalization accuracy to reduce diagonalization times during the GS self-consistency iterations, and some additional features for GS PARSEC. All imple-mentations that are directly related to the PARSEC functionalities used in this thesis are described in the following chapters and the related appendices.
Chapter 3
Self-interaction correction
Another feature which at least induced a semblance of popularity was the lecturer's intention to make clear the fundamental idea, ...
Erwin Schrödinger What is life? (1944)
Density functional theory and time-dependent density functional theory have gained pop-ularity because of their success in predicting and explaining properties of many dierent kinds of systems and their applicability to sizable systems at bearable computational cost.
However, standard density functionals may suer from well-known deciencies: incorrect dis-sociation limits [ZY98], wrong asymptotic behavior of the potential [PZ81], overestimation of electrical response properties [vGSG+99] and transport characteristics [TFSB05], incor-rect representation of charge-transfer (CT) states [DWHG03, Toz03], and problems with excitonic eects in conned systems [RL98, MCR01, ORR02, VOC06]. Deciencies of this kind have been attributed to the self-interaction error (SIE) of standard, explicitly density dependent functionals. The self-interaction correction (SIC) is a promising approach for curing these deciencies of DFT and TDDFT.
I provide insight into the self-interaction problem of DFT and correction ideas in Secs.
3.1 and 3.2. The SIC of Perdew and Zunger (PZ) is an explicitly orbital-dependent density functional. In KS DFT, it requires implementation via the optimized eective potential (OEP) or generalized OEP (GOEP) method introduced in Sec. 3.3. The unitary variance of PZ SIC is taken into account in the GOEP by additional unitary transformations (see Sec.
3.4). The impact of these transformations on the performance of SIC is discussed in Sec. 3.5.
Finally, I present in Sec. 3.6 one of the main results of this thesis: the TDDFT extension of GOEP SIC. An overview of the performance of this method on a wide range of test systems is given in Sec. 3.7 and Chap. 4.
3.1 The self-interaction problem
The self-interaction problem of DFT lies at the heart of the energy partitioning of KS theory. In the KS scheme, the Hartree energy represents the classical part of the Coulomb interaction, thus the xc energy needs to cover everything beyond the Hartree contribution.
The self-interaction problem of this energy partitioning manifests most clearly in a single-electron (se) system, where the single-electron is described by the single-particle wave function
17
ϕse(r) with densitynse(r) =|ϕse(r)|2. In this case, the Hartree energy does not vanish, but includes the Coulomb interaction of the single electron with itself. In the exact KS approach, this spurious self-interaction needs to be covered by the xc energy [PZ81], thus the sum of Hartree and xc energy contributions should vanish according to
EH[nse] +Exc[nse,0] = 0. (3.1) As a matter of fact, however, most xc density functional approximations Excapp do not cover this single-electron SIE, i.e., typically EH[nse] +Excapp[nse,0]6= 0.
The self-interaction error is well dened in the single-electron case, yet more dicult in a many-particle context. The PZ attempt towards dening many-electron self-interaction is based on an extension of the single-electron criterion of Eq. (3.1) [PZ81]. In this approach, the occupied orbitals from a DFT single particle approach are interpreted to represent the Nσ electrons of the system. Then, a functional is declared to be self-interaction free, when
X
σ=↑,↓
Nσ
X
j=1
[EH[njσ] +Excapp[njσ,0]] = 0, (3.2) where the orbital densities njσ(r) = |ϕjσ(r)|2. The ambiguity of this denition has led to alternative denitions of self-interaction in many-electron systems [RPC+06, MSCY06, KKM08]. One prominent criterion for the freedom of many-electron self-interaction is the straight line behavior of the total energy with respect to non-integer particle num-bers [RPC+06, MSCY06, KKM08] that is introduced in Sec. 2.4. Perdew [Per90] re-lated the SIE to the derivative discontinuity concept. The dependence of the total energy from standard density functionals on fractional electron numbers was investigated by Refs.
[MSCY06, VSP07] with the conclusion: None of the investigated standard xc density func-tionals strictly fullls the straight-line criterion.