First investigations of TDGSIC [Hof08] focused on the propagation stability question of Na5, a notoriously problematic system for RT propagation. Here, the propagation instability manifests in zero-force theorem violation and notable drifts of the total energy from its GS value although no external eld acts [MKvLR07, Mun07]. Reference [Hof08] indicates that propagation stability can be increased by increasing the numerical accuracy of the unitary optimization. However, using the initial guess of Eq. (3.19) puts this nding into a new perspective. The appendix of Pub4 demonstrates that TDGSIC schemes show much better stability than propagation with TDKLI-SIC even for the dicult case of Na5. For most applications, reasonably stable propagation of TDGSIC can be performed in a time window that is long enough for computing the dipole spectrum. However, as a warning, one should be aware that instability problems may occur and always check for stability issues.
Based on this reassuring nding, TDGSIC is ready for applications to dynamic situations and investigations of excitation features. I summarize the results on the performance of TDGSIC in the following.
Hydrogen chains are a transparent model system to study the response behavior of the potential for dierent approximations tovxc(r, t). For slowly varying external perturbations, one expects the TD xc potential to perform similar to GS xc potentials, i.e., that it coun-teracts the external eld (see Sec. 2.4). Pub4 demonstrates that the TD response to such an external potential parallels the response behavior known from GS xc potentials: vxc(r, t) of TDLDA follows the external eld, whereas TDGKLI-SIC exhibits a eld-counteracting behavior. At higher frequencies of the external perturbation, TDGKLI-SIC develops compli-cated features that can not be assigned to a simple frequency-dependent response behavior.
Similar studies of dierent approximations to TDGKLI-SIC are also presented in Pub4. In-terestingly, TDFOBO-SIC exhibits a response behavior very similar to TDGKLI-SIC. While a clear assignment of the energy-minimizing GSLA-SIC response was dicult in the GS case,
3.7. PERFORMANCE OF GENERALIZED SIC IN TDDFT 27 TDGSLA-SIC unambiguously follows the external eld even at elevated frequencies of the external perturbation. Moreover, using only Eq. (3.19) for the unitary transformation spoils the eld-counteracting behavior of the xc potential already at low frequencies of the external eld. Thus, the results of Pub4 show that the response term of the (G)KLI approximation and optimizing the unitary transformation at every instant of time are important for a proper eld-counteracting response behavior.
The TD response behavior studies are complemented in Pub4 by investigations of hy-drogen chain excitations energies. Whereas TDKLI-SIC results suer from propagation instability and do not improve upon TDLDA, TDGSIC notably shifts the lowest excitation energies to higher values. TDGKLI-SIC excitation energies are in close agreement to B3LYP but deviate from approximate coupled-cluster singles-and-doubles model excitation energies by at least 0.9 eV. In conclusion, TDGKLI-SIC and its TDFOBO-SIC approximation show promising performance on hydrogen chains. Yet, already static polarizabilities of hydrogen chains exhibit large deviations between OEP and GOEP [KK11] (for more background, see Appendix A.1) and a comparison between TDOEP and TDGOEP is not available. Because of the peculiar nature of hydrogen chains [vFdBvL+02], the performance of TDGSIC also needs to be assessed for real molecules.
Application to metal clusters in Pub4 shows that TDGSIC does not spoil the good accu-racy that already TDLDA reaches. However, TDGSIC improves in cases where (semi)local functionals exhibit systematic failures: In hydrogenated silicon clusters, quantum conne-ment and excitonic eects are known to play an important role [OCL97, RL98, RL00, ORR02, VOC02, VOC06]. Here, low-lying optical excitations are underestimated by standard func-tionals and TDKLI-SIC [MCR01], in particular for the very small clusters. Pub2 and Pub4 demonstrate that TDGSIC notably improves and yields excitation energies in good agree-ment with results from the GW Bethe-Salpeter equation approach. Lowest excitation ener-gies of oligo-acetylenes in Pub4 also improve upon TDLDA and reveal dierences between the dierent possible choices of the unitary transformation during time propagation. Last but not least, TDGSIC also gives promising results for static and dynamic CT situations as well as CT excitation energies. This is the topic of the next chapter.
Chapter 4
Charge transfer and charge-transfer excitation energies
The obvious inability of present-day physics and chemistry to account for such events is no reason at all for doubting that they can be accounted for by those sciences.
Erwin Schrödinger What is life? (1944)
A trustworthy description and theoretical prediction of charge transfer is one of the well-known and longstanding problems of (time-dependent) density functional theory [DWHG03, Toz03, Mai05, TFSB05, KBE06, TS07, KBY07, TS08, EVV09, LBBS12]. Typi-cally, energies of electronic excitations that exhibit CT character are underestimated when calculated with (semi)local xc functionals [DWHG03, Toz03, Mai05]. Transport proper-ties as, e.g., conductance and I-V characteristics of molecular electronic devices may be severely in error [TFSB05, KBE06, TS07, KBY07, TS08, LBBS12] when calculated from the Landauer-Büttiker approach [Lan57, Büt86] based on non-equilibrium Green's function the-ory in combination with DFT using standard xc functionals. Recently, also RT propagation has been used to study CT scenarios [KSA+05, CEVV06], but the reliability of results is limited by the quality of xc functional approximations. I provide more insight into the CT problem of (TD)DFT in Sec. 4.1 and complement this discussion by some solution ideas.
The CT failure is commonly related to the self-interaction error of (semi)local xc functionals [TFSB05, TS07, KBY07, TS08] and the lack of a derivative discontinuity in such approaches [TFSB05, KBE06, TS07, TS08, LBBS12]. Therefore, self-interaction correction is a promis-ing approach for improvpromis-ing the (TD)DFT description of CT phenomena. I discuss in Sec. 4.2 how generalized KS SIC introduced in Chap. 3 performs in static and dynamic CT situations and how it improves the description of CT excitation energies.