LR-TDDFT

The core element of LR-TDDFT is the relation between the dynamic (frequency dependent) polarizability of the system under investigation and the quantities derivable from Kohn-Sham equations. LR-TDDFT has been subject to many reviews (for fundamental aspects see the recent review by Gross and Marquard112, for the original description of the adaptation of LR-TDDFT to molecular systems see the review by Casida111). Before discussing the response of an embedded electron density, the key elements of LR-TDDFT will be provided here1. 

A second and more widely used approach for the computation of excitation energies within DFT is based on the linear-response formulation of the time-dependent perturbation of the electronic density. The basic quantity in linear response TDDFT (LR-TDDFT) is the time-dependent density-density response function [33]... 

In order to perform ah initio molecular dynamics in excited states, the forces on the atoms are computed on the fly. There are two different implementations of force calculations for the two TDDFT schemes, P-TDDFT and LR-TDDFT. In the first case, the excited state is obtained with the promotion of one electron from the highest occupied molecular orbital (HOMO) to a selected unoccupied molecular orbital (the lowest one, LUMO, in our case). The corresponding KS excited single-determinant configuration is taken for the computation of the electronic density, which is then used to compute the forces on the atoms according to the Hellmann-Feynman electrostatic theorem [35]... 

In case of LR-TDDFT, the forces on the nuclei are derived within the Tamm-DancofF approximation [36,37] from nuclear derivatives of the excited-state energies using the extended Lagrangian formalism introduced by Hutter [34], In general, LR-TDDFT MD simulations are about 70-90 times faster than P-TDDFT MD simulations. The LR-TDDFT scheme has also been combined with our QM/MM approach [38,39] in order to enable the calculation of excitation spectra [40-42] and excited-state dynamics in condensed-phase systems. 

Fig. 7.4 (a) Frontier orbitals of PSB5. (b) Hirshfeld charges (hydrogens summed into heavy atoms). S, and S2 calculated with LR-TDDFT. (c) Bond lengths in planar optimized S0 (black) and St (dotted) structures. LR-TDDFT values shown with crosses, CASPT2 values from Ref. [80] shown with circles. 

In summary, LR-TDDFT results for PSB5 are in disagreement with the experimental detection of photoinduced double bond isomerizations in similar compounds. The shape of the Sj PES is different from the one computed with CASSCF/CASPT2, P-TDDFT, or ROKS. This seems to concern mainly the bond... 

Since LR-TDDFT breaks down for the description of the photodynamics of PSB5 and P-TDDFT is computationally too demanding for the description of the full RPSB, we employ ROKS for the QM/MM description of the photoreaction in rhodopsin (see Section 7.5). 

Fig. 7.7 Frontier orbitals of formaldimine in the S0 (top) and in the S, (bottom) optimized structures. Geometry optimization in S, was carried out with LR-TDDFT. Angles are given in degree.

Fig. 7.8 LR-TDDFT simulations offormaldimine without temperature control (a) Time evolution of the S0 (black) and S, (red) energies. Two different trajectories are shown in blue and in green, (b) Relaxation of the angles OcNh (upper panel, full lines), >H3CNH (upper panel, dashed lines) and co, (lower panel) along the green and the blue trajectories from (a), (c) Qualitative PES diagram. The colors correspond to the trajectories shown in (a). Reproduced with permission from Mol. Phys. 2005,103, 963-981. Copyright Taylor Francis Ltd. http //www.tandf.co.uk/journals.

In summary, LR-TDDFT simulations of the excited-state dynamics of formaldi-mine show that the Sj PES minimum does not coincide with the crossing be-... 

In the case of the protonated Schiff base model, the forces computed from the LR-TDDFT Sj PES lead to a single bond rotation instead of double bond isomerization. This failure might be related to the local approximation of the exchange-correlation functional or, as suggested by the significantly different results obtained with P-TDDFT, to a breakdown of the linear response approximation. Further investigations in this respect are needed. P-TDDFT and ROKS correctly... 

A review of single-reference methods for large-molecule excited-state calculations noted that LR-TDDFT can treat systems with up to 200 to 300 first-row atoms and stated that LR-TDDFT is the most prominent method for the calculation of excited states of medium-sized and large molecules [A. Dreuw and M. Head-Gordon, Chem. Rev., 105, 4009 (2005)]. For more on TDDFT, see C. Ullrich, Time-Dependent Density-Functional Theory, Oxford, 2012. 

The Hohenberg-Kohn-Sham theory is basically a ground-state theory. LR-TDDFT can only be applied to certain kinds of excited states. (One can use DFT to calculate the lowest state of each symmetry for example, one can calculate the lowest singlet state and the lowest triplet state.)... 

Linear Response Time-dependent Density Functional Theory (LR-TDDFT)... 

In a DSSC, the light absorption efficiency of an organic sensitizer ean be eal-culated by LR-TDDFT because of its typically small size (less than 100 atoms). In LR-TDDFT, the electronic excitation frequencies, co, are determined by solving the non-Hermitian eigenvalue problem ... 

N3 dye has four carboxyl groups capable of taking a proton, making its valence state charges range from —4 to 0. In a LR-TDDFT study that accounts for aqueous solvation by means of a conductor-like polarizable continuum model (C-PCM), the fully protonated form, N3 , is found to be... 

As implied by Equation (4.11) and Equation (4.12), the computational cost of LR-TDDFT grows rapidly with increasing system size, because the size of the response matrices is proportional to the product of the numbers of occupied and unoccupied molecular orbitals, that is, (Voce x Aunocc- Moreover, the diffuse unoccupied orbitals usually require very large basis sets to satisfactorily converge the results. To circumvent these numerical challenges, a RT-TDDFT method was derived to track the response of an optical system upon external perturbation. In RT-TDDFT, an additional term, E r) x ]i r), is added to the system s quantum Hamiltonian, Hq, to reflect Kght-matter interactions ... 

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