Tamm-Dancoff TDDFT

Since DFT has essentially the same mean-field formalism as the HF theory and share much the same computational algorithm, it is not surprising that it has the excited-state counterparts corresponding to TDHF and CIS. They—TDDFT [83-88] and Tamm-Dancoff TDDFT [89], respectively - can be derived analogously to Section 2.2.1 with the only differences being in the definitions of the operator (now called the Kohn-Sham or KS Hamiltonian) and its derivative with respect to the density matrix [see Eq. (2-7)]. The latter is... 

It may not be obvious that the self interaction problem is present in TDDFT, but it can be made evident by comparing CIS and Tamm-Dancoff TDDFT for an excitation describable by just a pair of orbitals (xai = 1) ... 

The extension of TDDFT and Tamm-Dancoff TDDFT to crystalline polymers is straightforward within the formalisms of Section 2.2.2. Figure 2-7 summarizes the results of TDDFT calculations of the photoconduction, photoemission, and optical absorption thresholds (energy gap, ionization energy, and excitation energy) of polyethylene as a function of basis set [50], The Slater-Vosko-Wilk-Nusair functional [116, 117] is used, but the following conclusion is unaltered... 

The corrected Linear Response approach (cLR) consists in the use the TDDFT relaxed density and the corresponding apparent charges (7-38) into Eqs. (7-36) and (7-37) to obtain the first-order approximation to the state specific free energy of the excited state. The details of the implementation are described in Ref. [17], This corrected Linear Response computational scheme can be applied to the analogous of the Time Dependent Hartree-Fock approach either in the complete (Random Phase Approximation) or approximated (Tamm-Dancoff approximation or Cl singles, CIS) version. 

As it is not possible to obtain TDDFT-SS results, the results refer to CIS method. In fact, this method can be obtained from two points of view one is to consider the method as a standard Cl, in which the wave function of the excited state is constructed by single excitations from the HF determinant and thus a SS solvent response can be obtained the other is to consider CIS as the result of the Tamm-Dancoff approximation applied to the linear response equation based on the HF wave function. The two ways of looking at the CIS method give the same equations in vacuo, but, as discussed above, they differ for molecules in solution due to the nature of the effective Hamiltonian. 

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. 

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