Time-dependent basis functions

Here, the represent the general time-dependent basis functions of the standard MCTDH... 

In accord with Eq. (3.7) the solutions to Eq. (3.5) can be written as linear combinations of the time-dependent basis functions... 

There are two possible approaches to solution of the Schrbdinger equation with the model Hamiltonian (9.3). The first is based on direct solution using grid [17] and basis [18] methods or the so-called multiconfiguration time-dependent Har-tree method [19]. (The latter method is a combination of grid and basis set methods in the sense that the time-dependent basis functions are represented on a suitable grid). The second approach uses a Bom-Oppenheimer type separation between the motions of the light and heavy nuclei (the method of adiabatic separation between vibrational variables) [20]. In the second approach the wave function F(s,R) is written as ... 

The expansion (50) of the time-dependent state vector in terms of time-independent basis functions becomes impractical for vibronic-coupling problems with more than about 7 vibrational modes, especially if one or several modes possess large coupling parameters or if the excess energy is very large. For such more demanding applications the MCTDH scheme, which is based on an expansion of the time-dependent state vector in terms of variationally determined time-dependent basis functions, has proven... 

In the Schrodinger representation of the latter matrix element in (8.10), the molecular states are regarded as time-dependent basis functions exp( — i fcf/ft) fe> and exp( —i t/ft) w>, and the operator /t is considered to be time-independent. For present purposes, it is more illuminating to use the Heisenberg representation, in which the molecular states are the time-independent basis functions lfc>, m> and the operator is viewed as time-dependent. Since H i> = , i> for each of the molecular states i>> we have... 

In response theory, the basis functions %v(f) e usually chosen to be time independent for strong fields or coupled electron-nuclear dynamics, time-dependent basis functions can sometimes be more appropriate. 

To deal with the problem of using a superposition of functions, Heller also tried using Gaussian wave packets with a fixed width as a time-dependent basis set for the representation of the evolving nuclear wave function [23]. Each frozen Gaussian function evolves under classical equations of motion, and the phase is provided by the classical action along the path... 

Figure 3.49 Plot of the electronic coupling contributions reported for the ethane dimer [47], The dimer is a sandwich orientation of the molecules with various center-to-center separations. These calculations were carried out at the time-dependent density functional theory level (B3LYP functional and 6-31G basis set). The labeling of the curves is consistent with the text and the subscripts indicate the solvent. For the lower curve, VEoul indicates V °ul/n2.

J. Hutter (2003) Excited state nuclear forces from the Tamm-Dancoff approximation to time-dependent density functional theory within the plane wave basis set framework. J. Chem,. Phys. 118, p. 3928... 

However, the numerical treatment of such explicitly time-dependent basis states would be time consuming compared to the treatment of bound states. Thus we search for a further simplification of by investigating the asymptotic behavior of Coulomb wave functions [42]. For rAe tt the radial wave function z/j. / is nearly independent of e and may be considered constant for integration. For et tt the exponential function in equation (18) is nearly independent of e. In both cases in equation (18) may be replaced by... 

The treatment in terms of induced current is in the mainstream of modem development of the time-dependent density functional theory (TDDFT). Moreover, the current density formalism has been proposed [4] as a variant of TDDFT. The evolution of current density presents properly the response of electrons on an external field. In general words, such a strong basis is promising for a theoretical treatment of many aspects of ion interactions with atoms, molecules and solids. 

As the basis sets developed for glycine produced excellent results for fhat amino acid (see Table 5.1 of Ref. [31] for details), we chose to use the same basis also for fhe presenf calculations [68]. Preliminary calculations were also carried out at the level of time-dependent density functional theory (TD-DFT) using the PBE functional [56]. However, the results differed by less fhan 1% from fhe RPA result and are thus not reported here. 

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