The time-dependent discrete variable representation method in molecular dynamics

Non‐Adiabatic Effects in Chemical Reactions: Extended Born‐Oppenheimer Equations and its Applications

“…The grid points are propagated by classical equations of motion in a so-called fixed width approach for the basis set. For a derivation of these equations the reader is referred to [81,89,90]. As mentioned, the theory generates classical equations of motion for the center of the basis set or in the DVR representation the center of the DVR grid points.…”

The time-dependent discrete variable representation (TDDVR) method to the quantum dynamics on ethylene (C2H4+)

“…The formulation of the TDDVR methodology [6][7][8] has some special characteristics that make it unique dynamical method: (a) TDDVR is appealing from the computational point of view, and (b) it paves the blending of classical and quantum concepts with a new twist. Since GH basis functions are time -dependent and used as the primitive basis to introduce DVR representation, TDDVR has the following advantages: (a) An optimized set of asymmetrically dense grid-points are generated from the Hermite polynomial associated with the eigenfunction of a harmonic oscillator defined around the center of an initial wave packet, GWP [9].…”

“…With this respect, it is worthy to mention that the width parameters, as defined in the harmonic oscillator eigenfunctions are associated with the on -and off -diagonal elements of the Hamiltonian matrix in the quantum equation of motion, any non -linear "classical" propagation of the width6 not only increases inaccuracy in the quantum equation of motion but also brings the stiffness in the classical equation of motion. Thus, a fixed width approach either by introducing approximations [6,7] or by using time -independent width parameter10 is the obvious choice. On the contrary, the "classical" trajectory and its momentum appear with on -diagonal elements of the quantum equation of motion, may affect the convergence but not the final solution of the TDSE.…”

An adaptive pseudospectral method for wave packet dynamics

“…The algorithm can be related to the time-dependent discrete variable representation (TDDVR) developed by Billing and co-workers. [7][8][9] In one spatial dimension, the Hagedorn and TDDVR basis functions are identical up to a variable substitution. However, the Hagedorn basis is more flexible when used for problems with multiple degrees of freedom and provides a promising framework for more challenging quantum dynamics problems.…”

“…In the present paper, we introduce an adaptive scaling which makes the support of the basis follow the support of the wave packet. As noted by Billing and Adhikari,8 using the classical equations of motion for propagating the shape parameters of the basis is not in all cases a good choice. By using a basis which is better adapted to the solution one gets better accuracy with fewer basis functions.…”