Quantum-classical dynamics of scattering processes in adiabatic and diabatic representations

Abstract

We demonstrate the workability of a TDDVR based [J. Chem. Phys. 118, 5302 (2003)], novel quantum-classical approach, for simulating scattering processes on a quasi-Jahn-Teller model [J. Chem. Phys. 105, 9141 (1996)] surface. The formulation introduces a set of DVR grid points defined by the Hermite part of the basis set in each dimension and allows the movement of grid points around the central trajectory. With enough trajectories (grid points), the method converges to the exact quantum formulation whereas with only one grid point, we recover the conventional molecular dynamics approach. The time-dependent Schrodinger equation and classical equations of motion are solved self-consistently and electronic transitions are allowed anywhere in the configuration space among any number of coupled states. Quantum-classical calculations are performed on diabatic surfaces (two and three) to reveal the effects of symmetry on inelastic and reactive state-to-state transition probabilities, along with calculations on an adiabatic surface with ordinary Born-Oppenheimer approximation. Excellent agreement between TDDVR and DVR results is obtained in both the representations.

Quantum-mechanical wavepacket propagation in a sparse, adaptive basis of interpolating Gaussians with collocation

“…Both aspects become very easy in the scheme we propose here, due to a static basis-function grid at a fixed, optimal resolution. Given the tight relations between basis and grid representations via the discrete variable representation (DVR), it is not surprising that there is also a time-dependent DVR, 14,15 which additionally offers a ''smooth'' interpolation between (semi-)classical dynamics and the fully quantum case. Also related to moving grids are ''quantum trajectories'', [16][17][18] as modern realizations of Bohmian mechanics.…”

A rigorous approach to the formulation of extended Born‐Oppenheimer equation for a three‐state system

“…This adiabatic wavefunction is being propagated by using single surface BO approximate, approximate EBO, and rigorous EBO equations as functions of time with the help of numerically accurate TDDVR [36] approach and the respective wavefunction at t 3 ϱ is projected on the asymptotic eigenfunctions of the Hamiltonian to obtain the state-to-state vibrational transition probabilities at different energies. We perform all those dynamical calculations at total energies 1.25, 1.50, and 1.75 eV.…”

A quantum-classical simulation of the nuclear dynamics in NO 2 − and C6H 6 + with realistic model Hamiltonian

“…Since the detailed formulations of the different versions of TDDVR approach were presented elsewhere, [28][29][30][31][32][33][34][35][36][37][38][39] in order to bring the completeness of this article, we briefly demonstrate the relevant equations used for current perspective in the simplest but completely generalized way. The scheme propagates the DVR grid-points by utilizing the 'classical' equation of motion with a time-independent width parameter in the primitive basis set.…”

“…When this multidimensional multi-surface wavefunction matrix is substituted in the TDSE, we find that the timedependence of the expansion coefficients measure the quantum dynamics where the classical equation of motion for the central trajectory and its' momentum appear naturally; (b) time evolution of the amplitudes on the grid points comes through quantum dynamics and movement of the grid points is obtained from classical mechanics where grid points are associated with a distribution of 'classical' momenta; (c) the formulation has enough scope to incorporate the details associated with the respective surface through TDDVR basis set as well as timedependent coefficients while moving from one surface to another; (d) each TDDVR basis function is originated due to multiplication of the corresponding DVR basis with a plane wave which differs from one surface to another by its' parameter; (e) the eigenfunctions of harmonic oscillator constructs the DVR basis set; (f) the plane wave is defined by a classical trajectory and its' momentum. In this regard, it is worthy to mention that the formulation is based on time-independent width parameter [28][29][30][31][32][33][34] to bypass the inaccuracy 26 in the quantum equation of motion as well as the stiffness in the classical equation of motion due to time-dependent width parameter. When enough trajectories are included, the method is numerically exact, whereas with one grid point, one can recover the classical molecular dynamics approach.…”

“…The time dependent discrete variable representation (TDDVR) method [26][27][28][29][30][31][32][33][34][35][36][37][38][39] takes the advantages of both DVR and time-dependent basis. TDDVR is a very convenient formulation with the following attributes: (a) a very few number of optimized set of asymmetrically dense grid-points are used which are generated from the Hermite polynomial associated with the eigenfunction of a harmonic oscillator defined around the center of an initial wave packet, GWP; (b) the classical dynamics of the timedependent parameters of GWP dictates the movement of these unevenly spaced grid-points; (c) though the evaluation of KE matrices is needed once for the entire propagation, the diagonal potential energy (PE) matrix by which the electronic states are coupled has to be calculated at each time-step; (d) the independent contributions of different modes on a time-dependent amplitude motivate us to parallelize the algorithm which reduces the computational time remarkably.…”