A “classical” trajectory driven nuclear dynamics by a parallelized quantum‐classical approach to a realistic model Hamiltonian of benzene radical cation

Abstract

ABSTRACT:We explore the workability of a parallelized algorithm of time-dependent discrete variable representation (TDDVR) methodology formulated by involving "classical" trajectories on each DOF of a multi-mode multi-state Hamiltonian to reproduce the population dynamics, photoabsorption spectra and nuclear dynamics of the benzene radical cation. To perform such dynamics, we have used a realistic model Hamiltonian consists of five lowest electronic states (X 2 E 1g , B 2 E 2g , C 2 A 2u , D 2 E 1u , and E 2 B 2u ) which are interconnected through several conical intersections with nine vibrational modes. The calculated nuclear dynamics and photoabsorption spectra with the advent of our parallelized TDDVR approach show excellent agreement with the results obtained by multiconfiguration time-dependent Hartree method and experimental findings, respectively. The major focus of this article is to demonstrate how the "classical" trajectories for the different modes and the "classical" energy functional for those modes on each surface can enlight the time-dependent feature of nuclear density and its' nodal structure.

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

“…Y k and Z k matrices couple grid-points or basis functions of the kth mode only. This feature allows the parallelization of the algorithm, reduces computational time remarkably [37][38][39] and pave the possibility to pursue relatively large dimensional calculations with much less CPU time. The present dynamical calculation has been carried out both serially and parallelly by using ~2 × 10 7 number of TDDVR basis functions on each surface for four surface nine mode model Hamiltonian (4).…”

“…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.…”

“…37,39 Moreover, the classical path equations for the kth mode, those appear along with the quantum equation of motion, can be written as…”

“…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.…”

“…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. [37][38][39] The formulation has the following assumptions on the form of the molecular wavefunction: (a) the wavefunction on each surface is being expanded as sum of vectors, namely product-type multi-mode TDDVR basis functions with time-dependent coefficients. 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.…”