We present explicit forms of nonadiabatic coupling (NAC) elements of nuclear Schrodinger equation (SE) for a coupled three-state electronic manifold in terms of mixing angles of real electronic basis functions. If the adiabatic-diabatic transformation (ADT) angles are the mixing angles of electronic bases, ADT matrix transforms away the NAC terms and brings diabatic form of SE. ADT and NAC matrices are shown to satisfy a curl condition with nonzero divergence. We have demonstrated that the formulation of extended Born-Oppenheimer (EBO) equation from any three-state BO system is possible only when there exists a coordinate-independent ratio of the gradients for each pair of mixing angles. On the contrary, since such relations among the mixing angles lead to zero curl, we explore its validity analytically around conical intersection(s) and support numerically considering two nuclear-coordinate-dependent three surface BO models. Numerical calculations are performed by using newly derived diabatic and EBO equations and expected transition probabilities are obtained.

We investigate the molecular dynamics of pyrazine after excitation to the S2 electronic state by using the time-dependent discrete variable representation (TDDVR) method. The investigation has been carried out with a realistic 24-mode model Hamiltonian consisting of all the vibrational degrees of freedom of pyrazine molecule. First, we perform the simulation on a basic four-mode model, and then by including additional eight important modes and finally, by introducing 20 bath modes on the basic model. This sequential inclusion of bath modes demonstrates the effect of weak modes on the subsystem, where the calculations of energy and population transfer from basic model to the bath quantify the same effect. The spectral profile obtained by using TDDVR approach shows reasonably good agreement with the results calculated by quantum mechanical approach. It appears that the TDDVR approach for those large systems where quantum mechanical description is needed in a restricted region is a good compromise between accuracy and speed.

We have used the time-dependent discrete variable representation (TDDVR) method to simulate the photoabsorption spectrum of pyrazine. The time-dependent molecular dynamics of pyrazine after excitation to the S2 electronic state is considered as a benchmark to investigate the S2 absorption spectrum. We have carried out the dynamics on a basic four-mode model of pyrazine with the inclusion of five major modes as well as the rest of the vibrational modes as bath modes. Investigations reveal the effect of bath modes such as energy and population transfer from the subsystem to the bath. Calculated results demonstrate excellent agreement with traditional quantum-mechanical findings during the entire propagation and converge to the exact quantum results when enough gridpoints are used. It appears that TDDVR, as a numerical quantum dynamics methodology, is a good compromise between accuracy and speed.

ABSTRACT:The time-dependent discrete variable representation (TDDVR) of a wave function with grid points defined by the Hermite part of the Gauss-Hermite (G-H) basis set introduces quantum corrections to classical mechanics. The grid points in this method follow classical trajectory and the approach converges to the exact quantum formulation with sufficient trajectories (TDDVR points) but just with a single grid point; only classical mechanics performs the dynamics. This newly formulated approach (developed for handling time-dependent molecular quantum dynamics) has been explored to calculate vibrational transitions in the inelastic scattering processes. Traditional quantum mechanical results exhibit an excellent agreement with TDDVR profiles during the entire propagation when enough grid points are included in the quantum-classical dynamics.

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.

When a group of four states forms a subspace of the Hilbert space, i.e., appears to be strongly coupled with each other but very weakly interacts with all other states of the entire space, it is possible to express the nonadiabatic coupling (NAC) elements either in terms of s or in terms of electronic basis function angles, namely, mixing angles presumably representing the same sub-Hilbert space. We demonstrate that those explicit forms of the NAC terms satisfy the curl conditions--the necessary requirements to ensure the adiabatic-diabatic transformation in order to remove the NAC terms (could be often singular also at specific point(s) or along a seam in the configuration space) in the adiabatic representation of nuclear SE and to obtain the diabatic one with smooth functional form of coupling terms among the electronic states. In order to formulate extended Born-Oppenheimer (EBO) equations [J. Chem. Phys. 2006, 124, 074101] for a group of four states, we show that there should exist a coordinate independent ratio of the gradients for each pair of ADT/mixing angles leading to zero curls and, thereafter, provide a brief discussion on its analytical validity. As a numerical justification, we consider the first four eigenfunctions of the Mathieu equation to demonstrate the interesting features of nonadiabatic coupling (NAC) elements, namely, the validity of curl conditions and the nature of curl equations around CIs.

We are investigating the molecular dynamics of the butatriene cation after excitation from the ground state (X(2)B(2g)) to the first excited electronic state (A(2)B(2u)) by using the time-dependent discrete variable representation (TDDVR) method. The investigation is being carried out with a realistic 18-mode model Hamiltonian consisting of all the vibrational degrees of freedom of the butatriene molecule. First, we perform the simulation on a basic five mode model, and then by including additional thirteen modes as bath on the basic model. This sequential inclusion of bath modes demonstrates the effect of so called weak modes on the subsystem, where the calculations of energy and population transfer from the basic model to the bath quantify the same effect. The spectral profile obtained by using the TDDVR approach shows reasonably good agreement with the results calculated by the quantum mechanical approach/experimental measurement. It appears that the TDDVR approach for those large systems where quantum mechanical description is needed in a restricted region, is a good compromise between accuracy and speed.

We include the phonon modes originating from the three layers of Cu(100)/Cu(111) surface atoms on the dynamics of molecular [H(2)(v,j)/D(2)(v,j)] degrees of freedom (DOFs) through a mean field approach, where the surface temperature is incorporated into the effective Hamiltonian (potential) either by considering Boltzmann probability (BP) or by including the Bose-Einstein probability (BEP) factor for the initial state distribution of the surface modes. The formulation of effective potential has been carried out by invoking the expression of transition probabilities for phonon modes known from the "stochastic" treatment of linearly forced harmonic oscillator (LFHO). We perform four-dimensional (4D⊗2D) as well as six-dimensional (6D) quantum dynamics on a parametrically time and temperature-dependent effective Hamiltonian to calculate elastic/inelastic scattering cross-section of the scattered molecule for the H(2)(v,j)-Cu(100) system, and dissociative chemisorption-physisorption for both H(2)(v,j)-Cu(100) and D(2)(v,j)-Cu(111) systems. Calculated sticking probabilities by either 4D⊗2D or 6D quantum dynamics on an effective potential constructed by using BP factor for the initial state distribution of the phonon modes could not show any surface temperature dependence. In the BEP case, (a) both 4D⊗2D and 6D quantum dynamics demonstrate that the phonon modes of the Cu(100) surface affect the state-to-state transition probabilities of the scattered H(2) molecule substantially, and (b) the sticking probabilities due to the collision of H(2) on Cu(100) and D(2) on Cu(111) surfaces show noticeable and substantial change, respectively, as function of surface temperature only when the quantum dynamics of all six molecular DOFs are treated in a fully correlated manner (6D).