Discrete variable representation dynamics

Luckhaus D 2000 6D vibrational quantum dynamics generalized coordinate discrete variable representation and (a)diabatic contraction J. Chem. Phys. 113 1329—47... 

Often the actions of the radial parts of the kinetic energy (see Section IIIA) on a wave packet are accomplished with fast Fourier transforms (FFTs) in the case of evenly spaced grid representations [24] or with other types of discrete variable representations (DVRs) [26, 27]. Since four-atom and larger reaction dynamics problems are computationally challenging and can sometimes benefit from implementation within parallel computing environments, it is also worthwhile to consider simpler finite difference (FD) approaches [25, 28, 29], which are more amenable to parallelization. The FD approach we describe here is a relatively simple one developed by us [25]. We were motivated by earlier work by Mazziotti [28] and we note that later work by the same author provides alternative FD methods and a different, more general perspective [29]. 

Discrete Fourier transform (DFT), non-adiabatic coupling, Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 153-155 Discrete variable representation (DVR) direct molecular dynamics, nuclear motion Schrodinger equation, 364-373 non-adiabatic coupling, quantum dressed classical mechanics, 177-183 formulation, 181-183... 

Normally the TDSE cannot be solved analytically and must be obtained numerically. In the numerical approach we need a method to render the wave function. In time-dependent quantum molecular reaction dynamics, the wave function is often represented using a discrete variable representation (DVR) [88-91] or Fourier Grid Hamiltonian (FGH) [92,93] method. A Fast Fourier Transform (FFT) can be used to evaluate the action of the kinetic energy operator on the wave function. Assuming the Hamiltonian is time independent, the solution of the TDSE may be written... 

W. H. Miller, J. Chem. Phys. 97 2499 (1992). (c) W. H. Miller and T. Seideman, Cumulative and state-to-state reaction probabilities via a discrete variable representation— absorbing boundary condition Green s function, Time Dependent Quantum Molecular Dynamics Experiments and Theory (J. Broeckhove, ed., NATO ARW. (d) W. H. Miller, Accts. Chem. Res. 26 174 (1993). 

Keywords Ab initio molecular dynamics simulations Always stable predictor-corrector algorithm Associated liquids Basis set Bom-Oppenheimer molecular dynamics simulations Car-Parrinello molecular dynamics simulations Catalysis Collective variable Discrete variable representation Dispersion Effective core potential Enhanced sampling Fictitious mass First-principles molecular dynamics simulations Free energy surface Hartree-Fock exchange Ionic liquids Linear scaling Metadynamics Nudged elastic band Numerically tabulated atom-centered orbitals Plane waves Pseudopotential Rare event Relativistic electronic structure Retention potential Self consistent field SHAKE algorithm ... 

QUANTUM DYNAMICS OF SMALL SYSTEMS USING DISCRETE VARIABLE REPRESENTATIONS ... 

The methodology of molecular quantum dynamics applied to non-adiabatic systems is presented from a time-dependent perspective in Chap. 4. The representation of the molecular Hamiltonian is first discussed, with a focus on the choice of the coordinates to parametrize the nuclear motion and on the discrete variable representation. The multi-configuration time-dependent Hartree (MCTDH) method for the solution of the time-dependent Schrddinger equation is then presented. The chapter ends with a presentation of the vibronic coupling model of Kdppel, Domcke and Cederbaum and the methodology used in the calculation of absorption spectra. 

Molecular Dynamics with Discrete Variable Representation Basis Sets Techniques and Application to Liquid Water. 

This sequence of states is a discrete representation of the continuous dynamical trajectory starting from zo at time t = 0 and ending at z at time t = . Such a discrete trajectory may, for instance, result from a molecular dynamics simulation, in which the equations of motion of the system are integrated in small time steps. A trajectory can also be viewed as a high-dimensional object whose description includes time as an additional variable. Accordingly, the discrete states on a trajectory are also called time slices. 

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

A particularly usefid way to measure the effects of uncertainty in the dynamic response of structural systems is the so-called first-excursion probability. This probability is widely used in stochastic structural dynamics and measures the chances that one or more structural responses exceed a prescribed threshold level within the duration of a dynamical excitation (Soong and Grigoriu 1993). First-excursion probability estimation is particularly challenging as characterization of uncertain loading usually comprises stochastic processes whose discrete representation can involve hundreds or even thousands of random variables. Similarly, the number of possible failure criteria involved can be extremely large as well, i.e., there can be several responses of interest that must be controlled at a large number of discrete time instants. Hence, several different techniques have been proposed in order to estimate first-excursion probabilities. Among these, methods based on simulation (such as the Monte Carlo method and its more advanced variants) have been shown to be the most appropriate approach to compute these probabilities (Schueller et al. 2004). 

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