Schrodinger equation dynamics

We now illustrate the utility of Eq. (27) in relating the RWP dynamics based on the arccosine mapping, Eq. (16), to the usual time-dependent Schrodinger equation dynamics, Eq. (1). We carried out three-dimensional (total angular momentum 7 = 0) wave packet calculations for the... 

Balint-Kurti G G, Dixon R N and Marston C C 1992 Grid methods for solving the Schrodinger equation and time-dependent quantum dynamics of molecular photofragmentation and reactive scattering processes/of. Rev. Phys. Chem. 11 317—44... 

The discussion in the previous sections assumed that the electron dynamics is adiabatic, i.e. the electronic wavefiinction follows the nuclear dynamics and at every nuclear configuration only the lowest energy (or more generally, for excited states, a single) electronic wavefiinction is relevant. This is the Bom-Oppenlieimer approxunation which allows the separation of nuclear and electronic coordinates in the Schrodinger equation. 

Reactive atomic and molecular encounters at collision energies ranging from thermal to several kiloelectron volts (keV) are, at the fundamental level, described by the dynamics of the participating electrons and nuclei moving under the influence of their mutual interactions. Solutions of the time-dependent Schrodinger equation describe the details of such dynamics. The representation of such solutions provide the pictures that aid our understanding of atomic and molecular processes. 

In Section II, molecular dynamics within the BO approximation was introduced. As shown in Appendix A, the full nuclear Schrodinger equation is, however. 

Various kinds of mixed quantum-classical models have been introduced in the literature. We will concentrate on the so-called quantum-classical molecular dynamics (QCMD) model, which consists of a Schrodinger equation coupled to classical Newtonian equations (cf. Sec. 2). 

Rather than solve a Schrodinger equation with the Nuclear Hamiltonian (above), a common approximation is to assume that atoms are heavy enough so that classical mechanics is a good enough approximation. Motion of the particles on the potential surface, according to the laws of classical mechanics, is then the subject of classical trajectory analysis or molecular dynamics. These come about by replacing Equation (7) on page 164 with its classical equivalent ... 

The Schrodinger equation with its time-independent hamiltonian does not in fact constitute a dynamical theorem it is simply a description of the time-dependence of the probability field corresponding to steady states or equilibrium conditions. 

Imprecise boundaries. The basic concept of the state of a system is governed by two mutually incompatible laws, namely the Schrodinger equation for normal dynamics and the measurement process for interactions with macroscopic devices. It is not made clear where the applicability of one ends and the other begins. 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

A time-dependent Schrodinger equation with H replaced by/(H) can be used to infer dynamics information about H, with f H) being chosen for computational convenience. 

All of the methods for designing laser pulses to achieve a desired control of a molecular dynamical process require the solution of the time-dependent Schrodinger equation for the system interacting with the radiation field. Normally, this equation must be solved many times within an iterative loop. Different possible approaches to the solution of these equations are discussed in Section V. 

Schmidt number, multiparticle collision dynamics, real-time systems, 113-114 Schrodinger equation ... 

Most of the AIMD simulations described in the literature have assumed that Newtonian dynamics was sufficient for the nuclei. While this is often justified, there are important cases where the quantum mechanical nature of the nuclei is crucial for even a qualitative understanding. For example, tunneling is intrinsically quantum mechanical and can be important in chemistry involving proton transfer. A second area where nuclei must be described quantum mechanically is when the BOA breaks down, as is always the case when multiple coupled electronic states participate in chemistry. In particular, photochemical processes are often dominated by conical intersections [14,15], where two electronic states are exactly degenerate and the BOA fails. In this chapter, we discuss our recent development of the ab initio multiple spawning (AIMS) method which solves the elecronic and nuclear Schrodinger equations simultaneously this makes AIMD approaches applicable for problems where quantum mechanical effects of both electrons and nuclei are important. We present an overview of what has been achieved, and make a special effort to point out areas where further improvements can be made. Theoretical aspects of the AIMS method are... 

The development of an ab initio quantum molecular dynamics method is guided by the need to overcome two main obstacles. First, one needs to develop an efficient, yet accurate, method for solving the electronic Schrodinger equation for both ground and excited electronic states. Second, the quantum mechanical character of the nuclear dynamics must be addressed. (This is necessary for the description of photochemical and tunneling processes.) This section provides a detailed discussion of the approaches we have taken to solve these two problems. 

This hybrid approach can significantly extend the domain of applicability of the AIMS method. The use of interpolation significantly reduces the computational effort associated with the dynamics over most of the timescale of interest, while regions where the PESs are difficult to interpolate are treated by direct solution of the electronic Schrodinger equation during the dynamics. The applicability and accuracy of the method was tested using a triatomic model collisional quenching of Li(p) by H2 [125], which is discussed in Section III.A below. 

Bound-state photoabsorption, direct molecular dynamics, nuclear motion Schrodinger equation, 365-373... 

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... 

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