Schrodinger equation electron nuclear dynamics

Nuclear motion Schrodinger equation direct molecular dynamics, 363-373 vibronic coupling, adiabatic effects, 382-384 electronic states ... 

Schiff approximation, electron nuclear dynamics (END), molecular systems, 339—342 Schrodinger equation ... 

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be just as useful. The electron nuclear dynamics (END) theory [1-4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schrodinger equation are derived by employing the time-dependent variational principle (TDVP). 

Electron-Nuclear Dynamics (END) method, " where both the orbitals describing the electronic wave function and the nuclear degrees of freedom are described by expansion into a Gaussian basis set, which moves along with the nuclei. Such an approach in principle allows a complete solution of the combined nuclear-electron Schrodinger equation without having to invoke approximations beyond those imposed by the basis set. Inclusion of the electronic parameters in the dynamics, however, means that the fundamental time step is short, and this results in a high computational cost for even quite short simulations and simple wave functions. 

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. 

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. 

The basic problem is to solve the time-independent electronic Schrodinger equation. Since the mass of the electrons is so small compared to that of the nuclei, the dynamics of nuclei and electrons can normally be decoupled, and so in the Born-Oppenheimer approximation the many-electron wavefunction P and corresponding energy may be obtained by solving the time-independent Schrodinger equation in which the nuclear positions are fixed. We thus solve... 

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