Schrddinger equation

Feit M D and Fleck J A Jr 1983 Solution of the Schrddinger equation by a spectral method, energy levels of triatomic molecules J. Chem. Phys. 78 301-8... 

Tal-Ezer H and Kosloff R 1984 An accurate and efficient scheme for propagating the time dependent Schrddinger equation J. Chem. Phys. 81 3967-71... 

Dey B D, Askar A and Rabitz H 1998 Multidimensional wave packet dynamics within the fluid dynamical formulation of the Schrddinger equation J. Chem. Phys. 109 8770-82... 

Pack R T and Hirschfelder J O 1968 Separation of rotational coordinates from the W-electron diatomic Schrddinger equation J. Chem. Phys. 49 4009... 

Leforestier C et a/1991 A comparison of different propagation schemes for the time dependent Schrddinger equation J. Comput. Phys. 94 59... 

It is possible to parametarize the time-dependent Schrddinger equation in such a fashion that the equations of motion for the parameters appear as classical equations of motion, however, with a potential that is in principle more general than that used in ordinary Newtonian mechanics. However, it is important that the method is still exact and general even if the trajectories aie propagated by using the ordinary classical mechanical equations of motion. 

By substituting the expression for the matrix elements in Eq. (B.21), we get the final form of the Schrddinger equation within the diabatic representation... 

With the time-dependent Schrddinger equation written as... 

Projecting the nuclear solutions Xt( ) oti the Hilbert space of the electronic states (r, R) and working in the projected Hilbert space of the nuclear coordinates R. The equation of motion (the nuclear Schrddinger equation) is shown in Eq. (91) and the Lagrangean in Eq. (96). In either expression, the terms with represent couplings between the nuclear wave functions X (K) and X (R). that is, (virtual) transitions (or admixtures) between the nuclear states. (These may represent transitions also for the electronic states, which would get expressed in finite electionic lifetimes.) The expression for the transition matrix is not elementaiy, since the coupling terms are of a derivative type. 

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, dynamic phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schrddinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac 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 Schrddinger equation are derived by employing the time-dependent variational principle (TDVP). 

Interaction with light changes the quantum state a molecule is in, and in photochemistry this is an electronic excitation. As a result, the system will no longer be in an eigenstate of the Hamiltonian and this nonstationaiy state evolves, governed by the time-dependent Schrddinger equation... 

In this section, the basic theory of molecular dynamics is presented. Starting from the BO approximation to the nuclear Schrddinger equation, the picture of nuclear dynamics is that of an evolving wavepacket. As this picture may be unusual to readers used to thinking about nuclei as classical particles, a few prototypical examples are shown. 

Using the BO approximation, the Schrddinger equation describing the time evolution of the nuclear wave function, can be written... 

In this picture, the nuclei are moving over a PES provided by the function V(R), driven by the nuclear kinetic energy operator, 7. More details on the derivation of this equation and its validity are given in Appendix A. The potential function is provided by the solutions to the electronic Schrddinger equation. 

If V(R) is known and the mahix elements ffap ate evaluated, then solution of Eq. nO) for a given initial wavepacket is the numerically exact solution to the Schrddinger equation. 

The time-dependent Schrddinger equation governs the evolution of a quantum mechanical system from an initial wavepacket. In the case of a semiclassical simulation, this wavepacket must be translated into a set of initial positions and momenta for the pseudoparticles. What the initial wavepacket is depends on the process being studied. This may either be a physically defined situation, such as a molecular beam experiment in which the paiticles are defined in particular quantum states moving relative to one another, or a theoretically defined situation suitable for a mechanistic study of the type what would happen if. .. 

And using this equation the nuclear Schrddinger equation can be written in matiix form [54,179]... 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... 

As described above in Appendix A, within the BO approximation the nuclear Schrddinger equation is... 

In the derivation used here, it is clear that two approximations have been made—the configurations are incoherent, and the nuclear functions remain localized. Without these approximations, the wave function fonn Eq. (C.l) could be an exact solution of the Schrddinger equation, as it is in 2D MCTDH form (in fact is in what is termed a natural orbital form as only diagonal configurations are included [20]). 

A chemical reaction takes place on a potential surface that is determined by the solution of the electronic Schrddinger equation. In Section, we defined an anchor by the spin-pairing scheme of the electrons in the system. In the discussion of conical intersections, the only important reactions are those that are accompanied by a change in the spin pairing, that is, interanchor reactions. We limit the following discussion to these class of reactions. 

In other words, for calculating the second-order energy (the vibrational energy), we only have to keep the term to do with the interatomic distance. The other terms, then, will enter the total Schrddinger equation in higher orders. 

The most consequent and the most straightforwaid realization of such a concept has been carried out by Handy, Carter, and Rosmus (HCR) and their coworkers. The final form of the vibration-rotation Hamiltonian and the handling of the corresponding Schrddinger equation in the absence of the vibronic... 

Now, consider the general case of a V2 multiply excited degenerate vibrational level where V2 > 2, which is dealt with by solving the Schrddinger equation for the isotropic 2D harmonic oscillator with the Hamiltonian assuming the fonn [95]... 

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