Schrodinger equation bound motion

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

The first problem is the subject of modern quantum chemistry. Many efficient methods and computer programs are nowadays available to solve the electronic Schrodinger equation from first principles without using phenomenological or experimental input data (so-called ab initio methods Lowe 1978 Carsky and Urban 1980 Szabo and Ostlund 1982 Schmidtke 1987 Lawley 1987 Bruna and Peyerimhoff 1987 Werner 1987 Shepard 1987 see also Section 1.5). The potentials Vfe(Q) obtained in this way are the input for the solution of the nuclear Schrodinger equation. Solving (2.32) is the subject of spectroscopy (if the motion is bound) and... 

Equation (1-13) or its body-fixed equivalent is of little use for Van der Waals complexes, as it discriminates one nuclear coordinate, e.g. y = 1. Specific mathematical forms of Hamiltonians describing the nuclear motions in Van der Waals dimers have been developed (7). This point will be discussed in more details in Section 12.4. Here we only want to stress that whatever the mathematical form of the Hamiltonian is used to solve the problem of nuclear motions, the results will be the same, if the Schrodinger equation is solved exactly. However, in weakly bound complexes there is a hierarchy of motions due to the strong intramolecular forces which determine the internal vibrations of the molecules, and to much weaker intermolecular forces which determine their relative translations and rotations. This hierarchy allows to make a separation between the intramolecular vibrations with high frequencies and the intermolecular modes with much lower frequencies. Such a separation of the fast intramolecular vibrations and slow rotation-vibration-tunneling motions can be performed if a suitable form of the Hamiltonian for the nuclear motions in Van der Waals molecules is used. 

To formulate and solve the Schrodinger equation for a bound atom it is usual to express the relevant observables in the coordinate representation. Observables for the motion of an electron are represented by differential operators. 

The Born-Oppenheimer approximation allows us to decouple the electronic and nuclear motions of the free molecule of the Hamiltonian Hq. Solving the Schrodinger equation //o l = with respect to the electron coordinates r = r[, O, gives rise to the electronic states (r, R) = (r n(R)), n = 0,..., Ne, of respective energies En (R) as functions of the nuclear coordinates R, with the electronic scalar product defined as (n(R) n (R))r = j dr rj( r. R) T,-(r, R). We assume Ne bound electronic states. The Floquet Hamiltonian of the molecule perturbed by a field (of frequency co, of amplitude 8, and of linear polarization e), in the dipole coupling approximation, and in a coordinate system of origin at the center of mass of the molecule can be written as... 

Most of this chapter deals with the electronic Schrddinger equation for diatomic molecules, but this section examines nucleeu" motion in a bound electronic state of a diatomic molecule. From (13.10) and (13.11), the Schrodinger equation for nuclear motion in a diatomic-molecule bound electronic state is... 

Schrodinger equation. In such a unitary treatment, the population is conserved, so that the sum of populations in the lower and in the excited state remains constant. Complete sets of vibrational (bound + continuum) levels in the two electronic states are introduced. Therefore, quantum threshold effects (see Section 7.3.4.2) are automatically accounted for. In this chapter, only 5-wave scattering is considered, and one introduces a two-component radial wavefunction 9(R,t) describing the relative motion of the nuclei both in the lower electronic state ( I groimdC, 0) and in the excited state ( I excC, 0). 

By solving the nuclear H2 Schrodinger equation with the complex potential surface we obtained two distinctive types of resonance states which are presented in Fig. 3. The first one is associated with vibrationally bound motion of autoionizing molecule, i.e. H2 H2 + e. We will refer to these states as vibrationally-discrete autoionization-resonance states. The... 

In this case we have a one dimensional potential and only one quantum number, n, and dipole selection rules dictate An odd. An example of the photon spectrum observed in the forward direction for the injection of 54 Mey electrons along the (110) planar direction in Si is shown in Fig. 16.xhe bound state An = 1 transitions are evident up to n = 5>4 and at higher energies the An = 3 transitions are also evident. One can now compare the observed spectrum with calculations based on e.g. Hartree-Fock descriptions of the Si atom. This can be done directly through the solution for the onedimensional Schrodinger equation or one may work in momentum space and use the many-beam formulation of the Schrodinger equation for the transverse motion. The results of the many-beam calculations which use Doyle-Turner scattering factors derived from Hartree-Fock wave functions are compared with experimental results in Table II. 

It is the purpose of this review to discuss and illustrate the methods presently employed to obtain potential energy surfaces by approximate, but non-empirical solutions to Schrodinger s electronic equation. In addition to discussing the different levels of approximation employed in these ab initio calculations, we emphasize the type of chemical system (in terms of its electronic structure) to which each level of calculation may be expected to yield usable results, i.e. results with acceptable errors or with predictable bounds on the error. Our interest will be primarily in surfaces which have been determined for the prediction and understanding of chemical reactions. This will include a survey of those calculations which have concentrated on determining the reaction path, and the geometry and properties of the system at points on this path, as well as those in which an essentially complete surface has been determined. The latter type of calculation coupled with either classical or quantal treatments of the nuclear motion on such a surface provides a total theoretical prediction of a chemical reaction. This ultimate objective has been achieved in the case of the H + Ha exchange reaction. 

Whatever way it is intended to go in specifying the electronic coordinates, they must be specifiable as translationally invariant so that the centre of mass motion can be separated from Schrodinger s equation for the system. It is only the translationally invariant part of the Hamiltonian that can have a bound state spectrum and thus be relevant to both the scattering and the bound molecule problem. 

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