Schrodinger-type equations, coupled

More specifically we will present the recent advances for the numerical integration of the radial time-independent Schrodinger equation and related problems and of the coupled differential equations of the Schrodinger type. The radial Schrodinger equation has the form ... 

Theoretical and numerical results obtained for the radial Schrodinger equation and for the well known Woods-Saxon potential and for the coupled differential equations of the Schrodinger type show the efficiency of the new methods. 

Coupled Differential Equations. - There are many problems in theoretical physics and theoretical chemistry, atomic physics, physical chemistry, quantum chemistry and chemical physics which can be transformed to the solution of coupled differential equations of the Schrodinger type. 

The close-coupling differential equations of the Schrodinger type may be written in the form... 

It is easy for one to see that a real problem in theoretical physics, theoretical chemistry, atomic physics, quantum chemistry, physical chemistry and molecular physics which can be transformed to close-coupling differential equations of the Schrodinger type is the rotational excitation of a diatomic molecule by neutral particle impact. Denoting, as in ref. 113, the entrance channel by the quantum numbers (/, /), the exit channels by (/,//), and the total angular momentum by J =j + l =j + /, we find that... 

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. 

Projecting the nuclear solutions Xt(R) n 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 Schrodinger 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 Xi (R) Xm( )> is, (virtual) transitions (or admixtures) between the nuclear states. (These may represent transitions also for the electronic states, which would get expressed in finite electronic lifetimes.) The expression for the transition matrix is not elementary, since the coupling terms are of a derivative type. 

A type of analysis of molecular properties and molecular transformations which, unlike the Bom-Oppenheimer approximation, assumes that electronic states depend strongly on nuclear coordinates. If stationary electronic states are obtained as solutions of the Schrodinger equation for fixed nuclei, an accounting for vibronic coupling terms in the Hamiltonian (interaction of electrons with nuclear displacements) mixes these electronic states. This mixing is especially strong in the cases of electronic degeneracy (See Jahn-Teller Effect). 

If separability in a problem is not immediately apparent, a product form of the wave-function can be tested just the same. If application of the Hamiltonian to the product form [e.g., X(x)Y(t/)] yields an equation of the same type as Equation 8.13, the problem is separable. If the problem is not separable, the Schrodinger equation ends up as a coupled differential equation, and solution is usually more difficult. 

MORBID [9] is a variational approach for solving the Schrodinger equation for the motion of nuclei of triatomic molecule, and RENNER [10] is an extension of MORBID for electronic states subject to the Renner-coupling. These approaches have been applied for simulating spectra of a large number of triatomic molecules (see below). MORBID and RENNER assume that dipole moment surfaces are given in special analytical representations, which in this paper will be referred to as a MORBID-type and described in detail below. 

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