Vibronic Schrodinger equation

In 1925, before the development of the Schrodinger equation, Franck put forward qualitative arguments to explain the various types of intensity distributions found in vibronic transitions. His conclusions were based on an appreciation of the fact that an electronic transition in a molecule takes place much more rapidly than a vibrational transition so that, in a vibronic transition, the nuclei have very nearly the same position and velocity before and after the transition. 

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

If the solution of the zero-order Schrodinger equation [i.e., all terms in (17) except V(r, Rq) are neglected] yields an/-fold degenerate electronic term, the degeneracy may be removed by the vibronic coupling terms. If T) and T ) are the two degenerate wave functions, then the vibronic coupling constant... 

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p2A, being thus mostly pronounced in n electronic state. Renner developed the system of two coupled Schrodinger equations and solved it for n states in the harmonic approximation by means of the perturbation theory. 

The most consequent and the most straightforward 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 Schrodinger equation in the absence of the vibronic... 

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

A wavefunction or vibronic wavefunction T(r, Q) is a solution of the Schrodinger equation for the molecular Hamiltonian... 

Following the work of Robin and Day, and Allen and Hush, the first rigorous solution to the vibronic Schrodinger equation for a mixed valence system was formulated by Piepho, Krausz, and Schatz. This model, based on the valence bond approach, is known as the PKS model. It is a model that has been modified to include orbital vibronic constants. A fully equivalent molecular orbital model developed by Piepho " is easier to conceptualize, however. The Piepho model also was the first to take advantage of the concept of orbital vibronic constants developed by Bersuker for Jahn-Teller systems (Figure 4). The Jahn-Teller theory is the basis for the PKS model. 

The combination of modern electronic-structure theory with efficient numerical techniques for the solution of the vibronic Schrodinger equation... 

In our early work on multimode vibronic dynamics, a fourth-order predictor-corrector method has been used to integrate the time-dependent Schrodinger equation. Later, FOD schemes and a fourth-order Runge-Kutta method have also been employed. These techniques proved to be superior to the predictor-corrector method for example, the FOD scheme was found to be 3-5 times faster than the SOD integrator (the latter... 

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). 

The multi-peak of Raman shift corresponds to different orders of the Fourier coefficients in the numerical solutions of the Schrodinger equation of the vibronic systems. In bio-molecules, the characteristic vibration frequency of the hydrogen bond is about 200 cm. It is known that the hydrogen bond is actually composed of a lone pair on one side and a covalent bond on the other in the B+ p A "). The B is less electronegative than A of which the sp orbits... 

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