Overlap integral nuclear

The function G in eq 1 is the Franck-Condon factor which accounts for the contribution of nuclear degrees of freedom and represents the thermal average of the overlap integrals between nuclear wavefunctions with respect to conservation of energy, and is given by (2, 3, 8, 9)... 

Johnson and Messmer expanded the interpretation of the bonding overlap integral to a Jahn-Teller coupling parameter fi via the relation Sbond(EF) (m/My, where m, M are the electron and nuclear masses, respectively. According to this expression, there are two derived trends (1) for physically realistic values of d (d > 0.05 nm), Tc drops as d increases and (2) at constant d, Tc increases as S increases. 

In order to simplify the evaluation of overlap integrals between bound and continuum wavefunctions, it is advisable (although not necessary) to describe both wavefunctions by the same set of coordinates. Usually, the calculation of continuum, i.e., scattering, states causes far more problems than the calculation of bound states and therefore it is beneficial to use Jacobi coordinates for both nuclear wavefunctions. If bound and continuum wavefunctions are described by different coordinate sets, the evaluation of multi-dimensional overlap integrals requires complicated coordinate transformations (Freed and Band 1977) which unnecessarily obscure the underlying dynamics. 

R — oo. As we will show below, the partial cross sections for absorbing the photon and producing the diatomic fragment in vibrational channel n are proportional to the square modulus of the overlap of these continuum wavefunctions with the nuclear wavefunction in the electronic ground state (indicated by the shaded areas). Since the bound wavefunction of the parent molecule is rather confined, only a very small portion of the continuum wavefunctions is sampled in the overlap integral. 

The tensors are termed atomic axial tensors (AATs) 7 and are the electronic and nuclear components. Here, (dpG/dXXa)0 is the same derivative which occurred already in Equation (2.86). The electronic AAT, 7 is the overlap integral with the derivative (dif/G/dHp)0. The latter is defined via... 

The manifestation of the dipole-dipole approximation can be seen explicitly in Equation (3.134) as the R 6 dependence of the energy transfer rate. In Equation (3.134) the electronic and nuclear factors are entangled because the dipole-dipole electronic coupling is partitioned between k24>d/(td R6) and the Forster spectral overlap integral, which contains the acceptor dipole strength. Therefore, for the purposes of examining the theory it is useful to write the Fermi Golden Rule expression explicitly,... 

Cusachs reported (4) that the repulsive terms in the W-H model which assumes that electron repulsion and nuclear repulsion cancel nuclear-electron attraction, consist of one-electron antibonding terms only. Cusachs noted Ruedenberg s observation that the two-center kinetic energy integral is proportional to the square of the overlap integral rather than the first power. Cusachs used this to develop the approximation ... 

Use of overlap integrals and electrostatic potentials, essentially nuclear attraction integrals, when dealing with two electron repulsion integrals. 

In this sense, electronic spin-spin contact integrals can be computed as simple bilinear functions of overlap integrals, and electron repulsion integrals are expressible as bilinear fimctions of some sort of two electron nuclear attraction integrals. 

A more general definition of the FCWD includes overlap integrals of quantum nuclear modes. The definition given by Eq. [19] includes only classical solvent modes (superscript s ) for which these overlap integrals are identically equal to unity. An extension of Eq. [19] to the case of quantum intramolecular excitations of the donor-acceptor complex is given below in the section discussing optical Franck-Condon factors. 

In certain cases, the classical Marcus formula is not sufficient to explain the observed-dependence of the electron transfer rate on temperature or AG, which could indicate that it is necessary to use a Franck-Condon term in which the contribution of the nuclei is treated in quantum mechanical terms. In this treatment, the Franck-Condon term equals the thermally-weighted sum of the contributions from all possible vibrational states of the reactants, each multiplied by their Franck-Condon factor i.e. the square of the overlap integral of a nuclear wave function of the reactant with the nuclear wave function of the product state that has the same total energy. 

The charge mass, a fictitious quantity, should be chosen to be small enough to guarantee that the charges readjust very rapidly to changes in the nuclear degrees of freedom[l]. The Coulomb interaction, Ji>(r), for intramolecular pairs is taken to be the Coulomb overlap integral between Slater orbitals centered on each atomic site, with each orbital characterized by a principal quantum number, n, and an exponent 0(6]-... 

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