Radial overlap integrals

If the ligands axe not linear ligators, two en parameters are needed, (II) and It is sometimes possible, by inspection of the ligand, to set one or both of these equal to zero. Since eK is deemed to be proportional to the square of the appropriate overlap integral, the radial dependence of ex can be found by calculating the overlap integral Sx over a range of internuclear distances. 

A variety of empirical rules exist for choosing the exponent(s) for a set of polarization functions. If only a single set is desired, one possible choice is to make the maximum in tlie radial density function, equal to that for the existing valence set (e.g., the 3d functions that best overlap the 2p functions for a first-row atom - note that the radial density is used instead of the actual overlap integral because the latter, by symmetry, must be zero). 

The group overlap integrals are for Ni(CN ) . The atomic wave functions are 3d and 4s radial functions for nidsei from ret. ... 

If the radial overlap integrals g(E, n) are considered as functions of the translational energy Etrans = E — en, where the en are the vibrational energies of the product molecule, they do not depend on the vibrational quantum number n because the radial functions Xn Etrans,n) all solve... 

This ultrasimple classical theory is, of course, too crude for practical applications, especially for highly excited states of the parent molecule. Its usefulness gradually diminishes as the degree of vibrational excitation increases, i.e., as the initial wavefunction becomes more and more oscillatory. If both wavefunctions oscillate rapidly, they can be approximated by semiclassical WKB wavefunctions and the radial overlap integral of the bound and the continuum wavefunctions can subsequently be evaluated by the method of steepest descent. This leads to analytical expressions for the spectrum (Child 1980, 1991 ch.5 Tellinghuisen 1985, 1987). In particular, relation (13.2), which relates the coordinate R to the energy E, is replaced by... 

Z, in the central ion radial function as we saw above, this is nearly a more fundamental effect than the symmetry-restricted covalency represented by positive 63 or 65. Another is the overlap integral S3 and S5 between the appropriate central ion orbitals and ligand orbitals of eq. (20) introducing the normalization conditions... 

An approximate (non-SCF) molecular-orbital method involving extensive parametrization of the required integrals and with the overlap integral represented as a simple product of radial and angular terms... 

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