Generalized centrifugal potential

The Cqq, are generalized centrifugal potentials which are proportional to R 2 the proportionality constants depend on J,p,j, and Q, and they are explicitly given by Balint-Kurti and Shapiro (1981). The wavenumbers kj are defined in (3.21) and the potential coupling elements are given by... 

The Schrodinger equation thus is the same as for D = 3, except for the addition of the D-dependent centrifugal potential, and now the Jacobian for the dP function is J3. This is the preferred form for calculations employing conventional variational methods. It allows any existing computer code to be extended to D dimensions simply by adding the matrix elements for the centrifugal potential of Eq.(42). These results are readily generalized to the iV-body problem, for any mass distribution of the particles or form of the interaction potential [2]. 

We note that v/vi = hi, where h,- is the perpendicular distance of the ith particle from the flat ( base ) specifled by the other N — 1 particles. Therefore the nondifferential component of the kinetic energy is a generalized centrifugal potential Tcent which has the form... 

For any p, this generalized centrifugal potential has its minimum at B = 90 (so that it favors orthogonality) and diverges when B = 60 or 120 . The divergence signals the fact that the electronic coordinates for the reference electron have become linearly dependent on those of the other electron [22] (a generalization of the observation that three vectors separated pairwise by 120 are linearly dependent). 

The general analysis, while not difficult, is complicated however, the limiting case of the very elongated, essentially cylindrical drop is not hard to treat. Consider a section of the elongated cylinder of volume V (Fig. II-18h). The centrifugal force on a volume element is u rAp, where w is the speed of revolution and Ap the difference in density. The potential energy at distance r from the axis of revolution is then w r Apfl, and the total potential energy for the... 

It is generally accepted that the centrifugal sudden (CS) approximation is the most reliable approximate method. Its results are usually very close to those obtained by ab initio close coupling (CC) calculations. The integral and differential cross-sections of Ar inelastic scattering on nitrogen were performed for a few low-frequency rotational transitions and four different interaction potentials [205]. Much better agreement of CC with CS results was found than with IOS calculations performed in... 

Along with the frictional force on a bead, given by Eq. (9), there will be in general other forces F-, derived in part from the potential energy U that holds the polymer chain together, and in part from external sources, e.g., electric or centrifugal fields. The total force on the y th bead is thus Fj + F/. 

At low temperature the classical approximation fails, but a quantum generalization of the long-range-force-law collision theories has been provided by Clary (1984,1985,1990). His capture-rate approximation (called adiabatic capture centrifugal sudden approximation or ACCSA) is closely related to the statistical adiabatic channel model of Quack and Troe (1975). Both theories calculate the capture rate from vibrationally and rotationally adiabatic potentials, but these are obtained by interpolation in the earlier work (Quack and Troe 1975) and by quantum mechanical sudden approximations in the later work (Clary 1984, 1985). 

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