Multi-dimensional integral evaluation

A method of determination of the envelopes of vibronic molecular spectra has been obtained by a generalization of a similar approach known in the atomic spectroscopy. Its application to diatomic molecules is very easy and has been illustrated by several examples. A simple way of taking into account the Q-dependence of the molecular transition moments has been described. The implementation of the method to more complex cases, involving multidimensional potential hypersurfaces, is straightforward though requires some numerical effort connected with the evaluation of multi-dimensional integrals. 

J. H. Halton, On the Efficiency of Certain Quasi-Random Sequences of Points in Evaluating Multi-dimensional Integrals, Num. Math. 2,84-90 (1960). 

However, the methods involved are general if a multi-dimensional integral is to be evaluated it must be... 

In Fig. 2, the resulting canonical FTST rate constant determined from Eq. (11) is labelled HR for hindered rotor. As required by its derivation, this result is essentially Identical to the published [8] canonical FTST rate constant for the same PES. This earlier calculation involved time consuming numerical evaluations of multi-dimensional integrals, many of which in the canonical limit are done analytically in the derivations above. 

Note that Eqs. (5.4) and (5.5) are coupled via the mean fields and the matrix elements. In practice, the EOMs are decoupled during an update time step, by the so called constant mean-field approach. Nevertheless, the evaluation of the EOMs (5.4) and (5.5) is rather costly as the calculation of the mean fields and matrix elements involves multi-dimensional integrals over all physical DOF. A key ingredient of the MCTDH algorithm therefore is that the Hamiltonian operator can be expressed in terms of products of low-dimensional terms such that... 

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. 

A series of benchmark calculations of critical experiments has been performed to assess the effects that recent chanp es in the ENDF/B data files have had on calculated LMFBR parameters. Three well-documented critical assemblies were studied using standard methods of fast reactor analysis [two-dimensional (2-D) multi-group diffusion theory] with both Versions III and IV of ENOF/B. A review of the changes in the principal cross sections incorporated in the latest evaluation has been made and was used to interpret the changes in calculated integral parameters. 

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