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This article proposes a novel multivariate generalized autoregressive conditionally heteroscedastic (GARCH) model that incorporates the modified Cholesky decomposition for a covariance matrix in or...

In this paper we provide a systematic method of finding the higher order asymptotic expansion of the Keldysh integral using the hydrogen atom as an example. Comparisons with exact numerical integration shows that the second-order correction improves the the first-order results significantly. We extend this method to molecular ionisation in H and calculate the total ionisation rate. The results are in good agreement with the exact solution of the time-dependent Schrodinger equation.

Classical limits of the quantal centrifugal-sudden approximation to generalized cross-sections have been derived and compared with the exact classical results. Detailed comparisons between the classical centrifugal-sudden approximation (cCSA) values, the exact classical trajectory results, and the corresponding quantal calculations are then carried out for the Ar−N2 system, and contrasted with similar comparisons for He−N2. Above 150 K the cCSA is found to be accurate to within 1% for the temperature dependent viscosity and diffusion cross-sections and 5% for the viscomagnetic and depolarized Rayleigh light-scattering cross-sections.

Explicit identification of the quantum mechanical energy dependent effective cross sections with their classical counterparts is made. Detailed quantal and classical trajectory computations of these cross sections are then carried out for the interaction between HD and He; excellent agreement between the two calculations is obtained at high energies although, as expected, the agreement deteriorates at lower energies. A hybrid scheme which employs classical results at high and quantal results at low energies proves to be an accurate and practical method for studying the temperature dependence of macroscopic cross sections.