Multiplicative integral approximation

Van Alsenoy, C. 1988. Ab Initio Calculations on Large Molecules The Multiplicative Integral Approximation. J. Comput. Chem. 9, 620-626. 

For the first time, a systematic uniform ab initio study of endohedral complexes of C60, Si60 and Ge60 with monoatomic cations (Li+, Na+, K+ and Rb+) and anions (F-, Cl% Br and P) was performed. The 3-21 G level (3-21+G for the anions) was used. This could be achieved by application of the multiplicative integral approximation [86], as implemented in the program BRABO [87]. The sequence studied for the cages bears a close resemblance with isomorphic substitution in zeolites modifying its catalytic activity via bridging hydroxyls. 

A.H. Stroud, Approximate Calculation of Multiple Integrals, Prentice HaU, 1971. 

Two approximations are introduced, for simplicity, to perform the integration of Eq. (4-9) for the Taylor mechanism dispersion coefficient Ezr-First, the term included in the second integral of Eq. (4-9) is eliminated from the integral by taking an effective mean value Vn, (= dvt), with d being a mean of a. Second, the local liquid holdup is eliminated from the multiple integral by using the mean liquid holdup Cl. with a correction factor/of order unity (/al). 

Now the DK Hamiltonian may be calculated to the desired level of accuracy. Within our finite basis set approximation the multiple integral expressions occurring at the evaluation of the momentum space operators are reduced to simple matrix multiplications, which are computationally not very demanding. As soon as /foKHn has been evaluated within the chosen p -representation, it can be transformed back to the usual configuration space representation by applying the inverse transformation This Hamiltonian is then available for every variational procedure without any further modifications. 

With the goal of minimizing CPU time that is required to assemble the Fock matrix, VanAlsenoy proposed the multiplicative integral approxima-tion" o in which products of contracted Gaussians are approximated by linear combinations of 13 auxiliary functions. The resulting three-index integrals are used in fast construction of the Fock matrix, which is later corrected with the ERIs for which the error of approximation exceeds some predetermined threshold. 

A. H. Stroud, Approximate calculation of multiple integrals, (Prentice-Hall, Inc., Englewod Cliffs, N.J., 1971). 

In the general case, the nonlinear creep behavior could be described by a multiple integral representation using several kernels. The practical application of such equations is limited by the absence of a clearly defined strategy of creep kernel determination as well as relations between the kernel and resolvent. Various simplified versions of this approach have also been proposed. Usually it is assumed that the creep kernel and relaxation time are independent of the stress value. At the same time, it is known that a good approximation to nonlinear creep may be obtained by using the following equation ... 

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