A new method, of very general applicability and very easily programmed for an electronic computer, is proposed for the numerical integration of functions of many independent variables. This new method renders obsolete, in most applications, the commonly used Monte Carlo procedure and the more recent, original method of Haselgrove. In the new scheme the sample points are distributed systematically rather than at random and the ensemble of points forms a unique, closed, symmetrical pattern, which effectively fills the space of the multidimensional integration. The paper contains an extensive statistical-analytic treatment of the error characteristics of the new method, one that enables advance quantitative estimation of upper limits of error in the integration of various types of functions. For continuous functions with bounded first derivatives, the error is shown ultimately to disappear at least as rapidly as the inverse square of the number of sample points; moreover, for runs of practical length the error limits with the new scheme are smaller—by a factor ranging from 2 to perhaps 104 or more—than those of any previous general procedure. The method employs certain rational constants which govern the arrangement of sample points. Tables of such constants, suitably optimized, which will permit the integration of functions with up to 12 independent variables, are provided along with a discussion of a method by which such constants may be obtained.
A general form for the wavefunction of a single bound electron moving in the field of arbitrary fixed nuclei is proposed. The use of this wavefunction in the molecular Schrödinger equation results in the exact cancellation of singularities due to the electron—nuclear attractive potential. Illustrations of its application to the first few states of the hydrogen molecule—ion (H2+) as well as information about accuracy and convergence rate are presented.
The results of calculations on the ground state of the H3 system are presented, and potential-energy surfaces are constructed for the linear and isosceles triangular configurations. The path of minimum energy along these surfaces for the H+H2→H2+H reaction passes through a maximum 7.74 kcal above the reactants at a linear symmetric configuration with an H–H separation of 1.76 bohrs. Calculations on the ground states of He2+, He2++, the lowest 1Π state of equilateral triangular H3+, and a number of one-electron states of H3++ and He2+++ are also summarized.
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