Self consistent modified extended Hückel (SC-MEH) calculations on heavy metal systems. I. Platinum(II) tetragonal planar complexes with and without relativistic effects

The Self Consistent Modified Extended Hückel molecular orbital method had been applied to several square planar complexes of platinum (II). Calculations including both the limited 5d, 6s, 6p and extended 5s, 5p, 5d, 6s, 6p starting bases for platinum were made. It is shown that in PtCl ₄ ^2– both the nuclear quadrupole moment and minimum total energy vs. bond distance are calculated to be in good agreement with experiment, only with the extended platinum AO basis.Specific inclusion of relativistic parameters via a pseudo-relativistic approximation are shown to have a significant effect on the energy molecular energy levels, however no meaningful rationalization can be made without the simultaneous inclusion of ligand field parameters as well.Supported in part by a grant made available through the Cancer Association of Greater New Orleans. Use of the facilities of the Computer Research Center of the University of New Orleans is gratefully acknowledged.     

The global Markov property for lattice systems

We prove the global Markov property for lattice systems of classical statistical mechanics, with bounded spins and finite range interactions. The method uses the one developed by two of us to prove the global Markov property of Euclidean generalized random fields. The result shows that the systems considered have a transition matrix, which together with a distribution on a hyperplane, describes completely the system.     

A geometric method in nonlinear programming

A differential geometric approach to the constrained function maximization problem is presented. The continuous analogue of the Newton-Raphson method due to Branin for solving a system of nonlinear equations is extended to the case where the system is under-determined. The method is combined with the continuous analogue of the gradient-projection method to obtain a constrained maximization method with enforced constraint restoration. Detailed analysis of the global behavior of both methods is provided. It is shown that the conjugate-gradient algorithm can take advantage of the sparse structure of the problem in the computation of a vector field, which constitutes the main computational task in the methods.This is part of a paper issued as Stanford University, Computer Science Department Report No. STAN-CS-77-643 (Ref. 45), which was presented at the Gatlinburg VII Conference, Asilomar, California, 1977. This work was supported in part by NSF Grant No. NAT BUR OF ECON RES/PO No. 4369 and by Department of Energy Contract No. EY-76-C-02-0016.The main part of this work was presented at the Japan-France Seminar on Functional Analysis and Numerical Analysis, Tokyo, Japan, 1976. The paper was prepared in part while the author was a visitor at the Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 1976–77, and was completed while he was a visitor at the Computer Science Department, Stanford University, Stanford, California, 1977. He acknowledges the hospitality and stimulating environment provided by Professor G. H. Golub, Stanford University, and Professors N. J. Rose and C. D. Meyer, North Carolina State University.     

A new approach to variational CI calculations on systems containing non-interacting molecules

A new method to perform variational CI calculations on systems containing non-interacting molecules is presented.     

A formula related to irreducible diagram expansions

A formula is presented which can be regarded as the analytic basis for irreducible diagram expansions. It expresses the off-diagonal elements of the inverse of a matrix of operators by the off-diagonal elements and by diagonal elements of various inverses of the original operator. The formula can be obtained by purely analytic means without reference to statistical considerations. No infinite processes are involved if one deals with a finite matrix of operators.     

Estimation of vector ARMAX models

The asymptotic properties of maximum likelihood estimates of a vector ARMAX system are considered under general conditions, relating to the nature of the exogenous variables and the innovation sequence and to the form of the parameterization of the rational transfer functions, from exogenous variables and innovations to the output vector. The exogenous variables are assumed to be such that the sample serial covariances converge to limits. The innovations are assumed to be martingale differences and to be nondeterministic in a fairly weak sense. Stronger conditions ensure that the asymptotic distribution of the estimates has the same covariance matrix as for Gaussian innovations but these stronger conditions are somewhat implausible. With each ARMAX structure may be associated an integer (the McMillan degree) and all structures for a given value of this integer may be topologised as an analytic manifold. Other parameterizations and topologisations of spaces of structures as analytic manifolds may also be considered and the presentation is sufficiently general to cover a wide range of these. Greater generality is also achieved by allowing for general forms of constraints.     

A time-decomposition algorithm for a stiff linear two-point boundary-value problem and its application to a nonlinear optimal control problem

An algorithm is proposed to solve a stiff linear two-point boundary-value problem (TPBVP). In a stiff problem, since some particular solutions of the system equation increase and others decrease rapidly as the independent variable changes, the integration of the system equation suffers from numerical errors. In the proposed algorithm, first, the overall interval of integration is divided into several subintervals; then, in each subinterval a sub-TPBVP with arbitrarily chosen boundary values is solved. Second, the exact boundary values which guarantee the continuity of the solution are determined algebraically. Owing to the division of the integration interval, the numerical error is effectively reduced in spite of the stiffness of the system equation. It is also shown that the algorithm is successfully imbedded into an interaction-coordination algorithm for solving a nonlinear optimal control problem.The authors would like to thank Mr. T. Sera and Mr. H. Miyake for their help with the calculations.