Exponential matrix function evaluation

One idea (not that we really do that) is to apply the Taylor series expansion on the exponential function of A, and evaluate the state transition matrix with... 

The problem of evaluating matrix elements of the interelectron repulsion part of the potential between many-electron molecular Sturmian basis functions has the degree of difficulty which is familiar in quantum chemistry. It is not more difficult than usual, but neither is it less difficult. Both in the present method and in the usual SCF-CI approach, the calculations refer to exponential-type orbitals, but for the purpose of calculating many-center Coulomb and exchange integrals, it is convenient to expand the ETO s in terms of a Cartesian Gaussian basis set. Work to implement this procedure is in progress in our laboratory. 

We use an alternative to this method, that enables a fast and accurate evaluation of the two-center integrals. Analytical integration is possible when linear combinations of Gaussian Type Orbitals are used to describe the atomic states [1,9, 10]. Imperfect behavior of such gaussian functions at large distances does not affect the results, since the two-center matrix-elements (7) have an exponential decay for increasing intemuclear distances. For example, for integrals and expressed in cartesian coordinates, one has to evaluate expressions such as... 

The integration in Eq. (9) occurs near the acceptor, i.e. in the asymptotic region of the donor wave function. Therefore, the asymptotics H) is substituted in the matrix element (9) instead of the wave function ( E). The value of the matrix element Vtf is proportional to exp —ktda due to the exponential decay of (F E). The evaluation of the correction to the wave function 4G(F —rA,E) due to the interaction with the donor obviously shows that it is also proportional to exp —ktda T... 

The exponential function of the matrix can be evaluated through the power series expansion of exp(). c is the column vector whose elements are the concentrations c.. The matrix elements of the rate coefficient matrix K are the first-order rate constants K--. The system is called closed if all reactions and back reactions are included. Then K is of rank N- with positive eigenvalues, of which exactly one is zero. It corresponds to the equilibrium state, with concentrations redetermined by the principle of microscopic reversibility ... 

Our method makes it possible to use simple linear rules for exploring complicated nonlinear systems. A simple application is the study of connectivity among various chemical species in complicated reaction networks. In the simple case of homogeneous systems with time-invariant structure, the susceptibility matrix x = [Xhh ] = X depends only on the transit time and not on time itself. The matrix elements Xuu ( ) are proportional to the elements (t) of a Green function matrix G (t) = [G / (t)], which is the exponential of a connectivity matrix K, that is, G (t) = exp [tK]. It follows that from a response experiment involving a system with time-invariant structure, it is possible to evaluate the connectivity matrix, K, which contains information about the relations among the different chemical species involved in the reaction mechanism. The nondiagonal elements of the matrix K = Kuu I show whether in the reaction mechanism there is a direct connection between two species in particular, if Kuu 0, there is a connection from the species u to the species u the reverse connection, from u io u, exists if Ku u 0. 

This contribution examines current approaches to Coulomb few-body problems mainly from a methodological perspective, in contrast to recent reviews which have focused on the results obtained for benchmark problems. The methods under discussion here employ wavefunctions which explicitly involve all the interparticle coordinates and use functional forms appropriate to nonadiabatic systems in which all the particles are of comparable mass. The use of such wavefunctions for states of arbitrary angular symmetry is reviewed, and the kinetic-energy operator, written in the interparticle coordinates, is presented in a convenient form. Evaluation of the resultant angular matrix elements is discussed in detail. For exponentially correlated wavefunctions, problems of integral evaluation are surveyed, the relatively new analytical procedures are summarized, and relations among matrix elements are presented. The current status of Gaussian-orbital and Hylleraas methods is also reviewed. 

Since is g a full matrix we still have to evaluate the exponential function. If G... 

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