Kernel projector matrices

Chromatography Processing of Information Experimental Data Evaluation and Quality Control Kernel Projector Matrices Application to Leu -zervamicin Quality Control, Data Analysis Structural Chemistry Application of Mathematics X-Ray Crystallographic Analysis and Semiempirical Computations... 

The purpose of the fragment calculations presented here is to obtain an ab initio projector matrix, P, when ab initio calculations of an entire molecule are either not feasible or considered to be too time-consuming. In our calculations, a fragment consists of an inner core or kernel and a number... 

The difference studies shown in Figures 5 and 6 indicate that the electron density is well represented by the P matrix obtained from the fragment calculations. As has been noted, the method for developing a P matrix presented here differs from the one presented in Ref. 1. The P matrix formed here is more accurate than that formed from the R matrix in Ref. 1. It was not necessary to employ the Clinton equations here to improve the projector quality. In fact, use of the Clinton equations left the good quality of the P matrix unchanged. Even greater accuracy may accrue if more atoms were included in the neighborhoods of the kernels. 

Note that the term open system refers here to exchange of energy and phase with the environment, as the number of particles is conserved throughout. The reduced density matrix p t) evolves coherently under the influence of the nuclear Hamiltonian, Hnuc, and the non-adiahatic effects enter the equation via the dissipative Liouvillian superoperator jSfn- The latter is also termed memory kernel , as it contains information about the entire history of the environmental evolution and its interaction with the system. The definition of the memory kernel is by no means unique nor straightforward. One possible solution is to start from the microscopic Hamiltonian of the total system, eqn (1). Using the projector formalism, it is possible to separate the evolution of the system, i.e., the... 

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