Mixed Coordinate Formulation

1 Mixed Coordinate Formulation for Tree Structured Systems  

For this end we write the equations of motion in a somehow artificial manner We take the equations of motion of the unconstrained system in absolute coordinates, together with the trivial equation Oq = 0 and impose to this set of equations the equation 7(p, g) = 0 as a constraint by introducing Lagrange multipliers /x in the usual way. 

Differentiating (1.4.7b) twice with respect to time, we obtain 

Solving this equation starting with the last line, we first get 

Inserting this result into the first line and premultiplying by gives 

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The similar appearance of the quantum and classical Liouville equations has motivated several workers to construct a mixed quantum-classical Liouville (QCL) description [27 4]. Hereby a partial classical limit is performed for the heavy-particle dynamics, while a quantum-mechanical formulation is retained for the light particles. The quantities p(f) and H in the mixed QC formulation are then operators with respect to the electronic degrees of freedom, described by some basis states 4> ), and classical functions with respect to the nuclear degrees of freedom with coordinates x = x, and momenta p = pj — for example. 

In this volume dedicated to Yngve Ohm we feel it is particularly appropriate to extend his ideas and merge them with the powerful practical and conceptual tools of Density Functional Theory (6). We extend the formalism used in the TDVP to mixed states and consider the states to be labeled by the densities of electronic space and spin coordinates. (In the treatment presented here we do not explicitly consider the nuclei but consider them to be fixed. Elsewhere we shall show that it is indeed straightforward to extend our treatment in the same way as Ohm et al. and obtain equations that avoid the Bom-Oppenheimer Approximation.) In this article we obtain a formulation of exact equations for the evolution of electronic space-spin densities, which are equivalent to the Heisenberg equation of motion for the electtons in the system. Using the observation that densities can be expressed as quadratic expansions of functions, we also obtain exact equations for Aese one-particle functions. 

A vibronic coupling model for mixed-valence systems has been developed over the last few years (1-5). The model, which is exactly soluble, has been used to calculate intervalence band contours (1, 3, 4, 5), electron transfer rates (4, 5, 6) and Raman spectra (5, 7, 8), and the relation of the model to earlier theoretical work has been discussed in detail (3-5). As formulated to date, the model is "one dimensional (or one-mode). That is, effectively only a single vibrational coordinate is used in discussing the complete ground vibronic manifold of the system. This is a severe limitation which, among other things, prevents an explicit treatment of solvent effects which are... 

The theoretical approach is based on the solution to the mixed type linear/nonlinear generalized Schrodinger equation for spatiotemporal envelope of electrical field with account of transverse spatial derivatives and the transverse profile of refractive index. In the quasi-static approximation, this equation is reduced to the linear/nonlinear Schrodinger equation for spatiotemporal pulse envelope with temporal coordinate given as a parameter. Then the excitation problem can be formulated for a set of stationary light beams with initial amplitude distribution corresponding to temporal envelope of the initial pulse. 

MXs(diarsine)]a Either seven-coordinate, or mixed six-eight-coordinate ionic formulation 12... 

In a serendipitous fashion, a novel mixed valence tetranuclear copper(II)/copper(III) dithiocarbamate [2]catenane was prepared in near quantitative yield by partial chemical oxidation of a preformed dinuclear copper(II) naphthyl dtc macrocycle (Scheme 6).49 X-ray structure, magnetic susceptibility, ESMS and electrochemical studies all support the tetranuclear catenane dication formulation. The combination of the lability of copper(II) dtc coordinate bonds and favourable copper(II) dtc-copper(III) dtc charge transfer stabilisation effects are responsible for the high yielding formation of the interlocked... 

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