Hamiltonian dynamical systems vectors

Quantum mechanics involves two distinct sets of hypotheses—the general mathematical scheme of linear operators and state vectors with its associated probability interpretation and the commutation relations and equations of motion for specific dynamical systems. It is the latter aspect that we wish to develop, by substituting a single quantum dynamical principle for the conventional array of assumptions based on classical Hamiltonian dynamics and the correspondence principle. 

For AB + CD systems the most straightforward choice of coordinates to describe the dynamical system is the Jacobi coordinates corresponding to the diatom-diatom arrangement. As shown in Fig. 1, the three vectors (R,ri,r2) denote, respectively, the vector R from the center of mass (CM) of diatom AB to that of CD, the AB diatomic vector F, and the CD diatomic vector F2. The full Hamiltonian expressed in this set of coordinates is written as... 

Additionally, non-Hamiltonian d5mamics can be used in applications/ methodologies such as Path-integral MD, replica-exchange methods, variable transformation techniques, free energy dynamics methods, and other new applications. Generating these alternative statistical ensembles from simulation requires the use of extended systems or extended phase space [9]. In these systems, the simulations do not only include the N coordinate and momentum vectors that are needed to describe a classical Ai-particle system, but they also include a set of additional control or extended variables that are used to drive the fluctuations required by the ensemble of interest. 

For a complete treatment of a laser-driven molecule, one must solve the many-body, multidimensional time-dependent Schrodinger equation (TDSE). This represents a tremendous task and direct wavepacket simulations of nuclear and electronic motions under an intense laser pulse is presently restricted to a few bodies (at most three or four) and/or to a model of low dimensionality [27]. For a more general treatment, an approximate separation of variables between electrons (fast subsystem) and nuclei (slow subsystem) is customarily made, in the spirit of the BO approximation. To lay out the ideas underlying this approximation as adapted to field-driven molecular dynamics, we will consider from now on a molecule consisting of Nn nuclei (labeled a, p,...) and Ne electrons (labeled /, j,...), with position vectors Ro, and r respectively, defined in the center of mass (rotating) body-fixed coordinate system, in a classical field E(f) of the form Eof t) cos cot). The full semiclassical length gauge Hamiltonian is written, for a system of electrons and nuclei, as [4]... 

Having left the framework of field theory outlined in chapter 7 and thus having avoided any need for subsequent renormalization procedures, the mass and charge of the electron are now the physically observable quantities, and therefore do not bear a tilde on top. In contrast to quantum electrodynamics, the radiation field is no longer a dynamical degree of freedom in a many-electron theory which closely follows nonrelativistic quantum mechanics. Vector potentials may only be incorporated as external perturbations in the many-electron Hamiltonian of Eq. (8.62). From the QED Eqs. (7.13), (7.19), and (7.20), the Hamiltonian of a system of N electrons and M nuclei is thus described by the many-particle Hamiltonian of Eq. (8.66). In addition, we refer to a common absolute time frame, although this will not matter in the following as we consider only the stationary case. 

Consider a molecular system described by the Hamiltonian operator Ho(r,R) where r and R are vectors containing the electronic and nuclear coordinates, respectively. This Hamiltonian acts in the Hilbert space H spanned by the basis of eigenstates of Ho. The dynamics of the system in interaction with a periodic electric field e(0 = ocos(a f - - 0) with eq = eoe in the semiclassical dipole approximation is described by the TDSE... 

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