Time-dependent equation direct molecular dynamics

Motion in a protein may be modeled by computer using Isaac Newton s equation of motion, / = ma. This modeling requires the three-dimensional coordinates from an X-ray structure analysis as a starting point and some knowledge of interatomic potentials, so that only reasonable interatomic distances will be employed at all stages.Such molecular dynamics calculations lead to a prediction of where atoms will move in a short period of time, and result in the calculation of a time-dependent trajectory of all atoms. Initially each atom is moved in the direction of the force on it from other atoms and then, as each atom moves, its trajectory may change to accommodate this. In addition, this method aids in protein structure refinement,as was described in Chapter 10, although it is important to ensure that the model so refined still fits the electron density map. 

A reaction-path based method is described to obtain information from ab initio quantum chemistry calculations about the dynamics of energy disposal in exothermic unimolecular reactions important in the initiation of detonation in energetic materials. Such detailed information at the microscopic level may be used directly or as input for molecular dynamics simulations to gain insight relevant for the macroscopic processes. The semiclassical method, whieh uses potential energy surface information in the broad vicinity of the steepest descent reaction path, treats a reaction coordinate classically and the vibrational motions perpendicular to the reaction path quantum mechanically. Solution of the time-dependent Schroedinger equation leads to detailed predictions about the energy disposal in exothermic chemical reactions. The method is described and applied to the unimolecular decomposition of methylene nitramine. 

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]... 

The QM/MM Hamiltonian can be used to cany out Molecular Dynamics simulations of a complex system. In the case of liquid interfaces, the simulation box contains the solute and solvent molecules and one must apply appropriate periodic boundary conditions. Typically, for air-water interface simulations, we use a cubic box with periodic boundary conditions in the X and Y directions, whereas for liquid/liquid interfaces, we use a rectangle cuboid interface with periodic boundary conditions in the three directions. An example of simulation box for a liquid-liquid interface is illustrated in Fig. 11.1. The solute s wave function is computed on the fly at each time step of the simulation using the terms in the whole Hamiltonian that explicitly depend on the solute s electronic coordinates (the Born-Oppenheimer approximation is assumed in this model). To accelerate the convergence of the wavefunction calculation, the initial guess in the SCF iterative procedure is taken from the previous step in the simulation, or better, using an extrapolated density matrix from the last three or four steps [39]. The forces acting on QM nuclei and on MM centers are evaluated analytically, and the classical equations of motion are solved to obtain a set of new atomic positions and velocities. 

According to the coupling model, for neat polymers at the times appropriate for most experimental measurements, the slowing down of segmental relaxation gives rise to a correlation function having the form of equation (1). The stretch exponent is a measure of the strength of the intermolecular constraints on the relaxation. These constraints depend on molecular structure because the chemical structure determines the intermolecular interactions. However, the complexity of cooperative dynamics in dense liquids and polymers precludes direct calculation of P it is invariably deduced from experiment. An assumption fundamental to the model is that the time at which intermolecular cooperativity effects become manifest is independent of temperature. 

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