Cartesian momenta

Using the fundamental commutation relations among the cartesian coordinates and the cartesian momenta ... 

The Cartesian momenta can be expressed in terms of Cartesian velocities vxj, vyi, vzj by the familiar relationship... 

The interest here is in the energy levels of molecular systems. It is well known that an understanding of these energy levels requires quantum mechanics. The use of quantum mechanics requires knowledge of the Hamiltonian operator Hop which, in Cartesian coordinates, is easily derived from the classical Hamiltonian. Throughout this chapter quantum mechanical operators will be denoted by subscript op . If the classical Hamiltonian function H is written in terms of Cartesian momenta and of interparticle distances appropriate for the system, then the rule for transforming H to Hop is quite straightforward. Just replace each Cartesian momentum component... 

In fact, the result of Equation 3.43 not only applies to internal displacement coordinates but also to Cartesian displacements. The kinetic energy in terms of Cartesian coordinates (Equation 3.11) can easily be transformed into an expression in terms of Cartesian momenta (Equation 3.28)... 

Here H is the Hamiltonian function for one N atomic molecule (i) and s is its symmetry number. One might have expected this result immediately from the Kirkwood formulation for the classical canonical partition function. H is a function of the 3N Cartesian momenta and the 3N Cartesian coordinates of molecule i. 

Here KE(1) and KE(2) are classical kinetic energy expressions for isotopomer 1 and isotopomer 2 respectively, each containing terms for the kinetic energy of each atom in each of the three coordinates. For N-atomic molecules there are three Cartesian momenta for each atom, 3N Cartesian momenta for each molecule, and consequently 3NN Cartesian momenta for the N molecule system. The integrals in the numerator and denominator can thus be written as a product of 3 integrals of the type... 

The classical kinetic energy expression in terms of Cartesian momenta,... 

The A + relative velocity t rel is now added along the z axis with the restraint that the A + center of mass remain at rest. The space-fixed Cartesian momenta are then... 

In laboratory-based Cartesian coordinates T is only a funetion of the Cartesian momenta and is written as... 

Though the above Hamiltonian in Cartesian momenta and coordinates is certainly... 

At the start of the MD simulations for the mutated system, the initial values of the Cartesian momenta were randomly chosen in phase space. To determine the initial momenta, we chose imiformly distributed random numbers on the interval [-l/2,l/2], then multiplied these by an appropriate scaling constant. After a desired particle was obtained, classical trajectories were propagated for 50 ps above bulk melting point, and then the system was annealed by scaling the Cartesian momenta with a constant scaling factor until the temperature reached 10 K to find a steady state of the amorphous PE particles. The temperature of the particle T was calculated from... 

To obtain the average values of properties of the particles at a fixed temperature and examine dependence of the conformations on temperature, we used Nose-Hoover chain (NHC) constant temperature molecular dynamics [228]. The initial configmations of the steady state of the amorphous PE particle were used at a start of the NHC simulations the initial values of the Cartesian momenta were given random orientation in phase space with magnitudes chosen so that the total kinetic energy was the equipartition theorem expectation value [229]. The NHC quasi-Hamiltonian for this system can be written,... 

Classical trajectory calculations are used to develop an understanding of the experimentally observed behavior of collisions and surface interactions of nano-and micro-scale polymer particles on Si, C, and Al surfaces in a vacuiun. The PE particles were propagated up to 100 ps for those surfaces to investigate the selforganization of polymer particles (mechanical memory) via coUisions. At the start of classical trajectory, the initial values of the Cartesian momenta, p, were given randomly in the amorphous particle to set the initial temperature to 300 K, and the system was subjected to zero center-of-mass position and velocity. After the surface was set at 10 C from the closest atom of the particle in z coordinate, a desired initial translational velocity, was assigned to all atoms of the particle in z direction... 

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