Removing Translational Motion

It is possible (see, for example, J. Nichols, H. E. Taylor, P. Schmidt, and J. Simons, J. Chem. Phys. 92, 340 (1990) and references therein) to remove from H the zero eigenvalues that correspond to rotation and translation and to thereby produce a Hessian matrix whose eigenvalues correspond only to internal motions of the system. After doing so, the number of negative eigenvalues of H can be used to characterize the nature of the... 

The phase equilibrium between a liquid and a gas can be computed by the Gibbs ensemble Monte Carlo method. We create two boxes, where the first box represents the dense phase and the second one represents the dilute phase. Each particle in the boxes experiences a Lennard-Jones potential from all the other particles. Three types of motion will be conducted at random the first one is particle translational movement in each box, the second one is moving a small volume from one box and adding to the other box, the third one is removing a particle from one box and inserting in the other box. After many such moves, the two boxes reach equilibrium with one another, with the same temperature and pressure, and we can compute their densities. 

A more detailed decomposition of macromolecular dynamics that can be used not only for assessing convergence but also for other purposes is principal components analysis (PCA), sometimes also called essential dynamics (Wlodek et al. 1997). In PCA the positional covariance matrix C is calculated for a given trajectory after removal of rotational and translational motion, i.e., after best overlaying all structures. Given M snapshots of an N atom macromolecule, C is a 3N X 3A matrix with elements... 

This description is elaborated below with an idealized model shown in Figure 17. Imagine a molecule tightly enclosed within a cube (model 10). Under such conditions, its translational mobility is restricted in all three dimensions. The extent of restrictions experienced by the molecule will decrease as the walls of the enclosure are removed one at a time, eventually reaching a situation where there is no restriction to motion in any direction (i.e., the gas phase model 1). However, other cases can be conceived for a reaction cavity which do not enforce spatial restrictions upon the shape changes suffered by a guest molecule as it proceeds to products. These correspond to various situations in isotropic solutions with low viscosities. We term all models in Figure 17 except the first as reaction cavities even... 

Ultrasonic resonance in solids. The mechanical resonances of a freely suspended solid object are special solutions to the equations of motion that depend only on the density, elastic moduli and shape. These solutions determine all the possible frequencies at which such an object would ring if struck. Because a solid with No atoms in it has 6No degrees of freedom, there are 6No — 6 possible resonances (viz. six frequencies corresponding to three rigid rotations and three rigid translations are removed). Most such resonances cannot be detected as individual modes because dissipation in the solid broadens the higher frequency resonances to an extent that they overlap to form a continuum response. 

The liquid phase of molecular matter is usually isotropic at equilibrium but becomes birefringent in response to an externally applied torque. The computer can be used to simulate (1) the development of this birefringence —the rise transient (2) the properties of the liquid at equilibrium under the influence of an arbitrarily strong torque and (3) the return to equilibrium when the torques are removed instantaneously—the fall transient. Evans initially considered the general case of the asymmetric top (C2 symmetry) diffusing in three-dimensional space and made no assumptions about the nature of the rotational and translational motion other than those inherent in the simulation technique itself. A sample of 108 such molecules was taken, each molecule s orientation described by three unit vectors, e, Cg, and parallel to its principal moment-of-inertia axes. 

Six of these normal coordinates (five for a linear molecule) have a frequency eigenvalue identically equal to zero. These motions are translations and rotations of the molecule. Although the approach through Cartesian displacement coordinates is theoretically elegant, it is generally more practical to express the vibrational motions in terms of internal coordinates, such as bond stretches and distortions of bond angles. The method is discussed in detail in Chapter 4 of Wilson, Decius and Cross [57]. Since the distortions of the molecule can be described in terms of 3A — 6 of these internal coordinates there are no redundant dimensions to be removed when the analysis is complete. 

The in-1 vibrational frequencies, C0 (s), are obtained from normal-mode analyses at points along the reaction path via diagonalization of a projected force constant matrix that removes the translational, rotational, and reaction coordinate motions. The B coefficients are defined in terms of the normal mode coefficients, with those in the denominator of the last term determining the reaction path curvature, while those in the numerator are related to the non-adiabatic coupling of different vibrational states. A generalization to non-zero total angular momentum is available [59]. 

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