Atomic coordinates functions

All components of these vectors can in principle be explicitly written in terms of the generalized coordinates, provided that all atomic coordinate functions, ragU7i > < 2> <73Ar-6). appearing in Eq. (2.6) have been formulated. If the vibrational coordinates are purely geometrically defined, however, the vibrational s-vectors and their LS-components, sk> aF, are independent of the axis convention used in formulating -functions, contrary, of course, to their MS-components, sk> ag. Applying Eq. (2.36) we realize that the vibrational part of the G-matrix is also independent of the axis convention under these special conditions. 

The problem of perception complete structures is related to the problem of their representation, for which the basic requirements are to represent as much as possible the functionality of the structure, to be unique, and to allow the restoration of the structure. Various approaches have been devised to this end. They comprise the use of molecular formulas, molecular weights, trade and/or trivial names, various line notations, registry numbers, constitutional diagrams 2D representations), atom coordinates (2D or 3D representations), topological indices, hash codes, and others (see Chapter 2). 

A force field does not consist only of a mathematical eiqjression that describes the energy of a molecule with respect to the atomic coordinates. The second integral part is the parameter set itself. Two different force fields may share the same functional form, but use a completely different parameterization. On the other hand, different functional forms may lead to almost the same results, depending on the parameters. This comparison shows that force fields are empirical there is no "correct form. Because some functional forms give better results than others, most of the implementations within the various available software packages (academic and commercial) are very similar. 

For each combination of atoms i.j, k, and I, c is defined by Eq. (29), where X , y,. and Zj are the coordinates of atom j in Cartesian space defined in such a way that atom i is at position (0, 0, 0), atomj lies on the positive side of the x-axis, and atom k lies on the xy-plaiic and has a positive y-coordinate. On the right-hand side of Eq. (29), the numerator represents the volume of a rectangular prism with edges % , y ., and Zi, while the denominator is proportional to the surface of the same solid. If X . y ., or 2 has a very small absolute value, the set of four atoms is deviating only slightly from an achiral situation. This is reflected in c, which would then take a small absolute value the value of c is conformation-dependent because it is a function of the 3D atomic coordinates. 

View the contour map m several planes to see the general Torm of the distiibiiiioii. As long as you don t alter the molecular coordinates, you don t need to repeat th e wave function calculation. Use the left mouse button and the IlyperChem Rotation or Translation tools (or Tool icons ) to change the view of amolecnle without changing its atomic coordinates. 

Class II dependence for the activation of a chemical bond as a function of surface metal atom coordinative unsaturation is typically found for chemical bonds of a character, such as the CH or C-C bond in an alkane. Activation of such bonds usually occurs atop of a metal atom. The transition-state configuration for methane on a Ru surface illustrates this (Figure 1.13). 

The PES is not an analytic function of the atomic coordinates, because it diverges to infinity when any two atoms are at the same position. Using the Znj to describe the PES means that these singularities are banished to infinity, Zn —> oo, resulting in a much better behaved description of the... 

The order parameter can be defined in two different ways. It can be either a function of atomic coordinates or just a parameter in the Hamiltonian. Examples of both types of order parameters are given in Sect. 2.8.1 in Chap. 2 and illustrated in Fig. 2.5. This distinction is theoretically important. In the first case, the order parameter is, in effect, a generalized coordinate, the evolution of which can be described by Newton s equations of motion. For example, in an association reaction between two molecules, we may choose as order parameter the distance between the two molecules. Ideally, we often would like to consider a reaction coordinate which measures the progress of a reaction. However, in many cases this coordinate is difficult to define, usually because it cannot be defined analytically and its numerical calculation is time consuming. This reaction coordinate is therefore often approximated by simpler order parameters. 

The potential energy V of the elastomer is presumed to be given as a function of the atomic coordinates x (lwell-defined equilibrium shape, there must be equilibrium positions x for all atoms that are part of the continuous network. Expand the potential in a Taylor series about the equilibrium positions, and set the potential to zero at equilibrium, to obtain... 

Another type of restraining potential for ligand i (/ , ) is defined as a function of x, and A,. In this case, the restraining potential does not depend directly on the environment atom coordinates... 

Here Slater functions (a3 2/7r 1/2) exp( —a r — ra ) with the atom being centered at position vectors ra. The overlap between these functions is given by S. After an FT and integrating over momentum coordinates of one particle, the EMD of H2 molecule within VB and MO theory are derived as... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

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