Central force coordinates

If, however, one intends to use a general quadratic potential function, the applicability of which is as appropriate in any one coordinate system as in any other, the simplicity of the kinetic energy matrix elements in the central force coordinate system may favor the use of these coordinates for some molecules. 

One of the simplest approximations employed in reducing the number of independent force constants is the assumption of central forces. It is assumed that the forces holding the atoms in their ciiuilibrium positions act only along the lines joining pairs of atoms and that every pair of atoms is connected by such a force. This type of force function would result if the molecule were held together by purely ionic interactions. Also, this type of force yields only diagonal terms in the force constant matrix when the internal coordinate system is the complete set of interatomic distances (the central force coordinates). 

Cauchy function 276 Cauchy s ratio test 35-36 central forces 107,132-135 spherical harmonics 134-135 spherical polar coordinates 132-133 chain rule 37, 57, 160 character 153,195,197 orthogonality 197, 204 tables 198-200... 

Fig. D. The pair correlation function g0H and the running coordination number oh predicted by the central force model...

Cameron system of, 298 fundamental IR band of, table of, 174 rotational transitions of, table of, 168 Carbon-13 NMR, 356-357 Cartesian displacement coordinates, 236 Case (a) coupling, 188-189, 212 Case (b) coupling, 190-191,212 Cayley, A., 78, 387 Center of symmetry, 53 Central-force problem, 38 Centrifugal distortion in diatomics, 158,166-167 in polyatomics, 213, 216, 218 Chain rule, 20-21 Characters ... 

Consider the two-dimensional motion of a particle in a central force field. The Lagrange function in Cartesian coordinates is... 

A simple example is provided by the coordinates that may be used to describe the motion of a particle in a plane. As generalized coordinates one can use either Cartesian coordinates qi = x, q2 = y, or plane polar coordinates qi = r, q2 = central force the coordinate p becomes cyclic, while both x and y are non-cyclic. The number of cyclic coordinates thus depends on the choice of generalized coordinates, and for each problem there may be one particular choice for which all coordinates are cyclic. 

Since the fictitious particle moves in a central force field described by a spherically symmetric potential function U(r), its angular momentum is conserved. Therefore, the motion of the fictitious particle will be in a plane defined by the velocity and the radius vectors. The Lagrangian may then be conveniently expressed in polar coordinates as... 

For most purposes, the force between two atoms is expressed in terms of an interatomic potential, which depends primarily on the separation, r, between the atoms or other charged particles. If the dependence of the potential on other coordinates is neglected (central force approximation), then the force, F(r). and the potential, V(r), are related by... 

The long range attractive potential for a diatomic dissociation is given by V r) = r ", where n is the interaction parameter. Typically = 6 for a neutral molecule (recall the Lennard-Jones 6-12 potential). On the other hand, for an ionic dissociation into an ion and a polarizable neutral, n = 4. The Hamiltonian for such a two-body central force system can be expressed in the center of mass as a one-body problem with reduced mass p. and relative velocity v. In polar coordinates, this takes the form [(Eq. 2.15)]... 

Using the area velocity D, see Eq. (1.16), which is a constant of motion, D = A in the classical case of the central force problem, one derives straightforwardly the following relations in standard polar coordinates r,

particle motion is in a plane perpendicular to the angular momentum vector L)... 

Since F is a function of the r coordinate only, we have a one-particle central-force problem, and we may apply the results of Section 6.1. Using Eqs. (6.16) and (6.17), we have for the wave function... 

For crystals with a cubic structure, the elastic properties are characterized by three elastic moduli cj, Cj2, and C44. When considering the first two coordination spheres in the central-force approximation, these three moduli are sufficient to reproduce the whole phonon spectrum. When the next coordination spheres are considered, additional force constants must be known. The number of force constants which must be known depends on the number n of the spheres beii considered. 

Complete sots, 117 Complex conjugate of matrices, 295 Components of matrices, 297 Conforrnability of matrices, 298 Conjugate transpose matrices, 296 Constant matrix, 294 Coordinates, bonding, 56, 303 central force, 173 framework, 203 oup, 203... 

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