General central force field

Let us consider the description of a simple three-atom molecule by the three most frequently used spectroscopic parameterization schemes, the general central force field (GCF), the general valence force field (GVF) and the Urey-Bradley force field (UBF Fig. 2.14). 

The relationship (equation (5.81)) between M and L depends only on fundamental constants, the electronic mass and charge, and does not depend on any of the variables used in the derivation. Although this equation was obtained by applying classical theory to a circular orbit, it is more generally valid. It applies to elliptical orbits as well as to classical motion with attractive forces other than dependence. For any orbit in any central force field, the angular... 

The potential energy expressions used for force field calculations are all descendants of three basic types originating from vibrational spectroscopy (5) the generalized valence force field (GVFF), the central force field, and the Urey-Bradley force field. General formulations for the relative potential energy V in these three force fields are the following ... 

The final general form of the VFF- and UBFF-expressions (2) and (4) as modified for our purposes is thus as follows (Central force fields are less efficient than valence- and UB-force fields, and are not considered here.) ... 

The development of a general theory of systems with non-central force fields can be divided into two parts. First the many types of directional interaction that may occur have to be classified within a general mathematical framework and then approximate methods of evaluating the partition function have to be devised. This paper summarizes some of the results of a method developed by the author 2 with particular reference to its application to the properties of liquid mixtures. 

In an exposition which aims to encompass general systems and ensembles, it is appropriate to make use of the Hamiltonian version of dynamics. In this view forces do not appear explicitly and the dynamics of the system evolve so as to keep the Hamiltonian function constant. In Newtonian dynamics forces appear explicitly and molecules move as a response to the forces they experience. For our purposes, the Newtonian view is sufficient since we will illustrate the large scale computational aspects with simplest possible particles, atoms with spherical, central force fields. The same principles hold for molecules with internal degrees of freedom as well. 

Molecules with Central Force Fields. For these molecules equation (25) is directly applicable it is necessary to assume a functional form for (/ ), then to perform the required integration. From general observations of the properties of matter it is obvious that u is large positive when r is small. It is also obvious that = 0 when r = oo. Many equations to describe (r) between r = oo and r = small have been proposed and plots of some of them are shown in Figure 11. The hard sphere and soft sphere representations of (r) (curves a and b) allow for repulsive forces only. The other representations illustrated in Figure 11 allow additionally for... 

Central to the quality of any computational smdy is the mathematical model used to relate the structure of a system to its energy. General details of the empirical force fields used in the study of biologically relevant molecules are covered in Chapter 2, and only particular information relevant to nucleic acids is discussed in this chapter. 

The concept has been generalized in the ONIOM method to include several layers, for example using high level ab initio (e.g. CCSD(T)) in the central part, lower-level electronic structure theory (e.g. MP2) in an intermediate layer and a force field to treat the outer layer. 

Molecular-mechanics force fields distinguish between general and 1,3 non-bonded interactions. The obvious reason for this distinction is that the distance between ligands is affected when linked to the same central atom. Their final non-bonded separation depends, not only on ligand type, but also on the size of the central atom. In such a three-atom system the relevant parameters are the characteristic radius (rc) of the central atom, together with the... 

A central issue is the number of different atom types that are used in a particular force field. There is always a compromise between increasing the number to allow for the inclusion of more environmental effects (i.e., local electronic interactions) vs. the increase in the number of parameters to be determined to adequately represent a new atom type. In general, the more subtypes of atoms (how many different kinds of nitrogen, for example), the less likely that the parameters for a particular application will be available in the force field. The extreme, of course, would be a special atom type for each kind of atomic environment in which the parameters were chosen, so that the calculated properties of each molecule would simply reproduce the experimental observations. One major assumption, therefore, is that the force constants (parameters) and equilibrium values of the equations are functions of a limited number of atom types and can be transferred from one molecular environment to another. This assumption holds reasonably well where one may be primarily interested in geometric issues, but is not so valid in molecular spectroscopy. This had led to the introduction of additional equations, the so-called "cross-terms" which allow additional parameters to account for correlations between bond lengths and bond angles... 

In order to simplify the presentation only axially symmetric molecules are considered in detail. This restriction is not altogether necessary as other systems can be treated by similar methods with very similar results. In the next section a general method of separating the intermolecular field into a central-force part and directional terms of various angular symmetries is described. Simple models such as dipole-dipole forces to represent interactions between polar molecules correspond to particular terms of this expansion. The additional free energy due to the directional part of the field can then be estimated by a perturbation method, provided that the additional field is not too large. The method is applicable at any density and enables approximate theories of monatomic systems to be extended so as to apply to more realistic intermolecular fields. 

In the approach adopted in my first edition, the derivation and use of the general dynamic equation for the particle size distribution played a central role. This special form of a population balance equation incorporated the Smoluchowski theory of coagulation and gas-to-panicle conversion through a Liouville term with a set of special growth laws coagulation and gas-to-particle conversion are processes that take place within an elemental gas volume. Brownian diffusion and external force fields transport particles across the boundaries of the elemental volume. A major limitation on the formulation was the assumption that the panicles were liquid droplets that coalesced instantaneously after collision. 

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