Potential functions of one variable

The problem of determinacy results, on the one hand, from the trend of reducing the investigated functions and, on the other, from the fact that potential functions being fitted to experimental data are always approximate. Therefore, the information about the point of truncation of the Taylor series of a function having a critical point of presumed properties is essential. 

A solution of the technical problem described above will be discussed first for the simplest case of a function of one state variable, the number of parameters being unlimited. Hence, we examine potential functions of the form F(x c), c = (cx. ct), where x and c (i = 1. k) are real variables. To simplify notation we shall assume that the critical point is x = 0. 

Local properties of a function F(x) (in cases where it does not lead to misunderstanding we shall omit the dependence on c) depend on its first and second derivative. Three basic cases may be distinguished  

Case I signifies that a function V does not have a critical point at x = 0. Case II corresponds to a so-called nondegenerate critical point, in case III the point x = 0 will be called a degenerate critical point. Further investigations of local properties of a function F(x), aiming at the solution of problems 1-3 (determinacy, unfolding and classification), will account for the usefulness of distinguishing the basic cases I-III. 

It will become evident later that catastrophes are associated with degenerate critical points of functions only in this case may a change of differential type in a function (change in the set of its critical points — a catastrophe) take place on varying control parameters. We shall see that functions having points of type I or II are structurally stable, while 

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Catastrophes described by Thom potential functions of one variable... 

The analysis of potential functions of one variable will finish with the examination of the swallowtail catastrophe A4, to which corresponds the potential function V x a, b, c) = 1/5 x5 + aj3 x3 + b/2 x2 + cx (again, we introduced for convenience of notation renormalization of coefficients in the catastrophe function, cf. Table 2.2). 

B, employed in Section 2.2 to investigate potential functions of one variable. In the case of functions of two variables the catastrophe manifold M is given by the equation (cf. equation (2.1a))... 

In principle, it should be possible to obtain the electronic energy levels of the molecules as a solution of the Schrodinger equation, if inter-electronic and internuclear cross-coulombic terms are included in the potential energy for the Hamiltonian. But the equation can be solved only if it can be broken up into equations which are functions of one variable at a time. A simplifying feature is that because of the much larger mass of the nucleus the motion of the electrons can be treated as independent of that of the nucleus. This is known as the Bom-Oppen-heimer approximation. Even with this simplification, the exact solution has been possible for the simplest of molecules, that is, the hydrogen molecule ion, H + only, and with some approximations for the H2 molecule. 

To conclude our considerations of functions of one variable, we shall examine the properties of several potential functions given in Table 2.2. In analysing the properties of these functions and the catastrophes described by them, the catastrophe manifold M defined by equation (2.1a) will be helpful. In the case of a function of one state variable it takes the form... 

The sets M, S, B, defined for families of functions of one variable at the beginning of Section 2.2.4, will play an equally vital role in the analysis of catastrophes occurring in systems described by potential functions dependent on two state variables. For the case of two state variables, the sets will be defined in Section 2.3.6. 

As mentioned above, we shall not describe at this point the method of solving the problem of determinacy for functions of two variables (see Appendix, A2). We shall confine ourselves to providing a list of the simplest potential function, having at x = (0, 0) a regular point, a degenerate critical point, for which the problem of determinacy has been solved. In other words, addition of a perturbation to the functions listed in Table 2.4 must not convert a degenerate critical point into another degenerate critical point. Table 2.4 (functions of two variables) is a counterpart to Table 2.1 (functions of one variable). 

To a potential function in one variable (see Table 2.2) may be added, without a change in properties of the elementary catastrophe being described, the term M and to a potential function in two variables the term... 

The isotopic difference obviously depends exclusively on the even derivatives of the perturbing potential, which implies that a linear potential, corresponding to a constant force, gives rise to no isotope effect. This may be easily understood in the following way if two functions of a variable x, one parabolic (fi(x)) and the other linear (fz x)), i.e.,... 

Two central problems remain. One is that one needs the potential which governs the motion. In many-atom systems, even if the motion is confined to the ground electronic state, this potential is a function of the spatial configuration of all the atoms. It is therefore a function of many variables, so its analytical form is far from obvious, nor do we necessarily want to know it everywhere. Indeed, we really only want it at each point along the actual trajectory of the system (so that the forces can be computed and thereby the next point to which the system will move to can be determined). Such an approach has been implemented [25] and applied to many-atom systems, and an extension to a multi-electronic state dynamics will be important... 

This has again a truncatable structure the excess part g inherits its moment structure from/. Note that because of the normalization of the rm,g does not depend on the density p = p0. It is therefore normally a function of one less variable than f, unless f is already independent of p [as is the case in Flory-Huggins theory, Eq. (64), for example]. The chemical potentials p(a) follow from Eq. (Al) as... 

One of the potential advantages of transition metal complexes as homogeneous catalysts is the possibility of variation of the ligands and the correlation of catalytic activity with some function of this variable. Few catalytic systems have been examined in the necessary detail, however, and the hydro-formylation reaction is no exception. Much of the work in this area has therefore taken a more empirical approach to catalyst modification. The result has been the production of an extensive patent literature of little scientific value other than to suggest areas in need of further study. 

The symmetry coordinates show themselves to be particularly useful for the functional representation of the molecular potential. For example, the potential function of a X3-type molecule must be invariant with respect to the interchange of any internal coordinate ft, (/ = 1, 2, 3) hence it must be totally symmetric in relation to those coordinates. Thus, in terms of the coordinates Qi (/ =1,2, 3), such a function can only be written in terms of or totally symmetric combinations of Q2 and Q3. Such combinations may in fact be obtained by using the projection-operator technique.16"27 In fact, one can demonstrate16 27 that any totally symmetric function of three variables is representable in terms of the integrity basis,28... 

Each nAChR subunit is encoded by a different gene and any mutation in any of these genes that affects the expression or function of an nAChR could lead to disease or contribute to individual differences in risk for disease. In this section, one disease directly caused by mutations in nAChR subunit genes is discussed. In addition, the potential role of genetic variability in nAChR subunit genes in altering risk for disease is summarized. 

If we know one of the thermodynamic potentials as a function of the variables to which it corresponds, we can express all the other thermodynamic variables as a function of this one by using equations (4.29). 

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium... 

The problem of universal unfolding for the functions listed in Table 2.1 may be solved similarly to the problem of finding a form of the function g(x a, b) in (2.8). Universal unfoldings obtained in such a way are compiled in Table 2.2 (this is the first part of the Thom theorem, which pertains to potential functions in one state variable). 

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