Static theories

We can distinguish between static theories, which in essence give a description of the electron density, and dynamic theories, where an attempt is marie to measure the response of a molecule to (e.g.) an approaching N02" " ion. In recent years, the electrostatic potential has been used to give a simple representation of the more important features of molecular reactivity. It can be calculated quite easily at points in space ... 

The bond lengths for the dianions of I, III, IV and VII shown in Fig. 5 indicate that these dianions all have a fully-symmetrical nuclear arrangement and that the bond lengths of the peripheral bonds of these anions are fairly equivalent. This means, in agreement with the results obtained using the static theory, that addition of two more electrons to molecules I, III and VII to form their dianions results in a complete disappearance of the bond alternations existing in these molecules. Dianions of ss) III , and iv have been prepared and are known to be stable species. 

The results of our calculations based on both the static and dynamic theories show that most of the nonbenzenoid cyclic conjugated systems examined exhibit in a greater or lesser degree a marked double-bond fixation. The static theory indicates that even in benzene there exists a hidden tendency to distort into a skewed structure and that such a tendency is actually realized in [4n-f-2] annulenes larger than a certain critical size. In nonalternant hydrocarbons bond distortion is a rather common phenomenon. Fulvenes, fulvalenes and certain peri-condensed nonalternant hydrocarbons undergo a first-order bond distortion, and... 

The approaches are divided into those which do not take frequency into account (except for relaxation phenomena) and therefore are "static" theories and those which take frequency into account, generally by a scattering approach. The latter, "dynamic" theories, when reduced to the low frequency or quasi-static limit, usually compare favorably with the static theories. Some approaches take multiple scattering into account and cannot be solved in closed form. These require elaborate computer number crunching techniques (5). 

Static Theories. The simplest effective modulus theory is that which assumes the components behave as an ideal liquid mixture of one substance in another (6). In this case, both the density and the compressibility are additive properties of the corresponding quantities of the two materials depending upon the proportional amount of each substance in the mixture. Thus,... 

Another condition, related to the possibility of simply interpreting the temperature dependence of the wing profile, is that one point in the spectrum should be related to one single point of the potential curve. In the present case this is always verified except in the neighborhood of extrema of the difference AV between the potentials of the upper and lower states of the transition. However this region, which corresponds to the so-called satellite, is not amenable to interpretation by the quasi static theory and will be considered later on. 

The theories in this paper are first-principles statistical mechanics theories used to calculate static thermodynamic and molecular ordering properties (including solubilities of LCPs in various kinds of solvents) and dynamic properties (diffusion from Brownian motion). The diffusion of the LCP molecules constitutes a lower limit for the speed of processing of the LCPs. The static theory is used to calculate the packing of the bulky relatively rigid side chains of SS LCPs these calculations indicate that head-to-tail polymerization of the monomers of these SS LCPs will be very strongly favored. The intermolecular energies and forces calculated from the static theory are used in the dynamic theory. 

The basic idea of this dynamic theory is to use the site-site intermolecular energies and forces from the static theory presented earlier in this paper to calculate friction coefficients, etc., for analytic Brownian motion calculations for the molecules. From References 32-33, the aperiodic case of Brownian motion of a harmonically-bound particle is given by... 

Note that U and Fb are calculated from the intermolecular potential energies < > k and separation distances ak which are calculated in the static theory for the phase of lowest free energy as a function of T, P, system composition, and molecule chemical structures. Thus, U and Fb are determined by the details of the molecule chemical structures and the orientational and positional ordering of the molecules. < >b xb / b From t le harmonic oscillator equation. 

For some of the first theoretically designed candidate SS LCPs, new results are now presented for (1) enhanced solubilities of the molecules (compared with backbone LCPs) in nonpolymeric LC solvents calculated using the static theory, (2) diffusion (i.e., lower limit of processability) of the molecules calculated using the dynamic theory, and (3) head-to-tail polymerization of the monomers predicted from the packing of the bulky relatively rigid side chains of the molecules as calculated using the static theory. Melt processability of some SS LCPs is also discussed. 

Distortions due to magnetic and electric fields static theory 3.4.1 The Freedericksz effect... 

The simplest method of measuring the three elastic constants of a nematic liquid crystal is by studying the deformations due to an external magnetic field (Freedericksz and Tsvetkov, Zocher ). The geometry has to be so chosen that the orienting effect of the field conflicts with the orientations imposed by the surfaces with which the liquid crystal is in contact. To develop a static theory of such deformations we apply the equation of... 

The simplest model of this kind is the elastic bar theory studied by Ericksen [11]. He considers the static theory of bars governed by a one-dimensional, nonlinearly elastic stress-strain relation shown in Fig. 6. 

Frank, F. C. and Van der Merwe, J. H. (1949), One-dimensional dislocations I. Static theory. Proceedings of the Royal Society (London) A198, 205-216. 

The static theory of atomic forces, which has been considered almost exclusively up to now with the methods of quantum mechanics, needs the addition of a dynamics of a chemical reaction. The collision methods, used by several groups, do not appear to be a useful method for the treatment of chemical processes (Section 1). In what follows, a dynamical theory for the simplest cases of bimolecular reactions is developed (Sections 2 and 3), in which the problem of chemistry is expressed immediately, and clearly discussed with the help of elementary examples (Section 4). Sections 5 and 6 contain a perturbation approach, whic converges for small and large collision velocities and allows for a relatively simple sqiproximation method for the reaction velocity of non-adiabatic processes. On the other hand the theory contains a quantitative description of the connection to the adiabatic reaction process in the limit of low velocities or separated characteristics of the potential. In this way it yields a conditional justification for the application of potential theoretical representations to chemical processes and at the same time a fixation of the limits of such idealisations. 

---