Potentials polynomial expansions

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

As already mentioned, one of the main weaknesses of the simple reflection method is the fact that the electronic transition dipole moment, (or the transition dipole moment surface, TDMS for polyatomic molecules in Section 4) is assumed to be constant. This weakness will remain in the Formulae (12), (27) and (29) derived below. The average value of the square of the TDM (or TDMS) is then included in amplitude A and A = A /V. In Formulae (3), (3 ) and (3") the mass (or isotopologue) dependent parameters are p and the ZPE. In contrast, W and V., which define the upper potential, are mass independent. This Formula (3) is already known even if different notations have been used by various authors. As an example, Schinke has derived the same formula in his book [6], pages 81, 102 and 111. Now, the model will be improved by including the contribution of the second derivative of the upper potential at Re- The polynomial expansion of the upper potential up to second order in R - Re) can be expressed as ... 

In packed-bed, flow-through electrodes, concentration and potential variation within the bed can also give more than one steady state. The convective transport equation with axial dispersion, coupled with Ohm s law for the electrode potential, was solved recently (418) by polynomial expansion and orthogonal collocation within the bed, to determine multiplicity regions. 

The one-dimensional potential energy curve describes precisely the electronic term of a system only in the simple case of diatomic molecules. With A H bonds, where A is part of a multiatom molecule, the diatomic approximation is valid. But in the general case the potential energy as a function of the A H bond length, F(r), should be expressed using a polynomial expansion... 

Realize that dvo/dr)r=R and Kdy/dt)re]r=R are nonzero for potential flow around a cylinder and that the first-order term in the polynomial expansion for V(, does not vanish, but this first-order term is small relative to the leading zeroth-order term. Now the locally flat description of the equation of continuity allows one to calculate the radial velocity component. For example, integration from the nondeformable solid-liquid interface at y = 0, where Vy = 0, to any position y within the thin mass transfer boundary layer produces the following result ... 

Figure 16.7 Approximations to the Morse potential by expansion in a polynomial basis set vertical lines indicate the range used for the fitting...

For many potential energy surfaces with barriers, the dominant term in a Legendre polynomial expansion of the interaction potential is the leading term, that is. the spherically averaged potential ... 

At first sight this looks like nothing more than a polynomial expansion of the ideal gas law. However, it turns out to have real physical significance. Statistical mechanics shows that the second coefficient arises from the interaction of pairs of molecules, the third from the interaction of molecules three at a time, and so on. They can be calculated from known interaction potentials, or used to estimate such potentials from observed PVT behavior. The details can be found in most textbooks on statistical mechanics (for example McQuarrie, 2000, Chapter 12), and Prausnitz et al. (1999) give an extensive treatment of various commonly used formulations of these intermolecular forces and their use in equations of state. 

The free energy of the lamellar microstructure in an electric field is a function of orientation. Is it reasonable to expect an orientation-dependent disordering induced by an electric field This issue was considered theoretically by us [65]. The analysis begins with the thermodynamic potential of Fredrickson and Helfand [73], which is the Landau free energy density for a symmetric block copolymer written as a polynomial expansion in the amplitude. A, of the composition pattern [74] ... 

Figure 17.7 Lowest adiabatic potential energy surface for collinear geometries of H3 as fitted to 9477 DMBE points with a polynomial expansion up to power 20 in the basis b8. See text.

An approach which couples flexibility to a fairly simple analytical formulation is that suggested by Murrell and coworkers decomposing the potential energy in two and three body terms. These terms are then represented using polynomial expansions in physical coordinates damped by exponential or hyperbolic tangent factors. We have suggested... 

H( S) + CCl X E" ) and the dependence of the calculated rate constants on the interaction potential, V R,0). We compare results for two forms of the interaction potential, which we label as V2 R,0) and Vscf R )- Both of these can be conveniently expressed in terms of a Legendre polynomial expansion. V2 R 0) is similar to Eq.(l) but includes the contributions from the permanent quadrupole moment, 0, and static anisotropic polarizability, tt2, of the HCl molecule. 

Among the various analytical representations of a PES, those advantageous for use on supercomputers are polynomial expansions. For this reason, following Murrell et al. it is useful to divide the potential energy of a triatomic ABC system into its multi-body components... 

A third-degree polynomial expansion of the potential function has been fitted directly to a very large set of experimental frequencies. The distance is one of the fitted parameters. 

Realistic intermolecular interaction potentials for mesogenic molecules can be very complex and are generally unknown. At the same time molecular theories are often based on simple model potentials. This is justified when the theory is used to describe some general properties of liquid crystal phases that are not sensitive to the details on the interaction. Model potentials are constructed in order to represent only the qualitative mathematical form of the actual interaction energy in the simplest possible way. It is interesting to note that most of the popular model potentials correspond to the first terms in various expansion series. For example, the well known Maier-Saupe potential JP2 (Sfli )) is just the first nonpolar term in the Legendre polynomial expansion of an arbitrary interaction potential between two uniaxial molecules, averaged over the intermolecular vector r,-, ... 

Despite their simplicity, elastic network models are based on very solid and intuitive mathematical-physical concepts (like polynomial expansion of the potential energy function from the global minimum), and this has constituted one of the reasons for their success in the literature in the past years. Nonetheless, at the same time, their simplicity restricts their applicability to a relatively small set of studies. 

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