Fluctuating-polarizability model

Almost all the formalism and the approximation schemes of Sections II and III have a natural extension to systems of polarizable dipolar particles, but the precise details of the extension depend on the way polarizability is introduced into the Hamiltonian. We refer to the two quite distinct Hamiltonian models that have been most thoroughly developed in this context as the constant-polarizability model and the fluctuating-polarizability model. The dielectric behavior of the former was first systematically investigated from a statistical mechanical viewpoint by Kirkwood and by Yvon, who considered the model almost exclusively in the absence of permanent dipole moments. (Kirkwood S subsequently pioneered an exact formulation of the statistical mechanics of polar molecules, but largely as a separate enterprise that did not attempt to treat the polarizability exactly.) The general case of polar-polarizable particles remained only very partially developed ... 

In the fluctuating-polarizability model with harmonic >(p), the a of (4.8) is identified as the polarizability of an isolated particle. With terms of higher... 

The work by Hoye and Stell on the fluctuating-polarizability model regards molecules with different m and p as being molecules of different species. The route to s that has been exploited by Heye and Stell in this connection involves (2.26) and the appropriate generalizations of (2.25c) to (2.25e). When fluctuating polarizability is added, we have the probability density p, p that gives the probable distribution of particles with permanent moment /i and instantaneous induced moment p. This gives a distribution for fixed m of... 

The total electric field, E, is composed of the external electric field from the permanent charges E° and the contribution from other induced dipoles. This is the basis of most polarizable force fields currently being developed for biomolecular simulations. In the present chapter an overview of the formalisms most commonly used for MM force fields will be presented. It should be emphasized that this chapter is not meant to provide a broad overview of the field but rather focuses on the formalisms of the induced dipole, classical Drude oscillator and fluctuating charge models and their development in the context of providing a practical polarization model for molecular simulations of biological macromolecules [12-21], While references to works in which the different methods have been developed and applied are included throughout the text, the major discussion of the implementation of these models focuses... 

The polarizable fluctuating charge model in CHARMM results from the work of Patel, Brooks and co-workers [92, 214], The water model is based on the TIP4P-FQ model of Rick, Stuart and Berne [17], In the development of the force field the electronegativities and hardnesses were treated as empirical parameters and do not have any association with experimental or QM values, for example, from ionization energies and electron affinities of single atoms. 

One feature of the semiempirical models is that because the polarization is described by a set of coefficients that have a normalization condition, for example, Eq. [69], there will be no polarization catastrophe like there can be with dipole polarizable or fluctuating charge models. With a finite basis set, the polarization response is limited and can become only as large as the state with the largest dipole moment. 

Quite probably the answer to the second question will look not too much different from the expressions for the models that have been thoroughly analyzed here, but the establishment of this result may turn out to be tedious. We have seen in Section II how to handle rigid nonpolarizable particles of arbitrary symmetry using the formalism of Hoye and Stell. The addition of fluctuating polarizability has been considered by those authors only for molecules of cylindrical symmetry, but its extension to molecules of arbitrary symmetry is unlikely to raise fundamental problems. On the other hand, particles lacking cylindrical symmetry even in the nonpolarizable case are substantially more awkward to deal with than cylindrically symmetric particles. In treating the constant-polarizability case, Wertheim excludes all permanent multipoles beyond the dipole clearly the quadrupole at least must also be included to provide a realistic model for many real fluids of interest. 

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