Constant-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 constant-polarizability model, each particle is assumed to carry an induced polarization p that is instantaneously proportional to the local electric field E, that acts on it, with a fixed tensor constant of proportionality, so that... 

For simplicity we now focus on the constant-polarizability model for orientation-independent potentials and scalar a. Here one has [with = T(r,y) given by (4.12)]... 

We recall now that at p = p, (4.68) is exact for a lattice gas hence, on the basis of widely held notions of universality, p can also be assumed in a continuum-fluid computation without doing violence to the structure of the dominant singularities that emerge at the fluid critical point as t— 0, p = p. Since (4.71) follows without further assumptions from (4.64) and (4.68), (4.71) appears to be an appropriate expression for the study of critical behavior of S2 in the constant-polarizability model. It is especially useful if we note that there is gross similarity between the function O2 in (4.71) and the attractive part of a typical pair potential. To exploit this, consider the potential... 

In the constant-polarizability model, const=4.) Such an will also share the singularity of u as would an Sj of the more general form... 

For a given a the force constant ko can be chosen in a way that the displacement d of the Drude particle remains much smaller than the interatomic distance. This guarantees that the resulting induced dipole jl, is almost equivalent to a point dipole. In the Drude polarizable model the only relevant parameter is the combination q /ko which defines the atomic polarizability, a. It is... 

Im. Z i(E) can be considered as a product of an ionic excitation density of states and an energy-dependent coupling constant. In model calculations one can independently vary the shape and the band with of the denstiy of states and the strength of the coupling constant. In the present case we can only vary these parameters indirectly by changing the atomic number Z. Since the self-energy involves the polarizability of the ionic system there must be an oscillator-strength sum rule such that... 

In the liquid phase, calculations of the pair correlation functions, dielectric constant, and diffusion constant have generated the most attention. There exist nonpolarizable and polarizable models that can reproduce each quantity individually it is considerably more difficult to reproduce all quantities (together with the pressure and energy) simultaneously. In general, polarizable models have no distinct advantage in reproducing the structural and energetic properties of liquid water, but they allow for better treatment of dynamic properties. 

It is now well understood that the static dielectric constant of liquid water is highly correlated with the mean dipole moment in the liquid, and that a dipole moment near 2.6 D is necessary to reproduce water s dielectric constant of s = 78 T5,i85,i96 holds for both polarizable and nonpolarizable models. Polarizable models, however, do a better job of modeling the frequency-dependent dielectric constant than do nonpolarizable models. Certain features of the dielectric spectrum are inaccessible to nonpolarizable models, including a peak that depends on translation-induced polarization response, and an optical dielectric constant that differs from unity. The dipole moment of 2.6 D should be considered as an optimal value for typical (i.e.. 

Since nonpolar molecules do not have constant polarizability, what can we say about their dielectric anomaly The difference in the pair and triplet polarizabilities in real noble-gas molecules and our model molecules can be summarized in terms of S2 by saying that we would expect a similar sort of integral to provide a reasonable approximation to S2 at p, but with an 02(r) quite different from that given by (4.71b). In the case of helium, for example, where the linearityof Sj in p suggests that three-body effects are of less importance than in argon, Sj might be reasonably approximated at p by the expression... 

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. 

In the Drude polarizable model, the only relevant adjustable parameter is the combination q /KD that corresponds to the atomic polarizability. In the limit of large Kd, the treatment of induced polarization based on Drude oscillators is formally equivalent to a point-dipole treatment such as used by AMOEBA. In practice, the magnitude of Kd is commonly chosen to achieve small displacements of Drude particles from their corresponding atomic positions, as required to remain close to the point-dipole approximation for the induced dipole associated with the atom-Drude pair [150] while preserving a stable integration of the equation of motion with a reasonable time step. For a fixed force constant Kd the atomic polarizability is determined by the amount of chaise assigned to the Drude particle. In the current implementation, the classical Drude model introduces atomic polarizabilities only to non-hydrogen atoms for practical considerations, as discussed below. However, this is adequate to accurately reproduce molecular polarizabilties, as seen in a number of published studies [127,142,146]. 

Rigid non-polarizable models for water attempt to approximate, via a two-body interaction, the many-body polarization effects which are responsible for a substantial contribution to the properties of the condensed phase of highly polar fluids such as water [58], especially on the dielectric constant [59], by having a large effective dipole moment. While a two-body model might work well in approximating quasi-... 

Mata et al have determined the dynamic polarizability and Cauchy moments of liquid water by using a sequential molecular dynamics(MD)/ quantum mechanical (QM) approach. The MD simulations are based on a polarizable model of liquid water while the QM calculations on the TDDFT and EOM-CCSD methods. For the water molecule alone, the SOS/TDDFT method using the BHandHLYP functional closely reproduces the experimental value of a(ffl), provided a vibrational correction is assumed. Then, when considering one water molecule embedded in 100 water molecules represented by point charges, a(co) decreases by about 4%. This decrease is slightly reduced when the QM part contains 2 water molecules but no further effects are observed when enlarging the QM part to 3 or 4 water molecules. These molecular properties have then been employed to simulate the real and imaginary parts of the dielectric constant of liquid water. 

The two models, the Langevin and the polarizable PB equation with a < 0, are designed to represent the same phenomena, the lowering of a dielectric constant as the hydration stractures form around dissolved ions. The results, however, are not precisely comparable. Decrement of a dielectric constant near a wall are captured by both models, but density profiles are not comparable even quahtatively. Counterion profiles of the Langevin model are more concentrated, while those of the polarizable model are more dilute. Without exact simulation results, it is hard to know which model is accurate, hi recent work by Ma et al. [39] the Langevin PB equation with correlations has been solved and it yields a non-monotonic density profile with a valley at a wall followed by a peak further away from a wall. The depletion is, therefore, captured but immediately at a wall and not for the entire profile. A possible weak point of the negative polarizability model is the linearity assumption, -e —c. For homogenous solutions linearity breaks down for... 

Most QM/MM applications are devoted to biological questions. In this field QM/MM approaches are mandatory since the molecular nature of the enzymatic environment determines geometrical arrangements and the energetics of enzymatic processes. The use of continuum polarizable models is problematic since the dielectric constant, for example, is not well defined. For regions in the outer spheres... 

J. Alejandre, G.A. Chapela, H. Saint-Martin, N. Mendoza, A non-polarizable model of water that yields the dielectric constant and the density anomalies of the liquid TIP4Q. PCCP 13, 19728-19740 (2011)... 

To give a simple classical model for frequency-dependent polarizabilities, let me return to Figure 17.1 and now consider the positive charge as a point nucleus and the negative sphere as an electron cloud. In the static case, the restoring force on the displaced nucleus is d)/ AtteQO ) which corresponds to a simple harmonic oscillator with force constant... 

The Self-Consistent Reaction Field (SCRF) model considers the solvent as a uniform polarizable medium with a dielectric constant of s, with the solute M placed in a suitable shaped hole in the medium. Creation of a cavity in the medium costs energy, i.e. this is a destabilization, while dispersion interactions between the solvent and solute add a stabilization (this is roughly the van der Waals energy between solvent and solute). The electric charge distribution of M will furthermore polarize the medium (induce charge moments), which in turn acts back on the molecule, thereby producing an electrostatic stabilization. The solvation (free) energy may thus be written as... 

Figure 6-12. Model for Ihe Calculation of the van der Waals potential experienced by a single T6 molecule on a Tfi ordered surface. Each molecule is modeled as a chain of 6 polarizable spherical units, and the surface as 8-laycr slab, each layer containing 266 molecules (only pan of the cluster is shown). Tire model is based on X-ray diffraction and dielectric constant experimental data. The two configurations used for evaluating the corrugation of the surface potential are shown. Adapted with permission front Ref. [48].

The theoretical methods can be divided into two fundamental groups. The so-called continuum models are characterized by assuming that the medium is a structureless and polarizable dielectricum described only by macroscopic physical constants. On the other hand there are the so-called discrete models. The main advantage of... 

In Eq. (6) Ecav represents the energy necessary to create a cavity in the solvent continuum. Eel and Eydw depict the electrostatic and van-der-Waals interactions between solute and the solvent after the solute is brought into the cavity, respectively. The van-der-Waals interactions divide themselves into dispersion and repulsion interactions (Ed sp, Erep). Specific interactions between solute and solvent such as H-bridges and association can only be considered by additional assumptions because the solvent is characterized as a structureless and polarizable medium by macroscopic constants such as dielectric constant, surface tension and volume extension coefficient. The use of macroscopic physical constants in microscopic processes in progress is an approximation. Additional approximations are inherent to the continuum models since the choice of shape and size of the cavity is arbitrary. Entropic effects are considered neither in the continuum models nor in the supermolecule approximation. Despite these numerous approximations, continuum models were developed which produce suitabel estimations of solvation energies and effects (see Refs. 10-30 in 68)). 

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