Linear Interaction Approximation

The linear interaction approximation (LIA) was introduced by Aqvist and co-workers44"48 to calculate absolute binding free energies via MD simulations. A version of the LIA equation takes the following form... 

The linear interaction energy method allows the approximation of the free energy of binding using the relationships64... 

Figure 1. Diagram of the intensity / (W/cm2) vs. duration of laser pulse tp(s) with various regimes of interaction of the laser pulse with a condensed medium being indicated very qualitatively. At high-intensity and high-energy fluence 4> = rpI optical damage of the medium occurs. Coherent interaction takes place for subpicosecond pulses with tp < Ti, tivr. For low-eneigy fluence (4> < 0.001 J/cm2) the efficiency of laser excitation of molecules is very low (linear interaction range). As a result the experimental window for coherent control occupies the restricted area of this approximate diagram with flexible border lines.

In this chapter we consider the extension of continuum solvent models to nonlocal theories in the framework of the linear response approximation (LRA). Such an approximation is mainly applicable to electrostatic solute-solvent interactions, which usually obey the LRA with reasonable accuracy. The presentation is confined to this case. 

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

At this stage, it is convenient to assume a very low density of excited dipoles. In other words, we assume that the exciting external source is sufficiently weak so that at each instant the probability of finding a given dipole in an excited state is very small compared to 1. In this condition, the system satisfies the linear-response approximation. Since the elementary excitations are very dilute (i.e., the occupation numbers are very small), all statistics are equivalent. For the convenience of further calculations (e.g. interaction with photons), the operators B B are assumed to obey Bose statistics21,22 ... 

In the case of a linear interaction between neighboring lipid bilayers, Helfrich has demonstrated that the repulsive free energy due to confinement is inversely proportional to (7b2. While this result is strictly valid for a harmonic interaction potential (linear force), we assume that it can be extended to any interaction. We will examine later under what conditions this approximation is accurate. 

Linear Superposition Approximation for the Double-Layer Interaction of Particles at Large Separations... 

In Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

Comparison is made with the results for the two conventional models for hard plates given by Honig and Mul [11]. We see that the values of the interaction energy calculated on the basis of the Donnan potential regulation model lie between those calculated from the conventional interaction models (i.e., the constant surface potential model and the constant surface charge density model) and are close to the results obtained the linear superposition approximation. 

The first-order correction to the linear superposition approximation Vlsa is given by the sum of the second and third terms on the right-hand side of Eqs. (14.23), (14.29) and (14.34), each corresponding to the image interaction of one sphere with respect to the other sphere. We denote this image interaction by Vimage nd expressed it as... 

FIGURE 14.4 Comparison of the linear superposition approximation V lsa = 1 ° the image interaction correction Fimage = their sum ULSA + Fmage, and the full... 

Fig. 14.5), then the interaction energy shows a minimum. That is, the interaction force, which is attractive at large separations, may become repulsive at small separations. As Fig. 14.5 shows, the change in sign of the interaction force or the appearance of the extremum in the interaction energy occurs when the contribution of the image interaction correction exceeds that of the leading term (or the linear superposition approximation term). 

A simple approximate analytic expression for P ih) can be obtained using the linear superposition approximation (LSA) (Chapter 11). In this approximation, y h/2) in Eq. (15.34) is approximated by the sum of the asymptotic values of the two scaled unperturbed potentials ys(T) that is produced by the respective plates in the absence of interaction. For two similar plates. 

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