Linear response approximation solvation

The first version of the LIE method employed the linear response approximation to estimate the electrostatic part of the solvation/binding free energies. The linear response result for this component of the solvation... 

In order to actually examine the validity of the electrostatic linear response approximation for different types of solutes and solvents, Aqvist and Hansson13 investigated the relationships between electrostatic solvation... 

III. SOLVATION MECHANISMS WITHIN THE LINEAR RESPONSE APPROXIMATION... 

III. Solvation dynamics within the linear response approximation 213... 

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],... 

If AE can be considered to be a small perturbation in system properties, the solvation response can be estimated using the linear response approximation (LRA), which relates S(t) to the time correlation function (TCF) C0(t) of fluctuations SAE = AE - (AE) of AE in the unperturbed system [23],... 

The solution of these differential equations yields the total electrostatic potential at any point r. Assuming a linear response approximation the electrostatic component of solvation can be obtained as of the work necessary to generate the solvent reaction potential, which can be determined by simply computing the ratio of potential generated by the solute in vacuo to the total potential around the solute (Equation (4.34)). 

Eq. [33] according to the assumption of the classical character of this collective mode. Depending on the form of the coupling of the electron donor-acceptor subsystem to the solvent field, one may consider linear or nonlinear solvation models. The coupling term - Si -V in Eq. [32] represents the linear coupling model (L model) that results in a widely used linear response approximation. Some general properties of the bilinear coupling (Q model) are discussed below. 

The left-hand side of (15.26) is, by Eq. (15.23), a linear response approximation of the corresponding solvation energies difference. This makes it possible for us to write a linear response expression for the solvation function which is defined by... 

The nonequilibrium solvation function iS (Z), which is directly observable (e.g. by monitoring dynamic line shifts as in Fig. 15.2), is seen to be equal in the linear response approximation to the time correlation function, C(Z), of equilibrium fluctuations in the solvent response potential at the position of the solute ion. This provides a route for generalizing the continuum dielectric response theory of Section 15.2 and also a convenient numerical tool that we discuss further in the next section. 

Here, again, the coefficient 1/2 comes from the linear response approximation, i.e., that half of the solvation energy is spent on creating the induced charges. 

The time dependent solvation funetion S(t) is a directly observed quantity as well as a convenient tool for numerical simulation studies. The corresponding linear response approximation C(t) is also easily eomputed from numerical simulations, and can also be studied using suitable theoretical models. Computer simulations are very valuable both in exploring the validity of such theoretical calculations, as well as the validity of linear response theory itself (by comparing S(t) to C(t)). Furthermore they can be used for direct visualization of the solute and solvent motions that dominate the solvation process. Many such simulations were published in the past decade, using different models for solvents such as water, alcohols and acetonitrile. Two remarkable outcomes of these studies are first, the close qualitative similarity between the time evolution of solvation in different simple solvents, and second, the marked deviation from the simple exponential relaxation predicted by the Debye relaxation model (cf Eq. [4.3.18]). At least two distinct relaxation modes are... 

Approximate analytical theories of solvation dynamics are typically based on the linear response approximation and additional statistical mechanics or continuum electrostatic approximations to Cy(r). The continuum electrostatic approximation requires the frequency-dependent solvent dielectric response For example, the Debye model, for which e(a>) = + (cq - )/(l +... 

The nonequilibrium solvation dynamics following the (u = 12D) - (n = 0) transition are essentially independent of the solute s location. The very rapid subpicosecond relaxation is complete within a few hundred femtoseconds. The linear response approximation is satisfied in all locations except on the vapor side of the water liquid/vapor interface and on the organic side of the water/CC interface. The main reason for the deviation from linear response is the different contribution of the inertial component to the total relaxation, as the equilibrium relaxation is determined from the solvent fluctuations around the neutral solute. 

These considerations have to be applied to phenomena in which the external field has its origin in the solute (or, better, in the response of the solute to some stimulus). The characteristics of this field (behaviour in time, shape, intensity) strongly depend on the nature of the stimulus and on the properties of the solute. The analysis we have reported of the behaviour of the solvent under the action of a sinusoidal field can here be applied to the Fourier development of the field under examination. It may happen that the Fourier decomposition will reveal a range of frequencies at which experimental determinations are not available to have a detailed description of the phenomena an extension of the s(w) spectrum via simulations should be made. It may also happen that the approximation of a linear response fails in such cases the theory has to be revisited. It is a problem similar to the one we considered in Section 1.1.2 for the description of static nonlinear solvation of highly charged solutes. 

A different analysis applies to the LR approach (in either Tamm-Dancoff, Random Phase Approximation, or Time-dependent DFT version) where the excitation energies are directly determined as singularities of the frequency-dependent linear response functions of the solvated molecule in the ground state, and thus avoiding explicit calculation of the excited state wave function. In this case, the iterative scheme of the SS approaches is no longer necessary, and the whole spectrum of excitation energies can be obtained in a single run as for isolated systems. 

---