Linear response approximation models

LD model, see Langevin dipoles model (LD) Linear free-energy relationships, see Free energy relationships, linear Linear response approximation, 92,215 London, see Heitler-London model Lysine, structure of, 110 Lysozyme, (hen egg white), 153-169,154. See also Oligosaccharide hydrolysis active site of, 157-159, 167-169, 181 calibration of EVB surfaces, 162,162-166, 166... 

Figure 2. MD simulation results for SD in response to electronic excitation of Cl 53 in room-temperature acetonitrile (left panel) and CO2 liquids. The solvent models and thermodynamic states are as in Ref. " and the solute model parameters are from Ref Nonequilibrium solvent response, S(f), and linear response approximations to it for the solute in the ground, Co(t), and excited, Q (f), electronic states are shown.

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],... 

In the previous section we demonstrated numerically the validity of the linear response approximation for the hybridization tetrahedra. Now we can use these relations to perform the announced transition from the DMM model of molecular PES to a model dependent on molecular geometry. It is formally obtained by inserting eq. (3.133) into eq. (3.132) which yields ... 

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. 

In the case of the protonated Schiff base model, the forces computed from the LR-TDDFT Sj PES lead to a single bond rotation instead of double bond isomerization. This failure might be related to the local approximation of the exchange-correlation functional or, as suggested by the significantly different results obtained with P-TDDFT, to a breakdown of the linear response approximation. Further investigations in this respect are needed. P-TDDFT and ROKS correctly... 

Because first-order sensitivity coefficients are easier to calculate than higher order sensitivity coefficients, it is likely that the former may be used more frequently in guiding molecular design. However, first-order sensitivity theory can provide reliable predictions only when the sensitivities of the properties of interest are approximately linear with respect to the model parameters. This linear response limit is satisfied when the perturbations of model parameters are small. For certain applications, such as in protein engineering where one amino acid is mutated into another, the linear response approximation may fail to reliably predict the change in the properties of a protein resulting from a point mutation. It is therefore useful to examine in more detail how well first-order sensitivity theory performs in guiding such predictions. 

The interband and vibronic contributions to the infrared properties of the model system have been calculated in the adiabatic, linear-response approximation. Two possible schemes for the occupation of the band states by a number of electrons or holes Np = N/2 (corresponding to half a carrier per molecule) have been considered (i) the case of regular fermion particles with spin, where the lower band only is half filled, (ii) that of spinless fermions, with the lower band completely full. Although no electron correlation term is explicitly included in the Hamiltonian, the latter case represents the situation that is attained, when the on-site correlation of an Hubbard model is U t. 

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... 

To this date, no stable simulation methods are known which are successful at obtaining quantum dynamical properties of arbitrary many-particle systems over long times. However, significant progress has been made recently in the special case where a low-dimensional nonlinear system is coupled to a dissipative bath of harmonic oscillators. The system-bath model can often provide a realistic description of the effects of common condensed phase environments on the observable dynamics of the microscopic system of interest. A typical example is that of an impurity in a crystalline solid, where the harmonic bath arises naturally from the small-amplitude lattice vibrations. The harmonic picture is often relevant even in situations where the motion of individual solvent atoms is very anhaimonic in such cases validity of the linear response approximation can lead to Gaussian behavior of appropriate effective modes by virtue of the central limit theorem. ... 

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 +... 

Marcus s model assumes the validity of a linear response approximation and that a continuum electrostatic description of the interface is suitable for the purpose of calculating the activation free energy. Furthermore, to obtain expressions for the rate constant, the interface is assumed to be either a mathematically sharp plane or a broad homogeneous phase. Unfortunately, an insufficient... 

We present and analyze the most important simplified free energy methods, emphasizing their connection to more-rigorous methods and the underlying theoretical framework. The simplified methods can all be superficially defined by their use of just one or two simulations to compare two systems, as opposed to many simulations along a complete connecting pathway. More importantly, the use of just one or two simulations implies a common approximation of a near-linear response of the system to a perturbation. Another important theme for simplified methods is the use, in many cases, of an implicit description of solvent usually a continuum dielectric model, often supplemented by a simple description of hydrophobic effects [11]. 

The linear component of the LMS model, qi (i.e., one of the parameters of the polynomial), is approximately equivalent to the slope at low doses of the dose-response relationship between the tumor incidence and the dose. This linearity at low dose is a property of the formulation developed for the multistage model and is considered by proponents to be one of its important properties. This linear component of the polynomial, qi, is used to carry out low-dose extrapolation. The linear response at low doses is considered to be conservative with regard to risk, as the dose-response relationship at low doses may well be sublinear. Although supralinearity at low doses cannot be excluded, it is usually considered to be unlikely. 

A fiequently-used approximation in modeling SD and other dynamical processes in hquids is that of linear response When apphed to SD it corresponds to assuming that nonequilibrium response of the system to the perturbation AE turned on at r = 0 can be approximated in terms of equilibrium fluctuations of AE in the absence of the perturbation, i.e., for the system con-taiiting the solvent and the ground-state (subscript 0) chromophore ... 

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