Continuum models, definition

Barone, V. Cossi, M. Tomasi, J. A new definition of cavities for the computation of solvation free energies by the polarizable continuum model. J. Chem. Phys. 1997, 107, 3210-3221. 

However, in this formulation, G /8 is simply a fitted parameter, since there is no rigorous molecular definition for this form of the interfacial energy. In addition, this model suffers the same problem as do other continuum models, in that it contains parameters that must be determined from experimental data and that cannot be estimated a priori from molecular techniques. 

The results of the SCRF models depend strongly on the radius Rx used for the definition of the spherical interface between the solute and the solvent. Unfortunately, the dielectric theory does not provide an answer for the question of which value is appropriate for this radius. Owing to the implicit assumption of the dielectric continuum models that the electron density of the solute should be essentially inside the cavity, any value of Rx below a typical van der Waals (vdW) radius would not be meaningful. On the other hand, at least at the distance of the first solvent shell, i.e., typically at two vdW radii, we should be in the dielectric continuum region. However, there is no clear rationale for the right value between these two limits other than empirical comparison of the results with experimental data. Among others, the choice of spherical cavities which correspond to the liquid molar-solute volume has proved to be successful. 

Things are much simpler in continuum methods. Continuum methods in fact directly give free energies which can be collected in a function G(R) (which could be also called the FES) continuous over the R space and computationally well defined at every point of this space (as it is for the PES function in vacuo) In continuum models there are computational codes enabling the analytical calculation of derivatives (see also the contribution by Cossi and Rega in this book) necessary for the definition of TS and RP. We shall thus limit ourselves to the examination of G(R) obtained with continuum methods. 

Theoretical bases of continuum models including their mathematical formulation and numerical implementation have already been discussed in the previous chapter of this book. We have therefore restricted our review to the environment effects on the NMR observables, without going into the theory of continuum models. This contribution is divided into five sections. After the Introduction, the definitions of the NMR parameters are recalled in the second section. The third section is focused on methodological aspects of the calculation of the NMR parameters in continuum models. The fourth section reviews calculations of the solvent effects on the nuclear magnetic shielding constants and spin-spin coupling constants by means of continuum models, and the final section presents a survey on the perspectives of this field. 

Order parameters are usually derived from the measured spectral splittings through relationships such as Equation (2.269-b) thus the availability of good estimates of the magnetic tensors is an essential requirement to obtain accurate order parameters. The theoretical results suggest that the magnetic tensors obtained from calculations in a solvent, introduced by the polarizable continuum model, should definitely be a better choice than the tensors derived, as customary, from solid-state data or from calculations for molecules in a vacuum. 

A key issue in any continuum model is the definition of the solute-solvent interface, since it largely modulates the electrostatic contribution to the solvation free energy. Generally, cavities are built up from the intersection of atom-centred spheres, whose size is determined from fixed standard atomic radii [32-36], However, other strategies have been proposed, such as the use of variable atomic radii, whose values depend... 

Keeping in mind the intrinsic features associated with the definition of the cavity in the most popular QM-SCRF methods, it can be questioned what is the influence of the fine details of the cavity definition on the computed solvation free energies. This question has been investigated in a recent study by Takano and Houk [61], who have examined the dependence of the solvation free energies estimated for a series of 70 compounds, including neutral and charged species, on both the choice of the cavity and the level of theory used in computations within the framework of the conductor-like polarizable continuum model (CPCM). The mean absolute deviation (MAD) between calculated... 

Over the last years, the basic concepts embedded within the SCRF formalism have undergone some significant improvements, and there are several commonly used variants on this idea. To exemplify the different methods and how their results differ, one recent work from this group [52] considered the sensitivity of results to the particular variant chosen. Due to its dependence upon only the dipole moment of the solute, the older approach is referred to herein as the dipole variant. The dipole method is also crude in the sense that the solute is placed in a spherical cavity within the solute medium, not a very realistic shape in most cases. The polarizable continuum method (PCM) [53,54,55] embeds the solute in a cavity that more accurately mimics the shape of the molecule, created by a series of overlapping spheres. The reaction field is represented by an apparent surface charge approach. The standard PCM approach utilizes an integral equation formulation (IEF) [56,57], A variant of this method is the conductor-polarized continuum model (CPCM) [58] wherein the apparent charges distributed on the cavity surface are such that the total electrostatic potential cancels on the surface. The self-consistent isodensity PCM procedure [59] determines the cavity self-consistently from an isodensity surface. The UAHF (United Atom model for Hartree-Fock/6-31 G ) definition [60] was used for the construction of the solute cavity. 

The rationale of using hybrid simulation here is that a classic diffusion-adsorption type of model, Eq. (2), can efficiently handle large distances between steps by a finite difference coarse discretization in space. As often happens in hybrid simulations, an explicit, forward discretization in time was employed. On the other hand, KMC can properly handle thermal fluctuations at the steps, i.e., provide suitable boundary conditions to the continuum model. Initial simulations were done in (1 + 1) dimensions [a pseudo-2D KMC and a ID version of Eq. (2)] and subsequently extended to (2 + 1) dimensions [a pseudo-3D KMC and a 2D version of Eq. (2)] (Schulze, 2004 Schulze et al., 2003). Again, the term pseudo is used as above to imply the SOS approximation. Speedup up to a factor of 5 was reported in comparison with KMC (Schulze, 2004), which while important, is not as dramatic, at least for the conditions studied. As pointed out by Schulze, one would expect improved speedup, as the separation between steps increases while the KMC region remains relatively fixed in size. At the same time, implementation is definitely complex because it involves swapping a microscopic KMC cell with continuum model cells as the steps move on the surface of a growing film. 

Implicit solvent models have been the dominant class of multiscale continuum methods over recent years. However exciting new classes of multiscale continuum models have recently been developed. These new methods fall into the following categories, which are of similar definition and type to the interfaces used between the QM/MM and atomistic/CG levels ... 

Marcus model and diabatic states make more immediate (even if not strictly necessary) the introduction of the dynamical solvent coordinate, of which in Section 5.2 we have given a definition based on parameters of the continuum model, but other definitions are possible. Actually, the Russian school used this concept without giving formal definitions, at the best of our knowledge (several papers have been published in relatively minor Russian journals, with limited circulation in western countries in those years), and basing it on a description of the solvent as a continuum or a set of os-... 

It should be noticed that, in many theoretical works, the term solvent polarity is defined by the values of the relative electric permittivity, also called dielectric constant. However, such a definition is by no means precise. The existence of hydrogen bonds (H-bonds) between solute and solvent molecules is one of the important limitations of the use of the continuum models based on the theory of dielectrics. In modern physical chemistry of solutions in order to quantitatively describe the solvatochromism phenomenon various empirical scales of the polarity are used. The exhaustive reviews on this topic have been presented by Reichardt [1, 2],... 

The parabolic form of the Marcus surfaces was obtained from a linear response theory applied to a dielectric continuum model, and we are now in a position to verify this form by using the microscopic definition (16.76) of the reaction coordinate, that is, by verifying that ln(P(A)), where P X) is defined by (16.77), is quadratic in A. Evaluating PIA) is relatively simple in systems where the initial and final charge distributions po and pi are well localized at the donor and acceptor sites so that /)o(r) = - a) + <7b <5(r - rs) and pi (r) = - fa) +... 

In the seventies, most of the 37 papers (8-24) that we report are quantum chemical calculations, mainly on H502+ (8-14,20) or H30+(14-20) and a few on larger clusters with n=4-6 (8,9). However these last calculations are not accurate, obtained either from semi-empirical methods (8) or with small basis sets (DZ, 4-31G) and at the SCF level in ab initio calculations (9). The first accurate Cl calculations definitely establish the pyramidal geometry of the oxonium ion (15,16). The first ab initio determination of the barrier in H502+ appeared in 1970 (10). An attempt was made to study the effect of Cl on this barrier (11) and the abnormal polarizability of H502+ (12). At the end of this decade appeared the first Cl ab initio calculation on the excited states of H30+ (19) and the first CNDO calculations on excited states of larger clusters (20). In parallel to these quantum chemistry studies, a kinetic model (21) treats large systems with n=20 and 26, a polarisation model (22) is proposed, and a study on the liquid uses a continuum model (23). 

Continuum models are used almost exclusively in this book and are adequate down to the microscale. They fail on the nanoscale where it becomes necessary to directly address the particulate nature of matter. On the nanoscale, the diffusion times and Reynolds numbers shown in Table 16.1 become rather meaningless. Instead, the models must address the behavior of individual molecules. Thus, an alternative definition of the nanoscale is the scale at which continuum models must be replaced by molecular models. 

The formulation of the QM continuum models reduces to the definition of an Effective Hamiltonian, i.e. an Hamiltonian to which solute-solvent interactions are added in terms of a solvent reaction potential. This effective Hamiltonian may be obtained from the basic energetic quantity which has the thermodynamic status of free energy for the whole solute-solvent system and for this reason is called free energy functional, This energy... 

The description of the electrostatic solute-solvent interactions has represented, historically, the keystone of continuum models. Within the continuum electrostatic frameworks the solvent is substituted with a dielectric medium and the solute occupies a cavity C within the dielectric. The main aspects to consider is thus the definition of the macroscopic characteristics of the dielectric, i.e. the form of the dielectric constant and the definition of the cavity C. Following this analysis we can define different systems ... 

Trying to give a more theoretically-stated definition of continuum models, but still keeping the same conciseness of the original one reported above, we have to introduce some basic notions of classical electrostatics. 

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