Apparent surface charge approach

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. 

A different set of dynamical variables can be given by the use of continuum models based on the apparent surface charge approach (ASC). In the PCM (Aguilar et a/., 1993b Cammi and Tomasi, 1995a) the set of coordinates is reduced to a discrete number, related to the cavity shape and... 

These methods combine a QM representation of solute with a classical continuum description of the solvent [18-23]. The methodology is equivalent to that of classical continuum methods, except that a) the solute charge distribution is allowed to relax by the solvent reaction field, and b) the solute-solvent interaction is computed at the QM level. Most QM continuum methods work within the multipole or apparent surface charge approaches, even though other formalisms are also available [18-23]. The solvent reaction field is introduced into the solute Hamiltonian by means of a perturbation operator (R in equation 22) that couples the solvent reaction field to the solute charge distribution. At this point, it is worth noting that equation 22 is not lineal, since T and R are mutually dependent. This means that a self-consistent process in which both the wavefunction and the reaction field are treated simultaneously is required to solve equation 22. This is the reason why these methods are typically known as self-consistent reaction field (SCRF) methods. 

The classical problem of electrostatics. The electrostatic interaction energy ms can be calculated in several ways. Historically at least the most important approach is based on the multipole expansion of the interaction integral in equation (4). Other methods are based on the apparent surface charge approach, the image charge approximation as well as finite difference and finite element based techniques. 

The bottleneck of a calculation in solution is the evaluation of the polarization which, in the case of PCM, corresponds to the evaluation of the apparent surface charges. In particular, the bottleneck is represented by the evaluation of the products between the integral matrices of the electrostatic potential (matrix S in Equation (1.8.6)) or of the normal component of the electric field (matrix D in Equation (1.92)) and the apparent charges vector q. Thus the criterion we use to compare the standard and the simultaneous approach is based on the number of matrix products (Sq or D q) necessary in the whole optimization process. We also remind the reader that the dimension of the matrices is equal to the square of the number of the surface elements. 

Most of the quantum chemical calculations of the nuclear shielding constants have involved two classes of solvation models, which belong to the second group of models (n), namely, the continuum group (i) the apparent surface charge technique (ASC) in formulation C-PCM and IEF-PCM, and (ii) models based on a multipolar expansion of the reaction filed (MPE). The PCM formalism with its representation of the solvent field through an ASC approach is more flexible as far as the cavity shape is concerned, which permits solvent effects to be taken into account in a more accurate manner. 

Moving now to QM/continuum approaches, we shall limit our exposition to the so-called apparent surface charges (ASC) version of such approaches, and in particular to the family known with the acronym PCM (polarizable continuum model) [11], In this family of methods, the reaction potential Vcont defined in Eq. (1-2) has a form completely equivalent to the Hel part of the Z/qm/mm operator defined in Eq. (1-4), namely ... 

We shall start from methods similar to that previously described, characterized by the use of the apparent surface charge (ASC) description of the electrostatic interaction term Ve/, passing then to consider other continuum methods, which use a different description of Ve/.To complete the exposition we shall introduce, where appropriate, methods not based on the solution of a Schrodinger equation, and hence not belonging to the category of continuum effective Hamiltonian methods. We shall pass then to a selection of methods based on mixed continuum-discrete representation of the solvent, to end up with the indication of some approaches based on a full discrete representation of the solvent. 

The following exposition will be focused only on the approach based on the apparent surface charges, and in particular in the PCM version of such... 

The apparent surface charge (ASC) approach appears to be a quite versatile method to calculate the reaction potential > (r), using either a quantum or a classical description of the solute molecule. According to classical electrostatics, the reaction potential can be described at any point in space in terms of an apparent charge distribution, a, spread on the cavity surface. Calling o-(s) the apparent charge per unit area, at a point s of the cavity surface E, one may write... 

Within the approaches based on the numerical integration of the electrostatic problem, the so called apparent surface charges (ASC) method, is by far computationally faster. 

Equation (11.7) can be used to eliminate the exterior derivative of (p from Eq. (11.6). Then, given some initial approximation for rp (perhaps just tpf, which is known once the solute s wave function has been computed), one could compute the surface charge, and thus the reaction-field potential, without the need to perform any calculations outside of the solute cavity. For a QM solute, this procedure must then be iterated to self-consistency. The original PCM of Miertus, Scrocco, and Tomasi [60, 61] used precisely this approach this model is now known as D-PCM. It is less desirable than more modern PC Ms, owing to the need to compute the normal electric field, which may be subject to increased numerical noise relative to later formulations that involve only electrostatic potentials [77]. Perhaps more significantly, the formulation of this model has conflated the apparent and actual surface charge distributions, and corresponds to a neglect of volume polarization [13]. 

We can use Eq. [94] to obtain a condition under which the Debye-Hiickel equation may be reliably applied to a planar system. As the ratio of the apparent charge density to the actual charge density approaches unity, the ADH surface potential approaches the PB (and DH) value. If we solve Eq. [94] for the ratio cja/cjo and, using Eqs. [27] and [11], insert this into the value of the surface potential given by Eq. [28], we obtain the expressions... 

Kjellander and Marcelja have also used their results to show that the traditional PB approach can describe ion distributions quite well if an apparent rather than actual surface charge is This prompted... 

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