Apparent surface charge methods

In the PCM procedure Vei is expressed in terms of an apparent charge distribution a(s) which is spread on the cavity surface (Apparent Surface Charge method, ASC) ... 

Ab-initio methods, 1 Analytical gradients (PCM-LRCC), 42 Apparent charges operator, 4, 9 Apparent surface charges (ASC), 3 Apparent surface charges method (ASC), viii Atomic orbitals (AO), 6... 

In the previous contributions to this book, it has been shown that by adopting a polarizable continuum description of the solvent, the solute-solvent electrostatic interactions can be described in terms of a solvent reaction potential, Va expressed as the electrostatic interaction between an apparent surface charge (ASC) density a on the cavity surface which describes the solvent polarization in the presence of the solute nuclei and electrons. In the computational practice a boundary-element method (BEM) is applied by partitioning the cavity surface into Nts discrete elements and by replacing the apparent surface charge density cr by a collection of point charges qk, placed at the centre of each element sk. We thus obtain ... 

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 ... 

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. 

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. 

PCM formulates the basic electrostatic problem with the aid of an apparent surface charge (ASC) spread on the surface of a cavity in the solvent where M is accommodated. There are several methods to treat the same electrostatic problem, we quote here a classification drawn from an exhaustive review on this subject to which reference is made for more details [8j. 

In this method, originally developed in 1981 but then almost completely redefined in 1995, the apparent surface charge is expressed by the following classical electrostatic relation ... 

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... 

K. Wiberg et al., J. Phys. Chem., 99,9072 (1995)], but other workers recommend other values [C.-G. Zhan and D. M. Chipman, J. Chem. Phys, 109,10543 (1998)]. Since the solute s electronic wave function changes in each SCRF iteration, the size of the molecular cavity changes in each IPCM iteration. In the IPCM method, is calculated from apparent surface charges. The self-consistmt isodensity PCM (SCIPCM) method is a refinement of the IPCM method Foresman and Frisch, Chapter 10), which allows geometry optimization and vibrational-frequency calculations to be done for the solute molecule in solution. 

This chapter explores an alternative category of methods aimed at solving the same continuum electrostatics problem using an apparent surface charge (ASC), ct(s), induced at the cavity surface by polarization of the medium. Here, we use s e F to denote a point on the cavity surface, F, whereas r e R. The quantity a (s) is determined from /o(r) as described in Section 11.2 but exists only on F. Thus... 

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. 

With respect to other QM continuum models, the PCM method represents of the interaction operator Vi l ) (i.e. of the solvent reaction potential Va) in terms of an apparent surface charge (ASC) charge distribution a spread on the boundary F of the cavity (C) hosting the solute M. 

Methods bcused on apparent surface charges These methods have become very popular mainly due to the seminal work of the Pisa group [21,40-51]. In these techniques the reaction field generated in the solvent by the presence of the solute is treated by a set of apparent charges spread over the solute cavity. At the classical level the electrostatic contribution to solvation is determined by equation 14, which is rigorously derived from Laplace and Poisson equations. 

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. 

Cammi, R., 8c Tomasi, J. (1995). Remarks on the use of the apparent surface charges (ASC) methods in solvation problems Iterative versus matrix-inversion procedures and the renormalization of the apparent charges. Journal of Computational Chemistry, 16,1449-1458. 

ASC = apparent surface charge BEM = boundary-element method CPHF = coupled perturbed Hartree-Fock C/RF = classical reaction field GBA = generalized Bom approximation FDM = finite-difference method FEM = finite-element method MPE = multipole expansion PD = potential derived SOS = sum over states SPT = scaled particle theory. 

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