The IEF version of the PCM solvation method: an overview of a new method addressed to study molecular solutes at the QM ab initio level

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Abstract

The integral equation formalism (IEF) is a recent method (the grounds have been elaborated at the beginning of 1997) addressed to solve the electrostatic solvation problem at the QM level with the aid of apparent surface charges (ASC). IEF uses a new formalism of this problem, based on integral operators never used before in the chemical community and it manages to treat on the same footing linear isotropic solvent models, as well as anisotropic liquid crystals and ionic solutions. In this overview we emphasize the good performances of IEF at the lowest level of its potentialities, i.e. for isotropic solvents, as a new approach to compute solvation free energies and properties (dipole hyperpolarizabilities) of molecular solutes, as well as energy gradients for geometry optimization procedures. Finally we present a new IEF implementation of the nonequilibrium problem for electronic spectra which appears to be decidedly competitive with the previous more standard ASC formulations.

Introduction

Accurate ab initio quantum mechanical (QM) descriptions are necessary when one has to study properties of molecules immersed in a solvent. Even simple problems as the evaluation of the equilibrium constant of a process A⇌B often require an accurate description of the electronic wave function, and then of the energy, of both partners. The numerical experience collected in the last years [1], [2] indicates that in dealing with chemical equilibria in solution an accurate description of both energetic terms, the internal energy of the solute and the solvation energy, are necessary.

In passing to more complex problems that cannot be limited to the comparison of the energy of two separate molecules, both at the equilibrium, the need of accurate QM descriptions is even more compelling. Studies on reaction mechanisms, on spectroscopic phenomena, on the response properties of the solute molecules are examples of problems now within reach of the available computational methods, that definitely require a good description of the molecular system, as it is modified by interactions with the solvent.

At present the highest level of accuracy is given by methods using effective Hamiltonians in which the portion of matter of direct interest (called here the solute M) is described at a good level of accuracy while an effective potential Vint describing solute–solvent interactions is kept at a level of complexity not preventing the use of an accurate QM description of the solute. There are now numerous methods based on the effective Hamiltonian (for a review, see Tomas and Persico [7]); we quote here those elaborated by Rivail et al. [4], van Duijnen et al. [5], by Zerner et al. [6] and by Cramer and Truhlar [1], which have been all amply tested on problems of different nature.

The problem of balance between the level of accuracy in the description of the two components of the effective Hamiltonian, namely the molecular Hamiltonian H0 and the interaction potential Vint, is very delicate. Too detailed descriptions of the solvent, trying to take into explicit account the molecular discreteness of the medium (in an averaged distribution for equilibrium properties or with appropriate time-dependent distributions for nonequilibrium properties) rise the computational costs so rapidly to make impossible the use of accurate enough QM levels for M. There is very active work to improve methods of this kind (as the so-called combined QM/MM method [3]) but the task is formidable and at present the largest amount of computational results of good quality comes from another approach.

In this second approach the detailed microscopic description of the solvent is replaced by a continuous distribution, characterized by appropriate properties of global or local character. In these methods the problem of balance in the accuracy of the two components assumes a different aspect. The continuous description must be well tailored on the characteristics of the specific case under examination, with an appropriate tuning of all the elements of the model.

From the first QM elaborations of this model (about 20 years ago) it was clear that the continuous solvent distribution must describe with good accuracy the cavity, i.e the portion of space where the solute is accommodated. Numerical evidence clearly shows that this is a critical parameter: with simple cavities the calculations are simpler but the accuracy sharply decreases.

The use of cavities with irregular shape, modeled on the solute shape, imposes further constraints on the elaboration of Vint to be included in the effective Hamiltonian.

Let us consider, for clarity, just one component of the most used phenomenological partition of Vint: the electrostatic component Vel. This component—the most important for polar systems— may be defined by resorting to the solution of an electrostatic problem in which the dielectric properties of the continuous solvent distribution are exploited. There is a number of models addressed to problems of different type (later, we shall consider some among them): the simplest describes the solvent as a continuous Isotropic distribution with a Linear Dielectric response (ILD model). There are many strategies to solve the ILD solvation problem, but the combination of an irregular cavity with QM calculations of good quality makes extremely convenient the use of apparent surface charges (ASC). With this approach the solution of the combined PoissonΔV=−4πρand LaplaceΔV=0equations inside and outside the cavity can be reduced to an integral two dimensional problem on a close surface, generally solved with the aid of the boundary element method (BEM).

The ASC formulation was first inserted into an ab initio QM code about 20 years ago, with the formulation of the Polarizable Continuum Method (PCM) [8]. The same method is still used here. PCM was rapidly extended to other models of liquid systems (always keeping attention to maintain the necessary degree of accuracy in describing the solute distribution and the solvation effects) and to a large set of properties.

Some extensions required however to abandon the simple ASC approach to describe electrostatic effects: we quote here the case of liquid crystals we shall consider later. The good performances of the simple isotopic model introduced here encouraged us, and a number of other researchers, to elaborate more efficient and more extended computer codes based on this approach. Many new items have been added to PCM in the years and especially in recent times: an overview of the progresses done in the preceding six months has been elaborated in June 1997, and now, five months later, we consider that there is sufficient material to present another overview, limited to the performances of a single new procedure we have developed, the Integral Equation Formalism (IEF).

IEF has been first proposed [9], [10], [11] to reduce two solution models to an ASC formulation, namely those of a solute immersed in a liquid crystalline phase or in a salt solution, both requiring in their straightforward formulation an integration over the whole space.

We soon realized that there is convenience to use IEF instead of other standard or non standard PCM formulations for many properties of solutes, also in the ILD model. In the following pages we shall emphasize this aspect, presenting in a concise way those specific characteristics of IEF which induce such positive performances.

Section snippets

Basic aspects of the PCM procedure

It is convenient to summarize the basic aspects of the PCM procedure (thermodynamic definition of the system and of its energy, phenomenological partition of the solute solvent interaction, etc.) before passing to a more detailed presentation of the IEF procedure, compared, where necessary, with standard PCM or with other PCM versions.

The system we are now considering is an infinitely dilute solution at equilibrium. The primary energetic quantity to be considered is a free energy (Gibbs or

The electrostatic contribution

We examine here with more details the electrostatic problem (VintVel) as expressed in the IEF method.

Acknowledgements

The authors acknowledge the ‘Consiglio Nazionale delle Ricerche’ (CNR) for financial support.

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