QM/continuum approaches

The details on the operators introduced in the two schemes will be given below, here we only want to add that the addition of Henv to the solute Hamiltonian automatically leads to a modification of the solute wavefunction which has now to be determined by solving the effective Eq. (1-1). This can be done using exactly the same methods used for isolated molecules here in particular we shall mainly focus on the standard self-consistent field (SCF) approach (either in its Hartree-Fock or DFT formulation). Due to the presence of Hem the modified SCF scheme is generally known as self-consistent reaction field (SCRF). Historically the term SCRF has been coined for the QM/continuum approach but here, due the parallelism between the two schemes which will be made clear in the following sections, it will be used indistinctly for both. 

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 ... 

In the following section we shall show how all these specificities of the QM/MM and QM/continuum approaches will affect the quality of the description one can obtain applying them to the study of solvent effects on molecular response properties. 

The TDDFT-LR formulation has been generalized to both QM/MM and QM/continuum approaches. For example, in the PCM formulation of continuum models, the A and B terms in Eq. [5] become ... 

The analysis presented so far on the difrerent specificities of LR and SS descriptions of excitation processes within QM/continuum approaches also ap-phes to the polarizable QM/MM approaches. In those cases, however, the picture is simpler because there is no need to partition the polarization into dynamic and inertial terms as in continuum models, since the inertial (nuclear) degrees of freedom are considered expUcidy through the fixed multipolar expansion while the dynamic response is represented by the polarizable term, such as the induced dipoles in the ID formulation described earlier. 

The apphcation of QM/Continuum approaches to IR spectroscopy follows exacdy the same path commonly apphed to isolated molecules. The simulation of the spectra is obtained with two calculations one to obtain the equihbrium geometry of the solvated molecule and the other to compute the vibrational frequencies and intensities at the equihbrated geometry, as required by the standard harmonic approximation (HA). The frequencies (intensities) are calculated as the derivatives of the energy (dipole) with respect to nuclear displacements. The difference is that now the energy includes the solute—solvent interaction term and this imphes that in the calculation of its derivatives the variation of the molecular cavity has to be taken into account as it is generally anchored on the solute atoms. 

QM/MM approaches where the solute is QM and the solvent MM are in principle useful for computing the effect of the slow reaction field (represented by the solute point charges) but require a polarizable solvent model if electronic equilibration to the excited state is to be included (Gao 1994). With an MM solvent shell, it is no more possible to compute differential dispersion effects directly than for a continuum model. An option is to make the first solvent shell QM too, but computational costs for MC or MD simulations quickly expand with such a model. Large QM simulations with explicit solvent have appeared using the fast semiempirical INDO/S model to evaluate solvatochromic effects, and the results have been promising (Coutinho, Canute, and Zemer 1997 Coutinho and Canute 2003). Such simulations offer the potential to model solvent broadening accurately, since they can compute absorptions for an ensemble of solvent configurations. 

These averaging procedures introduce macroscopic parameters, temperature and density which are not present in the QM formulation of the problem given by the Hamiltonian of Equation (1.1). The use of macroscopic parameters is necessary for the description of molecular systems in a condensed phase, whether one uses a discrete or continuum approach. 

Molecular solutes described within QM continuum solvation models are characterized by an effective Hamiltonian which depends on the wavefunction of the solute itself. This makes the determination of the wavefunction a nonlinear QM problem. We have shown how the standard methods of modern quantum chemistry, developed for isolated molecules, have been extended to these solvation models. The development of QM continuum methods has reached a satisfactory stage for completely variational approaches (HF/DFT/MCSFC/VB). More progress is expected for continuous solvation model based on MP or CC wavefunction approaches. 

Within the QM continuum solvation framework, as in the case of isolated molecules, it is practice to compute the excitation energies with two different approaches the state-specific (SS) method and the linear-response (LR) method. The former has a long tradition [10-24], starting from the pioneering paper by Yomosa in 1974 [10], and it is related to the classical theory of solvatochromic effects the latter has been introduced few years ago in connection with the development of the LR theory for continuum solvation models [25-31],... 

The state-specific method solves the nonlinear Schrodinger equation for the state of interest (ground and excited state) usually within a multirefence approach (Cl, MCSCF or CASSCF descriptions), and it postulates that the transition energies are differences between the corresponding values of the free energy functional, the basic energetic quantity of the QM continuum models. The nonlinear character of the reaction potential requires the introduction in the SS approaches of an iteration procedure not present in parallel calculations on isolated systems. 

The quantities of interest in vibrational spectra are frequencies and intensities. Within the double harmonic approximation, vibrational frequencies and normal modes for solvated molecules are related, within the continuum approach, to free energy second derivatives with respect to nuclear coordinates calculated at the equilibrium nuclear configuration. The QM analogues for vibrational intensities , depend on the spectroscopy under study, but in any case derivative methods are needed. 

The approach which will be reviewed here has been formulated within the framework of the quantum mechanical polarizable continuum model (PCM) [7], Within this method, the effective properties are introduced to connect the outcome of the quantum mechanical calculations on the solvated molecules to the outcome of the corresponding NLO experiment [8], The correspondence between the QM-PCM approach and the semi-classical approach will also be discussed in order to show similarities and differences between the two approaches. 

By contrast, the description given by a continuum description does not require any knowledge of the solvent configuration around the solute as a structureless continuum dielectric is introduced instead. The response of such a dielectric to the presence of the solute is determined by its macroscopic properties (namely the dielectric constant and the refractive index) and thus it will be implicitly averaged. Contrary to what happens in a QM/MM approach, here a single calculation on a given solute contained within the continuum dielectric will be sufficient to get the correct picture of the solvated system. 

To try to reproduce the solvent effects on nitrogen nuclear shieldings of diazines and understand the physics beyond them we have applied the QM/continuum (PCM) and QM/MM(pol) (DPM) approaches described in the previous sections. 

In the QM/MM method the system is usually a priori divided into QM (the solute) and classical (MM, the solvent) parts, and an effective operator describes the interaction between the two subsystems. The solvent molecules are treated with a classical force field ( classical meaning that there are no elementary particles or quantum effects ) that opens the possibility to take a much larger number of solvent molecules into account. Optionally, the whole system can be embedded in a continuum, e.g., for taking large-range interactions into account. Similar to the continuum approach, the solute is separated from the solvent and its molecular properties are therefore well defined. The remaining problem is to find an accurate approximate representation of... 

NMR spectroscopy is a very useful tool for determining the local chemical surroundings of various atoms. Komin et al studied theoretically this for the adenine molecule of Fig. 20 both in vacuum and in an aqueous solution using different computational approaches. In all cases, density-functional calculations were used for the adenine molecule, but as basis functions they used either a set of localized functions (marked loc in Table 45) or plane waves (marked pw). Furthermore, in order to include the effects of the solvent they used either the polarizable continuum approach (marked PCM) or an explicit QM/MM model with a force field for the solvent and a molecular-dynamics approach for optimizing the structure (marked MD). In that case, the chemical shifts were calculated as averages over 40 snapshots from the molecular-dynamics simulations. Finally, in one case, an extra external potential from the solvent acting on the solute was included, too, marked by the asterisk in the table. 

The expressions of Vint which are now in use belong to two categories expressions based on a discrete distribution of the solvent, and expressions based on continuous distributions. The first approach leads to quite different methods. We quote here as examples the combined quantum me-chanics/molecular mechanics approach (QM/MM) which introduces in the quantum formulation computer simulation procedures for the solvent (see Gao, 1995, for a recent review), and the Langevin dipole model developed by Warshel (Warshel, 1991), which fits the gap between discrete and continuum approaches. We shall come back to the abundant literature on this subject later. 

The review is organized as follow. In section II we summarize the general basis of the QM continuum solvation models. In section III we present the formal aspects of the PCM, including the theory of the analytical derivatives of the energy. In section IV we present an overview of the PCM approach to the calculation of the properties of molecular solutes. 

Over the years, computational tools have been largely extended and developed to simulate IR and NMR spectroscopies of systems of increasing complexity and the QM/classical approaches are among the most popular ones especially in their continuum formulations. In particular, with the advent of DFT, hybrid DFT/continuum formulations have become the method of choice for simulating IR and NMR spectra of solvated systems. 

The QM-cluster and QM/MM approaches have been compared for some systems. A prime problem with the QM-cluster approach is that the selection of the QM system may be biased, because it is very hard to know whether all important residues have been included in the calculations and whether the continuum solvent is enough to model the surroundings. Several QM-cluster studies have shown alarming changes of the estimated energies as the QM model is increased, e.g. by 55 kj mol" for a minimal model and 27 kJ mol" for a 135-atom model, compared to a 220-atom model of aspartate... 

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