QM/continuum

In our discussion the usual Born-Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. 

The structure of yint depends, in general, on the nature of the solute-solvent interaction considered by the solvation model. As already noted in the contribution by Tomasi, a good solvation model must describe in a balanced way all the four fundamental components of the solute-solvent interaction electrostatic, dispersion, repulsion, charge transfer. However, we limit our exposition to the electrostatic components, this being components of central relevance, also for historical reason, for the development of QM continuum models. This is not a severe limitation. As a matter of fact, the QM problem associated with the solute-solvent electrostatic component defines a general framework in which all the other solute-solvent interaction components may be easily collocated, without altering the nature of the QM problem [5],... 

As said before, the nonlinear nature of the effective Hamiltonian implies that the Effective Schrodinger Equation (1.107) must be solved by an iterative process. The procedure, which represents the essence of any QM continuum solvation method, terminates when a convergence between the interaction reaction field of the solvent and the charge distribution of the solute is reached. 

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. 

We end our contribution to this editorial initiative of the European Academy of Science by recalling the leading role played by the European quantum chemistry community in the development of quantum mechanical (QM) solvation models. We cite, as an example, the first quantum chemistry continuum solvation code developed by Rivail s group in Nancy in 1973 [43], However, this initiative to address the solvation problem from a QM point of view was not entirely out-of-the-blue. It was, on the contrary, a response to the challenges stimulated by a limited number of scientists working in France and in Italy (Paris, Nancy, Pisa), with strong contact, in competition and also in collaboration to each others [44], More than 30 years later, the European quantum chemistry community is still at the forefront in the development of QM continuum solvation models, and we hope that the present contribution can be considered as a testimony of this activity. 

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 ... 

The brief descriptions given here for QM/MM(pol) (in its DPM formulation) and QM/continuum (in its PCM formulation) should make clear the parallelism of the two formulations both from a quantum-mechanical and a computational point of view. There are, however, fundamental differences which are worth being recalled here. 

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. 

In the previous sections we have briefly summarized the basic theory of QM/MM and QM/continuum methods showing their differences and similarities, now we can move on to describe their applications to the calculation of molecular response properties and the related spectroscopies for a generic 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. 

Before moving to the comparison between QM/continuum and QMZMM(pol) results, we briefly comment on differences between the previously QM-optimized clusters and the present MD-derived clusters. The hydrogen coordination number to... 

Figure 1-5. QM/continuum and QM/MM(pol) errors with respect to experiments for the N nuclear shielding of the three diazines...

This chapter reviews the recent progress of the TDDFT when coupled to quantum mechanical (QM) continuum solvation models. Although the discussion will be focused on a specific family of solvation models, namely the family of methods known with the acronym PCM (Polarizable Continuum Model) [6], most of the results can be straightforwardly extended to other classes of implicit solvation models [7, 8],... 

II) Self-Consistent Reaction Field (SCRF) Models (QM/Continuum)... 

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. 

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... 

In general, QM/classical models can be divided into two main families QM/continuum and QM/discrete models. They are both widely used and present different advantages and disadvantages. 

The QM/continuum models are characterized by a representation of the environment as a structureless dielectric, solely characterized by its macroscopic dielectric constant, e, which determines the environment polarization as a response to the presence of the In the most widespread of these... 

The QM/classical models introduced earUer are able to deal with this situation, by differentiating between fast and slow — or dynamical and inertial — environment responses. In particular, such a nonequihbrium formulation has been largely and successfully used in QM/continuum models where the different components of the polarization can be accounted for introducing a frequency-dependent dielectric function. 

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. 

The QM continuum solvation methods (QM/CSM) have a more simple physical and computational structure, as no explicit molecular degrees of freedom of the solvent enter into the calculation. The procedure is based on the definition of an effective Hamiltonian for the molecular solute, which is composed by the Hamiltonian of the isolated solute accompanied by a solute-solvent integral operators, with a nonlinear kernel, and describing the solute in the presence of the solvent reaction potential. The solution of the corresponding nonlinear Schrodinger equation, obtained at ab-initio QM level and with an iterative procedure, determines the properties of the molecular solute in the presence of the solvent, with a complete description of the solvent effects. 

As other QM continuum models, the PCM model requires the solution of two coupled problems an electrostatic classical problem for the determination of the solvent reaction potential Va induced by the total charge distribution and a quantum mechanical problem for the determination of the wavefunction I of the solute described by the effective QM Hamiltonian (1.1). The two problems are nested and they must be solved simultaneously. 

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