Polarizable continuum methods

These problems were partially solved through the inclusion of multipole expansions in ellipsoidal cavities [23] or through the use of the polarizable continuum method... 

In this contribution we will first outline the formalism of the ONIOM method. Although ONIOM has not yet been applied extensively to problems in the solvated phase, we will show how ONIOM has the potential to become a very valuable tool in both the explicit and implicit modeling of solvent effects. For the implicit modeling of solvent, we developed the ONIOM-PCM method, which combines ONIOM with the Polarizable Continuum Method (PCM). We will conclude with a case study on the vertical electronic transition to the it state in formamide, modeled with several explicit solvent molecules. 

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. 

CCI4. In the latter case, the solvents were treated with a polarizable continuum method (Fig. 6). 

In their study on solvent effects on the properties of oligothiophenes, Fig. 9, Meng et calculated also the dipole moment for chains consisting of 2-6 thiophene units, either in the gas phase or in either w-hexane or water. The calculations were carried through using the B3LYP density-functional method for the solute and the polarizable continuum method for the solvent. As Table 30 shows, the solvent leads to an increase of the dipole moment, but in all cases a clear even-odd oscillatory pattern is identified. The latter can be related to the zigzag-like structure of the systems, cf. Fig. 9. 

TD-DFT) they calculated the transition energies and dipole moments for NMA both in vacuum and in an aqueous solution. Moreover, in the treatment of the solvent they compared two different approaches, i.e., a polarizable-continuum method (COSMO) and a supermolecule approach. For the latter, the authors performed molecular-dynamics calculations using a force-field model and, subsequently, extracted a cluster containing the solute and 3 water molecules that form hydrogen bonds to the solute. Averages over 90 such configurations were ultimately determined. 

Kundrat and Autschbach calculated the relative total energies of these in the solvent. They used density-functional calculations for the solute and a polarizable continuum method for the solvent. Moreover, they compared the results obtained with two different basis sets. From the relative total energies, Kundrat and Autschbach could calculate the populations for a temperature of 293 K using a Boltzmann distribution, and compare those with available experimental information, see Table 41. Subsequently, by calculating the optical rotation for each rotamer and a wavelength of 589.3 nm (that of the sodium D line) and weighting the results with the Boltzmann factors, they obtained the values for the optical rotations of the... 

In an earlier work, the same group had used Hartree-Fock calculations in calculating An for the most stable isomer of some chiral organic molecules in vacuum.In the more recent work, ° they extended the study in several directions. At first, they applied the B3LYP density-functional method, whereby also correlation effects were included. Second, they compared gas-phase results with those obtained for a solution, whereby they applied the polarizable-continuum method for the treatment of the solvent. And third, for some of the larger molecules they included the effects of having a mixture of more different stable structures in an approach very similar to that we discussed above in section III I for the calculation of the optical rotation. 

For many chemical problems, it is crucial to consider solvent effects. This was demonstrated in our recent studies on the hydration free energy of U02 and the model reduction of uranyl by water [232,233]. The ParaGauss code [21,22] allows to carry out DKH DF calculations combined with a treatment of solvent effects via the self-consistent polarizable continuum method (PCM) COSMO [227]. If one aims at a realistic description of solvated species, it is not sufficient to represent an aqueous environment simply as a dielectric continuum because of the covalent nature of the bonding between an actinide and aqua ligands [232]. Ideally, one uses a combination model, in which one or more solvation shells (typically the first shell) are treated quantum-mechanically, while long-range electrostatic and other solvent effects are accounted for with a continuum model. Both contributions to the solvation free energy of U02 were... 

The effect of the solvent is usually modelled either by the use of the Onsager s self consistent reaction field (SCRF) [20] or by the polarizable continuum method (PCM) [21]. With regard to the relative stability of cytosine tautomers in aqueous solution, these methods provided results [14,15] which, in spite of some discrepancies, are in reasonable agreement with experimental data [3]. However, continuum-based methods do not explicitly take into consideration the local solvent-solute interaction which is instead important in the description of the proton transfer mechanism in hydrogen-bonded systems. A reasonable approach to the problem was recently proposed [22,23] in which the molecule of interest and few solvent molecules are treated as a supermolecule acting as solute, while the bulk of the solvent is represented as a polarizable dielectric. 

Key words Continuum models - Polarizable continuum method - Solvation... 

All the models above are finite in size and rely on explicit molecnlar additions subject to the same computational method as the species studied. Another way to take into account and model the real surrounding in an electrolyte is to use various continuum methods to implicitly mimic the effect of, e.g., the dielectric constant of the electrolyte. Popular since many years are different variants of the polarizable continuum methods (PCM) applicable to both ab initio and DFT methods and where parameters for a variety of different solvents exist, and with possibilities to tailor for special electrolytes. The use of continuum methods has demonstrated the importance of simulating solvent effects - especially the difference between the gas phase electronic energies and the free energies of solvation (AG) via PCM. The use of continuum methods can also be tweaked in various ways, e.g., in TD cycles to treat different dielectric constants for different parts of the cycle. 

As an example, values of the one- and two-electron reduction potential of pheophytin-a were calculated in N-N-dimethylformamide by a combination of quantum mechanical, statistical mechanical, and polarizable continuum methods (Mehta and Datta 2008). Two different computational methods gave -0.92 and -1.03 V for the one-electron, while -1.34 and -1.30 V for the two-electron reduction potential values, respectively. The observed one-and two-electron potentials are -0.90 and -1.25 V, respectively. Solution of the Poisson-Boltzmann equation gave for the reduction potential of pheophytin-a within the thylakoid membrane -0.58 V, which is in good agreement with the reported value of -0.61 V. 

The QM/MM method, and the polarizable continuum method as well, are usually considered as prototypical examples of the so-called multi-scale approaches. They combine two different description levels for the chemical system in both cases, a quantum part interacts with a classical part. Indeed, the QM/MM method can easily be extended to multi-scale schemes that include more than two description levels. Examples of three level schemes are the QM/MM/Continuum [47] and QM/QM7 MM approaches [48, 49]. In the later case, the system is divided into two QM parts, which may be described with the same or different methods, and a classical MM part. Dielectric continuum models for liquid interfaces are already available [43,50, 51] and a QM/MM/Continuum partition could be imagined in this case too, for instance to describe a solute-solvent cluster interacting with a polarizable dielectric medium. Here, however, we will focus on the QM/QM /MM partition. There is not a general scheme for this kind of approach and different algorithms can be employed to describe the interaction between subsystems. The main issue is the calculation of the interaction between two quantum subsystems that are described at QM (possibly different) theoretical levels. 

In the static approach the ground state equilibrium geometry is optimized, with solvent treated implicitly with a continuum method, for instance polarizable continuum method (PCM). Excited stales are then obtained as vertical transition form the equilibrium geometry... 

In summary, the supermolecule approach and polarizable continuum methods were used in concert to obtain solvation energies which take into account both specific solvent-solute effects and the electrostatic contribution to solvation. The... 

COSMO = conductor-like screening model PCM = polarizable continuum method QM/MM = quantum mecha-nics/molecular mechanics. 

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