Discrete polarizable method

The second term in Eq. (1-3) is the polarization interaction between the induced dipole moments and the electric field from the QM system. In the following exposition, the approach proposed by Mikkelsen, Kongsted and coworkers will be used [12,13,14] for such a version of the polarizable QM/MM scheme the acronym DPM (discrete polarizable method) has been introduced and will be used here. In the DPM Hpo1 can be expressed as... 

The theoretical methods can be divided into two fundamental groups. The so-called continuum models are characterized by assuming that the medium is a structureless and polarizable dielectricum described only by macroscopic physical constants. On the other hand there are the so-called discrete models. The main advantage of... 

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 ... 

It should be stressed that the relation between the SCRF and SAPT approaches is not obvious, as the former describes the solvation energetics in terms of the free energy of solvation at a finite temperature T, while in the latter one considers the interaction energy between the molecule of the solute and all molecules of the solvent at T = 0 K. One should also note that in the SCRF theory the solvent is modeled by a polarizable continuum, so the SCRF Hamiltonian is semiempirical. Still, by assuming a discrete equivalent of the SCRF Hamiltonian one can get approximate relations between SAPT and SCRF at T = 0 K. A SAPT analysis of the free energy of solvation AG within the SCRF method was reported in Ref. (234). It was shown that the free energy of solvation AG is given by,... 

Discrete dipole approximation. For particles with complex shape and/or complex composition, presently the only viable method for calculating optical properties is the discrete dipole approximation (DDA). This decomposes a grain in a very big number of cubes that are ascribed the polarizability a according to the dielectric function of the dust material at the mid-point of a cube. The mutual polarization of the cubes by the external field and the induced dipoles of all other dipoles is calculated from a linear equations system and the absorption and scattering efficiencies are derived from this. The method is computationally demanding. The theoretical background and the application of the method are described in Draine (1988) and Draine Flatau (1994). 

Method (III) attempts to include a discrete molecular description of the solvent structure around a central solute molecule. The solute molecule is described, again, through a QM calculation while the spatial distribution, charge distribution, polarizabilities etc. of the adjacent part of the solvent is represented by a parametrized molecular mechanics (MM) model. The parametrization may be achieved through high level calculations on the isolated solvent molecules. 

For a long time the finite oligomer approach was the only method available for determining linear and nonlinear polarizabilities of infinite stereoregular polymers. Recently, however, the problem of carrying out electronic band structure (or crystal orbital) calculations in the presence of static or frequency-dependent electric fields has been solved [115, 116]. A related discretized Berry phase treatment of static electric field polarization has also been developed for 3D solid state systems... 

The polarizable continuum model (PCM) by Tomasi and coworkers [77-79] was selected to describe the effects of solvent, because it was used to successfully investigate the effect of solvent upon the energetics and equilibria of other small molecular systems. The PCM method has been described in detail [80]. The solvents and dielectric constants used were benzene (s = 2.25), methylene chloride (g = 8.93), methanol (g = 32.0), and water (g = 78.4). Full geometry optimizations were carried out for the discrete and PCM models. To simultaneously account for localized hydrogen bonding and bulk solvation effects, PCM single-point energy calculations have been conducted on stationary points of the acrolein and butadiene reaction with two waters explicitly... 

Methods based on an ASC have a long history in quantum-mechanical (QM) calculations with continuum solvent [60, 61, 77], where they are generally known as polarizable continuum models (PCMs). However, PCMs have seen little use in the area of biomolecular electrostatics, for reasons that are unclear to us. In the QM context, such methods are inherently approximate, even with respect to the model problem defined by Poisson s equation, owing to the volume polarization that results from the tail of the QM electron density that penetrates beyond the cavity and into the continuum [13, 14, 89], The effects of volume polarization can be treated only approximately within the ASC formalism [14, 15, 89], For a classical solute, however, there is no such tail and certain methods in the PCM family do afford a numerically exact solution of Poisson s equation, up to discretization errors that are systematically eliminable. Moreover, ASC methods have been generalized to... 

An alternative to the use of finite differences or finite elements to discretize the differential operator is to use boundary element methods (BEM). " One of the most popular of these is the polarizable continuum model (PCM) developed originally by the Pisa group of Tomasi and co-workers. The main aspect of PCM is to reduce the electrostatic Poisson equation (1) into a boundary element problem with apparent charges (ASCs) on the solute cavity surface. 

Several studies have shown that solvent methods based on the polarizable continuum model (PCM) [23] meet successfully those requirements. They provide accurate predictions of the effect of the polarity of the solvent on the magnitude of the magnetic parameters in nitroxides and in other classes of organic free radicals. Contemporarily they allow taking into account the effect of solvent molecules specifically bound to the nitroxide through mixed discrete-continuum approaches [24]. 

The analysis of the properties of mesityl oxide in aqueous solution was based on the continuous and discrete models of the solvent. Here, we used the polarizable continuum model (PCM) [32, 33], and for the discrete model of the solvent, we performed the sequential use of quantum mechanics and molecular mechanics methods, S-QM/MM [20, 21]. In the S-QM/MM procedure, initially the liquid-phase configurations are sampled from molecular simulations, and after statistical analysis, only configurations with less than 10 % of statistical correlation are selected and submitted to quantum mechanical calculations. In our study, we used the Monte Carlo method (MC) with... 

These methods treat the solute at the classical discrete level, and the solvent is represented as a classical continuum dielectric. The free energy of solvation is computed as the addition of steric and electrostatic components (equation 2). The steric term can be partitioned into cavitation and van der Waals contributions, while some authors determine it as a whole from empirical relationships with the molecular surface (see refs. 20,21 and 23 for a more detailed explanation). The electrostatic contribution is computed from the classical theory of polarizable fluids [19-23,26], which assumes that the solvent is a dielectric continuum which reacts against the solute charge distribution. 

These methods combine a QM description of the solute with a classical treatment of the solvent, which can be represented as a polarizable continuum (SCRF methods) or as discrete classical particles (QM/MM methods). In both cases the solute wavefunction is allowed to relax by the effect of the solvent reaction field, which makes possible to account for polarization effects. Furthermore, changes in molecular properties induced by solvent can be easily determined from the wavefunction of the solute in solution, which is a clear advantage with respect to pure classical methods. 

Van Duijnen et al. in a set of papers have developed a version of a QM/classical approach in which classical solvent (water) molecules are treated as point polarizable dipoles. The portion of space with discrete water molecules is kept small and surrounded by a continuum dielectric. More attention is paid here to boundary conditions. The method makes use of a direct reaction field (RF) (in contrast with almost all other continuum methods which use an averaged RF) the average is given later with the aid of MC calculations, where the classical particles are also provided with a repulsive potential to avoid collapse of the particles. Not many details are given, however. 

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