Semi-continuum model

DR. P. P. SCHMIDT, (Oakland University) I m working on lithium. That s a pretty simple system, especially without its single 2s electron. And yet it s still complicated. There are several problems. One is how to model the effect of the solvent and whether, in fact, you want to throw away the modeling that has been used, namely the continuum approximation. I think the answer to that is no, you don t want to throw it away completely. But one of the notions which has been developing, and is now being tested in the thermodynamics of solvation and other problems, is what is called the semi-continuum model. In... 

Fueki K, Feng D-F, Kevan L. (1974) Application of the semi-continuum model to temperature effects on solvated electron spectra and relaxation rates of dipole orientation around an excess electron in liquid alcohols. J Phys Chem 78 393-398. 

The model using spherical harmonics expansions for the RF potential can be derived from Eq. (26) by introducing spherical boundary conditions. The procedure has already been outlined by this author [6] and will not be repeated here. Semi-continuum models. In this type of approach, the first solvation shell is represented in the supermolecule and, consequently, enters into the quantum chemical description. Basically, the radius of the sphere embedded in the continuum dielectric is much larger than for the desolvated solute. This model has been used in several occasions, e.g. solvated electron, electron transfer in solution. 

In the IPCM calculations, the molecule is contained inside a cavity within the polarizable continuum, the size of which is determined by a suitable computed isodensity surface. The size of this cavity corresponds to the molecular volume allowing a simple, yet effective evaluation of the molecular activation volume, which is not based on semi-empirical models, but also does not allow a direct comparison with experimental data as the second solvation sphere is almost completely absent. The volume difference between the precursor complex Be(H20)4(H20)]2+ and the transition structure [Be(H20)5]2+, viz., —4.5A3, represents the activation volume of the reaction. This value can be compared with the value of —6.1 A3 calculated for the corresponding water exchange reaction around Li+, for which we concluded the operation of a limiting associative mechanism. In the present case, both the nature of [Be(H20)5]2+ and the activation volume clearly indicate the operation of an associative interchange mechanism (156). 

The most common approach to solvation studies using an implicit solvent is to add a self-consistent reaction field (SCRF) term to an ab initio (or semi-empirical) calculation. One of the problems with SCRF methods is the number of different possible approaches. Orozco and Luque28 and Colominas et al27 found that 6-31G ab initio calculations with the polarizable continuum model (PCM) method of Miertius, Scrocco, and Tomasi (referred to in these papers as the MST method)45 gave results in reasonable agreement with the MD-FEP results, but the AM1-AMSOL method differed by a number of kJ/mol, and sometimes gave qualitatively wrong results. 

The MPE/MCSCF approach has been employed to study the interplay of solvent and conformation effects on the spin-spin coupling constants in methanol and methylamine [72], The simulated solvent effects are noticeable for the one-bond coupling constants and for some of the geminal coupling constants but negligible for 3/ee. The dielectric continuum effects have been found to depend considerably on the molecular conformation in the case of and 27ece. It is worth noting here that the MCSCF results have confirmed the conclusions drawn in ref. [80] from semi-empirical continuum model calculations. 

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. 

The most expensive part of a simulation of a system with explicit solvent is the computation of the long-range interactions because this scales as Consequently, a model that represents the solvent properties implicitly will considerably reduce the number of degrees of freedom of the system and thus also the computational cost. A variety of implicit water models has been developed for molecular simulations [56-60]. Explicit solvent can be replaced by a dipole-lattice model representation [60] or a continuum Poisson-Boltzmann approach [61], or less accurately, by a generalised Bom (GB) method [62] or semi-empirical model based on solvent accessible surface area [59]. Thermodynamic properties can often be well represented by such models, but dynamic properties suffer from the implicit representation. The molecular nature of the first hydration shell is important for some systems, and consequently, mixed models have been proposed, in which the solute is immersed in an explicit solvent sphere or shell surrounded by an implicit solvent continuum. A boundary potential is added that takes into account the influence of the van der Waals and the electrostatic interactions [63-67]. 

There have been many efforts for combining the atomistic and continuum levels, as mentioned in Sect. 1. Recently, Santos et al. [11] proposed an atomistic-continuum model. In this model, the three-dimensional system is composed of a matrix, described as a continuum and an inclusion, embedded in the continuum, where the inclusion is described by an atomistic model. The model is validated for homogeneous materials (an fee argon crystal and an amorphous polymer). Yang et al. [96] have applied the atomistic-continuum model to the plastic deformation of Bisphenol-A polycarbonate where an inclusion deforms plastically in an elastic medium under uniaxial extension and pure shear. Here the atomistic-continuum model is validated for a heterogeneous material and elastic constant of semi crystalline poly( trimethylene terephthalate) (PTT) is predicted. 

In the seventies, most of the 37 papers (8-24) that we report are quantum chemical calculations, mainly on H502+ (8-14,20) or H30+(14-20) and a few on larger clusters with n=4-6 (8,9). However these last calculations are not accurate, obtained either from semi-empirical methods (8) or with small basis sets (DZ, 4-31G) and at the SCF level in ab initio calculations (9). The first accurate Cl calculations definitely establish the pyramidal geometry of the oxonium ion (15,16). The first ab initio determination of the barrier in H502+ appeared in 1970 (10). An attempt was made to study the effect of Cl on this barrier (11) and the abnormal polarizability of H502+ (12). At the end of this decade appeared the first Cl ab initio calculation on the excited states of H30+ (19) and the first CNDO calculations on excited states of larger clusters (20). In parallel to these quantum chemistry studies, a kinetic model (21) treats large systems with n=20 and 26, a polarisation model (22) is proposed, and a study on the liquid uses a continuum model (23). 

Classical and semi-classical theories of electron transfer provide quantitative models for determining the reaction pathway. Of particular importance is the theory of nonequilibrium solvent polarization based on the dielectric continuum model.5 From these theories Eqs. 

The most primitive model that takes solvent effect into account is (dielectric) continuum model. A treatment combined with a quantum chemical calculations including semi-empirical methods began in early 1970 s. In this section we summarize the continuum methods applied to quantum chemical calculations. 

Balbuena et al. (1998) studied the reorientation times of water molecules over a range of temperatures including supercritical ones, employing semi-continuum molecular dynamics by means of the SPC/E water model (500 water molecules per ion). The reorientation times in bulk water relative to those in the first hydration shell, assuming a coordination number of Nco = 6, are Tiw/Tri = 0.20,0.47,0.65, and 0.90 for Na+, K+, Rb" ", and Cr, respectively, at 25 °C, showing faster reorientation as... 

But in principle one has to take into account the effect of the solvent on the HOMO energies in theoretical calculations. This is possible with the polarized continuum model (PCM) of solvents [275]. Furthermore, semi-empirical and ab initio HF calculations do not take into account the electron correlation energy Tcorr-To rectify this deficiency, one has to switch from semi-empirical or ah initio HF methods to density-functional theory (DFT) [277, 278] or pure post-HF methods like configuration interaction (Cl), many body perturbation theory (MBPT), or coupled cluster (CC) methods. 

For the simulation of more complex flows, one needs a constitutive equation or a rheological equation of state. Nearly all of the many equations that have been proposed over the past fifty years are basically empirical in nature, and only in the last twenty-five years have such models been developed on the basis of mean field molecular theories, e.g., tube models. Although the early models were often developed with a molecular viewpoint in mind, it is best to think of them as continuum models or semi-empirical models. The relaxation mechanisms invoked were crude, involving concepts such as network rupture or anisotropic friction without the molecular detail required to predict a priori the dependence of viscoelastic behavior on molecular structure. While these lack a firm molecular basis and thus do not have universal validity or predictive capability, they have been useful in the interpretation of experimental data. In more recent times, constitutive equations have been derived from mean field models of molecular behavior, and these are described in Chapter 11. We describe in this section a few constitutive equations that have proven useful in one or another way. More complete treatments of this subject are given by Larson [7] and by Bird et al. [8]. 

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