Semi-empirical polarizable

The molecular electronic polarizability is one of the most important descriptors used in QSPR models. Paradoxically, although it is an electronic property, it is often easier to calculate the polarizability by an additive method (see Section 7.1) than quantum mechanically. Ah-initio and DFT methods need very large basis sets before they give accurate polarizabilities. Accurate molecular polarizabilities are available from semi-empirical MO calculations very easily using a modified version of a simple variational technique proposed by Rivail and co-workers [41]. The molecular electronic polarizability correlates quite strongly with the molecular volume, although there are many cases where both descriptors are useful in QSPR models. 

Polarizabilities and hyperpolarizabilities have been calculated with semi-empirical, ah initio, and DFT methods. The general conclusion from these studies is that a high level of theory is necessary to correctly predict nonlinear optical properties. 

This result indicates that in strictly theoretical calculations, the f functions may almost as well be omitted unless they can be optimized for the London energy itself. For the purpose of semi-empirical calculations, however, the /A functions from the polarizability must be retained for the substitution in the London energy. The error for hydrogen atoms is only about 4 per cent, however, and there does not appear to be any reason that it would increase greatly in more complex systems. 

Conceptually, the self-consistent reaction field (SCRF) model is the simplest method for inclusion of environment implicitly in the semi-empirical Hamiltonian24, and has been the subject of several detailed reviews24,25,66. In SCRF calculations, the QM system of interest (solute) is placed into a cavity within a polarizable medium of dielectric constant e (Fig. 2.2). For ease of computation, the cavity is assumed to be spherical and have a radius ro, although expressions similar to those outlined below have been developed for ellipsoidal cavities67. Using ideas from classical electrostatics, we can show that the interaction potential can be expressed as a function of the charge and multipole moments of the solute. For ease... 

Fig. 2.2 Self-Consistent Reaction Field (SCRF) model for the inclusion of solvent effects in semi-empirical calculations. The solvent is represented as an isotropic, polarizable continuum of macroscopic dielectric e. The solute occupies a spherical cavity of radius ru, and has a dipole moment of p,o. The molecular dipole induces an opposing dipole in the solvent medium, the magnitude of which is dependent on e.

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. 

Molecular dynamic studies used in the interpretation of experiments, such as collision processes, require reliable potential energy surfaces (PES) of polyatomic molecules. Ab initio calculations are often not able to provide such PES, at least not for the whole range of nuclear configurations. On the other hand, these surfaces can be constructed to sufficiently good accuracy with semi-empirical models built from carefully chosen diatomic quantities. The electric dipole polarizability tensor is one of the crucial parameters for the construction of such potential energy curves (PEC) or surfaces [23-25]. The dependence of static dipole properties on the internuclear distance in diatomic molecules can be predicted from semi-empirical models [25,26]. However, the results of ab initio calculations for selected values of the internuclear distance are still needed in order to test and justify the reliability of the models. Actually, this work was initiated by F. Pirani, who pointed out the need for ab initio curves of the static dipole polarizability of diatomic molecules for a wide range of internuclear distances. 

The polarizability radial functions obtained in this work can be used for the construction of semi-empirical PES necessary for molecular dynamics studies. 

A precise quantitative theory of the mutual polarization of ions in molecules is not possible as long as one cannot take into account the inhomogeneity of the field of the polarizing ion and the dependence of the polarizability of the polarized ion on its surroundings. It is therefore attempted to correlate the observed dependence of the p values on r and the polarizability of the ions in a semi-quantitative and semi-empirical fashion. This proves to be successful for the alkali fluorides but explains only qualitatively why the degree of polarity is smaller for BaO than for SrO. 

A Semi-quantitative and Semi-empirical Approach 2.3.1. Constant Polarizabilities... 

Solvation Effects. Many previous accounts of the activity coefficients have considered the connections between the solvation of ions and deviations from the DH limiting-laws in a semi-empirical manner, e.g., the Robinson and Stokes equation (3). In the interpretation of results according to our model, the parameter a also relates to the physical reality of a solvated ion, and the effects of polarization on the interionic forces are closely related to the nature of this entity from an electrostatic viewpoint. Without recourse to specific numerical results, we briefly illustrate the usefulness of the model by defining a polarizable cosphere (or primary solvation shell) as that small region within which the solvent responds to the ionic field in nonlinear manner the solvent outside responds linearly through mild Born-type interactions, described adequately with the use of the dielectric constant of the pure solvent. (Our comments here refer largely to activity coefficients in aqueous solution, and we assume complete dissociation of the solute. The polarizability of cations in some solvents, e.g., DMF and acetonitrile, follows a different sequence, and there is probably some ion-association.)... 

The intensities of the infrared absorptions and of the inelastic scattered light (Raman) are determined by such electrical factors as dipole moments and polarizabilities. At the time of the pioneering studies on the infrared spectra of carbohydrates by the Birmingham school,7"11 calculations of the vibrational frequencies had been performed only for simple molecules of fewer than ten atoms.27,34,35 However, many tables of group frequencies, based on empirical or semi-empirical correlations between spectra and molecular structure, are available.32,34"37... 

Both the classical and quantum approaches ultimately lead to a model in which the polarizability is related to the ease with which the electrons can be displaced within a potential well. The quantum mechanical picture presents a more quantitative description of the potential well surface, but because of the number of electrons involved in nonlinear optical materials, theoreticians often use semi-empirical calculations with approximations so that quantitative agreement with experiment is not easily achieved. 

This method was first proposed by Cohen and Roothaan730 for the calculation of atomic polarizabilities, and used soon afterwards by Schweig74 within a semi-empirical framework for large conjugated dye molecules. 

Calculations of Higher Polarizabilities and Semi-empirical Calculations.—O Hare and Hurst77 have calculated the first hyperpolarizabilities of some first-row diatomics by the uncoupled method. They find that a, and to a greater extent 0, are very sensitive to the basis set. Table 8 shows the effect of basis set on a for LiCl. Gupta et al.7 have... 

Abdulnur84 has given upper and lower bounds to the various C coefficients in terms of various sum rules that are available either experimentally, semi-empirically, or theoretically. Similar rules have previously been given, but the present ones give narrower bounds than are obtained by the application of corresponding methods to the imaginary frequency polarizabilities. 

Depending on the type of interaction between an adsorbed particle and a solid state surface there are cases, where adsorption enthalpies can be calculated using empirical and semi-empirical relations. In the case of atoms with a noble-gas like ground-state configuration and of symmetrical molecules the binding energy (EB) to a solid surface can be calculated as a function of the polarizability (a), the ionization potential (IP), the distance (R) between the adsorbed atom or molecule and the surface, and the relative dielectric constants (e) (Method 9) [58-61] ... 

We discussed DRF in perspective with other methods, gave the theoretical background and addressed the implementation. In a short section on the validation of DRF we showed that we can treat a system with QM, MM or QM/MM without significant loss of accuracy. A set of examples of its application ranges from simple solvation energies, spectra to (hyper)polarizabilities and processes of excited states of molecules in solution. These examples employ DRF in combination with—ab initio or semi-empirical—conventional wave function and DFT techniques. 

Some other cross-conjugated structures of the type [131] have been characterized for their third-order polarizabilities (Bosshard et al., 1996). Computations have been performed for the tetra-substituted type [98] (Nalwa et al., 1995 Tomonari et al., 1997) at the semi-empirical and ab initio level. 

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