Induced polarization, calculation

Continuum models of solvation treat the solute microscopically, and the surrounding solvent macroscopically, according to the above principles. The simplest treatment is the Onsager (1936) model, where aspirin in solution would be modelled according to Figure 15.4. The solute is embedded in a spherical cavity, whose radius can be estimated by calculating the molecular volume. A dipole in the solute molecule induces polarization in the solvent continuum, which in turn interacts with the solute dipole, leading to stabilization. 

The second part of this paper concerns the choice of the atomic basis set and especially the polarization functions for the calculation of the polarizability, o , and the hyperpo-larizabiliy, 7. We propose field-induced polarization functions (6) constructed from the first- and second-order perturbed hydrogenic wavefunctions respectively for a and 7. In these polarization functions the exponent ( is determined by optimization with the maximum polarizability criterion. These functions have been successfully applied to the calculation of the polarizabilities, a and 7, for the He, Be and Ne atoms and the molecule. 

This calculation has shown the importance of the basis set and in particular the polarization functions necessary in such computations. We have studied this problem through the calculation of the static polarizability and even hyperpolarizability. The very good results of the hyperpolarizabilities obtained for various systems give proof of the ability of our approach based on suitable polarization functions derived from an hydrogenic model. Field—induced polarization functions have been constructed from the first- and second-order perturbed hydrogenic wavefunctions in which the exponent is determined by optimization with the maximum polarizability criterion. We have demonstrated the necessity of describing the wavefunction the best we can, so that the polarization functions participate solely in the calculation of polarizabilities or hyperpolarizabilities. 

Since the parameters used in molecular mechanics contain all of the electronic interaction information to cause a molecule to behave in the way that it does, proper parameters are important for accurate results. MM3(2000), with the included calculation for induced dipole interactions, should model more accurately the polarization of bonds in molecules. Since the polarization of a molecular bond does not abruptly stop at the end of the bond, induced polarization models the pull of electrons throughout the molecule. This changes the calculation of the molecular dipole moment, by including more polarization within the molecule and allowing the effects of polarization to take place in multiple bonds. This should increase the accuracy with which MM3(2000) can reproduce the structures and energies of large molecules where polarization plays a role in structural conformation. 

Several issues remain to be addressed. The effect of the mutual penetration of the electron distributions should be analyzed, while the use of theoretical densities on isolated molecules does not take into account the induced polarization of the molecular charge distribution in a crystal. In the calculations by Coombes et al. (1996), the effect of electron correlation on the isolated molecule density is approximately accounted for by a scaling of the electrostatic contributions by a factor of 0.9. Some of these effects are in opposite directions and may roughly cancel. As pointed out by Price and coworkers, lattice energy calculations based on the average static structure ignore the dynamical aspects of the molecular crystal. However, the necessity to include electrostatic interactions in lattice energy calculations of molecular crystals is evident and has been established unequivocally. 

Ab-initio SCF calculations on the water molecule in various model complexes such as (H20)2, Li+H20, and (Li+)2 H20 show a depletion of the density of the lone pair in the internuclear region for long bonds, while for short bonds, such as Li+ O < a 1.6 A and O(H) O < 2.6-2.7 A, the effect is found to be reversed (Hermansson 1985), in accordance with the observations on oxalic acid dihydrate. The induced polarization of the acceptor density towards the Li+ or H atom is apparently still present for the longer distances, but very diffuse and below the lowest contour of most maps (D. Feil, private communication). Exchange repulsion opposing the attractive effect becomes important for larger ions such as K+, for which the oxygen lone pair penetrates the ion s electron... 

The induced polarization is important in the calculation of molecular properties, such as the hyperpolarizability discussed earlier in this chapter, and for the prediction of molecular packing and macromolecular folding. The diffraction... 

Figure 18. The centers of mass of the induced electronic charge (xe), and induced polarization charge (x,) as a function of the amount of induced surface charge, in units of 10-3 e/(a.u.)3. The black filled circles show the calculated values of xe, and the solid black line is a quadratic fit to these values. The open circles indicate the calculated center of mass of the xs (equal to the edge position of an equivalent, classical uniform dielectric), and the line labeled Exp. dielectric edge indicates where they would need to be in order to reproduce the experimental compact capacity. The line labeled Shifted Oxygen dist. is the position of the oxygen surface layer as a function of charge, shifted downward by 2.4 a.u. From Ref. 52, by permission.

The calculated first hyperpolarizabilities (see Table 2-4) are surprisingly close to the experimental data, which is probably fortuitous because they were calculated without taking into account vibrational effect. These studies demonstrated also that the double-zeta basis set augmented by field-induced polarization functions, although sufficient for calculations of dipole and quadrupole moments of the studied molecules at the Kohn-Sham LDA level, is not sufficient in the case of hyperpolarizabilities. 

Here a and d are the number of atoms in the acceptor and the donor, respectively, Ry is the distance between atoms i and j and and are the van der Waals and electrostatic potentials, respectively. The van der Waals potential is often represented by a Lennard-Jones potential (Eq. 8) or by a Buckingham potential (Eq. 9). The parameters a, fi, y and o are obtained from solid-state crystal data. The leading term in the electrostatic potential is the Coulomb interaction (first term in Eq. 10), where D is the effective dielectric constant (usually < D <2). Other terms may be added to represent induced polarization, etc. [40]. The geometries of the two components of the cluster are obtained from microwave or electron diffraction data or from quantum chemical calculations. It is assumed that these geometries do not change upon adduct formation. An initial guess is made for the structure of the adduct, and then the relative positions of the two (or more) components are varied until a local energy minimum is obtained. 

The hcc s obtained at the B3LYP/EPR-II level are shown in Table 12. The calculated hcc s can be dissected into three terms a contribution due to the electronic and structural configurations assumed by the radicals in the gas phase (first column in Table 12) a contribution due to the solvent-induced polarization on the solute wave function without allowing any relaxation of the gas-phase geometry (direct solvent effect, second coliunn in Table 12), and a last contribution due to the solvent-induced geometry relaxation (indirect solvent effect, third column in Table 12). 

If it is assumed that the TDDs are oriented preferentially along a <110> axis, with a C2V site symmetry, the piezospectroscopic results can be explained satisfactorily on the basis of the stress-induced line shifts and polarizations calculated by Kaplyanskii [73], which are discussed in the next section. This led to propose the C2V site symmetry for the TDDs in silicon [137]. In expression (8.15), the non-zero components of the piezospectroscopic tensor for C2v centres, labelled as orthorhombic (or rhombic) I, are Axx (A2), Ayy (A2), Azz Wi) and Axy = Ayx (A3). These orthorhombic I centres have a C2 symmetry axis in the <100> direction and the 1 s —> 2po transitions have their transition dipole moment oriented along this axis for a Is state constructed from a pair of valleys along this axis, while the Is —> 2p transitions have their transition dipole moment oriented in a plane perpendicular to this axis. In a cubic crystal, a C2V centre has a sixfold orientational degeneracy represented by the six diagonals of a cube (see Fig. 8.16a). 

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