Energy polarization

We have already introduced the concept of ionic polarizability (section 1.8) and discussed to some extent the nature of dispersive potential as a function of the individual ionic polarizability of interacting ions (section 1.11.3). We will now treat another type of polarization effect that is important in evaluation of defect energies (chapter 4). 

Let us imagine subtracting a — e charge at a given point in the crystal lattice at that point, effective charge +e exerts polarization energy of the type 

The solution of the system shown in equations 1.156 to 1.159 leads to a large number of equations (cf Rittner et al., 1949 Hutner et al., 1949), and simplifying 

Mineral Formula Calculated Experimental Calculated Experimental 

The terms in equation 1.166 represent total ionic polarizability, composed of electronic polarizability a plus an additional factor a , defined as a displacement term, due to the fact that the charges are not influenced by an oscillating electric field (as in the case of experimental optical measurements) but are in a static field (Lasaga, 1980)  

If the charge separation takes place in a solvent, there is a polarization around the acceptor molecule during and after PIET. The relative dielectric constant, e, appears in the equation for the stabilization energy. Immediately after CT, the relevant dielectric constant is n, which is the dielectric constant for electronic polarization. If the solvent molecules have a dipole moment, the polarization around the donated charge will further increase, given time. 

Let 8i be the total electric field exerted by surrounding molecules at pixel i, aj the polarizability at pixel i, and p.i the dipole induced at pixel i by that field. The linear polarization energy is  

There are obvious difficulties in the definition of the distributed polarizabilities Oj [10]. The expedient adopted in the Pixel formulation uses atomic polarizabilities taken from standard repertories [11] (Table 12.3), and takes ai = ( /Zatom) otatom, where 

Atom Atomic radius, A Atomic polarizability, Electronegativity Ionization potential I , au  

Zatom and aatom are the atomic charge and polarizability of the atom to whose basin the pixel belongs. This procedure is at least partially justified by the fact that the sum of distributed ai s is equal to the total volume polarizability of the molecule. 

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This quantity is found to be related to the local polarization energy and is complementary to the MEP at the same point in space, making it a potentially very useful descriptor. Reported studies on local ionization potentials have been based on HF ab-initio calculations. However, they could equally well use semi-empirical methods, especially because these are parameterized to give accurate Koopmans theorem ionization potentials. 

The and Oj terms always contribute, regardless of the specific electric charge distributions ia the adsorbate molecules, which is why they are called nonspecific. The third nonspecific Op term also always contributes, whether or not the adsorbate molecules have permanent dipoles or quadmpoles however, for adsorbent surfaces which are relatively nonpolar, the polarization energy Op is small. 

When two or more molecular species involved in a separation are both adsorbed, selectivity effects become important because of interaction between the 2eobte and the adsorbate molecule. These interaction energies include dispersion and short-range repulsion energies (( ) and ( )j ), polarization energy (( )p), and components attributed to electrostatic interactions. 

The environmental (i.e., solvent and/or protein) free energy curves for electron transfer reactions can be generated from histograms of the polarization energies, as in the works of Warshel and coworkers [79,80]. 

As an example, here is an output from Gaussian 98 on CH3F (Figure 17.2). I forced the finite field method by choice of Polar = Enonly (Polar = Energy only) in the route. The geometry was first optimized and stored in a checkpoint file. 

Using this model in analogy with previous studies we can calculate a magnetic moment (fi) of the system with fixed Stoner exchange parameter Id and occupation of the d states. The total energy could then be calculated as the balance between the kinetic energy and the spin-polarization energy ... 

The vibrational relaxation of simple molecular ions M+ in the M+-M collision (where M = 02, N2, and CO) is studied using the method of distorted waves with the interaction potential constructed from the inverse power and the polarization energy. For M-M collisions the calculated values of the collision number required to de-excite a quantum of vibrational energy are consistently smaller than the observed data by a factor of 5 over a wide temperature range. For M+-M collisions, the vibrational relaxation times of M+ (r+) are estimated from 300° to 3000°K. In both N2 and CO, t + s are smaller than ts by 1-2 orders of magnitude whereas in O r + is smaller than t less than 1 order of magnitude except at low temperatures. 

The main difference between a molecule-molecule (M-M) collision and an ion-molecule (M+-M) collision is the presence of a polarization force in the latter system owing to the attraction between the static charge on M+ and the dipole moment induced on M. For a large inter molecular separation, the polarization energy is known as... 

Table IV. Calculated Values of the Polarization Energy Term...

The hard-core limiting forms of U(r) do not lead to physically acceptable results. We conclude that this is caused by a complete neglect of the effect of the attractive forces on the slope of the repulsive part in U(r). If the interaction energy is assumed as the sum of a Morse exponential function and the polarization energy evaluated at r = r°, the resulting transition probabilities appear useful for analyzing ion-molecule collisions. 

The effect of polarization has been neglected. This is usually permissible, since polarization energies rarely amount to more than one or two volt-electrons. 

And actually the two-determinant function will be symmetry-broken for a symmetric configuration when the resonanee energy is weaker than the polarization energy. This has been observed first by Ellinger et al. [36] in a problem with three electrons in two... 

The general expression of the polarization energy at center I located at the centroid of an LMO of a A molecule is ... 

Eo and E (Afi(i)) are respectively the electric fields generated by the permanent and induced multipoles moments. a(i) represents the polarisability tensor and Afi(i) is the induced dipole at a center i. This computation is performed iteratively, as Epoi generally converges in 5-6 iterations. It is important to note that in order to avoid problems at the short-range, the so-called polarization catastrophe, it is necessary to reduce the polarization energy when two centers are at close contact distance. In SIBFA, the electric fields equations are dressed by a Gaussian function reducing their value to avoid such problems. 

The inducing field responsible for the energy of the induced dipoles, Umd, has contributions from three terms the permanent or static field, Ustat, the induced dipole-induced dipole interaction, Udip, and the polarization energy, Upou... 

Evaluation of the polarization energies for the multi-centered QM system therefore only requires the derivation of values for ak and ak,. In the case of a single atom of charge qk, ak can be approximated by a coulomb radius chosen such that ... 

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