Polarizability matrix

The selection rules for Raman vibrational transitions are also readily derived from group theory. Here, the transition probability depends on integrals involving the components of the molecular polarizability matrix a. Since a is symmetric, it has only six independent components axx aw axi axr ayi aMf These six quantities can be shown14 to transform the same way the six functions... 

To complete the picture, calculating mutual polarizabilities relevant to the EHCF/ MM context can be found in [77]. The atom-atom mutual polarizability matrix II has a block-diagonal form ... 

The coefficient one-half at the diagonal interaction element in the above expression reflects the fact that in the HFR approximation for the closed electron shell system, only that half of the electron density residing at the a-th AO contributes to the energy shift at the same AO, which corresponds to the opposite electron spin projection. Then the expression for the renormalized mutual atomic polarizability matrix IIA can be obtained ... 

Molecular polarizability, a, is a measure of the ability of an external electric field, E, to induce a dipole moment, = aE, in the molecule. As such, it can be viewed as contributing to a model for induced dipole (dispersive) interactions in molecules. Because the polarizability is a tensor (matrix) quantity, there is the question of how to represent this in a scalar form. One approach is to use the average of the diagonal components of the polarizability matrix, (a x + otyy + Since the polarizability increases with size (and... 

Highest eigenvalue of bond-bond polarizability matrix (d ). )) Free valence (unitless), m) Localization energy for electrophilic attack. ") Localization energy for radical attack. o) Localization energy for nucleophilic attack. P) Energy of highest occupied molecular orbital. 

It has been shown [22] that by diagonalizing the bond-bond polarizability matrix, it is possible to ascertain the likely symmetry of distortion (eigenvectors) and the energy gain (eigenvalues, X) due to second-order Jahn—Teller distortion. In fact, the... 

By taking the Fourier transforms of the induced density, one is led to the following well-knownform for the polarizability matrix ... 

Since the density fluctuations are related to the potential (see Eq. 36) by the polarizability matrix, some elementary algebraic manipulations yield ... 

Summing this energy correction over the occupied states, one readily sees by interchanging I and m that pm in the last contribution at the r.h.s can be replaced by (pm - Pi)/2, and the summation over the states in this last contribution simply turns out to be half the polarizability matrix. The sum over the states in the first term at the r.h.s. is recognized to be the Fourier transform of the equilibrium density ... 

Since the dielectric formulation is not based on the weakness of the pseudopotential in the small wave vector region, one might be interested in examining the difference between both approaches for simple metals. Let us therefore calculate the polarizability matrix to second order in the pseudopotential. 

In terms of the wave functions of the valence electrons, the polarizability matrix is given by ... 

The matrix elements in the polarizability matrix can easily be evaluated ... 

Previously the present authors did not perform the summation over the conduction bands in the polarizability matrix but instead approximated this summation by means of a moment expansion. Results of this approximation for the phonon dispersion curves of Si have been published in the literature [2]. However in their present work presented in these proceedings the polarizability matrix is evaluated by means of a straightforward summation over all the conduction bands obtained from diagonali-zation of the Hamiltonian matrix. 

The present authors showed that it is very important to use a basis consisting of a sufficiently large number of plane waves. The summation over the conduction bands in the polarizability matrix should be performed using all bands calculated. In this way the first order wave functions are expanded in exactly the same basis as the one used for the expansion of the unperturbed wave function. 

The most recent of the "dire pj " methods concerng the evaluation of the elements of the e dielectric matrix it is based on tl fact that the modification in the electronic charge dmslty An(r) is related to the modification in total potential V (r) by the polarizability matrix x(q+G,q+G ). The method does ng fe u j.r achieving self-consistency and provides the elements of e j (q+G,q+G ) within the RPA approximation. After inversion of jthe e the combination of the two direct methods (for e and e ) seems to offer a possibility of switching on and off, at will, the exchange-correlation its effects on various physical properties could then be studied in detail. Beside tl original work Ref. 77,... 

The changes in molecular polarizability during vibrational transitions determine the intensities of Raman lines. The aj polarizability matrix element for a transition from a vibronic state m to a vibronic state n can be presented as follows [4,254-256]... 

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