The polarizability matrix

Assuming that the potential, acting upon an electron in the crystal, is known, one can examine the response of the electrons if the external 

From perturbation theory, it is well known that the first order correction to the wave function for an electron with unperturbed states 1 is given by  

In the following, a distinction has to be made between occupied and unoccupied states in the equilibrium configuration. Therefore we intro- 

By interchanging the indices I and m in the complex conjugated part, this expression is readily converted into  

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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... 

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,... 

Equation (A 1.6.94) is called the KHD expression for the polarizability, a. Inspection of the denominators indicates that the first temi is the resonant temi and the second temi is tire non-resonant temi. Note the product of Franck-Condon factors in the numerator one corresponding to the amplitude for excitation and the other to the amplitude for emission. The KHD fonnula is sometimes called the siim-over-states fonnula, since fonnally it requires a sum over all intennediate states j, each intennediate state participating according to how far it is from resonance and the size of the matrix elements that coimect it to the states i. and The KHD fonnula is fiilly equivalent to the time domain fonnula, equation (Al.6.92). and can be derived from the latter in a straightforward way. However, the time domain fonnula can be much more convenient, particularly as one detunes from resonance, since one can exploit the fact that the effective dynamic becomes shorter and shorter as the detuning is increased. 

In a normal Hartree-Fock job, the hyperpolarizability tensor is given only in the archive entry, in the section beginning HyperPolar=. This tensor is also in lower tetrahedral order, but expressed in the input (Z-matrix) orientation. (This is also true of the polarizability tensor within the archive entry.)... 

By making the approximation of setting matrix B to zero and A to Ao, the resolvent matrix becomes diagonal, every coupling between the crystal orbitals disappears and the polarizability reads ... 

Although a direct comparison between the iterative and the extended Lagrangian methods has not been published, the two methods are inferred to have comparable computational speeds based on indirect evidence. The extended Lagrangian method was found to be approximately 20 times faster than the standard matrix inversion procedure [117] and according to the calculation of Bernardo et al. [208] using different polarizable water potentials, the iterative method is roughly 17 times faster than direct matrix inversion to achieve a convergence of 1.0 x 10-8 D in the induced dipole. 

To calculate x( ) we have to calculate the polarizability P(t), which is related to the reduced density matrix p(f). [Here, for convenience, p is used instead of cr(f).] The reduced density matrix satisfies the Liouville equation ... 

This thermal stability comes with a price, however, as low polarizability of the covalent matrix imparts a modest dielectric constant of 5 for A12P055 compositions (although substitution of 33 at% La can boost this value to 8.5). This suboptimal value can be somewhat mitigated by high breakdown fields... 

To obtain Raman spectra one needs the trajectories of the pq tensor elements of the chromophore s transition polarizability. Actually, for the isotropic Raman spectrum one needs only the average transition polarizability. This depends weakly on bath coordinates and this, together with the weak frequency dependence of the position matrix element, was included in our previous calculations [13, 98, 121]. For the VV and VH spectra, others have implemented... 

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