Polarizability damping

While nonbonded atom pairs will typically not come within 1A of each other, it is possible for covalently bound pairs, either directly bounds, as in 1-2 pairs, or at the vertices of an angle, as in 1-3 pairs. Accordingly it may be considered desirable to omit the 1-2 and 1-3 dipole-dipole interactions as is commonly performed on additive force fields for the Coulombic and van der Waals terms. However, it has been shown that inclusion of the 1-2 and 1-3 dipole-dipole interactions is required to achieve anistropic molecular polarizabilites when using isotropic atomic polariz-abilites [50], For example, in a Drude model of benzene in which isotropic polarization was included on the carbons only inclusion of the 1-2 and 1-3 dipole-dipole interactions along with the appropriate damping of those interactions allowed for reproduction of the anisotropic molecular polarizability of the molecule [64], Thus, it may be considered desirable to include these short range interactions in a polarizable force field. 

Thole s polarizability parameters were selected to optimize the molecular polarizabilities for a set of 16 molecules. The method was later expanded to fit 52 molecules [146], It must be emphasized that this electric-field damping method is totally independent of the polarization scheme used. For the Drude and fluctuating charge methods only /i(r) is required, whereas for methods based on induced dipoles both /i(r) and /2(r) are necessary. In the context of the induced dipole model other models were proposed since the formula of Thole does not provide enough attenuation. For example, in Ref. [152] the field is evaluated using... 

Let e,- be the total electric field exerted by surrounding molecules at pixel i, a, the polarizability at pixel i, and p,- the dipole induced at pixel i by that field. The linear polarization energy is poi,j = —1/2 p, c, = —1/2 a, c,. The local pixel polarizability is approximated as a,- = (fj i/Zatom) atom> where Zatom and aatom are the atomic charge and polarizability of the atom to whose basin the pixel belongs (from standard repertories). The polarization energy at pixel i is then damped as... 

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

The quantity eoo represents the optical permittivity, which is determined by the electronic polarizability. The second term represents the ionic crystal lattice as a sum of N classical harmonic oscillators with eigenfrequencies ujj, damping constants Tj and oscillators strengths Sj. In order to fit Equation (5.7) to the observed far infrared spectra, these parameters are used... 

Equation (1) is, strictly speaking, not suitable for optical fields, which are rapidly varying in time. Even for linear polarization, the oscillation of the induced dipole moment may be damped (by material resonances) and thereby phase-shifted with respect to the oscillation of the external electric field. The usual way of expressing this phase shift is by considering the relationship between the Fourier components of the induced effect (oscillation of the induced dipole) and the stimulus (the electric field), with the damping and phase shift conveniently expressed by treating the terms involved as complex. Thus, the linear polarizability can be written as... 

These equations show the damped expression for the ath component of the electric field from the solute multipoles in ra on the solvent centre in rk where a polarizability is located observe that once more the Einstein summation convention has been used with respect to the Greek indices. The parameter c determines the range of the damply 1 Y... 

In Eq. (33), e, is the core polarizability arising from high-frequency transitions and y is the damping rate. The plasma frequency cop0 is given by... 

As to infrared spectroscopy - and the same holds good for other spectral ranges -the orientational order is readily observable in form of dichroism Being related to the molecular shape, the molecular polarizability is anisotropic as well. By the alignment of the molecules this anisotropy is transferred to the sample, however damped due to the imperfect order as described by the order parameters. As a consequence, the dielectric function and furthermore the (complex) refractive index are anisotropic, so that eventually (linear) dichroism and birefringence occur. 

The parameters C, D, E and F involve components of polarizability derivative tensors. Qjs is the displacive order parameter in the low-temperature phase and Qjq corresponds to which is present in the intermediate pseudo-phase. The damping constant T qjh) includes the effect of coupling between the hard mode and the reorientational relaxation of displaced lead atoms. 

By explicitly including the wavevector and branch index dependence of the radiation frequency, and unfolding the detail of the molecular state damping, we conclude that the mean polarizability as used in Eq. (23) is... 

Intermolecular dispersion energies are calculated as a sum of pixel-pixel terms in a London-type expression, involving the above defined distributed polarizabilities and an overall oscillator strength > Eos To avoid singularities (as before described) due to very short pixel-pixel distances in an inverse sixth-power formula, each term in the sum is damped, as it is shown here for the molecule A...molecule B interaction ... 

Note that for H(uj + uj ) = Eb it is necessary to take the damping of the biphonon into account in the expression for A(w + u/,0). In this case, along with the real part of the tensor Xije u>,uj), an imaginary part is also present. This corresponds, as is well known, to the occurrence of two-photon absorption that is accompanied, in the given case, by the excitation of a biphonon. As applied to excitons this question has been discussed by Hanamura (51) within the framework of a somewhat different approach. We refer to it here (see also Fly-tzanis (52)) because both the generation of a second harmonic and two-photon absorption are processes that are completely described by the nonlinear polarizability of the crystal found above with bound states taken into account. Actually, these processes can be investigated by a single method. 

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