Moment expansion polarizability

Spectroscopic applications usually require us to go beyond single-point electronic energy calculations or structure optimizations. Scans of the potential energy hypersurface or at least Taylor expansions around stationary points are needed to extract nuclear dynamics information. If spectral intensity information is required, dipole moment or polarizability hypersurfaces [202] have to be developed as well. If multiple relevant minima exist on the potential energy hyper surface, efficient methods to explore them are needed [203, 204],... 

By using the multipole expansion, we in fact replace the exact radial expansion coefficients A (7 ) in Eq. (1-124) by the approximate coefficients A poi(-K), which are power series in R 1. Closed expressions for the latter have been given149 161 in terms of the irreducible spherical tensors of multipole moments and polarizabilities. 

Since the single-center multipole expansion of the interaction energy is divergent, one could use a kind of multicenter expansion. One can hope that the multipole expansion will provide better results if multipole moments and polarizabilities localized at various points of a molecule are used instead of global multipole moments and polarizabilities. This idea forms the basis of the so-called distributed multipole analysis of the electrostatic, induction, and dispersion interactions between molecules187 195. 

The multicenter expansion of the induction energy in terms of the distributed multipole moments and polarizabilities can be obtained is a similar way starting from Eq. (1-87) rewritten as follows,... 

The moments and polarizabilities of molecules can be determined by indirect means. In collision experiments, the nature of the interaction is governed by the potential energy surface, itself a function of the molecular properties of the colliding partners. Usually the potential energy is written in a multipole expansion whereby the electrical properties are displayed in the long-range terms [38]. The potential that is generated must satisfy simultaneously... 

The perturbed total energies or other properties of the system can be written as an expansion in terms of moment and polarizability components (see Section I). If different values of the field strength or charge positions are used, a system of simultaneous equations can be written from the truncated series, and these equations are solved to find the unknown polarizabilities. The system of equations must be chosen sufficiently large to ensure that the truncation error is minimized, but sometimes it is not practical to carry out the number of finite-field calculations that this might call for. 

Finite-field methods were first used to calculate dipole polarizabilities by Cohen and Roothaan [66]. For a fixed field strength V, the Hamiltonian potential energy term for the interaction between the electric field and ith electron is just The induced dipole moment with the applied field can be calculated from the Hartree-Fock wavefunction by integrating the dipole moment operator with the one-electron density since this satisfies the Hellmann-Feyman theorem. With the usual dipole moment expansion. 

Equations (2) and (8) define the SAPT expansions of the interaction-induced dipole moment and polarizability,... 

The coefficients C can be computed from properties of monomers such as multipole moments and polarizabilities. The relevant formulas are obtained from the polarization series truncated at some finite order by replacing the potential V by its asymptotic expansion in powers of l/R. For the Coulomb potential 1 / r 1 —r21, such expansion has the form... 

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. 

This is often called the Cauchy moment expansion of the frequency-dependent polarizability and the sums S k) for even but negative values of k are called Cauchy moments. S (—2) in particular turns out to be proportional to the static polarizability... 

These moments are related to many physical properties. The Thomas-Kulm-Reiche sum rule says that. S (0) equals the number of electrons in the molecule. Other sum rules [36] relate S(2),, S (1) and. S (-l) to ground state expectation values. The mean static dipole polarizability is md = e-S(-2)/m,.J Q Cauchy expansion... 

Raman scattering has been discussed by many authors. As in the case of IR vibrational spectroscopy, the interaction is between the electromagnetic field and a dipole moment, however in this case the dipole moment is induced by the field itself The induced dipole is pj j = a E, where a is the polarizability. It can be expressed in a Taylor series expansion in coordinate isplacement... 

The second procedure, several aspects of which are reviewed in this paper, consists of directly computing the asymptotic value by employing newly-developed polymeric techniques which take advantage of the one-dimensional periodicity of these systems. Since the polarizability is either the linear response of the dipole moment to the field or the negative of the second-order term in the perturbation expansion of the energy as a power series in the field, several schemes can be proposed for its evaluation. Section 3 points out that several of these schemes are inconsistent with band theory summarized in Section 2. In Section 4, we present the main points of the polymeric polarization propagator approaches we have developed, and in Section 5, we describe some of their characteristics in applications to prototype systems. 

L-D. With polarizable charges obtained by A, B, or C combined with a truncated multipole expansion, including multipole moments up to some predetermined cutoff /, where l > 1 but not necessarily large enough for convergence. 

Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Refer to Fig. 12.1 and its caption for more detail. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion. Any such property, P, can be expressed in terms of a Taylor series expansion over the displacements of the coordinates from their equilibrium positions,... 

Hyper)polarizabilities are defined as the coefficients in the Taylor series expansion of the dipole moment - or the energy - in the presence of static and/or oscillating electric fields ... 

The dielectric constant is a macroscopic property of the material and arises from collective effects where each part of the ensemble contributes. In terms of a set of molecules it is necessary to consider the microscopic properties such as the polarizability and the dipole moment. A single molecule can be modeled as a distribution of charges in space or as the spatial distribution of a polarization field. This polarization field can be expanded in its moments, which results in the multipole expansion with dipolar, quadrupolar, octopolar and so on terms. In most cases the expansion can be truncated to the first term, which is known as the dipole approximation. Since the dipole moment is an observable, it can be described mathematically as an operator. The dipole moment operator can describe transitions between states (as the transition dipole moment operator and, as such, is important in spectroscopy) or within a state where it represents the associated dipole moment. This operator describes the interaction between a molecule and its environment and, as a result, our understanding of energy transfer. 

---