Taylor series polarizability

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 high-field output of laser devices allows for a wide variety of nonlinear interactions [17] between tire radiation field and tire matter. Many of tire initial relationships can be derived using engineering principles by simply expanding tire media polarizability in a Taylor series in powers of tire electric field ... 

The molecular quantities can be best understood as a Taylor series expansion. For example, the energy of the molecule E would be the sum of the energy without an electric field present, Eq, and corrections for the dipole, polarizability, hyperpolarizability, and the like ... 

As for the change of dipole moment, the change of polarizability with vibrational displacement x can be expressed as a Taylor series... 

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 mathematical formulation of the nonlinear polarization is unknown but a common approximation is to expand the polarizability as a Taylor series ... 

The dependence of the many-body polarizability of the liquid on the molecular coordinates can be expressed as a Taylor-series expansion that is entirely analogous to Equation (1) ... 

Four frequently used conventions exist for the definition of non-linear optical polarizabilities, leading to confusion in the realm of NLO. This has been largely clarified by Willets et al. (1992) and in their nomenclature we have used the Taylor series expansion (T convention), originally introduced by Buckingham (1967), where the factorials n are explicitly written in the expansion. Here the polarizabilities of one order all extrapolate to the same value for the static limit w— 0. /3 values in the second convention, the perturbation series (B), have to be multiplied by a factor of 2 to be converted into T values. This is the convention used most in computations following the sum-over-states method (see p. 136). The third convention (B ) is used by some authors in EFISHG experiments and is converted into the T convention by multiplication by a factor of 6. The fourth phenomenological convention (X) is converted to the T convention by multiplication by a factor of 4. 

Molecular electric properties give the response of a molecule to the presence of an applied field E. Dynamic properties are defined for time-oscillating fields, whereas static properties are obtained if the electric field is time-independent. The electronic contribution to the response properties can be calculated using finite field calculations , which are based upon the expansion of the energy in a Taylor series in powers of the field strength. If the molecular properties are defined from Taylor series of the dipole moment /x, the linear response is given by the polarizability a, and the nonlinear terms of the series are given by the nth-order hyperpolarizabilities ()6 and y). 

However, Equations (1) and (2) are approximations generally, polarizability cannot be regarded as a constant and the induced polarization is a non-linear function of field strength. This non-linearity becomes increasingly important at very intense electric fields. The non-linear dependence of a dipole moment on field can be expressed as a Taylor series as shown in Equations (3) and (4)... 

From the expansion of A( >,f ) in Taylor series for as small values of F as computationally possible, it is possible to obtain cycle-averaged quantities corresponding to measurable polarizabilities and hyperpolarizabilities, of ground, excited and autoionizing states, e.g.. Refs. [182,186]. Specifically, the computed energy shift for atomic states is written as... 

An expression for the induced polarization can be derived in terms of the normal coordinate and electric field by writing the polarizability as a Taylor series in the normal coordinate and retaining only the linear terms. For the ij component, then. 

Normal Raman spectroscopy probes the variations of the polarizability tensor with respect to the degrees of freedom, in the ground electronic state. When an electrical field is applied to a system the electron distribution is modified and the sample acquires an induced dipole moment as the barycenters of the charges are displaced. The polarizability tensor [a] defines the correspondence between the incident electrical field E and the induced dipole moment M = [a]E. The polarizability tensor can be expanded in a Taylor series analogous to Equation (8.8) ... 

Polarizability tensor invariants a R) and y R) for the family of the most stable configurations of the CH4-N2 complex can be also presented in the form of Taylor series in the vicinity of R = 6.84 ao. 

In molecules the polarizability a does not have a constant value since certain vibrations and rotations can cause a to vary. For example, during the vibration of a diatomic molecule, the molecular shape is alternately compressed and extended. Since the election cloud is not identical at the extremes of the vibration, a change, in polarizability results. For small displacements the polarizability can be expanded in a Taylor series as... 

Ho is die Hamiltonian for the unperturbed system and p is the dipole moment operator. From the perturbed wave functions evaluated at SCF or Cl level the dipole moment value p(f) is obtained. From the Taylor series expansion [ q. (10.1)] the coefficients a, p and y can be derived from calculations at a number of distortions along particular coordinate and field direcdons. Molecular polarizability is then obtained from the expression... 

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