Dipole linear polarizability

The linear polarizability, a, describes the first-order response of the dipole moment with respect to external electric fields. The polarizability of a solute can be related to the dielectric constant of the solution through Debye s equation and molar refractivity through the Clausius-Mosotti equation [1], Together with the dipole moment, a dominates the intermolecular forces such as the van der Waals interactions, while its variations upon vibration determine the Raman activities. Although a corresponds to the linear response of the dipole moment, it is the first quantity of interest in nonlinear optics (NLO) and particularly for the deduction of stracture-property relationships and for the design of new... 

The fundamental equation (1) describes the change in dipole moment between the ground state and an excited state jte expressed as a power series of the electric field E which occurs upon interaction of such a field, as in the electric component of electromagnetic radiation, with a single molecule. The coefficient a is the familiar linear polarizability, ft and y are the quadratic and cubic hyperpolarizabilities, respectively. The coefficients for these hyperpolarizabilities are tensor quantities and therefore highly symmetry dependent odd order coefficients are nonvanishing for all molecules but even order coefficients such as J3 (responsible for SHG) are zero for centrosymmetric molecules. Equation (2) is identical with (1) except that it describes a macroscopic polarization, such as that arising from an array of molecules in a crystal (10). 

In this expression, the Einstein convention of summation over repeated indices has been followed p0 is the permanent dipole moment, while al7, fiijk, and yiJkl are the tensorial elements of the linear polarizability, and the second- and third-order hyperpolarizabilities of the molecule, respectively. 

As an example of the connection between perturbation theory wave function corrections and polarizability, we now calculate the linear polarizability, ax. The states are corrected to first order in H. Since the polarization operator (Zx) is field independent, polarization terms linear in the electric field arise from products of the unperturbed states and their first-order corrections from the dipole operator. The corrected states are [12]... 

Electro- and magnetooptical phenomena in colloids and suspensions are widely used for structure and kinetics analysis of those media as well as practical applications in optoelectronics [143,144]. The basic theoretical model used to study optical anisotropy of the disperse systems is the noninteracting Brownian particle ensemble. In the frame of this general approximation, several special cases according to the actual type of particle polarization response to the applied field may be distinguished (1) particles with permanent dipole moments, (2) linearly polarizable particles, (3) nonlinearly polarizable particles, and (4) particles with hysteretic dipole moment reorientation. 

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... 

Periodic oscillations in this dipole can act as a source term in the generation of new optical frequencies. Here a is the linear polarizability discussed in Exps. 29 and 35 on dipole moments and Raman spectra, while fi and x are the second- and third-order dielectric susceptibilities, respectively. The quantity fi is also called the hyperpolarizability and is the material property responsible for second-harmonic generation. Note that, since E cos cot, the S term can be expressed as -j(l + cos 2 wt). The next higher nonlinear term x is especially important in generating sum and difference frequencies when more than one laser frequency is incident on the sample. In the case of coherent anti-Stokes Raman scattering (CARS), X gives useful information about vibrational and rotational transitions in molecules. 

Note that linearly polarizable point dipoles provide only an approximation to the true polarization response in two different ways. Eirst, polarization can include terms that are nonlinear in the electric field. Thus, Eq. [3] represents only the first term in an infinite series. 

Despite the many differences between the various polarizable models, it is encouraging to note that the most recent models seem to be converging on the same set of necessary features. A variety of successful models based on different formalisms all share many of the same characteristics.Regardless of the direction from which the models evolved, there is a growing consensus that accurate treatment of polarization requires (1) either diffuse charge distributions or some other type of electrostatic screening (2) a mixture of both monopoles and dipoles to represent the electrostatic charge distribution, and (3) only linear polarizability. 

The linear and non-linear polarizabilities of a molecule in solution differ from those of the isolated molecule in the gas phase since the molecular properties are modified by solute-solvent interactions. Some of these interactions are present even in the absence of externally applied static or optical fields. For molecules with a non-zero dipole moment fj in the electronic ground state the dominant interaction is usually due to the reaction field contribution The molecular dipole moment polarizes the solvent environment and thus generates a polarization field which interacts with the solute. This field is given by (88) (Boettcher, 1973 Wortmann and Bishop, 1998). 

For tetrahedral molecules the linear polarizability is isotropic, and the first non-zero permanent electric moment is the octupole Q = TlJe electric fields of these octupoles induce a dipole moment in any given molecule, whence 0 and we have by equation (241) to a suflBciently good approximation ... 

General Treatment of Fluctuational Processes. The previous treatment is good only as long as we deal with strongly dipolar substances and all other polarizational effects remain negligible. In the majority of substances, besides reorientation of permanent dipoles, one has to consider reorientation of the polarizability ellipsoids as well as statistical-fluctuational processes. In calculating the electric polarization (277), one has to include the term accounting for linear distortional polarizability of the dielectric (non-linear polarizabilities are dealt with below) ... 

Since pNA and most of the chromophores of interest have large dipole moments an important feature of the continuum models is the introduction of the reaction field. The pNA molecule at the centre of the cavity in the continuum induces a polarization on the surface of the cavity, which produces the reaction field acting on the central molecule. This reaction field changes the dipole moment of the pNA molecule via the linear polarizability. A self consistent procedure is required in which the effects of the reaction field and also the effects of the applied macroscopic fields modified by the internal field factors are included in a self-consistent determination of the molecular response within a specified quantum mechanical model. 

An axially symmetric molecule is characterized by its linear polarizability in the principal axes a x and a y = a" and a" = af/. It is a good approximation to assume that its second- and third-order polarizability tensors each have only one component and respectively, which is parallel to the z principal axis of the molecule. For linear and nonlinear optical processes, the macroscopic polarization is defined as the dipole moment per unit volume, and it is obtained by the linear sum of the molecular poiarizabilities averaged over the statistical orientational distribution function G(Q). This is done by projecting the optical fields on the molecular axis the obtained dipole is projected on the laboratory axes and orientational averaging is performed. The components of the linear and nonlinear macroscopic polarizabilies are then given by ... 

In this equation, po is the permanent dipole moment of the molecule, a is the linear polarizability, 3 is the first hyperpolarizability, and 7 is the second hyperpolarizability. a, and 7 are tensors of rank 2, 3, and 4 respectively. Symmetry requires that all terms of even order in the electric field of the Equation 10.1 vanish when the molecule possesses an inversion center. This means that only noncentrosymmetric molecules will have second-order NLO properties. In a dielectric medium consisting of polarizable molecules, the local electric field at a given molecule differs from the externally applied field due to the sum of the dipole fields of the other molecules. Different models have been developed to express the local field as a function of the externally applied field but they will not be presented here. In disordered media,... 

In this expression, po is the permanent dipole moment, a the linear polarizability and (3 the quadratic hyperpolarizability (origin of the NLO behavior), E being the electric field component of the light. The... 

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