Electric field polarizabilities

The linear response function [3], R(r, r ) = (hp(r)/hv(r ))N, is used to study the effect of varying v(r) at constant N. If the system is acted upon by a weak electric field, polarizability (a) may be used as a measure of the corresponding response. A minimum polarizability principle [17] may be stated as, the natural direction of evolution of any system is towards a state of minimum polarizability. Another important principle is that of maximum entropy [18] which states that, the most probable distribution is associated with the maximum value of the Shannon entropy of the information theory. Attempts have been made to provide formal proofs of these principles [19-21], The application of these concepts and related principles vis-a-vis their validity has been studied in the contexts of molecular vibrations and internal rotations [22], chemical reactions [23], hydrogen bonded complexes [24], electronic excitations [25], ion-atom collision [26], atom-field interaction [27], chaotic ionization [28], conservation of orbital symmetry [29], atomic shell structure [30], solvent effects [31], confined systems [32], electric field effects [33], and toxicity [34], In the present chapter, will restrict ourselves to mostly the work done by us. For an elegant review which showcases the contributions from active researchers in the field, see [4], Atomic units are used throughout this chapter unless otherwise specified. 

E is the strength of electric field. Polarizability determines the deformability of the electron cloud of a molecule by an external electric field, similar to the compliance of elastic deformation. Mathematically, Raman activity requires that the first derivative of polarizability with... 

Dielectrophoresis is the motion of polarizable particles that are suspended in an ionic solution and subjected to a spatially non-uniform electric field. Polarizability of particle relative to the suspending medium determines the basic direction of DEP force (positive/negative DEP), which also strongly depends on the frequency of the applied electric field. In the case of electric field with constant phase, time-averaged DEP force can be represented as [8]... 

Notation. The symbols a, f, y are used throughout to denote the electric field polarizability, first and second hyperpolarizabilities respectively, suitably qualified by frequency factors where necessary. The magnetizability is denoted by y and the nuclear screening tensor by a. The numerous but well-known acronyms specifying the computational procedures are used without definition. The possibly rather less well-known acronyms for the principal gauge invariant procedures are given in Table 1. 

The theoretical aspects of nanoparticle adsorption at ITIES was recently reviewed by Flatte et al. [355]. In particular, they discuss the effects that drive or hamper the localization at the interface, namely, competitive wetting, solvation of the charged nanoparticles, shift in the external electric field, polarizability drive, and line tension. A simple model is presented to account for these different contributions, and the results are shown in Figure 1.38. 

The charge redistribution that occurs when a molecule is exposed to an electric field is characterized by a set of constants called polarizabilities. In a imifonn electric field F, a component of the dipole moment is... 

Consider the interaction of a neutral, dipolar molecule A with a neutral, S-state atom B. There are no electrostatic interactions because all the miiltipole moments of the atom are zero. However, the electric field of A distorts the charge distribution of B and induces miiltipole moments in B. The leading induction tenn is the interaction between the pennanent dipole moment of A and the dipole moment induced in B. The latter can be expressed in tenns of the polarizability of B, see equation (Al.S.g). and the dipole-mduced-dipole interaction is given by... 

Flere, is the static polarizability, a is the change in polarizability as a fiinction of the vibrational coordinate, a" is the second derivative of the polarizability with respect to vibration and so on. As is usually the case, it is possible to truncate this series after the second tenn. As before, the electric field is = EQCOslnvQt, where Vq is the frequency of the light field. Thus we have... 

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 perturbation V = H-H appropriate to the particular property is identified. For dipole moments ( i), polarizabilities (a), and hyperpolarizabilities (P), V is the interaction of the nuclei and electrons with the external electric field... 

Polarization is usually accounted for by computing the interaction between induced dipoles. The induced dipole is computed by multiplying the atomic polarizability by the electric field present at that nucleus. The electric field used is often only that due to the charges of the other region of the system. In a few calculations, the MM charges have been included in the orbital-based calculation itself as an interaction with point charges. 

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 ... 

Polarizability (Section 4 6) A measure of the ease of distortion of the electric field associated with an atom or a group A fluonne atom in a molecule for example holds its electrons tightly and is very nonpolanzable Iodine is very polanz able... 

This result, called the Clausius-Mosotti equation, gives the relationship between the relative dielectric constant of a substance and its polarizability, and thus enables us to express the latter in terms of measurable quantities. The following additional comments will connect these ideas with the electric field associated with electromagnetic radiation ... 

Equations (10.17) and (10.18) show that both the relative dielectric constant and the refractive index of a substance are measurable properties of matter that quantify the interaction between matter and electric fields of whatever origin. The polarizability is the molecular parameter which is pertinent to this interaction. We shall see in the next section that a also plays an important role in the theory of light scattering. The following example illustrates the use of Eq. (10.17) to evaluate a and considers one aspect of the applicability of this quantity to light scattering. 

When monochromatic radiation falls on a molecular sample in the gas phase, and is not absorbed by it, the oscillating electric field E (see Equation 2.1) of the radiation induces in the molecule an electric dipole which is related to E by the polarizability... 

In the context of discussion of the Raman effect, Equation (5.43) relates the oscillating electric field E of the incident radiation, the induced electric dipole fi and the polarizability a by... 

Introduction Dielectrophoresis (DEP) is defined as the motion of neutral, polarizable matter produced by a nonimiform electric (ac or dc) field. DEP should be distinguished from electrophoresis, which is the motion of charged particles in a uniform electric field (Fig. 22-30). 

Formal Theory A small neutral particle at equihbrium in a static elecdric field experiences a net force due to DEP that can be written as F = (p V)E, where p is the dipole moment vecdor and E is the external electric field. If the particle is a simple dielectric and is isotropically, linearly, and homogeneously polarizable, then the dipole moment can be written as p = ai E, where a is the (scalar) polarizability, V is the volume of the particle, and E is the external field. The force can then be written as ... 

Polarizability describes the ease of distortion of the electron cloud of the attacking atom of the nucleophile. Again, since the S f2 process requires bond formation by an electron pair from flie nucleophile, the more easily distorted the electric field of the atom, the higher is its nucleophilicity. Polarizability increases going down and to the left in the periodic table. 

The interaction energy of a material having a polarizability a and an applied electric field is given by [97]... 

Another way of obtaining information about the distribution of electrons is by computing the polarizability. This property depends on the second derivative of the energy with respect to an electric field. We ll examine the polarizability of formaldehyde in Chapter 4. 

Gaussian can also predict some other properties dependent on the second and h er derivatives of the energy, such as the polarizabilities and hyperpolarizabilities. These depend on the second derivative with respect to an electric field, and are included automatically in every Hartree-Fock frequency calculation. 

Suppose now, that the material is a gas comprising N atoms each of polarizability o in a volume V. When an electric field E is applied, each atom acquires an induced dipole moment aE and so the polarization is... 

A more thorough analysis shows that one should not expect the electric dipole moment to remain constant, because real molecules have polarizability. The polarization of the dielectric in the electric field of the molecule itself gives rise to a reaction field, which tends to enhance the electrical asymmetry. 

Many ab initio packages use the two key equations given above in order to calculate the polarizabilities and hyperpolarizabilities. If analytical gradients are available, as they are for many levels of theory, then the quantities are calculated from the first or second derivative (with respect to the electric field), as appropriate. If analytical formulae do not exist, then numerical methods are used. 

We have seen above how to calculate polarizabilities and hyperpolarizabilities for statie electric fields. I will consider a simple oscillating electric field... 

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