Field-dependent dipole moment

Two polarization mechanisms are possible. If the molecules possess a permanent electric dipole moment pbp rm, each molecule can align its moment with the field direction by reorientation, producing a macroscopic dipole moment. Even if perm = 0 in the field-free limit, each molecule can achieve a field-dependent dipole moment pind by induction. The induced dipole moment is proportional to field strength, pind = a , where a is the electric polarizability of the molecule. In both cases, work must be performed on the system to achieve the macroscopic polarization. Molecules with large permanent dipole moments correspond to high k. 

Electric field dependent electronic polarization in a molecule yields a field dependent dipole moment ... 

Aside the SOS oaches, the (hyper)polarizabilities can be evaluated by differentiating die energy (Eo) or the pole moment with respect to the external electric fields. Since, in the electric dipole approximation, the field-dependent dipole moment is related to the energy through the following expression ... 

The numerical approach to the calculation of electric polarizabilities requires much higher accuracy of the total SCF energy than in standard applications. The convergence threshold on the density matrix elements was set up as 10In this work we are only interested in the longitudinal component, a, of the polarizability tensor gi, the z-axis being directed along tSe chain. The value of is obtain from the first derivative of the field-dependent dipole moment y(E)jwith respect to E in the limit of zero-field. 

Using the Hellmann-Feynman theorem it can be shown that the negative of the first derivative of the energy with respect to the applied field is given by the expectation value of the dipole operator. This is referred to as the field-dependent dipole moment and can be expressed as ... 

The simplified theory is adequate to obtain qualitative agreement with experiment [1,16]. Comparisons between the simplified and more advanced versions of the theory show excellent agreement for the dominant (electronic) contribution to the time-dependent dipole moment, except during the initial excitation, where the k states are coupled by the laser field [17]. The contributions to the dipole from the heavy holes and light holes are not included in the simplified approach. This causes no difficulty in the ADQW because the holes are trapped and do not make a major contribution to the dynamics [1]. This assumption may not be valid in the more general case of superlattices, as discussed below. 

The dielectric constant e of a gas sample depends on the total dipole moment induced in response to an applied electric field. The dipole moment is the vector sum of the partially oriented permanent dipoles pt which individual molecules i may possess, plus the field-induced dipoles a, E arising from the polarizability a of the molecules i, plus all interaction-induced dipoles fiik, plus the field-induced dipoles which arise from the interaction-induced polarizability ptk [93]. The dielectric constant e depends, therefore, on the density g of the gas, according to... 

The expectation value of the property A at the space-time point (r, t) depends in general on the perturbing force F at all earlier times t — t and at all other points r in the system. This dependence springs from the fact that it takes the system a certain time to respond to the perturbation that is, there can be a time lag between the imposition of the perturbation and the response of the system. The spatial dependence arises from the fact that if a force is applied at one point of the system it will induce certain properties at this point which will perturb other parts of the system. For example, when a molecule is excited by a weak field its dipole moment may change, thereby changing the electrical polarization at other points in the system. Another simple example of these nonlocal changes is that of a neutron which when introduced into a system produces a density fluctuation. This density fluctuation propagates to other points in the medium in the form of sound waves. 

On a molecular level, describes the linear response of the electric (d) and magnetic (m) dipole moment of a single chiral molecule to the presence of time-dependent electric (E) and magnetic (B) fields. We refer the reader to the books by Kauzmann [32] and Barron [1] and the review by Condon [31] for derivations of the following equations for the field-induced dipole moments ... 

Figure 25 (upper plot) A schematic plot of the enantiodiscriminator. The three levels of each enantiomer are resonantly coupled by three fields. The dipole moments of the two enantiomers have opposite signs, (middle plot) The time evolution of the population of the three levels. The D and L enantiomers start in the 1) state. At the end of the process one enantiomer is found in the 3) state and the other in the 1) state, (lower plot) The time-dependence of the eigenvalues of the Hamiltonian of Eq. (73). The population initially follows the E0) dark state. At t rthe population crosses over diabatically to ) for one enantiomer and to E+) for the other. 

Figure 5.40 shows the spectra computed from the time-dependent dipole moments in Fig. 5.39. Energy of the driving field is indicated within each panel. Fast Fourier transform in the range from t = 275 to 800 fs, a time range long enough after the pump pulse, is used to obtain S (w). t = 275 was chosen so as to avoid unnecessary complication due to the interaction during the pump.) In each panel, the solid (upper) and dashed (lower) curves show the spectra obtained with and without the pump pulse, respectively. The lower curves are plotted shifted as (log 5 — 6) for clearer comparison with the upper curves. Note the difference in scales for the energy axes.

Alternatively, the dipole-quadrupole polarizability can be obtained as derivative of the electric-field-gradient-dependent dipole moment... 

The polarization of the electric field is determined by the motion of the elementary charge producing the time-dependent dipole moment. If the electron is performing linear oscillations, equations (2.69) and (2.70) show that the radiation fields are also linearly polarized. In the more general case of elliptical or circular motion, the... 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

Assuming that the system has no pennanent dipole moment, tire existence ofP(t) depends on a non-stationary j induced by an external electric field. For weak fields, we may expand the polarization in orders of the perturbation. 

The same idea was actually exploited by Neumann in several papers on dielectric properties [52, 69, 70]. Using a tin-foil reaction field the relation between the (frequency-dependent) relative dielectric constant e(tj) and the autocorrelation function of the total dipole moment M t] becomes particularly simple ... 

The magnitude of the induced dipole moment depends on the electric field strength in accord with the relationship = nT, where ]1 is the induced dipole moment, F is the electric field strength, and the constant a is caHed the polarizabHity of the molecule. The polarizabHity is related to the dielectric constant of the substance. Group-contribution methods (2) can be used to estimate the polarizabHity from knowledge of the number of each type of bond within the molecule, eg, the polarizabHity of an unsaturated bond is greater than that of a saturated bond. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

This is not an SCRF model, as the dipole moment and stabilization are not calculated in a self-consistent way. When the back-polarization of the medium is taken into account, the dipole moment changes, depending on how polarizable the molecule is. Taking only the first-order effect into account, the stabilization becomes (a is the molecular polarizability, the first-order change in the dipole moment with respect to an electric field, Section 10.1.1). 

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