Electric dipole polarizability tensor, linear

Abstract The modified equation-of-motion coupled cluster approach of Sekino and Bartlett is extended to computations of the mixed electric-dipole/magnetic-dipole polarizability tensor associated with optical rotation in chiral systems. The approach - referred to here as a linearized equation-of-motion coupled cluster (EOM-CCl) method - is a compromise between the standard EOM method and its linear response counterpart, which avoids the evaluation of computationally expensive terms that are quadratic in the field-perturbed wave functions, but still yields properties that are size-extensive/intensive. Benchmark computations on five representative chiral molecules, including (P)-hydrogen peroxide, (5)-methyloxirane, (5 )-2-chloropropioniuile, (/ )-epichlorohydrin, and (75,45)-norbornenone, demonstrate typically small deviations between the EOM-CCl results and those from coupled cluster linear response theory, and no variation in the signs of the predicted rotations. In addition, the EOM-CCl approach is found to reduce the overall computing time for multi-wavelength-specific rotation computations by up to 34%. 

Linear Response (2nd rank tensor) Electric dipole polarizability, a, magnetic dipole susceptibihty, optical rotatory power, k, nuclear shielding tensor at... 

Comparing this with the classical expansion of a time-dependent dipole moment in Eq. (7.33) we can identify the frequency-dependent mixed electric dipole magnetic dipole polarizability tensor as a linear response function or polarization propagator... 

The frequency-dependent electric dipole polarizability is, next to the dipole moment, the most important effect characterizing the response of a molecule to electromagnetic radiation. It is a second-order symmetric tensor determined by the linear response fimction... 

Electronic polarizability is often included in force fields via the use of induced dipoles. Assuming that hyperpolarization effects are absent, the induced dipoles respond linearly relative to the electric field. In this case, the induced dipole p on an atom is the product of the total electric field E and the atomic polarizability tensor a. 

Here a designates the trace of the polarizability tensor of one molecule (l/47i o) times the factor of a represents the electric fieldstrength of the quadrupole moment q2. Other non-vanishing multipole moments, for example, octopoles (e.g., of tetrahedral molecules), hexadecapoles (of linear molecules), etc., will similarly interact with the trace or anisotropy of the polarizability of the collisional partner and give rise to further multipole-induced dipole components. 

The intensity of Rayleigh scattering and the linear Raman effect is governed by the polarizability tensor apa of a molecule and its derivatives with respect to the normal coordinates. When the electric field of the exciting radiation is very high, further terms in the expression for the induced dipole moment 104)... 

Another important class of forces, induction or polarization forces, involves permanent moments that induces multipoles in a polarizable species. Polarizability, a, measures the ability of an atomic or molecular species to develop an induced dipole moment, as a response to an applied electric field E. Within the limits of linear response theory, the induced dipole moment is given by the product of polarizability tensor times the electric field E. 

In this notation, the elements of the polarizability tensor are only proportional to the particle volume, similar to the atomic polarizability. is the dielectric constant of the surrounding medium. A dipole with linear response oscillates with the same frequency as the applied electric field and, hence, emits an electric field Es asymptotically given in the far field as... 

The vector of a permanent dipole moment Pe and polarizability tensor a y describe the linear (in field) electric and optical properties. The nonlinear properties are described by tensors of higher ranks (this depends on the number of fields included). For instance, the efficiency of mixing two optical waves of frequencies coi and CO2 is determined by polarization Py (co3) = E coi) Ey(co2) where E/(coi) and Ey((X)2)j are amplitudes of two interacting fields. Here is a third rank tensor of the electric hyperpolarizability. 

Let US Start examining the first contribution to the induced dipole liM in Eq. (3.1) which is linear in the inducing electric field E. The linear molecular polarizability tensor in the molecular coordinate system can be written, within the oriented gas model, as ... 

It can be shown that the trace of the tensor G, and hence the computed optical rotation of a sample of randomly oriented chiral molecules, is independent of the origin provided that the linear response function satisfies O Eq. 5.46 and that the commutator of O Eq. 5.47 is fulfilled. Consequently, approximate linear response calculations of the length gauge optical rotation depend on the chosen coordinate origin. On the other hand, the trace of the velocity gauge formulation of the electric dipole - magnetic dipole polarizability... 

At the sodium frequency, the optical rotation is rather small for most molecules. However, the individual diagonal elements of the mixed electric dipole-magnetic dipole polarizability maybe fairly large in absolute value, often canceling each other in the trace. This is illustrated for a few selected molecules in O Table 11-7. Consequently, the optical rotation is highly sensitive to numerical errors in the tensor components, because small residual errors in the individual tensor components, arising from the solution of the linear response equations, may lead to substantial errors in Another consequence of this cancellation is that is very... 

The summation runs over repeated indices, /r, is the i-th component of the induced electric dipole moment and , are components of the applied electro-magnetic field. The coefficients aij, Pijic and Yijki are components of the linear polarizability, the first hyperpolarizability, and the second hyperpolarizability tensor, respectively. The first term on the right hand side of eq. (12) describes the linear response of the incident electric field, whereas the other terms describe the nonhnear response. The ft tensor is responsible for second order nonlinear optical effects such as second harmonic generation (SHG, frequency AotAAin, frequency mixing, optical rectification and the electro-optic effect. The ft tensor vanishes in a centrosymmetric envirorunent, so that most second-order nonlinear optical materials that have been studied so far consists of non-centrosyrmnetric, one-dimensional charge-transfer molecules. At the macroscopic level, observation of the nonlinear optical susceptibility requires that the molecular non-symmetry is preserved over the physical dimensions of the bulk stmcture. 

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). 

For further progress it is necessary to specify how E varies with D, or how P depends on Ea. For this purpose, we introduce the constitutive relations D - e(T,V)E or P - ot0(T,V)F0, where e is the dielectric constant and a0 is a modified polarizability. (Conventionally, the polarizability is defined through the relation P - oE, but no confusion is likely to arise through the introduction of this variant.) Note several restrictions inherent in the use of these constitutive relations. First, the material under study is assumed to be isotropic. If this is not the case, e and c 0 become tensors. Second, the material medium must not contain any permanent dipole moments in the preceding constitutive relations P or E vanishes when E0 or D does. Third, we restrict our consideration to so-called linear materials wherein e or a0 do not depend on the electric field phenomena such as ferroelectric or hysteresis effects are thus excluded from further consideration. These three simplifications obviously are not fundamental restrictions but render subsequent manipulations more tractable. Finally, in accord with experimental information available on a wide variety of materials, e and aQ are considered to be functions of temperature and density assuming constant composition, these quantities vary with T and V. 

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,... 

Here, pa and ma are the electric and magnetic dipole moment functions, aap, Papy> YapyS 81 the polarizability and first and second hyperpolarizabilities, is the magnetizability, and, of the other terms, only the hypermagnetizability will be of interest (it relates to the Cotton-Mouton effect). The Greek subscripts a, p,... denote vector or tensor quantities and can be equal to the Cartesian coordinates x, y, or z. Einstein summation over these subscripts is implied both here and elsewhere. Differentiation of this expression with respect to F (or B) leads to an expression for the dipole moment (or polarization) of the species in the presence of the perturbing fields and it is clear that P, y, t), etc. will govern the non-linear terms in the induced electric (or magnetic) dipole moment - hence, non-linear optics. 

Fig. 12.4. The direction of the induced dipole moment may differ from the direction of the electric field applied (due to the tensor character of the polarizability and hyperpolarizabilities). Example the vinyl molecule in a planar conformation. Assume the following Cartesian coordinate system x (horizontal in the Figure plane), y tvertical in the Figure plane and (peipendicular to the Figure plane), and the external electric field E = (0, y, 0). Tlie compmient x rf the induced dipole mcxnent is equal to [within the accuracy of linear terms, Eq. (12.19)] = M.v — ft0.v yv y. I ind.y ftind, Due to the...

This formula pertains exclusively to the interaction of the molecular dipole (the permanent dipole plus the induced linear and non-linear response) with the electric field. As seen from (12.19), the induced dipole moment with the components may have a different direction from the applied electric field (due to the tensor character of the polarizability and hyperpolarizabilities). This is quite understandable, because the electrons wih move in a direction which will represent a compromise between the direction of the electric field which forces them to move, and the direction where the polarization of the molecule is easiest (Fig. 12.4). 

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