Hyperpolarizability

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. 

Where polarizability by orientation is concerned it has been established by experiment that dielectric constants e are changed, but only 

Buckingham and Pople (1955) have modified the Langevin-Born equations for electric birefringence to allow for a field-dependent polarizability such as (38)  

The final result, written as the molar Kerr constant (equation 28) of a gas at low pressure, contains the anisotropy and dipole terms 61 and 02 (equations 25 and 26), already provided by the classical treatment, plus a third term which should be invariant with temperature. This third term has particular interest when the apparent molar Kerr constants of spherically symmetric molecules, for which a = 0, are being considered if p, = 0, and b1=b2 = b3, then 01 and d2 both become zero, and the observed mK is simply 

Values of c, obtained by multiplying the mK s of Table 25 by 10-7 x 10-24, range from 9-1 x 10-36 to 78-1 x 10 36 e.s.u. for carbon tetrachloride c is 12-2 x 10-36. The low-field polarizability of the last-named molecule should not—by (38)—be detectably altered by fields 

Buckingham (1956c) expanded the molecular refraction of a compressed gas in powers of the molar volume V  

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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] ... 

Shelton D P and Rice J E 1994 Measurements and calculations of the hyperpolarizabilities of atoms and small molecules in the gas phase Chem. Rev. 94 3... 

Maroulis G 1999 On the accurate theoretical determination of the static hyperpolarizability of transbutadiene J. Chem. Phys. 111 583... 

B1.3.2.5 THE MICROSCOPIC HYPERPOLARIZABILITY TENSOR, ORIENTATIONAL AVERAGING, THE KRAMERS-HEISENBERG EXPRESSION AND DEPOLARIZATION RATIOS... 

B) THE MICROSCOPIC HYPERPOLARIZABILITY IN TERMS OF THE LINEAR POLARIZABILITY THE KRAMERS-HEISENBERG EQUATION AND PLACZEK LINEAR POLARIZABILITY THEORY OF THE RAMAN EFFECT... 

If we neglect pure dephasing, the general tensor element of the third order hyperpolarizability relates to those of the first order polarizability tensor according to... 

In fact, each linear polarizability itself consists of a sum of two temis, one potentially resonant and the other anti-resonant, corresponding to die two doorway events, and D, and the window events, and described above. The hyperpolarizability chosen in equation (B1.3.12) happens to belong to the generator. As noted, such tliree-coloiir generators caimot produce Class I spectroscopies (fiill quadrature with tliree colours is not possible). Only the two-colour generators are able to create the Class I Raman spectroscopies and, in any case, only two colours are nomially used for the Class II Raman spectroscopies as well. 

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... 

The susceptibility tensors give the correct relationship for the macroscopic material. For individual molecules, the polarizability a, hyperpolarizability P, and second hyperpolarizability y, can be defined they are also tensor quantities. The susceptibility tensors are weighted averages of the molecular values, where the weight accounts for molecular orientation. The obvious correspondence is correct, meaning that is a linear combination of a values, is a linear combination of P values, and so on. 

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 ... 

Polarizabilities and hyperpolarizabilities have been calculated with semi-empirical, ah initio, and DFT methods. The general conclusion from these studies is that a high level of theory is necessary to correctly predict nonlinear optical properties. 

Ah initio methods can yield reliable, quantitatively correct results. It is important to use basis sets with diffrise functions and high-angular-momentum polarization functions. Hyperpolarizabilities seem to be relatively insensitive to the core electron description. Good agreement has been obtained between ECP basis sets and all electron basis sets. DFT methods have not yet been used widely enough to make generalizations about their accuracy. 

There have been some attempts to compute nonlinear optical properties in solution. These studies have shown that very small variations in the solvent cavity can give very large deviations in the computed hyperpolarizability. The valence bond charge transfer (VB-CT) method created by Goddard and coworkers has had some success in reproducing solvent effect trends and polymer results (the VB-CT-S and VB-CTE forms, respectively). 

Ah initio calculations of polymer properties are either simulations of oligomers or band-structure calculations. Properties often computed with ah initio methods are conformational energies, polarizability, hyperpolarizability, optical properties, dielectric properties, and charge distributions. Ah initio calculations are also used as a spot check to verify the accuracy of molecular mechanics methods for the polymer of interest. Such calculations are used to parameterize molecular mechanics force fields when existing methods are insulficient, which does not happen too often. 

A number of molecular properties can be computed. These include ESR and NMR simulations. Hyperpolarizabilities and Raman intensities are computed using the TDDFT method. The population analysis algorithm breaks down the wave function by molecular fragments. IR intensities can be computed along with frequency calculations. 

The properties available include electrostatic charges, multipoles, polarizabilities, hyperpolarizabilities, and several population analysis schemes. Frequency correction factors can be applied automatically to computed vibrational frequencies. IR intensities may be computed along with frequency calculations. 

The hyperpolarizability is a tensor with eighteen elements We shall not go further into... 

Some vibrations which are both Raman and infrared inactive may be allowed in the hyper Raman effect. Indeed, the occasional appearance of such vibrations in Raman spectra in a condensed phase has sometimes been attributed to an effect involving the hyperpolarizability. 

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. 

The other major properties computed by a frequency job are the polarizability and hyperpolarizability tensors. Normally, the polarizability is printed at the end of the output, just before the archive entry ... 

In a normal Hartree-Fock job, the hyperpolarizability tensor is given only in the archive entry, in the section beginning HyperPolar=. This tensor is also in lower tetrahedral order, but expressed in the input (Z-matrix) orientation. (This is also true of the polarizability tensor within the archive entry.)... 

If you begin the route section with P rather than T, then additional information is printed at various points in the job. One of these items is a display of the polarizability and hyperpolarizability tensors much earlier in the output, just prior to the frequency results ... 

The next terms in the series, denoted. .. in equation 17.1 above, are called the dipole hyperpolarizabilities. The first one is and this also is a tensor. It has three indices, and the corresponding formula for the induced dipole, equation 17.3, becomes... 

There are in fact an infinite number of hyperpolarizabilities, and one occasionally comes across terms higher than fi. 

Just as the dipole changes in an external field, so do all the other moments, and we can develop a set of equations for the quadrupole polarizability (and hyperpolarizabilities), the octupole polarizability, and so on. These esoteric quantities are rarely met in chemistry. 

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... 

Frequency-dependent polarizability a and second hyperpolarizability y corresponding to various third-order nonlinear optical processes have been... 

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