Polarizability and Hyperpolarizability

Note that since both /z and E are vector quantities, a is a second-rank tensor. The elements of a can be computed through differentiation of Eqs. (9.1) and (9.2). The difference between die permanent electric dipole moment and that measured in the presence of an electric field is referred to as the induced dipole moment. 

It is possible to generalize this discussion in a useful way. Spectral measurements invariably assess how a molecular system changes in energy in response to some sort of external perturbation. The example presently under discussion involves application of an external electric field. If we write the energy as a Taylor expansion in some generalized vector perturbation X, we have 

Analogous quantities to die electric moments can be defined when the external perturbation takes the form of a magnetic field. In this instance die first derivative defines the permanent magnetic moment (always zero for non-degenerate electronic states), the second derivative the magnetizability or magnetic susceptibility, etc. 

Equation (9.32) is also useful to the extent it suggests die general way in which various spectral properties may be computed. The energy of a system represented by a wave function is computed as the expectation value of the Hamiltonian operator. So, differentiation of the energy with respect to a perturbation is equivalent to differentiation of the expectation value of the Hamiltonian. In the case of first derivatives, if the energy of the system is minimized with respect to the coefficients defining die wave function, the Hellmann-Feynman theorem of quantum mechanics allows us to write 

Note that H here is the complete Hamiltonian, that is, it presumably includes new terms dependent on the nature of X. It is occasionally the case that the integral on the r.h.s. of Eq. (9.33) can be readily evaluated. Indeed, it is choice of X = E that leads to the definition of the dipole moment operator presented in Eq. (9.1). 

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  

The tensor is given in lower-triangular format (i.e. a. standard orientation. The Approx polarizability line gives the results of the cruder polarizability estimate using sum-over-states perturbation theory, which is suggested by some older texts. 

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.) 

Exploring Chemistry with Electronic Structure Methods 

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  

In the preceding two sections we have discussed different ways in which a molecular system may respond to an electromagnetic field, first of all through excitation or ionization. Other responses include that the particles (electrons and nuclei) change positions due to these fields. This may, e.g., be quantified through the dipole moment, i.e. 

In the static case, the frequencies of all field components as well as of the dipole moment vanish, but in the dynamic case they may be non-zero, obeying, however, 

A formally more correct approach is to use time-dependent density-functional theory. In their approach, Jamorski et al. showed that the average polarizability 

Also van Gisbergen et al applied the time-dependent density-functional theory in calculating frequency-dependent hyperpolarizabilities of some small molecules. They considered different density functionals and compared their results with those of previous theoretical studies and of experiment. In Tables 16 and 17 we reproduce their results for Nj and CO. The molecules were assumed placed along the x axis, and the authors considered also the averaged hyperpolarizabilities given by Eq. (112). 

Chemical Modelling Applications and Theory, Volume 2 DYNAMIC MEAN POLARIZABILITY 

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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] ... 

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. 

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. 

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... 

Table 6 Effect of the polarization functions on the polarizabilities and hyperpolarizabilities of He...

Table 7 Polarizabilities and hyperpolarizabilities of H2 at = 1.4 au, with optimization of C in polarization functions (Cipl = l-l) from Ref. 6...

The present paper is aimed at developing an efficient CHF procedure [6-11] for the entire set of electric polarizabilities and hyperpolarizabilities defined in eqs. (l)-(6) up to the 5-th rank. Owing to the 2n+ theorem of perturbation theoiy [36], only 2-nd order perturbed wavefunctions and density matrices need to be calculated. Explicit expressions for the perturbed energy up to the 4-th order are given in Sec. IV. 

Dickson, R. M., Becke, A. D., 1996, Local Density-Functional Polarizabilities and Hyperpolarizabilities at the... 

Guan, J., Duffy, P., Carter, J. T., Chong, D. P., Casida, K., Casida, M. E., Wrinn, M., 1993, Comparison of Local-Density and Hartree-Fock Calculations of Molecular Polarizabilities and Hyperpolarizabilities , J. Chem. Phys., 98, 4753. 

Ruocco G, Sampoli M (1994) Computer-simulation of polarizable fluids - a consistent and fast way for dealing with polarizability and hyperpolarizability. Mol Phys 82(5) 875-886... 

We have shown in this paper the relationships between the fundamental electrical parameters, such as the dipole moment, polarizability and hyperpolarizability, and the conformations of flexible polymers which are manifested in a number of their electrooptic and dielectric properties. These include the Kerr effect, dielectric polarization and saturation, electric field induced light scattering and second harmonic generation. Our experimental and theoretical studies of the Kerr effect show that it is very useful for the characterization of polymer microstructure. Our theoretical studies of the NLDE, EFLS and EFSHG also show that these effects are potentially useful, but there are very few experimental results reported in the literature with which to test the calculations. More experimental studies are needed to further our understanding of the nonlinear electrooptic and dielectric properties of flexible polymers. 

Further work on long polyenes, including vibrational distortion, frequency dispersion effects and electron correlation, would be important for evaluating more accurate asymptotic longitudinal polarizabilities and hyperpolarizabilities. 

Mikkelsen, K., Luo, Y., Agren, H. and Joergensen, P. Solvent induced polarizabilities and hyperpolarizabilities of para-nitroaniline studied by reaction field linear response theory, J.Chem.Phys., 100 (1994), 8240-8250... 

The values for the dipoles, polarizabilities, and hyperpolarizabilities of the H2 series were obtained using (a) a 16-term basis with a fourfold symmetry projection for the homonuclear species and (b) a 32-term basis with a twofold symmetry projection for the heteronuclear species. These different expansion lengths were used so that when combined with the symmetry projections the resulting wave functions were of about the same quality, and the properties calculated would be comparable. A crude analysis shows that basis set size for an n particle system must scale as k", where k is a constant. In our previous work [64, 65] we used a 244-term wave function for the five-internal-particle system LiH to obtain experimental quality results. This gives a value of... 

M. Spassova and B. Champagne, aP-SOS, a Code for Evaluating Static and D5mamic Polarizabilities and Hyperpolarizabilities (Namur Sofia, 1996). 

JHC735>. The polarizability and hyperpolarizability of 1,2,3-triazole have been computed by the Hartree-Fock perturbation theory on an extended basis CNDO method <90JPC1755>. 

See [11] for a recent review of applications of even-tempered basis set to the calculation of accurate molecular polarizabilities and hyperpolarizabilities within the matrix Hartree-Fock approximation. In [11] the results finite basis set Hartree-Fock calculations are compared with finite difference Hartree-Fock calculations. 

As for the utility of different levels of theory for computing the polarizability and hyperpolarizability, the lack of high-quality gas-phase experimental data available for all but the smallest of molecules makes comparison between theory and experiment rather limited. As a rough rule of thumb, ab initio HE Iheory seems to do better for these properties than for dipole moments - at least there does not appear to be any particular systematic error. Semiempirical levels of theory are less reliable. DFT and correlated levels of MO theory do well, but it is not obvious for the latter that the improvement over HF necessarily justifies the cost, at least for routine purposes. 

In order to provide accurate static values for this purpose, we have recently undertaken a series of calculations of the polarizability and hyperpolarizability of the rare gases. An extensive basis set investigation was performed for Ne, and we shall consider these results in detail. We shall also discuss aspects of the correlation treatment and computational methodology. We begin by considering methods for the calculation of the polarizability and hyperpolarizability. 

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