Polarizabilities and Magnetizabilities

Pal Dahle, Kenneth Ruud, and Trygve Helgaker Department of Chemistry, University of Oslo P. 0. Box 1033 Blindern, N-0315 Oslo, 

Department of Chemistry and Biochemistry University of California, San Diego and 

The ab-initio calculation of molecular properties is a useful tool in chemical research, bridging the gap between observation and interpretation. The agreement between calculation and experiment may lend support to the theoretical interpretation of a given molecular property or help unravel the mechanistic reasons for a given molecular behavior, thereby assisting in the understanding, prediction, and design of molecules with specific properties. 

We do not here attempt to review the developments of methods for calculating molecular polarizabilities and magnetizabilities, nor will we attempt any review of theoretical results obtained for these properties. We will instead outline the approach for calculating molecular properties in general, and the polarizability and the magnetizability of closed-shell systems in particular, using ab-initio methods. We will also devote some time to discuss 

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We now proceed to consider second-order properties such as the polarizability and magnetizability tensors. Differentiating the first-order property Eq. 11, we obtain from the chain rule... 

We conclude that, for a fully variational wave function, only the first-order response of the wave function dX/dx is required to calculate the energy to second order. In particular, for the evaluation of polarizabilities and magnetizabilities, the second-order response of the wave function d2X/dx2 is not needed. 

The response equations are usually solved in some iterative manner, in which the explicit construction of Q is avoided, being replaced by the repeated construction of matrix-vector products of the form Q where v is some trial vector . In general, the solution of one set of response equations is considerably cheaper than the optimization of the wave function itself. Moreover, since the properties considered in this chapter involves at most three independent perturbations (corresponding to the three Cartesian components of the external field), the solution of the full set of equations needed for the evaluation of the molecular dipole-polarizability and magnetizability tensors is about as expensive as the calculation of the wave function in the first place. 

In view of the discussion of magnetizabilities and polarizabilities we have presented for the alkanes in the previous sections, let us finally compare the polarizability and magnetizability of some polyaromatic hydrocarbons. In Table 3, the polarizability and magnetizability components are listed for a representative set of aromatic systems. Not surprisingly, all three components increase with the size of the system. For the polarizability, the increase is most pronounced along the x axis—that is, along the main... 

In the present paper, we have discussed the ab-initio evaluation of the static polarizabilities and magnetizabilities of molecular systems, with emphasis on the principles underlying such calculations. With the recent widespread availability of powerful computers, these second-order molecular properties may nowadays be calculated a priori for large molecular systems, allowing us to explore, for instance, the relationship between the properties and molecular structure. Such calculations complement the experimental work in the area and may help in reassessing and improving on the empirical schemes... 

Electromagnetic energy Energy contained in electromagnetic fields and associated polarizable and magnetizable media. 

Thirdly, induced properties are those that measure the response of a system to an applied field. I will concentrate on polarizabilities (for which the external field is electric) and magnetizabilities (for which the apphed field is magnetic) in a later chapter. 

Table 3 The magnetizability and polarizability of some polyaromatic hydrocarbons as obtained using high-level ab-initio methods. Polarizabilities reported in atomic units, and magnetizabilities in ppm cgs. The molecules are confined to the xy plane, the x direction along the longest side of the...

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. 

Table 16-1. Energies, electric dipolar moments, net atomic populations, vibrational polarizabilities and mean vibrational molecular polarization, magnetizability and contributions thereto, isotropic g tensor and nuclear and electronic paramagnetic and diamagnetic contributions thereto, principal moments of inertia and rotational parameters calculated for H2 C N2 in seven structural isomers... 

A theoretical interpretation of CME has been given by Buckingham and Pople, who showed that the Cotton-Mouton constant can be expressed in terms of the anisotropy of the fourth-rank hypermagnetizability, the electric polarizability, and the magnetizability [20], and by Atkins and Miller [21],... 

The magnetizability (0 of a chemical species varies proportionately with its softness (5) and polarizabihty (a). A softer species is, therefore, more polarizable and hence more magnetizable [288]. 

A softer species is more polarizable" " " and more magnetizable" " . The inverse relationship" " between hardness and polarizability/ magnetizability provides two other electronic structure principles to complement the maximum hardness principle. The minimum polarizability principle" " " (MPP) states that, The natural direction of evolution of any system is towards a state of minimum polarizability and the statement of the minimum magnetizability principle" " (MMP) is, A stable config-uration/conformation of a molecule or a favorable chemical process is associated with a minimum value of the magnetizabihty . 

Among the molecular properties introduced above are the permanent electric dipole moment /xa and traceless electric quadrupole moment a(8, the electric dipole polarizability aajg(—w to) [aiso(to) = aaa(—or, o>)], the magnetizability a(8, the dc Kerr first electric dipole hyperpolarizability jBapy(—(o a>, 0) and the dc Kerr second electric-dipole hyperpolarizability yapys(— ( >, 0,0). The more exotic mixed hypersusceptibilities are defined, with the formalism of modern response theory [9]... 

Birefringences are mostly observed in condensed phases, especially pure liquids or solutions, since the strong enhancement of the effects allows for reduced dimensions (much shorter optical paths) of the experimental apparatus. Nowadays measurements of linear birefringences can be carried out on liquid samples with desktop-size instruments. Such measurements may yield information on the molecular properties, molecular multipoles and their polarizabilities. In some instances, for example KE, CME and BE, measurements (in particular of their temperature dependence) have been carried out simultaneously on some systems. From the combination of data, information on electric dipole polarizabilities, dipole and quadrupole moments, magnetizabilities and higher order properties were then obtained. 

Terms of higher order in the field amplitudes or in the multipole expansion are indicated by. . . The other two tensors in (1) are the electric polarizability ax and the magnetizability The linear response tensors in (1) are molecular properties, amenable to ab initio computations, and the tensor elements are functions of the frequency m of the applied fields. Because of the time derivatives of the fields involved with the mixed electric-magnetic polarizabilities, chiroptical effects vanish as a> goes to zero (however, f has a nonzero static limit). Away from resonances, the OR parameter is given by [32]... 

Some of those have long been known a is the electric dipole polarizability (a symmetric polar tensor of dimension P) in length formalism, k (an asymmetric axial tensor of dimension P t) is related to the optical activity, A is the mixed dipole-quadrupole polarizability, x is the magnetic susceptibility (or magnetizability), written as a sum of diamagnetic and paramagnetic components. 

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