Nonlocal Polarizability Densities and Their Relationships

Relation of Hyperpolarizability Derivative to Second Hyperpolarizability

Dr. Hunt, Dr. Xiaoping Li (post-doc, MSU) and I have recently discovered a fundamental relationship between the derivative of the hyperpolarizability of a molecule with respect to nuclear coordinates and the second hyperpolarizability density, the nuclear charge and the dipole propagator. This work is an extension of earlier work done by Dr. Hunt and others on the relationship between the derivative of the polarizability with respect to nuclear coordinates and the hyperpolarizability density. Dr. Hunt has also proved a relationship between the derivative of the dipole moment with respect to nuclear coordinates and the polarizability density. Hmmm! There appears to a pattern here. Though we have not proved it generally, there is a relationship between the derivative of the nth order hyperpolarizability with respect to nuclear coordinates and the (n+1)th order hyperpolarizability density, the nuclear charge and the dipole propagator.

Possible Applications in Nonlinear Optics

This work may assist in the screening of nonlinear optical materials. The hyperpolarizability derivative is related to nonresonant hyper-Raman scattering intensities. The form of the second hyperpolarizability we used is related to the intensities of DC-electric field second harmonic generation. This form is especially useful for nonpolar molecules or crystals whose unit cell has a center of inversion. If a material's hyper-Raman response can be measured, its hyperpolarizability derivative with respect to nuclear coordinate can be calculated. With the relation that we have proven, ideally one could then calculate the material's second hyperpolarizability and determine the material appropriateness in nonlinear optical elements. This determination is much easier said than done since the hyperpolarizability derivative is related to the integral of the second hyperpolarizability density and the dipole propagator and not just the second hyperpolarizability. The difference between a polarizability and a polarizability density is important and is explained below. We have published a paper in the Journal of Chemical Physics Vol. 103 pg. 6873 Oct 22, 1995.

Related Experimental Research

My group mate, Sandjaja Tjahajadiputra,is working with Dr. Gary Blanchard in our department to experimentally verify a relationship between the polarizability and the hyperpolarizability. If a relationship exists, the ability to screen materials for nonlinear optical properties is greatly enhanced. His research is going very well and is interesting since there is no proven relationship between these two quantities

Distinction between Polarizability and Polarizability Density

I should emphasize the distinction between the polarizability density and the polarizability. The polarizabiity is a property of the entire molecule. When an electric field is applied, the response of the entire molecule is what is being considered. A polarizability density is a property of only two points in the molecule. I'll call these points the field point and the response point. I'll define the field point as the point in the molecule where an incident electric field is perturbing the electronic structure of the molecule. The response point shall be defined as the that point in the molecule that is responding to the perturbation at the field. Thus the polarizability density is the quantity that determines the nonlocal response at the response point due to the perturbation applied at the source point. If we want to know the total response of an applied electric field at a single response point, we would need to sum the effects of all of the field points, i. e. integrate the polarizability density tensor contracted with the electric field vector over all of the field point space. Further if we wish the total response from all the response points, we would need to integrate our previous result over the response point space. Thus, the polarizability density integrated over all molecular space is the polarizability. The polarizability density is not, at the present, an observable quantity. The experimenter would have to examine the electronic response of a molecule within small regions within the molecule to construct an electronic response density. A difficult task indeed. However, Xiaoping Li and Chetan Ahuja, both of the Hunt research group, are performing ab initio calculations using delta function potentials in an attempt to calculate and, perhaps more importantly, visualize polarizability densities. So far their early work has produced some interesting plots of polarizability densities of water.

Ongoing Research

I'm now moving on to attempt to construct a nonlocal magnetizability theory that hopefully will give more physical insight into molecular magnetic properties such as chemical shift tensors and spin-spin coupling tensors. I am trying a quasi-relativistic approach to construct a nonlocal electromagnetizability density tensor using Dirac spinors and the Breit Hamiltonian. It's slow progress, but it is moving forward.

Also my group mate, Mark Champagne, and I are trying to explain intermolecular resonance forces using Dr. Hunt's polarizability densities theories. This project appears to be relatively simple; however, we're not that bright. (Or least I'm not that bright.)

Oh my! How disturbing ignorance is! But the pursuit of truth must go on.

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