Frequency-Dependent Polarizabilities General Theory

The general theory of time-dependent response functions has been described in many publications.2,18,19 The response is non-local in time and the Fourier transforms of the general time-dependent functions lead to the definitions of the frequency-dependent response functions which are the quantities most easily related to experimental measurements and potential applications. The notation 

The time-dependent field components are here taken to be the complex forms 

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We saw earlier that a very simple form of the dispersion energy is obtained from frequency-dependent polarizabilities at the so-called uncoupled Hartree-Fock level. The sum over states appearing in second order RS perturbation theory is simply a sum over (occupied and virtual) orbitals. A first improvement of this simple model is obtained by including apparent correlation [140], i.e. by using frequency-dependent polarizabilities obtained from the TDCHF method [36,141]. This method was initially proposed in the context of the multipole expansion, but could be generalized [142-146] to charge density susceptibility functions (or polarization propagators), which avoids the use... 

The application of the Lorentz-Lorenz equation gives a convincing demonstration of the general similarity of the linear response in gas and liquid but its application in the liquid introduces an approximation which has not yet been quantified. A more precise objective for the theory would be to calculate the frequency dependent susceptibility or refractive index directly. For a continuum model this may lead to a polarizability rigorously defined through the Lorentz-Lorenz equation as shown in treatments of the Ewald-Oseen theorem (see, for example Born and Wolf, plOO),59 but the polarizability defined in this way need not refer to one molecule and would not be precisely related to the gas parameters. 

It is apparent that non-linear-optical processes rely on a dynamic or frequency-dependent property. I will therefore, in general, restrict this article to calculations made at this level and, with one or two exceptions, I will consider only ab initio theory. Much work has been done on the static hyper-polarizabilities (as well as the static and dynamic polarizability a) but in order to make a direct connection with experiment, I have chosen to exclude this work. Also, again with one or two exceptions, I will only deal with molecules and exclude atoms. 

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