# The calculation of frequency-dependent polarizabilities as pseudo-energy derivatives

## Abstract

The definition of frequency-dependent polarizabilities α(- ω;ω),β(-2ω;ω,ω),β(-ω;ω,0), and β(0;ω, - ω) is discussed, and it is argued that the most convenient definitions are as energy derivatives, a pseudo-energy being defined as the expectation value of [H - i(∂/∂t)]. This definition outlines a straightforward procedure for obtaining frequency-dependent polarizabilities for all quantum chemistry methods including those which account for the effects of electron correlation. It is demonstrated at the self-consistent field level of theory that αλμ(-ω;ω) cos ωt may be considered as the derivative of the static dipole moment μ λ with respect to the strength Eωμ of a frequency-dependent field Eωμ cos ωt (as is usual), or as the derivative of an appropriately defined frequency-dependent dipole moment μμ cos ωt with respect to a static field E 0λ. In this way, polarizabilities may be determined from finite static field calculations on lower-order tensors. Therefore, α(- ω;ω) cos ωt is defined within second-order Møller-Plesset perturbation theory (MP2) as the second derivative of the MP2 energy with respect to one static and one frequency-dependent field. An analytic expression is given for αλμ(- ω;ω) at the MP2 level of theory. An MP2 frequency-dependent dipole expression is also defined, which if finite static field calculations are applied, gives the same values for αλμ(- ω;ω). MP2 values are reported for α(- ω;ω) of formaldehyde and ammonia for a range of frequency ω = 0.01-0.1 a.u. From comparison of the self-consistent field (SCF) and MP2 values of the frequency-dependent contribution to ᾱ(- ω;ω), it is concluded that it is appropriate to use an SCF frequency-dependent correction in conjunction with a static polarizability determined at a higher level of theory in order to obtain an accurate value for α(- ω;ω) of H 2CO in this frequency range. For ammonia, the frequency-dependent contribution to ā(- ω;ω) is more sensitive to electron correlation. Nevertheless, compared to the total polarizability ᾱ(- ω;ω), the error in the frequency-dependent contribution determined using the SCF method is small (∼2% at ω = 0.1 a.u.) © 1991 American Institute of Physics.