THE CALCULATION OF FREQUENCY-DEPENDENT POLARIZABILITIES AS PSEUDO-ENERGY DERIVATIVES

The definition of frequency-dependent polarizabilities alpha(-omega;omega), beta(-2-omega;omega,omega), beta(-omega;omega,0), and beta(0;omega, -omega) 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(partial/partial 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 alpha-lambda-mu(-omega;omega) cos omega-t may be considered as the derivative of the static dipole moment mu-lambda with respect to the strength E-omega-mu of a frequency-dependent field E-omega-mu cos omega-t (as is usual), or as the derivative of an appropriately defined frequency-dependent dipole moment mu-mu cos omega-t with respect to a static field E(o)lambda. In this way, polarizabilities may be determined from finite static field calculations on lower-order tensors. Therefore, alpha(-omega;omega) cos omega-t is defined within second-order Moller-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 alpha-lambda-mu(-omega;omega) 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 alpha-lambda-mu(-omega;omega). MP2 values are reported for alpha(-omega;omega) of formaldehyde and ammonia for a range of frequency omega = 0.01-0.1 a.u. From comparison of the self-consistent field (SCF) and MP2 values of the frequency-dependent contribution to alphaBAR(-omega;omega), 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 alphaBAR(-omega;omega) of H2CO in this frequency range. For ammonia, the frequency-dependent contribution to alphaBAR(-omega;omega) is more sensitive to electron correlation. Nevertheless, compared to the total polarizability alphaBAR(-omega;omega), the error in the frequency-dependent contribution determined using the SCF method is small (approximately 2% at omega = 0.1 a.u.)