Casimir-Polder integrals

This coupling is not convenient. As will become clear below it is better to first couple the transition moments on each center. The dispersion energy becomes a sum of which the summand can be expressed with the use of Eq. (2) as a Casimir-Polder integral... 

A very similar equation holds for the multipole-expanded induction energy. The difference is that the polarizabilities are static, so that there is no Casimir-Polder integral, and that one of the irreducible polarizabilities is replaced by a Clebsch-Gordan coupled product of permanent multipole moments. 

The problem of computing dispersion energies is reduced to the computation of polarizabilities for a sufficient number of frequencies, so that the Casimir-Polder integral can be obtained by numerical quadrature [93]. An alternative to this quadrature is the substitution of the product of the polarizabilities by a sum over Hartree-Fock orbitals... 

The key is the replacement of the energy denominator via the Casimir-Polder identity, which can be established by a simple contour integration ... 

London s eqn. (15) for the dipole-dipole dispersion energy is not a simple product of properties of the separate atoms. A partial separation was achieved in 1948 by Casimir and Polder who expressed the /r dispersion energy as the product of the polarizability of each molecule at the imaginary frequency iu integrated over u from zero to infinity. The polarizability at imaginary frequencies may be a bizarre property but it is a mathematically well behaved function that decreases monotonically from the static polarizability at m = 0 to zero as u—> oo. 

Casimir and Polder [33] have shown that this identity, which can be proved easily by contonr integration, can be used to express the long-range coefficient Ce in terms of frequency-dependent monomer polarizabilities... 

Clayfield, E. J., E. C. Lumb, and R H. Mackey. 1971. Retarded dispersion forces in colloidal particles—Exact integration of the Casimir and Polder equation. Journal of Colloid and Interface Science 37, no. 2 382-389. doi 10.1016/0021-9797(71)90306-7. 

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