Long-Range (Retarded) Intermolecular Forces

The Casimir and Polder retarded dipole—dipole energy of interaction between two ground‐state (non‐degenerate) atoms is expressed in terms of sine and cosine integrals. This result should be accurate for all interatomic separations R. In the range of moderately large separations (discussed in the preceding paper), where the charge distributions do not overlap and where R is small compared to λ/₀=(αΔε)⁻¹, the Casimir and Polder results can be expanded in the form $$E_{\mathrm{int}}{\mathrm{=R}}^{\text{−6}}{C}_6{\mathrm{+\alpha }}^2{R}^{\text{−4}}{W}_4{\mathrm{+\alpha }}^3{R}^{\text{−3}}{\mathrm{A+\alpha }}^4{R}^{\text{−2}}\mathrm{B+···.}$$ This expansion is only accurate for R/λ/₀<0.6. Here α is the fine‐structure constant. The R⁻⁶C₆ term is the usual London dispersion energy. The α²R⁻⁴W₄ term was obtained in the preceding paper by taking the expectation value of the Breit—Pauli Hamiltonian using the wavefunction for the two‐atom system corrected for the classical electrostatic dipole—dipole interactions. Thus, at least in the dipole—dipole approximation, the Breit—Pauli Hamiltonian gives the energy of interaction accurate through O(α²). For large values of R/λ/₀, the interaction energy is expanded in the series $$E_{\mathrm{int}}{\mathrm{=\alpha }}^{\text{−1}}{R}^{\text{−7}}{\mathrm{D+\alpha }}^{\text{−3}}{R}^{\text{−9}}{\mathrm{F+\alpha }}^{\text{−5}}{R}^{\text{−11}}\mathrm{G+···.}$$ This large R expansion is only accurate for R/λ/₀>5.0.