Dipole dispersion response

Model and range

Use atomic units and two separated systems $A,B$ in an ordinary spatial three-plane, with nondegenerate ground states, positive excitation gaps $\Delta_a,\Delta_b$, and zero permanent dipoles. Assume separation $R$ large compared with their sizes, negligible exchange, and propagation delay short compared with the relevant response times. The leading instantaneous interaction is

$$V=R^{-3}d_A^i T_{ij}d_B^j,\qquad T_{ij}=\delta_{ij}-3\hat R_i\hat R_j.$$

This is a conditional ground-state response calculation. A classical field application must supply its fluctuation state and normalization; zero-point excitation weight is not implicit in the geometry.

Spectral sum

First order vanishes. The second-order ground energy is

$$\Delta E^{(2)}=-\frac{C_6}{R^6},\qquad C_6=\sum_{a,b}\frac{ |\langle0_A|d_A^i|a\rangle T_{ij} \langle0_B|d_B^j|b\rangle|^2}{\Delta_a+\Delta_b}\ge0.$$

All denominators are positive, fixing the attractive sign. Sums include complete excited multiplets and, where necessary, continuum integrals; convergence is an assumption on the response. For isotropic systems each spectral tensor is proportional to $\delta_{ij}$ and $\operatorname{Tr}T^2=6$. Writing $D_a=\sum_i|\langle0|d^i|a\rangle|^2$ with degenerate states grouped gives

$$C_6=\frac23\sum_{a,b}\frac{D_aD_b}{\Delta_a+\Delta_b}.$$

The sign and power have a direct second-order oscillator derivation in the van der Waals teaching article.

Imaginary-frequency form

Define isotropic polarizability at imaginary frequency $iu$ by

$$\alpha(iu)=\frac23\sum_a\frac{\Delta_aD_a}{\Delta_a^2+u^2}.$$

Partial fractions, including the continuous equal-gap limit, give

$$\int_0^\infty\frac{ab\,du}{(a^2+u^2)(b^2+u^2)} =\frac{\pi}{2(a+b)},\qquad a,b>0.$$

Substitution into the spectral sum therefore proves

$$\boxed{C_6=\frac3\pi\int_0^\infty\alpha_A(iu)\alpha_B(iu)\,du.}$$

For a single-resonance response $\alpha_X(iu)=\alpha_X(0)\Omega_X^2/(\Omega_X^2+u^2)$,

$$C_6=\frac32\alpha_A(0)\alpha_B(0) \frac{\Omega_A\Omega_B}{\Omega_A+\Omega_B}.$$

For equal resonances this is $3\alpha_A(0)\alpha_B(0)\Omega/4$. Replacing the resonance by an ionization scale is an additional approximation. These equations give a target for a V2 response calculation once its actual modes, dipole vertex, and state have been determined.

• Continuum mode memory — retains exact response before taking a decay approximation.

• Gaussian contact profiles — supplies short-range spatial overlaps for comparison with this long-range expansion.

• Resonance and profile tools — overview and verification.

resonance_profile_checks.py checks finite examples by quadrature, matrix evolution, and independent algebra.