van der Waals interaction of atoms in circular Rydberg states

Analytical expressions for the constants of resolution in powers of inverse interatomic distance 1∕R are determined for the energy of the long-range interaction between two atoms A and B in their circular Rydberg states of equal principal quantum numbers n_A = n_B ≡ n ≫ 1, maximal angular momenta and magnetic quantum numbers l_A = l_B = |m_A| = |m_B| = n − 1 in the case of arbitrary orientation of the interatomic axis relative the axis of the m-quantization. Coefficients of the odd-power terms C_k∕R^k are derived for k = 5, 7, 9 from the first-order perturbation theory (PT) for the interaction Hamiltonian. The constant C₆ of the van der Waals interaction ΔE_vdW = −C₆∕R⁶, determined from the second-order PT, is resolved into irreducible components. The combinations of irreducible components determine coefficients at Legendre polynomials and/or cosine functions describing the dependence of C₆ on the angle between the interatomic and quantization axes. The asymptotic dependence on the principal quantum number C₆ ∝ n¹² is demonstrated analytically. Polynomials in powers of n are derived for extrapolation of calculated data to the case of arbitrary n. The transformation of attractive van der Waals force (C₆ > 0) for low-energy states n < 6 into repulsive force (C₆ < 0) for all higher-energy states of n ≥ 8 is demonstrated in numerical calculations. Graphical abstract: .