Energy of van der Waals and dipole-dipole interactions between atoms in Rydberg states

The van der Waals coefficient C₆(θ;nlJM) of two like Rydberg atoms in their identical Rydberg states |nlJM is resolved into four irreducible components called scalar Rₛₛ, axial (vector) Rₐₐ, scalar-tensor RsT=RTs, and tensor-tensor RTT parts in analogy with the components of dipole polarizabilities. The irreducible components determine the dependence of C₆(θ;nlJM) on the angle θ between the interatomic and the quantization axes of atoms. The spectral resolution for the biatomic Green's function with account of the most contributing terms is used for evaluating the components R_αβ of atoms in their Rydberg series of doublet states of the low angular momenta (²S, ²P, ²D, ²F). The polynomial presentations in powers of the Rydberg-state principal quantum number n taking into account the asymptotic dependence C₆(θ;nlJM)∝n¹¹ are derived for simplified evaluations of irreducible components. Numerical values of the polynomial coefficients are determined for Rb atoms in their n0.222222em0ex²S1/2, n0.222222em0ex²P1/2,3/2, n0.222222em0ex²D3/2,5/2, and n0.222222em0ex²F5/2,7/2 Rydberg states of arbitrary high n. The transformation of the van der Waals interaction law -C₆/R⁶ into the dipole-dipole law C₃/R³ in the case of close dipole-connected two-atomic states (the F\"orster resonance) is considered and the dependencies on the magnetic quantum numbers M and on the angle θ of the constant C₃(θ;nlJM) are determined together with the ranges of interatomic distances R, where the transformation appears.