The van der Waals interaction in one, two, and three dimensions Available to Purchase

Note that H_I is badly behaved when d = 1. For example, there will in general be an (exponentially small) overlap between the electron wavefunctions, which will make $⟨0|HI|0⟩$ divergent. A physical realization of a one-dimensional system would of course be embedded in three dimensional space and have a nonzero thickness, which would regularize the divergence. Because we expand H_I (in 1/R), we do not see this problem.

It might seem counterintuitive that for d = 1, 2 the leading term of ΔE₀ comes from a subleading term in the expansion of H_I. If we consider ordinary functions f and g we find that the leading term of f(g(ϵ)) comes from the leading term of g(ϵ). However, if g is vector (or function) valued, one can construct examples where this intuitive picture fails. In our example g corresponds to H_I as a function of 1/R and f corresponds to ΔE₀ (as a function of H_I).

By rotational symmetry H_A commutes with the angular part of the Laplacian $∇Sd−1,A$⁠. We thus have common eigenvectors $|ψAl,n⟩$ of H_A and $∇Sd−1,A$ such that $∇Sd−1,A|ψAl,n⟩=−l(l+d−2)|ψAl,n⟩$⁠. For fixed $|rA|$ it follows that $|ψAl,n⟩$ is a spherical harmonic, but for spherical harmonics we have $(IψAl,n)(rA)=ψAl,n(−rA)=(−1)lψAl,n(rA)$⁠. We conclude that $|ψAl,n⟩$ is also an eigenvector of $I$⁠, which means that $I$ and H_A commute.