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

Note that H I H_{I} is badly behaved when d = 1 d=1 . For example, there will in general be an (exponentially small) overlap between the electron wavefunctions, which will make ⟨ 0 | H I | 0 ⟩ \langle 0|H_{I}|0\rangle 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 H_{I} (in 1 / | 𝐑 | 1/|{\mathbf{R}}| ) we do not see this problem

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It might seem counter intuitive that for d = 1 , 2 d=1,2 the leading term of Δ ​ E 0 \Delta E_{0} comes from a subleading term in the expansion of H I H_{I} . If we consider ordinary functions f f and g g we find that the leading term of f ⁡ ( g ⁡ ( ϵ ) ) f(g(\epsilon)) comes from the leading term of g ⁡ ( ϵ ) g(\epsilon) . However, if g g is vector (or function) valued, one can construct examples where this intuitive picture fails. In our example g g corresponds to H I H_{I} as a function of 1 / | 𝐑 | 1/|{\mathbf{R}}| and f f corresponds to Δ ​ E 0 \Delta E_{0} (as a function of H I H_{I} )

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By rotational symmetry H A H_{A} commutes with the angular part of the Laplacian, ∇ S d − 1 , A \nabla_{S^{d-1},A} . We thus have common eigenvectors, | ψ A l , n ⟩ |\psi_{A}^{l,n}\rangle , of H A H_{A} and ∇ S d − 1 , A \nabla_{S^{d-1},A} such that ∇ S d − 1 , A | ψ A l , n ⟩ = − l ⁡ ( l + d − 2 ) ​ | ψ A l , n ⟩ \nabla_{S^{d-1},A}|\psi_{A}^{l,n}\rangle=-l(l+d-2)|\psi_{A}^{l,n}\rangle . For fixed | 𝐫 A | |{\mathbf{r}}_{A}| it follows that | ψ A l , n ⟩ |\psi_{A}^{l,n}\rangle is a spherical harmonic, but for spherical harmonics we have ( ℐ ​ ψ A l , n ) ​ ( 𝐫 A ) = ψ A l , n ​ ( − 𝐫 A ) = ( − 1 ) l ​ ψ A l , n ​ ( 𝐫 A ) (\mathcal{I}\psi_{A}^{l,n})({\mathbf{r}}_{A})=\psi_{A}^{l,n}(-{\mathbf{r}}_{A})=(-1)^{l}\psi_{A}^{l,n}({\mathbf{r}}_{A}) . We conclude that | ψ A l , n ⟩ |\psi_{A}^{l,n}\rangle is also an eigenvector of ℐ \mathcal{I} which means that ℐ \mathcal{I} and H A H_{A} commute

Cited in the paper.