Solutions of the Schrödinger equation for anisotropic dipole-dipole interaction plus isotropic van der Waals interaction

By generalizing Bo Gao's approach [Phys. Rev. A 58, 1728 (1998)] for solving the Schrödinger equation for an isotropic van der Waals (vdW) potential to the systems with a multi-scale anisotropic long-range interaction, we derive the solutions for the Schrödinger equation for an anisotropic dipole-dipole interaction plus an isotropic attractive vdW potential, i.e., ${C_d(1-3\cos^2θ)}/{r^3}-{C_6}/{r^6}$, which is projected to the subspace with angular momentum $l\leq l_{\rm cut}$, with $l_{\rm cut}$ being an arbitrary angular-momentum cutoff. Here $θ$ is the polar angle of the coordinate $\boldsymbol{r}$ and $r=|\boldsymbol{r}|$. The asymptotic behaviors of these solutions for $r\rightarrow 0$ and $r\rightarrow \infty$ are obtained. These results can be used in the research of collisions and chemical reactions between ultra-cold polar molecules in a static electric field. Our approach to derive the solutions can be applied to the systems with a general long-range potential $\sum_{λ= 2}^{λ_{\rm max}} {V_λ(θ,φ)}/{r^λ}$, with $φ$ being the azimuthal angle of $\boldsymbol{r}$, and thus can be used in various problems on molecule-molecule interaction.

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