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By generalizing Bo Gao's approach [Phys. Rev. A 58, 1728 (1998)] for solvingthe Schr\"{o}dinger equation for an isotropic van der Waals (vdW) potential tothe systems with a multi-scale anisotropic long-range interaction, we derivethe solutions for the Schr\"{o}dinger equation for an anisotropic dipole-dipoleinteraction plus an isotropic attractive vdW potential, i.e.,${C_d(1-3\cos^2\theta)}/{r^3}-{C_6}/{r^6}$, which is projected to the subspacewith angular momentum $l\leq l_{\rm cut}$, with $l_{\rm cut}$ being anarbitrary angular-momentum cutoff. Here $\theta$ is the polar angle of thecoordinate $\boldsymbol{r}$ and $r=|\boldsymbol{r}|$. The asymptotic behaviorsof these solutions for $r\rightarrow 0$ and $r\rightarrow \infty$ are obtained.These results can be used in the research of collisions and chemical reactionsbetween ultra-cold polar molecules in a static electric field. Our approach toderive the solutions can be applied to the systems with a general long-rangepotential $\sum_{\lambda= 2}^{\lambda_{\rm max}}{V_\lambda(\theta,\varphi)}/{r^\lambda}$, with $\varphi$ being the azimuthalangle of $\boldsymbol{r}$, and thus can be used in various problems onmolecule-molecule interaction.

Solutions of the Schr\"{o}dinger equation for anisotropic dipole-dipole interaction plus isotropic van der Waals interaction