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

Abstract

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., $C_d(1-3{\cos }^2\hat{I}¸)/r^3-C_6/r^6$, which is projected to the subspace with angular momentum $l\le l_{\mathrm{cut}}$, with $l_{\mathrm{cut}}$ being an arbitrary angular-momentum cutoff. Here $\hat{I}¸$ is the polar angle of the coordinate $\mathbf{r}$ and $r=|\mathbf{r}|$. The asymptotic behaviors of these solutions for $r\to 0$ and $r\to \infty$ 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 ${\sum }_{\hat{I}»=2}^{\hat{I}{»}_{\max }}{V}_{\hat{I}}»(\hat{I}¸,\ddot{I}†)/r^{\hat{I}}»$, with $\ddot{I}†$ being the azimuthal angle of $\mathbf{r}$, and thus can be used in various problems on molecule-molecule interaction.