On the products of bipolar harmonics

Abstract

The products of the two and three bipolar harmonics ${\mathcal{Y}}_{LM}^{{\ell }_1{\ell }_2}({\mathbf{r}}_{31},{\mathbf{r}}_{32})$ are represented as the finite sums of powers of the three relative coordinates $r_{32},r_{31}$ and $r_{21}$. The complete (angular+radial) integrals of the products of the two and three bipolar harmonics in the basis of exponential radial functions are expressed as finite sums of the auxiliary three-particle integrals ${\Gamma }_{n,k,l}(\alpha ,\beta ,\gamma )$. The formulas derived in this study can be used to accelerate highly accurate computations of the rotationally excited (bound) states in arbitrary three-body systems. In particular, we have constructed compact (400-term) variational wave functions for the triplet and singlet $2P(L=1)-$states in light two-electron atoms and ions. Highly accurate calculations (20 - 21 stable decimal digits in the total energy) of the triplet and singlet $2P(L=1)-$states in the two-electron Li${}^{+}$, Be${}^{2+}$, B${}^{3+}$ and C${}^{4+}$ ions are performed for the first time