On the products of bipolar harmonics

Abstract

The products of two and three bipolar harmonics ${\mathcal {Y}}^{\ell _1 \ell _2}_{LM}(\mathbf{r}_{31}, \mathbf{r}_{32})$ 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 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.