The products of the two and three bipolar harmonics ${ \cal Y}^{ \ell_1 \ell_2}_{ LM}({ \bf r}_{ 31}, { \bf r}_{ 32})$ 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
Categories: physics.atom-ph math-ph math.MP nucl-th