On the products of bipolar harmonics

TL;DR

This paper develops formulas for products of bipolar harmonics to improve the accuracy and efficiency of calculating bound states in three-body quantum systems, demonstrated through high-precision calculations of two-electron ions.

Contribution

It introduces new finite-sum representations for bipolar harmonic products and their integrals, enabling more precise and faster computations of three-body quantum states.

Findings

Derived formulas for bipolar harmonic products and integrals.

Constructed compact variational wave functions for two-electron ions.

Achieved highly accurate energy calculations with 20-21 decimal digits.

Abstract

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