Three-body potential and third virial coefficients for helium including relativistic and nuclear-motion effects

The non-additive three-body interaction potential for helium was computed using the coupled-cluster theory and the full configuration interaction method. The obtained potential comprises an improved nonrelativistic Born--Oppenheimer energy and the leading relativistic and nuclear-motion corrections. The mean absolute uncertainty of our calculations due to the incompleteness of the orbital basis set was determined employing complete-basis-set extrapolation techniques and was found to be 1.2%. For three helium atoms forming an equilateral triangle with the side length of 5.6~bohr our three-body potential amounts to $-$90.6~mK, with an estimated uncertainty of 0.5~mK. An analytic function, developed to accurately fit the computed three-body interaction energies, was chosen to correctly describe the asymptotic behavior of the three-body potential for trimer configurations corresponding to both the three-atomic and the atom-diatom fragmentation channels. For large triangles with sides $r_{12}$, $r_{23}$, and $r_{31}$, the potential takes correctly into account all angular terms decaying as $r_{12}^{-l} r_{23}^{-m} r_{31}^{-n}$ with $l+m+n \le 14$ for the nonrelativistic Born--Oppenheimer energy and $l+m+n \le 9$ for the post-Born--Oppenheimer corrections. We also developed a short-range analytic function describing the local behavior of the total uncertainty of the computed three-body interaction energies. Using both fits we calculated the third pressure and acoustic virial coefficients for helium and their uncertainties for a wide range of temperatures. The results of these calculations were compared with available experimental data and with previous theoretical determinations. The estimated uncertainties of present calculations are 3-5 times smaller than those reported in the best previous work.