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The non-additive three-body interaction potential for helium was computedusing the coupled-cluster theory and the full configuration interaction method.The obtained potential comprises an improved nonrelativistic Born--Oppenheimerenergy and the leading relativistic and nuclear-motion corrections. The meanabsolute uncertainty of our calculations due to the incompleteness of theorbital basis set was determined employing complete-basis-set extrapolationtechniques and was found to be 1.2%. For three helium atoms forming anequilateral triangle with the side length of 5.6~bohr our three-body potentialamounts to $-$90.6~mK, with an estimated uncertainty of 0.5~mK. An analyticfunction, developed to accurately fit the computed three-body interactionenergies, was chosen to correctly describe the asymptotic behavior of thethree-body potential for trimer configurations corresponding to both thethree-atomic and the atom-diatom fragmentation channels. For large triangleswith sides $r_{12}$, $r_{23}$, and $r_{31}$, the potential takes correctly intoaccount 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 ashort-range analytic function describing the local behavior of the totaluncertainty of the computed three-body interaction energies. Using both fits wecalculated the third pressure and acoustic virial coefficients for helium andtheir uncertainties for a wide range of temperatures. The results of thesecalculations were compared with available experimental data and with previoustheoretical determinations. The estimated uncertainties of present calculationsare 3-5 times smaller than those reported in the best previous work.

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