We report the implementation of a general-order relativistic coupled-cluster method for performing high-precision calculations of atomic and molecular properties. As a first application, the black-body radiation shift of the Al${}^{+}$ clock has been estimated precisely. The computed shift relative to the frequency of the $3s^2\hspace{0.5em}{}^1{S}_0^e\to 3s3p\hspace{0.5em}{}^3{P}_0^o$ clock transition given by $(-3.66\pm 0.60)\times {10}^{-18}$ calls for an improvement over the recent measurement with a reported result of $(-9\pm 3)\times {10}^{-18}$ [Phys. Rev. Lett. 104, 070802 (2010)].

• Received 6 October 2010

DOI:https://doi.org/10.1103/PhysRevA.83.030503