We report the implementation of a general-order relativistic coupled-cluster method for performing highprecision 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 The role of high-precision calculations of various properties of heavy atoms and molecules which support the stateof-the-art measurements has gained incredible importance in recent years. This is particularly true in the context of atomic clocks [1,2], probes of fundamental symmetry violations [3][4][5][6], and search for the variation in the fundamental constants [7]. The relativistic coupled-cluster (CC) method with single and double excitations (CCSD) supplemented by the important triple excitations has yielded reasonably accurate results [3][4][5]8]. However, an extension to this method by including higher excitations and its application to large systems are extremely challenging. The general nonrelativistic CC approach of Kállay and co-workers provides one of the most efficient routes to the incorporation of higher excitations by exploiting the features of the many-body diagrammatic techniques and using automated programming tools based on the strings of spin-orbital labels [9].In this Rapid Communication, we extend the general-order nonrelativistic CC work [9] to the relativistic framework and we also consider its extension to linear-response (LR) calculations which is the first application of its kind to atoms. The new relativistic CC code has a great potential for high-precision studies in several important areas of fundamental physics; to mention a few: atomic clocks, parity nonconservation, electric dipole moment due to parity, and time reversal violations. As a proof of principle, we have employed the method for the calculation of the black-body radiation (BBR) shift of the 3s + . This transition provides the basis for the most accurate atomic clock to date [1,7,10], for which the fractional frequency inaccuracy has recently been estimated as 8.6 × 10 −18 [7]. Although the size of the BBR shift in the Al + clock is smaller than those in most of the other ions considered for atomic clocks, the associated uncertainty in the estimated BBR shift is about 35% of the total uncertainty when calculated using static polarizabilities and oscillator strengths taken from different sources [11]. Other semiempirical configuration interaction (CI) calculations [12] and nonrelativistic results [13][14][15] agree with this estimate.The exact wave function in the single-reference (SR) CC theory involves an exponential parametrization of the formwhere |0 is the Dirac-Hartree-Fock (DHF) reference determinant, and the cluster operatorT can be decomposed aŝ T = n k=1T k , wherê