The electron affinity (EA) calculations based on the equation-of-motion coupled cluster method proved to be an efficient scheme in the treatment of potential energy curves (PECs) for alkali molecular ions, Me. The EA approach provides description of states obtained by an attachment of one electron to the reference, which for the Me is a doubly ionized Me system. The latter has a very concrete advantage in the calculations of the PECs, since it dissociates into the closed-shell fragments (Me → Me + Me); hence, the restricted Hartree-Fock reference can be used in the whole range of interatomic distances. In this work accurate PECs and spectroscopic constants are obtained for the six lowest-lying electronic states of the K ion. The relativistic effects are included by adding appropriate terms of the Douglas-Kroll Hamiltonian to the one-electron integrals.