  We develop a new method to describe properties of truly open-shell nuclei.
This method is based on single-reference coupled-cluster theory and the
equation-of-motion method with extensions to nuclei with $A\pm 2$ nucleons
outside a closed shell. We perform proof-of-principle calculations for the
ground states of the helium isotopes $^{3-6}$He and the first excited $2^+$
state in $^6$He. The comparison with exact results from matrix diagonalization
in small model spaces demonstrates the accuracy of the coupled-cluster methods.
Three-particle--one-hole excitations of $^4$He play an important role for the
accurate description of $^6$He. For the open-shell nucleus $^6$He, the
computational cost of the method is comparable with the coupled-cluster
singles-and-doubles approximation while its accuracy is similar to
coupled-cluster with singles, doubles and triples excitations.
