  Fourth-order many-body corrections to matrix elements for atoms with one
valence electron are derived. The obtained diagrams are classified using
coupled-cluster-inspired separation into contributions from n-particle
excitations from the lowest-order wavefunction. The complete set of
fourth-order diagrams involves only connected single, double, and triple
excitations and disconnected quadruple excitations. Approximately half of the
fourth-order diagrams are not accounted for by the popular coupled-cluster
method truncated at single and double excitations (CCSD). Explicit formulae are
tabulated for the entire set of fourth-order diagrams missed by the CCSD method
and its linearized version, i.e. contributions from connected triple and
disconnected quadruple excitations. A partial summation scheme of the derived
fourth-order contributions to all orders of perturbation theory is proposed.
