  We present a general-order spin-free formulation of the single-reference
closed-shell coupled-cluster method. We show that the working equations of a
fully biorthogonal contravariant projection formulation of the residual
equations, as near-universally used in closed-shell CCSD, can also be defined
at the CCSDT and CCSDTQ levels, despite singularities in the spin projection
manifolds. We describe permutation-group based techniques for obtaining and
simplifying the equations encountered in general second-quantization-based
methods; this includes a permutation group based approach of evaluating
second-quantized matrix elements into tensor contraction networks, and the use
of Portugal's double coset canonical representation technique [Int. J. Mod.
Phys. C 13, 859 (2002)] for eliminating redundant terms. A computer
implementation of our techniques is simple, because no operator-valued symbolic
algebra is required. Explicit working equation lists for closed-shell CCSD,
CCSDT, and CCSDTQ in the semi-biorthogonal formulation are provided. We also
release open-source computer programs for both deriving and numerically
evaluating these equations.

Chemical Physics