  In this paper, we report on a correctly scaling novel coupled cluster singles
and doubles (CCSD) implementation for arbitrary high-spin open-shell states.
The chosen cluster operator is completely spin-free, i.e. employs spatial
substitutions only. It is composed of our recently developed L\"owdin-type
operators (doi: 10.1063/5.0026762), which ensure (1) spin completeness and (2)
spin adaption i.e. spin purity of the CC wave function. In contrast to the
proof-of-concept matrix-representation-based implementation presented there,
the present implementation relies on second quantization and factorized tensor
contractions. The generated singles and doubles operators are embedded in an
equation generation engine. In the latter, Wick's theorem is used to derive
prefactors arising from spin integration directly from the spin-free full
contraction patterns. The obtained Wick terms composed of products of Kronecker
deltas are represented by special non-antisymmetrized Goldstone diagrams.
Identical (redundant) diagrams are identified by solving the underlying graph
isomorphism problem. All non-redundant graphs are then automatically translated
to locally -- one term at a time -- factorized tensor contractions. Finally,
the spin-adapted and spin-complete (SASC) CCS and CCSD variants are applied to
a set of small molecular test systems. Both correlation energies and amplitude
norms hint towards a reasonable convergence of the SASC-CCSD method for a BCH
series truncation of order four. In comparison to spin orbital CCSD, SASC-CCSD
leads to slightly improved correlation energies with differences of up to 0.886
mE_H (0.98% w.r.t. spin orbital CCSD(T)).
