  The analytic gradient theory for both iterative and non-iterative
coupled-cluster approximations that include connected quadruple excitations is
presented. These methods include, in particular, CCSDT(Q), which is an analog
of the well-known CCSD(T) method which starts from the full CCSDT method rather
than CCSD. The resulting methods are implemented in the CFOUR program suite,
and pilot applications are presented for the equilibrium geometries and
harmonic vibrational frequencies of the simplest Criegee intermediate,
CH$_2$OO, as well as to the isomerization pathway between dimethylcarbene and
propene. While all methods are seen to approximate the full CCSDTQ results well
for "well-behaved" systems, the more difficult case of the Criegee intermediate
shows that CCSDT(Q), as well as certain iterative approximations, display
problematic behavior.
