  We demonstrate how redox control of intra-molecular quantum interference in
phase-coherent molecular wires can be used to enhance the thermopower (Seebeck
coefficient) S and thermoelectric figure of merit ZT of single molecules
attached to nanogap electrodes. Using first principles theory, we study the
thermoelectric properties of a family of nine molecules, which consist of
dithiol-terminated oligo(phenylene-ethynylenes) (OPEs) containing various
central units. Uniquely, one molecule of this family possesses a conjugated
acene-based central backbone attached via triple bonds to terminal sulfur atoms
bound to gold electrodes and incorporates a fully conjugated hydroquinone
central unit. We demonstrate that both S and the electronic contribution ZelT
to the figure of merit ZT can be dramatically enhanced by oxidizing the
hydroquinone to yield a second molecule, which possesses a cross-conjugated
anthraquinone central unit. This enhancement originates from the conversion of
the pi-conjugation in the former to cross-conjugation in the latter, which
promotes the appearance of a sharp anti-resonance at the Fermi energy.
Comparison with thermoelectric properties of the remaining seven conjugated
molecules demonstrates that such large values of S and ZelT are unprecedented.
We also evaluate the phonon contribution to the thermal conductance, which
allows us to compute the full figure of merit ZT = ZelT/(1 +
\k{appa}p/\k{appa}el), where \k{appa}p is the phonon contribution to the
thermal conductance and \k{appa}el is the electronic contribution. For
unstructured gold electrodes, \k{appa}p/\k{appa}el >> 1 and therefore
strategies to reduce \k{appa}p are needed to realise the highest possible
figure of merit.
