  If design principles for controlling quantum interference in single molecules
could be elucidated and verified, then this will lay the foundations for
exploiting such effects in nanoscale devices and thin-film materials.When the
core of a graphene-like polyaromatic hydrocarbon (PAH) is weakly coupled to
external electrodes by atoms i and j, the single-molecule electrical
conductance sigma-ij depends on the choice of connecting atoms i,j.
Furthermore, conductance ratios sigma-ij/sigma-lm corresponding to different
connectivities i,j and l,m are determined by quantum interference within the
PAH core. In this paper, we examine how such conductance ratios change when one
of the carbon atoms within the "parent" PAH core is replaced by a heteroatom to
yield a "daughter" molecule. For bipartite parental cores, in which
odd-numbered sites are connected to even-numbered sites only, the effect of
heteroatom substitution onto an odd-numbered site is summarized by the
following qualitative rules:
  (a) When i and j are odd, both parent and daughter have low conductances (b)
When i is odd and j is even, or vice versa both parent and daughter have high
conductances (c) When i,j are both even, the parent has a low conductance and
the daughter a high conductance. These rules are verified by comparison with
density-functional calculations on naphthalene, anthracene, pyrene and
anthanthrene cores connected via two different anchor groups to gold
electrodes.
