  We develop a general theory for thermal transport in anharmonic systems under
the weak system-bath coupling approximation similar to the quantum master
equation formalism. A current operator is derived, which is valid not only in
the steady state, but in the transient regime as well. Here we focus on the
effects of anharmonicity on the steady-state thermal conductance of a mono and
diatomic molecular junctions. We also study molecules being confined in a
double-well potential. We find that when the molecules have a non-linear
on-site potential the low-temperature thermal conductance is dramatically
affected by the strength of non-linearity, whereas for the diatomic molecule
connected by an anharmonic spring the strength of anharmonicity plays almost no
role in the low-temperature regime. In case of the molecules confined in a
double-well potential we find that the height of the barrier greatly affects
the thermal conductance; once the molecules can feel the effect of the barrier
we observe negative differential thermal conductance at both high and low
temperatures.
