  The study of open system quantum dynamics has been transformed by the
hierarchical equations of motion (HEOM) method, which gives the exact dynamics
for a system coupled to a harmonic bath at arbitrary temperature and
system-bath coupling strength. However in its standard form the method is only
consistent with the weak-coupling quantum master equation at all temperatures
when many auxiliary density operators are included in the hierarchy, even when
low temperature corrections are included. Here we propose a new low temperature
correction scheme for the termination of the hierarchy based on Zwanzig
projection which alleviates this problem, and restores consistency with the
weak-coupling master equation with a minimal hierarchy. The utility of the new
correction scheme is demonstrated on a range of model systems, including the
Fenna-Metthews-Olson complex. The new closure is found to improve convergence
of the HEOM even beyond the weak-coupling limit and is very straightforward to
implement in existing HEOM codes.
