  The time-convolutionless (TCL) quantum master equation provides a powerful
tool to simulate reduced dynamics of a quantum system coupled to a bath. The
key quantity in the TCL master equation is the so-called kernel or generator,
which describes effects of the bath degrees of freedom. Since the exact TCL
generators are usually hard to calculate analytically, most applications of the
TCL generalized master equation have relied on approximate generators using
second and fourth order perturbative expansions. By using the hierarchical
equation of motion (HEOM) and extended HEOM methods, we present a new approach
to calculate the exact TCL generator and its high order perturbative
expansions. The new approach is applied to the spin-boson model with different
sets of parameters, to investigate the convergence of the high order expansions
of the TCL generator. We also discuss circumstances where the exact TCL
generator becomes singular for the spin-boson model, and a model of excitation
energy transfer in the Fenna-Matthews-Olson complex.
