  A nonperturbative theory is developed, aiming at an exact and efficient
evaluation of a general quantum system interacting with arbitrary bath
environment at any temperature and in the presence of arbitrary time-dependent
external fields. An exact hierarchical equations of motion formalism is
constructed on the basis of calculus-on-path-integral algorithm, via the
auxiliary influence generating functionals related to the interaction bath
correlation functions in a parametrization expansion form. The corresponding
continued-fraction Green's functions formalism for quantum dissipation is also
presented. Proposed further is the principle of residue correction, not just
for truncating the infinite hierarchy, but also for incorporating the small
residue dissipation that may arise from the practical difference between the
true and the parametrized bath correlation functions. The final
residue-corrected hierarchical equations of motion can therefore be used
practically for the evaluation of arbitrary dissipative quantum systems.
