The HEOM Formalism

Modified Zusman Equation versus HEOM 13.3.1 The HEOM Formalism... 

We turn now to the HEOM formalism. We will demonstrate that it not just recovers the ZE, but also leads to a formally simple but physically significant modification. To proceed we focus on the following extended singleexponential form of bath correlation function (see eqn (13.2) for the parameters) ... 

The HEOM formalism for the bath correlation the form of eqn (13.23) reads ... 

The dynamics quantities in the HEOM formalism are a set of well-defined auxiliary density operators (ADOs), (p (r) = 0,1,...,, in which Po(t) = p t) is just the reduced system density operator. The hierarchy construction resolves not just system-bath coupling strengths but, more importantly, also memory time (l/yo) scales. ... 

This form could be rather general, provided the initial system-bath factorization ansatz, Px( ) = p(0)Pb. is applicable and the system Hamiltonian is time-independent. In contact with the HEOM formalism of eqn (13.27), we show in section 13.4.2 that the memory dissipation kernel in eqn (13.47), H(r), is the time-domain counterpart of a self-energy in the hierarchical Liouville space. 

In relating eqn (13.47) to the HEOM formalism presented in section 13.3, we start with the fact that the aforementioned initial system bath factorization ansatz corresponds to the initial ADOs of p (0) = p(0) 6, as inferred from the HEOM construction. Following the standard algebra, for example, in... 

Their evaluation goes by inward steps in a recursive and alternating manner, i.e. n and Q 5), with n = M,..., 0, initiated by setting n = 0 thus +D( ) = ( + Xeff)- via eqn (13.59) at a sufficiently large M by convergence. Interestingly, as the continued fraction always converges, the results above may have also mathematically proved this remarkable feature for the HEOM formalism. 

A hierarchical equation of motion (HEOM) [60, 61] is developed where for noninteracting systems, such as TDKS reference system in the present case, the hierarchy terminates exactly at the second-tier without any approximation [61], Within the TDDFT-NEGF-HEOM formalism, the Liouville-von Neumann equation reads,... 

The key quantity in quantum dissipative dynamics is the reduced system density operator, ps(t) = trBPT(0> Ihe bath-subspace trace over the total composite density operator. It is worth mentioning here that the harmonic bath described above assumes rather Gaussian statistics for thermal bath influence. Realistic anharmonic environments usually do obey Gaussian statistics in the thermodynamic mean field limit. For general treatment of nonperturbative and non-Markovian quantum dissipation systems, HEOM formalism has now emerged as a standard theory. It is discussed in the next section. 

For entangled iV-state transfer processes, the hierarchy Green s functions formalism involves tensors each having elements. As far as the kinetics regime is concerned, however, many interesting rate processes proceed practically in a step-wise manner. In this case the kinetics rate matrix could be determined by the individual rates between two states. Remarkably, the hierarchy Green s functions for the modified ZE/HEOM is analytically solvable for two-state systems. The resulting analytical expression of rate resolution, K/,j[s), for individual elementary a) b rate process can therefore be used to construct the NxN kinetics rate matrix. 

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