Hierarchical equations of motion

In the remainder of this chapter, we now focus on one specific mefhod nof discussed so far, fhe Liouville-von Neumann approach in TDDFT. This scheme falls under point 2.2.2 from above as the contacts are treated as semi-infinite leads. In contrast to the approach by Kurth et al. [41], the basic variable is the Kohn-Sham one-particle density matrix rather than the Kohn-Sham orbitals. We begin with a discussion on the formal foundations of the method, before the relevant equations of motion are presented. Special emphasis is also laid on the exact treatment of the contacts in the hierarchical equation of motion approach. 

Hierarchical Equation of Motion for Reduced Single-Electron Density Matrix Based Liouville-von Neumann Equation... 

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,... 

Jin J, Zheng X, Yan YJ (2008) Exact dynamics of dissipative electronic systems and quantum transport hierarchical equations of motion approach. J Chem Phys 128 234703 Mathews J, Walker R (1970) Mathematical methods of physics. Benjamin, New York Croy A, Saalmann U (2009) Partial fraction decomposition of the Fermi function. Phys Rev B 80 073102. doi 10.1103/PhysRevB.80.073102. http //link.aps.org/doi/10.1103/ PhysRevB.80.073102... 

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