Extension of the Cumulant Definition for Low Particle Number Systems

Cumulants are usually interpreted as the connected components of density matrices, but this interpretation fails and practical problems arise when the rank n of cumulants is larger than the number of particles (N) in the system. In that case, cumulants defined in the traditional way become disconnected. To solve this problem, the definition of cumulants is extended by introducing a simple modulation factor. The modified cumulants reduce to the conventional definition, but they vanish when N < n. Using the modified definition also eliminates the error in the approximation of density matrices by low-rank cumulants, when N < n. The problem assumes a slightly different form when we work with active space-based theories, and it can be solved by a similar approach. Another problem with cumulants, due to spin coupling [Herbert, Int J Quantum Chem 2007, 107, 703], can be solved via the introduction of a similar modulation factor. A related yet more serious issue, termed as the local particle number constraint problem, is also discussed. (C) 2010 Wiley Periodicals, Inc. Int J Quantum Chem 111: 3541-3547, 2011