fragmentslog1rho

Fragmentslog1rho is a term used in fragment-based approaches to quantum chemistry and quantum information theory to describe a fragment-level logarithmic functional of the one-particle reduced density matrix, ρ. It arises in methods that partition a larger system into smaller fragments to analyze local electronic structure and inter-fragment correlations. The construct provides a compact, entropy-like descriptor that can be computed from standard reduced density matrices produced by electronic-structure calculations.

For a system divided into fragments F, the fragment reduced density matrix ρ_F is obtained by tracing out the degrees of freedom outside F. The fragmentslog1rho value for fragment F is defined by applying the matrix logarithm to ρ_F and taking the trace, i.e., fragmentslog1rho(F) = Tr[ log(ρ_F) ]. In practice, ρ_F may have zero eigenvalues, so a small regularization term is added to ensure numerical stability, yielding Tr[ log(ρ_F + ε I_F) ].

The resulting quantity behaves as an entropy-like measure, capturing the occupancy distribution within a fragment and, indirectly, inter-fragment correlations. It is used as a diagnostic in fragmentation methods to guide partitioning, compare different fragmentation schemes, or monitor convergence between fragment-based approximations and full-system calculations. The concept is closely related to the log-determinant of ρ_F and to other entropy-like functionals derived from reduced density matrices.

Limitations include sensitivity to basis choice and fragmentation, potential singularities from zero eigenvalues, and interpretation that depends on the regularization scheme. It is most meaningful when used alongside other descriptors such as fragment entropies, mutual information, or correlation energies. See also density matrices, von Neumann entropy, and log-determinant measures.