Reduced Density Matrices

Gustavo E. Massaccesi, P. Capuzzi, Ofelia B. Oña +4 more

The N-representability problem of reduced density matrices represents a fundamental challenge in electronic structure theory. In this work, we focus on the N-representability of transition reduced density matrices and propose a practical approach to embed a p-body transition reduced density matrix (RDM) of an N-particle system into a (p + 1)-body RDM of an (N + 1)-particle system. This embedding allows us to apply a unitary evolution algorithm based on a recently developed adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to determine the N-representability of reduced density matrices. The algorithm works by effectively applying a sequence of unitary transformations to a given (N + 1)-particle initial state in such a way that the distance of a projection of its corresponding (p + 1)-body RDM (embedded p-body transition RDM) to a target is minimized. Our methodology enables the purification of transition reduced density matrices and the reconstruction of approximate wave functions involved in the studied transitions, providing an effective strategy to correct and refine transition density matrices. We validate our approach with numerical simulations on N-particle systems, including a three-electron model system and the H3 molecule, demonstrating its robustness and accuracy.

Jonathon P. Misiewicz, Justin M. Turney, Henry F. Schaefer

Reduced density matrix cumulants play key roles in the theory of both reduced density matrices and multiconfigurational normal ordering. We present a new, simpler generating function for reduced density matrix cumulants that is formally identical with equating the coupled cluster and configuration interaction ansätze. This is shown to be a general mechanism to convert between a multiplicatively separable quantity and an additively separable quantity, as defined by a set of axioms. It is shown that both the cumulants of probability theory and the reduced density matrices are entirely combinatorial constructions, where the differences can be associated with changes in the notion of "multiplicative separability" for expectation values of random variables compared to reduced density matrices. We compare our generating function to that of previous works and criticize previous claims of probabilistic significance of the reduced density matrix cumulants. Finally, we present a simple proof of the generalized normal ordering formalism to explore the role of reduced density matrix cumulants therein. While the formalism can be used without cumulants, the combinatorial structure of expressing RDMs in terms of cumulants is the same combinatorial structure on cumulants that allows for a simple extended generalized Wick's theorem.

Jerzy Ciosłowski

Preface. I: Properties of Reduced Density Matrices. 1. RDMs: How Did We Get Here? A.J. Coleman. 2. Some Theorems on Uniqueness and Reconstruction of Higher-Order Density Matrices M. Rosina. 3. Cumulant Expansions of Reduced Densities, Reduced Density Matrices, and Green's Functions P. Ziesche. 4. On Calculating Approximate and Exact Density Matrices R. Erdahl, B. Jin. II: The Contracted Schroedinger Equation. 5. Density Equation Theory in Chemical Physics H. Nakatsuji. 6. Critical Questions Concerning Iterative Solution of the Contracted Schroedinger Equation C. Valdemoro, L.M. Tel, E. Perez-Romero 7. Cumulants and the Contracted Schroedinger Equation D.A. Mazziotti.. III: Density Matrix Functional Theory. 8. Natural Orbital Functional Theory S. Goedecker, C.J. Umrigar. 9. The Pair Density in Approximate Density Functional Theory: The Hidden Agent N.T. Maitra, K. Burke. 10. Functional N-representability in Density Matrix and Density Functional Theory: An Illustration for Hooke's Atom E.V. Ludena, V. Karasiev, A. Artemiev, D. Gomez.. IV: Electron Intracule and Extracule Densities. 11. Intracule and Extracule Densities: Historical Perspectives and Future Prospects E. Valderrama, J.M. Ugalde, R.J. Boyd. 12. Topology of Electron Correlation J. Cioslowski. 13. Electron-Pair Densities of Atoms T. Koga. Index.