Reduced Density Matrix Cumulants: The Combinatorics of Size-Consistency and Generalized Normal Ordering

Abstract

Reduced density matrix cumulants play key roles in the theory of both reduced density matrices and multiconfigurational normal ordering, but the underlying formalism has remained mysterious. We present a new, simpler generating function for reduced density matrix cumulants that is formally identical to 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 reduced density matrices are entirely combinatorial constructions, where the differences can be associated to 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 the simplest proof to date of the Generalized Normal Ordering formalism to explore the role of reduced density matrix cumulants therein.

Challenges for variational reduced-density-matrix theory with three-particle N-representability conditions

“…[66][67][68][69][70] The 2-cumulant has the following nice properties that enhance its utility as a correlation metric: (i) its Frobenius norm is invariant to unitary transformations and (ii) the trace and square of the Frobenius norm of the 2-cumulant are additive. 67,69,71 To our knowledge, no studies have considered the information contained in the three-body cumulant RDM (the 3cumulant, 3 λ ) as a measure of correlation in many-body systems. This quantity contains information regarding pure three-body correlation effects, the magnitude of which can be quantified via the Frobenius norm of the 3-cumulant…”

Assessing the orbital-optimized unitary Ansatz for density cumulant theory

“…For exact wavefunctions, γ pq rs is multiplicatively separable, not additively separable, i.e., not size-consistent. 10 We may decompose it into size-consistent tensors with…”

“…It may be checked manually that if γ pq rs is multiplicatively separable, λ pq rs must be zero. 10 These size-consistent tensors are called RDM cumulants and denoted with λ. Because the 1RDM equals its cumulant, we shall usually refer to it with γ, as done in (5).…”

“…Therefore, the set of γ so constructed from d is precisely the set of solutions to (9). Equations ( 9) and (10) are merely necessary for γ and λ to be pure n-representable, not sufficient, but we shall not need sufficiency.…”

Analytic gradients for the single-reference driven similarity renormalization group second-order perturbation theory

“…Numerous efforts have been made to reduce the cost of high-order density matrices. For example, building and storing the 4-RDM can be avoided using a cumulant decomposition [29][30][31][32][33][34][35] and subsequently neglecting contributions from the 4-body density cumulant. This approach has lent itself to efficient and robust implementations of CASPT2 that can handle up to thirty active orbitals.…”