Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Chemical Theory and Computation

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

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

Published Aug 31, 2020• 19 citations
Publisher

Abstract

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.

DOI
10.1021/acs.jctc.0c00422

Related papers

Cumulant decomposition of reduced density matrices, multireference normal ordering, and Wicks theorem: A spin-free approach

K. R. Shamasundar

We propose a spin-free approach to the cumulant decomposition of reduced density matrices of singlet and spin-rotation or SU(2) invariant ensemble of nonsinglet states as in [W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 110, 2800 (1999); 116, 4787 (2002)]. We provide a simple recursive procedure to obtain expressions which relate different spin components of spin-orbital reduced density matrices and cumulants of such states to the spin-free counterparts. These results are used to set up a spin-summation procedure to arrive at a definition of spin-free cumulants of any order. Alternatively, an analytic formula for the spin-free form resulting from a spin summation involving product of two spin-orbital cumulants is derived and its utility in spin-free cumulant decomposition of reduced density matrices is demonstrated. This leads to suitable definitions of spin-free analog of multireference normal ordering and the associated Wicks theorem. The results of this formulation are expected to be useful in investigations of spin-free multireference internally contracted coupled-cluster methods where cumulant approximations to the active reduced density matrices are employed.

The Journal of Chemical Physics•Nov 4, 2009•49 citations

Cumulant expansion of the reduced density matrices

Werner Kutzelnigg, Debashis Mukherjee

k -particle cumulants λk (for 2⩽k⩽n) corresponding to the k-particle reduced density matrices γk for an n-fermion system are defined via a generating function. The two-particle cumulant λ2 describes two-particle correlations (excluding exchange), λ3 genuine three-particle correlations etc. The properties of these cumulants are analyzed. Conditions for vanishing of certain λk are formulated. Necessary and sufficient for λ2=0 is the well-known idempotency condition γ2=γ for γ≡γ1. For λ3=0 to hold, a general necessary condition is Tr{2γ3−3γ2+γ}=0, for three special forms of the wave function (arbitrary two-electron state, antisymmetrized product of strongly orthogonal geminals on antisymmetrized geminal power wave function of extreme type) 2γ3−3γ2+γ=0 turns out to be necessary and sufficient. For a multiconfiguration self-consistent field wave function the only nonvanishing matrix elements of the cumulants are those where all labels refer to active (partially occupied) spin orbitals. Spin-free cumulants Λk corresponding to the spin-free reduced density matrices Γk are also defined and analyzed. The main interest in the density cumulants is in connection with the recently formulated normal ordering and the corresponding Wick theorem for arbitrary reference functions, but they are also useful for an analysis of electron correlation.

The Journal of Chemical Physics•Feb 8, 1999•291 citations

Spinfree formulation of reduced density matrices, density cumulants and generalised normal ordering

Werner Kutzelnigg, K. R. Shamasundar, Debashis Mukherjee

The k-particle reduced density matrices γ k for a singlet state or a spin-orientation averaged ensemble are invariant under the spin rotation group SU(2), i.e. they are independent of spin, even if they are expressed in terms of spin orbitals. The γ k contain the same information as their spinfree counterparts Γ k expressed in terms of spinfree orbitals. However, the relation between the two quantities is, by no means, elementary. The summation over spin is particularly demanding and surprisingly complicated for the k-particle density cumulants λ k and for the formulation of a generalised normal ordering with respect to an arbitrary reference state, including the generalised Wick theorem. Plausible extrapolations from small k turned out to be unjustified and some published expressions for k = 4 need to be revised. A compact general result, valid for all k has not yet been found.

Molecular Physics•Feb 10, 2010•54 citations

Cumulant approach to the direct calculation of reduced density matrices: A critical analysis

Marcel Nooijen, Mark Wladyslawski, Anirban Hazra

In this paper we address a number of topics regarding the contracted Schrödinger equation (CSE): (i) The cumulant expansion of reduced density matrices (RDMs), and the reduction of the CSE to a connected form. (ii) The comparison of the resulting formalism to coupled cluster theory. (iii) The direct calculation of the two-particle RDM in terms of the cumulants, without explicitly imposing any N-representability conditions. We explore a number of approximations, the most elaborate of which is the 3,5-CSE in which the cumulant expansion is carried out to the three-particle level, and the only approximation made is to equate the four- and five-particle cumulants to zero. Overall the numerical results we present for selected small model systems are found to be disappointing and we provide a critical analysis of the perceived shortcomings of our cumulant approach to the CSE.

The Journal of Chemical Physics•Feb 28, 2003•42 citations

Correlated holes and their relationships with reduced density matrices and cumulants

Roberto C. Bochicchio, Alicia Torre, Luis Laín

This paper describes a matrix formulation for the correlated hole theory within the framework of the domain-averaged model in many electron systems (atoms, molecules, condensed matter, etc.). General relationships between this quantity and one-particle reduced density matrices for any independent particle or correlated state functions are presented. This formulation turns out to be suitable for computational purposes due to the straightforward introduction of cumulants of two-particle reduced density matrices within the quantum field structure. Numerical calculations in selected simple molecular systems have been performed in order to determine preliminary correlated values for such a quantity.

The Journal of Chemical Physics•Feb 18, 2005•14 citations

Many-Electron Densities and Reduced Density Matrices

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.

•Nov 2, 2012•155 citations
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.
We propose a spin-free approach to the cumulant decomposition of reduced density matrices of singlet and spin-rotation or SU(2) invariant ensemble of nonsinglet states as in [W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 110, 2800 (1999); 116, 4787 (2002)]. We provide a simple recursive procedure to obtain expressions which relate different spin components of spin-orbital reduced density matrices and cumulants of such states to the spin-free counterparts. These results are used to set up a spin-summation procedure to arrive at a definition of spin-free cumulants of any order. Alternatively, an analytic formula for the spin-free form resulting from a spin summation involving product of two spin-orbital cumulants is derived and its utility in spin-free cumulant decomposition of reduced density matrices is demonstrated. This leads to suitable definitions of spin-free analog of multireference normal ordering and the associated Wicks theorem. The results of this formulation are expected to be useful in investigations of spin-free multireference internally contracted coupled-cluster methods where cumulant approximations to the active reduced density matrices are employed.
k -particle cumulants λk (for 2⩽k⩽n) corresponding to the k-particle reduced density matrices γk for an n-fermion system are defined via a generating function. The two-particle cumulant λ2 describes two-particle correlations (excluding exchange), λ3 genuine three-particle correlations etc. The properties of these cumulants are analyzed. Conditions for vanishing of certain λk are formulated. Necessary and sufficient for λ2=0 is the well-known idempotency condition γ2=γ for γ≡γ1. For λ3=0 to hold, a general necessary condition is Tr{2γ3−3γ2+γ}=0, for three special forms of the wave function (arbitrary two-electron state, antisymmetrized product of strongly orthogonal geminals on antisymmetrized geminal power wave function of extreme type) 2γ3−3γ2+γ=0 turns out to be necessary and sufficient. For a multiconfiguration self-consistent field wave function the only nonvanishing matrix elements of the cumulants are those where all labels refer to active (partially occupied) spin orbitals. Spin-free cumulants Λk corresponding to the spin-free reduced density matrices Γk are also defined and analyzed. The main interest in the density cumulants is in connection with the recently formulated normal ordering and the corresponding Wick theorem for arbitrary reference functions, but they are also useful for an analysis of electron correlation.
The k-particle reduced density matrices γ k for a singlet state or a spin-orientation averaged ensemble are invariant under the spin rotation group SU(2), i.e. they are independent of spin, even if they are expressed in terms of spin orbitals. The γ k contain the same information as their spinfree counterparts Γ k expressed in terms of spinfree orbitals. However, the relation between the two quantities is, by no means, elementary. The summation over spin is particularly demanding and surprisingly complicated for the k-particle density cumulants λ k and for the formulation of a generalised normal ordering with respect to an arbitrary reference state, including the generalised Wick theorem. Plausible extrapolations from small k turned out to be unjustified and some published expressions for k = 4 need to be revised. A compact general result, valid for all k has not yet been found.
In this paper we address a number of topics regarding the contracted Schrödinger equation (CSE): (i) The cumulant expansion of reduced density matrices (RDMs), and the reduction of the CSE to a connected form. (ii) The comparison of the resulting formalism to coupled cluster theory. (iii) The direct calculation of the two-particle RDM in terms of the cumulants, without explicitly imposing any N-representability conditions. We explore a number of approximations, the most elaborate of which is the 3,5-CSE in which the cumulant expansion is carried out to the three-particle level, and the only approximation made is to equate the four- and five-particle cumulants to zero. Overall the numerical results we present for selected small model systems are found to be disappointing and we provide a critical analysis of the perceived shortcomings of our cumulant approach to the CSE.
This paper describes a matrix formulation for the correlated hole theory within the framework of the domain-averaged model in many electron systems (atoms, molecules, condensed matter, etc.). General relationships between this quantity and one-particle reduced density matrices for any independent particle or correlated state functions are presented. This formulation turns out to be suitable for computational purposes due to the straightforward introduction of cumulants of two-particle reduced density matrices within the quantum field structure. Numerical calculations in selected simple molecular systems have been performed in order to determine preliminary correlated values for such a quantity.
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.