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Physical Review A

Reduced-density-matrix cumulants and correlation

John E. Harriman

Published Mar 21, 2007• 13 citations
Publisher

Abstract

Spin components for the Grassmann product of the one-electron reduced density matrix with itself are compared with those of the two-electron reduced density matrix for the same state. It is found that the Grassmann product introduces some inappropriate behaviors which must be removed by the cumulant but are not what is usually thought of as correlation.

DOI
10.1103/physreva.75.032513

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The Journal of Chemical Physics•Nov 7, 2006•82 citations
Spin components for the Grassmann product of the one-electron reduced density matrix with itself are compared with those of the two-electron reduced density matrix for the same state. It is found that the Grassmann product introduces some inappropriate behaviors which must be removed by the cumulant but are not what is usually thought of as correlation.
Some properties of Grassmann products of one-electron operators and of cumulants and related quantities for two-electron reduced density matrices (2-RDMs) are reported. The results suggest that any physical significance of the Grassmann product plus cumulant decomposition must be based on the physics of the system described, not just on the mathematical structure. The Grassmann product plus cumulant decomposition of the 2-RDM is resolved into components based on unitary invariants. Simple examples using mathematically acceptable but physically unlikely wave functions show that a Grassmann product differing from the usual one may better approximate the 2-RDM, and that the usual Grassmann product, cumulant, and 2-RDM can be mutually proportional.
We propose a novel interpretation of the reduced density matrix (RDM) and its cumulant that combines linear and exponential parametrizations of the wavefunction. Any n-particle RDM can be written as a weighted average of "configuration interaction" amplitudes. The corresponding n-particle cumulant is represented in terms of two types of contributions: "connected" (statistical averages of substitution amplitudes) and "disconnected" (cross-correlations of substitution amplitudes). A diagonal element of n-RDM represents the average occupation number of the orbital n-tuple. The diagonal elements of 2- and 3-cumulants take particularly elegant forms in the natural spin-orbital basis: they represent the covariances (correlated fluctuations) of the occupation numbers of the orbital pair and triples, respectively. Thus, the diagonal elements of the cumulants quantify the correlation between the orbital occupation numbers. Our interpretation is used to examine the weak to strong correlation transition in the "two electrons in two orbitals" problem.
The formulation of the density matrix functional theory (DMFT) in which the correlation component U of the electron–electron repulsion energy is expressed in terms of a model two-electron density cumulant matrix (a.k.a. the 2-cumulant) and the tensor of two-electron integrals g is critically analyzed. The dependence of on both the vector of occupation numbers n and g derived from the corresponding vector (1) of natural spinorbitals, and its concomitant invariance to the nature of the spin-independent interparticle interaction potential that enters the definition of g are emphasized. In the case of spin-polarized systems, (and thus U ) is found to be a function of not only n and g, but also of the vector and the matrix, where is the spin-flip operator. The presence of spin polarization imposes additional constraints upon, including a simple condition that, when satisfied, assures the underlying wavefunction being an eigenstate of both the and operators with the eigenvalues and, respectively. This feature allows targeting electronic states with definite multiplicities, which is virtually impossible in the case of the Kohn–Sham implementation of density functional theory. Among the four possible pairing schemes for the natural spinorbitals that give rise to approximations employing with only two independent indices, three are found to result in unphysical constraints even for spin-unpolarized systems, whereas the failure of the fourth one turns out to be precipitated by the presence of spin polarization. Consequently, any implementation of DMFT based upon “two-index” is shown to be generally unsuitable for spin-polarized systems (and incapable of yielding the spin-parallel components of U for the spin-unpolarized ones). A clear distinction is made between the genuine 1-matrix functionals that are defined for arbitrary N -representable 1-matrices and general energy expressions that depend on auxiliary quantities playing the role of fictitious 1-matrices subject to additional (often unphysical) constraints.
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.
Several measures of electron correlation are compared based on two criteria: (i) the presence of a unique mapping between the reduced variables in the measure and the many-electron wave function and (ii) the linear scaling of the measure and its variables with system size. We propose the squared Frobenius norm of the cumulant part of the two-particle reduced density matrix (2-RDM) as a measure of electron correlation that satisfies these criteria. An advantage of this cumulant-based norm is its ability to measure the correlation from spin entanglement, which is not contained in the correlation energy. Alternative measures based on the 2-RDM, such as the von Neumann entropy, do not scale linearly with system size. Properties of the measures are demonstrated with Be, F(2), HF, N(2), and a hydrogen chain.