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.
A new entropy measure is proposed for the molecular electronic structure theory. The corresponding many-electron, Shannon-like measure reflects quantum entanglement between electronic subsystems with different spins. The related Renyi entropy is introduced, and its connection with the previously considered collectivity analysis is discussed. Investigation of the second-order reduced density matrix (2-RDM) allows one to define an irreducible, size-consistent component of the 2-RDM entropy and to find the irreducible component of the 2-RDM squared norm. Specific computations on several small molecules are carried out at the full configuration interaction level, providing new insights into non-classical, long-range electron correlations and quantum entanglement in these systems. The local spin analysis and the spin-correlator analysis reveal nontrivial spin structure underlying the entangled states.
Recent ultrafast spectroscopy experiments have demonstrated the entanglement of chromophores in photosynthetic light harvesting. Here we apply the cumulant parts of reduced density matrices (RDMs) to measure the statistical dependence and entanglement between electronic quantum domains in light harvesting. Because the cumulant RDMs are invariant to one-electron unitary transformations, they provide a measure of electron correlation and entanglement that is independent of the orbital basis set. Specifically, we apply the cumulant to a three-chromophore subsystem of the Fenna-Matthews-Olson complex, which has been shown to exhibit similar entanglement as the full system. Time-dependent Frobenius norms of the cumulant
p-RDMs for $2\ensuremath{\le}p\ensuremath{\le}6$ reveal correlation and entanglement in groupings of
p chromophores. The results show that the entanglement of pairs of chromophores is significantly more important than the entanglement in higher groupings of the chromophores. Data from the model are generally consistent with recent findings from ultrafast spectroscopy. Beyond their application to light harvesting, the Frobenius norms of the cumulant RDMs provide a useful measure of statistical dependence, correlation, and entanglement between electronic quantum domains with applications to molecules and materials.
Off-diagonal long-range order (ODLRO) in the two-electron reduced density matrix (2-RDM) has long been recognized as a mathematical characteristic of conventional superconductors. The large eigenvalue of the 2-RDM has been shown to be a useful measure of this long-range order. The 2-RDM can be represented as the sum of a connected (cumulant) piece and an unconnected piece. In this work, we show that the cumulant 2-RDM also has a large eigenvalue in the limit of ODLRO. The largest eigenvalue of the cumulant 2-RDM, we prove, is bounded from above by
N. In the limit of extreme pairing, such as Cooper pairing, the largest eigenvalue and the trace of the cumulant 2-RDM approach their extreme values of
N and $\ensuremath{-}N$, respectively. While the trace of the cumulant 2-RDM, which is computable from only a knowledge of the 1-RDM, can reflect ODLRO, it alone does not appear to be a sufficient criterion. The large eigenvalue of the cumulant 2-RDM, we show, implies the large eigenvalue of the 2-RDM and, hence, is a natural measure of ODLRO that vanishes in the mean-field limit.
An improved parametrization of the two-electron reduced density matrix (2-RDM) [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)] was recently shown to yield energies and properties that are markedly better than those calculated by traditional ab initio methods of similar computational scaling. In this paper a family of such energy functionals, generalizing the ones obtained previously, is derived through the use of (i)
p-particle contraction relations based on the contraction of the cumulant expansions of
p-particle RDMs and (ii) Cauchy-Schwarz relations that arise from an important set of
N-representability constraints known as the two-positivity conditions. The 2-RDMs are explicitly parameterized in terms of the first-order part of the cumulant 2-RDM and, for the inclusion of single excitations, a second-order part of the 1-RDM. In contrast to earlier formulations based on the coefficients from configuration interaction with single and double excitations (CISD), the cumulant-based parametric 2-RDM methods, from the properties of cumulants, are rigorously size extensive. We also show that writing the energy functionals in terms of correlated 1-RDMs and cumulant 2-RDMs reduces the computational cost of the parametric 2-RDM methods to that of CISD. Applications are made to ground-state energies of several molecules, equilibrium bond distances, and frequencies of HF,
F2, and CO, the relative energy of the cis and trans isomers of
HO3${}^{\ensuremath{-}}$, and the HCN-HNC isomerization reaction. For bond breaking in hydrogen fluoride the improved and more efficient parametric 2-RDM methods yield energies with similar accuracies at both equilibrium and nonequilibrium geometries in 6-31G** and polarized valence quadruple-$\ensuremath{\zeta}$ basis sets. Computed 2-RDMs very nearly satisfy well-known
N-representability conditions.
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.
Strong electron correlation in materials is explored within a class of model Hamiltonians that treat only pair interactions between electrons. The model is unique among typical spin Hamiltonians in that it does not have an effective mean-field reference wave function. The ground-state wave functions from all Hamiltonians in the model have the same one-electron reduced density matrix (1-RDM); consequently, one-electron theories such as the Hartree–Fock and density functional theories are inapplicable. In contrast, the ground-state two-electron reduced density matrix (2-RDM) has a one-to-one mapping to the ground-state wave function. For a range of lattices including linear, ladder, and square topologies, we variationally compute the 2-RDM subject to constraints, known as N -representability conditions, that are necessary for the 2-RDM to represent an N -electron ensemble density matrix. We find that for all model Hamiltonians the 2-RDM is accurately computed as long as the D, Q, and G N -representability conditions are supplemented with the T 2 condition. Energies, orbital and geminal occupation numbers, and correlation functions are computed. The model has applications to superconductivity as well as more general pairing phenomena in electronic systems. Effective methods are needed to treat strong electron correlation arising in the study of materials including the study of high-temperature superconductivity and phase transitions.