Two-electron Reduced Density Matrix Research Articles

Universal Generalization of Density Functional Theory for Static Correlation.

A major challenge for density functional theory (DFT) is its failure to treat static correlation, yielding errors in predicted charges, band gaps, van der Waals forces, and reaction barriers. Here we combine one- and two-electron reduced density matrix (1- and 2-RDM) theories with DFT to obtain a universal O(N^{3}) generalization of DFT for static correlation. Using the lowest unitary invariant of the cumulant 2-RDM, we generate a 1-RDM functional theory that corrects the convexity of any DFT functional to capture static correlation in its fractional orbital occupations. Importantly, the unitary invariant yields a predictive theory by revealing the dependence of the correction's strength upon the trace of the two-electron repulsion matrix. We apply the theory to the barrier to rotation in ethylene, the relative energies of the benzynes, as well as an 11-molecule, dissociation benchmark. By inheriting the computational efficiency of DFT without sacrificing the treatment of static correlation, the theory opens new possibilities for the prediction and interpretation of significant quantum molecular effects and phenomena.

Influence of the N-representability conditions on the variational determination of two-electron reduced density matrices for ground and excited N-electron states in the doubly occupied configuration interaction space.

In this work, we analyze the effectiveness of different sets of well-known necessary N-representability conditions that are imposed on the variational determination of two-electron reduced density matrices corresponding to the ground and excited states of N-electron Hamiltonians within doubly occupied configuration interaction subspaces. The two-electron reduced density matrix elements are determined so that they are suitably related to those of higher-order reduced density matrices and other associated matrices, up to a given order, constituting positive semidefinite matrices. These conditions up to two-, three-, and four-order have been sequentially imposed on the variational calculation of the expectation values of the dispersion operator corresponding to several Hamiltonian models, obtaining the electronic spectra. These results have been contrasted with those arising from the full configuration interaction method, illuminating the sufficiency/insufficiency of the variational N-representability conditions applied.

Challenges for Variational Reduced-Density-Matrix Theory: Total Angular Momentum Constraints.

The variational two-electron reduced density matrix (v2RDM) method is generalized for the description of total angular momentum (J) and projection of total angular momentum (MJ) states in atomic systems described by nonrelativistic Hamiltonians, and it is shown that the approach exhibits serious deficiencies. Under ensemble N-representability constraints, v2RDM theory fails to retain the appropriate degeneracies among various J states for fixed spin (S) and orbital angular momentum (L), and for fixed L, S, and J, the manifold of MJ states is not necessarily degenerate. Moreover, a substantial energy error is observed for a system for which the two-electron reduced density matrix is exactly ensemble N-representable; in this case, the error stems from violations in pure-state N-representability conditions. Unfortunately, such violations do not appear to be good indicators of the reliability of energies from v2RDM theory in general. Several states are identified for which energy errors are near zero and yet pure-state conditions are clearly violated.

Variational determination of the two-electron reduced density matrix within the doubly occupied configuration interaction framework: Treatments of triplet N-electron systems.

In this work, we perform variational calculations of two-electron reduced density matrices corresponding to open-shell N-electron systems within the framework of the doubly occupied configuration interaction treatment, traditionally limited to studies of closed-shell systems. This has allowed us to provide a satisfactory description of molecular systems in triplet states following two methods. One of them adds hydrogen atoms at an infinite distance of the triplet system studied, constituting a singlet supersystem. The energies and reduced density matrices of the triplet system are obtained by removing the contributions of the added atoms from the singlet supersystem results. The second procedure involves variational determination of the two-electron reduced density matrices corresponding to the triplet systems by means of adequate couplings of basis-set functions. Both models have been studied by imposing N-representability conditions on the reduced density matrix calculations. Results obtained from these methods for molecular systems in triplet ground states are reported and compared with those provided by benchmark methods.

Resolving correlated states of benzyne with an error-mitigated contracted quantum eigensolver

The simulation of strongly correlated many-electron systems is one of the most promising applications for near-term quantum devices. Here we use a class of eigenvalue solvers (presented in Phys. Rev. Lett. 126, 070504 (2021)) in which a contraction of the Schr\"odinger equation is solved for the two-electron reduced density matrix (2-RDM) to resolve the energy splittings of ortho-, meta-, and para-isomers of benzyne ${\textrm C_6} {\textrm H_4}$. In contrast to the traditional variational quantum eigensolver, the contracted quantum eigensolver solves an integration (or contraction) of the many-electron Schr\"odinger equation onto the two-electron space. The quantum solution of the anti-Hermitian part of the contracted Schr\"odinger equation (qACSE) provides a scalable approach with variational parameters that has its foundations in 2-RDM theory. Experimentally, a variety of error mitigation strategies enable the calculation, including a linear shift in the 2-RDM targeting the iterative nature of the algorithm as well as a projection of the 2-RDM onto the convex set of approximately $N$-representable 2-RDMs defined by the 2-positive (DQG) $N$-representability conditions. The relative energies exhibit single-digit millihartree errors, capturing a large part of the electron correlation energy, and the computed natural orbital occupations reflect the significant differences in the electron correlation of the isomers.

Cooper-pair condensates with nonclassical long-range order on quantum devices

An important problem in quantum information is the practical demonstration of non-classical long-range order on quantum computers. One of the best known examples of a quantum system with non-classical long-range order is a superconductor. Here we achieve Cooper pairing of qubits on a quantum computer to represent superconducting or superfluid states. We rigorously confirm the quantum long-range order by measuring the large $O(N)$ eigenvalue of the two-electron reduced density matrix. The demonstration of maximal quantum long-range order is an important step towards more complex modeling of superconductivity and superfluidity as well as other phenomena with significant quantum long-range order on quantum computers.

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

The direct variational optimization of the two-electron reduced density matrix (2RDM) can provide a reference-independent description of the electronic structure of many-electron systems that naturally capture strong or nondynamic correlation effects. Such variational 2RDM approaches can often provide a highly accurate description of strong electron correlation, provided that the 2RDMs satisfy at least partial three-particle N-representability conditions (e.g., the T2 condition). However, recent benchmark calculations on hydrogen clusters [N. H. Stair and F. A. Evangelista, J. Chem. Phys. 153, 104108 (2020)] suggest that even the T2 condition leads to unacceptably inaccurate results in the case of two- and three-dimensional clusters. We demonstrate that these failures persist under the application of full three-particle N-representability conditions (3POS). A variety of correlation metrics are explored in order to identify regimes under which 3POS calculations become unreliable, and we find that the relative squared magnitudes of the cumulant three- and two-particle reduced density matrices correlate reasonably well with the energy error in these systems. However, calculations on other molecular systems reveal that this metric is not a universal indicator for the reliability of the reduced-density-matrix theory with 3POS conditions.

Quantum simulation of molecules without fermionic encoding of the wave function

Molecular simulations generally require fermionic encoding in which fermion statistics are encoded into the qubit representation of the wave function. Recent calculations suggest that fermionic encoding of the wave function can be bypassed, leading to more efficient quantum computations. Here we show that the two-electron reduced density matrix (2-RDM) can be expressed as a unique functional of the unencoded N-qubit-particle wave function without approximation, and hence, the energy can be expressed as a functional of the 2-RDM without fermionic encoding of the wave function. In contrast to current hardware-efficient methods, the derived functional has a unique, one-to-one (and onto) mapping between the qubit-particle wave functions and 2-RDMs, which avoids the over-parametrization that can lead to optimization difficulties such as barren plateaus. An application to computing the ground-state energy and 2-RDM of H4 is presented.

Conductance Switching in an Organometallic Single-Electron Transistor Using Current-Constrained Reduced-Density Matrix Theory.

We report switching of molecular conductance at finite bias in a binuclear organometallic complex and its cation which were previously experimentally analyzed at low voltages to see the signature of Kondo resonance. The variational reduced density matrix theory is applied to show that the system is strongly multireferenced especially in its charged form. We also study the molecular conductance of both forms using recently developed current-constrained two-electron reduced density matrix theory which is capable of handling strong electronic correlation. We compare the results against an uncorrelated 1-electron reduced density matrix version of conductance calculations using Hartree-Fock molecular orbitals. We observe that despite quantitative disagreements, the qualitative trend in the conductance is correctly predicted to be favorable for the cationic partner by both methods. We explain the results using the inherently high density of states for the low-lying excited states in the cationic partner which is also replicable from uncorrelated electronic structure methods. Our results not only indicate that the low-bias conductance trend is maintained even beyond the Kondo regime and produces quantitative agreement with that of the experiment but also identifies important physical markers that are responsible for the high conductance of the charged species.

Correlation-driven phenomena in periodic molecular systems from variational two-electron reduced density matrix theory.

Correlation-driven phenomena in molecular periodic systems are challenging to predict computationally not only because such systems are periodically infinite but also because they are typically strongly correlated. Here, we generalize the variational two-electron reduced density matrix (2-RDM) theory to compute the energies and properties of strongly correlated periodic systems. The 2-RDM of the unit cell is directly computed subject to necessary N-representability conditions such that the unit-cell 2-RDM represents at least one N-electron density matrix. Two canonical but non-trivial systems, periodic metallic hydrogen chains and periodic acenes, are treated to demonstrate the methodology. We show that while single-reference correlation theories do not capture the strong (static) correlation effects in either of these molecular systems, the periodic variational 2-RDM theory predicts the Mott metal-to-insulator transition in the hydrogen chains and the length-dependent polyradical formation in acenes. For both hydrogen chains and acenes, the periodic calculations are compared with previous non-periodic calculations with the results showing a significant change in energies and increase in the electron correlation from the periodic boundary conditions. The 2-RDM theory, which allows for much larger active spaces than are traditionally possible, is applicable to studying correlation-driven phenomena in general periodic molecular solids and materials.

Accurate singlet-triplet gaps in biradicals via the spin averaged anti-Hermitian contracted Schrödinger equation.

The accurate description of biradical systems, and in particular the resolution of their singlet-triplet gaps, has long posed a major challenge to the development of electronic structure theories. Biradicaloid singlet ground states are often marked by strong correlation and, hence, may not be accurately treated by mainstream, single-reference methods such as density functional theory or coupled cluster theory. The anti-Hermitian contracted Schrödinger equation (ACSE), whose fundamental quantity is the two-electron reduced density matrix rather than the N-electron wave function, has previously been shown to account for both dynamic and strong correlations when seeded with a strongly correlated guess from a complete active space (CAS) calculation. Here, we develop a spin-averaged implementation of the ACSE, allowing it to treat higher multiplicity states from the CAS input without additional state preparation. We apply the spin-averaged ACSE to calculate the singlet-triplet gaps in a set of small main group biradicaloids, as well as the organic four-electron biradicals trimethylenemethane and cyclobutadiene, and naphthalene, benchmarking the results against other state-of-the-art methods reported in the literature.

Dual-cone variational calculation of the two-electron reduced density matrix

The computation of strongly correlated quantum systems is challenging because of its potentially exponential scaling in the number of electron configurations. Variational calculation of the two-electron reduced density matrix (2-RDM) without the many-electron wave function exploits the pairwise nature of the electronic Coulomb interaction to compute a lower bound on the ground-state energy with polynomial computational scaling. Recently, a dual-cone formulation of the variational 2-RDM calculation was shown to generate the ground-state energy, albeit not the 2-RDM, at a substantially reduced computational cost, especially for higher $N$-representability conditions such as the T2 constraint. Here we generalize the dual-cone variational 2-RDM method to compute not only the ground-state energy but also the 2-RDM. The central result is that we can compute the 2-RDM from a generalization of the Hellmann-Feynman theorem. Specifically, we prove that in the Lagrangian formulation of the dual-cone optimization the 2-RDM is the Lagrange multiplier. We apply the method to computing the energies and properties of strongly correlated electrons -- including atomic charges, electron densities, dipole moments, and orbital occupations -- in an illustrative hydrogen chain and the nitrogen-fixation catalyst FeMoco. The dual variational computation of the 2-RDM with T2 or higher $N$-representability conditions provides a polynomially scaling approach to strongly correlated molecules and materials with significant applications in atomic and molecular and condensed-matter chemistry and physics.

Prediction of the Existence of LiCH: A Carbene-like Organometallic Molecule.

Carbenes comprise a well-known class of organometallic compounds each consisting of a neutral, divalent carbon and two unshared electrons. Carbenes can have singlet or triplet ground states, each giving rise to a distinct reactivity. Methylene (CH2), the parent hydride, is well-known to be bent in its triplet ground state. Here, we predict the existence of LiCH, a carbene-like organometallic molecule. Computationally, we treat the electronic structure with parametric and variational two-electron reduced density matrix (2-RDM) methods, which are capable of capturing multireference correlation typically associated with the singlet state of a diradical. Similar to methylene, LiCH is a triplet ground state with a predicted 15.8 kcal/mol singlet-triplet gap. However, unlike methylene, LiCH is linear in both the triplet state and the lowest excited singlet state. Furthermore, the singlet state is found to exhibit strong electron correlation as a diradical. In comparison to dissociation channels Li + CH and Li+ + CH-, the LiCH was found to be stable by approximately 77 kcal/mol.

Variational determination of the two-electron reduced density matrix within the doubly occupied configuration interaction scheme: An extension to the study of open-shell systems.

This work proposes to describe open-shell molecules or radicals using the framework of the doubly occupied configuration interaction (DOCI) treatments, so far limited to closed-shell system studies. The proposal is based on considering molecular systems in singlet states generated by adding extra hydrogen atoms located at infinite distance from the target radical system. The energy of this radical is obtained by subtracting the energies of the dissociated hydrogen atoms from that provided by the two-electron reduced density matrix corresponding to the singlet state system in the DOCI space, which is variationally calculated by imposing a set of N-representability conditions. This method is numerically assessed by describing potential energy curves and reduced density matrices in selected ionic and neutral open-shell systems in the doublet spin symmetry ground state.

Size-extensive seniority-zero energy functionals derived from configuration interaction with double excitations.

The doubly occupied configuration interaction (DOCI) approach can provide an accurate black-box description of nondynamic electron correlation at a computational cost that increases combinatorially with the system size. Remarkably, a pair coupled-cluster doubles (pCCD) approach (also known as the antisymmetrized product of one-reference orbital geminals) can reproduce DOCI energies with only a quadratic number of wave function parameters, and, when neglecting the cost associated with any two-electron integral transformations, these parameters can be determined at a cubic computational cost. Other simpler seniority-zero approaches derived from size-extensive modified configuration interaction doubles functionals can also provide approximations to DOCI energies at similar computational costs. We develop seniority-zero formulations of the coupled-electron pair approximation, the averaged coupled-pair functional, averaged quadratic coupled-cluster, and the parametric two-electron reduced density matrix (p2RDM) approach. These methods are Hermitian and thus offer several potential advantages over pCCD theory, including a reduction in the number of variable parameters and simplified definitions of reduced density matrices. Of the methods investigated, only the pair p2RDM (pp2RDM) approach yields energies that are comparable in quality to pCCD and DOCI. For the molecular systems investigated, pp2RDM-derived RDMs are found to be better approximations to DOCI ones than those obtained from pCCD.

Reduced Density Matrix-Driven Complete Active Apace Self-Consistent Field Corrected for Dynamic Correlation from the Adiabatic Connection.

The recently proposed multireference adiabatic connection (AC) formalism [Pernal, Phys. Rev. Lett. 120, 013001 (2018)] is applied to recover dynamic electron correlation effects lacking in variational two-electron reduced density matrix (v2RDM)-driven complete active space self-consistent field theory (CASSCF). The AC approach is validated by computing potential energy curves for the dissociation of molecular nitrogen and the symmetric double dissociation of H2O while enforcing two sets of approximate N-representability conditions in the underlying v2RDM-driven CASSCF calculations (either two-particle or two-particle plus partial three-particle conditions). The AC yields smaller absolute errors than second-order N-electron perturbation theory (NEVPT2) at all molecular geometries for both sets of the N-representability conditions considered. The efficacy of the approach for thermochemistry is also assessed for a set of 31 small-molecule reactions. When imposing partial three-particle N-representability conditions, mean and maximum unsigned errors in reaction energies from the AC are superior to those from NEVPT2.

Active-Space Pair Two-Electron Reduced Density Matrix Theory for Strong Correlation

An active-space variational calculation of the two-electron reduced density matrix (2-RDM) is derived and implemented where the active orbitals are correlated within the pair approximation. The pair approximation considers only doubly occupied configurations of the wave function, which enables the calculation of the 2-RDM at a computational cost of [Formula: see text]. Calculations were performed both with the pair active-space configuration interaction (PASCI) method and the pair active-space self-consistent field (PASSCF) method. The latter includes a mixing of the active and inactive orbitals through unitary transformations. The active-space pair 2-RDM method is applied to the nitrogen molecule, the p-benzyne diradical, a newly synthesized biscobalt complex, and the nitrogenase cofactor FeMoco. The FeMoco molecule is treated in a [120,120] active space. Fractional occupations are recovered in each of these systems, indicating the presence of strong electron correlation.

Entangled Electrons Drive a Non-superexchange Mechanism in a Cobalt Quinoid Dimer Complex

A central theme in chemistry is the understanding of the mechanisms that drive chemical transformations. A well-known, highly cited mechanism in organometallic chemistry is the superexchange mechanism in which unpaired electrons on two or more metal centers interact through an electron pair of the bridging ligand. We use a combination of novel synthesis and computation to show that such interactions may in fact occur by a more direct mechanism than superexchange that is based on direct quantum entanglement of the two metal centers. Specifically, we synthesize and experimentally characterize a novel cobalt dimer complex with benzoquinoid bridging ligands and investigate its electronic structure with the variational two-electron reduced density matrix method using large active spaces. The result draws novel connections between inorganic mechanisms and quantum entanglement, thereby opening new possibilities for the design of strongly correlated organometallic compounds whose magnetic and spin properties have applications in superconductors, energy storage, thermoelectrics, and spintronics.

Global Hybrid Multiconfiguration Pair-Density Functional Theory.

A global hybrid extension of multiconfiguration pair-density functional theory (MC-PDFT) is developed. Using a linear decomposition of the electron-electron repulsion term, a fraction λ of the nonlocal exchange interaction, obtained from variational two-electron reduced-density matrix (v2RDM)-driven complete active-space self-consistent field (CASSCF) theory, is combined with its local counterpart, obtained from an on-top pair-density functional. The resulting scheme (called λ-MC-PDFT) inherits the benefits of MC-PDFT (e.g., its simplicity and the resolution of the symmetry dilemma) and, when combined with the v2RDM approach to CASSCF, requires only polynomially scaling computational effort. As a result, λ-MC-PDFT can efficiently describe static and dynamical correlation effects in strongly correlated systems. The efficacy of the approach is assessed for several challenging multiconfigurational problems, including the dissociation of molecular nitrogen, the double dissociation of a water molecule, and the 1,3-dipolar cycloadditions of ozone to ethylene and ozone to acetylene in the O3ADD6 benchmark set.

An adiabatic connection for doubly-occupied configuration interaction wave functions.

An adiabatic connection (AC) is developed as an electron correlation correction for doubly occupied configuration interaction (DOCI) wave functions. Following the work of Pernal [Phys. Rev. Lett. 120, 013001 (2018)], the working equations of the approach, termed AC-DOCI, are rooted in the extended random phase approximation (ERPA) and require knowledge of only the ground-state two-electron reduced density matrix (2RDM) from the DOCI. As such, the AC is naturally suited to modeling electron correlation in variational 2RDM (v2RDM)-based approximations to the DOCI. The v2RDM-driven AC-DOCI is applied to the dissociation of molecular nitrogen and the double dissociation of water; the approach yields energies that are similar in quality to those from second-order multireference perturbation theory near equilibrium, but the quality of the AC-DOCI energy degrades at stretched geometries. The exact adiabatic connection path suggests the assumption that the one-electron reduced-density matrix is constant along the AC path is invalid at stretched geometries, but this deficiency alone cannot explain the observed behavior. Rather, it appears that the ERPA's single-particle-transition ansatz cannot, in general, provide good approximations to the 2RDM along the AC path. The AC-DOCI is also applied to a set of 45 reaction energies; for these systems, the approach has an average accuracy that is comparable to that of single-reference second-order many-body perturbation theory.