One-electron Reduced Density Matrix Research Articles

Development of a joint refinement model for the spin-resolved one-electron reduced density matrix using different data sets.

The paper describes a joint refinement model of the spin-resolved one-electron reduced density matrix using simultaneously magnetic structure factors and magnetic directional Compton profiles. The model is guided by two strategies: (i) variation of basis functions and (ii) variation of the spin population matrix. The implementation for a finite system is based on an expansion of the natural orbitals on basis sets. To show the potential benefits brought by the joint refinement model, the paper also presents the refinement results using magnetic structure factors only. The joint refinement model provides very satisfactory results reproducing the pseudo-data. In particular, magnetic Compton profiles have a strong effect not only on the off-diagonal elements of the spin-resolved one-electron reduced density matrix but also on its diagonal elements.

Two-point weighted density approximations for the kinetic energy density functional

We construct a model for the one-electron reduced density matrix that is symmetric and which satisfies the diagonal of the idempotency constraint and then use this model to evaluate the kinetic energy. This strategy for designing density functionals directly addresses the N-representability problem for kinetic energy density functionals. Results for atoms and molecules are encouraging, especially considering the simplicity of the model. However, like all of the other kinetic energy functionals in the literature, quantitative accuracy is not achieved.

Reduced density matrix embedding. General formalism and inter-domain correlation functional

An embedding method for a one-electron reduced density matrix (1-RDM) is proposed. It is based on partitioning of 1-RDM into domains and describing each domain in the effective potential of the other ones. To assure N-representability of the total 1-RDM N-representability and strong-orthogonality conditions are imposed on the domains. The total energy is given as a sum of single-domain energies and domain-domain electron interaction contributions. Higher than two-body inter-domain interaction terms are neglected. The two-body correlation terms are approximated by deriving inter-domain correlation from couplings of density fluctuations of two domains at a time. Unlike in most density embedding methods kinetic energy is treated exactly and it is not required that densities pertaining to the domains are only weakly overlapping. We propose to treat each domain by a corrected perfect-pairing functional. On a few examples it is shown that the embedding reduced density matrix functional method (ERDMF) yields excellent results for molecules that are well described by a single Lewis structure even if strong static intra-domain or dynamic inter-domain correlation effects must be accounted for.

Strong Electron Correlation in Materials from Pair-Interacting Model Hamiltonians

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 cond...

Communication: satisfying fermionic statistics in the modeling of open time-dependent quantum systems with one-electron reduced density matrices.

For an open, time-dependent quantum system, Lindblad derived the most general modification of the quantum Liouville equation in the Markovian approximation that models environmental effects while preserving the non-negativity of the system's density matrix. While Lindblad's modification is correct for N-electron density matrices, solution of the Liouville equation with a Lindblad operator causes the one-electron reduced density matrix (1-RDM) to violate the Pauli exclusion principle. Consequently, after a short time, the 1-RDM is not representable by an ensemble N-electron density matrix (not ensemble N-representable). In this communication, we derive the necessary and sufficient constraints on the Lindbladian matrix within the Lindblad operator to ensure that the 1-RDM remains N-representable for all time. The theory is illustrated by considering the relaxation of an excitation in several molecules F2, N2, CO, and BeH2 subject to environmental noise.

Generalized Pauli conditions on the spectra of one-electron reduced density matrices of atoms and molecules

The Pauli exclusion principle requires the spectrum of the occupation numbers of the one-electron reduced density matrix (1-RDM) to be bounded by one and zero. However, for a 1-RDM from a wave function, there exist additional conditions on the spectrum of occupation numbers, known as pure $N$-representability conditions or generalized Pauli conditions. For atoms and molecules, we measure through a Euclidean-distance metric the proximity of the 1-RDM spectrum to the facets of the convex set (polytope) generated by the generalized Pauli conditions. For the ground state of any spin symmetry, as long as time-reversal symmetry is considered in the definition of the polytope, we find that the 1-RDM's spectrum is pinned to the boundary of the polytope. In contrast, for excited states, we find that the 1-RDM spectrum is not pinned. Proximity of the 1-RDM to the boundary of the polytope provides a measurement and classification of electron correlation and entanglement within the quantum system. For comparison, this distance to the boundary of the generalized Pauli conditions is also compared to the distance to the polytope of the traditional Pauli conditions, and the distance to the nearest 1-RDM spectrum from a Slater determinant. We explain the difference in pinning in the ground- and excited-state 1-RDMs through a connection to the $N$-representability conditions of the two-electron reduced density matrix.

Excitation energies from range-separated time-dependent density and density matrix functional theory

Time-dependent density functional theory (TD-DFT) in the adiabatic formulation exhibits known failures when applied to predicting excitation energies. One of them is the lack of the doubly excited configurations. On the other hand, the time-dependent theory based on a one-electron reduced density matrix functional (time-dependent density matrix functional theory, TD-DMFT) has proven accurate in determining single and double excitations of H(2) molecule if the exact functional is employed in the adiabatic approximation. We propose a new approach for computing excited state energies that relies on functionals of electron density and one-electron reduced density matrix, where the latter is applied in the long-range region of electron-electron interactions. A similar approach has been recently successfully employed in predicting ground state potential energy curves of diatomic molecules even in the dissociation limit, where static correlation effects are dominating. In the paper, a time-dependent functional theory based on the range-separation of electronic interaction operator is rigorously formulated. To turn the approach into a practical scheme the adiabatic approximation is proposed for the short- and long-range components of the coupling matrix present in the linear response equations. In the end, the problem of finding excitation energies is turned into an eigenproblem for a symmetric matrix. Assignment of obtained excitations is discussed and it is shown how to identify double excitations from the analysis of approximate transition density matrix elements. The proposed method used with the short-range local density approximation (srLDA) and the long-range Buijse-Baerends density matrix functional (lrBB) is applied to H(2) molecule (at equilibrium geometry and in the dissociation limit) and to Be atom. The method accounts for double excitations in the investigated systems but, unfortunately, the accuracy of some of them is poor. The quality of the other excitations is in general much better than that offered by TD-DFT-LDA or TD-DMFT-BB approximations if the range-separation parameter is properly chosen. The latter remains an open problem.

Analysis of self-consistency effects in range-separated density-functional theory with Møller-Plesset perturbation theory

Range-separated density-functional theory combines wave function theory for the long-range part of the two-electron interaction with density-functional theory for the short-range part. When describing the long-range interaction with non-variational methods, such as perturbation or coupled-cluster theories, self-consistency effects are introduced in the density functional part, which for an exact solution requires iterations. They are generally assumed to be small but no detailed study has been performed so far. Here, the authors analyze self-consistency when using Møller-Plesset-type (MP) perturbation theory for the long range interaction. The lowest-order self-consistency corrections to the wave function and the energy, that enter the perturbation expansions at the second and fourth order, respectively, are both expressed in terms of the one-electron reduced density matrix. The computational implementation of the latter is based on a Neumann series which, interestingly, even though the effect is small, usually diverges. A convergence technique, which perhaps can be applied in other uses of Neumann series in perturbation theory, is proposed. The numerical results thus obtained show that, in weakly bound systems, self-consistency can be neglected since the long-range correlation does not affect the density significantly. Although MP is not adequate for multireference systems, it can still be used as a reliable analysis tool. Though the density change is not negligible anymore in such cases, self-consistency effects are found to be much smaller than long-range correlation effects (less than 10% for the systems considered). For that reason, a sensible approximation might be to update the short-range energy functional term while freezing its functional derivative, namely, the short-range local potential, in the wave function optimization. The accuracy of such an approximation still needs to be assessed.

The Fukui matrix: a simple approach to the analysis of the Fukui function and its positive character

The Fukui matrix is introduced as the derivative of the one-electron reduced density matrix with respect to a change in the number of electrons under constant external potential. The Fukui matrix extends the Fukui function concept: the diagonal of the Fukui matrix is the Fukui function. Diagonalizing the Fukui matrix gives a set of eigenvectors, the Fukui orbitals, and accompanying eigenvalues. At the level of theory used, there is always one dominant eigenvector, with an eigenvalue equal to 1. The remaining eigenvalues are either zero or come in pairs with eigenvalues of the same magnitude but opposite sign. Analysis of the frontier molecular orbital coefficient in the eigenvector with eigenvalue 1 gives information on the quality of the frontier molecular orbital picture. The occurrence of negative Fukui functions can be easily interpreted in terms of the nodal character of the dominant eigenvector versus the characteristics of the remaining eigenvectors and eigenvalues.

Electronegativity and Electron Currents in Molecular Tunnel Junctions

Electronegativity is shown to control charge transfer, energy level alignments, and electron currents in single molecule tunnel junctions, all of which are described through the density matrix. Currents calculated from the one-electron reduced density matrix correct to second order in electron−electron correlation are identical to currents obtained from the one-electron Green’s function corrected to second order in electron self-energy. A tight binding model of hexa-1,3,5-triene-1,6-dithiol bonded between metal electrodes is introduced, and the effect of analytically varying electron−electron correlation on electron currents and electronegativity is examined. The model analysis is compared to electronic structure descriptions of a gold−hexatriene (approximated by different exchange−correlation functionals) and Hartree−Fock states as zeroth-order approximations to the one-electron Green’s function. Comparison between the model calculations and the electronic structure treatment allows us to relate the abil...

Long-range density-matrix-functional theory: Application to a modified homogeneous electron gas

We propose a method that employs functionals of the one-electron reduced density matrix (density matrix) to capture long-range effects of electron correlation. The complementary short-range regime is treated with density functionals. In an effort to find approximations for the long-range density-matrix functional, a modified power functional is applied to the homogeneous electron gas with Coulomb interactions replaced by their corresponding long-range counterparts. For the power $\ensuremath{\beta}=1/2$ and the range-separation parameter $\ensuremath{\omega}=1/{r}_{s}$, the functional reproduces the correlation and the kinetic correlation energies with a remarkable accuracy for intermediate and large values of ${r}_{s}$. Analysis of the Euler equation corresponding to this functional reveals correct ${r}_{s}$ expansion of the correlation energy in the limit of large ${r}_{s}$. The first expansion coefficient is in very good agreement with that obtained from the modified Wigner-Seitz model.

Nonequilibrium, steady-state electron transport with N-representable density matrices from the anti-Hermitian contracted Schrödinger equation

We study molecular conductivity for a one-electron, bath-molecule-bath model Hamiltonian. The primary quantum-mechanical variable is the one-electron reduced density matrix (1-RDM). By identifying similarities between the steady-state Liouville equation and the anti-Hermitian contracted Schrödinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. A 75, 022505 (2007)], we develop a way of enforcing nonequilibrium, steady-state behavior in a time-independent theory. Our results illustrate the relationship between current and voltage in molecular junctions assuming that the total number of electrons under consideration can be fixed across all driving potentials. The impetus for this work is a recent study by Subotnik et al. that also uses the 1-RDM to study molecular conductivity under different assumptions regarding the total number of electrons [J. E. Subotnik et al., J. Chem. Phys. 130, 144105 (2009)]. Unlike calculations in the previous study, our calculations result in 1-RDMs that are fully N-representable. The present work maintains N-representability through a bath-bath mixing that is related to a time-independent relaxation of the baths in the absence of the molecule, as governed by the ACSE. A lack of N-representability can be important since it corresponds to occupying energy states in the molecule or baths with more than one electron or hole (the absence of an electron) in violation of the Pauli principle. For this reason the present work may serve as an important, albeit preliminary, step in designing a 2-RDM/ACSE method for studying steady-state molecular conductivity with an explicit treatment of electron correlation.

Comment on “Momentum density and spatial form of correlated density matrix in model two-electron atoms with harmonic confinement”

In a recent paper A. Akbari, N. H. March, and A. Rubio [Phys. Rev. A 76, 032510 (2007)] investigated the one-electron reduced density matrix and the momentum density of several two-electron model atoms, including Hooke's atom. The method used by the authors for deriving an integral form of the momentum density is in fact the most suited for deriving a closed-form expression of the exact reciprocal form factor, in a special case. This function is important inasmuch as it reflects the off-diagonal side of the exact reduced density matrix.

Electronic current density expanded in natural orbitals

Quantum electronic currents can be determined directly from off-diagonals of the one electron reduced density matrix. Behaviour of the density matrix in a natural orbital expansion is well understood, whereas very little is known about the current density expressed in any orbital basis. Given the relationship of the current to the density matrix, it seems plausible that improved approximations to the reduced density matrix should lead to better descriptions for the current density. To examine how improved approximations to the reduced density matrix are related to the current density, an upper limit to the error resulting from truncating a natural orbital expansion of the current is found and related to the density matrix and kinetic energy expressed in the same basis. As a consequence, it is confirmed that quantum transport formulations leading to an accurate one-electron reduced density matrix will deliver controllable and well-behaved approximations for quantum electronic currents.

Determination of a one-electron reduced density matrix using a coupled pseudo-atom model and a set of complementary scattering data

A possible model of one-electron reduced density matrices is presented, adapted from the Hansen-Coppens pseudo-atomic description of electron density [Hansen & Coppens (1978). Acta Cryst. A34, 909-913]. Potential benefits from a joint refinement of the model from X-ray diffraction and deep inelastic scattering data are illustrated.

Reduced-density-matrix cumulants and correlation

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.

Grand-canonical-ensemble representability problem for the one-electron reduced density matrix

Grand-canonical-ensemble representability problem for the one-electron reduced density matrix

Coupled-perturbed density-matrix functional theory equations. Application to static polarizabilities

Starting from the variational equations for the natural occupation numbers and the recently proposed eigenequations for the natural spin-orbitals, we derive coupled-perturbed density-matrix equations that furnish a linear response of the one-electron reduced density matrix to a static perturbation when the total energy is a functional of the one-electron reduced density matrix. Cases when some occupation numbers achieve exactly 0 or 1 or when the total number of the particles in a system is not preserved are taken into consideration. The scheme is applied to computing static polarizabilities from two simple density-matrix functionals. The behavior of the functionals is erratic and they provide only little or no improvement over the coupled-perturbed Hartree-Fock results.

BestN-term approximation in electronic structure calculations I. One-electron reduced density matrix

We discuss best N -term approximation spaces for one-electron wavefunctions and reduced density matrices ρ emerging from Hartree-Fock and density functional theory. The approximation spaces for anisotropic wavelet tensor product bases have been recently characterized by Nitsche in terms of tensor product Besov spaces. We have used the norm equivalence of these spaces to weighted spaces of wavelet coefficients to proof that both and ρ are in for all with . Our proof is based on the assumption that the possess an asymptotic smoothness property at the electron-nuclear cusps.

New constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix

Three strict constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix (the 1-matrix) are obtained by combining its invariance and stationary properties with the extended Koopmans' theorem. The constraints relate the quantities derived from the functional pertaining to an N-electron system with those of its (N-1)-electron congener. Together with the N-representability requirement for the 1-matrix of the congener, identities involving the electron-electron repulsion energies of the two systems and their derivatives with respect to the 1-matrices seriously narrow down the choices for potential approximate density-matrix functionals. This fact is well illustrated in the case of two-electron systems, where the validity of the new constraints is confirmed and found to originate from a nontrivial cancellation among different terms. Thus, the constraints provide a new tool for the construction and testing of new functionals that complements the previously known conditions such as the reproduction of the homogeneous gas energies and momentum distributions, convexity, and the N-representability of the associated 2-matrices.