One-electron Reduced Density Matrices Research Articles

N-representability violations in truncated equation-of-motion coupled-cluster methods.

One-electron reduced density matrices (1RDMs) from equation-of-motion (EOM) coupled-cluster with single and double excitations (CCSD) calculations are analyzed to assess their N-representability (i.e., whether they are derivable from a physical N-electron state). We identify EOM-CCSD stationary states whose 1RDMs violate either ensemble-state N-representability conditions or pure-state conditions known as generalized Pauli constraints. As such, these 1RDMs do not correspond to any physical N-electron state. Unphysical states are also encountered in the course of time-dependent EOM-CC simulations; when an external field drives transitions between a pair of stationary states with pure-state N-representable 1RDMs, the 1RDM of the time-dependent state can violate ensemble-state conditions. These observations point to potential challenges in interpreting the results of time-dependent EOM-CCSD simulations.

N-Representable one-electron reduced density matrices reconstruction at non-zero temperatures

This article retraces different methods that have been explored to account for the atomic thermal motion in the reconstruction of one-electron reduced density matrices from experimental X-ray structure factors (XSF) and directional Compton profiles (DCP). Attention has been paid to propose the simplest possible model, which obeys the necessary N-representability conditions, while accurately reproducing all available experimental data. The deconvolution of thermal effects makes it possible to obtain an experimental static density matrix, which can directly be compared with theoretical 1-RDM (reduced density matrix). It is found that above a 1% statistical noise level, the role played by Compton scattering data becomes negligible and no accurate 1-RDM is reachable. Since no thermal 1-RDM is available as a reference, the quality of an experimentally derived temperature-dependent matrix is difficult to assess. However, the accuracy of the obtained static 1-RDM, through the performance of the refined observables, is strong evidence that the Semi-Definite Programming method is robust and well adapted to the reconstruction of an experimental dynamical 1-RDM.

Construction of explicitly correlated one-electron reduced density matrices.

A general construction of an ensemble N-representable one-electron reduced density matrix Γ1(r1→';r→1) is presented. Unlike the conventional spectral representation, it explicitly incorporates the recently derived discontinuity in the fifth derivative of Γ1(r1→';r→1) with respect to |r1→'-r→1|. Its practical relevance in the context of the density-matrix functional theory is discussed.

Mean-field density matrix decompositions.

We introduce new and robust decompositions of mean-field Hartree-Fock and Kohn-Sham density functional theory relying on the use of localized molecular orbitals and physically sound charge population protocols. The new lossless property decompositions, which allow for partitioning one-electron reduced density matrices into either bond-wise or atomic contributions, are compared to alternatives from the literature with regard to both molecular energies and dipole moments. Besides commenting on possible applications as an interpretative tool in the rationalization of certain electronic phenomena, we demonstrate how decomposed mean-field theory makes it possible to expose and amplify compositional features in the context of machine-learned quantum chemistry. This is made possible by improving upon the granularity of the underlying data. On the basis of our preliminary proof-of-concept results, we conjecture that many of the structure-property inferences in existence today may be further refined by efficiently leveraging an increase in dataset complexity and richness.

Satisfying fermionic statistics in the modeling of non-Markovian dynamics with one-electron reduced density matrices.

Treatment of Markovian, many-electron dynamics from the solution of the Lindblad equation for the 1-electron reduced density matrix requires additional constraints on the bath operators to maintain fermion statistics. Recently, we generalized Lindblad's formalism to non-Markovian dynamics through an ensemble of Lindbladian trajectories. Here we show that the fermion statistics of non-Markovian dynamics can be enforced through analogous constraints on the bath operators of each Lindbladian trajectory in the ensemble. To illustrate, we apply the non-Markovian method to three distinct systems of two fermions in three levels. While the electrons violate the fermion statistics without the constraints, correct fermion behavior is recovered with the constraints.

Sparsity of the wavefunction from the generalized Pauli exclusion principle.

Electron occupations that arise from pure quantum states are restricted by a stringent set of conditions that are said to generalize the Pauli exclusion principle. These generalized Pauli constraints (GPCs) define the boundary of the set of one-electron reduced density matrices (1-RDMs) that are derivable from at least one N-electron wavefunction. In this paper, we investigate the sparsity of the Slater-determinant representation of the wavefunction that is a necessary, albeit not sufficient, condition for its 1-RDM to lie on the boundary of the set of pure N-representable 1-RDMs or in other words saturate one of the GPCs. The sparse wavefunction, we show, is exact not only for 3 electrons in 6 orbitals but also for 3 electrons in 8 orbitals. For larger numbers of electrons and/or orbitals in the lowest spin state, the exact wavefunction does not generally saturate one of the GPCs, and hence, the sparse representation is typically an approximation. Because the sparsity of the wavefunction is a necessary but not sufficient condition for saturation of one of the GPCs, optimization of the sparse wavefunction Ansatz to minimize the ground-state energy does not necessarily produce a wavefunction whose 1-RDM exactly saturates one of the GPCs. While the sparse Ansatz can be employed with arbitrary orbitals or optimized orbitals, in this paper, we explore the Ansatz with the natural orbitals from full configuration interaction, which yields an upper bound to the ground-state energy that equals the exact energy for a given basis set if the full-configuration-interaction wavefunction saturates the Ansatz's GPC. With calculations on the boron isoelectronic sequence, the dinitrogen cation N2+, hydrogen chains, and cyclic conjugated π systems, we examine the quality of the sparse wavefunction Ansatz from the amount of correlation energy recovered.

Noise-assisted energy transfer from the dilation of the set of one-electron reduced density matrices

Noise-assisted energy transfer can be explained geometrically in terms of the set of one-electron reduced density matrices (1-RDMs) [R. Chakraborty and D. A. Mazziotti, Phys. Rev. A 91, 010101(R) (2015)]. In this paper, we examine the geometric picture of quantum noise for the seven-chromophore Fenna-Matthews-Olson (FMO) complex. Noise expands the feasible set of orbital occupation trajectories to the target state through the violation of the pure-state N-representability conditions on the 1-RDM, known as the generalized Pauli constraints. While the generalized Pauli constraints are not explicitly known for seven-electron systems, we are able to treat a seven-exciton model of the FMO complex through the use of generalized Pauli constraints for p qubits which are known for arbitrary p. In the model, we find that while dephasing noise alone produces a trajectory of ensemble states that neither expands the set of 1-RDMs nor reaches the reaction center, the inclusion of both dephasing and dissipation expands the set of 1-RDMs and exhibits an efficient energy transfer to the reaction center. The degree to which the noise expands the set of 1-RDMs, violating the generalized Pauli constraints, is quantified by the distance of the 1-RDM outside its pure set to the distance of the 1-RDM inside its ensemble set. The geometric picture of energy transfer has applications to general quantum systems in chemistry and physics.

One-electron reduced density matrices of strongly correlated harmonium atoms.

Explicit asymptotic expressions are derived for the reduced one-electron density matrices (the 1-matrices) of strongly correlated two- and three-electron harmonium atoms in the ground and first excited states. These expressions, which are valid at the limit of small confinement strength ω, yield electron densities and kinetic energies in agreement with the published values. In addition, they reveal the ω(5/6) asymptotic scaling of the exchange components of the electron-electron repulsion energies that differs from the ω(2/3) scaling of their Coulomb and correlation counterparts. The natural orbitals of the totally symmetric ground state of the two-electron harmonium atom are found to possess collective occupancies that follow a mixed power/Gaussian dependence on the angular momentum in variance with the simple power-law prediction of Hill's asymptotics. Providing rigorous constraints on energies as functionals of 1-matrices, these results are expected to facilitate development of approximate implementations of the density matrix functional theory and ensure their proper description of strongly correlated systems.

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.

Exchange-Dispersion Energy: A Formulation in Terms of Monomer Properties and Coupled Cluster Treatment of Intramonomer Correlation.

A new formulation of the second-order exchange-dispersion energy of symmetry-adapted perturbation theory is presented. The proposed approach allows to study - for the first time for many-electron monomers - the effect of intramonomer electron correlation on this energy correction. In the new formalism the exchange-dispersion energy is expressed in terms of properties of interacting molecules, such as dynamic density-matrix susceptibilities and one-electron reduced density matrices. The resulting formula has been used to obtain the main (noncumulant) part of the exchange-dispersion energy for monomers described by coupled cluster theory limited to single and double excitations. A density fitting approach has been applied in order to reduce the computational effort for obtaining coupled cluster density-matrix susceptibilities. The new coupled cluster exchange-dispersion energy has been compared with other available values of this interaction energy component, obtained with monomers treated on the level of Hartree-Fock or density-functional theories.

Projected gradient algorithms for Hartree-Fock and density matrix functional theory calculations

We present projected gradient algorithms designed for optimizing various functionals defined on the set of N-representable one-electron reduced density matrices. We show that projected gradient algorithms are efficient in minimizing the Hartree-Fock or the Muller-Buijse-Baerends functional. On the other hand, they converge very slowly when applied to the recently proposed BBk (k=1,2,3) functionals [O. Gritsenko et al., J. Chem. Phys. 122, 204102 (2005)]. This is due to the fact that the BBk functionals are not proper functionals of the density matrix.

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.

Partial fourth order electron propagator theory

Third order diagonal self-energies are augmented with fourth order terms that are easily determined from by-products of a third order calculation. Comparisons are made with the outer valence approximation, a method for estimating fourth and higher order terms. The number of operations needed for all of these procedures is less than the number required for the limited two electron integral transformation. Improved one-electron reduced density matrices are another useful by-product of these calculations. The present methods are applied to calculating the ionization energies and dipole moments of the anions F−, OH−, NH, Cl−, SH−, PH2−, and CN−. The basis sets used in these calculations have been shown to be sufficiently flexible in a variety of calculations.