One-electron Reduced Density Matrix Research Articles

N-representable one-electron reduced density matrix reconstruction with frozen core electrons.

Recent advances in quantum crystallography have shown that, beyond conventional charge density refinement, a one-electron reduced density matrix (1-RDM) satisfying N-representability conditions can be reconstructed using jointly experimental X-ray structure factors and directional Compton profiles (DCP) through semidefinite programming. So far, such reconstruction methods for 1-RDM, not constrained to idempotency, have been tested only on a toy model system (CO2). In this work, a new method is assessed on crystalline urea [CO(NH2)2] using static (0 K) and dynamic (50 K) artificial experimental data. An improved model, including symmetry constraints and frozen core-electron contribution, is introduced to better handle the increasing system complexity. Reconstructed 1-RDMs, deformation densities and DCP anisotropy are analysed, and it is demonstrated that the changes in the model significantly improve the reconstruction quality, even when there is insufficient information and data corruption. The robustness of the model and the strategy are thus shown to be well adapted to address the reconstruction problem from actual experimental scattering data.

Short-range screened density matrix functional for proper descriptions of thermochemistry, thermochemical kinetics, nonbonded interactions, and singlet diradicals.

Common one-electron reduced density matrix (1-RDM) functionals that depend on Coulomb and exchange-only integrals tend to underestimate dynamic correlation, preventing reduced density matrix functional theory (RDMFT) from achieving comparable accuracy to density functional theory in main-group thermochemistry and thermochemical kinetics. The recently developed ωP22 functional introduces a semi-local density functional to screen the erroneous short-range portion of 1-RDM functionals without double-counting correlation, potentially providing a better treatment of dynamic correlation around equilibrium geometries. Herein, we systematically evaluate the performance of this functional model, which consists of two parameters, on main-group thermochemistry, thermochemical kinetics, nonbonded interactions, and more. Tests on atomization energies, vibrational frequencies, and reaction barriers reveal that the ωP22 functional model can reliably predict properties at equilibrium and slightly away from equilibrium geometries. In particular, it outperforms commonly used density functionals in the prediction of reaction barriers, nonbonded interactions, and singlet diradicals, thus enhancing the predictive power of RDMFT for routine calculations of thermochemistry and thermochemical kinetics around equilibrium geometries. Further development is needed in the future to refine short- and long-range approximations in the functional model in order to achieve an excellent description of properties both near and far from equilibrium geometries.

Machine learning electronic structure methods based on the one-electron reduced density matrix

The theorems of density functional theory (DFT) establish bijective maps between the local external potential of a many-body system and its electron density, wavefunction and, therefore, one-particle reduced density matrix. Building on this foundation, we show that machine learning models based on the one-electron reduced density matrix can be used to generate surrogate electronic structure methods. We generate surrogates of local and hybrid DFT, Hartree-Fock and full configuration interaction theories for systems ranging from small molecules such as water to more complex compounds like benzene and propanol. The surrogate models use the one-electron reduced density matrix as the central quantity to be learned. From the predicted density matrices, we show that either standard quantum chemistry or a second machine-learning model can be used to compute molecular observables, energies, and atomic forces. The surrogate models can generate essentially anything that a standard electronic structure method can, ranging from band gaps and Kohn-Sham orbitals to energy-conserving ab-initio molecular dynamics simulations and infrared spectra, which account for anharmonicity and thermal effects, without the need to employ computationally expensive algorithms such as self-consistent field theory. The algorithms are packaged in an efficient and easy to use Python code, QMLearn, accessible on popular platforms.

1-Matrix functional for long-range interaction energy of two hydrogen atoms.

The leading terms in the large-R asymptotics of the functional of the one-electron reduced density matrix for the ground-state energy of the H2 molecule with the internuclear separation R are derived thanks to the solution of the phase dilemma at the R → ∞ limit. At this limit, the respective natural orbitals (NOs) are given by symmetric and antisymmetric combinations of "half-space" orbitals with the corresponding natural amplitudes having the same amplitudes but opposite signs. Minimization of the resulting explicit functional yields the large-R asymptotics for the occupation numbers of the weakly occupied NOs and the C6 dispersion coefficient. The highly accurate approximates for the radial components of the p-type "half-space" orbitals and the corresponding occupation numbers (that decay like R-6), which are available for the first time thanks to the development of the present formalism, have some unexpected properties.

Comparison of Density-Matrix Corrections to Density Functional Theory.

Density functional theory (DFT), one of the most widely utilized methods available to computational chemistry, fails to describe systems with statically correlated electrons. To address this shortcoming, in previous work, we transformed DFT into a one-electron reduced density matrix theory (1-RDMFT) via the inclusion of a quadratic one-electron reduced density matrix (1-RDM) correction. Here, we combine our 1-RDMFT approach with different DFT functionals as well as Hartree-Fock to elucidate the method's dependence on the underlying functional selection. Furthermore, we generalize the information density matrix functional theory (iDMFT), recently developed as a correction to the Hartree-Fock method, by incorporating density functionals in place of the Hartree-Fock functional. We relate iDMFT mathematically to our approach and benchmark the two with a common set of functionals and systems.

Partial-wave decomposition of the one-electron properties of the LiH molecule computed with explicitly correlated basis sets

Two one-electron properties of the LiH molecule in its ground state at the internuclear distance of 3.015 bohr, namely the number of electrons and their kinetic energy, are decomposed into contributions due to projections (described by the quantum number m) of individual angular momenta onto the molecular axis. For sufficiently accurate electronic wavefunctions obtained with explicitly correlated Gaussian basis sets, the results of these partial-wave decompositions are found to obey the asymptotics valid at that stem from the off-diagonal cusp in the one-electron reduced density matrix. It turns out that the kinetic energy may be accurately extrapolated to the complete-basis-set limit, even if only the contributions pertaining to small values of are available.

N-representable one-electron reduced density matrix reconstruction at non-zero temperatures

<i>N</i>-representable one-electron reduced density matrix reconstruction at non-zero temperatures

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.

First Ionization Energy as the Asymptotic Limit of the Average Local Electron Energy.

The first vertical ionization energy of an atom or molecule is encoded in the rate of exponential decay of the exact natural orbitals. For natural orbitals represented in terms of Gaussian basis functions, this property does not hold even approximately. We show that it is nevertheless possible to deduce the first ionization energy from the long-range behavior of Gaussian-basis-set wave functions by evaluating the asymptotic limit of a quantity called the average local electron energy (ALEE), provided that the most diffuse functions of the basis set have a suitable shape and location. The ALEE method exposes subtle qualitative differences between seemingly analogous Gaussian basis sets and complements the extended Koopmans theorem by being robust in situations where the one-electron reduced density matrix is ill-conditioned.

Off-diagonal derivative discontinuities in the reduced density matrices of electronic systems.

An explicit expression relating the magnitude of the fifth-order off-diagonal cusp in the real part of the one-electron reduced density matrix to the "on-top" two-electron density is derived in a rigorous manner from the behavior of the underlying electronic wavefunction at the electron-electron coalescence. The implications of the presence of this cusp upon electronic structure calculations of quantum chemistry and solid-state physics, including the limits imposed upon their accuracy, are elucidated. In particular, the power-law decay of the occupation numbers of the natural orbitals is demonstrated for 1S states of systems composed of arbitrary even numbers of electrons. The practical importance of analogous off-diagonal cusps in many-electron reduced density matrices is briefly discussed.

One-Electron Reduced Density Matrix Functional Theory of Spin-Polarized Systems.

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) [Formula: see text] and the tensor of two-electron integrals g is critically analyzed. The dependence of [Formula: see text] on both the vector of occupation numbers n and g derived from the corresponding vector [Formula: see text](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, [Formula: see text] (and thus U) is found to be a function of not only n and g, but also of the vector [Formula: see text] and the matrix [Formula: see text], where [Formula: see text] is the spin-flip operator. The presence of spin polarization imposes additional constraints upon [Formula: see text], including a simple condition that, when satisfied, assures the underlying wavefunction being an eigenstate of both the [Formula: see text] and [Formula: see text] operators with the eigenvalues [Formula: see text] and [Formula: see text], 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 [Formula: see text] 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" [Formula: see text] 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.

Inferring the one-electron reduced density matrix of molecular crystals from experimental data sets through semidefinite programming.

Constructing a quantum description of crystals from scattering experiments is of great significance to explain their macroscopic properties and to evaluate the pertinence of theoretical ab initio models. While reconstruction methods of the one-electron reduced density matrix have already been proposed, they are usually tied to strong assumptions that limit and may introduce bias in the model. The goal of this paper is to infer a one-electron reduced density matrix (1-RDM) with minimal assumptions. It has been found that the mathematical framework of semidefinite programming can achieve this goal. Additionally, it conveniently addresses the nontrivial constraints on the 1-RDM which were major hindrances for the existing models. The framework established in this work can be used as a reference to interpret experimental results. This method has been applied to the crystal of dry ice and provides very satisfactory results when compared with periodic ab initio calculations.

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.

Bilinear Constraints upon the Correlation Contribution to the Electron-Electron Repulsion Energy as a Functional of the One-Electron Reduced Density Matrix.

Perturbative analysis of the functional U[n, ψ] that yields the correlation component U of the electron-electron repulsion energy in terms of the vectors ψ(1) and n of the natural spinorbitals and their occupation numbers (the 1-matrix functional) facilitates examination of the flaws inherent to the present implementations of the density matrix functional theory. Recognizing that the practical usefulness of any approximate 1-matrix functional hinges upon its capability of exactly reproducing the leading contribution to U at the limit of vanishing electron-electron interactions gives rise to asymptotic bilinear constraints for the (exact or model) 2-cumulant that enters the expression for U. The asymptotic behavior of certain blocks of is found to be equally important. These identities, which are obtained for both the single-determinantal and a model multideterminantal cases, take precedence over the linear constraints commonly enforced in the course of approximate construction of such functionals. This observation reveals the futility of designing sophisticated approximations tailored for the second-order contribution to while neglecting proper formulation of the respective first-order contribution that in the case of the so-called JKL-only functionals requires abandoning the JK-dependence altogether. It has its repercussions not only for the functionals of the PNOF family but also for the expressions involving only the L-type two-electron repulsion integrals (in the guise of their exchange counterparts) that account only for the correlation effects due to electrons with antiparallel spins and are well-defined only for spin-unpolarized and high-spin systems (yielding vanishing U for the latter).

Active Space Selection Based on Natural Orbital Occupation Numbers from n-Electron Valence Perturbation Theory.

Efficient and robust approximations to the full configuration interaction (full-CI) method such as the density matrix renormalization group (DMRG) and the full-CI quantum Monte Carlo (FCIQMC) algorithm allow for multiconfigurational self-consistent field (MC-SCF) calculations of molecules with many strongly correlated electrons. This opens up the possibility to treat large and complex systems that were previously untractable, but at the same time it calls for an efficient and reliable active space selection as the choice of how many electrons and orbitals enter the active space is critical for any multireference calculation. In this work we propose an Active Space Selection based on 1st order perturbation theory (ASS1ST) that follows a "bottom-up" strategy and utilizes a set of quasi-natural orbitals together with sensible thresholds for their occupation numbers. The required quasi-natural orbitals are generated by diagonalizing the virtual and internal part of the one-electron reduced density matrix that is obtained from strongly contracted n-electron valence perturbation theory (SC-NEVPT) on top of a minimal active space calculation. Self-consistent results can be obtained when the proposed selection scheme is applied iteratively. Initial applications on four chemically relevant benchmark systems indicate the capabilities of ASS1ST. Eventually, the strengths and limitations are critically discussed.

Evolution of entanglement in electron–electron collisions

The collision between two electrons is treated in a time-dependent three-dimensional approach involving a two-electron wave packet, which is composed of stationary solutions of the two-electron Schroedinger equation with Coulomb interaction. As a quantitative measure of the entanglement between the two electrons we employ a renormalized linear entropy, which is readily obtainable from the one-electron reduced density matrix in real space. Starting with a non-entangled initial state consisting of two separate one-electron packets, the scattering process is visualized on a femtosecond time scale by a sequence of snap shots of the one-electron density. The corresponding evolution of the linear entropy is shown for several energy values up to 200 eV (per electron in the center-of-mass system). The maximal entanglement, which is characterized by the asymptotic value of the linear entropy, is found to decrease strongly with increasing energy. The entropy for anti-parallel spins turns out to be almost the same as for parallel spins, which means that spin entanglement is vastly dominated by spatial entanglement.

Entanglement and entropy in electron–electron scattering

Treating Coulomb scattering of two free electrons in a stationary approach, we explore the momentum and spin entanglement created by the interaction. We show that a particular discretisation provides an estimate of the von Neumann entropy of the one-electron reduced density matrix from the experimentally accessible Shannon entropy.For spinless distinguishable electrons the entropy is sizeable at low energies, indicating strong momentum entanglement, and drops to almost zero at energies of the order of 10 keV when the azimuthal degree of freedom is integrated out, i.e. practically no entanglement and almost pure one-electron states. If spin is taken into account, the entropy for electrons with antiparallel spins should be larger than in the parallel-spin case, since it embodies both momentum and spin entanglement. Surprisingly, this difference, as well as the deviation from the spin-less case, is extremely small for the complete scattering state. Strong spin entanglement can however be obtained by post-selecting states at scattering angle π/2.

Exact and approximate adiabatic connection formulae for the correlation energy in multireference ground and excited states

Recently it has been shown how to employ the adiabatic connection (AC) formalism to obtain correlation energy for multireference wavefunctions [K. Pernal, Phys. Rev. Lett. 120, 013001 (2018)]. Approximations to the exact AC formulation have been based on assuming that a one-electron reduced density matrix is constant along the AC path and by employing the extended random phase approximation. In this paper, the importance of these approximations is examined by comparing approximate AC integrands with their exact counterparts obtained for the hydrogen molecule in its ground and excited states. Encouraging results obtained for H2 indicate that AC is a viable and promising approach to a correlation energy problem not only for ground but also for excited states of electronic systems.

Joint refinement model for the spin resolved one-electron reduced density matrix of YTiO3 using magnetic structure factors and magnetic Compton profiles data.

In this paper, we propose a simple cluster model with limited basis sets to reproduce the unpaired electron distributions in a YTiO3 ferromagnetic crystal. The spin-resolved one-electron-reduced density matrix is reconstructed simultaneously from theoretical magnetic structure factors and directional magnetic Compton profiles using our joint refinement algorithm. This algorithm is guided by the rescaling of basis functions and the adjustment of the spin population matrix. The resulting spin electron density in both position and momentum spaces from the joint refinement model is in agreement with theoretical and experimental results. Benefits brought from magnetic Compton profiles to the entire spin density matrix are illustrated. We studied the magnetic properties of the YTiO3 crystal along the Ti-O1-Ti bonding. We found that the basis functions are mostly rescaled by means of magnetic Compton profiles, while the molecular occupation numbers are mainly modified by the magnetic structure factors.

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.