Electronic Density Matrices Research Articles

Incorporating memory into propagation of 1-electron reduced density matrices

For any linear system with unreduced dynamics governed by invertible propagators, we derive a closed, time-delayed, linear system for a reduced-dimensional quantity of interest. This method does not target dimensionality reduction: rather, this method helps shed light on the memory-dependence of 1-electron reduced density matrices in time-dependent configuration interaction (TDCI), a scheme to solve for the correlated dynamics of electrons in molecules. Though time-dependent density functional theory has established that the 1-electron reduced density possesses memory-dependence, the precise nature of this memory-dependence has not been understood. We derive a symmetry/constraint-preserving method to propagate reduced TDCI electron density matrices. In numerical tests on two model systems (H2 and HeH+), we show that with sufficiently large time-delay (or memory-dependence), our method propagates reduced TDCI density matrices with high quantitative accuracy. We study the dependence of our results on time step and basis set. To implement our method, we derive the 4-index tensor that relates reduced and full TDCI density matrices. Our derivation applies to any TDCI system, regardless of basis set, number of electrons, or choice of Slater determinants in the wave function.

Statistical learning for predicting density–matrix‐based electron dynamics

We consider the problem of learning density‐dependent molecular Hamiltonian matrices from time series of electron density matrices, all in the context of Hartree–Fock theory. Prior work developed a solution to this problem for small molecular systems with density and Hamiltonian matrices of size at most 6 × 6. Here, using a battery of techniques, we scale prior methods to larger molecular systems with, for example, 29 × 29 matrices. This includes systems that either have more electrons or are expressed in large basis sets such as 6‐311++G**. Scaling the method to larger systems enhances its relevance for realistic applications in chemistry and physics. To achieve this scaling, we apply dimensionality reduction, ridge regression and analytic computation of Hessians. Through the combination of these techniques, we are able to learn Hamiltonians by minimizing an objective function that encodes local propagation error. Importantly, these learned Hamiltonians can then be used to predict electron dynamics for thousands of steps: When we use our learned Hamiltonians to numerically solve the time‐dependent Hartree–Fock equation, we obtain predicted dynamics that are in close quantitative agreement with ground truth dynamics. This includes field‐off trajectories similar to the training data and field‐on trajectories outside of the training data.

Signatures of Through‐Space Charge Transfer in Two‐Photon Absorption of Paracyclophane Derivatives

Third order polarizability, (γ) taken from the collective electronic oscillator (CEO) method was used to calculate the two‐photon absorption (TPA) of tetrastyryl‐[2,2]paracyclophane derivatives with different through‐space charge transfer configurations considering various donor and acceptor combinations at the terminal styryl groups. For the virtually same linear absorption, different TPA spectra were obtained. For controlling and fine‐tuning frequency and cross‐sections of TPA the through‐space charge transfer interactions can be used. The results are explained by the electronic density matrices corresponding to governing oscillators in one‐ and two‐photon absorption and the ground state. It is indicated that for the studied systems mainly the lowest four oscillators are responsible for the TPA cross‐sections rather than a simple effective three‐state model.

A comparative study of ionospheric IRIEup and ISP assimilative models during some intense and severe geomagnetic storms

Three-dimensional (3-D) electron density matrices, computed in the Mediterranean area by the IRI climatological model and IRIEup and ISP nowcasting models, during some intense and severe geomagnetic-ionospheric storms, were ingested by the ray tracing software tool IONORT, to synthesize quasi-vertical ionograms. IRIEup model was run in different operational modes: (1) assimilating validated autoscaled electron density profiles only from a limited area which, in our case, is the Mediterranean sector (IRIEup_re(V) mode); (2) assimilating electron density profiles from a larger region including several stations spread across Europe: (a) without taking care of validating the autoscaled data in the assimilation process (IRIEup(NV)); (b) validating carefully the autoscaled electron density profiles before their assimilation (IRIEup(V)).The comparative analysis was carried out comparing IRI, IRIEup_re(V), ISP, IRIEup(NV), and IRIEup(V) foF2 synthesized values, with corresponding foF2 measurements autoscaled by ARTIST, and then validated, at the truth sites of Roquetes (40.80°N, 0.50°E, Spain), San Vito (40.60°N, 17.80°E, Italy), Athens (38.00°N, 23.50°E, Greece), and Nicosia, (35.03°N, 33.16°E, Cyprus). The outcomes demonstrate that: (1) IRIEup_re(V), performs better than ISP in the western Mediterranean (around Roquetes); (2) ISP performs slightly better than IRIEup_re(V) in the central part of Mediterranean (around Athens and San Vito); (3) ISP performance is better than the IRIEup_re(V) one in the eastern Mediterranean (around Nicosia); (4) IRIEup(NV) performance is worse than the IRIEup(V) one; (5) in the central Mediterranean area, IRIEup(V) performance is better than the IRIEup_re(V) one, and it is practically the same for the western and eastern sectors.Concerning the overall performance, nowcasting models proved to be considerably more reliable than the climatological IRI model to represent the ionosphere behaviour during geomagnetic-ionospheric storm conditions; ISP and IRIEup(V) provided the best performance, but neither of them has clearly prevailed over the other one.

Generalization of the Kohn-Sham system that can represent arbitrary one-electron density matrices

Density functional theory is currently the most widely applied method in electronic structure theory. The Kohn-Sham method, based on a fictitious system of non-interacting particles, is the work horse of the theory. The particular form of the Kohn-Sham wavefunction admits only idem-potent one electron density matrices whereas wavefunctions of correlated electrons in post-Hartree-Fock methods invariably have fractional occupation numbers. Here we show that by generalizing the orbital concept, and introducing a suitable dot-product as well as a probability density a non-interacting system can be chosen that can represent the one-electron density matrix of any system, even one with fractional occupation numbers. This fictitious system ensures that the exact electron density is accessible within density functional theory. It can also serve as the basis for reduced density matrix functional theory. Moreover, to aid the analysis of the results the orbitals may be assigned energies from a mean-field Hamiltonian. This produces energy levels that are akin to Hartree-Fock orbital energies such that conventional analyses based on Koopmans theorem are available. Finally, this system is convenient in formalisms that depend on creation and annihilation operators as they are trivially applied to single determinant wavefunctions.

A Comprehensive Analysis in Terms of Molecule-Intrinsic, Quasi-Atomic Orbitals. III. The Covalent Bonding Structure of Urea.

The analysis of molecular electron density matrices in terms of quasi-atomic orbitals, which was developed in previous investigations, is quantitatively exemplified by a detailed application to the urea molecule. The analysis is found to identify strong and weak covalent bonding interactions as well as intramolecular charge transfers. It yields a qualitative as well as quantitative ab initio description of the bonding structure of this molecule, which raises questions regarding some traditional rationalizations.

High-temperature superconductivity and long-range order in strongly correlated electronic systems

A selective review of the question of how repulsive electron correlations might give rise to off-diagonal long-range order (ODLRO) in high-temperature superconductors is presented. The article makes detailed explanations of the relevance to superconductivity of reduced electronic density matrices and how these can be used to understand whether ODLRO might arise from Coulombic repulsions in strongly correlated electronic systems. Time-reversed electron pairs on alternant Cuprate and the iron-based pnictide and chalcogenide lattices may have a weak long-range attractive tail and much stronger short-range repulsive Coulomb interaction. The long-range attractive tail may find its origin in one of the many suggested proposals for high-Tc superconductivity and thus has an uncertain origin. A phenomenological Hamiltonian is invoked whose model parameters are obtained by fitting to experimental data. A detailed summary is given of the arguments that such interacting electrons can cooperate to produce a superconducting state in which time-reversed pairs of electrons effectively avoid the repulsive hard-core of the Coulomb interaction but reside on average in the attractive well of the long-range potential. Thus, the pairing of electrons itself provides an enhanced screening mechanism. The alternant lattice structure is the key to achieving robust high-temperature superconductivity with dx2-y2 or sign alternating s-wave or s± condensate symmetries in cuprates and iron-based compounds. Some attention is also given to the question first raised by Leggett as to where the Coulombic energy is saved in the superconducting transition in cuprates. A mean-field-type model in which the condensate density serves as an order parameter is discussed. Many of the observed trends in the thermal properties of cuprate superconductors are reproduced giving strong support for the proposed model for high-temperature superconductivity in such strongly correlated electronic systems. © 2015 Wiley Periodicals, Inc.

Vibrational Properties of Hydrogen-Bonded Systems Using the Multireference Generalization to the "On-the-Fly" Electronic Structure within Quantum Wavepacket ab Initio Molecular Dynamics (QWAIMD).

We discuss a multiconfigurational treatment of the "on-the-fly" electronic structure within the quantum wavepacket ab initio molecular dynamics (QWAIMD) method for coupled treatment of quantum nuclear effects with electronic structural effects. Here, multiple single-particle electronic density matrices are simultaneously propagated with a quantum nuclear wavepacket and other classical nuclear degrees of freedom. The multiple density matrices are coupled through a nonorthogonal configuration interaction (NOCI) procedure to construct the instantaneous potential surface. An adaptive-mesh-guided set of basis functions composed of Gaussian primitives are used to simplify the electronic structure calculations. Specifically, with the replacement of the atom-centered basis functions positioned on the centers of the quantum-mechanically treated nuclei by a mesh-guided band of basis functions, the two-electron integrals used to compute the electronic structure potential surface become independent of the quantum nuclear variable and hence reusable along the entire Cartesian grid representing the quantum nuclear coordinates. This reduces the computational complexity involved in obtaining a potential surface and facilitates the interpretation of the individual density matrices as representative diabatic states. The parametric nuclear position dependence of the diabatic states is evaluated at the initial time-step using a Shannon-entropy-based sampling function that depends on an approximation to the quantum nuclear wavepacket and the potential surface. This development is meant as a precursor to an on-the-fly fully multireference electronic structure procedure embedded, on-the-fly, within a quantum nuclear dynamics formalism. We benchmark the current development by computing structural, dynamic, and spectroscopic features for a series of bihalide hydrogen-bonded systems: FHF(-), ClHCl(-), BrHBr(-), and BrHCl(-). We find that the donor-acceptor structural features are in good agreement with experiments. Spectroscopic features are computed using a unified velocity/flux autocorrelation function and include vibrational fundamentals and combination bands. These agree well with experiments and other theories.

Modeling biophysical and biological properties from the characteristics of the molecular electron density, electron localization and delocalization matrices, and the electrostatic potential.

The electron density and the electrostatic potential are fundamentally related to the molecular hamiltonian, and hence are the ultimate source of all properties in the ground- and excited-states. The advantages of using molecular descriptors derived from these fundamental scalar fields, both accessible from theory and from experiment, in the formulation of quantitative structure-to-activity and structure-to-property relationships, collectively abbreviated as QSAR, are discussed. A few such descriptors encode for a wide variety of properties including, for example, electronic transition energies, pKa's, rates of ester hydrolysis, NMR chemical shifts, DNA dimers binding energies, π-stacking energies, toxicological indices, cytotoxicities, hepatotoxicities, carcinogenicities, partial molar volumes, partition coefficients (log P), hydrogen bond donor capacities, enzyme–substrate complementarities, bioisosterism, and regularities in the genetic code. Electronic fingerprinting from the topological analysis of the electron density is shown to be comparable and possibly superior to Hammett constants and can be used in conjunction with traditional bulk and liposolubility descriptors to accurately predict biological activities. A new class of descriptors obtained from the quantum theory of atoms in molecules' (QTAIM) localization and delocalization indices and bond properties, cast in matrix format, is shown to quantify transferability and molecular similarity meaningfully. Properties such as “interacting quantum atoms (IQA)” energies which are expressible into an interaction matrix of two body terms (and diagonal one body “self” terms, as IQA energies) can be used in the same manner. The proposed QSAR-type studies based on similarity distances derived from such matrix representatives of molecular structure necessitate extensive investigation before their utility is unequivocally established. © 2014 The Author and the Journal of Computational Chemistry Published by Wiley Periodicals, Inc.

Reduced purities as measures of decoherence in many-electron systems

A hierarchy of measures of decoherence for many-electron systems that is based on the purity and the hierarchy of reduced electronic density matrices is presented. These reduced purities can be used to characterize electronic decoherence in the common case when the many-body electronic density matrix is not known and only reduced information about the electronic subsystem is available. Being defined from reduced electronic quantities, the interpretation of the reduced purities is more intricate than the usual (many-body) purity. This is because the nonidempotency of the r-body reduced electronic density matrix that is the basis of the reduced purity measures can arise due to decoherence or due to electronic correlations. To guide the interpretation, explicit expressions are provided for the one-body and two-body reduced purities for a general electronic state. Using them, the information content and structure of the one-body and two-body reduced purities is established, and limits on the changes that decoherence can induce are elucidated. The practical use of the reduced purities to understand decoherence dynamics in many-electron systems is exemplified through an analysis of the electronic decoherence dynamics in a model molecular system.

A study of the relationships between unpaired electron density, spin-density and cumulant matrices

This work describes the derivation of simple relationships between the density matrix of effectively unpaired electrons and the spin-density matrix in N-electron systems. The link between both devices turns out to be the one-electron matrix arising from the diagonal contraction of the cumulant matrix corresponding to the second-order reduced density matrix. We study some features of this contracted matrix, showing its usefulness to describe the electronic correlation. Numerical determinations performed in selected systems with different spin symmetries confirm the theoretical predictions.

Comment on “Curvy-steps approach to constraint-free extended-Lagrangian ab initio molecular dynamics, using atom-centered basis functions: Convergence toward Born–Oppenheimer trajectories” [J. Chem. Phys. 121, 11542 (2004)

The curvy-extended-Lagrangian molecular-dynamics (ELMD) approach [J. M. Herbert and M. Head-Gordon, J. Chem. Phys. 121, 11542 (2004)] is similar to atom-centered density-matrix propagation (ADMP) [H. B. Schlegel, J. M. Millam, S. S. Iyengar, G. A. Voth, A. D. Daniels, G. E. Scuseria, and M. J. Frisch, J. Chem. Phys. 114, 9758 (2001); S. S. Iyengar, H.B. Schlegel, J.M. Millam, G.A. Voth, G.E. Scuseria, and M.J. Frisch, ibid.115, 10291 (2001); H.B. Schlegel, S.S. Iyengar, X. Li, J.M. Millam, G.A. Voth, G.E. Scuseria, and M.J. Frisch, ibid. 117, 8694 (2002); S.S. Iyengar, H.B. Schlegel, G.A. Voth, J.M. Millam, G.E. Scuseria, and M.J. Frisch, Israel J. Chem. 42, 191 (2002)] and based on Car-Parrinello [Phys. Rev. Lett. 55, 2471 (1985)] extended-Lagrangian [H.C. Andersen, J. Chem. Phys. 72, 2384 (1980)] molecular dynamics. Similarities between curvy-ELMD and ADMP arise from using unconverged electronic single-particle density matrices within Gaussian basis functions as dynamical variables. Curvy-ELMD differs from ADMP in not requiring idempotency to be explicitly enforced. In this Comment, we address several misleading remarks in Refs. 1 [J.M. Herbert and M. Head-Gordon, J. Chem. Phys. 121, 11542 (2004)] and 8 [J.M. Herbert and M. Head-Gordon, J. Chem. Phys. (submitted)].

Self-consistent density matrix algorithm for electronic structure and excitations of molecules and aggregates

An ab initio density matrix algorithm for electronic structure computations of many-electron systems is proposed. The reduced single-electron density matrices are derived by mapping the density functional theory nonlinear optical response functions onto an effective multilevel system. These density matrices are then used as a zeroth order iteration into self-consistent equations whose solution should yield the exact energies and the complete set of (transition and diagonal) single-electron density matrices. Higher order (n electron) density matrices are not computed explicitly. The linear and nonlinear optical response functions may be obtained at a low computational cost. Application is made to constructing an exciton Hamiltonian for molecular aggregates using density matrices of isolated molecules, avoiding electronic structure calculations of the entire aggregate.

Time-resolved x-ray Raman spectroscopy of photoexcited polydiacetylene oligomer: A simulation study

Off-resonant x-ray diffraction provides a novel real-space and real-time probe of electronic and vibrational dynamics in optically excited molecules. The entire manifold of valence electronic excitations may be monitored through the dependence of the x-ray Raman peaks on the scattering wave vector Δk and energy Δω. The electronic excitation energies and transition density matrices of a polydiacetylene oligomer, computed using the time-dependent Hartree–Fock collective electronic oscillator algorithm, are used to simulate the Raman signals and illustrate their information content.

Direct diabatization of electronic states by the fourfold way. II. Dynamical correlation and rearrangement processes

Diabatic representation of coupled potential energy surfaces and their scalar couplings provides a compact and convenient starting point for dynamics calculations carried out in either the adiabatic or diabatic representation. In a previous paper we presented a general, path-independent scheme, called the fourfold way, for calculating diabatic surfaces and their scalar couplings from adiabatic surfaces and electronic density matrices such that the manifold of diabatic states spans the variationally optimized space of a finite number of adiabatic states. In the present paper we extend that scheme in these ways: (1) We show how to include dynamical electronic correlation energy by multireference perturbation theory or configuration interaction based on a complete active reference space. (2) We present a more general strategy for treating rearrangements. (3) We present consistency criteria for testing the validity of the assumptions for a particular choice of reference geometries, diabatic molecular orbital (DMO) ordering, dominant configuration-state-function lists, and choice(s) for reference DMO(s) for systems involving rearrangements. The first extension is illustrated by multiconfiguration quasidegenerate perturbation theory (MC-QDPT) calculations on LiF, and all three extensions are illustrated by MC-QDPT calculations on the reaction Li(2 2S,2 2P)+HF→LiF+H.

Excited-State Molecular Dynamics Simulations of Conjugated Oligomers Using the Electronic Density Matrix

Vibrational motions in optically excited polyacetylene and polydiacetylene oligomers are calculated using potential surfaces obtained from the collective electronic oscillators (CEO) technique. The role of the effective conjugation coordinate (ECC) in the relaxation processes following an impulsive vertical excitation from the ground state is demonstrated. Real-space analysis of the electronic transition density matrices shows the charge and bond-order redistribution taking place upon photoexcitation.

Ultrafast dynamics of carrier–LO phonon system in high electric field in polar semiconductors

The dynamics of carriers excited by an ultrashort laser pulse is simulated numerically in the presence of a strong electric field. The carrier density matrices (DMs) are treated within the equation of motion method by taking interaction with longitudinal optical (LO) phonons into account. It is shown that the temporal evolution of an electron DM exhibits a strong modification owing to LO phonon emissions in addition to interferences between the interband polarization and the optical pulse field in the external strong electric field.

Electronic Structure-Factor, Density Matrices, and Electron Energy Loss Spectroscopy of Conjugated Oligomers

The electronic dynamic structure factor S(q,ω) of conjugated finite-size oligomers is computed using the time-dependent Hartree−Fock (TDHF) approach. The resonances observed in electron scattering are analyzed using the Bloch representation of the single-electron transition density matrices for the corresponding polymers. The separation of relative and center-of-mass motions of electron−hole pairs, which is expected to hold for infinite chains, is shown to apply even for relatively small molecules, paving the way for a band-structure picture of the quasiparticles. Signatures of the exciton coherence sizes in the momentum dependence of the structure factor are analyzed.

An open-shell method for neutral vacancy in silicon and diamond

The restricted Hartree-Fock-Roothaan method with closed and open electronic shells projected by electron density matrices and the quasi-molecular large-unit-cell (LUC) model have been applied to calculate the electronic structure of monovacancy and semivacancy in the neutral charge state in the totally symmetric atomic configuration with relaxation and symmetry-lowering distortions. The difference in the energies of states with total spins of 1 and 3/2 for a neutral monovacancy is determined within the ΔSCF approximation.

The electronic structure of a silicon divacancy calculated by the open shell method

The restricted Hartree-Fock-Roothaan method with projection of the electron density matrices for a filled and an open electron shell and the quasimolecular large unit cell model are used to calculate the electronic structure of a silicon divacancy in the charge states 0, ±1 in the fully-symmetric atomic configuration with relaxation and with symmetry-lowering distortions. It is shown that the Jahn-Teller effect is bimodal and that there is a vibronic coupling between the electron orbital doublet and the resonant mode and pairing mode of the distortion.