Two-electron Model Research Articles

Removing Basis Set Incompleteness Error in Finite-Temperature Electronic Structure Calculations: Two-Electron Systems

Removing Basis Set Incompleteness Error in Finite-Temperature Electronic Structure Calculations: Two-Electron Systems

Electron correlation in two-electron atoms: A Bohmian analysis of high-order harmonic generation in high-frequency domain

Abstract In studying interactions between intense laser fields and atoms or molecules, the role of electron correlation effects on the dynamical response is an important and pressing issue to address. Utilizing Bohmian mechanics (BM), we have theoretically explored the two-electron correlation characteristics while generating high-order harmonics in xenon atoms subjected to intense laser fields. We initially employed Bohmian trajectories to reproduce the dynamics of the electrons and subsequently utilized time-frequency analysis spectra to ascertain the emission time windows for high-order harmonics. Within these time windows, we classified the nuclear region Bohmian trajectories and observed that intense high-order harmonics are solely generated when paired Bohmian particles (BPs) concurrently appear in the nuclear region and reside there for a duration within a re-collision time window. Furthermore, our analysis of characteristic trajectories producing high-order harmonics led us to propose a two-electron re-collision model to elucidate this phenomenon. The study demonstrates that intense high-order harmonics are only generated when both electrons are in the ground state within the re-collision time window. This work discusses the implications of correlation effects between two electrons and offers valuable insights for studying correlation in multi-electron high-order harmonic generation.

Bioinspired Tetranuclear Manganese Cubane Complex as an Efficient Molecular Electrocatalyst for Two-Electron Water Oxidation Towards Hydrogen Peroxide.

Stable homogeneous two-electron water oxidation electrocatalysts are highly demanded to understand the precise mechanism and reaction intermediates of electrochemical H2O2 production. Here we report a tetranuclear manganese complex with a cubane structure which can electrocatalyze water oxidation to hydrogen peroxide under alkaline and neutral conditions. Such a complex demonstrates an optimal Faradaic efficiency (FE) of 87 %, which is amongst (if not) the highest FE(H2O2) of reported homogeneous and heterogeneous electrocatalysts. In addition, active species were identified and co-catalysts were excluded through ESI-MS characterization. Furthermore, we identified water binding sites and isolated one-electron oxidation intermediate by chemical oxidation of the catalyst in the presence of water substrates. It is evident that efficient proton-accepting electrolytes avoid rapid proton building-up at electrode and substantially improve reaction rate and selectivity. Accordingly, we propose a two-electron catalytic cycle model for water oxidation to hydrogen peroxide with the bioinspired molecular electrocatalyst. The present work is expected to provide an ideal platform to elucidate the two-electron WOR mechanism at the atomic level.

Solid–liquid heterojunction UV photoelectrochemical photodetector based on WO3 nanosheets and acidic electrolyte

The acid-resistant tungsten trioxide photoelectrochemical solid–liquid ultraviolet (UV) photodetector uses a thin film of tungsten trioxide as the photoelectrode, forming a stable heterojunction with the electrolyte. This study employed band theory and double electron layer theory to analyze the mechanisms underlying the effect of pH on the redox potential and photocurrent, utilizing the ion product constant of water and the Nernst equation. By applying the principles of energy band theory and the two-electron layer model, the electron transfer process was analyzed and explained. These findings hold significant promise for enhancing solid–liquid heterojunction UV photodetectors. Tungsten trioxide has fast response and high sensitivity under extreme conditions. The device performance of WO3 nanosheets fabricated by annealing at 300 °C for one hour is excellent, including a rise time of 0.7 s, decay time of 6.8 s, photosensitivity of 1.90, and photoresponsivity of 2.31 mA/W. 0.5M sulfuric acid produced the highest photocurrent (5.46 μA) and sensitivity (14.07). This material has potential applications in optoelectronics, catalysis, sensing, water treatment, and air purification.

Extended N-centered ensemble density functional theory of double electronic excitations.

A recent work (arXiv:2401.04685) has merged N-centered ensembles of neutral and charged electronic ground states with ensembles of neutral ground and excited states, thus providing a general and in-principle exact (so-called extended N-centered) ensemble density functional theory of neutral and charged electronic excitations. This formalism made it possible to revisit the concept of density-functional derivative discontinuity, in the particular case of single excitations from the highest occupied Kohn-Sham (KS) molecular orbital, without invoking the usual "asymptotic behavior of the density" argument. In this work, we address a broader class of excitations, with a particular focus on double excitations. An exact implementation of the theory is presented for the two-electron Hubbard dimer model. A thorough comparison of the true physical ground- and excited-state electronic structures with that of the fictitious ensemble density-functional KS system is also presented. Depending on the choice of the density-functional ensemble as well as the asymmetry of the dimer and the correlation strength, an inversion of states can be observed. In some other cases, the strong mixture of KS states within the true physical system makes the assignment "single excitation" or "double excitation" irrelevant.

Capturing the electron-electron cusp with the coupling-constant averaged exchange-correlation hole: A case study for Hooke's atoms.

In density-functional theory, the exchange-correlation (XC) energy can be defined exactly through the coupling-constant (λ) averaged XC hole n̄xc(r,r'), representing the probability depletion of finding an electron at r' due to an electron at r. Accurate knowledge of n̄xc(r,r') has been crucial for developing XC energy density-functional approximations and understanding their performance for molecules and materials. However, there are very few systems for which accurate XC holes have been calculated since this requires evaluating the one- and two-particle reduced density matrices for a reference wave function over a range of λ while the electron density remains fixed at the physical (λ = 1) density. Although the coupled-cluster singles and doubles (CCSD) method can yield exact results for a two-electron system in the complete basis set limit, it cannot capture the electron-electron cusp using finite basis sets. Focusing on Hooke's atom as a two-electron model system for which certain analytic solutions are known, we examine the effect of this cusp error on the XC hole calculated using CCSD. The Lieb functional is calculated at a range of coupling constants to determine the λ-integrated XC hole. Our results indicate that, for Hooke's atoms, the error introduced by the description of the electron-electron cusp using Gaussian basis sets at the CCSD level is negligible compared to the basis set incompleteness error. The system-, angle-, and coupling-constant-averaged XC holes are also calculated and provide a benchmark against which the Perdew-Burke-Ernzerhof and local density approximation XC hole models are assessed.

Classification and quantitative characterisation of the excited states of π-conjugated diradicals.

Diradicals are of high current interest as emerging materials for next generation optoelectronic applications. To tune their excited-state properties it would be greatly beneficial to have a detailed understanding of the wave functions of the different states involved but this endeavour is hampered by formal and practical barriers. To tackle these challenges, we present a formal analysis as well as concrete results on diradical excited states. We start with a detailed investigation of the available states of a two-orbital two-electron model viewed from both the valence-bond and molecular orbital perspectives. We highlight the presence of diradical and zwitterionic states and illustrate their connections to the states found in closed-shell molecules. Subsequently, we introduce practical protocols for analysing states from realistic multireference computations applying these to the para-quinodimethane (pQDM) molecule. The analysis reveals four different categories of states - diradical, zwitterionic, HOMO-SOMO as well as biexciton - while also providing insight into their energetics and optical properties. Twisting the CH2 groups allows us to interconvert between the closed- and open-shell forms of pQDM illustrating the connection between the states in both forms. More generally, we hope that this work will lay the foundations for a more powerful rational design approach to diradicals for photophysical applications.

Basis-set correction based on density-functional theory: Linear-response formalism for excited-state energies.

The basis-set correction method based on density-functional theory consists in correcting the energy calculated by a wave-function method with a given basis set by a density functional. This basis-set correction density functional incorporates the short-range electron correlation effects missing in the basis set. This results in accelerated basis convergences of ground-state energies to the complete-basis-set limit. In this work, we extend the basis-set correction method to a linear-response formalism for calculating excited-state energies. We give the general linear-response equations as well as the more specific equations for configuration-interaction wave functions. As a proof of concept, we apply this approach to the calculations of excited-state energies in a one-dimensional two-electron model system with harmonic potential and a Dirac-delta electron-electron interaction. The results obtained with full-configuration-interaction wave functions expanded in a basis of Hermite functions and a local-density-approximation basis-set correction functional show that the present approach does not help in accelerating the basis convergence of excitation energies. However, we show that it significantly accelerates basis convergences of excited-state total energies.

Nuclear-Electron Correlation Effects and Their Photoelectron Imprint in Molecular XUV Ionisation.

The ionisation of molecules by attosecond XUV pulses is accompanied by complex correlated dynamics, such as the creation of coherent electron wave packets in the parent ion, their interplay with nuclear wave packets, and a correlated photoelectron moving in a multi-centred potential. Additionally, these processes are influenced by the dynamics prior to and during the ionisation. To fully understand and subsequently control the ionisation process on different time scales, a profound understanding of electron and nuclear correlation is needed. Here, we investigate the effect of nuclear–electron correlation in a correlated two-electron and one-nucleus quantum model system. Solving the time-dependent Schrödinger equation allows to monitor the correlation impact pre, during, and post-XUV ionisation. We show how an initial nuclear wave packet displaced from equilibrium influences the post-ionisation dynamics by means of momentum conservation between the target and parent ion, whilst the attosecond electron population remains largely unaffected. We calculate time-resolved photoelectron spectra and their asymmetries and demonstrate how the coupled electron–nuclear dynamics are imprinted on top of electron–electron correlation on the photoelectron properties. Finally, our findings give guidelines towards when correlation resulting effects have to be incorporated and in which instances the exact correlation treatment can be neglected.

Magnetism of competing high-spin/low-spin states in Ba2NiO2(AgSe)2 and related two-orbital two-electron systems

We discuss the thermodynamic and magnetic properties of a competing high-spin/low-spin two-orbital two-electron model on a square lattice possibly relevant to the nickelates such as ${\mathrm{Ba}}_{2}{\mathrm{NiO}}_{2}{(\mathrm{AgSe})}_{2}$. We focus on parameter regimes where a high-spin ($S=1$) and a low-spin ($S=0$) state are energetically close to each other and discuss various exchange processes in such a system. The model we study is motivated by, but different from, the Kondo-necklace model proposed for the system recently [Jin et al., Phys. Rev. Research 2, 033197 (2020)]. Although there are similarities between the two models in terms of the ground-state phases and their symmetries, the detailed properties of the different phases and the nature of phase transitions are entirely different and should be easy to distinguish from an experimental point of view.

Low-Threshold Nanolaser Based on Hybrid Plasmonic Waveguide Mode Supported by Metallic Grating Waveguide Structure.

A high Q-factor of the nanocavity can effectively reduce the threshold of nanolasers. In this paper, a modified nanostructure composed of a silver grating on a low-index dielectric layer (LID) and a high-index dielectric layer (HID) was proposed to realize a nanolaser with a lower lasing threshold. The nanostructure supports a hybrid plasmonic waveguide mode with a very-narrow line-width that can be reduced to about 1.79 nm by adjusting the thickness of the LID/HID layer or the duty ratio of grating, and the Q-factor can reach up to about 348. We theoretically demonstrated the lasing behavior of the modified nanostructures using the model of the combination of the classical electrodynamics and the four-level two-electron model of the gain material. The results demonstrated that the nanolaser based on the hybrid plasmonic waveguide mode can really reduce the lasing threshold to 0.042 mJ/cm2, which is about three times lower than the nanolaser based on the surface plasmon. The lasing action can be modulated by the thickness of the LID layer, the thickness of the HID layer and the duty cycle of grating. Our findings could provide a useful guideline to design low-threshold and highly-efficient miniaturized lasers.

On-top description of the effect of excitation on electron correlation with quasiparticles

A spectrum of $\mathrm{\ensuremath{\Delta}}$-correlon quasiparticles is proposed to describe the excitation effect on electron correlation. The proposed quasiparticle description is based on the comparison of the pair-correlation functions of the excited and ground states at the special coalescence points of the configurational space. The complex form of the $\mathrm{\ensuremath{\Delta}}$-correlon wave function allows us to naturally separate regions with the excitation suppression of correlation (ESC $\mathrm{\ensuremath{\Delta}}$-correlon), where correlation is stronger in the ground state, from those with the excitation enhancement of correlation (EEC $\mathrm{\ensuremath{\Delta}}$-correlon), where correlation is stronger in the excited state. The proposed $\mathrm{\ensuremath{\Delta}}$-correlon approach is applied to characterize the important single excitations with the competition of ``anticorrelation squeezing'' and ``enhanced-correlation spreading'' of the two-electron distribution. The former feature characterizes single valence excitations of the ionic nature and this leads to the prevailing of the ESC effect in this type of excitations. In turn, the enhanced-correlation spreading characterizes single Rydberg excitations, which leads to the prevailing of the EEC effect in this type of excitations. The proposed $\mathrm{\ensuremath{\Delta}}$-correlon quasiparticles are studied both analytically and numerically. The analytical expressions for the on-top correlation functions are obtained with the two-determinantal two-electron model of the active site of an excitation. The two-dimensional contour plots of the ESC and EEC $\mathrm{\ensuremath{\Delta}}$-correlons obtained for the lowest vertical valence ionic and Rydberg excitations in the prototype molecules ${\mathrm{H}}_{2}, {\mathrm{N}}_{2}$, and ${\mathrm{C}}_{2}{\mathrm{H}}_{4}$ from the highly correlated wave functions confirm, in general, the trends in the relative correlation strength deduced from the model.

Comparative study of classical theoretical descriptions of the ionization of atoms induced by few-cycle laser pulses

We have investigated the performance of classical models in the description of the single and double ionization of argon induced by 4-fs, 750-nm linearly polarized laser pulses by comparing the calculated quantities with existing experimental data, primarily the results of the systematic measurements of K\"ubel at al. [Phys. Rev. A 93, 053422 (2016)], in the intensity range (1--10)$\phantom{\rule{0.16em}{0ex}}\ifmmode\times\else\texttimes\fi{}\phantom{\rule{0.16em}{0ex}}{10}^{14}\phantom{\rule{0.28em}{0ex}}{\mathrm{Wcm}}^{\ensuremath{-}2}$. The comparison involved different one- and two-electron models based on the classical ensemble approximation. The analysis covered single- and double-ionization probabilities and their ratios, longitudinal momentum distributions of the ${\mathrm{Ar}}^{+}$ and ${\mathrm{Ar}}^{2+}$ recoiled ions, and the carrier-envelope-phase-dependent asymmetry of the yield of the ${\mathrm{Ar}}^{+}$ and ${\mathrm{Ar}}^{2+}$ ions.

Approximate density matrix functionals applied to hetero-atomic bond dissociation

A two-orbital two-electron diatomic model resembling LiH is used to investigate the differences between the exact L\"owdin-Shull and approximate Hartree-Fock-Bogoliubov and Baerends-Buijse density matrix functionals in the medium- to long-distance dissociation region. In case of homolytic dissociation (one electron on each atom), the approximate functionals fail to generate the correct energy due to a compromise between the Hartree-Fock component (which favors partial charge transfer) and the strong correlation component (which hampers charge transfer). The exact functional is able to generate the physically correct answer by enforcing the equi-charge distribution of the bonding and antibonding orbitals. Besides, the approximate functionals also have issues in correctly describing heterolytic dissociation (two electrons on one atom) due to the strong correlation component hampering charge transfer. In this work, we propose a new scheme in which the homolytic dissociation problem for approximate functionals is avoided by adding a Lagrange multiplier that enforces equi-charge distribution of the bonding and antibonding orbitals. The symmety based nature of the findings implies that they are most likely transferable to other cases in which one uses an approximate one-particle method in conjunction with a symmetrical particle-hole correction factor.

Vibronic and Spin-Orbit Couplings of 3Π and 3Σ+ Electronic States in Linear Triatomic Molecules

A two-electron model describing vibronic and spin-orbit couplings of 3Π and 3Σ+ electronic states in linear triatomic molecules is proposed. The analysis is based on the use of an ab initio Breit–Pauli form of the spin-orbit coupling operator. Electronic Hamiltonian symmetry operators are shown to involve both spatial operations (acting on electronic coordinates) and matrix operations (acting on electron spins). In our analysis, we take into account only deformation π-modes, and the resulting 9 × 9 vibronic matrix actually describes the relativistic Renner pseudo-effect (3Π + 3Σ+) × π. The eigenvalues of the vibronic matrix (i.e., potential energy surfaces) have an axial symmetry, however, they cannot be represented by analytical expressions. The calculated vibronic Hamiltonian contains six electrostatic and 16 spin-orbit real parameters.

Non-LTE analysis of K I in late-type stars

Context. Older models of Galactic chemical evolution (GCE) predict [K/Fe] ratios as much as 1 dex lower than those inferred from stellar observations. Abundances of potassium are mainly based on analyses of the 7698 Å resonance line, and the discrepancy between GCE models and observations is in part caused by the assumption of local thermodynamic equilibrium (LTE) in spectroscopic analyses. Aims. We study the statistical equilibrium of K I, focusing on the non-LTE effects on the 7698 Å line. We aim to determine how non-LTE abundances of potassium can improve the analysis of its chemical evolution, and help to constrain the yields of GCE models. Methods. We construct a new model K I atom that employs the most up-to-date atomic data. In particular, we calculate and present inelastic e+K collisional excitation cross-sections from the convergent close-coupling (CCC) and the B-Spline R-matrix (BSR) methods, and H+K collisions from the two-electron model (LCAO). We constructed a fine, extended grid of non-LTE abundance corrections based on 1D MARCS models that span 4000 < Teff∕K < 8000, 0.50 < log g < 5.00, − 5.00 < [Fe/H] < + 0.50, and applied the corrections to potassium abundances extracted from the literature. Results. In concordance with previous studies, we find severe non-LTE effects in the 7698 Å line. The line is stronger in non-LTE and the abundance corrections can reach approximately − 0.7 dex for solar-metallicity stars such as Procyon. We determine potassium abundances in six benchmark stars, and obtain consistent results from different optical lines. We explore the effects of atmospheric inhomogeneity by computing for the first time a full 3D non-LTE stellar spectrum of K I lines for a test star. We find that 3D modeling is necessary to predict a correct shape of the resonance 7698 Å line, but the line strength is similar to that found in 1D non-LTE. Conclusions. Our non-LTE abundance corrections reduce the scatter and change the cosmic trends of literature potassium abundances. In the regime [Fe/H] ≲−1.0 the non-LTE abundances show a good agreement with the GCE model with yields from rotating massive stars. The reduced scatter of the non-LTE corrected abundances of a sample of solar twins shows that line-by-line differential analysis techniques cannot fully compensate for systematic LTE modelling errors; the scatter introduced by such errors introduces a spurious dispersion to K evolution.

Inelastic O+H collisions and the O I 777 nm solar centre-to-limb variation

The O I777 nm triplet is a key diagnostic of oxygen abundances in the atmospheres of FGK-type stars; however, it is sensitive to departures from local thermodynamic equilibrium (LTE). The accuracy of non-LTE line formation calculations has hitherto been limited by errors in the inelastic O+H collisional rate coefficients; several recent studies have used the Drawin recipe, albeit with a correction factorSHthat is calibrated to the solar centre-to-limb variation of the triplet. We present a new model oxygen atom that incorporates inelastic O+H collisional rate coefficients using an asymptotic two-electron model based on linear combinations of atomic orbitals, combined with a free electron model based on the impulse approximation. Using a 3D hydrodynamicSTAGGERmodel solar atmosphere and 3D non-LTE line formation calculations, we demonstrate that this physically motivated approach is able to reproduce the solar centre-to-limb variation of the triplet to 0.02 dex, without any calibration of the inelastic collisional rate coefficients or other free parameters. We infer log ϵO= 8.69 ± 0.03 from the triplet alone, strengthening the case for a low solar oxygen abundance.

Exact-exchange optimized effective potential and memory effect in time-dependent density functional theory

The memory effect in time-dependent density functional theory (TDDFT) is important in simulating many time-dependent physical processes, and its implementation in real time has been a longstanding challenge, thus limiting most of TDDFT applications to either adiabatic or linear-response regime. In this paper, we conduct the non-adiabatic calculations for a one-dimensional two-electron Helium model in a triplet state using the recently formulated Sturm-Liouville-type time-local equation for the time-dependent optimized effective potential (TDOEP) with the exact exchange functional, and the results agree with the exact time-dependent Schrodinger equation solutions. It is also found that the time-dependent dipole moment and probability density calculated from the TDOEP approach are more accurate than those from the adiabatic time-dependent Krieger-Li-Iafrate (TDKLI) approximation and the adiabatic local spin density approximation. Specifically, the non-adiabatic and memory-dependent terms in the time-local TDOEP equation correctly describe the time-dependent structure of exchange-correlation potential and yield the probability density evolution. These findings should provide important insights toward future studies on memory effects in TDDFT.

Binding of two-electron metastable states in semiconductor quantum dots under a magnetic field

Applying a strong enough magnetic field results in the binding of few-electron resonant states. The mechanism was proposed many years ago but its verification in laboratory conditions is far more recent. In this work we study the binding of two-electron resonant states. The electrons are confined in a cylindrical quantum dot which is embedded in a semiconductor wire. The geometry considered is similar to the one used in actual experimental setups. The low-energy two-electron spectrum is calculated numerically from an effective-mass approximation Hamiltonian modelling the system. Methods for binding threshold calculations in systems with one and two electrons are thoroughly studied; in particular, we use quantum information quantities to assess when the strong lateral confinement approximation can be used to obtain reliable low-energy spectra. For simplicity, only cases without bound states in the absence of an external field are considered. Under these conditions, the binding threshold for the one-electron case is given by the lowest Landau energy level. Moreover, the energy of the one-electron bounded resonance can be used to obtain the two-electron binding threshold. It is shown that for realistic values of the two-electron model parameters it is feasible to bind resonances with field strengths of a few tens of tesla.

How Important Are Radical Mechanisms in Frustrated Lewis Pair Chemistry?

Reaction mechanism is the foundation of understanding chemical reactivity. In this issue of Chem , Stephan and co-workers overturn long-held assumptions regarding the reactivity of frustrated Lewis pairs by providing compelling evidence that radical mechanisms might operate in some cases.