Two-electron Wave Research Articles

Exact time-evolving scattering states in open quantum-dot systems with an interaction: discovery of time-evolving resonant states

We study exact time-evolving many-electron states of an open double quantum-dot system with an interdot Coulomb interaction. A systematic construction of the time-evolving states for arbitrary initial conditions is proposed. For any initial states of one- and two-electron plane waves on the electrical leads, we obtain exact solutions of the time-evolving scattering states, which converge to known stationary scattering eigenstates in the long-time limit. For any initial states of localized electrons on the quantum dots, we find exact time-evolving states of a new type, which we refer to as time-evolving resonant states. In contrast to stationary resonant states, whose wave functions spatially diverge and not normalizable, the time-evolving resonant states are normalizable since their wave functions are restricted to a finite space interval due to causality. The exact time-evolving resonant states enable us to calculate the time-dependence of the survival probability of electrons on the quantum dots for the system with the linearized dispersions. It decays exponentially in time on one side of an exponential point of resonance energies while, on the other side, it oscillates during the decay as a result of the interference of the two resonance energies.

A triradical-containing trinuclear Pd(II) complex: spin-polarized electronic transmission, analog resistive switching and neuromorphic advancements.

Neuromorphic computation has emerged as a potential alternative to subvert the von Neumann bottleneck issue in conventional computing. In this context, the development of resistive switching-based memristor devices mimicking various synaptic functionalities has engendered paramount attention. Here, we report a triradical-containing trinuclear Pd(II) cluster with a cyclohexane-like framework constituted by the Pd-Se coordination motif displaying facile memristor property with neuromorphic functionality as a thin-film device. The metal-ligand complex (complex 1) possessed an St = 1/2 ground state by experiencing a spin-frustrated-type magnetic coupling phenomenon amongst the three ligand-based organic radicals (SR = 1/2), coordinated to the Pd(II) ions. Three reversible one-electron reduction waves countered with a one-electron and one two-electron reversible oxidation waves were noticed in the cyclic voltammogram of the complex, confirming electrons accepting and releasing capacity of the complex at low potentials, i.e., within +0.2 V to -1.1 V. Employing the radical-containing complex 1 as the active thin-film sandwiched between two orthogonal electrodes, resistive switching based memristor property with biological synaptic actions were successfully emulated. Intriguingly, the artificial neural network (ANN) simulated efficient pattern recognition demonstrated using the recorded potentiation and depression curves from the device, which is a step ahead for the hardware realization of neuromorphic computing. The performance of the ANN on MNIST data with reduced image resolution has further been evaluated. Density functional theory (DFT)-based theoretical calculation predicted that the spin-polarized electronic transmission substantiated the memristive property in the neutral complex 1.

A benzoquinone-imidazole hybrid organic anolyte for aqueous redox flow batteries.

A benzoquinone derivative annelated by two imidazole rings was investigated as an organic anolyte of aqueous redox flow batteries. The anolyte showed a high solubility of 0.18 M in 1 M KOH aqueous solution and exhibited a one-step two-electron reversible redox wave with a half-wave potential of -0.59 V VS. SHE. An aqueous redox flow cell comprising the benzoquinone-imidazole hybrid as the anolyte and potassium ferrocyanide as the catholyte showed an operating voltage of ∼1.1 V and minimum capacity fading of over 220 cycles.

Doubly differential cross sections for ionisation in proton–helium collisions at intermediate energies: energy and angular distribution of emitted electrons

Using the two-centre wave-packet convergent close-coupling approach, we continue our study of the proton–helium collision system. This method uses a correlated two-electron wave function to describe the helium target and discretises the continuum using wave-packet pseudostates. The cross section differential in the electron-emission energy and emission angle is calculated for incident-projectile energies in the intermediate range from 70 to 300 keV, where coupling between various channels and electron–electron correlation effects are important. We also apply an alternative, simpler approach that reduces the target to an effective single-electron system. Overall, the present results from both methods agree well with the available experimental data. This positions both implementations of the two-centre wave-packet convergent close-coupling approach well to further study other doubly differential, as well as fully differential, cross sections of single ionisation in proton–helium collisions.Graphical abstract

Dynamics of Correlated Double-Ionization of Two-Electron Quantum Dots in Laser Fields.

Using an ab initio, time-dependent calculational method, we study the non-linear dynamics of a two-electron quantum dot in the presence of ultrashort Thz laser pulses. The analysis of the contribution of the various partial waves to two-electron joint radial and energy distribution patterns revealed strongly correlated electron ejection channels. In the double-ionization process, regardless of the photon energy, the two-electron wave packets are born and remain concentrated until the pulse's peak; at later times, and depending on the photon energy of the field, distinctly different patterns emerge. Our calculations also showed the gradual transition of the radial and energy patterns from a single-peak to a doubly peaked structure, associated with the direct and the sequential double-ionization mechanisms, respectively.

Investigation the fully differential cross section of helium atom single ionization by fast protons impact with three and four-body formalism

In the present work, the fully differential cross section of atomic helium ionization by protons impact at high energy ranges is calculated. The calculations of the fully differential cross section have been performed by using the three-body formalism in the first order Born-Faddeev approximation with an active electron model and four body formalism with the first Born approximation. In order to investigate the collision dynamics accurately, the first order Born-Faddeev approximation has been performed by using the single parameter and Hartree Fock as single-electron wave functions and in the first order Born approximation, the two-electron wave functions of Hylleraas and Silverman are used with considering the electron correlation. The fully differential cross sections at different ejected electron energies and momentum transfers are compared with the available experiment and theoretical results. The results show that the effects of electronic correlation are considerable at high impact energy range and ionization process.‎

Reducing the Quantum Many-Electron Problem to Two Electrons with Machine Learning.

An outstanding challenge in chemical computation is the many-electron problem where computational methodologies scale prohibitively with system size. The energy of any molecule can be expressed as a weighted sum of the energies of two-electron wave functions that are computable from only a two-electron calculation. Despite the physical elegance of this extended "aufbau" principle, the determination of the distribution of weights─geminal occupations─for general molecular systems has remained elusive. Here we introduce a new paradigm for electronic structure where approximate geminal-occupation distributions are "learned" via a convolutional neural network. We show that the neural network learns the N-representability conditions, constraints on the distribution for it to represent an N-electron system. By training on hydrocarbon isomers with only 2-7 carbon atoms, we are able to predict the energies for isomers of octane as well as hydrocarbons with 8-15 carbons. The present work demonstrates that machine learning can be used to reduce the many-electron problem to an effective two-electron problem, opening new opportunities for accurately predicting electronic structure.

Gaussian product rule for two-electron wave functions.

The Gaussian product rule for two-electron wave functions is introduced. The two-electron Gaussian product rule enables a new way for solving two-electron integrals. The solution is demonstrated with an example of the two-center two-electron integral in solid harmonic Gaussian basis. The solution is obtained by expanding inverse inter-electron separation and integrating in spherical coordinates. The resulting integral separates into four integrals, three of which are straightforward to solve. The remaining integral can be solved with Boys-like functions. It is demonstrated that the solution can deliver results with accuracy comparable with that of the McMurchie-Davidson scheme.

Dependence of Quinone Electrochemistry on Solvent Polarity and pH

We studied the polarographic reduction of quinones in aqueous/organic mixtures with dielectric constants (ε) from 78.4 down to 47. Added organic co-solvents were either protic or aprotic. The increase in polarographic half-wave potential, E 1/2, with declining ε was successfully fit to the Born equation down to ε ≈ 55. Cu2+ and ubiquinone0 were reduced in a single two-electron step that was more ε-sensitive when the added organic co-solvent was aprotic. Naphtho- and anthraquinone were reduced in two successive one-electron steps that were influenced identically by protic or aprotic organic co-solvents. The product of the first reduction wave was neutral semiquinone (·QH), which surprisingly, had to be protonated to ·QH2 + before reduction in the second wave. Except for r H+, radii derived from fitting our E 1/2 vs ε results to the Born equation were too small; in other words, non-electrostatic effects destabilized the oxidized species, greatly enhancing the Born electrostatic increase in E 1/2 with declining ε. Additionally, for ε < 55, we observed deviation from the Born equation, which may be due to changes in solvent structure and dynamics, and solvent-solute interactions. Finally, we studied quinones incorporated into phosphatidyl choline sonicated bilayer vesicles: Ubiquinone0 had two distinct irreversible two-electron reduction waves, one due to a population bound at the membrane surface, and another (whose E 1/2 was negatively shifted by 150 mV) due to a population localized in the membrane interior. Ubiquinone10 had a single irreversible two-electron reduction wave that was 250 mV more negative than the UQ0 membrane-interior population.

Synthesis, Structure and Properties of Donor-acceptor-type [4]Dendralenes

[4]Dendralene and its analogs possessing both strongly electron-donating 1,3-dithiol-2-ylidenes and electron-accepting dicyanomethylidenes (2b–4b) were successfully synthesized. X-ray structure analysis revealed that 2b adopts a nonplanar structure, in which half units of 2b are significantly inclined to each other. Cyclic voltammograms of 2b–4b consisted of one pair of two-electron oxidation waves and one pair of one-electron reduction waves.

Proton-helium collisions at intermediate energies: Singly differential ionization cross sections

We investigate the four-body proton-helium scattering problem using the two-center wave-packet convergent close-coupling approach. The approach uses a correlated two-electron wave function for the helium target. The continuum is discretized using wave-packet pseudostates. Calculations of ionization cross sections differential in the electron emission energy, in the emission angle, as well as in the scattered-projectile angle are performed in the intermediate energy region where coupling between various channels and electron-electron correlation effects are important. The results agree well with experimental data, where available. Moreover, our calculations reveal an interesting interplay between direct ionization and electron capture into the continuum. In particular, we demonstrate that the ionization cross section differential in the angle of the ejected electron is dominated by electron capture into the continuum for ejection into small angles, while ejection into large angles is purely due to direct ionization. It is concluded that the two-center wave-packet convergent close-coupling approach can provide accurate singly differential cross sections for ionization in proton-helium collisions. For comparison, a recently developed method that reduces the target to an effective single-electron system is also used. Somewhat unexpectedly, the results of the effective one-electron method exhibit a very good level of agreement with the full two-electron ones for all three singly differential cross sections. Therefore, we also conclude that, at least for the purpose of calculating the singly differential cross sections for single ionization of helium, an effective one-electron treatment of the target suffices.

Differential study of proton-helium collisions at intermediate energies: Elastic scattering, excitation, and electron capture

In a series of papers we present a comprehensive investigation of the four-body proton-helium differential scattering problem using the two-center wave-packet convergent close-coupling approach in the intermediate energy region where coupling between various channels is important. The approach uses correlated two-electron wave functions for the helium target. For comparison, a recently developed method that reduces the target to an effective single-electron system is also used. In this paper we present calculations of angular differential cross sections for elastic-scattering, target excitation, and electron-capture processes. The results of the two-electron and effective single-electron methods exhibit a good level of agreement. They also agree well with available experimental data. It is concluded that both versions of the wave-packet convergent close-coupling approach are capable of providing a realistic differential picture of all interdependent and interconnected binary processes taking place in proton-helium collisions at intermediate energies.

Cluster decomposition principle and two-electron wave function of the Cooper pair in the BCS superconducting state

We present the explicit forms of the maximum eigenvalue and the corresponding eigenfunction for the second-order reduced density matrix (RDM2) of the BCS superconducting state (SS). Using these quantities, we deal with two topics in the present paper. As the first topic, it is shown that the cluster decomposition principle holds in the BCS-SS. This proof gives a theoretical foundation that the abnormal density can be chosen as the order parameter of the SS. As the second topic, it is shown that such an eigenfunction is spin singlet and spatially extends isotopically, and further that the mean distance of two electrons which consists of the above eigenfunction is in a good agreement with Pippard’s coherence length. This means that maximum geminal of the RDM2 of the BCS-SS can be regarded as the Cooper pair itself which are condensed to the same energy level in a number of

Nondipole effects in interference patterns of a two-electron wave

When the photoionization is described beyond the dipole approximation, the electron in atoms and molecules will experience a spatially dependent light phase. As demonstrated in a recent experiment on one-photon double ionization of ${\mathrm{H}}_{2}$ [S. Grundmann et al., Science 370, 339 (2020)], the resulting phase difference modifies the electron interference pattern. By solving the full-dimensional time-dependent Schr\"odinger equation beyond the dipole approximation, the present work provides corresponding ab initio calculations that have been missing so far, to the best of our knowledge. Due to the two-electron nature of this process, one may wonder about the roles played by the electron-electron correlation in the shifting of the interference fringes. Indeed, we show that the modification of the interference pattern occurs at the two-electron level, which is independent of the energy partition between the two electrons. Our numerical results agree excellently with the prediction of a simple model, in which the entangled electron pair is launched by light with a spatially dependent phase.

Full configuration interaction simulations of exchange-coupled donors in silicon using multi-valley effective mass theory

Donor spins in silicon have achieved record values of coherence times and single-qubit gate fidelities. The next stage of development involves demonstrating high-fidelity two-qubit logic gates, where the most natural coupling is the exchange interaction. To aid the efficient design of scalable donor-based quantum processors, we model the two-electron wave function using a full configuration interaction method within a multi-valley effective mass theory. We exploit the high computational efficiency of our code to investigate the exchange interaction, valley population, and electron densities for two phosphorus donors in a wide range of lattice positions, orientations, and as a function of applied electric fields. The outcomes are visualized with interactive images where donor positions can be swept while watching the valley and orbital components evolve accordingly. Our results provide a physically intuitive and quantitatively accurate understanding of the placement and tuning criteria necessary to achieve high-fidelity two-qubit gates with donors in silicon.

Synthesis and Properties of Fused Extended Tetrathiafulvalene Donors with Dithienylmethylene Spacer and Application to Organic Rechargeable Batteries

Abstract Several derivatives of 1,3-dithiole[3]dendralene derivatives with two thiophene spacers (2) were synthesized. The results of an X-ray structure analysis of an analog without fused TTF units and a DFT calculation of the unsubstituted-2 (2Aa) suggested that 2Aa had a structure significantly distorted between the central 1,3-dithiole (DT) ring and one of the thiophene rings. Cyclic voltammograms of tetrakis(n-hexylsulfanyl) derivatives (2Be and 2Ce) consisted of one pair of two-electron redox waves and five pairs of one-electron redox waves in correspondence with the presence of seven redox-active DT rings. The tetrakis(methylsulfanyl) or bis(ethylenedithio) derivative (2Bc or 2Bd) worked as a positive electrode material for lithium-ion batteries, and the 2Bc/Li and 2Bd/Li cells exhibited the first discharge capacities of 133–138 mAh g−1.

Synthesis, Structure, and Electrochemical Properties of Extended Tetrathiafulvalene Dimers Linked by Flexible Butylene Chain

Abstract Tetrathiafulvalene derivatives bridged by butylene units were successfully synthesized. X-ray structure analysis of the dimeric EBDT 1 revealed that C–C bonds have staggered arrangement and all substituents are arranged anti to each other in the butylene bridge. Cyclic voltammograms of the molecules 1 and the fused TTF analog 2 consisted of two-pairs of two-electron and four pairs of two-electron redox waves, respectively. X-ray structure analysis and calculation of overlap integrals of a dicationic salt of unsubstituted-1 (1Aa) indicated that the EBDT units are stacked with strong dimerization. Coin-type cells using the molecule 2 (2Ce) as a positive electrode material showed initial discharge capacity of 179 mAh g−1, initial energy density of 600 mWh g−1, and an average voltage of 3.35 V.

Collinear configuration of the helium atom and two-electron ions

Collinear configurations of the helium-like atomic systems, relevant, e.g., for the quasifree mechanism of the double photoionization of helium, are studied, parameterized by the single scalar parameter −1≤λ≤1 (“collinear parameter”) where λ=0 corresponds to the electron–nucleus (e −−n) coalescence and λ=1 corresponds to the electron–electron (e −−e) coalescence. In general, λ>0 corresponds to the n–e–e configuration, and λ<0 to the e–n–e configuration. Simple mathematical representations of the expectation values of the Dirac delta function relevant for the collinear configurations are derived and calculated from fully three-body dynamics without approximation for the two-electron atomic wave functions with nuclear charge 1≤Z≤5. Simple formulas for calculating the expectation values of the kinetic and potential energy operators in collinear configurations are derived. Unusual physical properties of the n-e-e collinear configurations found for certain ranges of λ are presented. The first few angular Fock coefficients for collinear configurations are derived as functions of λ. Highly accurate model wave functions describing the ground states of the two-electron atoms with a collinear arrangement of the particles are constructed. All results are illustrated in tables and figures.

Synthesis of Peripherally Arylated Tetrathiafulvalenes Extended with an Anthraquinoid Spacer via Pd-Catalyzed C–H Arylation and Construction of a Double-Helical Cobalt-Based Metal-Organic Framework

Peripherally arylated tetrathiafulvalenes with an anthraquinoid spacer (TTFAQs) have been synthesized by using palladium-catalyzed direct C–H arylation of the 1,3-dithiole rings. Electrochemical analysis by cyclic voltammetry has revealed that the new tetraarylated TTFAQs show one pair of simultaneous two-electron transfer waves as the parent TTFAQ does. The hydrolysis of the tetra(p-ethoxycarbonylphenyl)-substituted derivative affords the corresponding tetracarboxylic acid, which forms a new double-helical metal-organic framework upon complexation with cobalt(III) nitrate.

Two-electron selective coupling in an edge-state based conditional phase shifter

We investigate the effect of long-range Coulomb interaction on the two-electron scattering in the integer quantum Hall regime at bulk filling factor 2. A parallel version of the Split-Step Fourier method evolves the exact two-particle wave function in a 2D potential background reproducing the effect of depleting gates in a realistic heterostructure, with the charge carrier represented by a localized wavepacket of edge states. We compare the spatial shift induced by Coulomb repulsion in the final two-electron wave function for two indistinguishable electrons initialized in different configurations according to their Landau index, and analyze their bunching probability and the effect of screening. We finally prove the feasibility of the present operating regime as a two-qubit conditional phase shifter to generate entanglement from product states.