Correlation energy of two-electron systems
Correlation energy of two-electron systems
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Correlation energy of two-electron systems
Comparison of hartree-fock orbital with first natural spin orbital for two-electron system
We investigate the basis-set-size dependence for quantities related to interacting electrons in the canonical ensemble. Calculations are performed using exact diagonalization (finite temperature full configuration interaction method) on two-electron model systems–the uniform electron gas (UEG) and the helium atom. Our data reproduce previous observations of a competition for how the internal energy converges between the ground-state correlation energy and the high-temperature kinetic energy. We explore how this can be related to component parts of the internal energy including kinetic, exchange, and correlation energies and show there is surprising nuance in how this can be broken down into mostly monotonically converging quantities. We also show that separation of the free energy into a free energy with/without correlation allows for monotonic convergence with basis set size due to the variational principle. We find that the free energy convergence matches the previously observed convergence properties of the internal energy. We discuss the free energy divergence that happens when converging a finite basis analytical hydrogen atom to the complete basis set limit and compare this to the energies of a helium atom in a large periodic box. Reducing the box size, we saw convergence trends for the helium atom that were similar to the UEG.
Correlated closed- and open-shell functions considering 1/Z expansion for obtaining corresponding first-order wave functions and second-order energy
Cross sections for electron scattering on the ground state of LiH are calculated using the molecular convergent close-coupling method. The fixed-nuclei approximation is utilized, and calculations are performed at the mean internuclear separation of 3.06 𝑎 0 for projectile energies ranging from 0.1 to 500 eV. Here, a model-potential approach is utilized to treat the LiH molecule as a two-electron system, and good agreement is found with the literature for various properties of the structure model. Cross sections are presented for elastic scattering, excitation to several electronic states, and ionization. Comparison is made with existing results, and qualitative agreement is found with small close-coupling calculations at low projectile energies. Good agreement is found with first-order results at high projectile energies.
Partial separation of variables applied to two p-electron calculations - spin orbit, angular momentum, energy transfer, atomic excitation
Symmetric Euler angle decomposition of two electron fixed-nucleus problem - quantum mechanics considerations of angular momentum, parity, and kinetic energy
Optimum selection of Euler angles for expansion in eigenfunctions of angular momentum of two identical particles in fixed nucleus field
Central field wave functions for oscillator strength computations in two-electron helium system
Materials utilized by novel energy systems are often studied using weakly correlated mean-field theories. However, if these systems incorporate heavy elements, relativistic effects must be included. Therefore, a Kramers unrestricted coupled cluster with singles and doubles excitation formalism within a molecular mean-field exact two-component framework (X2C mmf ) using a four-component Dirac–Hartree–Fock (DHF) reference state is presented. The exact X2C mmf transformed normal-order Hamiltonian incorporates all one-electron and two-electron (2e) contributions from the Coulomb, Gaunt, and Breit operators and is used with the equation of motion method to calculate the excitation energies of the alkali group of elements. Using this framework, the effects of 2e Gaunt and Breit integrals are studied. Results demonstrate growing contributions from these integrals to the generated X2C mmf mean-fields and electronic fine structure calculations with increasing atomic number. Overall, this paper outlines the method, its effect within the X2C mmf approach, and lays the foundation for future theoretical development of relativistic calculations within this framework.
In this work, we detail an implementation, suitable for calculations on highly correlated ionizing systems, of a modified finite element discrete variable representation (FE-DVR) appended with a Gauss-Radau-Laguerre element. The appended element includes exterior complex scaling (ECS) to impose outgoing wave boundary conditions on treatments of processes involving continuum electrons. In this “infinite range” ECS (irECS), the complications that introduce reflections from the end of the grid when the last ECS finite element has finite range are avoided by the use of the Laguerre-weighted exponentially decaying tails, while outgoing wave boundary conditions are still imposed via the ECS transformation. For highly correlated systems in the absence of strong external fields we find that accurate two-electron integrals are essential in this modified FE-DVR. To accurately compute the two-electron integrals over the entire ECS contour, we present a detailed examination of the implications from the boundary terms that arise in a solution of Poisson’s equation with the Radau-Laguerre basis. A boundary term correction is necessary, and when included, the Radau-Laguerre DVR can accurately describe highly correlated states such as the doubly excited states of helium over the entire ECS contour.
Molten fluoride salts have long been the subject of investigation given their application in a variety of industrial processes. Oxides are a common exogeneous impurity within these systems, and can lead to many operational and regulatory concerns. In recent studies, the electro-oxidation of dissolved metallic oxides has been attributed to a single step oxygen evolution reaction despite voltametric evidence suggesting otherwise. Here, we first use square wave voltammetry to confirm that the oxidation of dissolved oxides in fluorides proceeds via a two-electron transfer reaction and then demonstrate that the presence of a peroxide-mediated redox reaction better explains the electrochemical data. Numerical simulations were then used to support the determination of improved diffusion coefficients to enable electroanalytical measurements of the oxide concentration across at a variety of temperatures. A linear Arrhenius relationship was observed when the experimental data was corrected for ohmic resistance effects. Concentration measurements using the corrected square wave voltammetry data demonstrated high accuracy across a large range of concentrations, while the uncorrected data showed plateauing and high errors. In total, this work serves to rectify previously misconstrued electrochemical data of oxides in molten salts and demonstrate how accurate, in situ concentration measurements can be achieved in real-world systems.
Reliable transport and thermodynamic data for multivalent ions in complex molten salts are scarce, limiting model fidelity for electrorefining and impurity control. Here, we report a comprehensive electrochemical characterization of SnCl₂ in LiCl–KCl–CaCl₂ (50.5–44.2–5.3 mol%) at 685 K, including the effects of Ni 2+ and Cu + impurities. Using cyclic voltammetry (CV), chronoamperometry (CA), and chronopotentiometry (CP), we quantified Sn 2+ and Ni 2+ diffusion with exceptional agreement across methods: Sn 2+ averaged (1.03 ± 0.10) × 10 −5 cm 2 s −1 , and Ni 2+ averaged (0.75 ± 0.19) × 10 −5 cm 2 s −1 . The tight confidence-interval overlap across CV, CA, and CP strengthens confidence in these values and is uncommon in molten chloride studies. Open-circuit-potential measurements provided standard apparent reduction potentials that closely match LiCl–KCl literature, indicating minimal shift with CaCl₂ present. The Sn 2+ /Sn couple behaves as a reversible two-electron soluble–insoluble process at 685 K; the Sn 4+ /Sn 2+ couple transitions to soluble–soluble behavior near 788 K, which may correlate with the decomposition of surface bound chlorostannates, though direct characterization remains to be established. In mixed systems, Cu+/Cu overlaps Sn 2+ /Sn, limiting Cusingle bondSn electroseparation, whereas the larger potential gap between Ni 2+ /Ni and Sn 2+ /Sn supports selective Ni removal. These internally consistent transport and thermodynamic data establish a validated basis for process modeling and optimization of Sn electrorefining and impurity management in LiCl–KCl–CaCl₂.
Increasing interest in alternative methods for fuel generation and chemical synthesis has resulted in an increased focus on the development of electrocatalysts for energy relevant small molecule transformations, such as the oxidation of methanol. Partial methanol oxidation is a crucial step in the generation of the commodity chemical formaldehyde, and its complete oxidation to carbon dioxide can also serve as the anodic reaction in direct methanol fuel cells. We report a coelectrocatalytic system comprised of an oxo-centered triruthenium cluster (Ru 3 O) as the catalyst, with the electro-generated N-phthalimido-N-oxyl (PINO) radical species acting as a redox mediator. Only a mild Brønsted base, 2,6-lutidine, is required to achieve an electrocatalytic response. The cocatalytic system demonstrates remarkable cooperativity, shifting the oxidation potential of MeOH (Ep) less positive by ca. 0.5 V compared to the intrinsic response of the Ru 3 O complex. Controlled potential electrolysis on a model substrate, 4-trifluoromethylbenzyl alcohol, demonstrates selective production of the two-electron, two-proton aldehyde product with a Faradaic efficiency of 79 ± 11% at a rate of 3.14 s –1 . The rate of cocatalysis is 50-fold greater than the intrinsic activity of Ru3O and 26-fold greater than that of PINO alone under otherwise identical conditions. Mechanistic studies reveal the oxidation of a Ru 3 O–alkoxide species as the potential-determining step, while two possible rate-determining steps are identified depending on the substrate. A preference for sterically uninhibited electron-rich benzyl alcohol substrates suggests that a H atom transfer from the Ru 3 O–alkoxide adduct to PINO is rate-determining, while the lack of an observed kinetic isotope effect using deuterated MeOH suggests the oxidation of the Ru 3 O–alkoxide species is both rate- and potential-determining for cocatalysis.
Nanoarchitectonics offers a systematic approach to creating an artificial framework by integrating different multiscale components such as semiconductor lattices and biological substances. However, most nanoarchitectonic abiotic-biotic hybrid systems have intrinsic limitations in imparting nonequilibrium biological features into semiconductor lattices at the nanoscale. Here, in this study, we report a new nanoarchitectonic system integrating bismuth oxychloride (BiOCl) nanosheets with purple membrane (PM) patches. PM is an archaeal subcellular fraction capable of unidirectionally transferring photogenerated charge carriers to its surroundings at the nanoscale independent of archaeal metabolism and retaining this dynamic functionality after isolation from living archaea. Microscopy, spectroscopy, electrochemical, and synchrotron X-ray scattering analyses verify that the nanoarchitectonic hybridization between BiOCl and PM generates a vertical heterostructure, thereby enhancing photogenerated charge-carrier dynamics and enabling the associated photocatalytic capacity. The resulting PM-BiOCl hybrid nanosheets efficiently convert dioxygen into hydrogen peroxide through a two-electron and two-proton transfer process under ambient conditions while simultaneously converting ethylene glycol into value-added chemicals. This study presents a nanoarchitectonic approach that leverages the photogenerated charge-carrier dynamics of the archaeal subcellular fractions to modulate the optoelectronic and catalytic capacity limitations of semiconductors.
Battery technologies beyond Li-ion are likely needed for extensive integration of grid-scale storage. The rechargeable Zn-MnO 2 chemistry has the potential for high sustainability, high safety, and low cost, using Earth-abundant basis materials. In an alkaline electrolyte, the MnO 2 cathode can cycle reversibly if modified by including a Bi-containing additive, although the cycling mechanism remains mostly unknown. This work presents an account of the intermediate species involved in the electrochemical transformation from layered δ-MnO 2 to Mn(OH) 2 and back. During charge, a disordered intermediate with a structure resembling layered β-MnOOH exists stably for an extended period, corresponding to a regime known to have unexpected electrochemical activity of Bi. During discharge, β-MnOOH exists only briefly and is never the majority material, revealing that the cycling mechanism is asymmetric. In conclusion, these findings represent a significant advance in mechanistic knowledge and can enable engineering to develop the system for commercial use.
We present a simple relativistic exact 2-component (X2C) Hamiltonian that models two-electron picture-change effects using Lehtola’s superposition of atomic potentials (SAP) [S. Lehtola, J. Chem. Theory Comput. 15, 1593−1604 (2019)]. The SAP-X2C approach retains the low cost and technical simplicity of the popular 1-electron X2C (1eX2C) predecessor but is significantly more accurate and has a well-defined thermodynamic limit, making it applicable to extended systems (such as large molecules and periodic crystals). The assessment of the SAP-X2C-based Hartree−Fock total and spinor energies, spin−orbit splittings, equilibrium bond distances, and harmonic vibrational frequencies suggests that SAP-X2C is similar to the more complex atomic meanfield (AMF) X2C counterparts in its ability to approximate the 4-component Dirac−Hartree−Fock reference.
The use of lanthanide complexes for catalytic dinitrogen reduction is a new development in homogeneous catalysis. Density functional theory calculations on our recently reported cerium phenolate catalyst [K 2 Ce 2 (sol) 4 (mTP) 2 ] (mTP = {(OC 6 H 2 - 2- t Bu-4-Me) 2 CH} 2 - 1,3-C 6 H 4 ; sol = OMe 2 here; THF in the experiment) have been undertaken to elucidate the reduction, activation and silylation steps at the bound dinitrogen molecule, in the presence of the reductant, potassium metal (K 0 ) and the electrophile Me 3 SiCl (TMSCl). Out of the total of six electron reductions required to cleave the N 2 , the first two-electron reduction step was found to be highly disfavoured unless potassium cations (K + ) are included, upon which the step is rendered strongly exergonic; N–Si bond formation at the two-electron stage is predicted to be unfavourable. The three-electron-reduced N 2 -adduct is found to be at the reductive activation limit in the absence of added electrophiles, which can form N-element bonds and lower the overall charge. Added electron density beyond three-electron reduction no longer localises on N 2 , preventing formal N 2 4− formation. A pathway in which both K 0 and Me 3 SiCl work in concert was modelled, and six sequential reduction–silylation steps were calculated, showing how the N–N bond is cleaved after the third reduction, eventually releasing two equivalents of N(SiMe 3 ) 3 , and regenerating the starting complex with the highest barrier of any step being 22 kcal mol −1 . We establish alkali metal coordination and coupled electron–electrophile transfer as key factors in the design of rare-earth-mediated dinitrogen functionalisation.