Second-order Exchange Research Articles

Tensor hypercontraction for fully self-consistent imaginary-time GF2 and GWSOX methods: Theory, implementation, and role of the Green's function second-order exchange for intermolecular interactions.

We present an efficient MPI-parallel algorithm and its implementation for evaluating the self-consistent correlated second-order exchange term (SOX), which is employed as a correction to the fully self-consistent GW scheme called scGWSOX (GW plus the SOX term iterated to achieve full Green's function self-consistency). Due to the application of the tensor hypercontraction (THC) in our computational procedure, the scaling of the evaluation of scGWSOX is reduced from O(nτnAO5) to O(nτN2nAO2). This fully MPI-parallel and THC-adapted approach enabled us to conduct the largest fully self-consistent scGWSOX calculations with over 1100 atomic orbitals with only negligible errors attributed to THC fitting. Utilizing our THC implementation for scGW, scGF2, and scGWSOX, we evaluated energies of intermolecular interactions. This approach allowed us to circumvent issues related to reference dependence and ambiguity in energy evaluation, which are common challenges in non-self-consistent calculations. We demonstrate that scGW exhibits a slight overbinding tendency for large systems, contrary to the underbinding observed with non-self-consistent RPA. Conversely, scGWSOX exhibits a slight underbinding tendency for such systems. This behavior is both physical and systematic and is caused by exclusion-principle violating diagrams or corresponding corrections. Our analysis elucidates the role played by these different diagrams, which is crucial for the construction of rigorous, accurate, and systematic methods. Finally, we explicitly show that all perturbative fully self-consistent Green's function methods are size-extensive and size-consistent.

Impact of Second-Order Leader-Member Exchange on Employee Advocacy in the Chinese Hotel Industry

Impact of Second-Order Leader-Member Exchange on Employee Advocacy in the Chinese Hotel Industry

Write error rate analysis of field-free spin-orbit torque switching in conically magnetized free layer nanomagnet

Enhancing the performance of magnetic random access memories (MRAMs) is crucial, considering speed, energy efficiency, and endurance. Spin-orbit torque-based MRAMs offer ultrafast operation and enhanced reliability. Still, the energy efficiency and external magnetic field requirement for deterministic switching of nanomagnets with perpendicular magnetic anisotropy (PMA) are a significant hurdle. To address these issues, one of the proposed solutions employs a conically magnetized free layer magnetic tunnel junction device featuring second-order PMA. This approach can minimize the required switching current density by leveraging the interplay between first- and second-order PMA energies. Additionally, coupling with exchange bias from the antiferromagnet can eliminate the need for an external field entirely. Our analysis investigates the impact of current density (magnitude and rise/fall time), first- and second-order PMA fields, exchange bias, and field-like torque on the device's switching performance. By optimizing the perpendicular magnetic anisotropies, we report improvement in the write error rate from 10−4 to 10−7. Our findings hold promise for magnetic memory applications.

Fully Dynamic G3W2 Self-Energy for Finite Systems: Formulas and Benchmark.

Over the years, Hedin's GW self-energy has been proven to be a rather accurate and simple approximation to evaluate electronic quasiparticle energies in solids and in molecules. Attempts to improve over the simple GW approximation, the so-called vertex corrections, have been constantly proposed in the literature. Here, we derive, analyze, and benchmark the complete second-order term in the screened Coulomb interaction W for finite systems. This self-energy named G3W2 contains all the possible time orderings that combine 3 Green's functions G and 2 dynamic W. We present the analytic formula and its imaginary frequency counterpart, with the latter allowing us to treat larger molecules. The accuracy of the G3W2 self-energy is evaluated on well-established benchmarks (GW100, Acceptor 24, and Core 65) for valence and core quasiparticle energies. Its link with the simpler static approximation, named SOSEX for static screened second-order exchange, is analyzed, which leads us to propose a more consistent approximation named 2SOSEX. In the end, we find that neither the G3W2 self-energy nor any of the investigated approximations to it improve over one-shot G0W0 with a good starting point. Only quasi-particle self-consistent GW HOMO energies are slightly improved by addition of the G3W2 self-energy correction. We show that this is due to the self-consistent update of the screened Coulomb interaction, leading to an overall sign change of the vertex correction to the frontier quasiparticle energies.

Significant contributions of second-order exchange terms in GW electron–hole interaction kernel for charge-transfer excitations

The GW electron–hole interaction kernel, which includes two second-order exchange terms in addition to the first-order direct and exchange terms considered in the conventional GW + Bethe–Salpeter method, is applied to 10 two-molecular systems and six thermally activated delayed fluorescence (TADF) molecules in which inter- and intramolecular charge transfer excitations are expected to occur. The contributions of the two second-order exchange terms are almost zero for intermolecular charge transfer excitations and ∼0.75 eV for intramolecular charge transfer excitations according to our exciton analysis method with exciton wave functions. For TADF molecules, we found that the second-order exchange terms are more significant than the first-order exchange terms, and the contributions—even for local-type and delocalized-type excitations—are not negligibly small. We revealed that the two second-order exchange terms are proportional to the molecular size, the exciton binding energy, and the electron–hole overlap strength for intramolecular charge-transfer excitations. We believe that our findings are indispensable for further considerations of the GW method in the future.

Assessment of the Second-Order Statically Screened Exchange Correction to the Random Phase Approximation for Correlation Energies.

With increasing interelectronic distance, the screening of the electron–electron interaction by the presence of other electrons becomes the dominant source of electron correlation. This effect is described by the random phase approximation (RPA) which is therefore a promising method for the calculation of weak interactions. The success of the RPA relies on the cancellation of errors, which can be traced back to the violation of the crossing symmetry of the 4-point vertex, leading to strongly overestimated total correlation energies. By the addition of second-order screened exchange (SOSEX) to the correlation energy, this issue is substantially reduced. In the adiabatic connection (AC) SOSEX formalism, one of the two electron–electron interaction lines in the second-order exchange term is dynamically screened (SOSEX(W, vc)). A related SOSEX expression in which both electron–electron interaction lines are statically screened (SOSEX(W(0), W(0))) is obtained from the G3W2 contribution to the electronic self-energy. In contrast to SOSEX(W, vc), the evaluation of this correlation energy expression does not require an expensive numerical frequency integration and is therefore advantageous from a computational perspective. We compare the accuracy of the statically screened variant to RPA and RPA+SOSEX(W, vc) for a wide range of chemical reactions. While both methods fail for barrier heights, SOSEX(W(0), W(0)) agrees very well with SOSEX(W, vc) for charged excitations and noncovalent interactions where they lead to major improvements over RPA.

Development of the Bethe-Salpeter method considering second-order corrections for a GW electron-hole interaction kernel

We derive two second-order exchange terms in the $GW$ electron-hole interaction kernel. The contributions of these terms have been neglected in the conventional $GW +$ Bethe-Salpeter method, and we implement them in an all-electron mixed basis program. To reveal the effect of these terms, we apply them to 28 molecules of Thiel's benchmark set and compare the ${S}_{1}$ excitation energies with those obtained from the conventional $GW +$ Bethe-Salpeter method. In addition, using the exciton analysis method with exciton wave functions, we estimate the expectation values for each term in the $GW$ electron-hole interaction kernel. The contribution of the two second-order exchange terms is approximately 0.1--0.2 eV at the exciton states of the $n\ensuremath{\rightarrow}{\ensuremath{\pi}}^{*}$ transition; however, the contributions are smaller for the $\ensuremath{\pi}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{*}$ and $\ensuremath{\pi}\ensuremath{\rightarrow}$ Rydberg transitions. Our findings reveal that the errors of the conventional $GW +$ Bethe-Salpeter method are potentially reduced by considering these terms; however, the extent of the corrections is insufficient for the underestimated excitation energies. We believe that our findings are a significant step towards advancing the conventional $GW +$ Bethe-Salpeter method.

Exploring the statically screenedG3W2correction to theGWself-energy: Charged excitations and total energies of finite systems

Electron correlation in finite and extended systems is often described in an effective single-particle framework within the $GW$ approximation. Here, we use the statically screened second-order exchange contribution to the self-energy ($G3W2$) to calculate a perturbative correction to the $GW$ self-energy. We use this correction to calculate total correlation energies of atoms, relative energies, as well as charged excitations of a wide range of molecular systems. We show that the second-order correction improves correlation energies with respect to the RPA and also improves relative energies for many, but not all considered systems. While the full $G3W2$ contribution does not give consistent improvements over $GW$, taking the average of $GW$ and $GW + G3W2$ generally gives excellent results. Improvements over quasiparticle self-consistent $GW$, which we show to give very accurate charged excitations in small and medium molecules by itself, are only minor. $G_0W_0$ quasiparticle energies evaluated with eigenvalue and orbitals from range-separated hybrids, however, are tremendously improved upon: The second-order corrected $G_0W_0$ outperforms all existing $GW$ methods for the systems considered herein and also does not come with substantially increased computational cost compared to $G_0W_0$ for systems with up to 100 atoms.

The GW Miracle in Many-Body Perturbation Theory for the Ionization Potential of Molecules.

We use the GW100 benchmark set to systematically judge the quality of several perturbation theories against high-level quantum chemistry methods. First of all, we revisit the reference CCSD(T) ionization potentials for this popular benchmark set and establish a revised set of CCSD(T) results. Then, for all of these 100 molecules, we calculate the HOMO energy within second and third-order perturbation theory (PT2 and PT3), and, GW as post-Hartree-Fock methods. We found GW to be the most accurate of these three approximations for the ionization potential, by far. Going beyond GW by adding more diagrams is a tedious and dangerous activity: We tried to complement GW with second-order exchange (SOX), with second-order screened exchange (SOSEX), with interacting electron-hole pairs (W TDHF), and with a GW density-matrix (γ GW ). Only the γ GW result has a positive impact. Finally using an improved hybrid functional for the non-interacting Green’s function, considering it as a cheap way to approximate self-consistency, the accuracy of the simplest GW approximation improves even more. We conclude that GW is a miracle: Its subtle balance makes GW both accurate and fast.

Superexchange-induced valley splitting in two-dimensional transition metal dichalcogenides: A first-principles study for rational design

Monolayer transition metal dichalcogenides (TMDs) with spin-valley coupling are a well-studied class of two-dimensional materials with potential for novel optoelectronics applications. Breaking time-reversal symmetry via an external magnetic field or supporting magnetic substrate can lift the degeneracy of the band gaps at the inequivalent $K$ and ${K}^{\ensuremath{'}}$ high symmetry points, or valleys, in the monolayer TMD Brillouin zone, a phenomenon known as valley splitting. However, reported valley splittings thus far are modest, and a detailed structural and chemical understanding of valley splitting via magnetic substrates is lacking. Here we probe the underlying physical mechanism with a series of density functional theory (DFT) calculations of magnetic atoms with varying coverage on the surface of prototypical monolayer ${\mathrm{WSe}}_{2}$ and ${\mathrm{MoS}}_{2}$ TMDs. Near-valence band edge energies for variable magnetic atom height, lateral registry, and magnetic moment are calculated with DFT, and trends are rationalized with a model Hamiltonian with second-order spin-dependent exchange coupling. From our analysis, we demonstrate how large valley splittings may be achieved and that the valley splitting can be understood with a superexchange mechanism, which strongly depends on overlaps of TMD Bloch states at the valley extrema with the localized $d$ states of the magnetic atom, as well as the out-of-plane component of the magnetic moment of the magnetic atom. Our calculations provide a basis for understanding prior measurements of valley splitting and suggest routes for enhancing valley splitting in future systems of interest.

Insight Into The Spin-Vibronic Problem of a Mixed Valence Magnetic Molecular Cell for Quantum Cellular Automata.

The effects of the vibronic coupling in quantum cellular automata (QCA) based on the square planar mixed valence (MV) molecular cells comprising four paramagnetic centers (spin cores) and two excess mobile electrons are analyzed in the important particular case when the Coulomb energy gap between the ground antipodal diagonal-type two-electron configurations and the excited side-type configurations considerably exceeds both the one-electron transfer parameter (strong U-limit) and the vibronic stabilization energy. Under such conditions the developed model involves the second-order double exchange, the Heisenberg-Dirac-Van Vleck (HDVV) exchange and the vibronic coupling of the excess electrons with the molecular B1g -vibration composed of four full-symmetric local vibrations. The latter interaction is shown to significant amplify the ability of the electric field produced by the driver-cell to polarize the excess electrons in the working cell, which can be termed "the effect of the vibronic enhancement of the cell-cell interaction". This effect leads to a redetermination of the conditions for switching between different spin-states, as well as to a significant change in the shapes of the cell-cell response functions. The obtained results demonstrate the importance of the vibronic coupling in all aspects (such as description of a free cell and cell-cell response) of the theory of molecular QCA based on MV clusters.

Nonapproximated third-order exchange induction energy in symmetry-adapted perturbation theory.

The exchange terms in symmetry-adapted perturbation theory (SAPT) are normally calculated within the so-called S2 or single exchange approximation, which approximates the all-electron antisymmetrizer by interchanges of at most one electron pair between the interacting molecules. This approximation is typically very accurate at the van der Waals minimum separation and at larger intermolecular distances but begins to deteriorate at short range. Nonapproximated expressions for the second-order SAPT exchange corrections have been derived some time ago by Schäffer and Jansen [Mol. Phys. 111, 2570 (2013)]. In this work, we extend Schäffer and Jansen's formalism to derive and implement a nonapproximated expression for the third-order exchange-induction correction. Numerical tests on several representative noncovalent databases show that the S2 approximation underestimates the exchange-induction contributions in both second and third orders. This underestimation is very similar in relative terms, but the larger absolute values of the third-order exchange-induction effects, and their near complete cancellation with the corresponding induction energies, make the third-order errors more severe. In the worst-case scenario of interactions involving ions, the breakdown of the S2 approximation can result in a qualitatively wrong, attractive character of SAPT total energies at short range {as first observed by Lao and Herbert [J. Phys. Chem. A 116, 3042 (2012)]}. As expected, the inclusion of the full third-order exchange-induction energy in place of its S2-approximated counterpart restores the correct, repulsive short-range behavior of the SAPT potential energy curves computed through the third order.

Absence of Spontaneous Spin Symmetry Breaking in 1D and 2D Quantum Ferromagnetic Systems with Bilinear and Biquadratic Exchange Interactions

Some measurements have shown that the second-order exchange interaction is non-negligible in ferromagnetic compounds whose microscopic interactions are described by means of half-odd integer quantum spins. In these spin systems the ground state is either ferromagnetic or antiferromagnetic when the bilinear exchange interaction is dominant. Instead, in ferromagnetic systems characterized by bilinear and biquadratic exchange interactions of comparable magnitude, the energy minimum occurs when spins are in a canting ground-state. To this aim, a one-dimensional (1D) quantum spin chain and a two-dimensional (2D) lattice of quantum spins subjected to periodic boundary conditions are modeled via the generalized quantum Heisenberg Hamiltonian containing, in addition to the isotropic and short-range bilinear exchange interaction of the Heisenberg type, a second-order interaction, the isotropic and short-range biquadratic exchange interaction between nearest-neighbors quantum spins. For these 1D and 2D quantum systems a generalization of the Mermin–Wagner–Hohenberg theorem (also known as Mermin–Wagner–Berezinksii or Coleman theorem) is given. It is demonstrated, by means of quantum statistical arguments, based on Bogoliubov’s inequality, that, at any finite temperature, (1) there is absence of long-range order and that (2) the law governing the vanishing of the order parameter is the same as in the bilinear case for both 1D and 2D quantum ferromagnetic systems. The physical implications of the absence of a spontaneous spin symmetry breaking in 1D spin chains and 2D spin lattices modeled via a generalized quantum Heisenberg Hamiltonian are discussed.

Can the Double Exchange Cause Antiferromagnetic Spin Alignment?

The effect of the double exchange in a square-planar mixed-valence dn+1−dn+1−dn−dn–type tetramers comprising two excess electrons delocalized over four spin cores is discussed. The detailed analysis of a relatively simple d2−d2−d1−d1–type tetramer shows that in system with the delocalized electronic pair the double exchange is able to produce antiferromagnetic spin alignment. This is drastically different from the customary ferromagnetic effect of the double exchange which is well established for mixed-valence dimers and tetramers with one excess electron or hole. That is why the question “Can double exchange cause antiferromagnetic spin alignment?” became the title of this article. As an answer to this question the qualitative and quantitative study revealed that due to antiparallel directions of spins of the two mobile electrons which give competitive contributions to the overall polarization of spin cores, the system entirely becomes antiferromagnetic. It has been also shown that depending on the relative strength of the second-order double exchange and Heisenberg–Dirac–Van Vleck exchange the system has either the ground localized spin-triplet or the ground delocalized spin-singlet.

Performance and Scope of Perturbative Corrections to Random-Phase Approximation Energies

It has been suspected since the early days of the random-phase approximation (RPA) that corrections to RPA correlation energies result mostly from short-range correlation effects and are thus amenable to perturbation theory. Here we test this hypothesis by analyzing formal and numerical results for the most common beyond-RPA perturbative corrections, including the bare second-order exchange (SOX), second-order screened exchange (SOSEX), and approximate exchange kernel (AXK) methods. Our analysis is facilitated by efficient and robust algorithms based on the resolution-of-the-identity (RI) approximation and numerical frequency integration, which enable benchmark beyond-RPA calculations on medium- and large-size molecules with size-independent accuracy. The AXK method systematically improves upon RPA, SOX, and SOSEX for reaction barrier heights, reaction energies, and noncovalent interaction energies of main-group compounds. The improved accuracy of AXK compared with SOX and SOSEX is attributed to stronger screening of bare SOX in AXK. For reactions involving transition-metal compounds, particularly 3d transition-metal dimers, the AXK correction is too small and can even have the wrong sign. These observations are rationalized by a measure α̅ of the effective coupling strength for beyond-RPA correlation. When the effective coupling strength increases beyond a critical α̅ value of approximately 0.5, the RPA errors increase rapidly and perturbative corrections become unreliable. Thus, perturbation theory can systematically correct RPA but only for systems and properties qualitatively well captured by RPA, as indicated by small α̅ values.

On the Relation between Equation-of-Motion Coupled-Cluster Theory and the GW Approximation.

We discuss the analytic and diagrammatic structure of ionization potential (IP) and electron affinity (EA) equation-of-motion coupled-cluster (EOM-CC) theory, in order to put it on equal footing with the prevalent GW approximation. The comparison is most straightforward for the time-ordered one-particle Green's function, and we show that the Green's function calculated by EOM-CC with single and double excitations (EOM-CCSD) includes fewer ring diagrams at higher order than does the GW approximation, due to the former's unbalanced treatment of time-ordering. However, the EOM-CCSD Green's function contains a large number of vertex corrections, including ladder diagrams, mixed ring-ladder diagrams, and exchange diagrams. By including triple excitations, the EOM-CCSDT Green's function includes all diagrams contained in the GW approximation, along with many high-order vertex corrections. In the same language, we discuss a number of common approximations to the EOM-CCSD equations, many of which can be classified as elimination of diagrams. Finally, we present numerical results by calculating the principal charged excitations energies of the molecules contained in the so-called GW100 test set [ J. Chem. Theory Comput. 2015 , 11 , 5665 - 5687 ]. We argue that (in molecules) exchange is as important as screening, advocating for a Hartree-Fock reference and second-order exchange in the self-energy.

Zeolite P synthesis based on fly ash and its removal of Cu(II) and Ni(II) ions

Zeolite P was synthesized through hydrothermal method based on a kind of Class C fly ash (FA). X-ray diffraction (XRD), scanning electron microscopy (SEM), and Brunauer–Emmett–Teller (BET) were used to analyze and characterize the synthetic sample. The kinetics and thermodynamics of copper and nickel ions removed by the zeolite samples were experimentally explored in detail. The results of kinetic treatment showed the second-order exchange second-order saturation model (SESSM) can well describe the removal process of copper ions, while the first-order empirical kinetic model (FEKM) is the best kinetic model for nickel ions. Langmuir and Freundlich isotherms were used to fit the equilibrium concentration of Cu(II) or Ni(II) under certain conditions. Whether for copper or nickel ion, the Langmuir model is in good agreement with the experimental equilibrium concentration. The apparent theoretical removal capacities for Cu(II) and Ni(II) can reach to 138.1 mg·g− 1 and 77.0 mg·g− 1, respectively.

Fast Collisional Lipid Transfer Among Polymer-Bounded Nanodiscs

Some styrene/maleic acid (SMA) copolymers solubilise membrane lipids and proteins to form polymer-bounded nanodiscs termed SMA/lipid particles (SMALPs). Although SMALPs preserve a lipid-bilayer core, they appear to be more dynamic than other membrane mimics. We used time-resolved Förster resonance energy transfer and small-angle neutron scattering to determine the kinetics and the mechanisms of phospholipid transfer among SMALPs. In contrast with vesicles or protein-bounded nanodiscs, SMALPs exchange lipids not only by monomer diffusion but also by fast collisional transfer. Under typical experimental conditions, lipid exchange occurs within seconds in the case of SMALPs but takes minutes to days in the other bilayer particles. The diffusional and second-order collisional exchange rate constants for SMALPs at 30 °C are kdif = 0.287 s−1 and kcol = 222 M−1s−1, respectively. Together with the fast kinetics, the observed invariability of the rate constants with probe hydrophobicity and the moderate activation enthalpy of ~70 kJ mol−1 imply that lipids exchange through a “hydrocarbon continuum” enabled by the flexible nature of the SMA belt surrounding the lipid-bilayer core. Owing to their fast lipid-exchange kinetics, SMALPs represent highly dynamic equilibrium rather than kinetically trapped membrane mimics, which has important implications for studying protein/lipid interactions in polymer-bounded nanodiscs.

Nonlocal energy-optimized kernel: Recovering second-order exchange in the homogeneous electron gas

In order to remedy some of the shortcomings of the random phase approximation (RPA) within adiabatic connection fluctuation-dissipation (ACFD) density functional theory, we introduce a short-ranged, exchange-like kernel that is one-electron self-correlation free and exact for two-electron systems in the high-density limit. By tuning a free parameter in our model to recover an exact limit of the homogeneous electron gas correlation energy, we obtain a nonlocal, energy-optimized kernel that reduces the errors of RPA for both homogeneous and inhomogeneous solids. Using wave-vector symmetrization for the kernel, we also implement RPA renormalized perturbation theory for extended systems, and demonstrate its capability to describe the dominant correlation effects with a low-order expansion in both metallic and nonmetallic systems. The comparison of ACFD structural properties with experiment is also shown to be limited by the choice of norm-conserving pseudopotential.

Two-dimensional second-order quadrupolar exchange powder spectra for nuclei with half-integer spins. Calculations and an experimental example using oxygen NMR

Two-dimensional chemical exchange spectra are determined for powders containing half-integer quadrupolar nuclei that evolve under the second-order quadrupolar interaction. For simple exchange processes it is shown that the calculated exchange patterns depend sensitively on the jump angles, the tensor parameters, and the rotation axes characterizing the geometry of the underlying molecular motion. The inclusion of other interactions is discussed. To demonstrate the feasibility of the experimental method a two-dimensional second-order quadrupolar exchange spectrum was recorded for a clathrate hydrate using oxygen NMR. Simulations based on a motional model involving a six-site jump capture the essential features of the experimental results.