Two-electron Wave Functions Research Articles

Theoretical study of differential cross sections for the ionization of helium by fast proton impact

Abstract We present the angular distribution of the ejected electron for single ionization of He by fast proton impact. A four-body formalism of the three-Coulomb wave is applied to calculate the triple differential cross sections at several impact energies in the scattering, perpendicular and azimuthal planes. Moreover, the three-body formalism of three-Coulomb, two-Coulomb and first Born approximation models has also been used to study the many-body effect on electron emission and the validity of the models. In the three-Coulomb wave model, the final state wave function incorporates distortion due to the three-body mutual Coulombic interaction. In this formalism, we use an uncorrelated and correlated Born initial state, which consists of a plane wave for the incoming projectile times a two-electron bound state wavefunction of the helium atom representing the 1s2(1S) state. But, in the case of the three-body formalism, the initial state wavefunction consists of a long-range Coulomb distortion for the incoming projectile and one active electron of the He atom described by the Roothaan–Hartree–Fock wavefunction. The structure with a single or two peaks with unequal intensity is observed in the angular distributions of the triple differential cross sections for the different kinematic conditions. In addition, the influence of static electron correlations is investigated using different bound state wavefunctions for the ground state of the He target. In the four-body formalism, the present computations are very fast by reducing a nine-dimensional integral to a two-dimensional real integral. Despite the simplicity and speed of the proposed quadrature, the comparison shows that the obtained results are in reasonable agreement with the experiment and are compatible with those of other theories.

Coupled cluster-inspired geminal wavefunctions.

Electron pairs have an illustrious history in chemistry, from powerful concepts to understanding structural stability and reactive changes to the promise of serving as building blocks of quantitative descriptions of the electronic structure of complex molecules and materials. However, traditionally, two-electron wavefunctions (geminals) have not enjoyed the popularity and widespread use of the more standard single-particle methods. This has changed recently, with a renewed interest in the development of geminal wavefunctions as an alternative to describing strongly correlated phenomena. Hence, there is a need to find geminal methods that are accurate, computationally tractable, and do not demand significant input from the user (particularly via cumbersome and often ill-behaved orbital optimization steps). Here, we propose new families of geminal wavefunctions inspired by the pair coupled cluster doubles ansatz. We present a new hierarchy of two-electron wavefunctions that extends the one-reference orbital idea to other geminals. Moreover, we show how to incorporate single-like excitations in this framework without leaving the quasiparticle picture. We explore the role of imposing seniority restrictions on these wavefunctions and benchmark these new methods on model strongly correlated systems.

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

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.

Two-electron wavefunctions are matrix product states with bond dimension three

We prove the statement in the title for a suitable (wavefunction-dependent) choice of the underlying orbitals and show that 3 is optimal. Thus, for two-electron systems, the quantum chemistry density matrix renormalization group (QC-DMRG) method with bond dimension 3 combined with fermionic mode optimization exactly recovers the full configuration-interaction (FCI) energy.

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.

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

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.

High-order harmonic generation of para-helium and ortho-helium

We theoretically study the high-order harmonic generation (HHG) of para-helium and ortho-helium with different symmetric spatial configurations. Our time-dependent Schrödinger equation simulations clearly show that the HHG spectra of para-helium and ortho-helium have a distinct difference in the harmonic amplitude. By performing a detailed time-frequency wavelet analysis of the HHG spectra, the distinction between the HHG spectra of para-helium and ortho-helium is ascribed to the different excitation cross-section of the cation of helium. Furthermore, in terms of a simple recollision model, this dependence of the excitation cross-section of the cation on the symmetry of the two-electron wavefunction of helium is revealed.

Solution of two-electron Schrödinger equations using a residual minimization method and one-dimensional basis functions

Distinctive from conventional electronic structure methods, we solve the Schrödinger wave equations of the helium atom and its isoelectronic ions by employing one-dimensional basis functions to separate components. We use full two-electron six-dimensional operators and wavefunctions represented with real-space grids where the refinement of the latter is carried out using a residual minimization method. In contrast to the standard single-electron approach, the current approach results in exact treatment of repulsion energy and, hence, more accurate electron correlation within five centihartrees or better included, with moderate computational cost. A simple numerical convergence between the error to accurate results and the grid-spacing size is found. The obtained two-electron Schrödinger wavefunction that contains vast and elaborating information for the radial correlation function and common one-dimensional functions shows the electron correlation effect on one-electron distributions.

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.

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.

Total cross sections in five methods for two-electron capture by alpha particles from helium: CDW-4B, BDW-4B, BCIS-4B, CDW-EIS-4B and CB1-4B

This work is on quantum-mechanical four-body distorted wave theories for double electron capture in collisions between fast heavy multiply charged ions and heliumlike atomic systems. The five widely used distorted wave methods of the first- and second-order in the perturbation series expansions are compared with the available experimental data on alpha –He collisions. These are the four-body boundary-corrected first Born (CB1-4B), the boundary-corrected continuum intermediate state (BCIS-4B), the Born distorted wave (BDW-4B), the continuum distorted wave (CDW-4B) and the continuum distorted wave-eikonal initial state (CDW-EIS-4B) methods.We address the complete breakdown of the CDW-EIS-4B method at all impact energies within its expected validity domain (100–10000 keV). Further, the relative performance is evaluated of the second-order theories with and without the eikonalization of the two-electron Coulomb wavefunctions for double continuum intermediate states. Finally, at all the considered intermediate and high energies, the practical aspects of the studied five methods are investigated by protracted evaluations of the convergence rates of total cross sections as a function of the number of quadrature points per axis in numerical computations of multi-dimensional (3D-5D) integrals.

A Simple Numerical Matrix Method for Accurate Triplet-1s2s 3S1 Energy Levels of Some Light Helium-like Ions

Triplet-1s2s 3S1 energy levels of some light Helium-like ions (Li+ to Ne8+) were numerically calculated using a truncated-matrix method implemented in a simple Mathematica code. The two-electron wave functions were expanded in finite number of basis states consisting of hydrogenic s-orbitals. Here, the Hamiltonian matrix used was of size 15×15, 25×25 and 35×35. From the results, triplet-1s2s 3S1 energies of the ions were in good agreement with other advanced and highly accurate calculations in the literature. Comparisons with one of the most accurate calculations in the literature showed that our results were very accurate with the largest error in energy being only 0.135 % for Li+ ion and the smallest one being approximately 0.018 % for Ne8+ ion energy, when 35 basis states were used in our calculations. Errors from our calculations were also much lower than those from the geometrical model.

Full configuration-interaction calculations of the angular quantities for helium atom

Abstract The angular quantities 〈 θ 12 〉 and 〈 cos θ 12 〉 for helium atom in both ground and singly-excited states are calculated in the framework of full configuration interaction. The two-electron system wave functions are expanded in terms of the product of Slater-type orbitals. The convergence and accuracy of reported results are examined in detail. From the comparison with previous work, it is shown that the present work establishes the most accurate prediction of the quantity 〈 θ 12 〉 for helium atom up to the present. Radial quantities of 〈 r 〉 , 〈 r > 〉 , 〈 r 〉 , and 〈 r 12 〉 are also reported and compared with previous calculations.

The influence of the electron correlations on dynamic of two-electron charge exchange between ions and polar molecules

Introduction. In the theory of electron capture the exchange interactions at large internuclear distances R between projectile and target particles are the most important ones, since they are associated with the largest cross sections. However, their accurate calculation in the asymptotic region of large R by standard ab initio techniques still requires significant computational effort. In the case of two-electron exchange interaction the situation is much complicated due to necessity of the usage of analytic approximation for the two-electron wave function which must have correct asymptotic (at large R ) behavior in whole configuration space of the electronic coordinates. This requirements can be satisfied by using the method of surface integrals and the Green’s function formalism. The present studies devoted to construction of asymptotically correct two-electronic wave function and two-electronic exchange interaction of the system which consist of the atomic ion and polar molecule. Purpose. To construct a closed analytical expression for two-electron exchange interaction potential, which defines the charge-exchange processes between the polar molecule and multicharged ion. Results . Within the framework of a semiclassical approach and approximation of a coulomb-dipole potential the closed analytical expression of the two-electron exchange interaction responsible for two-step mechanism of charge-exchange between the polar molecule and multicharged ion has been obtained. Conclusion . We have constructed the semiclassical representation for two-electron wave function of the quasimolecular system which consist of polar molecule and atomic ion for large separation R between heavy particles. Using obtained results a closed analytical form (in terms of full elliptic integrals) of the leading asymptotic term of exponentially small two-electron exchange interaction between the polar molecule and atomic ion has been obtained. The derived expression for exchange interaction can be used for study of the two-electron transfer processes in collision of polar molecules with atomic ions.

Imaging the square of the correlated two-electron wave function of a hydrogen molecule

The toolbox for imaging molecules is well-equipped today. Some techniques visualize the geometrical structure, others the electron density or electron orbitals. Molecules are many-body systems for which the correlation between the constituents is decisive and the spatial and the momentum distribution of one electron depends on those of the other electrons and the nuclei. Such correlations have escaped direct observation by imaging techniques so far. Here, we implement an imaging scheme which visualizes correlations between electrons by coincident detection of the reaction fragments after high energy photofragmentation. With this technique, we examine the H2 two-electron wave function in which electron–electron correlation beyond the mean-field level is prominent. We visualize the dependence of the wave function on the internuclear distance. High energy photoelectrons are shown to be a powerful tool for molecular imaging. Our study paves the way for future time resolved correlation imaging at FELs and laser based X-ray sources.