Two-electron Wave Functions Research Articles

Visualization of two-body electron-densities and wavefuctions for several molecules

The purpose of this paper is to propose a visualization method of two-electron wavefuctions. This method is based on a simple idea that the two-electron wavefunction maps on the direct product of two-electron coordinates or basis-sets. The obtained graphical image helps us to understand many-body problems intuitively. As examples, we apply this method to H 2 and H 4.

Magnetic ordering in strongly correlated-electron uranium systems: Consequences of two kinds of f-electron-band–electron states

Magnetic ordering involves the electronic behavior globally; and for uranium-based systems, the hybridization-induced effects dominate over the Coulomb exchange effects in determining the magnetic ordering. Therefore, as long as the hybridization is treated as acting between properly exchange-symmetrized two-electron wave functions, the effects of exchange can be incorporated in the one-electron exchange-correlation potential. As a consequence of the necessary exchange symmetrization, there are essentially two kinds of f electrons, localized magnetic and itinerant nonmagnetic. This has enabled us to make absolute material-specific predictions of alloying or high-pressure effects on magnetic ordering in uranium strongly correlated-electron (SCE) systems using local-density approximation input into many-electron dynamics. Experimentally, the alloying effects can be dramatic, e.g., in UxLa1−xS the magnetic ordering abruptly disappears at about 55% uranium. The theory is quite successful in its detailed absolute predictions, and this has important implications for the overall understanding of electronic behavior in SCE systems including heavy fermion systems. The key conclusion is that strengthening the hybridization, as kinematically restricted by exchange symmetry, leads to a chemical-environment-dependent sharp phase transition in SCE systems with dramatic observable consequences. This phase transition is associated with the elimination of the localized-magnetic transition-shell electrons (f electrons for light actinide and cerium-based SCE materials, d electrons for transition-metal–oxide-based SCE materials).

Visualization of two-body electron densities and wave functions of magnetic molecules

Visualization methods of two-electron wave functions, electron–electron correlation plot, proposed previously are developed for magnetic molecules. This method can provide intuitive understanding of the two-body wave function and is a useful tool to compare characteristics of various approximation methods for many-body problems such as restricted Hartree–Fock (RHF), unrestricted Hartree–Fock (UHF), (GHF), and configuration interaction (CI). Our visualization method is fully explained on the basis of the two-body density matrix. Applications of these methods are performed for trans-butadiene and trimethylene methane, which are typical examples of nonmagnetic and magnetic organic molecules, respectively. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 645–654, 1999

Lattice, time-dependent approach for electron-hydrogen scattering

A time-dependent approach for treating electron-hydrogen scattering is reported that utilizes a fully correlated two-electron wavefunction represented on a three-dimensional lattice using the basis-spline collocation method. The lattice, time-dependent approach obviates the need for consideration of the three-body Coulomb boundary conditions, avoids the use of severe approximations such as those of perturbation theory for slow collisions, and provides a relatively dense representation of the one- and two-electron continua. Probabilities for excitation and ionization are computed by projection onto lattice eigenstates of the H atom. Partial cross sections for excitation and ionization are obtained and compared with results of other theoretical methods for the 1S and 3S channels.

Configuration-interaction Hartree-Fock calculations for two-electron atoms using a pseudopotential

We have developed an approach to study doubly excited Rydberg states of atoms or ions with two optically active electrons outside an ionic closed-shell core. The interaction between the core and one valence electron is modeled by an accurate semiempirical pseudopotential. For any given parity and total angular momentum, single configuration Hartree-Fock calculations are performed to build a basis set of numerical two-electron wave functions. Configuration-interaction calculations then provide the energy positions of the correlated states and their compositions in terms of the single configuration basis set. Atomic properties, such as autoionization linewidths, can be derived. Results concerning neutral barium are reported to illustrate the approach. They are compared with available experimental and theoretical data, and discussed. Energy positions of the bound and autoionizing $J=0$ even-parity and $J=1$ odd-parity states of barium below the $5d{}^{2}{D}_{5/2}$ threshold are predicted as well as associated autoionization linewidths. The symmetrical ${\mathrm{ns}}^{2} (n=7\ensuremath{-}11),$ ${7p}^{2},$ and ${6d}^{2}$ configurations are also studied. The overall agreement is satisfactory and especially good for high-lying doubly excited Rydberg states.

Calculations of doubly excited states of beryllium-like ions using the stabilization method

The resonance energies and widths for doubly excited states of beryllium-like ions are determined by calculating the density of resonance states using the stabilization method. A model potential is used to represent the interaction between the outer electrons and the core electrons. Products of Slater orbitals are used to represent the two-electron wavefunctions. We present results for the two lowest doubly excited states for Z = 5-10, and Z = 12. Comparisons are made with other calculations in the literature.

Magnetic-field effects on the doubly excited resonance states of

Magnetic-field effects on the Feshbach resonances below the H (N = 2) threshold and the shape resonance of lying above the H (N = 3) threshold are investigated theoretically using a method of complex-coordinate rotation. Products of Slater-type orbitals (STOs) are used to represent the two-electron wavefunctions, with being employed for individual electrons. Block matrices with up to (K-states) are used. Convergence behaviour for the resonance parameters (resonance energy and width) is examined by using different values of . When the external magnetic field is turned on, we examine the field effects on the resonance energies and the autoionization widths. A physical interpretation of our results is given.

Electric-field effects on the1Poshape resonance above theH(N=2)threshold ofH−

Electric-field effects on the ${}^{1}{P}^{o}$ shape resonance of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ lying above the $\mathrm{H}(N=2)$ threshold is investigated theoretically using a method of complex-coordinate rotation. Products of Slater-type orbitals are used to represent the two-electron wave functions, with block matrices up to ${L}_{\mathrm{max}}=6$ ($I$ states) are used when the external field is turned on. Comparisons are made with the available experimental and theoretical results.

Doubly excited states ofH−in parallel electric and magnetic fields

Doubly excited ${}^{1}{S}^{e}$ and ${}^{1}{D}^{e}$ Feshbach resonances of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ lying below the $\mathrm{H}(N=3)$ threshold under the influences of external electric and magnetic fields in parallel directions are investigated theoretically using a method of complex-coordinate rotation. Products of Slater-type orbitals are used to represent the two-electron wave functions, with ${l}_{\mathrm{max}}=9$ being employed for the individual electron. Block matrices with up to ${L}_{\mathrm{max}}=5$ including ${}^{1}{S}^{e},$ ${}^{1}{P}^{o},$ ${}^{1}{D}^{e},$ ${}^{1}{F}^{o},$ ${}^{1}{G}^{e},$ and ${}^{1}{H}^{o}$ states are used. Convergence behavior for the resonance parameters (resonance energy and width) is examined by using different values of ${L}_{\mathrm{max}}.$ Results for the combined electric and magnetic field effects on the ${M}_{L}=0$ components of the ${}^{1}{S}^{e}(1)$ and ${}^{1}{D}^{e}(1)$ Feshbach resonances are reported.

Polaron and bipolaron formation in a cubic perovskite lattice

The Rice-Sneddon model for BaBiO$_3$ is a nice model Hamiltonian for considering the properties of polarons and bipolarons in a three-dimensional oxide crystal. We use exact diagonalization methods on finite samples to study the stability and properties of polarons and bipolarons. Because polarons, when they form, turn out to be very well-localized, we are able to converge accurately our calculations for two-electron bipolaron wavefunctions, accounting for the Coulomb interaction without approximation. Some of our results are compared with and interpreted by reference to the variational method of Landau and Pekar. We calculate both electronic and vibrational excitations of the small polaron solutions, finding a single vibrational state localized with the full symmetry of the polaron, which has its energy significantly increased. Both on-site (Hubbard) and long-range Coulomb repulsion are included in the bipolaron calculation, but due to the high degree of localization, the long-range part has only a small influence. For a reasonable on-site repulsion $U$ equal to 2 times the band width $W$, bipolaron formation is significantly suppressed; there is a large window of electron-phonon coupling where the polaron is stable but the bipolaron decays into two polarons.

Double ionization of atoms by multiply charged ions and a strong electromagnetic field: Effects of correlation in the continuum

A nonstationary theory of double ionization of two-electron atoms in collisions with multiply charged ions or by an intense electromagnetic field is developed. An approach that permits investigating both problems by a single method is formulated. A two-electron continuum wave function that takes into account the interaction of the electrons with the atomic nucleus and the external ionizer as well as with one another is obtained as a product of Coulomb waves with modified Sommerfeld parameters. The computational results obtained for the double ionization of helium atoms by multiply charged ions are in good quantitative agreement with the existing experimental data.

Magnetic-field effects on the 1P o shape resonance of H − above the H( N = 2) threshold

Magnetic-field effects on the 1P o shape resonance of H − lying above the H( N = 2) threshold are investigated theoretically using a method of complex-coordinate rotation. Products of Slater-type orbitals (STO) are used to represent the two-electron wave functions, with l max = 9 being employed for individual electrons. Block matrices with up to L max = 7 (K-states) are used. Convergence behaviour for the resonance parameters (resonance energy and width) is examined by using different values of L max. Magnetic-field effects on both the M L = 0 and M L = ± 1 components of the 1P o shape resonance are investigated.

Mackey Theorem and Two-Electron Wave Function of a Multi-Centre System

The two-electron wave function in a system of many equivalent atoms is investigated group-theoretically. It is shown that the classification of different types of two-electron (two-hole) localizations can be made by the double-coset decomposition of the symmetry group with respect to the local subgroup, and that the group appearing in the Mackey theorem can be used for the additional classification of states. The Mackey theorem on symmetrized squares and the generalized Frobenius reciprocity theorem are applied to the construction of two-electron states in octahedral symmetry.

Strong electric-field effects on the doubly-excited intrashell states of He below the He + ( N =2) threshold

The method of complex-coordinate rotation is used to investigate electric-field effects on the doubly-excited 2s2p1P0, 2p21De, 2p23Pe and 2s2p3P0 states of He. Strong electric-field strengths up to F = 0.02 Ry are used in our present study. Products of Slater orbitals are used to represent the two-electron wave functions, with lmax = 8 being employed for the individual electron. Block matrices with up to Lmax = 5 (H-states) are used to investigate the convergence behavior for the resonance parameters (resonance energy and width). When the external electric field is turned on, “classic” Stark effect is observed for the M = 0 and M = ±1 components of the two singlet-spin states and for the M = ±1 components of the triplet-spin states. Comparisons are made with other calculations when available.

Computation of exchange probability in one-dimensional electron-hydrogen scattering.

We present numerical ab initio calculations for a quantum system that is a one-dimensional analog of electron-hydrogen scattering at low energies. This approach allows for an exact treatment of electron correlation. The exchange probability can be computed directly from the full unsymmetrized two-electron wave function. One can thus determine without approximation the relative importance of the direct versus the exchange process. We find that, in this low-energy regime, the exchange probability can be quite high and can exceed the probability of the direct process. We then take into account the exchange probability in a calculation of the single-electron effective potential for the free electron. \textcopyright{} 1996 The American Physical Society.

Quantum-classical hybrid approach to helium double photoionization.

The photoionization process is divided into (A) the absorption of the photon by one electron and (B) the correlated motion of the electron pair leading to singly or doubly ionized helium. We relate (A) to the total cross section to be calculated analytically in the quasiclassical reflection approximation as known from molecular problems. For (B) the two-electron wave function is propagated with the semiclassical version of Feynman's path integral to separate single and double ionizing events. A probe for (B) is the ratio between double and single ionization. The results for (A) and (B) and for absolute ionization cross sections obtained by combining (A) and (B) are in good agreement with different experiments that cover together a range of photon energy from the double-ionization threshold to several hundred eV.

اشتقاق الرتبة الخامسة للنظرية متعددة القلقلة لمعاملات الطاقة للجزيئات متعددة الذرات

A multiperturbation theory has been developed for molecular systems. In the present paper we extend this theory to fifth order in the energy. The "bare-nucleus" hydrogenic function is chosen as the zero-order wave function rather than the more customary hartree-fock function. With this choice the multiperturbation wave functions are independent of the nuclear charges and of the total number of nuclear centers and electrons for the molecule and are thus completely transferable to other systems. Making the simplest possible choice, we describe an n-electron, m-center polyatomic molecule as n "hydrogenic" electrons on a single center perturbed by electron-electron and electron-nucleus coulomb interactions. With this choice of zero-order Hamiltonian (H0) the first-order wave function for any polyatomic molecule will consist entirely of two-electron, one-center and one-electron, two-center first-order wave functions. These are exactly transferable from calculations on He-like and H2-like systems. To calculate the first-order and second order correction for the wave function of any polyatomic molecule, we need the first-order and second-order correction for a two-electron atomic wave function, the first-order and second-order correction for a one-electron diatomic molecular wave function and some additional mixed second-order corrections. The wave functions necessary will be two-center, one-electron at most. The second-order wave function for a polyatomic molecule contains additional contributions which cannot be obtained from the simple subsystems, but represent multiple perturbation contributions which are two electron diatomic, and one-electron triatomic in character. The expressions for the multiperturbation energy-expansion coefficients through fifth order are derived.

Time-dependent approach to electron scattering and ionization in the s-wave model.

The time-dependent Schroedinger equation is integrated for continuum states of two-electron atoms in the framework of the {ital s}-wave model, in which both electrons are restricted to having vanishing individual orbital angular momenta. The method is suitable for studying the time evolution of correlations in the two-electron wave functions and yields probabilities for elastic and inelastic electron scattering and for electron-impact ionization. The spin-averaged probabilities for electron-impact ionization of hydrogen in the {ital s}-wave model reproduce the shape of the experimentally observed integrated ionization cross section remarkably well for energies near and above the maximum.

Electric-field effects on the 2p2 3Pe and 3Po states of H- below the H(N=2) threshold

A theoretical investigation of static electric-field effects on the 2p2 3Pe state and the doubly-excited 3Po states below the H(N=2) threshold is carried out using the method of complex-coordinate rotation. Products of Slater orbitals are used to represent the two-electron wavefunctions, with lmax=8 employed for individual electrons. Block matrices with up to Lmax=4 (G-states) are used to investigate the convergence behaviours for the resonance parameters (resonance energy and width). When the external electric field is turned on, the Stark effect is observed for the M=+or-1 components of the two nearly-degenerate 2p2 3Pe and 2s3p 3Po (2) states. Results for electric-field effects on the lowest 2s2p 3Po(1) state are also given, as well as effects on the 2s3s 3Se(1) and 2s4s 3Se(2) states.

Electric-field effects on doubly excited states of H- between the N=3 and N=4 thresholds of hydrogen atoms.

A theoretical investigation of electric-field effects on the doubly excited resonances of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ between the H(N=3) and H(N=4) thresholds is made using the method of complex-coordinate rotation. Products of Slater orbitals are used to represent the two-electron wave functions, while ${\mathit{l}}_{\mathrm{max}}$=8 is employed for an individual electron. Block matrices up to ${\mathit{L}}_{\mathrm{max}}$=7 (K states) are used. Convergence behaviors for the resonance parameters (resonance energy and width) are examined by using different values of ${\mathit{L}}_{\mathrm{max}}$. Results for the electric-field effects on the Feshbach $^{1}$${\mathit{S}}^{\mathit{e}}$(1), $^{1}$${\mathit{P}}^{\mathit{o}}$(1), $^{1}$${\mathit{D}}^{\mathit{e}}$(1), $^{1}$${\mathit{F}}^{\mathit{o}}$(1), and $^{1}$${\mathit{G}}^{\mathit{e}}$(1) resonances are given, as well as for the effects on the $^{1}$${\mathit{D}}^{\mathit{e}}$ shape resonance lying above the H(N=3) threshold.