Local Exchange Potential Research Articles

Fermi surface pressure dependence in potassium

The pressure dependence of the Fermi surface of K has been calculated using the Korringa–Kohn–Rostoker (KKR) method and local exchange potentials adjusted to simulate the effect of a nonlocal operator. Agreement with the experimental data is obtained. Some effective masses have also been calculated.

Elastic scattering of slow electrons from Xe atoms

The spin polarisation and differential cross section of unpolarised electrons elastically scattered from Xe atoms in the energy range 25-300 eV have been calculated fully relativistically for several local density approximations of the effective single-particle potential, starting from a relativistic Hartree-Fock electron density of the target. Comparison with experimental data shows that good overall agreement below 100 eV is also obtained by using an energy-dependent local exchange potential.

Polarisation and exchange effects on elastic scattering of electrons from helium

The authors examine the effects of the polarisation potential and of local exchange potentials in the adiabatic exchange approximation for low-energy elastic scattering of electrons from helium atoms. They conclude that the most reliable results are obtained by including only the dipole part of the polarisation potential and by treating exchange exactly. They present detailed results for phaseshifts, differential, total elastic and momentum transfer cross sections. These agree well with other calculations and with experimental measurements.

Effective exchange potentials for electronically inelastic scattering

We propose new methods for solving the electron scattering close coupling equations employing equivalent local exchange potentials in place of the continuum-multiconfiguration-Hartree–Fock-type exchange kernels. The local exchange potentials are Hermitian. They have the correct symmetry for any symmetries of excited electronic states included in the close coupling expansion, and they have the same limit at very high energy as previously employed exchange potentials. Comparison of numerical calculations employing the new exchange potentials with the results obtained with the standard nonlocal exchange kernels shows that the new exchange potentials are more accurate than the local exchange approximations previously available for electronically inelastic scattering. We anticipate that the new approximations will be most useful for intermediate-energy electronically inelastic electron–molecule scattering.

Characteristic X-Rays of Highly Charged Ions

The significance of X-ray energy shift data for highly charged ions is discussed. It is shown that relativistic self-consistent-field calculations lead to the same qualitative results for local and non-local electron exchange potentials. In addition to X-ray energy shift data for all ionization stages of selected elements, analogous data for the Kα1-, Kβ1- and Lα1-energy shifts for selected stages of ionization for all elements in the range Z ⩽ 70 are shown.

Elastic scattering of slow electrons by CH4and H2O using a local exchange potential and new polarisation potential

A parameter-free polarisation potential is proposed for the elastic scattering of electrons by polyatomic molecules, based on the method of Pople and Schofield and the work of Temkin. Cross sections are obtained for CH4 and H2O using this polarisation potential and a Hara local exchange potential. Agreement with experiment is good.

Local exchange and correlation potentials for electron scattering at solid surfaces

Atomic local exchange and correlation potentials for use in LEED, photoelectron diffraction and near edge EXAFS calculations are investigated by comparing the calculated total and differential electron scattering cross sections for Ar, Kr and Xe atoms with experimental data. Four potentials are considered, namely the Xα potential with α = 0.7 (a value appropriate to ground state properties), the Xα potential with a reduced α value of 0.4; the Hara potential and the potential of Lee and Beni. At energies of 100 eV and above, all four potentials give an adequate description of the data. At lower energies ranging from 10 to 60 eV, the Hara potential and the Xα ( α = 0.4) potential remain adequate but the Xα ( α = 0.7) potential and the Lee and Beni potential give very poor correspondence with the data. Since the Lee and Beni potential is theoretically an improvement over the Hara potential within the Local Density Approximation (LDA) we conclude that the LDA itself is inadequate for the description of the scattering of low energy electron from atoms and that the success of the Hara potential must be due to a fortunate cancellation of errors.

Local exchange and correlation potentials for electron scattering at solid surfaces

Local exchange and correlation potentials for electron scattering at solid surfaces

Use of local exchange potentials in inelastic scattering

We have investigated the use of local exchange potentials for approximating the solution of the exchange integro-differential equation. It is found that judging the accuracy of such potentials by their ability to produce correct elastic scattering phase shifts may not be sufficient for inelastic scattering. This is illustrated through a model calculation of electron excitation of the 2p state of hydrogen.

Relativistically corrected impulse approximation in the presence of a local central potential

An impulse approximation with relativistic corrections is derived to incorporate the fact that each constituent nucleon of the target nucleus is moving in an average local central potential generated by the other nucleons. The constraint imposed by current conservation is described. An application of the resultant impulse approximation to reanalyze symmetry tests in the $A=12$ nuclei induces a few subtle changes but does not upset the standard interpretation of these experimental results.NUCLEAR REACTIONS Impulse approximation with relativistic corrections, derived in the presence of a local central potential, constrained by current conservation; symmetry tests in the $A=12$ nuclei.

Atomic exchange potentials and angle resolved photoemission

The usefulness of angle-resolved photoemission from core states of adatoms for surface structure studies has been questioned because of the total mismatch between theory and experiment for normal emission from the 4d levels of Te on Ni(100) as a function of photon energy. We point out that the theory was based on the use of an Xα potential for Te with a α value appropriate for ground state properties. We show that such a potential is unsuitable for scattering problems and that much lower a values are required to give a reasonable description of sharp peaks in the atomic photoionization cross section as a function of photon energy. We show that Xα calculations of the phase shifts give much improved correspondence with full Hartree-Fock calculations when α is reduced below its usual range of 1 to 23, and that a corresponding improvement in the matching of experimental and calculated photoelectron azimuthal distributions is found. A realistic exchange potential must be energy dependent, decreasing to zero at high enough energies. We show that the simplest reasonable approximation to the local density exchange potential, i.e. the Hartree-Fock approximation for the uniform electron gas, resolves the problem of the normal emission data from Te 4d on Ni(100).

Use of local exchange potentials in the calculation of photoionization and electron-ion scattering

Several local approximations to exchange are investigated. These are compared against exact static-exchange results for $e$,${\mathrm{Li}}^{+}$ scattering and for the photoionization of the $2S$ and $2P$ state of Li. All except one are found to be fairly accurate for these cases. Calculations are also performed for photoionization of the $5D$ state of Cs, which exhibits an $f$-wave resonance. All exchange models studied for this case (including the Slater $X\ensuremath{\alpha}$) are found to give widely differing results on the position and width of this resonance. Calculations are also performed for the photoionization of the Cs $6P$ state and compared against experimental results.

Momentum eigenfunctions in the complex momentum plane. IV. The construction of local potential functions. Perturbation series

Expansion of exchange amplitudes, in inverse powers of a momentum, as a means of generating local exchange potentials has been studied further. The first term in such expansions varies inversely as the square of the momentum. Huo [J. Chem. Phys. 67, 5133 (1977)] obtained a local exchange potential by replacing a momentum by its average value obtained from a semiclassical model. The Xα potential with α=1 was obtained. The present research shows that mometum squared should be averaged and this leads to the Xα potential with α=20/27. The potentials were tested numerically for beryllium and although the potential with α=20/27 was superior in some instances to that with α=1, this was not always the case. In fact, for general all around use, α=1 provided best results. This apparent contradiction has been resolved by extending the expansion of exchange amplitude by one more term and using the semiclassical model to obtain the averages. The Xα potential is again obtained but with a α=80/81, a value differing negligibly from unity. Using the known behavior of exchange amplitudes in the complex momentum plane, a method is developed for testing the convergence of expansions in inverse powers of the momentum. The power series expansions are shown to be divergent. This provides a plausible explanation for the infinities at r=0 observed by Huo and shows that inclusion of more and more terms cannot lead to an exact amplitude. Although divergent, the series are asymptotic, i.e., they hold more accurately as the momentum increases as long as only a few terms are used. Therefore, the formulas obtained by Huo are approximately correct since she showed that the average momenta are large. Some different method must be found, however, for improving the accuracy. Two expansions in series of orthogonal polynomials are developed for which the first term gives a local exchange potential. The convergence region of each series is determined.

Resonances, scattering theory, and rigged Hilbert spaces

The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,’’in,’’ and ’’out’’ eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ’’out’’ eigenvector family are the same as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ’’complete’’ sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ’’out’’ eigenvectors. The free, ’’in’’ and ’’out’’ eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee–Friedrichs model and to the scattering of a spinless particle by a local central potential.

Electron collisions with polar molecules: exchange and polarisation in elastic scattering by HCl

The local free-electron-gas exchange potential is shown to be not only superior to the orthogonalisation approximation to exchange, but also to yield results in good agreement with those from an essentially exact treatment of exchange. The imposition of orthogonalisation in addition to the local exchange potential yields a further small improvement in the results. The use of these two approximations in combination appears to be a promising technique. A polarisation potential cut-off at about the position of the H atom yields results for the 2 Sigma eigenphase sum that have resonant behaviour more or less where expected on the basis of measured vibrational excitation cross sections. The elastic differential cross section has a strong peak in the backward direction, with important implications for resonant vibrational excitation.

Comparison of local exchange potentials for electron–N2 scattering

We consider vibrationally and electronically elastic electron scattering by N2 at 2–30 eV impact energy. We consider static, static-exchange, and static–exchange-plus-polarization potentials, Cade–Sales–Wahl and INDO/1s wave functions, and semiclassical exchange and Hara free-electron–gas exchange potentials. We show that the semiclassical exchange approximation is too attractive at low energy for N2. We show quantitatively by consideration of partial and total integral cross sections how the effects of approximations to exchange become smaller as the incident energy is increased until these differences are about 8% for the total integral cross section at 30 eV.

Spatial localization of quantum states and physical meaning of the matrix elements of the resolvent operator

We investigate how the second-sheet singularities of the matrix elements of the resolvent operator depend on the dense set of states on which they are evaluated. We illustrate the fact that, for a given Hamiltonian, one can choose a dense set of states such that the corresponding matrix elements of the resolvent can be continued across the spectrum into the lower half-plane and exhibit therein singularities at arbitrary preassigned locations. This shows that, given a resonant dynamics, some further physical input is needed in addition to the assignement of the total Hamiltonian, in order to identify privileged dense sets of states for which the singularities of the analytic continuation of the matrix elements of the resolvent have a physical meaning as characterizing the unstable states of the system. We develop these considerations explicitly in the examples of a nonrelativistic spinless particle subject to a local central potential or to a degenerate interaction. In the above cases, we show that a privileged set can either be chosen to be the set of states which are entire functions in the representation in which the free Hamiltonian is diagonal or, equivalently, the set of spatially localized states. The close connection with the preparation procedure of an unstable state as a localization process is discussed.

Photoionisation of CO

Calculations are performed for the photoelectric cross sections and angular distributions for the CO molecule. The method used is the single-centre coupled partial-wave method, in which the potential is the sum of the static potential and a local exchange potential for an electron in the field of CO+. Comparisons are made with recent experimental data.

Numerical tests of a local exchange potential for atomic systems

Abstract Berrondo and Goscinski have proposed a functional dependence of the exchange energy kernel on the electron density. Their approximate exchange energy kernel is compared with the exact exchange energy kernel obtained from Hartee—Fock wavefunctions for neon, argon, and zinc. The Berrondo—Goscinski functional is asymptotically accurate beyond the outermost maximum of the radial distribution function, but serious divergences occur in the core.

Electron-electron interaction in simple metals

We calculate ${V}_{\mathrm{ee}}$, the effective screened interaction between two electrons in a simple metal. The result is derived by a self-consistent perturbation theory based on Hartree-Fock theory. Exchange and correlation are treated in local approximation. We consider the lattice to be a smooth but elastically deformable, charged medium in which the electrons move. Our results is a generalization of the Fr\ohlich interaction to include exchange and correlation. The effective interaction is spin dependent and may be written ${V}_{\mathrm{ee}}(q,\ensuremath{\omega})={V}_{0}(q,\ensuremath{\omega})\ensuremath{-}J(q,\ensuremath{\omega}){\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}}_{2}$. Explicit expressions for ${V}_{0}$ and $J$ are given in terms of the lattice stiffness and the same local exchange and correlation potentials of the electron gas which are needed to calculate the dielectric function and the spin susceptibility.