Compton line shapes for hartree–fock wave functions

Recent papers on electron-N/sub 2/ scattering by Rumble, Stevens, and Truhlar (J. Phys. B 17, 3151 (1984)) and Weatherford, Brown, and Temkin (Phys. Rev. A 35, 4561 (1987)) have suggested that Hartree-Fock (HF) wave functions may not be accurate for calculating potentials for use in studying electron-molecule collisions. A comparison of results for electron-H/sub 2/ scattering using both correlated and HF wave functions is presented. It is found that for both elastic and inelastic collisions and for all energies considered (up to 10 eV) the HF wave functions yield results in excellent agreement with those obtained from the more accurate wave functions.

The reduced local Hartree–Fock energy for the neutral atoms Li through to Al have been calculated using the Hartree–Fock wave functions of Clementi and Roetti. These wave functions were found to give rather accurate values of the reduced local energy in the medium range region (r values 1∼12 a.u.9), but inaccurate values close to the nucleus. The reduced local energy was found in each case to exhibit oscillations about the ’’exact’’ total Hartree–Fock energy as a function of the configuration space coordinate r, indicating some cancellation of opposing errors in the wave functions used for the calculation of the total energy. A method is outlined for determining Hartree–Fock wave functions which not only satisfy the customary condition that the total energy be minimized, but in addition, are locally accurate. The criterion which is used to monitor the local accuracy is the reduced local energy. The reduced local energy is incorporated into a global accuracy constraint on the wave function, and the energy is determined subject to the minimization of this additional constraint. Calculations are performed for various helium wave functions. The five-term Clementi–Roetti wave function is refined, maintaining essentially the same total energy, but with considerable improvement in local accuracy. The Clementi (1965) wave function for helium is also refined. A ten-term function for helium is devised which has improved accuracy at the local level. Comparison of expectation values of r1k using the different wave functions is made, in order to assess the utility of the reduced local energy as a criterion for refinement of such values.

We have calculated differential, integral, and momentum-transfer cross sections for vibrational-rotational excitation as well as for pure rotational excitation, pure vibrational excitation, and elastic scattering for electron collisions with N2 at 5–50 eV impact energy. Scattering calculations were carried out in the fixed-nuclei approximation and excitation cross sections obtained using the sudden approximation. The interaction potential used in these calculations consists of three terms: static and local-exchange potentials, calculated from extended-basis-set Hartree–Fock wave functions as functions of internuclear distance, and a semiempirical polarization potential. Two different local exchange potentials were employed. The results are compared to each other, to previous calculations, and to experiment.

Hartree—Fock wavefunctions are presented for the LiH(X1Σ+), BeH(X2Σ+), BH(X1Σ+), CH(X2Πr), NH(X3Σ−), OH(X2Πi), and HF(X1Σ+) molecules. These are the analytic self-consistent-field wavefunctions obtained from the solutions of the Hartree—Fock—Roothaan equations. Large sets of Slater-type functions centered on both nuclei were used as the expansion basis, and extensive optimization of the orbital exponents has been carried out. The total energies obtained for Re(exptl) are −7.98731, −15.15312, −25.13137, −38.27935, −54.97806, −75.42083, and −100.07030 hartrees, respectively, for the AH hydrides listed above. The first ionization potentials, which are obtained from the Hartree—Fock energy differences between AH and AH+ systems, are 7.02, 8.14, 8.45, 10.08, 12.82, 11.44, and 14.54 eV, respectively. In addition, potential curves, spectroscopic constants, and certain other energetic quantities are presented. Crude estimates of the correlation energy of the first-row hydrides are made and such quantities are compared within the series and with their respective united and separated atoms. These results suggest that the changes in correlation energy of AH relative to the correct united atom is independent of which hydride is involved and the change is small.

A block-localized wave function method is introduced to evaluate the electronic delocalization effect in molecules. The wave function for the hypothetical and strictly localized structure is constructed based on the assumption that all electrons and primitive basis functions can be divided into several subgroups; each localized molecular orbital is expanded in terms of primitive orbitals belonging to only one subgroup. The molecular orbitals belonging to the same subgroup are constrained to be mutually orthogonal, while those belonging to different subgroups are free to overlap. The final block-localized wave function at the Hartree–Fock level is expressed by a Slater determinant. In this manner, the energy difference between the Hartree–Fock wave function and the block-localized wave function can be generally defined as the electronic delocalization energy. The method is applied to two cases. The first concerns the resonance stabilization in the allyl ions. We find that the vertical resonance energies for the planar cation and anion are −45.7 (or −44.7) and −46.7 (or −48.2) kcal/mol at the HF/6-31G* (or 6-31+G*) level, respectively. Their rotational barriers are decomposed in terms of conjugation, hyperconjugation, steric effect, and pyramidalization. The n→σ* negative hyperconjugation in the staggered allyl anion is very strong and stabilizes the system by as much as −13 kcal/mol. The second concerns the hyperconjugation effect in propene. Our calculations suggest that the theoretical hyperconjugation energy in propene is about −5 kcal/mol, which is close to the experimental estimate (−2.7 kcal/mol) derived from the hydrogenation heats of propene and ethylene. Comparisons between the results based on the present block-localized wave function method and those based on the natural bond orbital method are presented and discussed. The examples demonstrate that the block-localized wave function method can be employed as a useful model to analyze chemical bondings and intuitive concepts.

The half‐projected Hartree–Fock wave function (HPHF) is one of simplest models for introducing some electronic correlation effects. In this model, the wave function is built up with only two Slater determinants. This simple form suggests its application for the direct determination of singlet excited states. On the other hand, because the HPHF model does not mix singlet and triplet states with Ms = 0, it can be used for determining independently singlet and triplet states without any mutual contamination. In the present work, we applied this model to determine nine electronic states of the lithium molecule; one of them exhibits even the same symmetry of the fundamental one. For this purpose, the 6‐311G(d) basis was used. Potential energy curves were determined and some spectroscopic constants derived. The numerical results were compared with the available experimental data, as well as with other theoretical values. © 1995 John Wiley & Sons, Inc.

Classical and quantum mechanical rainbow-scattering of fast He atoms from a KCl(0 0 1) surface

High-resolution K{alpha} x-ray spectra of Ca, Cr, and Fe induced by (0.7{endash}1.5)-MeV protons and of Ti induced by (0.7{endash}4)-MeV protons were measured by a flat crystal spectrometer. From the relative yields of the KL{sup 1} satellites, which are the result of multiple inner-shell ionization, the average L-shell ionization probabilities in near-central collisions were determined. The effects of the rearrangement of the inner-shell holes prior to the K x-ray emission and the changes of the fluorescence yields due to the multiple ionization were taken into account in the evaluation of the ionization probabilities. The direct Coulomb ionization probabilities in the near-central collisions were determined from the relative yields of the KL{sup 1} satellites after subtracting the shake contribution. The values obtained were compared with semiclassical approximation calculations exploiting relativistic hydrogenic and Hartree-Fock wave functions. The importance of a realistic atomic description using Hartree-Fock wave functions was demonstrated. {copyright} {ital 1997} {ital The American Physical Society}

A collective model for inner shell ionization of very heavy targets

The recent development of the technique of inelastic x-ray scattering to study the electron momentum distribution in the scatterer is surveyed. The simple relationship between the electron momenta and the Compton line shape in the impulse approximation is derived, and the validity of that approximation is discussed in the light of recent measurements. Current areas of research are reviewed.

250. Preparation of translucent coating for As 2S 3 glass : I F Kopinets et al, Neorg Mater, 8 (9), 1972, 1551–1553 ( in Russian)

Using the projected Hartree-Fock (HF) wave functions, a theoretical formulation is given for calculating the electromagnetic properties of nuclei, the nuclear form factors in electron scattering, $\mathrm{ft}$ values in $\ensuremath{\beta}$ decay, and the reduced widths of nuclear levels in direct reactions. The formulation is applied to calculate the quadrupole and magnetic moments, the electromagnetic transition probabilities, and the lifetimes of nuclear levels of a large number of nuclei ranging from $^{19}\mathrm{F}$ to $^{37}\mathrm{Cl}$ in the $2s\ensuremath{-}1d$ shell. The good agreement between the calculated and the experimental values indicates a remarkable success of the projected HF wave functions in predicting the electromagnetic properties of nuclei.

Self-consistent-field wavefunctions near the Hartree–Fock limit have been calculated by the Roothaan expansion method for OF(2π) at several internuclear separations; the total energy at the Hartree–Fock minimum (1.321 Å) was found to be − 174.19502 hartree. Based on the assumption that the correlation energy was approximately the same as for O2−, and also by interpolating the correlation energy of OF across the CF, NF, OF, and F2 sequence, the binding energy D(OF) was deduced to be 3.0+0.3−0.8 eV. A Dunham analysis of the potential-energy curve gave values for the various spectroscopic constants. In addition, the ionization potential and electron affinity for the vertical processes, and the dipole and quadrupole moments of OF+(3Σ), OF(2π), and OF−(1Σ) were also calculated.

In the study of the ground‐state binding energy of atoms, relatively more attention has been paid to the gradient expansion of the kinetic energy term of the Thomas–Fermi–Dirac model than to the gradient expansion of the exchange term of this model. Recently Shih and Shih, Murphy, and Wang have focused on this problem and calculated the first gradient expansion correction to the Dirac exchange term by making use of electron densities constructed from Hartree–Fock wave functions. In previous work, aimed to introduce the shell structure of atoms via a variational procedure using the energy density functional formalism, electron densities have been obtained for the Na atom within the Thomas–Fermi–Dirac model with and without the Weizsäcker and Hodges gradient expansion corrections to the kinetic energy term. In this paper we make use of the respective electron densities and calculate the first gradient expansion corrections to the Dirac exchange term. The results show that, for the Na atom, the magnitude of this correction is about 1% of the magnitude of the total binding energy.

Fast atom diffraction during grazing scattering from a MgO(001) surface