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Hartree–Fock method

Puigmolina, E. Ressouche, O. Rutty, and J. Gillon and P. Becker , Magnetization Densities in Material Science , pp. Coppens, Z. Su, and P. Becker , Analysis of charge and spin densities , pp. Brown, A. Capiomont, B. Gillon, and J. Bell, J. Burke, and A. Gillet, P. Becker, and P. Cortona , Joint refinement of a local wave-function model from Compton and Bragg scattering data , Physical Review B , vol.

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Keith , Aimall version Gatti and S. Bader , Atoms in molecules , Gatti , Abstract , Zeitschrift f?? Tang, E. Sanville, and G. Gatti, G. Macetti, and L. Presti , Insights on spin delocalization and spin polarization mechanisms in crystals of azido copper II dinuclear complexes through the electron spin density Source Function , Acta Crystallographica Section B Structural Science, Crystal Engineering and Materials , vol. Aroyo , International tables for crystallography, volume a, space-group symmetry. Boguslawski, M. Jacob, and. Jacob and M.

Komarek, H. Roth, M. Cwik, W. Stein, J. Baier et al. Mochizuki and M. Akimitsu, H. Ichikawa, N. Eguchi, T. Miyano, M. Nishi et al. Ito, N. Tuji, F. Nevertheless, considerable care is needed in choosing them if accurate surface geometries are to be obtained, since the size and direction of rumpling in such.

In binary ionic crystals the surface ion having the greater polarizability tends to relax outwards relative to the other. Unfortunately, it is not always the case that potential parameters derived to reproduce bulk properties are applicable at the surface. Positive shell charges, for example, which sometimes emerge from bulk parameter-fitting procedures, produce the opposite sign of rumpling to negative shell charges. Such problems have been extensively discussed by Martin and Bilz,18 and by de Wette, Kress, and Schroder. To study the surface and interfacial properties of layered-oxide films, Hartree-Fock calculations were performed on model slabs that were infinitely periodic in two dimensions and consisted of a stack of up to seven atomic layers parallel to the exposed surface.

Such structures bypass the unphysical approximations related to the definition of the electrostatic boundary conditions inherent in cluster calculations, which are frequently used to model surfaces.

Vasp Molecular Dynamics Output

Both individual slabs of MgO and NiO and composite layered structures were considered. Experimental determinations and theoretical calculations of the surface geometry of NiO This, of course, implies the neglect of the Mo substrate upon which the NiO film is grown, a feature of our calculations which may be justified as follows: 1 The MgO surface dipole will lead to an image potential28'29 in the metal, that is, charge density will be shifted around near the NiO-Mo interface, creating an equal and opposite dipole to prevent the existence of a long-range electric field in the metal.

To some extent this will mimic the presence of a second layer of MgO on the opposite face of the NiO film, as in our model. This implies that electron transfer from the substrate and between the two oxide films is greatly inhibited. Note that this mechanism does not operate for a positively charged NiO film, and so surface-charging effects associated with XPS can be quenched by electron transfer from the metal.

Furthermore, the original experimental data did not depend significantly on the thickness of the NiO film. It will be shown that it is not necessary to invoke the metal substrate to explain the spectroscopic observations. The calculations were performed using the crystal program, a well-established package that has been used to study a wide range of crystalline materials. Recent modifications to the code32 were used to perform open-shell calculations within the unrestricted Hartree-Fock formalism.

There have been several previous studies of MgO surfaces using the crystal program, including treatments of the surface,21 the surface,33 and surface-adsorbate interactions. As might be expected, the error associated with using the same basis set for ine-quivalent surface and bulk atoms was found to be negligible. Reproduction of the bulk properties of magnetic insulators such as NiO using the methods of one-electron band theory has been a traditionally difficult theoretical problem.

This is because the occupation of localized electron levels in these materials has an enormous effect on the energies of the unoccupied levels, producing a gap in the one-electron band spectrum. This feature is not reproduced in methods using a local approximation to the exchange, such as the local-spin-density approxima-. It has been shown by us in previous work36,37,51 that in Hartree-Fock HF theory the strong on-site Coulomb interactions responsible for the wide-band-gap insulating nature of these materials are correctly treated, which resolves the apparent conflict between band theory and experiment.

At present, therefore, methods based on the Hartree-Fock approximation appear to be a useful starting point for the study of the ground state of magnetic insulators.

Hartree-Fock method

The Ni and O all-electron Gaussian atomic-orbital basis sets used in this work were the same as those used in our previous studies of bulk NiO see, e. Analytic gradients are not yet available within the crystal code, and thus all geometry optimizations were done numerically using repeated line searches. Care was taken to carry out the calculations with the lattice parameters in the plane of the slab corresponding to the equilibrium structure of the bulk material at the same level of theory.

This ensured that the surface would not distort in unphysical ways in response to an inappropriate lattice parameter. Other studies39 on the ZnO surface have shown that the inclusion of a posteriori correlation corrections to the total energy in crystal appear to have little effect on computed surface geometries. Recent work by Causa and Zupan using a modified version of the crystal program40,41 has shown that there is little consistent improvement to HF geometries using correlation corrections either at the Hartree-Fock, hybrid Hartree-Fock density-functional, or Kohn-Sham density-functional levels.

The results of full surface geometry optimizations for each model slab are reported in Table III. The structure of three-layer MgO is similar to that found in an early crystal study21 of the MgO surface, with minor differences due to the larger basis set and higher computational tolerances used in the present calculations. The small positive rumple also found in NiO differs in sign from that obtained both in the shell model calculations of Tasker and Duffy,26 and in the LEED study of Prutton et al. Increasing layer numbers refer to increasing depth below the surface.

The figures for rumpling and relaxation are defined in the text, and are given both in A and as a percentage of the bulk layer spacing. The total-energy data were also used to estimate surface formation energies, which are reported in Table IV, together with other values from the literature for comparison.

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Our data were calculated from. Agreement with existing experimental and theoretical work shown in the table is good. The inclusion of more diffuse d valence functions in the basis set considerably reduces the calculated surface energies relative to those reported in the earlier crystal study of two- and three-layer MgO slabs. The results of a distributed multipole analysis of the electronic charge density in the surface region is given in Table V for various slabs.

The ionic charges and the dipole and quadrupole moments that are not constrained by symmetry to be zero z and 2z2—x2—y29 respectively are reported. The definition of these quantities within the crystal program is discussed extensively in Ref. The analysis in Table V shows clearly that for all systems considered the form of the charge density in all non-surface layers is very similar to that in the bulk. The only change of any size occurs in the surface MgO layer, where a slight reduction in the ionicity of both components of around 0.

Furthermore, there is no evidence of any large transfer of charge from MgO over-layers to underlying NiO, in contrast to the suggestion of Burke and Goodman. All surface dipoles are negative outward as a result of electron density spilling out into the vacuum. The surface quadrupoles in NiO are such that both the metal and oxygen ions are slightly expanded in a direction normal to the surface. By contrast, in MgO and the composite systems, the surface Mg ions are contracted in this direction.

Smaller charge deformations also occur at the interface between MgO and NiO in the layered slabs. The scale is logarithmic so as to enhance detail near the surface; many of the features from Table V can be seen. In the study reported in Ref.

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Calculated total and surface energy per cell for various slabs, compared with ex-. Calculated electronic charges, dipoles, and quadrupoles for all symmetry-inequivalent ions in various model slabs relaxed to equilibrium geometry a. MgO films of varying thickness were then deposited on the oxide surface and XPS spectra recorded as a function of coverage.

The Ni XPS spectrum was by far the most well-characterized experimentally, since the local environment of Mg and O ions changes a great deal as MgO is grown from fractions of a monolayer to multilayers. An examination of the published spectrum1 reveals that around half of the apparent shift appears to develop after deposition of the first MgO monolayer. However, the precise functional form of the shift with coverage is presumably subject to some degree of uncertainty as changes in shape of the peak as a function of coverage may occur, and the number of MgO monolayers is not precisely determined.

It is perhaps simpler to say merely that there is an increase in the binding energy of the Ni core levels after depositing MgO on the NiO surface. The explanation given for the Ni shift was as follows. The results of our calculations shown in Table V imply that such a charge transfer does not take place. The work of Duffy, Hoare, and Tasker43 has shown that in ionic crystals, where long-range electrostatic fields can be maintained, the effect of the surface double layer is.

These arguments may be used to construct an alternative explanation of Burke and Goodman's XPS results, by examining the effect of changes in surface geometry on the XPS data. The calculated relaxations and rumplings are small, but are sufficient to produce significant changes in the electrostatic potential inside the slab. This can be demonstrated as follows.

Integration over the whole surface gives a potential V at this point of. Vis therefore independent of the depth below the surface. Surface dipoles caused by reconstruction of atoms in the surface layer thus give rise to a constant potential in the interior of the sample, which may be important in calculations of core-level binding-energy shifts.

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One might think of the physical basis of this effect in XPS as follows. When an ejected photoelectron travels from inside the sample through a negative-outward electrostatic dipole barrier at the surface, the electron will be retarded and consequently emerge with a lower kinetic energy. This is equivalent to a greater measured binding energy. Hence any effect which changes the magnitude of the surface dipole will affect the observed XPS spectrum.

The important point in the case of the layered oxide system is that, according to our calculations, the total rumpling and the resultant surface dipole increase when MgO is deposited on the surface Table III. The effect of. The map refers to the difference between the charge density from the periodic calculation and that of a superposition of spherical ionic densities. The plane shown is that through the ion centers and perpendicular to the [] direction in the conventional unit cell. The large roughly spherical ions are oxygen ions, the aspherical ion in the central three layers is Ni, and the smaller ion in the outer two layers is Mg.

Solid, dashed, and dot-dashed lines refer to positive, negative, and zero values, respectively. It is apparent that moving from the bare nickel oxide surface to two overlayers of MgO gives a binding-energy increase in the Ni core levels of around 0. This difference in rumpling might be accentuated in the real system due to the very slight discrepancy in the lattice constants of the two oxides, which is ignored in our treatment, and the presence of surface disorder. A comparison with calculations where the surface is merely a truncation of the bulk geometry is shown on the same plot.

In Fig. Since no change in surface dipole is involved in this case, there is little effect on the eigenvalue spectrum. It is important to note that core-level shifts of this type should be detectable with overlayer films no more than a few monolayers thick, since it is changes in the surface double layer that appear to produce large chemical shifts. For relatively thick films the surface geometry should be independent of the number of layers.

A further practical point is that substrate core levels of Ni, in this case can only be detected with XPS for relatively low MgO coverages. In real systems, surface defects and disorder will affect the surface double layer. Shell model calculations45 have suggested that nonplanar surfaces of MgO tend to be. Calculated Hartree-Fock eigenvalues at the T point for the Ni 2pz levels of the central three Ni layers, as a function of MgO coverage.

Results for relaxed filled points and unrelaxed unfilled points surface geometries are compared. Calculated Hartree-Fock eigenvalues for the Ni 2pz levels in the five-layer NiO slab, for differing amounts of surface relaxation. No rumpling of the surface was carried out.

The energy scale is as in Fig. It can be surmised that such large distortions might be responsible for a proportion of the observed shifts. Shifts in core-level binding energies for ions in the outermost layers relative to those of bulk ions are also of interest. Such surface core-level shifts are more difficult to compute accurately within the Hartree-Fock approximation because the final-state contribution is expected to be different at sites in the surface layer compared to other layers. However, the initial-state contribution to the SCLS depends partly on changes in the local charge distribution and partly on the difference in the Madelung potential at the surface site.

Hence electrons at surface cation sites are more deeply bound and electrons at surface anion sites less deeply bound than in the bulk. The calculated potential in the layer next to the surface is found to be almost indistinguishable from that in deeper layers. In Figs. The opposite sign of. This is presumably due to local-charge-density effects; a decrease in ionicity of the surface atoms, for example, will tend to decrease the shifts due solely to the Madelung potential.

Table V shows that a redistribution of charge occurs at the surface, consistent with a decrease in ionicity of both components. Surface core-level shifts in metals are often interpreted in terms of the narrowing of the valence-band width of atoms at the surface due to their reduced coordination, with consequent redistribution of charge. The density of states of the oxygen valence levels for surface and bulk atoms is shown in Fig.

O 2p and O 2s projected density of states for valence bands of ions in the bulk and surface layers of a MgO three-layer slab. Agreement with available experimental data is reasonably good, and clearly the method is an attractive one for the study of surface struc-. Binding-energy shifts based on Hartree-Fock eigenvalues for the Ni 2p levels in the layered oxides were also calculated.

Shifts due to changes in the surface and interface geometries of the composite slabs relative to the pure NiO surface were shown to be of the correct order of magnitude and direction to account for the XPS results of Burke and Goodman. Our results do not support the hypothesis of MgO to NiO charge transfer suggested by these authors to account for their spectra. In conclusion, our calculations suggest that small surface geometry changes can have significant effects on the initial states of ions in oxide materials.

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