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The computational advantage and accuracy of the Harris method is linked to the simplicity and adequacy of the reference-density model. In an earlier paper, we investigated one way the Harris functional could be extended to systems outside the limits of weakly interacting atoms by making the charge density of the interacting atoms self-consistent within the constraints of overlapping spherical atomic densities. In the present study, a method is presented for augmenting the interacting atom charge densities with symmetrized partial-wave expansions on each atomic site. The added variational freedom of the partial waves leads to a scheme capable of giving exact results within a given exchange-correlation approximation while maintaining many of the desirable convergence and stability properties of the original Harris method. Incorporation of the symmetry of the cluster in the partial-wave construction further reduces the level of computational effort. This partial-wave cluster method is illustrated by its application to the dimer ${\mathrm{C}}_{2}$, the hypothetical atomic cluster ${\mathrm{Fe}}_{6}$${\mathrm{Al}}_{8}$, and the benzene molecule.

A method based on steepest-descent algorithms is described for calculating one-electron occupation numbers in finite-atomic-cluster calculations. Unlike zero-temperature Fermi-Dirac statistics, the technique can be used to determine fractional occupation numbers in those cases where they give the lowest self-consistent-field energy. Applications to the iron atom and an iron-aluminum cluster illustrate the principles of the method.

A fully self-consistent series of nonlocal (gradient) density-functional calculations has been carried out using the augmented-Gaussian-orbital method to determine the magnitude of gradient corrections to the potential-energy curves of the first-row diatomics, Li{sub 2} through F{sub 2}. Both the Langreth-Mehl-Hu and the Perdew-Wang gradient-density functionals were used in calculations of the binding energy, bond length, and vibrational frequency for each dimer. Comparison with results obtained in the local-spin-density approximation (LSDA) using the Vosko-Wilk-Nusair functional, and with experiment, reveals that bond lengths and vibrational frequencies are rather insensitive to details of the gradient functionals, including self-consistency effects, but the gradient corrections reduce the overbinding commonly observed in the LSDA calculations of first-row diatomics (with the exception of Li{sub 2}, the gradient-functional binding-energy error is only 50--12 % of the LSDA error). The improved binding energies result from a large differential energy lowering, which occurs in open-shell atoms relative to the diatomics. The stabilization of the atom arises from the use of nonspherical charge and spin densities in the gradient-functional calculations. This stabilization is negligibly small in LSDA calculations performed with nonspherical densities.

The rapid variation of charge and spin densities in atoms and molecules provides a severe test for local-density-functional theory and for the use of gradient corrections. In the study reported in this paper, we use the Langreth, Mehl, and Hu (LMH) functional and the generalized gradient approximation (GGA) of Perdew and Yue to calculate s-d transition energies, 4s ionization energies, and 3d ionization energies for the 3d transition-metal atoms. These calculations are compared with results from the local-density functional of Vosko, Wilk, and Nusair. By comparison with experimental energies, we find that the gradient functionals are only marginally more successful than the local-density approximation in calculating energy differences between states in transition-metal atoms. The GGA approximation is somewhat better than the LMH functional for most of the atoms studied, although there are several exceptions.

Structural characterization of rare earth adsorption at surfaces or interfaces of $\ensuremath{\beta}{\text{-Si}}_{3}{\text{N}}_{4}$ grains within silicon nitride ceramics has recently been reported by three different groups using $Z$-contrast scanning transmission electron microscopy (STEM) imaging. Here we report the electronic structure basis for these observations and discuss the origin of similarities and differences among the lanthanides characterized in that work. Along with the features that are well described by a first-principles cluster and surface slab models, we identify those differences in the experiment and theory that warrant further investigation. Stereochemical bonding factors are found to determine adsorption site preferences as opposed to ionic size effects. The set of possible bond sites is a characteristic of the $\ensuremath{\beta}{\text{-Si}}_{3}{\text{N}}_{4}$ interface; however the strength of the rare earth--interface bonding is determined by the electronic structure of the nitride surface and the specific adsorbate. This is the principal factor controlling the effects of dopants on the $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\beta}$ phase transformation and on the $\ensuremath{\beta}{\text{-Si}}_{3}{\text{N}}_{4}$ grain growth at high temperature as well as the subsequent microstructure of the ceramic.

An efficient approach to extending the spatial scale of electronic structure calculations is described in this work. The method is formulated as a combination of the ``interacting fragments'' concept of Harris [Phys. Rev. B 31, 1770 (1985)] and the divide and conquer (D) method of Yang [Phys. Rev. Lett. 66, 1438 (1991); Phys. Rev. A 44, 7823 (1991)], which recognizes the intrinsic locality of electron bonding and is devised to optimize the total electron charge density within an approximate representation of partitioned components. Beginning with a brief review of D concepts, we report results from this method using the D as an ``embedding'' method for coupling an atomic cluster to its extended environment. The convergence properties as implemented within the self-consistent partial-wave (SCPW) linear variational method are illustrated through various applications. In particular, results from a study of the adsorption of La atoms at the prism plane of $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Si}}_{3}{\mathrm{N}}_{4}$ demonstrate the practicality of the SCPW using D as an embedding technique.

The simplified energy functional of Harris has given results of useful accuracy for systems well outside the limits of weakly interacting fragments for which the method was originally proposed. In the present study, we discuss the source of the frequent good agreement of the Harris energy with full Kohn-Sham self-consistent results. A procedure is described for extending the applicability of the scheme to more strongly interacting systems by going beyond the frozen-atom fragment approximation. A gradient-force expression is derived, based on the Harris functional, which accounts for errors in the fragment charge representation. Results are presented for some diatomic molecules, illustrating the points of this study.

Describing materials properties and behavior over increasing scales of dimension and complexity requires an optimal balance of completeness and accuracy in solving the local density equations. In this study, the convergence properties of a set of schemes that aim to achieve increasing accuracy are systematically examined according to the hierarchical approximations upon which they are based. Specifically, the Harris density functional (HDF) and related schemes that express the total energy in terms of atomic densities and limited self-consistency are compared within a single consistent framework. Convergence of the HDF energy relative to input density is first tested by carrying out calculations within the non-self-consistent atom fragment and self-consistent atom fragment (SCAF) approximations and then by supplementing the SCAF density by increasing numbers of partial waves about each atomic site using the self-consistent partial wave (SCPW) method. The construct of the SCPW method, that solves the local density equations with controlled precision according to the number of partial waves in the site density expansions, enables this study. The rapid convergence of structural properties with an increasing number of partial waves on each site, sometimes even with only $L=0$ partial waves, provides additional justification for HDF-based tight-binding and molecular dynamics methods where the interatomic potentials are obtained from the superposition of atomiclike densities. The convergence of ground state structural properties is demonstrated by application to the set of molecules: carbon monoxide, water, orthosilicic acid $({\mathrm{H}}_{4}{\mathrm{SiO}}_{4})$, formamide $({\mathrm{HCONH}}_{2})$, iron pentacarbonyl $[\mathrm{Fe}(\mathrm{CO}{)}_{5}]$, and dimanganese decacarbonyl $[{\mathrm{Mn}}_{2}(\mathrm{CO}{)}_{10}]$.

It is shown that the rare earths (REs) have different tendencies to segregate to the grain surfaces in silicon nitride ceramics, and this alters the filling of adsorption sites on the prismatic plane surfaces. Here, the adsorption and binding strengths of RE segregants at prismatic grain surfaces are determined by first-principles calculations. The adsorption of three rare earths (La, Gd, and Lu) is expected to be distinctly different at the predicted adsorption sites, and this is confirmed by atomic-resolution scanning transmission electron microscopy images. This combination of theory and experiment provides a basis for understanding the adsorption behavior of RE elements in silicon nitride ceramics.

Rare earth (RE) and group III oxide additions are frequently used to optimize densification during the processing of ceramics. Silicon nitride ceramics frequently serve as model cases, and in these systems the effects of rare earths are important. Additions often determine the morphology of {beta}-Si{sub 3}N{sub 4} crystallites that grow in the multiphase ceramic, thereby affecting the microstructure and mechanical toughness of the ceramic. The influence of different rare earths has recently been experimentally characterized in terms of their effects on grain growth aspect ratios. In the study reported here, a new energy parameter is introduced that provides a first-principles based understanding of these effects. Grain growth aspect ratios measured for various RE additions in silicon nitride correlate well with corresponding differential binding energies (DBE) calculated within the partial wave self-consistent field atomic cluster model. The DBE provides a second-difference measure of relative site stabilities of RE vs Si atoms in regions of variable O/N content. The physical mechanism that underlies anisotropic grain growth is found to originate from the site competition between REs and Si for bonding at {beta}-Si{sub 3}N{sub 4} interfaces and within the O-rich glass. The different segregation strengths exhibited by rare earth elements in oxynitride glassesmore » are simply a reflection of their different local chemistries in O, N environments. Elements that segregate to the prism planes of the embedded {beta}-Si{sub 3}N{sub 4} grains impede the attachment of Si-based silicon nitride growth units, and the extent of this limitation leads to the observed grain growth anisotropy.« less

The nanoscale intergranular films that form in the sintering of ceramics often occur as adherent glassy phases separating the crystalline grains in the ceramic. Consequently, the properties of these films are often equal in importance to those of the constituent grains in determining the ceramic's properties. The measured characteristics of the silica-rich phase separating the crystalline grains in ${\mathrm{Si}}_{3}{\mathrm{N}}_{4}$ and many other ceramics are so reproducible that ${\mathrm{SiO}}_{2}$ has become a model system for studies of intergranular films (IGF's). Recently, the influence of fluorine and chlorine dopants in ${\mathrm{SiO}}_{2}$-rich IGF's in silicon nitride was precisely documented by experiment. Along with the expected similarities between the halogens, some dramatically contrasting effects were found. But the atomic-scale mechanisms distinguishing the effects F and Cl on IGF behavior have not been well understood. First-principles density functional calculations reported here provide a quantum-level description of how these dopant-host interactions affect the properties of IGF's, with specific modeling of F and Cl in the silica-rich IGF in silicon nitride. Calculations were carried out for the energetics, structural changes, and forces on the atoms making up a model cluster fragment of an ${\mathrm{SiO}}_{2}$ intergranular film segment in silicon nitride with and without dopants. Results show that both anions participate in the breaking of bonds within the IGF, directly reducing the viscosity of the ${\mathrm{SiO}}_{2}$-rich film and promoting decohesion. Observed differences in the way fluorine and chlorine affect IGF behavior become understandable in terms of the relative stabilities of the halogens as they interact with Si atoms that have lost one if their oxygen bridges.

The Hohenberg-Kohn-Sham density-functional equations in the local spin-density approximation (LSDA) have been solved with essentially no loss of accuracy for dimers of the first row of the Periodic Table with the use of a fully-self-consistent spin-polarized Gaussian-orbital approach. Spectroscopic constants (binding energies, equilibrium separations, and ground-state vibrational frequencies) have been derived from the calculated potential-energy curves. Intercomparison of results obtained using the exchange-correlation functionals of Slater (scaled exchange or X..cap alpha..), Gunnarsson and Lundqvist (GL), and Vosko, Wilk, and Nusair (VWN) permits assessment of the relative merits of each and serves to identify general shortcomings in the LSDA. Basic trends are similar for each functional, but the treatment of the spin dependence of the exchange-correlation energy in the GL and VWN functionals yields a variation of the binding energy across the series which is more systematic than that in the X..cap alpha.. approximation. Agreement between the present results and those of Dunlap, Connolly, and Sabin in the X..cap alpha.., approximation confirms the accuracy of the variational charge-density-fit procedure used in the latter work. The refinements in correlation treatment within the VWN functional are reflected in improvements in binding energies which are only slight for most dimers in the series. Thismore » behavior is attributed to the error remaining in the exchange channel within the LSDA and demonstrates the necessity for self-interaction corrections for more accurate binding-energy determinations. Within the current LSDA, absolute accuracies of the VWN functional for the first-row dimers are within 2.3 eV for binding energies, 0.07 a.u. for bond lengths, and approx.200 cm/sup -1/ for vibrational frequencies.« less

The local-density-functional description of chemical bonding in first-row homonuclear dimers is analyzed in terms of both static and dynamic orbital forces. The dynamic orbital force of the i-italicth molecular orbital is equal to the negative of the derivative of the one-electron energy epsilon-c/sub i-italic/ with respect to the p-italicth nuclear coordinate, i.e., -partialepsilon-c/sub i-italic//partialX/sub p/. The static orbital force is also equal to a derivative of the one-electron energy, but the differentiation is carried out with the orbital held fixed, i.e., (-partialepsilon-c/sub i-italic//partialX/sub p/)psi. It is shown that the static force is the orbital's contribution to the total Hellmann-Feynman force, whereas the dynamic force describes the change in the total force due to change in the orbital's occupation number. The chemical bond force in the first-row dimers is observed to be a delicate balance between bonding and antibonding orbital forces. Most of the bond force comes from the 2sigma/sub g-italic/ orbital and to a lesser extent from the 1..pi../sub u-italic/ state. The polarization of the core orbitals in N/sub 2/, O/sub 2/, and F/sub 2/ is found to originate indirectly through their interaction with the 3sigma/sub g-italic/ orbital. The dynamic orbital force gives accurate predictions about the change of equilibriummore » bond distances accompanying electronic ionization and excitation. The formalism and results are related to earlier Hartree-Fock studies.« less

The electronic energy bands and wave functions of cubic $3C$-SiC have been calculated in the first-principles Hartree-Fock-Slater model, making use of the discrete variational method. A comparison is made between experimental optical data, calculated indirect transitions, and theoretical results for ${\ensuremath{\epsilon}}_{2}(E)$ and reflectivity. The valence-band density of states is found to be in reasonably good agreement with x-ray emission spectra.

The fully self-consistent implementation of the nonlocal-density functional of Langreth and Mehl has been used in conjunction with the augmented Gaussian method to calculate the potential-energy curves and dissociation energies of N/sub 2/, Li/sub 2/, O/sub 2/, and F/sub 2/. The nonlocal functional gives ground-state potential-energy curves in very good agreement with experiment. We compare these nonlocal results with other local-density calculations, and examine the effect of nonlocal-potential self-consistency on the dissociation energy.

The principle of augmentation, used to introduce inner-atom core structure into slowly varying basis functions, is applied to Gaussian orbitals to define a new basis set for highly accurate total-energy calculations for atomic clusters within the density-functional formalism. Diffuse Gaussian-orbital tails are matched continuously and differentiably to inner-atom numeric radial functions at the atomic-sphere radius. Major advantages of Gaussian-orbital basis sets are acquired without the need for numerous Gaussians of large exponent for the core region. The numeric functions used inside the atom permit essentially exact solutions for that region. Procedures are described which recover use of the efficient integral algorithms for the Gaussian-orbital-tail matrix elements. The interactions over the structured inner-atom region are treated by efficient integrand smoothing and integration procedures for the sphere. The new augmented Gaussian basis removes the primary limitations on the use of Gaussian orbitals for heavy atoms. As an illustration the method is applied to the copper dimer in an all-electron framework within the local-spin-density approximation (LSDA). The calculated binding energy, equilibrium separation, and first ionization potential of ${\mathrm{Cu}}_{2}$ are within 2% of experiment within the $X\ensuremath{\alpha}$ model. Excitation energies are better described within more recent refined exchange-correlation functionals. These all-electron results show the LSDA model predicts a slightly contracted bond length for ${\mathrm{Cu}}_{2}$, consistent with bulk LSDA calculations for the $3d$ transition-metal series.

Results of a first-principles, all-numerical linear variational energy-band calculation are presented for a Cu (001) monolayer. The electronic structure is discussed in terms of the changes in d bonding which accompany the descent in symmetry in going from the bulk to the surface. The relation of the splittings and ordering of the levels to those of the bulk follow expectations based on simple d-bonding considerations. Results are compared with those from other studies and some discrepancies among earlier works are resolved.

A study of some factors which influence the position and shape of the resonance bands of adsorbates on surfaces is presented with emphasis upon the case of oxygen chemisorbed on aluminum. A twofold approach has been taken to determine the relative roles of local and delocalized electron effects: the energy-band structure for a thin film geometry is correlated with the eigenvalue spectrum from a "surface-molecule" cluster model. Oxygen resonance features are found to depend mainly upon a bond mechanism involving orbitals directed parallel to the surface such that the local bonding in the cluster model provides a reasonable description of the resonance position. Delocalization of the substrate electrons in the plane is found to provide a bulklike background against which the localized resonance is observed.

The virial expression valid in density-functional theory contains an integral over the gradient of the effective exchange-correlation potential. The significance of this term in correcting the noninteracting kinetic and electrostatic energies is discussed in this work. From the gradient integral in the local density approximation an expression is derived for the kinetic-energy part of the exchange-correlation energy density, and its connection with the derivation of von Barth and Williams is discussed. To illustrate the application of these developments to the molecular case results are presented from Gaussian-orbital cluster calculations for H/sub 2/. In this study the forces in the dimer are calculated using the Hellmann-Feynman and virial theorems, and results are compared with the force obtained by direct interpolation from the binding-energy curve. The usefulness of the various methods for calculating the force is discussed.

The accurate correlation energies for the para- and ferro-magnetic states of the electron liquid calculated by Ceperley and Alder were recently used by Vosko, Wilk, and Nusair to produce a new correlation-energy density of increased accuracy and proper limiting behavior in the metallic density regime (${r}_{s}\ensuremath{\le}6$). In the present work, the correlation potential in the local-spin-density approximation (LSDA) is derived from the correlation-energy-density representation of Vosko et al. Characteristics of the new exchange-correlation model are compared with those of the LSDA model of Gunnarsson and Lundqvist. Specific comparison is made between these models and exact results in the treatment of atomic and molecular hydrogen. Since the new treatment of correlation primarily affects the region of small ${r}_{s}$, which is exchange dominated, correlation corrections are small compared with errors in the exchange energy. Thus, in light atoms the improved correlation model leads to a reduced cancellation of error between exchange and correlation energies, emphasizing the necessity for improved exchange treatment. For more homogeneous systems, the model should offer real improvement. The present results obtained with precise treatment of correlation within the prescription of Vosko et al. serve to define the present limitations of the LSDA and indicate the importance of nonlocal corrections, particularly for atoms.

The x-ray $K$-emission spectrum and the soft-x-ray absorption spectrum are calculated for diamond and compared with experiment. The influence of transition matrix elements, calculated in a one-electron approximation, is discussed. Binding-energy corrections on the core and valence states are estimated and used to explain observed shifts between band transition energies and experimental data. Approximate methods are used to study localized valence relaxation effects on the transition matrix elements in emission.

Contemporary electronic-structure methods avoid shape approximations but in doing so encounter the difficult problem of integral evaluation over complicated interstitial volumes. In this paper, we present a simple and efficient technique for applying rapidly convergent Gaussian product formulas to general interstitial regions. Like the recent methods of Boerrigter, te Velde, and Baerends, it is based upon partitioning space into Voronoi cells and atomic spheres. In the present work, introduction of a general pseudospherical local-coordinate system unifies the integration procedures and effects a simplified approach. A systematic procedure is derived for determining the number of Gaussian points required for a specified level of numerical precision.

Errors in kinetic and exchange contributions to the molecular bonding energy are assessed for approximate density functionals by reference to near-exact Hartree-Fock values. From the molecular calculations of Allan et al. and of Lee and Ghosh, it is demonstrated that the density-gradient expansion does not accurately describe the noninteracting kinetic contribution to the bonding energy, even when this expansion is carried to fourth order and applied in its spin-density-functional form to accurate Hartree-Fock densities. In a related study, it is demonstrated that the overbinding of molecules such as N/sub 2/ and F/sub 2/, which occurs in the local-spin-density (LSD) approximation for the exchange-correlation energy, is not attributable to errors in the self-consistent LSD densities. Contrary to expectations based upon the Gunnarsson-Jones nodality argument, it is found that the LSD approximation for the exchange energy can seriously overbind a molecule even when bonding does not create additional nodes in the occupied valence orbitals. LSD and exact values for the exchange contribution to the bonding energy are displayed and discussed for several molecules.

A method for solving Poisson's equation as a set of finite-difference equations is described for an arbitrary localized charge distribution expanded in a partial-wave representation. The procedure is an extension of the widely used technique developed by Loucks for spherically symmetric charge densities. In the present development, the higher partial-wave components of the density lead to an l dependence in the boundary conditions and the finite-difference equations. The potential components for l>0 can be calculated recursively with the same degree of accuracy and efficiency as for the l=0 case. This procedure requires the additional computational step of evaluating a one-dimensional integral of each density component to determine the multipole moment, required by the boundary condition at large r. Specific modifications required to adapt the Loucks technique for this more general charge density are described, and illustrative results are given for a model density with components rho/sub l/(r)=C/sub l/r/sup l/el-italicr, which can also be treated analytically.

The electronic band structure of graphite has been calculated from an ab initio variational approach using a linear-combination of atomic-orbitals (LCAO) basis of Bloch states, including nonspherical terms in the one-electron crystal potential. Matrix elements of the Hamiltonian are evaluated directly without any tight-binding approximations. The optical transitions deduced from the energy bands calculated using a single-layer crystal model agree nicely with recent polarized-light reflectance measurements. Details of the band structure are calculated for the three-dimensional Brillouin zone and related to the results obtained using the single-layer crystal structure. The results are encouraging, not only from the standpoint that the method employed is an ab initio approach with no special a priori assumptions, but also because the band structure is quite insensitive to the particulars of the crystal potential function.

The recently developed discrete-variational method (DVM) for the energy-band problem has been combined with the Korringa---Kohn---Rostoker (KKR) procedure to adapt the latter to treat general one-electron crystal potentials. Orbitals obtained from a KKR calculation for the muffin-tin average of the full Bloch Hamiltonian are used as trial variational functions for treating the complete problem within the framework of the DVM. Non-muffin-tin corrections originating from all regions of the unit cell are explicitly included by diagonalization of the relevant secular matrix in this basis set. This combination of the DVM and KKR methods to a large extent achieves the main advantages of each scheme and promises to comprise an efficient procedure for calculating the electronic structure of crystal compounds. With this method there is no limitation to a single-type basis set; for some purposes, it may be useful to supplement the KKR orbitals with other trial functions. The high accuracy that is achieved allows definitive conclusions to be drawn concerning the effects of the non-muffin-tin corrections on electronic structure. Illustrative results given for an application of the method to paramagnetic nickel demonstrate this point. An extension of the method is described by which the non-muffin-tin potential and charge-density corrections can be included in the multiple-scattering approach to the electronic structure of molecules.

A general variational method for efficiently calculating energy bands and charge densities in solids is presented; the method can be viewed as a weighted local-energy procedure or alternately as a numerical integration scheme. This rapidly convergent procedure circumvents many of the difficulties associated with the evaluation of matrix elements of the Hamiltonian in an arbitrary basis and treats the general nonspherical potential with no more complication than the usual "muffin-tin" approximation. Thus the band structure of ionic and covalent materials can be calculated with realistic crystal potentials. As an example, the method is applied to the one-electron model Hamiltonian with a nonspherical local potential, using a linear combination of atomic orbitals basis. Matrix elements of the Hamiltonian are evaluated directly without decomposition into atomic basis integrals; no "tight-binding" approximations are made. Detailed calculations are presented for the band structure and charge density of bcc lithium which demonstrate the feasibility of our method, and reveal the sensitivity of the energy bands to nonspherical and exchange components of the crystal potential. Various prescriptions for the construction of crystal potentials are considered, and convenient least-squares expansions are described. The extension of these methods to nonlocal potentials such as are encountered in the Hartree-Fock self-consistent-field procedure is discussed.

Fine structure in the energy distribution of secondary electrons "back-scattered" from a graphite crystal surface is resolved and shown to be consequence of inelastic electron-electron scattering in which the dominant process is the population of final states above the vacuum level by electron-hole pair production via screened-Coulombic interaction between the incident primary electrons and the valence electrons in the solid. The scattering theory of Kane is applicable and emphasizes that features due to one-electron density of final states should be resolvable in experimental secondary-electron-emission spectra of crystals. Experimental results are presented, which provide strong support for this view. Previous measurements on graphite have been extended and weak secondary-electron-emission structure, resolved in the second derivative of the energy-distribution spectrum, is reported for kinetic energies, $10\ensuremath{\lesssim}{E}_{\mathrm{kin}}\ensuremath{\lesssim}40$ eV. Maxima are observed at 16.2, 22.2, 29.2, 31.2, 36.2, and 40.7 \ifmmode\pm\else\textpm\fi{} 0.5 eV above the Fermi energy. Details are presented of a first-principles high-energy band-structure calculation of graphite extending over a 80-eV energy range. The observed spectral features correlate closely with final-density-of-states maxima as predicted by the theory.

In presenting their layer multiple-scattering approach to the energy-band problem for thin films, Kar and Soven reported that the ordering of levels in their calculations for a copper monolayer differed from previous results obtained by Kasowski. Some question also arose in relating the d-level splittings at the surface to the order of the bulk E/sub g/ and T/sub 2g/ levels. In this Comment, it is suggested that these discrepancies arise from an inconsistency in coordinate systems used by Kar and Soven. It is shown that within a consistent system, the level splittings and orderings in Kar and Soven's results follow expectations based on simple bonding considerations. The resulting good agreement with other works supports the essential accuracy of the multiple-scattering approach of Kar and Soven.