invited papers
Hole counts from Xray absorption spectra
^{a}Dept. of Physics, University of Washington, Seattle, WA 981951560, USA
^{*}Correspondence email: alex@phys.washington.edu
The interpretation of Xray absorption spectra in terms of electronic structure has long been of interest. Hole counts derived from such spectra are often interpreted in terms of freeatom occupation numbers or Mülliken counts. It is shown here, however, that renormalizedatom (RA) and cellular counts are better choices to characterize the configuration of occupied electron states in molecules and condensed matter. A projectionoperator approach is introduced to subtract delocalized states and to determine quantitatively such hole counts from Xray absorption spectra. The described approach is based on multiplescattering theory (MST) and on atomic calculations of a smooth transformation relating the Xray absorption spectra to local projected densities of states (LDOS). Theoretical tests for the s and d electrons in transition metals show that the approach works well.
Keywords: hole counts; Mülliken counts; electronic structure; Xray absorption spectra; fine structure.
1. Introduction
l projected (LDOS). In the 1970s, the Pt white line was related to the number of d holes (Brown et al., 1977). Later, a quantitative procedure to obtain such hole counts was proposed (Mansour et al., 1984). These methods are based on a tightbinding linear combination of atomic orbitals (LCAO) picture and thus essentially give occupation numbers in terms of Mülliken counts. However, the procedure is complicated by the presence of contributions to the spectra from both the localized atomic orbitals and the delocalized atomic continuum states. Their separation is typically performed with an ad hoc stepfunction subtraction of delocalized states. Also, the procedure needs an a priori estimate of the number of holes or the reduced matrix element for the p → d transition. Thus a better approach is desirable.
directly probes the unoccupied excited states of a system. The determination of hole counts from has been of interest for many years, since the spectrum is directly related to the finalstate angularmomentumThe number of holes can be calculated theoretically in several ways, as reviewed by Müller et al. (1998). These authors criticized cellular counts, since the separation of space into cells is ambiguous, and argued in favor of a Mülliken analysis, based on atomic orbitals. However, as discussed below, from deep core levels is not sensitive to the density matrix beyond the cell boundaries and thus cannot precisely determine such Mülliken counts, which depend on the density matrix throughout the range of a given orbital. Also, the LCAO picture has a fundamental problem connected with nonzero amplitude of atomic orbitals at neighboring sites. This can be regarded as a violation of the Pauli principle (Hodges et al., 1972), since the orbital tails do not satisfy the Schrödinger equation in the potential near the neighbors. As a result, the total number of such d electrons per site is not 10. To cope with this problem, Hodges et al. (1972) suggested the renormalizedatom (RA) method, which has many advantages with regard to transitionmetal properties.
In the present paper, we develop a procedure to extract hole counts from ab initio atomic calculations of a smooth transformation relating the spectrum to the LDOS.
and compare them to directly calculated RA, cellular and Mülliken counts. Our approach is based in part on2. Theory
The central quantity of interest here is the electron density matrix . We begin with the expression for within multiplescattering theory (MST) and then outline its connection to e.g. Mülliken and RA counts, from and to separate the LDOS into localized and delocalized parts. Such a separation of data is ambiguous, owing to the presence of localized–delocalized cross terms.
Next we discuss the use of projection operators to obtain hole counts,The transformation from the μ(E) to the LDOS ρ(E), i.e. the number of electrons per unit energy, can be derived within MST (Müller & Wilkins, 1984). Starting from the `golden rule' μ(E) = ∑_{f}〈fdc〉^{2}δ(E − E_{f}), one obtains a linear relation between μ(E) and the electrondensity matrix (E), i.e. μ(E) = M_{Γ}(E), where M_{Γ} is a linear operator containing dipole matrix elements and final state broadening, and the density matrix is given as a sum over angular momenta L = (l, m),
where χ = ∑_{m}χ_{L,L}/(2l + 1), and l depends on the initial core orbital momentum as l = l_{c} + 1 since it represents the dominant absorption channel. To simplify the relationship between and LDOS, we note that from MST both the μ(E) = μ_{0}(1 + χ) and the LDOS (E) = ρ_{0}(1 + χ) have similar decompositions in terms of the smooth atomic background μ_{0} and fine structure or χ. Consequently,
The ratio ρ_{0}(E)/μ_{0}(E) is essentially an atomic quantity which can be calculated by any code based on MST or even appropriately modified atomic codes. Strictly speaking, both μ_{0} and ρ_{0} refer to embedded atomic quantities, i.e. the effective atomic states defined by the local molecular potential at a given site. MST is based on the separation of space into cells (Voronoi polyhedra) and thus the number of holes (or the change in the number of holes) corresponds to spaceseparation counts. Such a space separation is not unique and therefore one faces difficulty in interpreting the results.
Using equation (4), one can transform the experimental spectrum into the normalized LDOS above the However, this unoccupied LDOS contains contributions both from localized atomiclike states and from delocalized atomic continuum states. A projection operator can then be used to obtain the localized contribution to the LDOS. In particular, the Mülliken count is given by projection of all occupied states onto a particular atomic orbital,
The main disadvantage of Mülliken counts is that if one sums the occupied and unoccupied 3d states, one obtains a total count that is different from 10, as a result of the overlap of the 3d orbital with orbitals centered on neighbors. Indeed, our estimate for the total Mülliken d count in Fe, Co and Ni is 10.4 (Table 1). Therefore, one faces the problem of separating tail contributions from Mülliken counts to obtain physically meaningful total 3d counts. The situation is even worse for s and p electrons, which extend much further.

The RA method (Hodges et al., 1972) has been proposed to overcome the above difficulty. In this method, each is set to zero outside a given radius R_{0}, and is multiplied by a constant factor inside to recover unit normalization. The RA states are therefore better approximations to embedded atom states inside a cell. Also, the wave functions used in the projection operator become essentially orthonormal to one another and the tail contribution is much smaller. The RA counts thus overcome the problem of nonorthogonality with the LCAO method, and also justify the use of the MST formula = ρ_{0}(1 + χ). The error in total RA count for occupied and unoccupied states was found to be an order of magnitude smaller for each d_{j} subshell than the corresponding Mülliken count (Table 1). Our calculations were performed with the selfconsistent realspace multiplescattering code FEFF8 (Ankudinov et al., 1998). The total number of 3d electrons obtained with RA counts is 9.95 for Fe, Co, Ni and an integration up to 100 eV above the edge. The 4s RA count is 1.90 for all tested cases, with integration up to 400 eV. The error mostly arises from inaccuracy of our grid for the integration to that high energy. Thus, with the RA method one can easily transform hole counts to meaningful electron counts and this transformation is valid for localized d and f electrons, as well as for highly delocalized s and p electrons.
The RA counts depend on the choice of the embedded atomic radius R_{0}. The most natural choice is that for which the total count for occupied and unoccupied d states is equal to 10. Thus unlike the ambiguous cellular counts, the RA method fixes the radius of cells. The present work shows that this radius is very close to the Norman radius r_{N}, i.e. the radius for which the net charge enclosed is the (Norman, 1974). With such a definition of a cell, both s and d total counts per site are simultaneously close to the numbers expected from the Pauli principle. Charge transfer also has a natural definition as the net charge within the Norman sphere, which can be separated unambiguously into s, p, d and f electron contributions.
Table 1 summarizes our results for delectron counts. The first three columns show the occupation numbers for cellular, Mülliken and RA counts. All of them show an expected change of about one d electron from Fe to Co and from Co to Ni. The cellular and RA counts are very close to each other, while the Mülliken count is about one d electron larger. The fractional counts can be interpreted as the wave function being a mixture of d^{n}s and d^{n+1} states. Note that the RA counts are much closer to those expected from atomic multiplet calculations, where Ni is regarded as having a d^{8} configuration (de Groot, 1994). For Pt we find that the calculated ratio of d_{5/2} holes to d_{3/2} holes is 2.8, in agreement with the value of 2.9 calculated by Mattheiss & Dietz (1980), thus updating an earlier estimate (Brown et al., 1977). The total number of d holes for Pt according to the RA count is 1.8, which is also consistent with the results of Mattheiss & Dietz (1980).
Table 2 presents the selectron counts determined by the cellular and RA methods. We could not make a reliable estimate of the Mülliken counts in this case, since the 4s orbital extends significantly beyond the nearestneighbor distance. Note that for s electrons, the occupied RA and cellular counts are also practically equal. The RA total count is 1.9 for all cases, which shows that the RA method works well with highly nonlocalized s and p orbitals.

3. RA counts from XAS
In this section we describe a procedure to obtain hole counts from l → l − 1 transition, and thus estimate the error in hole count caused by its inevitable presence in the measured signal.
and apply it to a theoretically calculated spectrum. Such a theoretical test is necessary, both to check the validity of the procedure and also to estimate the possible systematic error of the method. For example, in theoretical calculations, one can include or exclude theIn order to separate the localized and continuum contributions, it is natural to use the projection operator P_{at} which projects onto a particular (free or renormalized atom), i.e. = φ_{at}〉〈φ_{at}. In theoretical calculations, it is also possible to project onto any particular orbital character: e.g. to atomic t_{2g} or e_{g} orbitals. These orbitals result from splitting of d states by a cubic crystal field, and their occupation and ordering is often used to explain properties of 3d elements. However, such a projection cannot be performed for an experimental spectrum. Within the dipole approximation, only six irreducible operators can be extracted from polarizationdependent (Ankudinov & Rehr, 1995) and the number of holes is one of them. A direct separation of the spectrum into localized (i.e. projected atomic) and delocalized (i.e. continuum) parts cannot be made unambiguously. If the projection operator is used to separate the final state wave function, the spectrum can be expressed as
where the cross term is nonzero. Previous procedures (Brown et al., 1977) ignore μ_{cross}. But the cross term has the largest contribution exactly where localized and delocalized contributions are comparable and the ad hoc separation of μ_{cross} or its neglect can lead to erroneous results. In contrast to the separation of LDOS into localized atomic and continuum parts is completely unambiguous. By substituting (E) = (E) + (1 − )(E) in equation (1) for the density matrix, we can decompose the total LDOS (E) into localized and delocalized parts:
In this case, the cross term is exactly zero, as a result of the definition of the projection operator.
The projection operator acts only on the radial part in equation (1) and both the localized and the delocalized contributions should have the same χ, which is, therefore, the same as for Thus one can transform the spectrum directly into the localized part of the LDOS using atomic ratios:
This equation provides the basis for our projectionoperator procedure to obtain the occupied E)/μ^{0}(E) to obtain the contribution to the LDOS from localized electrons. The integral over energy of this contribution gives the number of holes in the for a particular orbital character. Unfortunately, because of the dipole selection rule, the determination of the number of s holes, especially when the d shell is partially occupied, is not reliable. However, one can obtain from an estimate of the number of p, d and f electrons in terms of RA counts.
counts from experimental Thus, one simply multiplies the normalized experimental data by the theoretically calculated atomiclike ratio (Our procedure, which can be generalized for various experimental applications, has several steps, as follows.
For final d and f states, one can obtain separately the number of electrons with total angular momentum j = l ± 1/2 when the corresponding L_{2,3} or M_{4,5} edges are well separated. Only the total number of p electrons can be estimated from K and L_{1} edges.
Numerical tests of the first four steps of this approach for 3d transition metals show that typically the number of holes obtained from is underestimated by about 10–20%. The main source of this reduction stems from the strong effect of corehole lifetime broadening on white lines, as verified by observing that the reduction vanishes when Γ_{ch} is reduced to zero. This also provides a way of calculating the reduction factor of step (v). To understand the effect, suppose that all states below E_{0} are localized and all states above are delocalized. Then as a result of the corehole broadening, the localized LDOS will leak above E_{0}, while the delocalized will leak below E_{0}. However, since for the case of white lines the localized LDOS is larger, the integration to the ideal localized–delocalized separation energy will give a reduced number of localized states. The theoretically calculated reduction factor for the dhole count due to Γ_{ch} broadening is 1.14 for Fe, 1.16 for Co, 1.07 for Ni, 1.04 for Pt d_{3/2}, and 1.18 for Pt d_{5/2} holes.
In the analysis of Xray et al., 2000). In that case one can use the smooth ratio of equation (4) instead of equation (6). As one can see from Tables 1 and 2, the cellular counts within a Norman sphere are very close to the RA counts. Their agreement means that the shape of d and s orbitals practically does not change within the Norman radius. The smoothness is important since, for example, multiplet effects can lead to an additional splitting of the order of 5 eV. But with a smooth ρ_{0}/μ_{0}, the error in hole counts caused by this effect is expected to be smaller than for the transformation to the localized LDOS ρ_{loc}/μ_{0}, which has a significant rise at the border between localized and delocalized states. As one can see from the holecount calculations, such a procedure can lead to a significant error.
(XMCD), one can avoid the use of such a reduction coefficient if the goal is to obtain spin and orbital moments in terms of cellular counts (NesvizhskiiIn Fig. 1, we demonstrate the separation of data into localized and delocalized contributions. The directly calculated total spectrum is represented by a solid line. The delocalized contribution is obtained by transforming data into LDOS, subtracting the localized LDOS contribution and transforming the delocalized LDOS back to an spectrum. As expected, at high energies, the delocalized contribution dominates, while near the edge, practically all absorption comes from the localized part. For comparison, we also present an ad hoc two edgestep modeling of the delocalized contribution, which is often employed in XMCD analysis and clearly gives a poor approximation to the unambiguous delocalized contribution calculated by FEFF8.
In the procedure of Brown et al. (1977), the difference between Pt L_{2} and L_{3} is interpreted as solely caused by the localized contribution to the data. If this were the case, the division of the difference signal by the square of the p → d matrix element would yield the difference in the number of d_{3/2} and d_{5/2} holes. However, this analysis must be modified for the presence of the cross term, because the difference in d_{3/2} and d_{5/2} occupation will show up both in localized and in crossterm contributions. Our procedure specifically avoids the crossterm problem by transforming spectra into LDOS, and hence yields, we believe, a more reliable analysis of occupation numbers.
Fig. 2 represents the application of this procedure to Pt L_{2,3} data. The FEFF8 calculations agree well with Pt as shown by Ankudinov et al. (2001, Fig. 1 therein). Even though the peak positions are not changed by our procedure, the ratio of peak intensities in the total LDOS is significantly altered compared to the spectrum. The localized LDOS is mostly limited to the region of first two peaks, i.e. about 40 eV above the edge. By inspection, one sees that the ratio of 5d_{5/2} to 5d_{3/2} hole counts should be around 3.
Several manyelectron effects may be important for the extraction of hole counts. For example, the usual dipole matrix elements, which are not corrected with respect to local screening of the Xray electric field, are not accurate for most 3d metals, as the screening effect modifies the L_{3}/L_{2} whiteline ratio (Schwitalla & Ebert, 1998). Our procedure can account for this effect by including screening in the calculated μ_{0}. Multiplet splitting and are neglected in our singleparticle derivation. However, such effects are not expected to change the counts significantly, since such splitting is expected to transfer spectral density only within about 5 eV of the where the ratio is smooth. Thus one expects a greater cancellation of these effects in equation (4) than in equation (6). Equation (4) can be used to obtain S_{z} and L_{z} from XMCD, but is inapplicable to the problem of hole determination.
4. Conclusions
The present work updates previous techniques for holecount determination. We have found that RA counts are more appropriate than Mülliken counts for the interpretation of both electronic structure and μ^{0} and a calculated correction factor to account for lifetime broadening. As a result, physically meaningful p, d and f counts can all be obtained from experimental data to an accuracy of a few percent.
Noteworthy is the fact that occupied RA counts are very close to cellular counts when the chosen cell has the volume of a Norman sphere. We have introduced a quantitative technique, based on the use of projection operators, to subtract atomic delocalized states and hence to determine the in terms of RA counts, from In this approach, one transforms the measured spectrum to a localized projected LDOS, using a calculated atomic ratio /Acknowledgements
This work was supported in part by DOE grants DEFG0398ER45718, DEFG0397ER45623, and by UOP LLC.
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