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Phys Rev Lett. Author manuscript; available in PMC 2017 July 28.

Published in final edited form as:

Published online 2017 March 3. doi: 10.1103/PhysRevLett.118.096402

PMCID: PMC5532736

NIHMSID: NIHMS869036

Yufeng Liang,^{1} John Vinson,^{2} Sri Pemmaraju,^{1} Walter Drisdell,^{3} Eric L. Shirley,^{2} and David Prendergast^{1}

The publisher's final edited version of this article is available at Phys Rev Lett

Constrained-occupancy delta-self-consistent-field (ΔSCF) methods and many-body perturbation theories (MBPT) are two strategies for obtaining electronic excitations from first-principles. Using the two distinct approaches, we study the O 1*s* core excitations that have become increasingly important for characterizing transition-metal oxides and understanding strong electronic correlation. The ΔSCF approach, in its current single-particle form, systematically underestimates the pre-edge intensity for chosen oxides, despite its success in weakly correlated systems. By contrast, the Bethe-Salpeter equation within MBPT predicts much better lineshapes. This motivates one to reexamine the many-electron dynamics of X-ray excitations. We find that the single-particle ΔSCF approach can be rectified by explicitly calculating many-electron transition amplitudes, producing X-ray spectra in excellent agreement with experiments. This study paves the way to accurately predict X-ray near-edge spectral fingerprints for physics and materials science beyond the Bethe-Salpether equation.

X-ray absorption spectroscopy (XAS) is a powerful characterization technique in physics, chemistry, and materials science, owing to its element specificity and orbital selectivity. With the help of density-functional theory (DFT), the interpretation of XAS is greatly facilitated by simulating spectral fingerprints for hypothetical structures from first-principles. Satisfactory X-ray absorption spectra have been simulated across a wide range of systems from small molecules [1, 2] to condensed matter [3–9] and even complex interfaces [10].

In this letter, we introduce a novel first-principles theory for simulating X-ray absorption spectra extending the constrained-occupancy delta-self-consistent-field (ΔSCF) method [6, 7, 9]. The ΔSCF method assumes a fixed core-hole potential and employs only single-particle orbitals for finding transition amplitudes. It is generally thought this theory cannot capture certain many-electron effects as incorporated in many-body perturbation theories, such as the Bethe-Salpeter Equation (BSE) [11–14]. However, if the one-body transition amplitude in the ΔSCF method is recast into a many-body formalism, we show that the method can in fact capture many-electron effects in X-ray excitations. Furthermore, we outline how the excitation spectrum can be enriched by considering multiple electron-hole (*e*-*h*) excitations, permitting extensions beyond the BSE, on the level of the Mahan-Noziéres-De Dominicis (MND) theory[15, 16].

For our examples, we choose the O *K* edges (1*s* → *np* transitions) of transition-metal oxides (TMOs) to illustrate the utility of the many-body ΔSCF method. The study of the O K edges for TMOs is fueled by the quest for next-generation energy materials, for rechargeable battery cathodes [17–20], fuel cells [21, 22], water-splitting catalysts [23–25], and transparent conductive layers [26]. Many of these materials are TMOs with complex chemical properties due to their *d* orbitals. Additionally, XAS has been employed to understand electron correlations inherent in TMOs, such as metal-insulator transitions [27–29], high-*T _{c}* superconductivity [30, 31], and emergent phenomena at perovskite interfaces [32]. XAS can also serve as powerful guides to advance theories for correlated electron systems, including the DFT+U method [33], dynamical mean-field theory [34], and exact diagonalization approaches [35, 36]. In almost all of the aforementioned examples [17, 19–21, 23–28, 30–33, 35, 36], there are measurements at the O

Five TMOs are selected for benchmarking: the rutile TiO_{2}, VO_{2}, and CrO_{2} as well as the corundum *α*-Fe_{2}O_{3} and the perovskite SrTiO_{3}. They vary greatly in structure, band gap, or magnetism. The rutile VO_{2} (> 340*K*) and CrO_{2} are metallic, whereas TiO_{2} and SrTiO_{3} are insulating. CrO_{2} is ferromagnetic (FM) while Fe_{2}O_{3} is antiferromagnetic (AFM). The O *K* edges from previous experiments [23, 27, 37–39] are shown in Fig. 1(a). These spectra are angularly averaged except for CrO_{2}, where the polarization is perpendicular to the magnetization axis [38]. The TM-3*d*-O-2*p* hybridization manifests as sharp double peaks around 530 eV, which result from the *t*_{2}* _{g}*-

A comparison of experimental O *K* edges (a) with the simulated spectra by the FCH approach (b) and BSE (c). The pre-edge regions are covered by shaded areas. Spectra are normalized according to the *e*_{g} peak intensity. The severely underestimated *t*_{2}_{g} peaks **...**

The X-ray absorbance *σ*(*ω*) is obtained from Fermi’s golden rule

$$\sigma (\omega )\propto \omega \sum _{f}\mid \langle {\mathrm{\Psi}}_{f}\mid \mathit{\epsilon}\xb7\mathit{R}{\mid {\mathrm{\Psi}}_{i}\rangle \mid}^{2}\delta ({E}_{f}-{E}_{i}-\hslash \omega )$$

(1)

where ** ε** and

In the ΔSCF core-hole approach the core-excited atom is treated as a single impurity with one electron removed from the excited core level. Depending on whether or not the X-ray photo-electron is included, it is termed as an excited-electron and core-hole (XCH) or full core-hole (FCH) calculation [6, 7, 9]. One places the core-excited atom in a sufficiently large supercell and then finds the equilibrated electron density using constrained-occupancy DFT. This equilibrated state, with the presence of a core hole, is referred to as the *final state*[13], and that of the pristine system as the *initial state*. The working approximation is to use single Kohn-Sham orbitals for finding the transition amplitude

$$\langle {\mathrm{\Psi}}_{f}\mid \mathit{\epsilon}\xb7\mathit{R}\mid {\mathrm{\Psi}}_{i}\rangle \approx S\langle {\stackrel{\sim}{\psi}}_{f}\mid \mathit{\epsilon}\xb7\mathit{r}\mid {\psi}_{h}\rangle $$

(2)

where * _{f}*’s are the unoccupied orbitals in the final state (with tilde) and

To account for strong electron correlations in TMOs, the DFT+U theory [33] is employed where the DFT energy is captured by the Perdew-Burke-Ernzerhof (PBE) functional, and the on-site Coulomb potential *U* are from Ref. [42]. We interpret the DFT+U orbital energies as quasiparticle (QP) energies and perform FCH rather than XCH calculations so as not to favor any particular occupation. More numerical details can be found in the Supplemental Material [56].

Strikingly, the FCH approach systematically underestimates the intensity of the near-edge peak associated with the *t*_{2}* _{g}* manifold for all selected TMOs (Fig. 1(b)). The

In the BSE formalism [12, 43, 44], the final state is assumed to be a superposition of effective *e-h* pairs and the matrix elements are calculated as

$$\langle {\mathrm{\Psi}}_{f}\mid \mathit{\epsilon}\xb7\mathit{R}\mid {\mathrm{\Psi}}_{i}\rangle =\sum _{c}{A}_{c}^{f\ast}\langle {\psi}_{c}\mid \mathit{\epsilon}\xb7\mathit{r}\mid {\psi}_{h}\rangle $$

(3)

where *ψ _{c}*’s are the

As is shown in Fig. 1(c), the BSE substantially improves on the O *K*-edge line shapes. The “*t*_{2}* _{g}*” peak intensity is retrieved for each investigated TMO, particularly for CrO

In fact, the core-hole effects predicted by the ΔSCF method in these oxides are counterintuitive. Typically, excitonic effects tend to sharpen the absorption edge due to *e-h* attraction [5, 11, 12, 43]. In Fig. 2, we show the core-hole effects from ΔSCF by comparing the initial- and final-state spectra. The core-hole attraction does redshift the spectra by over 1 eV in both the TM 3*d* pre-edge and the 4*sp* [48] region (near 544 eV). However, the FCH core-hole effect substantially reduces the *t*_{2}* _{g}* peak intensity. Similarly underestimated pre-peak intensity was encountered before [49–51] but no satisfactory explanation has been provided to date.

With this comparison, we can now explain why the one-body ΔSCF formalism tends to underestimate the peak intensity. The single-particle approximation in Eq. (2), *S** _{f}*|

To alleviate the deficiencies of the one-body ΔSCF method, we seek a better approximation to Ψ* _{f}*|

Within the independent-KS-orbital approximation, the ground state |Ψ* _{i}* can be constructed as a Slater determinant built from the

The single-particle orbitals necessary for constructing |Ψ* _{i}* and |Ψ

$${A}_{c}^{f}=\text{det}\phantom{\rule{0.16667em}{0ex}}\left[\begin{array}{ccccc}{\xi}_{{f}_{1},1}& {\xi}_{{f}_{1},2}& \cdots & {\xi}_{{f}_{1},N}& {\xi}_{{f}_{1},c}\\ {\xi}_{{f}_{2},1}& {\xi}_{{f}_{2},2}& \cdots & {\xi}_{{f}_{2},N}& {\xi}_{{f}_{2},c}\\ \vdots & \phantom{\rule{0.16667em}{0ex}}& \ddots & \phantom{\rule{0.16667em}{0ex}}& \vdots \\ {\xi}_{{f}_{N+1},1}& {\xi}_{{f}_{N+1},2}& \cdots & {\xi}_{{f}_{N+1},N}& {\xi}_{{f}_{N+1},c}\end{array}\right]$$

(4)

Within the independent-KS-orbital approximation, the energy of |Ψ* _{f}* can be found by summing up the energies of the occupied KS orbitals in the

$${E}_{f}=\sum _{j=1}^{N+1}{\stackrel{\sim}{\epsilon}}_{{f}_{j}}$$

(5)

The energy difference *E _{f}* −

Determinant expressions similar to Eq. (4) were also obtained in previous work [52–54] but they are rarely applied in a solid-state context from first-principles. Thus it is of great interest to examine whether the many-electron ΔSCF formalism in Eq. (4) and (5) can reproduce the correct lineshapes for the investigated TMOs.

Evaluating the many-electron
${A}_{c}^{f}$ seems formidable at first glance because a solid contains many electrons, which leads to a combinatorially huge number of final states. To facilitate the calculation, we regroup the final-state configurations according to the number of *e-h* pairs excited. For example, we denote a *single* configuration with one core-excited *e-h* pair as *f*^{(1)} = (1, 2, ···, *N*, *f _{N}*

Schematic of the excitation configurations in the *final-state* space. *h* indicates the single core level while and indicate the empty and occupied orbtial spaces respectively.

Despite a large number of excitation configurations, not all of them contribute equally to the near-edge XAS, particularly near the absorption onset. First, for an insulator, Ω* _{f}* will increase proportionally with the number

We reexamine two extreme cases: the insulating TiO_{2} and the metallic CrO_{2}. Fig. 4 shows the XAS calculated from the many-body formalism as in Eq. (4) and (5). The orbitals used for calculating the transformation coefficients *ξ _{ij}* in
${A}_{c}^{f}$ are obtained from the same set of ground-state and FCH calculations [56] as previously described. For simplicity, only the states at the Γ-point of the supercell Brillouin zone are used for producing the XAS. Remarkably, the simulated XAS lineshapes (Fig 4) with the many-electron formalism are in excellent agreement with experiments. In particular for CrO

We find calculated XAS for TiO_{2} converges at the order of *f*^{(1)}. As expected, the contributions from *f*^{(2)} (Fig 4(c)) appear at higher energies due to the sizable band gap, and are much weaker due to reduced wavefunction overlap. However, it is more challenging to achieve numerical convergence in the metallic CrO_{2}. While the first peak is retrieved mainly from *f*^{(1)} excitations, the correct peak-intensity ratio can only be reproduced when the next-order, *f*^{(2)}, is taken into account, which substantially increases the intensity of the absorption feature ~ 4 eV above onset (Fig 4(b)). Meanwhile, we find *f*^{(3)} can be neglected up to the first 8 eV (Fig 4(d)).

Both the BSE (Fig. 1) and the many-electron formalism (Fig 4) yield very similar spectra for TiO_{2}, but the many-electron formalism produces a spectrum for CrO_{2} in closer agreement with experiment than the BSE. The experimental peak-intensity ratio is 1.7; 1.6 from the many-electron formalism; but only 1.3 from the BSE. We attribute this improvement the explicit inclusion of many-electron response to the core-hole potential via the transition amplitude
${A}_{c}^{f}$ [Eq. (4)]. Since the Slater determinants are exact solutions to the independent-electron model, the many-electron formalism has taken into account all possible many-electron processes in the MND theory. In addition to the *e-h* ladder diagrams in the BSE, the MND theory also considers non-ladder diagrams, such as diagrams with crossing Coulomb lines and bubble diagrams that account for the orthogonality catastrophe [15, 16, 52]. Instead of adopting a diagrammatic approach, the many-electron formalism considers theses processes via incorporating response of each electron to the core hole in the determinant amplitude
${A}_{c}^{f}$. Therefore, the many-electron formalism goes beyond the two-particle correlation in the BSE. For insulators like TiO_{2}, the many-electron formalism produces results very similar to the BSE because the non-ladder diagrams are not important near the absorption edge. However, this is not the case for CrO_{2} since multiple *e-h*-pair production is more likely in a metal and the non-ladder diagrams become significant for determining the near-edge lineshape. The significance of the multiple electron response can also be appreciated from the fact that the CrO_{2} spectrum converges at a higher-order *f*^{(}^{n}^{)}.

At last, we briefly discuss the validity of the independent-KS-orbital approximation. On the *f*^{(1)} level, there is a one-to-one correspondence between the many-electron configurations and the empty single-particle final-state orbitals, and the latter are typically good approximations to QP band topology and wave-functions [43, 55]. This approximation may break down if there are one or more valence *e-h* pairs. For example, valence *e-h* pairs in the *f*^{(2)} configurations of an insulator may hybridize to form bound excitons; the two excited electrons in the *f*^{(2)} configurations may interact through strong on-site Coulomb repulsion. In the case of CrO_{2}, however, the strong metallic screening can largely reduce *e-h* binding energy, maintaining the independent-KS-orbital approximation. As for strongly correlated effects, it will be worthwhile to further consider embedded models based on the current determinant formalism.

We have shown that the BSE and a newly developed many-electron ΔSCF approach are highly predictive for O *K*-edge fingerprints of TMOs. The systematic underestimation of the peak-intensity ratio within the original one-body ΔSCF approach is attributed to the absence of many-electron response in this formalism. We have demonstrated how to rectify these shortcomings by (1) extending the wavefunctions in the original ΔSCF to a many-electron form (2) calculating the determinant form of the transition amplitude. This many-electron formalism is transferable and not at all peculiar to TMOs. We leave the discussion of shakeup effects, improvement of efficiency, and other examples to near-future work.

Theoretical and computational work was performed by Y. L. and D. P. at The Molecular Foundry, which is supported by the Office of Science, Office of Basic Energy Sciences, of the United States Department of Energy under Contact No. DE-AC02-05CH11231. W. D. was supported by the Joint Center for Artificial Photosynthesis, a DOE Energy Innovation Hub, supported through the Office of Science of the U.S. Department of Energy (Award No. DE-SC0004993). We acknowledge fruitful discussion with Chunjing Jia (Y. L.) and Bill Gadzuk (J. V.). We also would like to acknowledge the referees for giving very patient, professional, and thoughtful comments in the review process. Computations were performed with the computing resources at the National Energy Research Scientific Computing Center (NERSC).

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