Tomographic reconstruction of oxygen orbitals in lithium-rich battery materials

The electrification of heavy-duty transport and aviation will require new strategies to increase the energy density of electrode materials1,2. The use of anionic redox represents one possible approach to meeting this ambitious target. However, questions remain regarding the validity of the O2−/O− oxygen redox paradigm, and alternative explanations for the origin of the anionic capacity have been proposed3, because the electronic orbitals associated with redox reactions cannot be measured by standard experiments. Here, using high-energy X-ray Compton measurements together with first-principles modelling, we show how the electronic orbital that lies at the heart of the reversible and stable anionic redox activity can be imaged and visualized, and its character and symmetry determined. We find that differential changes in the Compton profile with lithium-ion concentration are sensitive to the phase of the electronic wave function, and carry signatures of electrostatic and covalent bonding effects4. Our study not only provides a picture of the workings of a lithium-rich battery at the atomic scale, but also suggests pathways to improving existing battery materials and designing new ones. High-energy X-ray Compton measurements and first-principles modelling reveal how the electronic orbital responsible for the reversible anionic redox activity can be imaged and visualized, and its character and symmetry determined.

The electrification of heavy-duty transport and aviation will require new strategies to increase the energy density of electrode materials 1,2 . The use of anionic redox represents one possible approach to meeting this ambitious target. However, questions remain regarding the validity of the O 2− /O − oxygen redox paradigm, and alternative explanations for the origin of the anionic capacity have been proposed 3 , because the electronic orbitals associated with redox reactions cannot be measured by standard experiments. Here, using high-energy X-ray Compton measurements together with first-principles modelling, we show how the electronic orbital that lies at the heart of the reversible and stable anionic redox activity can be imaged and visualized, and its character and symmetry determined. We find that differential changes in the Compton profile with lithium-ion concentration are sensitive to the phase of the electronic wave function, and carry signatures of electrostatic and covalent bonding effects 4 . Our study not only provides a picture of the workings of a lithium-rich battery at the atomic scale, but also suggests pathways to improving existing battery materials and designing new ones.
The electrification of heavy-duty vehicles and aviation will require batteries with much higher specific energy than that of existing lithium-ion batteries 1,2 . Although substantial progress has been made with lithium metal anodes 5 and silicon 6,7 , there is a need to move beyond the existing layered cathodes to meet this demanding goal. Lithium-rich oxides present a promising class of cathode materials in this regard, with their high capacity of about 300 mAh g −1 (refs. [8][9][10][11][12][13][14]. The anionic redox mechanism underlying their electrochemical operation, however, has been difficult to understand fully with known probes and techniques. Experimental techniques capable of directly probing the oxygen activity include X-ray photoemission spectroscopy (XPS) 15 , soft X-ray absorption (XAS) 16 and X-ray resonant inelastic scattering (RIXS) 17 . However, the signal from the oxygen redox reaction in the bulk sample can be masked by surface effects in these techniques. It has also proven difficult to detect substantial oxygen activity 18 via X-ray Raman spectroscopy, even though it is more bulk sensitive.
First-principles density functional theory (DFT)-based modelling of lithium-rich battery materials is sensitive to subtle correlation effects 19,20 , and the development of robust design principles-with the eventual aspiration of identifying an inverse design paradigm-requires an iterative and reliable experiment-theory loop at the atomic scale 21 . In this way, we can be in a position to undertake direct and sharpened comparisons between theoretical predictions and corresponding experimental results in microscopic details of the cathode electronic structure, thereby accelerating the feedback loop between theory, spectroscopy and battery design in order to identify the origins of the anomalous cathode capacity.
Previous studies of cathode materials 4,22 have demonstrated the efficacy of Compton scattering spectroscopy in unravelling the relationship between the key characteristics of battery performance and the nature of the electronic orbitals involved in lithium intercalation reactions. The advantages of X-ray Compton scattering over standard techniques are enhanced penetration depth and high bulk sensitivity, which in turn allow in situ and in operando measurements of batteries [23][24][25][26] . As the Compton cross-section 27 is very small even at 100 keV, radiation damage is usually negligible. X-ray Compton scattering is thus uniquely suited for probing working batteries both at the macroscopic scale, by mapping of the lithium distribution, and at the atomic level, by extracting characteristics of redox orbitals through tomographic reconstructions 28,29 . An important proof-of-principle of the efficacy of the Compton scattering technique in advancing the design rules involves high-T c cuprate superconductors, where a combined experiment-theory study provided a comprehensive understanding of the redox orbitals associated with doped holes consistent with the complex phase diagram of these strongly correlated materials 30 .
Tracking the occupation of oxygen 2p orbitals is crucial in solving the puzzle of lithium-rich cathodes. This goal can be achieved by exploring the exemplar binary system 10,14 of Li 2 TiO 3 and LiMnO 2 , given by the formula Li x Ti 0.4 Mn 0.4 O 2 (LTMO). Here, we use high-energy X-ray Compton scattering spectra together with first-principles computations to extract the redox orbital, which stabilizes the oxidized oxygen atoms that emerge as the charging process proceeds in the LTMO cathode. We address fundamental issues regarding the nature of the Article key non-bonding oxygen states and the roles of Coulomb repulsion and covalent character in order to gain microscopic insight into the mechanisms at play in lithium-rich battery materials, which are still under debate 3 . We definitively show how anionic redox reactions in LTMO involve oxygen ions, resulting in a substantial enhancement of the battery capacity. Until now, the anionic redox activity in cathode has been difficult to characterize and quantify in both experiments and simulations. We present an analysis based on X-ray Compton experiments through which we are able to quantify electrostatic repulsion and oxygen character in lithium-rich cathodes. Our method will be suitable for the future optimization of anionic redox materials.
Compton scattering provides a quantum mechanical description of electronic states in materials through the reconstruction of wavefunctions from experimental spectra 27,31 . This technique is sensitive to the bonding properties of materials 32 ; it can detect weak bonds such as the hydrogen bond in ice; and it can differentiate these bonds from purely electrostatic interactions 33 . Measured Compton profiles in LTMO (Extended Data Fig. 1) contain features that can be used to characterize the redox orbitals 4 . The contribution of the redox orbitals can be extracted by considering the Compton profile difference (CPD) between two samples with different lithium concentrations. Figure 1 shows the CPD corresponding to the difference in profiles of Li x Ti 0.4 Mn 0.4 O 2 for lithium concentrations (x) of 0.8 and 0.4. The valence Compton profiles for the two lithium concentrations are normalized here to the corresponding total number of valence electrons of the system. The area under the CPD thus gives the difference in the number of electrons for the two lithium concentrations, and we renormalize the CPD to one when we consider it as a projection of the momentum density of a redox orbital 30 . This normalization to one allows comparison of shape of the CPD for redox orbitals in various cathode materials 4,22,28 .
In previous work 4 , the redox orbital associated with lithium intercalation in the spinel Li x Mn 2 O 4 (LMO) was observed to have mostly oxygen 2p character. A negative excursion was also seen between 1 and 4 atomic units (a.u.) in the CPD (for x = 1.079 and 0.496), and was accounted for in terms of the transfer of some 3d electrons of manganese from localized to less-localized 3d states. This transfer induces a modulation of about 0.16 electrons in the manganese 3d shell. The resulting delocalization could be explained by the formation of the covalent Mn-O bond displayed in the partial density of states calculated within the DFT 4 . Figure 1 reveals the presence of oxygen 2p anionic states in momentum space in LTMO. The long tail in the CPD can be explained if about 0.19 3d manganese electrons are transferred from the more-itinerant to less-itinerant 3d states (see the fit in Extended Data Fig. 2). This localization is produced by the Coulomb repulsion between an occupied state at the oxygen site and the 3d states in the neighbouring manganese ions, without the formation of a substantial covalent bond. Clearly, this anionic redox paradigm is different from the usual process examined in spinel LMO 4,22 , where oxygen 2p states mix with transition metal 3d states through the formation of the covalent bond between the oxygen and transition metal atoms. It is crucial to understand the difference between the anionic redox mechanisms in LTMO and LMO, as the number of electrons-and thus the capacity-from the mixed oxygen 2p and transition metal 3d states remains the same, independent of the oxygen character of these states. Therefore, in contrast with anionic oxygen 2p redox orbitals, oxygen redox participation through bonding with a transition metal is not an effective way to increase capacity 9 .
Our first-principles CPD model agrees well with our experiment, and thus our theoretical model contains the right level of correlation effects 20 to capture the main features of the electronic structure and the corresponding distribution of electron momentum density in LTMO. Clearly, the CPD highlights modifications in electron occupancy near the Fermi energy that are associated with lithium insertion or extraction, by eliminating the contributions of irrelevant electrons 4 . The oxygen 2p orbital contributes to the electron momentum density at low momenta, while the contribution of manganese 3d orbitals extends to high momenta (see Fig. 1 inset). Note that when the lithium atoms are completely ionized in the cathode material, they do not carry any valence electrons and cannot contribute to the valence Compton profile. The experimental profile is still somewhat narrower at small momenta. The reason for this is not clear, but it could be due to a slight delocalization of states induced by the presence of contacts between the neighbouring LTMO powder grains-an effect that is not included in our simulations. Related delocalization effects have been observed in the Doppler spectra in positron annihilation studies of semiconducting nanoparticles 34 .  We decided to measure Compton scattering profiles at x = 0.8 and 0.4, as we found theoretically that the energy level of the dominant oxygen state in LTMO lies around the Fermi energy and that its properties could be captured with these two lithium concentrations (Extended Data Fig. 3). The partial density-of-states (PDOS) obtained from our DFT calculations for x = 0.4 (Fig. 2) shows the presence of a peak of a localized hole state right above the Fermi level. These localized oxygen 2p holes point in the direction of a lithium-atom vacancy along the Li-O-Li axis 15 . In principle, a π-type interaction exists between the oxygen 2p and manganese 3d states 12 . Crystal orbital overlap population (COOP) analysis (Extended Data Fig. 4) indicates that the localized oxygen 2p states near the Fermi level involve an antibonding interaction with the manganese 3d states. However, when the lithium ion fills the vacancy and the companion electron occupies the oxygen 2p orbital, the π-type interaction is weak and almost non-bonding, as also shown in ref. 12 . Thus, the oxygen 2p orbital near the Fermi level appears to be orphaned. Figure 3 presents two-dimensional electron momentum densities (2D-EMDs) for the redox orbitals, which are one-dimensional integrals along a crystallographic axis of the three-dimensional EMD. These maps give information about the extent to which solid-state orbitals are modified by the chemical bonds. Experimentally, 2D-EMDs can be extracted by measuring Compton profiles for a number of different directions of the X-ray scattering vector; the resulting spectra can be used to reconstruct the 2D-EMDs and to investigate their differences with different dopant concentrations 30 . However, in our case the LTMO cathode is composed of a polycrystalline powder sample, and therefore it was natural for us to base our analysis on spherically averaged experimental Compton profiles. Nevertheless, we have been able to extract the angular dependence of the redox orbital through a careful strategy in which theoretical EMD simulations were used to model the spherically averaged experimental profile. We invoked Bayes' theorem 32,35 to justify this reconstruction process, yielding a good fit with the experimental data, where effects of electron correlations could be varied in the modelling 28 .
Recall that various electronic states carry their own angular characters that can be exploited for their detection. Along this line, features in Fig. 3a reveal the presence of repulsive Coulomb (non-bonding) interactions between the oxygen 2p and manganese t 2g orbitals. Sharp box-like features (coloured light blue) in the high-momentum region are associated with Fermi surface breaks related to the t 2g band. The repulsive Coulomb interaction produces a substantial delocalization of oxygen 2p orbitals in momentum space (or localization in real space) in comparison with the redox orbital in the spinel material Li x Mn 2 O 4 (Fig. 3b). In the spinel case, the oxygen 2p orbitals are modified by the covalent bond involving the e g states on the manganese atoms, which induces them to localize in momentum space (or to delocalize in real space). The corresponding e g contribution (coloured in green) is also more localized in the momentum space. The pure manganese t 2g character shown in Fig. 3c could be extracted by performing magnetic Compton scattering experiments 29 . This visualization of the repulsive Coulomb interaction between the oxygen 2p orbital with the t 2g states could be used with other lithium-rich cathode materials to describe the chemical hardness 13 of the localized non-bonding oxygen 2p states (see Fig. 3d). Displacement of the t 2g electrons owing to Coulomb interaction produced by the occupation of the orphaned oxygen 2p state is a key parameter for understanding the origin of the voltage hysteresis 10,13 , as this quantity describes an energy penalty. Therefore, our ability to estimate this number provides a useful descriptor for understanding the working of the battery, and paves the way to improving existing lithium-rich materials and designing new high-capacity cathodes.

Article
We have provided conclusive evidence in support of the anionic redox mechanism in LTMO, and have ruled out alternative explanations based on the oxidation number of manganese. In particular, we have visualized the orphaned oxygen 2p redox orbitals, which are responsible for achieving higher energy densities by moving beyond the limit set by the transition metal content. We have shown that the key interaction between the oxygen 2p and manganese t 2g orbitals is the Coulomb repulsion, which localizes about 0.19 manganese t 2g electrons per lithium. Therefore, because of the absence of the covalent bond in favour of Coulomb repulsion between the orphaned oxygen 2p and manganese t 2g electrons, we can adduce that there is no redox involvement of manganese in LTMO over the investigated lithium doping range. Visualizing the interaction between the oxygen 2p and transition metal t 2g electrons, to estimate the number of electrons displaced by the Coulomb interaction, provides useful descriptors for designing stable, high-capacity oxygen-redox electrode materials. Our study enables a direct visualization of the orbitals involved in the anionic redox processes, and offers quantitative analyses based on new descriptors derived from the electron momentum density.

Online content
Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-021-03509-z.

Compton experiments
Compton profile measurements were carried out using a Cauchois-type X-ray spectrometer on the BL08W beamline at the SPring-8 synchrotron facility ( Japan) [36][37][38] . Incident X-rays emitted from a multipole wiggler were used. The incident X-ray energy was tuned to 114.56 keV with a bent-type Si(400) crystal. The size of the incident X-ray beam was 2 mm 2 at the sample position. The sample pellet covered by the laminate film was arranged in the vacuum chamber. The scattering angle was fixed at 165°. The Compton scattered X-rays, which transmitted Ge(620), were measured using a 2D position-sensitive detector. The raw Compton profiles were corrected for absorption, analyser and detector efficiencies, scattering cross-section, possible double-scattering contributions and X-ray background 32 . Corrected Compton profiles were then normalized to the total number of valence electrons after the core electron contribution was subtracted. The core electron configurations were taken to be as follows: lithium 1s; titanium 1s, 2s, 2p, 3s and 3p; manganese 1s, 2s, 2p, 3s and 3p; and oxygen 1s and 2s. The valence electrons were estimated by subtracting core electrons from the full electronic configurations of each atom. The overall momentum resolution in the measurements was 0.14 atomic units (a.u.) (full-width-at-half-maximum).

Sample preparation
The polycrystalline sample of Li 1.2 Ti 0.4 Mn 0.4 O 2 was prepared as in ref. 10

First-principles calculations
First-principles calculations were performed using the pseudopotential projector-augmented-wave method 39 as implemented in the Vienna ab-initio simulation package (VASP) 40,41 , with a kinetic energy cutoff of 600 eV for the plane-wave basis set. The exchange-correlation functional was treated within the generalized gradient approximation (GGA) 42 with a correlation correction given by the parameter U = 5 eV (ref. 20 ) on the manganese d orbitals, which gives results similar to those given by the recently constructed strongly-constrained-and-appropriately-normed (SCAN) meta-GGA 43 functional. The supercell used in LTMO calculations was Li 30x Ti 12 Mn 12 O 60 , where lithium, titanium and manganese ions are randomly distributed on the cationic sites and the oxygen ions occupy the anionic sites. The number of lithium atoms in the supercell (Li 30x ) was adjusted with respect to lithium concentrations (x). These disordered structures at different lithium concentrations (x) were generated with a Monte Carlo algorithm implemented in the special quasirandom structure (SQS) code 44 , which is available in the open-source ATAT toolkit 45 . For energy optimization and DOS calculations, a Monkhorst-Pack k-point mesh of 3 × 1 × 7 was used to sample the Brillouin zone of the supercell. The equilibrium positions of the ions were calculated via structural optimization, where the internal degrees of freedom were allowed to vary until the residual forces on each atom were less than 0.05 eV A −1 . All calculations considered ferromagnetic ordering of the ground state with default VASP magnetic moments.
We calculated the 3D electron momentum density (EMD), ρ(p), and the spherically averaged Compton profiles, J(p), of the valence electrons from the Kohn-Sham orbitals as in ref. 46 . For this purpose, we first calculated the 3D-EMD over a dense k-point mesh to accurately capture the fine structure in the momentum density. The J(p) is then obtained from the spherically averaged 3D-EMD, ρ(p), using the formula: p ∞ This scheme was used recently 28 to study the Compton profile of lithium iron phosphate LiFePO 4 , an exemplar cathode battery material.

Experimental Compton profiles
The experimental Compton profiles shown in Extended Data Fig. 1 code the information about all occupied electronic orbitals in the material through their EMDs 27 . A spherically averaged Compton profile is related to a spherical average of the EMD as in equation (1). Extended Data Fig. 1 shows that the width of the Compton profile narrows with lithium concentration. Therefore, measures of this width can be used to estimate the state of charge of the cathode 23 .
Note that signatures of the redox orbitals in the Compton profiles of LTMO shown in Extended Data Fig. 1 are difficult to see straightforwardly because the contribution of the redox orbitals sits on a large 'mountain' of filled states. The contribution of the redox orbitals can, however, be isolated by taking the difference between the Compton profiles for two different lithium concentrations (x). This is the reason that the CPD in Fig. 1 is, in fact, the signature of the redox orbitals of LTMO in the momentum space. With our spherically averaged experimental Compton profiles (Extended Data Fig. 1), we thus obtain the experimental CPD data shown in Fig. 1, which represents the integrated spherically averaged EMD associated with the redox orbitals.

Atomic orbital Compton profiles and fitting
Extended Data Figure 2 shows our curve-fitting analysis of the CPD. This analysis is based on use of three distinct model profiles: an atomic lithium 2s Compton profile 47 , an atomic oxygen 2p Compton profile 47,48 and a Compton profile modulation, called the Coulomb profile, which accounts for effects of 3d orbital localization in real space due to Coulomb interaction at manganese sites 49 . The combination of these model profiles results in the fitting curve (red line). The atomic Compton profiles are calculated using Slater-type orbitals, which are characterized by an effective exponent, Z eff . For example, the Slater-type orbital of xy symmetry is given by ψ xy = Nxye −Z eff r , where N is the normalization factor and r is the radial distance. We allowed variational freedom to Z eff in optimizing the fit. For the manganese 3d orbital, values of Z eff (4.3 and 3.0) have been chosen to reproduce the shape of the experimentally observed long tail in the CPD. Manganese 3d orbital Compton profiles corresponding to these Z eff values are shown in the inset of Extended Data Fig. 2. The manganese 3d Compton profile for Z eff = 4.3 (green curve in inset) is more localized in real space (less localized in momentum space) than that for Z eff = 3 (purple curve in inset), and their difference gives the Coulomb profile (yellow line in Extended Data Fig. 2).
The fitting procedure for the manganese 3d orbital localization is further justified by an analysis of the reciprocal form factor 50 , B(r), which is the Fourier transform of the CPD. The experimental and fitted B(r) shapes agree very well over distances near the origin that correspond to the range of the 3d orbitals (Extended Data Fig. 5). We also estimate the B(r) for the experimental Coulomb profile by subtracting the B(r) of the atomic oxygen 2p orbital from the experimental B(r). This procedure allows us to determine an exponent for the oxygen 2p orbital (Z eff = 2.12) by minimizing the distance between the experimental Coulomb profile B(r) (black dashed line in Extended Data Fig. 5) and the B(r) contributions of the model Coulomb profile (yellow line in Extended Data Fig. 5). Note that both manganese and oxygen are embedded in the cathode material. The modification of the exponent Z eff for the 3d orbitals is an effective way to mimic the mechanism proposed in ref. 12 . In this mechanism, Coulomb correlation effects related to the occupation of oxygen 2p states lead to an increase in the Z eff for the manganese 3d electrons involved in a π-type interaction. Consequently, two values of Z eff can coexist in the manganese 3d shell of LTMO.
In order to check the extent to which the lithium 2s orbital contributes to the CPD, we carried out a Bader charge analysis 51 by taking a lithium concentration of x = 0.8, and found that no more than 1.2% of 2s electrons remain on lithium atoms in this case, indicating that the lithium atoms are essentially fully ionized. This small lithium 2s character (Z eff = 1.28) marginally contributes to the fitting curve below 1 a.u., as seen from the cyan curve in Extended Data Fig. 2. Notably, the lithium contribution has little effect on our estimate of the manganese 3d electrons displaced via the Coulomb interaction, which is the main purpose of the fit. The reason is that the lithium 2s momentum density is largely localized at low momenta, while the Coulomb profile contributes mainly in the high-momentum region.
The fitted curve shows the presence of a concave structure (dip) at around 1.5 a.u. in the CPD in Extended Data Fig. 2. We attribute this dip feature as originating from the localization of manganese 3d states. The reason is that the model Coulomb profile (yellow line in Extended Data Fig. 2) is rather flat up to about 1 a.u., and begins to develop a slope starting around 1.5 a.u. When the Coulomb profile is combined with the atomic oxygen 2p Compton profile (blue line in Extended Data Fig. 2), we obtain the dip feature around 1.5 a.u in the total profile (red curve). Moreover, as the Coulomb profile is a modulation in momentum space produced by Coulomb repulsion between the electrons, it contributes to the long tail in the CPD. This modulation integrates to zero, but its absolute value is two times the number of electrons displaced via Coulomb repulsion in the presence of an electron occupying the oxygen 2p orbital, where the CPD is normalized to one.

PDOS and COOP analysis
The importance of the oxygen 2p orbital can be seen with reference to the PDOS in Extended Data Fig. 3. The oxygen 2p orbital for a lithium concentration of x = 0.4 lies above the Fermi level (see main text), while for x = 0.8 this orbital is located below the Fermi energy. As the Compton profile includes only the momentum density of occupied electrons, the CPD codes the difference in the occupation of the oxygen 2p orbital between x = 0.8 and x = 0.4. The contribution of the oxygen 2p orbital to the CPD then follows immediately from the dominant oxygen 2p character of the PDOS. The COOP analysis of the oxygen 2p orbital of x = 0.4 (Extended Data Fig. 4) confirms the antibonding character of this orbital in terms of a π-type interaction with manganese t 2g orbitals, in agreement with ref. 12 . We carried out our COOP analysis using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program 52 .

Data availability
The experimental data and theoretical simulations that support the findings of this study are available from the corresponding authors upon reasonable request. was obtained analytically to take into account the effects of localization of the atomic manganese 3d orbitals using normalized Slater-type orbitals. The inset shows manganese 3d Compton profiles for two different effective values of Z eff . The fitting curve (red) and ∆J are both normalized to one. The experimental error bars were obtained from a statistical analysis of the curves in Extended Data Fig. 1 and have a total length of 2σ.