Direct visualization of orbital electron occupancy

Orbital is one of the primary physical parameters that determine materials’ properties. Currently, experimentally revealing the electron occupancies of orbitals under the control of external field remains demonstrated the visualization of the real-space orbital occupancy by choosing LiCoO 2 as a prototype. Through multipole modelling of the accurately measured structure factors, we found the opposite changes of Co t 2g and e g orbital occupancies under different electrochemical states which can be well-correlated with the CoO 6 octahedra distortion. This robust method provides a feasible route to quantify the real-space orbital occupancy on small-sized particles, and opens up a new avenue for exploring the orbital origin of physical properties for functional materials.

Materials' properties are usually tuned by physical parameters including lattice, charge, orbital and spin 1,2 . The lattice composed of the periodic arrangement of atoms is the structural basis to understand the physical properties of materials. The electrons have three attributes including charge, spin, and orbital, which give rise to a wide range of functional properties of materials such as electrical, magnetic, optical, and thermal properties along with the underlying crystal lattice 3,4 .
Understanding the origin of these distinctive functionalities in materials therefore critically relies on our ability to accurately measure both electron's degrees of freedom and lattice for revealing their complex interactions. Over the years, the remarkable progress in the microscopy, spectroscopy and diffraction has enabled to probe the lattice, charge and spin with high precision and sensitivity, which has made significant contributions to reveal fundamental physical mechanisms for the relevant functional properties that have made up the cornerstone of current electric devices (such as field-effect transistors, magnetic random-access memory and piezoelectric transducers) [5][6][7][8][9] . By contrast, much less advancement has been made on the detection and characterization of electron orbital on a specific atom in the materials. Although it is well known that orbital degree of freedom plays an important role in many novel physical phenomena (e.g., high temperature superconductivity, colossal magnetoresistance, metal-insulator transition, and topological states of matter as well as in (electro)chemistry [10][11][12] . Revealing the underlying real-space orbital physics will not only deepen our understanding on the emergent functionalities of materials and (electro)chemical processes, but also provide a new knob to tune the properties of functional materials.
Theoretically, density functional theory (DFT) has been developed to reveal the ground-state electron density and successfully explains the origin of the unique functionalities and predicts new properties for various materials 13 . Although great advance has been made on quantifying bonding interactions and electronic structures, it is limited by the computer processing power which can only calculate several unit cells at ground-state and may miss some chemical interactions under the external field in practice 14,15 . Over the past decades, improvements of methodology have made it possible to measure the charge density of materials. Although the orbital sensitivity of scanning tunneling microscopy (STM) with a well-controlled calibration of tip-sample distance has been demonstrated 16,17 , it can only visualize orbital relying on the freshly cleaved and atomically sharp surfaces under the ultra-high vacuum environment 18 . Thus, there are serious limitations in applications. Because the electron orbital represents the shape of the electron cloud in a solid, the most intuitive way to resolve the orbital configuration is to accurately measure the electron density distribution around the bonding atoms, which are coded in the structure factors through Fourier transformation. Compared with other methods which can be employed for structure-factor measurement, single-crystal X-ray diffraction (SCXRD) is the primary approach. Unfortunately, it suffers from defects, extinction effect, and absorption that degrade the measurement accuracy for low-order structure factors which carry most information of the valence electron density, especially for functional materials containing heavy elements 19 . Instead, electron diffraction is extinction-free and more sensitive at low scattering angles compared with XRD, which makes it more suitable for low-order reflections 20,21 . Besides, quantitative convergent-beam electron diffraction (QCBED) with nanometer-sized electron beam can be used to obtain structure factors from perfect crystal regions so that the dynamic diffraction theory for perfect crystal can be applied. Based on the manybeam dynamic diffraction theory, the intensity distribution of CBED patterns can be obtained through solving the Schrödinger equation under the periodic potential field in crystals, which is compared to experimental patterns to get low-order structure factors [22][23][24][25] . In addition, low temperature (100 K), short-wavelength (0.41343 Å) and high resolution (sinθ/λmax ≥ 1.2 Å -1 ) synchrotron powder X-ray diffraction (SPXRD) with negligible absorption effect can accurately measure the high-order structure factors and Debye-Waller factors of small-sized particles which are not accessible by SCXRD. Moreover, the acquisition of electron diffraction and SPXRD at the same temperature ensures that the thermal vibrations are the same, making the experimental results more accurate and precise. However, to the best of our knowledge, the combination of QCBED and SPXRD has not been realized. What is more, the quantitative topological analysis of the refined electron density based on the results from QCBED has also not been performed. Combing the strengths of QCBED and SPXRD, it becomes possible to accurately measure the shape of the valence electron clouds around bonding atoms and quantify the orbital occupancy to uncover the origin of properties for functional materials beyond the scope of the crystal lattice and charge under the control of external fields.
Here, we use LiCoO2 as a model material and apply the method described above to successfully resolve a mystery in its electrochemical properties from the orbital point-of-view. As is well-known, LiCoO2 is the first cathode material used for commercial LIBs and still dominates in the portable electronics market because of many unique advantages including high electron conductivity, high volumetric energy density as well as excellent cycle life 26 . However, only limited voltage can be applied to extract no more than 60% Li to maintain a reasonable cycle-reversibility. Much work focused on the charge compensation mechanism of LiCoO2 during electrochemical charging and found that oxygen is involved in the redox reaction in highly charged LiCoO2 27 . In addition, there have been many effective strategies to improve the cycle stability of LiCoO2 at high voltage [28][29][30] . can be accurately extracted from SPXRD through Rietveld refinement. In order to reduce systematic errors, low temperature (100K) was used for all data collection to minimize the thermal diffuse scattering and anharmonicity that contributes to the background and high-order Bragg reflections, and finally to enhance the signal-to noise ratio of diffraction data 32 . The low-order structure factors were accurately measured by QCBED at 100K to ensure that the X-ray and electron measurements were done at almost the same temperature. In the meantime, low-temperature measurements can reduce the beam damage during CBED acquisition. The initial structure parameters obtained by Rietveld refinement from SPXRD are employed for Bloch wave calculation, during which the thickness, beam direction and structure factors are treated as refinable parameters. The refinement was made by comparing the experimental intensity profile across CBED systematic rows with the calculated intensity using a goodness of fit criterion 20 . It is worth noting that these experimental structure factors are model independent 33 . Fig. 2 displays the five low-order structure factors measurements including (003), (01 ̅ 1), (006), (012), and (01 ̅ 4) for LiCoO2, where good agreement between the experimental intensity and the calculated one is reached. All refined low-order structure factors at different SOC are listed in Table S1-S4.
Atom-centered multipole expansion 19 , which is based on the spherical harmonic functions, has been proven to be successful to describe the real space non-spherical electron density, in which the electron density of each atoms is described as: where and are the core and valence electron densities, respectively. and ± are the population parameters of valence electron density and spherical harmonic density ± , respectively. and ′ are valence-shell contraction-expansion parameters. is the radial function. This method implicitly assigns each density fragment to the centered nucleus. Therefore, the shape of the observed electron density can be flexibly fitted by a sum of non-spherical pseudoatomic densities. These consist of a spherical-atom (or ion) electron density obtained from multi- the multipole modeling electron density 31 . The electron density topological analysis on BCPs illustrated in Table 1, indicates that the Co-O interaction is the closed-shell interaction 36  shows that the valence electrons occupy the t2g orbital rather than the eg orbital of the Co atom in the CoO6 octahedra, which agrees with the 3d-orbital populations of Co as seen in Table 2. However, with decreasing Li content, the accumulation and depletion of the valence electron density in terms of the t2g and eg orbital, respectively, becomes more and more inapparent as seen from Fig. 3B to Since the 3d-orbital electrons can be described in terms of atomic orbitals 39 : = ( ) ± , ( ) is the radial function, ± is the spherical harmonic function and is the population parameter of atomic orbital. This expression equals to the valence part of Eq.1.

Thus, the relationship between d-orbital occupancies and multipole population parameters is casted
in the form of a 15 × 15 matrix, and reduce to smaller size under different site symmetries 39 . In LiCoO2, the site symmetry of Co is 3 ̅ with a 4 × 4 matrix which converts the multipole populations to the eg and t2g orbital occupancies 40 , as shown in Table 2 What's more, the orbital rehybridization is related to the distortion of CoO6 octahedra ( Table S5). Especially when x ≥ 0.6, the total valence electrons of Co increase and that of O decrease, in line with the dramatic changes of lattice and electronic structure of the CoO6 framework ( Table 2 and   Table S5). These variations aggravate the structure degradation and lead to the irreversible capacity fading of LiCoO2 at high voltage. The authors declare no competing interests.

Methods
Sample preparation. Pristine polycrystalline LiCoO2 powder was bought from Alfa with a purity of 99.5%.
The delithiated samples were prepared using the electrochemical method. The high-loading electrodes (LiCoO2 loading mass ~100 mg) were charged/discharged to different states of charge in a Swagelok cell, with Li metal as the counter electrode (1M LiPF6 in ethylene, dimethyl carbonate).
Subsequently, the charged Swagelok cells were disassembled, and the obtained powder samples were washed three times with dimethyl carbonate before drying. The cell assembling/disassembling and powder washing/drying were carried out in an argon-filled glove-box.
The TEM samples were fabricated using focus ion beam (FIB) milling, FEI Helios 600i. To prevent surface damage from ion milling, a 40 nm thick carbon layer was deposited using thermal evaporation. On the top of the particle, a regular Pt protection layer was deposited with standard settings using an electron-beam at 5 kV, 86 pA for 300 nm thickness following by an ion-beam at To reduce the surface damage and the thickness of amorphous, low-energy focused Ar ion milling was conducted using Fischione 1040 NanoMill system.   Through multipole refinement and the quantitative topological analysis of the electron distribution, we can measure the occupancies of 3d orbital.  Table S5 for details). , ∇ and ∇ 2 denote the electron density as well as its gradient and Laplacian, respectively. The Hess eigenvectors is defined by the diagonalization of the symmetric matrix of the nine second derivatives of . Furthermore, 1 and 3 are the Hess eigenvalues perpendicular and parallel to the bond path at the critical point, respectively.