# Dynamic Neural Fields for Learning Atlases of 4D Fetal MRI Time-series

Zeen Chi1,2 Zhongxiao Cong1,211footnotemark: 1 Clinton J. Wang2 Yingcheng Liu2
Esra Abaci Turk3,4 P. Ellen Grant3,4 S. Mazdak Abulnaga2,4,5
Polina Golland2 Neel Dey2
1School of Information Science and Technology, ShanghaiTech University 2MIT
CSAIL
3Fetal-Neonatal Neuroimaging & Developmental Science Center, Boston Children’s
Hospital
4 Harvard Medical School 5Massachusetts General Hospital
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Equal contribution. Work
done while visiting MIT CSAIL.

###### Abstract

We present a method for fast biomedical image atlas construction using neural
fields. Atlases are key to biomedical image analysis tasks, yet conventional
and deep network estimation methods remain time-intensive. In this preliminary
work, we frame subject-specific atlas building as learning a neural field of
deformable spatiotemporal observations. We apply our method to learning
subject-specific atlases and motion stabilization of dynamic BOLD MRI time-
series of fetuses in utero. Our method yields high-quality atlases of fetal
BOLD time-series with $\sim$5-7$\times$ faster convergence compared to
existing work. While our method slightly underperforms well-tuned baselines in
terms of anatomical overlap, it estimates templates significantly faster, thus
enabling rapid processing and stabilization of large databases of 4D dynamic
MRI acquisitions. Code is available at https://github.com/Kidrauh/neural-
atlasing.

##  1 Introduction

Given biomedical image observations, constructing image atlases enables
morphometric analyses and registration to a common coordinate system. Current
conventional [6, 14, 16, 24, 26, 27] and deep learning methods [9, 10, 11, 31,
32] for atlas building yield high-quality atlases with accurate registration
at the cost of significant computation time. These computational costs
compound further when given subject-specific image time-series (e.g.,
longitudinal repeats) where a new atlas must be constructed for each subject
to enable motion stabilization and standardized analyses.

In the context of fetal image analysis, in-utero BOLD MRI time series can
track changes in fetal and placental oxygenation under induced maternal
hyperoxia to identify dysfunction and monitor fetal and maternal well-being
[2, 15, 25, 29]. However, the inter-timepoint motion caused by fetal movement
and maternal breathing necessitates nonlinear registration of the time series
to a common coordinate system for each individual subject to stabilize motion
prior to any analysis. To that end, this work presents a method for fast
subject-specific spatiotemporal atlasing.

We formulate atlas estimation as the learning of compactly-parameterized
dynamic neural fields [20, 21, 23, 28] to represent both the atlas and image-
to-atlas deformations. Using our proposed neural representation and training
strategy, we rapidly construct high-fidelity subject-specific atlases and
stabilize the motion present in BOLD MR images of fetuses in utero to enable
improved analyses of key BOLD time series-based fetal and maternal biomarkers
[29].

Figure 1: Architecture. Our method constructs neural fields for volume
registration and intensity estimation, which warp observations to an atlas
space and learn the atlas parameters, respectively.

##  2 Methods

Learning Neural Fields. Fig. 1 presents our method consisting of networks for
image-to-atlas deformation and atlas estimation. We use three neural fields,
each parameterized as a multi-resolution hash encoding followed by a small MLP
[19] for efficient processing. We further use stationary velocity fields (SVF)
to ensure diffeomorphic deformations [3, 4, 17]. The atlas is produced by an
encoder-decoder where the encoder consists of time-invariant (static) and
time-variant (intensity) functions that allow small changes in atlas
appearance to account for subtle topological changes.

Given spatial $\mathbf{x}=(x,y,z)$ and temporal $t\in T$ coordinates, the
registration field $\Psi_{\mathbf{R}}:\mathbb{R}^{4}\mapsto\mathbb{R}^{3}$
computes velocities $\mathbf{v(x)}$ which integrate to yield a diffeomorphic
displacement field $\mathbf{u(x)}$ between an image at time $t$ and the atlas,
such that the deformation between them is
$\bm{\varphi}(\mathbf{x})=\mathbf{u(x)}+\mathbf{x}$. On warping the image
coordinates into the atlas space, we query $\bm{\varphi}(\mathbf{x})$ from the
static field $\Psi_{\mathbf{S}}:\mathbb{R}^{3}\mapsto\mathbb{R}^{n}$ to get
the feature vector $\mathbf{v}_{static}\in\mathbb{R}^{n}$ encoding time-
invariant latent atlas features. We then query $(\bm{\varphi}(\mathbf{x}),t)$
from an intensity field
$\Psi_{\mathbf{I}}:\mathbb{R}^{4}\mapsto\mathbb{R}^{n}$ that yields
$\mathbf{v}_{intensity}\in\mathbb{R}^{n}$ encoding the latent intensity
differences between $\bm{\varphi}(\mathbf{x})$ in the atlas and $\mathbf{x}$
in the original image. An MLP
$\Psi_{\mathbf{D}}:\mathbb{R}^{n}\mapsto\mathbb{R}$ then decodes the fused
latent features and yields the estimated intensity $\hat{I}(\mathbf{x},t)$ of
the original image.

Table 1: Quantitative results of baseline comparisons (top) and ablations (bottom) studying registration performance (via local normalized cross-correlation and weighted dice), deformation qualities (via deformation magnitude, avg. Jacobian determinant, and folding ratio), and runtimes. |  LNCC ($\uparrow$) |  Wt. Dice ($\uparrow$) |  $\lVert\mathbf{u(x)}\rVert_{2}$ ($\downarrow$) | $|J_{\bm{\varphi}}|$ |  % folds ($\downarrow$) |  Runtime ($\downarrow$)
---|---|---|---|---|---|---
Unaligned | 0.392(0.073) | 0.80(0.05) | - | - | - | -
SyGN [7] | 0.528(0.075) |  0.91(0.02) | 0.0227(0.0035) | 1.000(0.000) | 0 | 12hrs / 96-core CPU
AtlasMorph [10] | 0.531(0.079) | 0.90(0.02) |  0.0083(0.0014) | 1.004(0.003) | 0 | 16hrs / A6000 GPU
Ours |  0.579(0.081) | 0.88(0.02) | 0.0183(0.0067) | 1.004(0.013) | 0.01(0.01) |  2.2hrs / A6000 GPU
(- SVF) | 0.503(0.081) | 0.85(0.04) | 0.0096(0.0021) | 1.006(0.010) | 0.04(0.02) | 1.1hrs / A6000 GPU
(- Divergence) | 0.579(0.078) | 0.87(0.02) | 0.0200(0.0063) | 1.013(0.012) | 0.06(0.04) | 1.5hrs / A6000 GPU
(- Intensity field) | 0.578(0.083) | 0.88(0.02) | 0.0209(0.0086) | 1.000(0.018) | 0.01(0.01) | 2.2hrs / A6000 GPU

Losses. We use the $L_{1}$ reconstruction objective
$\mathcal{L}_{rec}=\frac{1}{|\Omega|}\sum_{\mathbf{x}\in\Omega}|I(\mathbf{x},t)-\hat{I}(\mathbf{x},t)|$
where $\Omega$ is the spatial coordinates and $I$ and $\hat{I}$ are ground
truth and estimated intensities of the image, respectively. To encourage
smooth, locally-rigid, and central deformations, we develop the regularizer
$\mathcal{L}_{def}=\lambda_{1}\frac{1}{|\Omega|}\sum_{\mathbf{x}\in\Omega}\lVert\mathbf{u(x)}\rVert_{2}+\lambda_{2}\mathcal{L}_{div}+\lambda_{3}\lVert\mathbf{\bar{u}(x)}\rVert_{2}^{2}$,
where $\mathbf{\bar{u}(x)}$ is the moving average of displacement vectors [10]
and
$\mathcal{L}_{div}=\frac{1}{|\Omega|}\sum_{\mathbf{x}\in\Omega}|\mathrm{div}(\mathbf{u(x)})|^{2}$
is the divergence loss [30] that encourages locally-rigid deformations which
are essential to properly model fetal motion. To reduce folds in the computed
deformations, we use the negative Jacobian loss $\mathcal{L}_{jac}$ [18],
which reduces the number of negative elements in the determinant of the
Jacobian of the deformation. For intensity estimation, we use $L_{1}$
regularization $\mathcal{L}_{int}$ on $\mathbf{v}_{intensity}$ to limit
temporal appearance changes, and use total variation regularization
$\mathcal{L}_{tv}=\mathrm{tv}(\mathbf{v}_{static})+\mathrm{tv}(\mathbf{v}_{intensity})$
on $\mathbf{v}_{static}$ and $\mathbf{v}_{intensity}$ to encourage piecewise-
constant and sharp-edged atlases both spatially and temporally. Our overall
objective is
$\mathcal{L}(F)=\mathcal{L}_{rec}+\mathcal{L}_{def}+\lambda_{jac}\mathcal{L}_{jac}+\lambda_{int}\mathcal{L}_{int}+\lambda_{tv}\mathcal{L}_{tv}$
where $\lambda_{1}=10^{-3},\lambda_{2}=5\times
10^{-4},\lambda_{3}=0.1,\lambda_{jac}=1,\lambda_{int}=0.05$, and
$\lambda_{tv}=0.1$, chosen via grid search on two validation subjects.

Atlas Inference. To construct the final atlas (the single time-invariant
template) representing the entire time-series, we directly query
$(\mathbf{x},t)$ from the trained atlas encoder-decoder network (Fig. 1 right,
intensity estimation). We first calculate the static feature vector
$\mathbf{v}_{static}$ and the intensity feature vectors
$\mathbf{v}_{intensity}$ at each time step $t$ and then decode
$\mathbf{v}_{static}+\frac{1}{T}\sum_{t=1}^{T}\mathbf{v}_{intensity}$ using
$\Psi_{\mathbf{D}}$.

Figure 2: Given an arbitrarily chosen subject, we illustrate the mid-timepoint
of the time-series, the temporal linear average, and fetal atlases produced by
SyGN [7], AtlasMorph [10], and our method. Atlasmorph creates undesirable
checkerboard artifacts (indicated by red arrows).

##  3 Experiments

Data and Baselines. We use 11 dynamic BOLD MRI time-series of in utero fetal
subjects (2 for tuning hyperparameters and modeling decisions and 9 for held-
out testing) with a time-series length of 78 to 146 time points per subject.
Due to fetal motion and maternal breathing, there is a need for registration
of all images to a common unbiased subject-specific representation [1]. Each
image is resampled to $112\times 112\times 80$ at $3mm^{3}$ isotropic
resolution. As we use an intensity-based reconstruction loss, we use adaptive
histogram equalization [22] for inputs to our model to balance contributions
from bright and dark BOLD regions such as the amniotic fluid and fetal body,
respectively. We use SyGN [7] and AtlasMorph [10] as representative
conventional and deep network baselines, with local normalized cross-
correlation (LNCC) [5] as a registration loss which is locally-adaptive and
intensity scale-invariant by design. AtlasMorph and our method are trained on
a single NVIDIA RTX A6000 GPU and SyGN is optimized on a server CPU using 96
hyperthreaded cores.

Evaluation. Atlas building evaluation is subtle and involves trade-offs
between registration accuracy, deformation quality, and runtime [11]. To
measure performance, we follow [13] and randomly select 50 MRI pairs for each
subject and compose image-to-atlas and atlas-to-image warps to calculate LNCC
and multi-region Dice coefficients [12]. Our segmentation labels correspond to
the placenta, amniotic fluid, fetal body, fetal brain, and fetal eyes and are
generated by an in-house segmentation network. To assess deformation quality,
we calculate the average displacement $L_{2}$ norm between the atlas and
images with a lower value indicating improved template centrality, the mean
determinant of the Jacobian matrix $J_{\bm{\varphi}}(p)$ w.r.t. the input
voxel $p$, and the ratio of deformation folds.

Results. Table 1 reports LNCC and the weighted average Dice scores and
deformation statistics comparisons between the baselines and our model. All
methods produce invertible deformations. The proposed model achieves best-in-
class LNCC but lags behind slightly in terms of Dice score (i.e., anatomical
overlap). In terms of runtime, our proposed model converges $5.5-7.4\times$
faster than baselines yielding high-fidelity templates (see Fig. 2) with
smooth and invertible deformations. However, if the tuned baselines are
optimized to convergence, they currently yield improved anatomical overlap.
Ablations removing the SVF formulation, the divergence loss, and
$\Psi_{\mathbf{I}}$ all worsen performance.

##  4 Conclusions and Future Directions

We demonstrate that dynamic neural fields learn atlases of 4D fetal BOLD MRI
time-series significantly faster than current methods. These speed gains are
especially relevant to subject-specific atlas building of large collections of
subjects imaged using 4D dynamic MRI. Currently, our preliminary work finds
that well-tuned baselines optimized for longer still achieve better
registration overlap in terms of Dice. This performance gap points to several
future directions: (1) Fetal BOLD MRI time series are temporally sampled at
only $\sim$0.28 frames per second (FPS) as compared to conventional video (24+
FPS) for which existing work on dynamic neural fields was developed. This
gives rise to large, erratic motion between consecutive timepoints, and may
require modification to existing positional encoding functions which assume
temporal smoothness. (2) High-performing mono-modal biomedical image
registration frameworks typically use LNCC [8] as a registration loss.
However, due to the scale and shift-invariant formulation of LNCC, neural
regression networks trained with LNCC require significant regularization to
guide them towards non-degenerate solutions, which we find can introduce
significant artifacts in the estimated atlas. Future work may seek to mitigate
this tradeoff by constraining the optimization space of the network or using
data-driven priors.

##  5 Acknowledgements

We gratefully acknowledge funding from NIH NIBIB 5R01EB032708, NIH NICHD
R01HD100009, NIH NIA 5R01AG064027, and NIH NIA 5R01AG070988. We thank all the
participants for providing the data in the study.

## References

  * Abulnaga et al. [2022] S. M. Abulnaga, S. I. Young, K. Hobgood, E. Pan, C. J. Wang, P. E. Grant, E. Abaci Turk, and P. Golland.  Automatic segmentation of the placenta in bold mri time series.  In _Perinatal, Preterm and Paediatric Image Analysis: 7th International Workshop, PIPPI 2022, Held in Conjunction with MICCAI 2022, Singapore, September 18, 2022, Proceedings_ , pages 1–12. Springer, 2022.
  * Aimot-Macron et al. [2013] S. Aimot-Macron, L. Salomon, B. Deloison, R. Thiam, C. Cuenod, O. Clement, and N. Siauve.  In vivo mri assessment of placental and foetal oxygenation changes in a rat model of growth restriction using blood oxygen level-dependent (bold) magnetic resonance imaging.  _European radiology_ , 23:1335–1342, 2013.
  * Arsigny et al. [2006] V. Arsigny, O. Commowick, X. Pennec, and N. Ayache.  A log-euclidean framework for statistics on diffeomorphisms.  In _Medical Image Computing and Computer-Assisted Intervention–MICCAI 2006: 9th International Conference, Copenhagen, Denmark, October 1-6, 2006. Proceedings, Part I 9_ , pages 924–931. Springer, 2006.
  * Ashburner [2007] J. Ashburner.  A fast diffeomorphic image registration algorithm.  _Neuroimage_ , 38(1):95–113, 2007.
  * Avants et al. [2008] B. B. Avants, C. L. Epstein, M. Grossman, and J. C. Gee.  Symmetric diffeomorphic image registration with cross-correlation: evaluating automated labeling of elderly and neurodegenerative brain.  _Medical image analysis_ , 12(1):26–41, 2008\.
  * Avants et al. [2009] B. B. Avants, N. Tustison, G. Song, et al.  Advanced normalization tools (ants).  _Insight j_ , 2(365):1–35, 2009.
  * Avants et al. [2010] B. B. Avants, P. Yushkevich, J. Pluta, D. Minkoff, M. Korczykowski, J. Detre, and J. C. Gee.  The optimal template effect in hippocampus studies of diseased populations.  _Neuroimage_ , 49(3):2457–2466, 2010.
  * Avants et al. [2011] B. B. Avants, N. J. Tustison, G. Song, P. A. Cook, A. Klein, and J. C. Gee.  A reproducible evaluation of ants similarity metric performance in brain image registration.  _Neuroimage_ , 54(3):2033–2044, 2011.
  * Chen et al. [2021] L. Chen, Z. Wu, D. Hu, Y. Pei, F. Zhao, Y. Sun, Y. Wang, W. Lin, L. Wang, G. Li, et al.  Construction of longitudinally consistent 4d infant cerebellum atlases based on deep learning.  In _Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part IV 24_ , pages 139–149. Springer, 2021.
  * Dalca et al. [2019] A. V. Dalca, M. Rakic, J. Guttag, and M. R. Sabuncu.  Learning conditional deformable templates with convolutional networks.  _NeurIPS: Neural Information Processing Systems_ , 2019.
  * Dey et al. [2021] N. Dey, M. Ren, A. V. Dalca, and G. Gerig.  Generative adversarial registration for improved conditional deformable templates.  In _Proceedings of the IEEE/CVF international conference on computer vision_ , pages 3929–3941, 2021.
  * Dice [1945] L. R. Dice.  Measures of the amount of ecologic association between species.  _Ecology_ , 26(3):297–302, 1945.
  * Ding and Niethammer [2022] Z. Ding and M. Niethammer.  Aladdin: Joint atlas building and diffeomorphic registration learning with pairwise alignment.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 20784–20793, 2022.
  * Kuklisova-Murgasova et al. [2011] M. Kuklisova-Murgasova, P. Aljabar, L. Srinivasan, S. J. Counsell, V. Doria, A. Serag, I. S. Gousias, J. P. Boardman, M. A. Rutherford, A. D. Edwards, et al.  A dynamic 4d probabilistic atlas of the developing brain.  _NeuroImage_ , 54(4):2750–2763, 2011.
  * Luo et al. [2015] J. Luo, E. A. Turk, T. Hahn, M. Teulon Gonzalez, B. Gagoski, C. Bibbo, A. Palanisamy, C. Tempany, A. Torrado-Carvajal, N. Malpica, et al.  Human placental and fetal response to maternal hyperoxygenation in iugr pregnancy as measured by bold mri.  In _Proceedings of the 23rd Annual Meeting of ISMRM, Toronto, Ontario, Canada_ , page 633, 2015.
  * Makropoulos et al. [2016] A. Makropoulos, P. Aljabar, R. Wright, B. Hüning, N. Merchant, T. Arichi, N. Tusor, J. V. Hajnal, A. D. Edwards, S. J. Counsell, et al.  Regional growth and atlasing of the developing human brain.  _Neuroimage_ , 125:456–478, 2016.
  * Modat et al. [2012] M. Modat, P. Daga, M. J. Cardoso, S. Ourselin, G. R. Ridgway, and J. Ashburner.  Parametric non-rigid registration using a stationary velocity field.  In _2012 IEEE Workshop on Mathematical Methods in Biomedical Image Analysis_ , pages 145–150. IEEE, 2012.
  * Mok and Chung [2020] T. C. Mok and A. C. Chung.  Large deformation diffeomorphic image registration with laplacian pyramid networks.  In _Medical Image Computing and Computer Assisted Intervention–MICCAI 2020: 23rd International Conference, Lima, Peru, October 4–8, 2020, Proceedings, Part III 23_ , pages 211–221. Springer, 2020.
  * Müller et al. [2022] T. Müller, A. Evans, C. Schied, and A. Keller.  Instant neural graphics primitives with a multiresolution hash encoding.  _ACM Transactions on Graphics (ToG)_ , 41(4):1–15, 2022.
  * Park et al. [2021a] K. Park, U. Sinha, J. T. Barron, S. Bouaziz, D. B. Goldman, S. M. Seitz, and R. Martin-Brualla.  Nerfies: Deformable neural radiance fields.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 5865–5874, 2021a.
  * Park et al. [2021b] K. Park, U. Sinha, P. Hedman, J. T. Barron, S. Bouaziz, D. B. Goldman, R. Martin-Brualla, and S. M. Seitz.  Hypernerf: A higher-dimensional representation for topologically varying neural radiance fields.  _arXiv preprint arXiv:2106.13228_ , 2021b.
  * Pizer et al. [1987] S. M. Pizer, E. P. Amburn, J. D. Austin, R. Cromartie, A. Geselowitz, T. Greer, B. ter Haar Romeny, J. B. Zimmerman, and K. Zuiderveld.  Adaptive histogram equalization and its variations.  _Computer vision, graphics, and image processing_ , 39(3):355–368, 1987.
  * Pumarola et al. [2021] A. Pumarola, E. Corona, G. Pons-Moll, and F. Moreno-Noguer.  D-nerf: Neural radiance fields for dynamic scenes.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 10318–10327, 2021.
  * Rueckert et al. [1999] D. Rueckert, L. I. Sonoda, C. Hayes, D. L. Hill, M. O. Leach, and D. J. Hawkes.  Nonrigid registration using free-form deformations: application to breast mr images.  _IEEE transactions on medical imaging_ , 18(8):712–721, 1999.
  * Schöpf et al. [2012] V. Schöpf, G. Kasprian, P. C. Brugger, and D. Prayer.  Watching the fetal brain at ‘rest’.  _International Journal of Developmental Neuroscience_ , 30(1):11–17, 2012.
  * Schuh et al. [2015] A. Schuh, M. Murgasova, A. Makropoulos, C. Ledig, S. J. Counsell, J. V. Hajnal, P. Aljabar, and D. Rueckert.  Construction of a 4d brain atlas and growth model using diffeomorphic registration.  In _Spatio-temporal Image Analysis for Longitudinal and Time-Series Image Data: Third International Workshop, STIA 2014, Held in Conjunction with MICCAI 2014, Boston, MA, USA, September 18, 2014, Revised Selected Papers 3_ , pages 27–37. Springer, 2015.
  * Serag et al. [2012] A. Serag, V. Kyriakopoulou, M. A. Rutherford, A. D. Edwards, J. V. Hajnal, P. Aljabar, S. J. Counsell, J. Boardman, and D. Rueckert.  A multi-channel 4d probabilistic atlas of the developing brain: application to fetuses and neonates.  _Annals of the BMVA_ , 2012(3):1–14, 2012.
  * Song et al. [2022] L. Song, A. Chen, Z. Li, Z. Chen, L. Chen, J. Yuan, Y. Xu, and A. Geiger.  Nerfplayer: A streamable dynamic scene representation with decomposed neural radiance fields.  _arXiv preprint arXiv:2210.15947_ , 2022.
  * Sørensen et al. [2013] A. Sørensen, D. Peters, C. Simonsen, M. Pedersen, B. Stausbøl-Grøn, O. B. Christiansen, G. Lingman, and N. Uldbjerg.  Changes in human fetal oxygenation during maternal hyperoxia as estimated by bold mri.  _Prenatal diagnosis_ , 33(2):141–145, 2013.
  * Tretschk et al. [2021] E. Tretschk, A. Tewari, V. Golyanik, M. Zollhöfer, C. Lassner, and C. Theobalt.  Non-rigid neural radiance fields: Reconstruction and novel view synthesis of a dynamic scene from monocular video.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 12959–12970, 2021.
  * Zhang et al. [2016] Y. Zhang, F. Shi, G. Wu, L. Wang, P.-T. Yap, and D. Shen.  Consistent spatial-temporal longitudinal atlas construction for developing infant brains.  _IEEE transactions on medical imaging_ , 35(12):2568–2577, 2016.
  * Zhao et al. [2021] F. Zhao, Z. Wu, L. Wang, W. Lin, S. Xia, G. Li, and U. B. C. P. Consortium.  Learning 4d infant cortical surface atlas with unsupervised spherical networks.  In _Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part II 24_ , pages 262–272. Springer, 2021.


# Soft-Chemical Synthesis, Structure Evolution, and Insulator-to-Metal
Transition in a Prototypical Metal Oxide, $\lambda$-RhO2

Juan R. Chamorro  Materials Department and Materials Research Laboratory,
University of California, Santa Barbara, California 93106, USA
<EMAIL_ADDRESS>Julia L. Zuo  Materials Department and Materials Research
Laboratory, University of California, Santa Barbara, California 93106, USA
Euan N. Bassey  Materials Department and Materials Research Laboratory,
University of California, Santa Barbara, California 93106, USA  Aurland K.
Watkins  Materials Department and Materials Research Laboratory, University of
California, Santa Barbara, California 93106, USA  Guomin Zhu  Materials
Department and Materials Research Laboratory, University of California, Santa
Barbara, California 93106, USA  Arava Zohar  Materials Department and
Materials Research Laboratory, University of California, Santa Barbara,
California 93106, USA  Kira E. Wyckoff  Materials Department and Materials
Research Laboratory, University of California, Santa Barbara, California
93106, USA  Tiffany L. Kinnibrugh  X-ray Science Division, Advanced Photon
Source, Argonne National Laboratory, 9700 S. Cass Ave, Argonne, Illinois
60439, USA  Saul H. Lapidus  X-ray Science Division, Advanced Photon Source,
Argonne National Laboratory, 9700 S. Cass Ave, Argonne, Illinois 60439, USA
Susanne Stemmer  Materials Department and Materials Research Laboratory,
University of California, Santa Barbara, California 93106, USA  Raphaële J.
Clément  Materials Department and Materials Research Laboratory, University of
California, Santa Barbara, California 93106, USA  Stephen D. Wilson  Materials
Department and Materials Research Laboratory, University of California, Santa
Barbara, California 93106, USA  Ram Seshadri  Materials Department and
Materials Research Laboratory, University of California, Santa Barbara,
California 93106, USA<EMAIL_ADDRESS>
###### Abstract

$\lambda$-RhO2, a prototype 4d transition metal oxide, has been prepared by
oxidative delithiation of spinel LiRh2O4 using ceric ammonium nitrate.
Average-structure studies of this RhO2 polytype, including synchrotron powder
X-ray diffraction and electron diffraction, indicate the room temperature
structure to be tetragonal, in the space group $I4_{1}/amd$, with a first-
order structural transition to cubic $Fd\bar{3}m$ at $T$ = 345 K on warming.
Synchrotron X-ray pair distribution function analysis and 7Li solid state
nuclear magnetic resonance measurements suggest that the room temperature
structure displays local Rh–Rh bonding. The formation of these local dimers
appears to be associated with a metal-to-insulator transition with a non-
magnetic ground state, as also supported by density functional theory-based
electronic structure calculations. This contribution demonstrates the power of
soft chemistry to kinetically stabilize a surprisingly simple binary oxide
compound.

##  1 Introduction

In the presence of strong spin-orbit coupling, as expected in 4d and 5d
transition metals, a d5 electronic configuration can give rise to an effective
j = 1/2 ground state with enhanced correlations, paving the way for exotic
long-range macroscopic quantum states such as in $\alpha$-RuCl3 1, 2, 3, 4, 5,
Sr2IrO4 6, 7, 8, 9, Sr3Ir2O7 10, 11, 12, and A2IrO3 (A = Li, Na) 13, 14, 15,
16, 17, 18, 19, 20. Compounds containing tetravalent rhodium (Rh4+, 4d5),
which is isoelectronic to Ru3+ and is the 4d analogue of Ir4+, should possess
electronic and magnetic properties similar to those observed in other d5
compounds. However, studies of the chemistry and physics of Rh4+ in the solid
state phase space remain rather limited due to the relatively greater
stability of the trivalent state (Rh3+, 4d6). In RhO6 octahedra (the most
common coordination environment in rhodium oxides, owing to cation size
effects), Rh3+ assumes the low-spin d6 configuration, with fully filled t2g
states. This stable electronic configuration results in an enhancement of the
Rh-O bond strength that is comparatively destabilized in Rh4+, which instead
possesses a hole in the low-energy t2g manifold.

A search of the Inorganic Crystal Structure Database (ICSD) yields a total of
around 266 oxide compounds containing Rh3+, versus 30 unique oxides containing
Rh4+. The difficulty in synthesizing oxide compounds with Rh4+ originates from
the high oxidative potential required to oxidize Rh3+ to Rh4+, as well as the
predilection for RhO2 to vaporize21, 22. These issues have been addressed by
performing syntheses under high oxygen pressures that can stabilize higher
oxidation states and arrest volatilization, and have yielded a number of
rhodium(IV) oxides such as ARhO3 (A = Ca, Sr, and Ba) 23, 24, 25, Sr3Rh2O7 26,
and Sr4Rh3O10 27. In fact, the synthesis of RhO2 in either its rutile 28 or
cubic 29 forms requires high oxygen pressure in order to form.

Of the few reported Rh4+ oxide compounds in the literature, only a small
number have been demonstrated to possess phenomena similar to other
aforementioned d5 materials, such as in the correlated electron metal Sr2RhO4
30, 31, 32, spin-glass Li2RhO3 33, 34, 35, and the mixed-valent, Rh3+/Rh4+
spinel LiRh2O4 36, 37, 38. In contrast to other d5 systems, however, spin-
orbit coupling appears to play a near negligible role in establishing the
ground state of Rh4+ oxides, and other single-ion instabilities, such as Jahn-
Teller distortions, often play a much larger part. This is the case for mixed-
valent LiRh2O4, which crystallizes at room temperature in the prototypical
cubic Fd$\bar{3}$m spinel structure, but distorts at low temperature into a
tetragonal cell owing to a Jahn-Teller instability of the Rh4+ octahedra, and
then to an orthorhombic cell on further cooling due to charge-ordering 37, 38.
While many rhodium oxide spinels exist, including CoRh2O4 39, 40, NiRh2O4 41,
42, and CuRh2O4 40, 43, LiRh2O4 is the only one possessing any Rh4+.

The Rh cations in LiRh2O4 form a three-dimensional pyrochlore network, and
thus interactions between them can be destabilized by geometric frustration. A
total topotactic delithiation of LiRh2O4 to Rh2O4 would result in a pyrochlore
network of exclusively Rh4+, which would be a potential j = 1/2 Rh analog of
the RE2Ir2O7 (RE = Y, Pr$-$Lu) pyrochlore iridates. Some of these pyrochlore
iridates display metal-to-insulator transitions 44, 45, 46 arising from strong
interactions between Ir cations, and possess non-trivial band topologies that
give rise to various exotic states such as Weyl semimetal 47, 48, 49, 50 and
Luttinger liquid 51, 52, 53 states. In order to synthesize Rh2O4, we sought to
topotactically remove Li+ cations from LiRh2O4 using electrochemical cells and
chemical agents. Initial tests using these cells and common solution-based
delithiating oxidants such as Br2 and I2 in acetonitrile did not reveal
obvious changes to the crystallographic structure, as determined by X-ray
diffraction (XRD). We therefore turned to other chemical oxidants with
oxidation potentials greater than those of Br2 and I2, in order to overcome
the aforementioned Rh${}^{3+}\rightarrow$ Rh4+ redox barrier.

In this work, we report the topotactic, oxidative delithiation of LiRh2O4 to
form a new rhodium (IV) oxide, $\lambda$-RhO2. Fashioned after $\lambda$-MnO2,
which was also obtained via soft, chemical delithiation of LiMn2O4 spinel in
acid54, we have employed the use of ceric ammonium nitrate (NH4)2Ce(NO3)6 to
remove nearly all of the lithium cations in LiRh2O4 topotactically, retaining
the parent spinel architecture. 7Li solid-state nuclear magnetic resonance
(NMR) measurements indicate that the nominal lithium content is 84(1)% reduced
as compared to the parent compound. To our knowledge, this is the first
reported application of ceric ammonium nitrate, a powerful shelf-stable
oxidizer with an oxidizing potential superior to that of elemental chlorine,
for the topotactic oxidative delithiation of an extended solid. Our results
indicate that $\lambda$-RhO2 is a metal at high temperatures in excess of T =
350 K and crystallizes in the cubic Fd$\bar{3}$m space group, while it
undergoes a hysteretic metal-to-insulator transition on cooling that reduces
the average structure to tetragonal I41/amd. This transition is accompanied by
the formation of short-range Rh-Rh dimers, formed through direct metal-metal
bonding, and results in a non-magnetic ground state. This work expands the
ARh2O4 rhodium oxide spinel phase space to the extreme limit where A is
occupied by a vacancy.

##  2 Results and Discussion

###  2.1 Topotactic Oxidative Delithiation Reactions

Oxidative delithiation of LiRh2O4 was performed using ceric ammonium nitrate
(CAN), a reagent commonly used in other fields of synthetic and industrial
chemistry 55, 56, 57, 58, but rarely used in synthetic solid state chemistry.
It is a powerful, one electron oxidizing agent that is driven by the single
electron process Ce4+ \+ e- $\rightarrow$ Ce3+, which has a reduction
potential of E∘ = 1.61 V 55. This potential is higher than that of other
commonly used powerful oxidizers for the topotactic deintercalation of
extended solids, such as Cl2 and Br2 (E∘ = 1.36 and 1.08 V, respectively) 59,
and is on par with permanganates (MnO4-, 1.51$-$1.67 V). CAN is a non-
hygroscopic, shelf-stable compound that can be readily handled in air and is
soluble in aqueous and organic solvents. Its use in low temperature chimie
douce reactions remains largely unexplored in the synthetic materials
chemistry literature, and our findings present a case where CAN can be
employed to stabilize a rare, tetravalent oxidation state in rhodium.

This article is focused primarily on the nearly-fully-delithiated end member
of the Li1-xRh2O4 phase space, $\lambda$-RhO2 (Li0.1(1)Rh2O4). This compound
is synthesized using an excess amount of CAN in water, whereas other
Li1-xRh2O4 compounds are synthesized by selecting a target CAN concentration
in each solution per mol of LiRh2O4. The proposed chemical equation for the
oxidative delithiation reaction is:

LiRh2O4 \+ x(NH4)2CeIV(NO3)6 $\rightarrow$ Li1-xRh2O4 \+ x(NH4)2CeIII(NO3)5 \+
xLiNO3

Based on this reaction, one mole of CAN is needed to fully delithiate LiRh2O4.
We prepared samples of Li1-xRh2O4 at x = 0.1 intervals, and performed
synchrotron X-ray diffraction measurements at T = 400 K, as discussed in more
depth further in this text, in order to track their lattice parameters as a
function of targeted delithiation. At this temperature, every sample had a
cubic structure, and the cubic lattice parameter as a function of the relative
CAN/LiRh2O4 amount is shown in Figure 1(a).

Figure 1: (a) Lattice parameter of cubic Li1-xRh2O4 as a function of x, where
x is the relative molar amount of CAN to LiRh2O4 in each reaction. The trend
is linear and intercepts with the $\lambda$-RhO2 lattice parameter at x = 1.5.
(b)-(d) Results of operando X-ray diffraction measurements. The cell voltage
and lattice parameter of Li1-xRh2O4 increases on charging (removal of Li+ from
the lattice) up to approximately x = 0.45, beyond which the voltage and
lattice parameter begins to decrease.

The lattice parameter of the cubic Li1-xRh2O4 cell increases approximately
linearly with increasing targeted x. However, $\lambda$-RhO2, prepared with
CAN in excess, has a cubic lattice parameter at T = 400 K that is achieved
when using a 1.5CAN:1.0LiRh2O4 molar ratio, suggesting that the above
delithiation reaction is incomplete. Further work is required to understand
the reaction mechanism, including the thermodynamic and kinetic barriers faced
under these strong oxidizing conditions.

We also attempted to synthesize Li1-xRh2O4 phases with variable Li content
using other reagents and electrochemistry. Reactions in either Br2 or I2
solutions in acetonitrile, common oxidative deintercalation agents 60, 61, 62,
63, did not yield any noticeable differences in diffraction patterns of
Li1-xRh2O4 targeted samples vs. LiRh2O4, nor significant variations in
measurements of the low temperature physical properties. We also prepared an
electrochemical cell of LiRh2O4 vs. Li-metal in order to test electrochemical
delithiation. Cells were discharged to 1 V (lithiation) and then charged to 3
V (delithiation). As shown in Figure 1(b)$-$(d), the voltage of the cell
quickly approaches an extended plateau at 4.5 V with a voltage profile similar
to that observed in Mn- and Ni- containing spinels, albeit at a lower voltage
64. The removal of lithium from Li1-xRh2O4 results in an increase in the
lattice parameter upon delithiation, marked by a shift in the diffraction
peaks to lower angles. However, at approximately Li0.55Rh2O4, both the voltage
and lattice parameter begin to anomalously decrease, likely due to a breakdown
of the cell electrolyte. As such, the electrochemical method employing an
LiPF6 electrolyte is insufficient in removing more than 45% Li from the
structure. Results from refinements of the operando X-ray diffraction
measurements are shown in the supplementary information (Figure S1) and
further indicate that $\lambda$-RhO2 cannot be obtained electrochemically, as
revealed by the absence of XRD reflections associated with the tetragonal
$\lambda$-RhO2 phase, discussed in more detail below.

###  2.2 Average Structure, and Structure Evolution

Rietveld refinements were performed on X-ray diffraction data sets collected
on $\lambda$-RhO2 obtained via chemical delithiation at the 11-BM beamline at
Argonne National Laboratory between T = 100 and 400 K. Data and fits for these
two temperatures are shown in Figure 2, along with the cubic and tetragonal
structures of $\lambda$-RhO2.

Figure 2: (a),(c) Synchrotron X-ray diffraction patterns collected at T = 400
K and T = 100 K, respectively, along with Rietveld refinement fits. (b) The
structure of cubic $\lambda$-RhO2, demonstrating the three-dimensional
pyrochlore network of Rh cations. (d) The low-temperature, I41/amd structure,
where only the shortest Rh-Rh distances are shown. Rh cations are shown as
displacement ellipsoids to highlight the anisotropy of the refined
displacement parameters.

At T = 400 K, $\lambda$-RhO2 forms in the prototypical cubic Fd$\bar{3}$m
spinel structure. Refinements of anisotropic displacement parameters for
rhodium and oxygen do not result in a significant increase of the goodness of
fit, implying that in the cubic phase, displacement parameters are likely
isotropic. A structural phase transition occurs between T = 320 $-$ 340 K
which reduces the average structure from cubic to tetragonal I41/amd. Fit
parameters can be found in the supplementary information (Tables S1 and S2).

A consequence of the cubic-to-tetragonal structural phase transition is the
formation of rhodium xy-chains along either a or b. In the cubic phase, the
nearest-neighbor Rh$-$Rh distance is 3.002(1) $\mathrm{\AA}$, whereas in the
tetragonal phase, Rh$-$Rh intrachain distances become 2.898(2) $\mathrm{\AA}$
and Rh$-$Rh interchain distances become 3.016(3) $\mathrm{\AA}$. This
distortion is likely due to a Jahn-Teller distortion of the low-energy Rh4+
$t_{2g}$ orbital manifold, where the orbital degeneracy is lifted through a
lowering of the $d_{xz}$ and $d_{yz}$ orbitals relative to $d_{xy}$.
Refinements of the rhodium anisotropic displacement parameters indicate a
predilection for Rh displacements within the xy-chains, with a maximal
displacement B22 = 0.862(1) $\mathrm{\AA}$, as opposed to B11 = 0.287(2)
$\mathrm{\AA}$. Figure 2(d) shows the low temperature tetragonal structure of
$\lambda$-RhO2, where the xy-chains have been highlighted as well as the
anisotropic displacement parameters that are larger within the chain axes vs.
any other direction.

Figure 3: (a) Synchrotron X-ray diffraction patterns collected at various
temperatures down to T = 100 K. (b) The phase fractions of both cubic and
tetragonal structures of $\lambda$-RhO2. Approximately 8.4% of the cubic phase
remains at low temperature.

The structural phase transition is hysteretic in temperature, as demonstrated
in Figure 3. Based on the first derivative of the phase fractions as a
function of temperature, the transition is centered around $T_{W}$ = 345 K on
warming and $T_{C}$ = 329 K on cooling. The ccubic/acubic =
ctetragonal/$\sqrt{2}$atetragonal ratio at T = 100 K, 1.08, indicates an 8%
departure from cubic symmetry across the transition. All samples show a
remnant cubic phase at the lowest measured temperatures, on the order of
8-10%, regardless of synthetic conditions. As demonstrated later with
complementary solid-state NMR results, this remnant cubic phase could be
related to a fraction of the sample that is not delithiated.

Long-range structural phase transitions have been observed in other spinel
systems with electronic degrees of freedom on cations on the B-site, such as
CuIr2S4 65, 66 and MgTi2O4 67, 68, 69. In these compounds, which possess
active spin, orbital, and charge degrees of freedom, structural transitions
are observed due to the formation of molecular, non-magnetic units at low
temperature, such as Ir3+$-$ Ir4+ octamers in the former 70, 71 and helical
Ti3+$-$ Ti3+ dimers in the latter 68, 69. In both of these compounds, a single
phase transition occurs as in $\lambda$-RhO2 (hysteretic in the former and
non-hysteretic in the latter), whereas two phase transitions are observed in
LiRh2O4. It is also instructive to compare these findings to those in the
spinel magnetite Fe3O4, where a single transition (Verwey transition) is
observed near T = 120 K that is also the result from a complex coupling of Fe
electronic degrees of freedom 72, 73, 74. At the local scale, strong short-
range correlations have been observed in both CuIr2S4 75 and MgTi2O4 76 that
preceed the long-range structural phase transition that are suggestive of
dynamic short range fluctuations arising from electronic correlations. These
have also been suggested in LiRh2O4 at temperatures T${}_{CO}<$ T, and we
discuss these in comparison to $\lambda$-RhO2 in more detail in the following
local structure section.

Figure 4: Electron microscopy characterization of $\lambda$-RhO2 at room
temperature. (a) HAADF-STEM image of of a single crystallite of
$\lambda$-RhO2, demonstrating sample homogeneity. (b) High-resolution TEM
image of the single crystallite near the edge. Inhomogeneities can be observed
near the edges of crystallites, possibly due to both cubic and tetragonal
substructures owing to disparate Li content. (c) Electron diffraction pattern
showing the tetragonal phase along [011]. (d) HAADF image of a single
crystallite along the [011] zone axis.

In order to further study the average structure of $\lambda$-RhO2, we employed
transmission electron microscopy (TEM) and high-angle annular dark-field
(HAADF) imaging measurements on polycrystalline samples, the results of which
are shown in Figure 4. Position averaged convergent beam electron diffraction
(PACBED) was performed to accurately determine the zone axis for high-
resolution imaging. The bulk of each crystallite was found to be homogeneous
and give rise to a well-defined diffraction pattern (Figures 4(a), (c)), with
well defined and identifiable crystal planes (Figure 4(d)). The diffraction
patterns lack any noticeable features beyond the main Bragg reflections.

Near the edges of the crystallites, non-uniformities are observed that depart
from the homogeneous bulk. These homgeneities could be due to either an
inhomogeneous distribution of Li throughout the particles, or a redistribution
of Rh near the edges. The presence of lithium ultimately determines whether
the structure is expected to be cubic or tetragonal, especially at room
temperature, and the inhomogeneity observed by electron diffraction could
arise from an admixture of cubic and tetragonal Li1-xRh2O4 phases near the
edges of the crystallites, as in Figure 4(b). This hypothesis is supported by
NMR observations discussed below.

###  2.3 Local Structure Measurements

Total scattering measurements were performed on $\lambda$-RhO2 at the 11-ID-B
beamline at Argonne National Laboratory between temperatures of T = 100 and
400 K through the use of a liquid nitrogen cryostream. Analyses of this data
up to high-Q, in this case Qmax = 18 $\mathrm{\AA}$, allows for the extraction
of the pair distribution function (PDF), which can provide information about
atom-atom correlations via a Fourier transform of total scattering data, shown
in Figure 5. It offers a window into the local structure in materials and
permits the study of distortions and structural transformations at the local
bonding scale.

Figure 5: (a) Temperature-dependent X-ray pair distribution function
measurements, demonstrating a dramatic change in the local structure of
$\lambda$-RhO2 across the $T_{W}$ = 345 K on warming and $T_{C}$ = 329 K
structural phase transition. (b) Measured PDF patterns for $\lambda$-RhO2,
demonstrating the dimerization peak around 2.64 $\mathrm{\AA}$ that arises on
cooling. (c)$-$(d) Measured vs. simulated PDF patterns for the low and high
temperature phases of $\lambda$-RhO2. While the Fd$\bar{3}$m fit appears to
match the data reasonably well at T = 400 K, the I41/amd fit that captures the
average structure does not match the local structure well.

As can be observed in Figure 5(a), the PDF patterns of $\lambda$-RhO2 change
intensely across the structural phase transition above room temperature at
nearly all r length scales. This is in agreement with a long-range structural
phase transition, as the new low temperature cell is expected to possess
vastly different atom-atom correlations compared to the high temperature cubic
structure. However, as can be readily observed in Figure 5(c), unlike the
cubic fit in Figure 5(d), the tetragonal I41/amd cell that reasonably fits the
diffraction data cannot properly fit the PDF data. This suggests that the
local structure of $\lambda$-RhO2 differs from the average structure and
suggests the presence of short-range correlations with a limited correlation
length.

One pronounced difference between the observed low-T PDF pattern (collected at
T = 100 K) and the simulated tetragonal PDF pattern (Figure 5(c)) is the
presence of a peak that is centered around 2.64 $\mathrm{\AA}$, as
demonstrated in Figure 5(b). A similar peak has been observed in the parent
compound LiRh2O4 in the 2.70$-$2.75 $\mathrm{\AA}$ range 37, 38, as well as in
CuIr2S4 70, 77, and has been attributed in both cases to short-range Rh$-$Rh
dimerization (Ir$-$Ir in CuIr2S4). In LiRh2O4, in particular, this peak
emerges in the PDF patterns even above TJT (where LiRh2O4 is cubic), implying
that dimerization is favored at the local scale regardless of the average
structure. Dimerization in LiRh2O4 persists on cooling through both the
orbital ordering (Jahn-Teller distortion) and charge ordering transitions,
though it reaches a constant maximum value below the latter, implying a likely
coupling of the spins to the charge degrees of freedom.

The dimerization peak observed in $\lambda$-RhO2, which can be observed
clearly in Figure 5 (b), differs from that in LiRh2O4 in that it does not
appear above the long-range phase transition observed in average structure and
physical properties measurements, indicating that dimers are not present in
the high-temperature cubic structure, as shown in Figure 5(d). This implies
that these dimers are primarily the result of the formation of Rh-Rh metal-
metal bonds at low temperature, assisted by a Jahn-Teller distortion of the
RhO6 octahedra on cooling. Since $\lambda$-RhO2 possesses mostly Rh4+ (Rh3.95+
assuming Li0.1Rh2O4), there are no charge degrees of freedom that could give
rise to a long-range charge ordered phase such as in LiRh2O4.

Given that these dimers appear only in the local structure and not in the
average, it is possible they are either dynamically fluctuating or static and
disordered. The observation of large anisotropic displacement parameters for
Rh in the synchrotron X-ray diffraction data, however, suggests that the
dimers are only present at the local scale. In our refinements of the global
structure, they manifest as large displacement parameters along the xy-chains,
and form locally along these chains on cooling. Given that the chains are made
of Rh4+ species with an effective spin Seff = 1/2 , they are likely
susceptible to an orbitally driven Peierls instability.

7Li solid-state NMR measurements were performed on $\lambda$-RhO2 and LiRh2O4
in order to further probe the local structure around Li+, as well as to
quantify any remnant amount of Li+ cations within $\lambda$-RhO2 after
chemical delithiation . The room temperature 7Li NMR spectrum collected on
LiRh2O4 exhibits two resonances at 0 ppm (T2-weighted integral 3%) and 50 ppm
(97%), as shown in Figure 6(a). The 0 ppm peak is assigned to diamagnetic
impurities (e.g., Li2CO3, LiOH, LiNO3, and Li2O) at the surface of the LiRh2O4
particles. The 50 ppm peak likely corresponds to Li in the LiRh2O4 structure,
which at room temperature occupies solely the tetrahedrally-coordinated
A-site, and therefore gives rise to a single resonance. This relatively large
shift may arise from the Knight shift interaction, as well as the Fermi
contact interaction, between the unpaired electrons on Rh3+/4+ cations and the
vacant Li+ s-orbitals78.

Figure 6: 7Li NMR of (a) LiRh2O4 and (b) $\lambda$-RhO2, recorded at 2.35 T
under a MAS speed of 60 kHz, corresponding to a sample temperature of 318 K.
(c) Variable-temperature 7Li NMR of $\lambda$-RhO2 (2.35 T field, MAS speed 60
kHz) between 291 and 349 K. (d) The three local Li environments in
$\lambda$-RhO2 are indicated, with the Rh$-$Rh dimer position indicated for
the two tetrahedral interstitial sites Lia and Lic, as well as the octahedral
interstitial site Lib. (e) Illustration of the migration of Li from the
surface-based diamagnetic species into bulk $\lambda$-RhO2, with schematic
lithiation gradients below, at and above the transition temperature.

Delithiation of LiRh2O4 to $\lambda$-RhO2 results in a complex 7Li NMR
spectrum comprising at least five overlapping resonances at roughly 0 (46%), 7
(13%), 25 (20%), 52 (11%), and 62 ppm (10%), as shown in Figure 6(b). The peak
at 0 ppm is again assigned to diamagnetic species, and the higher intensity of
this resonance in the data obtained on $\lambda$-RhO2 as compared to LiRh2O4
is expected, as the synthesis likely results in the nucleation of Li salts at
the surface of the $\lambda$-RhO2 particles during chemical delithiation. The
peak at ca. 50 ppm is assigned to Li in an LiRh2O4-like (i.e. a pristine-like)
environment. Integration of this 50 ppm resonance suggests that about 10$-$11%
of the Li in the $\lambda$-RhO2 sample occupies tetrahedral A-sites in
LiRh2O4-like domains.

The three remaining signals at 7, 25, and 62 ppm have approximate integral
ratios of 1:2:1, suggesting that the resonances at 7 ppm and 62 ppm correspond
to the tetrahedral Li interstitial sites (Wyckoff site 4a), and the shift at
25 ppm corresponds to octahedral Li interstitials (Wyckoff site 8c) in
$\lambda$-RhO2, respectively. These are shown in Figure 6(d), where Lia and
Lic correspond to the tetrahedral Li cations and Lib corresponds to the
octahedral Li. We propose that the origin of the two unique tetrahedral
resonances Lia and Lic stems from the proximity of these sites to the Rh-Rh
dimers that form along the xy-chains. By analogy with the shifts of Li species
near Ru-Ru dimers in Li2RuO3 79, 80, 81, it is expected that Li occupying a
tetrahedral site between dimers will experience a smaller shift than Li
occupying a site at the edges of the dimers, as the Rh orbitals involved in
Rh-Rh metal-metal bond formation are partially spin-quenched, resulting in a
smaller effective electron spin moment for the Li nucleus to interact with,
while those pointing away from the dimer will contain a larger effective spin
moment. The absolute integrals of the 7Li NMR spectra (sample mass-normalized
and T2-weighted) obtained on the LiRh2O4 and $\lambda$-RhO2 samples suggest
that the chemically-delithiated sample contains approximately 16(1)% of the Li
in the pristine sample.

Variable-temperature NMR measurements were also performed on LiRh2O4 (Figure
S2) and $\lambda$-RhO2 samples (Figure 6(c)). These measurements were carried
out at sample temperatures between T = 291 and 349 K. In LiRh2O4, the 50 ppm
shift remains temperature-independent, suggesting metallic behavior and a
shift that is dominated by the Knight shift interaction (Figure S2(a)). For
the $\lambda$-RhO2 sample, the extensive overlap between the aforementioned
Lia, Lib, Lic, and Li in LiRh2O4 resonances prevents us from tracking subtle
chemical shift variations with temperature, but to a first approximation,
those shifts do not exhibit a significant temperature dependence.
Interestingly, while the overall 7Li NMR signal intensity obtained from the
$\lambda$-RhO2 sample remains roughly constant with temperature, the relative
intensities of the Lia, Lib, Lic, Li in LiRh2O4, and diamagnetic resonances
vary drastically between 349 K and 327 K (Figure S2(e)). Over this temperature
range, the intensity of the Lia resonance increases at the expense of the
diamagnetic signal as temperature increases. This suggests Li chemical
exchange or a transfer of Li population from the diamagnetic Li-containing
salts accumulated at the surface of the $\lambda$-RhO2 particles to the bulk,
and in particular to the tetrahedrally-coordinated Lia environments of the
spinel structure. The onset of this redistribution of Li populations appears
to occur between 322 and 331 K, which is concomitant with the phase
transformation from I41/amd to Fd$\bar{3}$m. Hence, we speculate that the
large quantity of Li in diamagnetic surface species (presumably generated
during synthesis) is driven into bulk Rh2O4 on heating, resulting in an
increased Li content in the outer layers of the $\lambda$-RhO2 particles near
room temperatures (Figure 6(e)). We also observe a drop in intensity of all
signals apart from the diamagnetic Li from 322 to 327 K, which we tentatively
ascribe to a shortened $T_{2}$, analogous to the loss in signal seen in LiCoO2
near the metal-to-insulator transition temperature82. Whilst the precise
origin of this enhanced transverse dephasing of nuclear magnetization is
unclear, it appears connected to this transition; this poses an interesting
avenue for future research. We suggest that the origin of the phase
transformation from tetragonal I41/amd to cubic Fd$\bar{3}$m is mediated by
the changing Li composition over this temperature range. Those results are
consistent with our observations in TEM measurements, where the bulk of the
crystallites appeared homogeneous and with minimal disorder, as opposed to the
crystallite edges which appeared greatly disordered and likely containing
admixtures of both cubic Li1-xRh2O4 and tetragonal $\lambda$-RhO2.

In summary, the presence of short Rh-Rh bonds is evidenced in both PDF and
NMR, despite the predicted average I41/amd structure for $\lambda$-RhO2 that
does not accommodate these bonds. Given this disparity between the average and
local structures, these dimers are either static and disordered, or
dynamically fluctuating on a timescale comparable to the diffraction
measurement. In LiRh2O4, this has been suggested in the temperature range of
$T_{CO}<T<T_{JT}$, where a dimerization peak appears in the PDF despite an
average tetragonal I41/amd structure 37, 38. Alternatively, the presence of a
small amount of remaining Li post-delithiation within the $\lambda$-RhO2
structure could prevent a long-range phase transition to an ordered state due
to the disorder induced by this Li, thus Rh-Rh bonding is only observed in
local probes.

###  2.4 Physical Properties Measurements.

Figure 7: Temperature-dependent physical properties measurements of
$\lambda$-RhO2, as well as the normalized dimer peak area from PDF
measurements. (a) dc magnetic susceptibility collected under an applied field
of $\mu_{0}H$ = 1 T. (b) Resistivity measurement under zero applied field on
cooling and warming. (c) Normalized dimer peak area from PDF measurements,
demonstrating it to be simultaneous with the metal-to-insulator transition and
transition to a non-magnetic state. (d) Heat capacity measurement under zero
applied field on cooling and warming, demonstrating the transition.

Physical properties measurements were carried out on $\lambda$-RhO2 and
LiRh2O4, including temperature-dependent magnetic susceptibility, electrical
resistivity, and heat capacity, as shown in Figure 7.

The dc magnetic susceptibility of $\lambda$-RhO2, shown in Figure 7(a), shows
a drop at TM-W = 342 K on warming and TM-C = 325 K on cooling, consistent with
the transition in diffraction data. This transition is indicative of a
transition into a non-magnetic state, such as in LiRh2O4 below the charge
ordering transition. In contrast to LiRh2O4, however, $\lambda$-RhO2 shows
only one, hysteretic transition. The formation of a non-magnetic state is
consistent with the formation of Rh-Rh dimers, as the individual S = 1/2 Rh4+
cations pair up to form Seff = 0 singlets. Curie-Weiss behavior was not
identified at any temperature region, including above T = 400 K (Figure S3 in
the supplementary information), suggesting the absence of localized moments.

Measurements of electrical resistivity, shown in Figure 7(b), demonstrate a
metal-to-insulator transition on cooling across the transition. The
resistivity increases by five orders of magnitude below the transition, but
cannot be fit to either an exponentially activated or variable range hopping
model. Given the fact that $\lambda$-RhO2 is a metastable compound, annealing
or sintering pellets for measurements at high temperatures is not a
possibility. As such, measurements were performed solely on cold-pressed
pellets, which likely affects the measurements through poor conductivity
across grain boundaries. Nevertheless, as can be seen in the inset, the
resistivity increases with increasing temperature above the transition,
suggesting a metallic state.

A metal-to-insulator transition occurs in LiRh2O436, as well as in the spinels
CuIr2S4 65, 66, 83, MgTi2O4 67, 84, and LiV2O4 (under pressure and doped)85,
86, 87. A mechanism that has been proposed to explain the transition in such
compounds is the orbitally-driven Peierls state, whereby strong anisotropic
orbital interactions between adjacent sites establish quasi-one-dimensional
chains within the pyrochlore lattice88. These chains are then individually
susceptible to a Peierls instability, and the pyrochlore network distorts to
form either complex molecular clusters, such as the octamers in CuIr2S4 70, or
dimers, such as in MgTi2O4 68, 89. This mechanism likely also explains the
observed transition in $\lambda$-RhO2, though we do not observe a long-range
crystal structure with dimers for this compound. Figure 7(c) shows the
normalized dimer peak area from PDF measurements, demonstrating that the onset
of the metal-to-insulator and the magnetic-to-non-magnetic transitions are
concomitant with the formation of short-range dimers. It appears as though
dimerization is strongly favored at the local scale through local bonding
interactions. However, we do not rule out a lower symmetry average structure
that could accommodate either dimers or some larger n-unit non-magnetic
cluster arrangement, such as the octamer ground state in CuIr2S4 70 or trimers
in RuP 90.

In theory, one would expect $\lambda$-RhO2 and LiRh2O4 to display behavior
closer to CuIr2S4 than to MgTi2O4 and LiV2O4, owing to the general similarity
of Rh/Ir chemistry and the comparatively strong spin-orbit coupling in Rh vs.
Ti/V. However, in both LiRh2O4 and $\lambda$-RhO2, spin-orbit coupling appears
to play a smaller role in relation to the Jahn-Teller instability of the low-
energy Rh4+ t2g orbital manifold. As such, a cubic-to-tetragonal phase
transition occurs in both that establishes a ground state structure with Rh-Rh
chains along the a\- or b-crystallographic directions (xy-chains).
Dimerization can then naturally occur along these chains composed of half-
filled Rh4+ via an orbitally-driven Peierls state 88, 75, 91, 92. In the
following section, we examine the impact of correlations and spin-orbit
coupling on the electronic structure via calculations based on density
functional theory.

###  2.5 Electronic Structure

Calculations based on density functional theory (DFT) were performed on
LiRh2O4 and $\lambda$-RhO2 using experimentally-derived structural parameters.
$\lambda$-RhO2 was treated as having no lithium on the A-site. Calculations
were done on both the cubic Fd$\bar{3}$m and tetragonal I41/amd cells with and
without structural relaxation using the the Perdew-Burke-Ernzerhof (PBE)
exchange functional93.

Based on PBE-relaxed structures, the energy per formula unit of rutile RhO2 is
more than 0.5 eV lower than that of both cubic and tetragonal $\lambda$-RhO2.
The greater computed thermodynamic stability of the rutile phase is consistent
with $\lambda$-RhO2 being a kinetically trapped phase accessible only through
low-temperature, oxidative delithiation. This is further corroborated by the
observation that heating $\lambda$-RhO2 in air above T = 500 K results in
decomposition to rutile RhO2 and Rh2O3.

Figure 8: (a) Electronic band structure of $\lambda$-RhO2, calculated using
PBE and $U_{\mathrm{eff}}$ = 4 eV. Orange and blue bands correspond to spin-up
and spin-down states, respectively. (b) Density of states plot, demonstrating
a gap at the Fermi level along with strong spin-polarization, likely due to
the absence of dimers in the calculation.

Structural relaxations of the cubic and tetragonal phases of $\lambda$-RhO2
were performed with and without Hubbard $U_{\mathrm{eff}}$ values applied to
the Rh 4$d$ bands and/or spin-orbit coupling (SOC). The results, shown in the
supplementary information (Table S3), reveal that the tetragonal structure is
only stabilized relative to the cubic one upon inclusion of a
$U_{\mathrm{eff}}>$ 1 eV term. However, the inclusion of an SOC term results
in a destabilization of the tetragonal phase. These results suggest that the
tetragonal phase is enabled by correlations that drive the formation of the
xy-chains, which exist on a higher energy scale compared to spin-orbit
coupling.

A gap in the calculated band structure for the low-temperature tetragonal
phase can be observed for $U_{eff}\geq$ 3 eV, as demonstrated in the
supplementary information (Figure S4). However, these calculations do not
include any dimerization, which would naturally provide a gap-opening
mechanism. As such, the predicted band structures in the absence of an
applied, non-zero U, shown in Figure 8, predict metallicity and substantial
spin-polarization. Applying a non-zero $U_{\mathrm{eff}}$ offers an avenue
toward localization, though strong spin polarization is still observed. Our
findings suggest that the spin polarization observed in our calculations stems
from a natural instability toward either dimerization or magnetic order, and
that the low temperature insulating state observed in experiment is likely the
result of dimer-induced localization. Naturally, our calculations do not
incorporate dimers, as these calculations have been performed on the long-
range structures derived from fits to average structure measurements. Should a
lower-symmetry, long-range structure exist that accomodates for the Rh-Rh
dimers in $\lambda$-RhO2, calculations of its band structure would likely
relieve this apparent inconsistency between DFT and experiment.

We conclude our discussion of the structure and properties of $\lambda$-RhO2
by considering it in comparison to other spinel compounds, especially CuIr2S4
and MgTi2O4, the only other spinels that have been found to undergo metal to
non-magnetic insulator transitions without doping or external pressure, as
well as the aforementioned pyrochlore iridates. Naively, one would expect both
LiRh2O4 and $\lambda$-RhO2 to display similar behavior as in CuIr2S4 rather
than MgTi2O4. LiRh2O4, in particular, possesses Rh3+/4+ just as CuIr2S4 has
Ir3+/4+. However, SOC plays a much larger role in establishing the single-ion
physics of Ir compounds compared to Rh compounds, and as our results suggest,
SOC indeed plays a near negligible role in establishing the ground state of
$\lambda$-RhO2. As such, these systems are closer to MgTi2O4 where SOC plays a
minor role in relation to the single-ion orbital instability and gives rise to
a long-range dimerized ground state. $\lambda$-RhO2, therefore, presents an
avenue toward the study of competing interactions in a 4d transition metal
with a rare oxidation state.

##  3 Conclusion

$\lambda$-RhO2 represents a platform to study the interplay of orbital and
spin degrees of freedom of 4d5 cations on a pyrochlore lattice. We have
synthesized this new Rh4+ oxide using ceric ammonium nitrate, a heavily
understudied, powerful oxidizer with wide-ranging future applications in the
low temperature, oxidative deintercalation of extended solids. Our
measurements indicate the presence of short-range Rh-Rh dimers that arise from
metal-metal bonding at the local scale that do not crystallize in the long-
range average structure. These dimers arise across a hysteretic phase
transition at $T_{W}$ = 345 K on warming and $T_{C}$ = 329 K on cooling, which
is concurrent to a metal-to-insulator transition and a magnetic to non-
magnetic transition. Our results inspire the search for other possible quantum
materials in frustrated lattices made up of transition metals with uncommon,
high oxidation states.

##  4 Methods

Synthesis. Polycrystalline samples of $\lambda$-RhO2 were prepared using soft
chemical techniques. First, LiRh2O4 spinel precursor was synthesized in
evacuated silica tubes using stoichiometric amounts of Li2O2 and Rh2O3, as
previously reported 36. Physically separated PbO2, which decomposes at high
temperatures via PbO2 $\rightarrow$ PbO + $\frac{1}{2}$O2, was used to
generate high oxygen pressures within reaction tubes. Powders were
characterized via in-house X-ray diffraction.

LiRh2O4 powders were then stirred in aqueous solutions of ceric ammonium
nitrate for 48 hours, followed by vacuum filtration and drying in air.
Stoichiometric amounts of ceric ammonium nitrate were used to target specific
stoichiometries with formula Li1-xRh2O4. A ten-fold molar excess of ceric
ammonium nitrate was used to synthesize the end-member, $\lambda$-RhO2.

X-ray structural characterization. High resolution synchrotron powder X-ray
diffraction (XRD) data was collected at the 11-BM beamline at the Advanced
Photon Source (APS) at Argonne National Laboratory, using an incident
wavelength of 0.4590443 $\mathrm{\AA}$. Data were collected at temperatures
ranging from T = 100 K to 400 K. Powder XRD data was also collected in-house
using a Panalytical Empyrean diffractometer employing Cu K$\alpha$ X-rays in a
Bragg-Brentano geometry.

Pair distribution function (PDF) datasets were collected at the 11-ID-B
beamline at APS, using an incident wavelength of 0.2116 $\mathrm{\AA}$. Data
were collected at temperatures ranging from T = 100 K to 400 K, and PDF
patterns were extracted with a maximum momentum transfer of $Q_{max}=18$
$\mathrm{\AA}$. Modeling and fitting of the XRD and PDF data was performed
using TOPAS Academic.

Physical properties measurements. Magnetic susceptibility measurements were
carried out on powder samples in a Quantum Design Magnetic Property
Measurement System (MPMS3). Resistivity and heat capacity measurements were
performed using a Quantum Design 14 T Dynacool Physical Property Measurement
System (PPMS). Resistivity measurements were performed via the four probe
method on cold pressed pellets of polycrystalline sample.

Transmission Electron Microscopy. TEM samples were made by first preparing a
suspension through the mixing of $\lambda$-RhO2 powder in water, and then drop
casting the particle suspension on a TEM grid. The grid was subsequently dried
in air below 80 ∘C for approximately 2 minutes before being inserted in a
Spectra 200 ThermoFisher Scientific transmission electron microscope equipped
with an ultra-high-brightness gun source (X-CFEG) and six-fold astigmatism
probe aberration corrector. The operating voltage is 200 kV, with a probe
convergence semi-angle of 30 mrad. The detector semi-angle was set between 25
mrad to 200 mrad (camera length: 160 mm).

7Li solid state nuclear magnetic resonance. Powder samples of LiRh2O4 and
$\lambda$-RhO2 were loaded into 1.3 mm diameter ZrO2 magic angle spinning
(MAS) rotors. 7Li NMR spectra were referenced to liquid LiCl in H2O (1 M) at 0
ppm and acquired on a Bruker AVANCE (2.35 T) using a Bruker 1.3 mm MAS probe,
a $\pi$/2 pulse length of 0.45 $\mu$s, and an MAS frequency of 60 kHz for
“room temperature” spectra (i.e., no external heating or cooling applied) or
50 kHz for variable-temperature spectra. Rotor-synchronized Hahn-echo pulse
sequences ($\pi$/2–$\tau$– $\pi$–$\tau$–acq.) were used to obtain spectra, the
intensities of which were scaled by sample mass and number of scans. The
recycle delay (2.5s; at least 5$T_{1}$) was set such that the bulk,
paramagnetically shifted signal was recorded quantitatively and the
diamagnetic signal due to surface-based impurities was suppressed; additional
spectra with a longer recycle delay (25s) were recorded such that the
diamagnetic signal was also recorded quantitatively. Sample temperatures were
obtained from internal calibration of the 79Br shift of KBr 94.

Electrochemistry and operando X-ray diffraction.  Electrochemistry experiments
were performed by casting electrodes made from a 80:10:10 (wt %) ratio of
LiRh2O4 : conductive carbon (TIMCAL Super P) : polyvinylidene fluoride (PVDF).
The PVDF was first dissolved in N-methylpyrrolidone and mixed in a FlackTek
speed mixer at 2000 rpm for 5 minutes. The conductive carbon and LiRh2O4 were
ground in a mortar and pestle for 10 minutes and then added to the viscous
mixture, forming a slurry. The slurry was mixed in the speed mixer for 10
minutes and later cast using a 200 $\mu$m blade. After 3 hours, the cast
slurry was dried in a vacuum oven at 80 ∘C overnight. The electrodes were
punched into 10 mm diameter disks with loading between 2 and 3 mg cm-2. The
electrodes were brought into an Ar-filled glovebox (H2O $<$ 0.1 ppm and O2 $<$
0.1 ppm) and assembled into Swagelok cells or Hohsen coin cells for
electrochemical testing. A glass fiber separator (Whatman GF/D) was soaked in
1 M LiPF6 in EC/DMC 50/50 v/v (Sigma-Aldrich) electrolyte, and a polished Li
foil was used as the counter and reference electrode. Cells were discharged to
1 V and charged to 3 V using BioLogic potentiostats (VMP1 and VMP3). All
measurements were carried out at room temperature.

Operando X-ray diffraction measurements were collected using a custom
Swagelok-type cell with a Be window approximately 250 $\mu$m thick, allowing
X-ray penetration into the cell while cycling. A pattern was collected every
20 minutes during cycling at a C/15 rate.

Density functional theory. First-principles electronic structure calculations
were performed using the Vienna ab Initio Simulation Package (VASP) version
5.4.4. All calculations employed the Perdew-Burke-Ernzerhof (PBE) functional
and projector-augemented wave potentials based on the v5.4 recommendations
(Rh_pv, O). The plane-wave energy cutoff was set to 520 eV and a 9 $\times$ 9
$\times$ 9 $\Gamma$-centered _k_ -point mesh was used to avoid incompatibility
problems associated with the primitive cells of body-centered tetragonal
structures. Electronic band structure and density of states calculations were
performed on both the experimentally-derived structure (unrelaxed) and
geometrically-optimized structures (relaxed) in which forces were converged
within 10$-$5 eV/Å. A _k_ -point path for the band structure was generated
using the AFLOW online tool. All calculations had an energy convergence better
than 10$-$6 eV.

J.R.C. acknowledges support through the NSF MPS-Ascend Postdoctoral Fellowship
(DMR-2137580). J.R.C, S.S., and S.D.W acknowledge support by the U.S.
Department of Energy (DOE), Office of Basic Energy Sciences, Division of
Materials Sciences and Engineering under Grant No. DE-SC0020305. E.N.B. and
R.J.C. acknowledge and are grateful to the Spectroscopy Facility at UC Santa
Barbara. S.S. and G.Z. acknowledge support by the U.S. Department of Energy
under Grant No. DEFG02-02ER45994. The research reported here made use of the
shared facilities of the Materials Research Science and Engineering Center
(MRSEC) at UC Santa Barbara: NSF DMR-2308708. The UC Santa Barbara MRSEC is a
member of the Materials Research Facilities Network (www.mrfn.com). Use was
made of the computational facilities administered by the Center for Scientific
Computing at the CNSI and MRL (an NSF MRSEC; DMR-2308708) and purchased
through NSF CNS-1725797. This work was also supported by the National Science
Foundation (NSF) through Enabling Quantum Leap: Convergent Accelerated
Discovery Foundries for Quantum Materials Science, Engineering and Information
(Q-AMASE-i): Quantum Foundry at UC Santa Barbara (DMR-1906325). This research
used resources of the Advanced Photon Source, a U.S. Department of Energy
(DOE) Office of Science user facility operated for the DOE Office of Science
by Argonne National Laboratory under Contract No. DE-AC02-06CH11357.

## References

  * Plumb et al. 2014 Plumb, K. W.; Clancy, J. P.; Sandilands, L. J.; Shankar, V. V.; Hu, Y. F.; Burch, K. S.; Kee, H.-Y.; Kim, Y.-J. $\alpha$-RuCl3 : A spin-orbit assisted Mott insulator on a honeycomb lattice. _Phys. Rev. B_ 2014, _90_ , 041112(R), DOI: 10.1103/physrevb.90.041112
  * Banerjee et al. 2017 Banerjee, A.; Yan, J.; Knolle, J.; Bridges, C. A.; Stone, M. B.; Lumsden, M. D.; Mandrus, D. G.; Tennant, D. A.; Moessner, R.; Nagler, S. E. Neutron scattering in the proximate quantum spin liquid $\alpha$-RuCl3. _Science_ 2017, _356_ , 1055–1059, DOI: 10.1126/science.aah6015
  * Banerjee et al. 2018 Banerjee, A. et al. Excitations in the field-induced quantum spin liquid state of $\alpha$-RuCl3. _npj Quantum Mater._ 2018, _3_ , 8, DOI: 10.1038/s41535-018-0079-2
  * Do et al. 2017 Do, S.-H.; Park, S.-Y.; Yoshitake, J.; Nasu, J.; Motome, Y.; Kwon, Y. S.; Adroja, D. T.; Voneshen, D. J.; Kim, K.; Jang, T.-H.; Park, J.-H.; Choi, K.-Y.; Ji, S. Majorana fermions in the Kitaev quantum spin system $\alpha$-RuCl3. _Nat. Phys._ 2017, _13_ , 1079–1084, DOI: 10.1038/nphys4264
  * Yadav et al. 2016 Yadav, R.; Bogdanov, N. A.; Katukuri, V. M.; Nishimoto, S.; van den Brink, J.; Hozoi, L. Kitaev exchange and field-induced quantum spin-liquid states in honeycomb $\alpha$-RuCl3. _Sci. Rep._ 2016, _6_ , 37925, DOI: 10.1038/srep37925
  * Kim et al. 2008 Kim, B. J.; Jin, H.; Moon, S. J.; Kim, J.-Y.; Park, B.-G.; Leem, C. S.; Yu, J.; Noh, T. W.; Kim, C.; Oh, S.-J.; Park, J.-H.; Durairaj, V.; Cao, G.; Rotenberg, E. Novel $J_{\mathrm{eff}}$ = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr2IrO4. _Phys. Rev. Lett._ 2008, _101_ , 076402, DOI: 10.1103/physrevlett.101.076402
  * Kim et al. 2009 Kim, B. J.; Ohsumi, H.; Komesu, T.; Sakai, S.; Morita, T.; Takagi, H.; Arima, T. Phase-Sensitive Observation of a Spin-Orbital Mott State in Sr2IrO4. _Science_ 2009, _323_ , 1329–1332, DOI: 10.1126/science.1167106
  * Kim et al. 2012 Kim, J.; Casa, D.; Upton, M. H.; Gog, T.; Kim, Y.-J.; Mitchell, J. F.; van Veenendaal, M.; Daghofer, M.; van den Brink, J.; Khaliullin, G.; Kim, B. J. Magnetic Excitation Spectra of Sr2IrO4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors. _Phys. Rev. Lett._ 2012, _108_ , 177003, DOI: 10.1103/physrevlett.108.177003
  * Zwartsenberg et al. 2020 Zwartsenberg, B.; Day, R. P.; Razzoli, E.; Michiardi, M.; Xu, N.; Shi, M.; Denlinger, J. D.; Cao, G.; Calder, S.; Ueda, K.; Bertinshaw, J.; Takagi, H.; Kim, B. J.; Elfimov, I. S.; Damascelli, A. Spin-orbit-controlled metal–insulator transition in Sr2IrO4. _Nat. Phys._ 2020, _16_ , 290–294, DOI: 10.1038/s41567-019-0750-y
  * Nagai et al. 2007 Nagai, I.; Yoshida, Y.; Ikeda, S. I.; Matsuhata, H.; Kito, H.; Kosaka, M. Canted antiferromagnetic ground state in Sr3Ir2O7. _J. Condens. Matter Phys._ 2007, _19_ , 136214, DOI: 10.1088/0953-8984/19/13/136214
  * Boseggia et al. 2012 Boseggia, S.; Springell, R.; Walker, H. C.; Boothroyd, A. T.; Prabhakaran, D.; Collins, S. P.; McMorrow, D. F. On the magnetic structure of Sr3Ir2O7: an x-ray resonant scattering study. _J. Condens. Matter Phys._ 2012, _24_ , 312202, DOI: 10.1088/0953-8984/24/31/312202
  * Mazzone et al. 2022 Mazzone, D. G. et al. Antiferromagnetic excitonic insulator state in Sr3Ir2O7. _Nat. Commun._ 2022, _13_ , 913, DOI: 10.1038/s41467-022-28207-w
  * Biffin et al. 2014 Biffin, A.; Johnson, R. D.; Choi, S.; Freund, F.; Manni, S.; Bombardi, A.; Manuel, P.; Gegenwart, P.; Coldea, R. Unconventional magnetic order on the hyperhoneycomb Kitaev lattice in $\beta$-Li2IrO3: Full solution via magnetic resonant x-ray diffraction. _Phys. Rev. B_ 2014, _90_ , 205116, DOI: 10.1103/physrevb.90.205116
  * Takayama et al. 2015 Takayama, T.; Kato, A.; Dinnebier, R.; Nuss, J.; Kono, H.; Veiga, L.; Fabbris, G.; Haskel, D.; Takagi, H. Hyperhoneycomb Iridate $\beta$-Li2IrO3 as a Platform for Kitaev Magnetism. _Phys. Rev. Lett._ 2015, _114_ , 077202, DOI: 10.1103/physrevlett.114.077202
  * Williams et al. 2016 Williams, S. C.; Johnson, R. D.; Freund, F.; Choi, S.; Jesche, A.; Kimchi, I.; Manni, S.; Bombardi, A.; Manuel, P.; Gegenwart, P.; Coldea, R. Incommensurate counterrotating magnetic order stabilized by Kitaev interactions in the layered honeycomb $\alpha$-Li2IrO3. _Phys. Rev. B_ 2016, _93_ , 195158, DOI: 10.1103/physrevb.93.195158
  * Halloran et al. 2022 Halloran, T.; Wang, Y.; Li, M.; Rousochatzakis, I.; Chauhan, P.; Stone, M. B.; Takayama, T.; Takagi, H.; Armitage, N. P.; Perkins, N. B.; Broholm, C. Magnetic excitations and interactions in the Kitaev hyperhoneycomb iridate $\beta$-Li2IrO3. _Phys. Rev. B_ 2022, _106_ , 064423, DOI: 10.1103/physrevb.106.064423
  * Singh and Gegenwart 2010 Singh, Y.; Gegenwart, P. Antiferromagnetic Mott insulating state in single crystals of the honeycomb lattice material Na2IrO3. _Phys. Rev. B_ 2010, _82_ , 064412, DOI: 10.1103/physrevb.82.064412
  * Liu et al. 2011 Liu, X.; Berlijn, T.; Yin, W.-G.; Ku, W.; Tsvelik, A.; Kim, Y.-J.; Gretarsson, H.; Singh, Y.; Gegenwart, P.; Hill, J. P. Long-range magnetic ordering in Na2IrO3. _Phys. Rev. B_ 2011, _83_ , 220403(R), DOI: 10.1103/physrevb.83.220403
  * Choi et al. 2012 Choi, S. K.; Coldea, R.; Kolmogorov, A. N.; Lancaster, T.; Mazin, I. I.; Blundell, S. J.; Radaelli, P. G.; Singh, Y.; Gegenwart, P.; Choi, K. R.; Cheong, S.-W.; Baker, P. J.; Stock, C.; Taylor, J. Spin Waves and Revised Crystal Structure of Honeycomb Iridate Na2IrO3. _Phys. Rev. Lett._ 2012, _108_ , 127204, DOI: 10.1103/physrevlett.108.127204
  * Comin et al. 2012 Comin, R.; Levy, G.; Ludbrook, B.; Zhu, Z.-H.; Veenstra, C. N.; Rosen, J. A.; Singh, Y.; Gegenwart, P.; Stricker, D.; Hancock, J. N.; van der Marel, D.; Elfimov, I. S.; Damascelli, A. Na2IrO3 as a Novel Relativistic Mott Insulator with a 340-meV Gap. _Phys. Rev. Lett._ 2012, _109_ , 266406, DOI: 10.1103/physrevlett.109.266406
  * Alcock and Hooper 1960 Alcock, C. B.; Hooper, G. W. Thermodynamics of the gaseous oxides of the platinum-group metals. _Proc. R. Soc. A_ 1960, _254_ , 551–561, DOI: 10.1098/rspa.1960.0040
  * Jacob and Prusty 2010 Jacob, K.; Prusty, D. Thermodynamic properties of RhO2. _J. Alloys Compd._ 2010, _507_ , L17–L20, DOI: 10.1016/j.jallcom.2010.07.179
  * Yamaura et al. 2009 Yamaura, K.; Shirako, Y.; Kojitani, H.; Arai, M.; Young, D. P.; Akaogi, M.; Nakashima, M.; Katsumata, T.; Inaguma, Y.; Takayama-Muromachi, E. Synthesis and Magnetic and Charge-Transport Properties of the Correlated 4d Post-Perovskite CaRhO3. _J. Am. Chem. Soc._ 2009, _131_ , 2722–2726, DOI: 10.1021/ja8091906
  * Li et al. 2017 Li, Y.; Cheng, J.; Alonso, J. A.; Goodenough, J. B.; Zhou, J. High-Pressure Synthesis, Crystal Structure, and Magnetic and Transport Properties of a Six-Layered SrRhO3. _Inorg. Chem._ 2017, _56_ , 8187–8194, DOI: 10.1021/acs.inorgchem.7b00864
  * Chamberland and Anderson 1981 Chamberland, B.; Anderson, J. The preparation and crystal structure of a BaRhO3 polytype. _J. Solid State Chem._ 1981, _39_ , 114–119, DOI: 10.1016/0022-4596(81)90309-1
  * Yamaura et al. 2002 Yamaura, K.; Huang, Q.; Young, D. P.; Noguchi, Y.; Takayama-Muromachi, E. Crystal structure and electronic and magnetic properties of the bilayered rhodium oxide Sr3Rh2O7. _Phys. Rev. B_ 2002, _66_ , 134431, DOI: 10.1103/physrevb.66.134431
  * Yamaura et al. 2004 Yamaura, K.; Huang, Q.; Young, D. P.; Takayama-Muromachi, E. Crystal Structure and Magnetic Properties of the Trilayered Perovskite Sr4Rh3O10: A New Member of the Strontium Rhodate Family. _Chem. Mater._ 2004, _16_ , 3424–3430, DOI: 10.1021/cm0491072
  * Shannon 1968 Shannon, R. Synthesis and properties of two new members of the rutile family RhO2 and PtO2. _Solid State Commun._ 1968, _6_ , 139–143, DOI: 10.1016/0038-1098(68)90019-7
  * Shirako et al. 2014 Shirako, Y.; Wang, X.; Tsujimoto, Y.; Tanaka, K.; Guo, Y.; Matsushita, Y.; Nemoto, Y.; Katsuya, Y.; Shi, Y.; Mori, D.; Kojitani, H.; Yamaura, K.; Inaguma, Y.; Akaogi, M. Synthesis, Crystal Structure, and Electronic Properties of High-Pressure PdF2-Type Oxides MO2 (M = Ru, Rh, Os, Ir, Pt). _Inorg. Chem._ 2014, _53_ , 11616–11625, DOI: 10.1021/ic501770g
  * Shimura et al. 1992 Shimura, T.; Itoh, M.; Nakamura, T. Novel two-dimensional conductor Sr2RhO4. _J. Solid State Chem._ 1992, _98_ , 198–200, DOI: 10.1016/0022-4596(92)90086-b
  * Perry et al. 2006 Perry, R. S.; Baumberger, F.; Balicas, L.; Kikugawa, N.; Ingle, N. J. C.; Rost, A.; Mercure, J. F.; Maeno, Y.; Shen, Z. X.; Mackenzie, A. P. Sr2RhO4: a new, clean correlated electron metal. _New J. Phys._ 2006, _8_ , 175–175, DOI: 10.1088/1367-2630/8/9/175
  * Battisti et al. 2020 Battisti, I.; Tromp, W. O.; Riccò, S.; Perry, R. S.; Mackenzie, A. P.; Tamai, A.; Baumberger, F.; Allan, M. P. Direct comparison of ARPES, STM, and quantum oscillation data for band structure determination in Sr2RhO4. _npj Quantum Mater._ 2020, _5_ , 91, DOI: 10.1038/s41535-020-00292-4
  * Todorova and Jansen 2010 Todorova, V.; Jansen, M. Synthesis, Structural Characterization and Physical Properties of a New Member of Ternary Lithium Layered Compounds - Li2RhO3. _Z. Anorg. Allg. Chem._ 2010, _637_ , 37–40, DOI: 10.1002/zaac.201000349
  * Luo et al. 2013 Luo, Y.; Cao, C.; Si, B.; Li, Y.; Bao, J.; Guo, H.; Yang, X.; Shen, C.; Feng, C.; Dai, J.; Cao, G.; an Xu, Z. Li2RhO3: A spin-glassy relativistic Mott insulator. _Phys. Rev. B_ 2013, _87_ , 161121(R), DOI: 10.1103/physrevb.87.161121
  * Khuntia et al. 2017 Khuntia, P.; Manni, S.; Foronda, F. R.; Lancaster, T.; Blundell, S. J.; Gegenwart, P.; Baenitz, M. Local magnetism and spin dynamics of the frustrated honeycomb rhodate Li2RhO3. _Phys. Rev. B_ 2017, _96_ , 094432, DOI: 10.1103/physrevb.96.094432
  * Okamoto et al. 2008 Okamoto, Y.; Niitaka, S.; Uchida, M.; Waki, T.; Takigawa, M.; Nakatsu, Y.; Sekiyama, A.; Suga, S.; Arita, R.; Takagi, H. Band Jahn-Teller Instability and Formation of Valence Bond Solid in a Mixed-Valent Spinel Oxide LiRh2O4. _Phys. Rev. Lett._ 2008, _101_ , 086404, DOI: 10.1103/physrevlett.101.086404
  * Knox et al. 2013 Knox, K. R.; Abeykoon, A. M. M.; Zheng, H.; Yin, W.-G.; Tsvelik, A. M.; Mitchell, J. F.; Billinge, S. J. L.; Bozin, E. S. Local structural evidence for strong electronic correlations in spinel LiRh2O4. _Phys. Rev. B_ 2013, _88_ , 174114, DOI: 10.1103/physrevb.88.174114
  * Shiomi et al. 2022 Shiomi, M.; Kojima, K.; Katayama, N.; Maeda, S.; Schneeloch, J. A.; Yamamoto, S.; Sugimoto, K.; Ohta, Y.; Louca, D.; Okamoto, Y.; Sawa, H. Charge-ordered state satisfying the Anderson condition in LiRh2O4 arising from local dimer order. _Phys. Rev. B_ 2022, _105_ , L041103, DOI: 10.1103/physrevb.105.l041103
  * Cascales and Rasines 1984 Cascales, C.; Rasines, I. The spinels CoRh2O4 and Co2RhO4. _Mater. Chem. Phys._ 1984, _10_ , 199–203, DOI: 10.1016/0254-0584(84)90048-8
  * Ge et al. 2017 Ge, L.; Flynn, J.; Paddison, J. A. M.; Stone, M. B.; Calder, S.; Subramanian, M. A.; Ramirez, A. P.; Mourigal, M. Spin order and dynamics in the diamond-lattice Heisenberg antiferromagnets CuRh2O4 and CoRh2O4. _Phys. Rev. B_ 2017, _96_ , 064413, DOI: 10.1103/physrevb.96.064413
  * Horiuti and Miyahara 1964 Horiuti, S.; Miyahara, S. Tetragonal Distortion of NiRh2O4. _J. Phys. Soc. Japan_ 1964, _19_ , 423–424, DOI: 10.1143/jpsj.19.423
  * Chamorro et al. 2018 Chamorro, J. R.; Ge, L.; Flynn, J.; Subramanian, M. A.; Mourigal, M.; McQueen, T. M. Frustrated spin one on a diamond lattice in NiRh2O4. _Phys. Rev. Mater._ 2018, _2_ , 034404, DOI: 10.1103/physrevmaterials.2.034404
  * Dollase and O'Neill 1997 Dollase, W. A.; O'Neill, H. S. C. The Spinels CuCr2O4 and CuRh2O4. _Acta Crystallogr. C Struct. Chem._ 1997, _53_ , 657–659, DOI: 10.1107/s0108270197000486
  * Matsuhira et al. 2011 Matsuhira, K.; Wakeshima, M.; Hinatsu, Y.; Takagi, S. Metal-Insulator Transitions in Pyrochlore Oxides Ln2Ir2O7. _J. Phys. Soc. Japan_ 2011, _80_ , 094701, DOI: 10.1143/jpsj.80.094701
  * Witczak-Krempa et al. 2014 Witczak-Krempa, W.; Chen, G.; Kim, Y. B.; Balents, L. Correlated Quantum Phenomena in the Strong Spin-Orbit Regime. _Annu. Rev. Condens. Matter Phys._ 2014, _5_ , 57–82, DOI: 10.1146/annurev-conmatphys-020911-125138
  * Wan et al. 2011 Wan, X.; Turner, A. M.; Vishwanath, A.; Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. _Phys. Rev. B_ 2011, _83_ , 205101, DOI: 10.1103/physrevb.83.205101
  * Yang et al. 2011 Yang, K.-Y.; Lu, Y.-M.; Ran, Y. Quantum Hall effects in a Weyl semimetal: Possible application in pyrochlore iridates. _Phys. Rev. B_ 2011, _84_ , 075129, DOI: 10.1103/physrevb.84.075129
  * Sushkov et al. 2015 Sushkov, A. B.; Hofmann, J. B.; Jenkins, G. S.; Ishikawa, J.; Nakatsuji, S.; Sarma, S. D.; Drew, H. D. Optical evidence for a Weyl semimetal state in pyrochlore Eu2Ir2O7. _Phys. Rev. B_ 2015, _92_ , 241108(R), DOI: 10.1103/physrevb.92.241108
  * Li et al. 2021 Li, Y.; Oh, T.; Son, J.; Song, J.; Kim, M. K.; Song, D.; Kim, S.; Chang, S. H.; Kim, C.; Yang, B.-J.; Noh, T. W. Correlated Magnetic Weyl Semimetal State in Strained Pr2Ir2O7. _Adv. Mater._ 2021, _33_ , 2008528, DOI: 10.1002/adma.202008528
  * Liu et al. 2021 Liu, X. et al. Magnetic Weyl Semimetallic Phase in Thin Films of Eu2Ir2O7. _Phys. Rev. Lett._ 2021, _127_ , 277204, DOI: 10.1103/physrevlett.127.277204
  * Ohtsuki et al. 2019 Ohtsuki, T.; Tian, Z.; Endo, A.; Halim, M.; Katsumoto, S.; Kohama, Y.; Kindo, K.; Lippmaa, M.; Nakatsuji, S. Strain-induced spontaneous Hall effect in an epitaxial thin film of a Luttinger semimetal. _PNAS_ 2019, _116_ , 8803–8808, DOI: 10.1073/pnas.1819489116
  * Mandal and Freire 2021 Mandal, I.; Freire, H. Transport in the non-Fermi liquid phase of isotropic Luttinger semimetals. _Phys. Rev. B_ 2021, _103_ , 195116, DOI: 10.1103/physrevb.103.195116
  * Nikolić et al. 2022 Nikolić, P.; Xu, Y.; Ohtsuki, T.; Nakatsuji, S.; Drichko, N. Weyl-Luttinger phase transition in pyrochlore iridates revealed by Raman scattering. 2022; https://arxiv.org/abs/2204.13722, DOI: 10.48550/ARXIV.2204.13722
  * Hunter 1981 Hunter, J. C. Preparation of a new crystal form of manganese dioxide: $\lambda$-MnO2. _J. Solid State Chem._ 1981, _39_ , 142–147, DOI: 10.1016/0022-4596(81)90323-6
  * Nair and Deepthi 2007 Nair, V.; Deepthi, A. Cerium(IV) Ammonium Nitrate - A Versatile Single-Electron Oxidant. _Chem. Rev._ 2007, _107_ , 1862–1891, DOI: 10.1021/cr068408n
  * Molander 1992 Molander, G. A. Application of lanthanide reagents in organic synthesis. _Chem. Rev._ 1992, _92_ , 29–68, DOI: 10.1021/cr00009a002
  * Xiao et al. 2000 Xiao, J.-P.; Wang, Y.-L.; Jia, X.-S.; Wang, X.-Y.; Wang, H. The Application of Ceric Ammonium Nitrate in the Synthesis of Carbodiazones. _Synth. Commun._ 2000, _30_ , 1807–1812, DOI: 10.1080/00397910008087226
  * Hwu and King 2010 Hwu, J. R.; King, K.-Y. Versatile Reagent Ceric Ammonium Nitrate in Modern Chemical Synthesis. _ChemInform_ 2010, _33_ , no–no, DOI: 10.1002/chin.200245265
  * Van 2005 _Corrosion: Materials_ ; ASM International, 2005; pp 665–671, DOI: 10.31399/asm.hb.v13b.a0006542
  * Miyazaki et al. 1983 Miyazaki, S.; Kikkawa, S.; Koizumi, M. Chemical and electrochemical deintercalations of the layered compounds LiMO2 (M = Cr, Co) and NaM’O2 (M’ Cr, Fe, Co, Ni). _Synth. Met._ 1983, _6_ , 211–217, DOI: 10.1016/0379-6779(83)90156-x
  * Neilson and McQueen 2012 Neilson, J. R.; McQueen, T. M. Bonding, Ion Mobility, and Rate-Limiting Steps in Deintercalation Reactions with ThCr2Si2-type KNi2Se4. _J. Am. Chem. Soc._ 2012, _134_ , 7750–7757, DOI: 10.1021/ja212012k
  * Wizansky et al. 1989 Wizansky, A. R.; Rauch, P. E.; Disalvo, F. J. Powerful oxidizing agents for the oxidative deintercalation of lithium from transition-metal oxides. _J. Solid State Chem._ 1989, _81_ , 203–207, DOI: 10.1016/0022-4596(89)90007-8
  * Chamorro and McQueen 2018 Chamorro, J. R.; McQueen, T. M. Progress toward Solid State Synthesis by Design. _Acc. Chem. Res._ 2018, _51_ , 2918–2925, DOI: 10.1021/acs.accounts.8b00382
  * Casas-Cabanas et al. 2016 Casas-Cabanas, M.; Kim, C.; Rodríguez-Carvajal, J.; Cabana, J. Atomic defects during ordering transitions in LiNi0.5Mn1.5O4 and their relationship with electrochemical properties. _J. Mater. Chem. A_ 2016, _4_ , 8255–8262, DOI: 10.1039/c6ta00424e
  * Furubayashi et al. 1994 Furubayashi, T.; Matsumoto, T.; Hagino, T.; Nagata, S. Structural and Magnetic Studies of Metal-Insulator Transition in Thiospinel CuIr2S4. _J. Phys. Soc. Japan_ 1994, _63_ , 3333–3339, DOI: 10.1143/jpsj.63.3333
  * Hagino et al. 1994 Hagino, T.; Seki, Y.; Nagata, S. Metal - insulator transition in CuIr2S4 : Comparison with CuIr2Se4. _Phys. C: Supercond. Appl._ 1994, _235-240_ , 1303–1304, DOI: 10.1016/0921-4534(94)91876-7
  * Isobe and Ueda 2002 Isobe, M.; Ueda, Y. Observation of Phase Transition from Metal to Spin-Singlet Insulator in MgTi2O4 with S=1/2 Pyrochlore Lattice. _J. Phys. Soc. Japan_ 2002, _71_ , 1848–1851, DOI: 10.1143/jpsj.71.1848
  * Schmidt et al. 2004 Schmidt, M.; Ratcliff, W.; Radaelli, P. G.; Refson, K.; Harrison, N. M.; Cheong, S. W. Spin Singlet Formation in MgTi2O4: Evidence of a Helical Dimerization Pattern. _Phys. Rev. Lett._ 2004, _92_ , 056402, DOI: 10.1103/physrevlett.92.056402
  * Leoni et al. 2008 Leoni, S.; Yaresko, A. N.; Perkins, N.; Rosner, H.; Craco, L. Orbital-spin order and the origin of structural distortion in MgTi2O4. _Phys. Rev. B_ 2008, _78_ , 125105, DOI: 10.1103/physrevb.78.125105
  * Radaelli et al. 2002 Radaelli, P. G.; Horibe, Y.; Gutmann, M. J.; Ishibashi, H.; Chen, C. H.; Ibberson, R. M.; Koyama, Y.; Hor, Y.-S.; Kiryukhin, V.; Cheong, S.-W. Formation of isomorphic Ir3+ and Ir4+ octamers and spin dimerization in the spinel CuIr2S4. _Nature_ 2002, _416_ , 155–158, DOI: 10.1038/416155a
  * Ishibashi et al. 2001 Ishibashi, H.; Sakai, T.; Nakahigashi, K. X-ray diffraction study on spinel compound CuIr2S4 with metal-insulator transition. _J. Magn. Magn. Mater._ 2001, _226-230_ , 233–234, DOI: 10.1016/s0304-8853(00)00638-7
  * VERWEY 1939 VERWEY, E. J. W. Electronic Conduction of Magnetite (Fe3O4) and its Transition Point at Low Temperatures. _Nature_ 1939, _144_ , 327–328, DOI: 10.1038/144327b0
  * Anderson 1956 Anderson, P. W. Ordering and Antiferromagnetism in Ferrites. _Phys. Rev._ 1956, _102_ , 1008–1013, DOI: 10.1103/physrev.102.1008
  * Iizumi et al. 1982 Iizumi, M.; Koetzle, T. F.; Shirane, G.; Chikazumi, S.; Matsui, M.; Todo, S. Structure of magnetite (Fe3O4) below the Verwey transition temperature. _Acta Crystallogr. B._ 1982, _38_ , 2121–2133, DOI: 10.1107/s0567740882008176
  * Bozin et al. 2019 Bozin, E. S.; Yin, W. G.; Koch, R. J.; Abeykoon, M.; Hor, Y. S.; Zheng, H.; Lei, H. C.; Petrovic, C.; Mitchell, J. F.; Billinge, S. J. L. Local orbital degeneracy lifting as a precursor to an orbital-selective Peierls transition. _Nat. Commun._ 2019, _10_ , 3638, DOI: 10.1038/s41467-019-11372-w
  * Torigoe et al. 2018 Torigoe, S.; Hattori, T.; Kodama, K.; Honda, T.; Sagayama, H.; Ikeda, K.; Otomo, T.; Nitani, H.; Abe, H.; Murakawa, H.; Sakai, H.; Hanasaki, N. Nanoscale ice-type structural fluctuation in spinel titanates. _Phys. Rev. B_ 2018, _98_ , 134443, DOI: 10.1103/physrevb.98.134443
  * Božin et al. 2011 Božin, E. S.; Masadeh, A. S.; Hor, Y. S.; Mitchell, J. F.; Billinge, S. J. L. Detailed Mapping of the Local Ir4+ Dimers through the Metal-Insulator Transitions of CuIr2S4 Thiospinel by X-Ray Atomic Pair Distribution Function Measurements. _Phys. Rev. Lett._ 2011, _106_ , 045501, DOI: 10.1103/physrevlett.106.045501
  * Grey and Dupré 2004 Grey, C. P.; Dupré, N. NMR Studies of Cathode Materials for Lithium-Ion Rechargeable Batteries. _Chem. Rev._ 2004, _104_ , 4493–4512, DOI: 10.1021/cr020734p
  * Reeves et al. 2019 Reeves, P. J.; Seymour, I. D.; Griffith, K. J.; Grey, C. P. Characterizing the Structure and Phase Transition of Li2RuO3 Using Variable-Tempearture 17O and 7Li NMR Spectroscopy. _Chem. Mater._ 2019, _31_ , 2814–2821, DOI: 10.1021/acs.chemmater.8b05178
  * Kimber et al. 2014 Kimber, S. A. J.; Mazin, I. I.; Shen, J.; Jeschke, H. O.; Streltsov, S. V.; Argyriou, D. N.; Valentí, R.; Khomskii, D. I. Valence bond liquid phase in the honeycomb lattice material Li2RuO3. _Phys. Rev. B_ 2014, _89_ , 081408(R), DOI: 10.1103/physrevb.89.081408
  * Park et al. 2016 Park, J. et al. Robust singlet dimers with fragile ordering in two-dimensional honeycomb lattice of Li2RuO3. _Sci. Rep._ 2016, _6_ , 25238, DOI: 10.1038/srep25238
  * Ménétrier et al. 1999 Ménétrier, M.; Saadoune, I.; Levasseur, S.; Delmas, C. The insulator-metal transition upon lithium deintercalation from LiCoO2: electronic properties and 7Li NMR study. _J. Mater. Chem._ 1999, _9_ , 1135–1140, DOI: 10.1039/A900016J
  * Nagata et al. 1994 Nagata, S.; Hagino, T.; Seki, Y.; Bitoh, T. Metal-insulator transition in thiospinel CuIr2S4. _Phys. B: Condens. Matter._ 1994, _194-196_ , 1077–1078, DOI: 10.1016/0921-4526(94)90868-0
  * Zhu et al. 2014 Zhu, Y.-Y.; Wang, R.-J.; Wang, L.; Liu, Y.; Xiong, R.; Shi, J.; An, L.-H.; Sun, D.-H. Transport Behavior in Spinel Oxide MgTi2O4. _Chin. Phys. Lett._ 2014, _31_ , 097201, DOI: 10.1088/0256-307x/31/9/097201
  * Browne et al. 2020 Browne, A. J.; Pace, E. J.; Garbarino, G.; Attfield, J. P. Structural study of the pressure-induced metal-insulator transition in LiV2O4. _Phys. Rev. Mater._ 2020, _4_ , 015002, DOI: 10.1103/physrevmaterials.4.015002
  * Kawakami et al. 1986 Kawakami, K.; Sakai, Y.; Tsuda, N. Metal-Insulator Transition in LixZn1-xV2O4. _J. Phys. Soc. Japan_ 1986, _55_ , 3174–3180, DOI: 10.1143/jpsj.55.3174
  * Onoda et al. 1997 Onoda, M.; Imai, H.; Amako, Y.; Nagasawa, H. Spin fluctuation and the transport mechanism in vanadium oxide spinels with a metal-insulator transition. _Phys. Rev. B_ 1997, _56_ , 3760–3771, DOI: 10.1103/physrevb.56.3760
  * Khomskii and Mizokawa 2005 Khomskii, D. I.; Mizokawa, T. Orbitally Induced Peierls State in Spinels. _Phys. Rev. Lett._ 2005, _94_ , 156402, DOI: 10.1103/physrevlett.94.156402
  * Yang et al. 2008 Yang, H. X.; Zhu, B. P.; Zeng, L. J.; Tian, H. F.; Ma, C.; Shi, J.; Li, J. Q. Structural modulation in the orbitally induced Peierls state of MgTi2O4. _J. Condens. Matter Phys._ 2008, _20_ , 275230, DOI: 10.1088/0953-8984/20/27/275230
  * Hirai et al. 2022 Hirai, D.; Kojima, K.; Katayama, N.; Kawamura, M.; Nishio-Hamane, D.; Hiroi, Z. Linear Trimer Molecule Formation by Three-Center–Four-Electron Bonding in a Crystalline Solid RuP. _J. Am. Chem. Soc._ 2022, _144_ , 17857–17864, DOI: 10.1021/jacs.2c06173
  * Britto et al. 2015 Britto, S.; Leskes, M.; Hua, X.; Hébert, C.-A.; Shin, H. S.; Clarke, S.; Borkiewicz, O.; Chapman, K. W.; Seshadri, R.; Cho, J.; Grey, C. P. Multiple Redox Modes in the Reversible Lithiation of High-Capacity, Peierls-Distorted Vanadium Sulfide. _J. Am. Chem. Soc._ 2015, _137_ , 8499–8508, DOI: 10.1021/jacs.5b03395
  * Hiroi 2015 Hiroi, Z. Structural instability of the rutile compounds and its relevance to the metal–insulator transition of VO2. _Prog. Solid. State Chem._ 2015, _43_ , 47–69, DOI: 10.1016/j.progsolidstchem.2015.02.001
  * Perdew et al. 2008 Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E.; Constantin, L. A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. _Phys. Rev. Lett._ 2008, _100_ , 136406, DOI: 10.1103/physrevlett.100.136406
  * Thurber and Tycko 2009 Thurber, K. R.; Tycko, R. Measurement of sample temperatures under magic-angle spinning from the chemical shift and spin-lattice relaxation rate of 79Br in KBr powder. _J. Magn. Reson._ 2009, _196_ , 84–87, DOI: 10.1016/j.jmr.2008.09.019

Figure 9: *

Table of contents graphic.