Redox Mechanisms, Structural Changes, and Electrochemistry of the Wadsley–Roth LixTiNb2O7 Electrode Material

The TiNb2O7 Wadsley–Roth phase is a promising anode material for Li-ion batteries, enabling fast cycling and high capacities. While already used in commercial batteries, many fundamental electronic and thermodynamic properties of LixTiNb2O7 remain poorly understood. We report on an in-depth first-principles study of the redox mechanisms, structural changes, and electrochemical properties of LixTiNb2O7 as a function of Li concentration. First-principles electronic structure calculations reveal an unconventional redox mechanism upon Li insertion that results in the formation of metal–metal bonds. This metal dimer redox mechanism has important structural consequences as it results in a shortening of cation-pair distances, which in turn affects lattice parameters of the host and thereby alters Li site preferences as the Li concentration is varied. The new insights about redox mechanisms in TiNb2O7 and their effect on the structure and Li site preferences provide guidance on how the electrochemical properties of a promising class of anode materials can be tailored by exploiting the tremendous structural and chemical diversity of Wadsley–Roth phases.


INTRODUCTION
Lithium-ion batteries continue to dominate the secondary energy storage space.A large fraction of energy storage demand originates in the automotive sector, 1−3 where electric motors show an efficiency of over 90% and enable the elimination of CO 2 , CO, NO x , and SO x exhaust emitted by internal combustion engines. 4Despite their benefits and widespread usage, current lithium-ion battery technologies that rely on graphite anodes possess lower charge rates, power densities, and operating temperature ranges than are needed for future generations of electric vehicles. 5,6−19 The TiNb 2 O 7 compound is a commercialized Wadsley−Roth phase that can reversibly cycle while achieving a capacity of 340 mA h g −1 . 16,20−23 TiNb 2 O 7 was first demonstrated to intercalate lithium by Cava in 1983 7 and was first cycled against lithium in 2011. 11TiNb 2 O 7 has since displayed the capability to supply high power densities at high charge rates. 24,25espite an increased interest in Wadsley−Roth phases as anodes in Li-ion batteries, [14][15][16]18,26,27 there is still a limited understanding of the electronic, thermodynamic, and structural properties of these phases as a function of the degree of Li insertion.This paper examines the full lithiation profile of TiNb 2 O 7 using first-principles statistical mechanics calculations to understand the redox mechanisms and their effect on the structural properties and Li site preferences. We discoveran unconventional redox mechanism upon Li insertion into TiNb 2 O 7 that is not localized on individual transition-metal cations but instead occurs on the bonding states that arise from the hybridization between the t 2g orbitals of edge-sharing transition-metal cations.The filling of these bonding states leads to a shortening of metal−metal distances, which has structural consequences for the host more broadly, thereby modifying the Li-site preference as a function of Li concentration.The metal−metal dimer redox mechanism, enabled by crystallographic features shared by all Wadsley− Roth phases, is likely to be relevant for a broad family of promising anode materials for Li-ion batteries.

METHODS
First-principles electronic structure calculations were performed with the density functional theory (DFT) using the generalized gradient approximation (GGA) parametrization of Perdew, Burke, and Ernzerhof (PBE) 28 as implemented in the Vienna ab initio simulation package (VASP). 29,30Several benchmark calculations were also performed with the strongly constrained and appropriately normed (SCAN) meta GGA. 31 Past studies have shown that SCAN is able to accurately describe the electronic structure of transition-metal oxides in high oxidation states when compared to predictions of the random phase approximation. 32Interactions between core and valence electrons were treated with the projector augmented wave (PAW) method. 33,34A plane-wave energy cutoff of 650 eV and a reciprocal space discretization of 25 K-points per Å −1 was used.All calculations were performed spin-polarized with magnetic moments initialized in a ferromagnetic state.The systematic enumeration of different Livacancy orderings over the interstitial sites of the TiNb 2 O 7 Wadsley− Roth crystal structure was performed with the clusters approach to statistical mechanics (CASM) 35,36 simulation package.Finite temperature electrochemical properties were calculated with CASM by applying Monte Carlo simulations to cluster expansion Hamiltonians trained to a large DFT data set of formation energies. 35,37ncertainty quantification was performed within a Bayesian framework 38−40 by sampling multiple cluster expansion Hamiltonians from a posterior distribution and calculating electrochemical properties for each sampled cluster expansion. 41A standard deviation of 1 meV/atom on the calculated formation energies was used in the Gaussian likelihood distribution, and a standard deviation of 49 meV was assumed for the Gaussian prior distribution of each effective cluster interaction (ECI), the coefficients of the cluster expansion.The mean of the prior distribution of the ECI was chosen to ensure that the predicted ground states and low energy structures are consistent with the DFT predictions as described by Ober et al. 41

TiNb 2 O 7
Wadsley−Roth Host Structure.Wadsley− Roth phases are a family of chemistries with crystal structures derived from the perovskite-like ReO 3 phase. 7,42They consist of infinitely long blocks of n × m corner-sharing transitionmetal oxygen octahedra.The blocks can tile space in different ways with octahedra at the peripheries of each block sharing edges with octahedra of neighboring blocks.Figure 1 shows the Wadsley−Roth structure adopted by TiNb 2 O 7 , which is made of 3 × 3 blocks.
The unit cell of the TiNb 2 O 7 Wadsley−Roth structure of Figure 1 has nine transition-metal sites.There are a total of 44 symmetrically distinct ways of arranging Ti and Nb over the cation sites of the primitive unit cell of TiNb 2 O 7 .The arrangement with the lowest energy as predicted with DFT-PBE is shown in Figure 1. 15,43The higher oxidation state Nb 5+ cations prefer to occupy the corner-sharing octahedra at the center of the block, while lower oxidation state cations, such as Ti 4+ , tend to segregate to the edge-sharing octahedra at the peripheries of each block.Experimentally synthesized forms of TiNb 2 O 7 exhibit some degree of disorder among Ti and Nb because the phase is cooled from high temperatures.Nevertheless, neutron diffraction studies have shown that Ti preferentially occupies edge sites, while Nb preferentially occupies the corner-sharing site at the center of the block. 44e use the lowest energy Ti−Nb ordering of Figure 1 as a model to explore redox mechanisms and Li-site preferences upon Li insertion.
The TiNb 2 O 7 Wadsley−Roth crystal can host Li-ions in four types of interstitial sites coordinated by oxygen. 15These are shown in Figure 1.Two groups of interstitial sites are pyramidally coordinated by five oxygen ions.These are termed P s and P s ′.They differ by the number of transition-metal cations that share edges with the sites; the P s sites share six edges with the neighboring transition metals while the P s ′ sites share seven.There are six P s sites and two P s ′ sites within each unit cell of TiNb 2 O 7 .The TiNb 2 O 7 host can also accommodate Li in horizontal and vertical window sites, labeled W h and W v , respectively.Both sites are coordinated by four oxygen ions in a square planar configuration.There are four W h and four W v sites per unit cell of TiNb 2 O 7 residing in the blocks, as illustrated in Figure 1.

Li Insertion into TiNb 2 O 7 at
Zero Kelvin.The energies of 937 symmetrically distinct orderings of Li and vacancies over the interstitial sites of TiNb 2 O 7 (i.e., P s , P s ′, W v and W h ) were calculated with DFT-PBE.In each of these structures, the TiNb 2 O 7 host has the lowest energy Ti−Nb ordering, as shown in Figure 1.The different Li-vacancy configurations were enumerated with CASM 36 and include different orderings within the primitive unit cell and in super cells, in which the b axis along the block lengths are doubled or quadrupled.Figure 2a shows the calculated formation energies as a function of the Li concentration.Many Li-vacancy arrangements reside on the convex hull.However, with the exception of three ordered phases at x = 0.66, x = 2.833, and x = 3.5, most are weakly stable and very close in energy to other structures that have similar Li-vacancy arrangements and compositions.
The voltage at 0 K is linearly related to the slope of the convex hull as a function of lithium concentration. 37The 0 K voltage profile is shown in Figure 2b.Each ordered phase with an energy on the convex hull appears as a step in the 0 K voltage curve, while each plateau corresponds to a two-phase coexistence between a pair of ordered phases. 37,45,46The voltage curve, which starts at 2 V upon insertion of a dilute concentration of Li, shows that TiNb 2 O 7 can intercalate more than 5 Li per formula unit, while maintaining a positive voltage.
−49 Figure 2c plots the number of each type of Li site that is occupied in each ground-state ordering.The pyramidal P s sites fill initially and have the highest occupancy at all Li concentrations.For the Ti and Nb ordering shown in Figure 1, the vertical window sites begin to fill at x = 0.66.Both the vertical and horizontal window sites, W v and W h , fill steadily beyond x = 0.66, but the largest fraction of Li continue to fill the pyramidal P s sites until they are close to saturated at around x = 3.5.Close to half the vertical window sites are filled beyond x = 2.5, which is consistent with neutron diffraction studies. 50The pyramidal P s ′ sites, which share seven edges with neighboring transitionmetal cations, only start to fill gradually past x = 2.  insulator.At this composition, the d 0 Ti 4+ and Nb 5+ cations are in their maximum oxidation states.Figure 3a, shows that DFT-PBE predicts a large band gap in the electronic density of states (DOS) of TiNb 2 O 7 .The gap separates the filled valence bands, which primarily have an oxygen p character, and the bottom of the empty conduction bands, which are derived from the t 2g orbitals of the octahedrally coordinated transition metals (i.e., the d xy , d xz , and d yz orbitals in an octahedral environment).This is consistent with past calculations by Catti et al. 50and Griffith et al. 15 The addition of Li to TiNb 2 O 7 leads to a reduction of the formal oxidation states of the Ti and Nb cations.At low Li concentrations, DFT-PBE predicts that the electrons donated to the host by Li tend to delocalize over the different transition metals, with a slight enhancement in the occupancy of the d xy orbital on the central Nb site. 15,51This is shown in Figure 3b for LiTiNb 2 O 7 .There is some degree of spin polarization, with the spin up DOS (positive) having more states below the Fermi level than the spin down DOS (negative).Figure 3b shows that the donated electrons fill the bottom of the conduction band without substantially altering the DOS of the conduction bands of pristine TiNb 2 O 7 .Figure 3c  The redox mechanism changes qualitatively with increasing Li concentration beyond x = 1.This is evident in Figure 4 for Li 1.66 TiNb 2 O 7 .New states emerge below the Fermi level (labeled δ in Figure 4b) that are disconnected from the more itinerant bands derived from the t 2g states.Figure 4a plots the electronic charge density due to the breakaway peaks in the DOS of Li 1.66 TiNb 2 O 7 (Figure 4b), clearly showing that the charge density associated with these states is concentrated between a pair of edge-sharing Nb cations.An analysis of the local projected density of states of each of the two Nb atoms of the pair shows that their d xy orbitals have energies that coincide with the peaks below the Fermi level in Figure 4b.The electronic charge density due to the remaining DOS that extends from the top of the breakaway peaks up to the Fermi level is shown in Figure 4c.These states have a combined electronic charge density that is more uniformly distributed throughout the crystal and more centered around individual transition-metal cations than between metal cations.
The enhanced electron charge density between the pair of edge-sharing Nb in Figure 4a is consistent with a filled bonding  state that arises when the d xy orbitals of a pair of edge-sharing transition-metal cations hybridize, as schematically illustrated in Figure 5. 52−54 The hybridization between edge-sharing d xy orbitals leads to metal−metal dimer formation and generates bonding states that have a lower energy than the unhybridized d xy orbitals. 52The bonding states are therefore favorable redox centers to accommodate the electrons donated by Li to the host.It is clear in the DOS of Figure 4b that a pair of electrons with opposite spin fills the bonding state.The filling of the bonding state of a hybridized metal−metal bond will lead to a shortening of the distance between the neighboring transitionmetal cations. 54This is indeed predicted to occur, with the distance between the pair of Nb cations that form a metalmetal bond contracting from a value of 3.09 Å at x = 1 to 2.69 Å at x = 1.66.
Additional metal−metal bonds form between edge-sharing octahedra, as electrons are added to Li x TiNb 2 O 7 upon further insertion of Li.At x = 2.5, for example, a complex of metal− metal bonds become evident in the charge density plot of Figure 6.Each edge-sharing pair of transition-metal cations with an enhanced charge density along the bond axis also has a shortened metal−metal distance, which is consistent with the filling of the bonding states that arises from metal−metal dimer formation.Several transition-metal cations even participate in two metal−metal bonds.A transition metal with two edgesharing cation neighbors can form a separate bond with each neighbor with orthogonal onsite t 2g orbitals. 54iNb 2 O 7 starts out as an insulator, but its electronic conductivity increases dramatically upon the addition of Li. 15 This is consistent with the calculated electronic density of states plots of Figures 3a,b, 4b, and 6b, which show a sizable electronic density of states at the Fermi-level.This suggests the presence of itinerant electrons and more metallic behavior.Even in the presence of localized metal−metal bonds, which have electronic densities of states below the Fermi level (Figures 4b and 6b), there continues to be a large density of states at the Fermi level corresponding to more itinerant electrons.
While the DOS plot of Figure 3 shows that there is some degree of spin polarization at x = 1, the DOS plots of Figures 4   + , where g = 2 and s = 1/2.The light blue line connects the magnetic moment per Li for the ground states of Li x TiNb 2 O 7 .Figure 7 shows that at dilute concentrations the electrons donated to the host adopt a spinpolarized configuration, resulting in a net magnetic moment of the crystal.However, above x = 1, the magnetic moment per Li ion decreases to negligible values.It is above this concentration that the bonding states of the metal−metal dimers are predicted to accommodate the electrons donated by Li to the host.Figure 7 also shows the magnetic moment per Li (in units of μ B ) as measured experimentally by Griffith et al. 15 The qualitative agreement between the calculated and measured magnetic moments per Li ion is very good.For comparison, Figure 7 also shows the calculated magnetic moment per Li in the ground-state configurations as calculated with DFT-SCAN.The same trend is predicted, and the values are similar to those predicted with DFT-PBE.

Effect of Redox Mechanism on
Structure.The pristine TiNb 2 O 7 crystal structure has highly distorted MO 6 octahedra (M = Ti or Nb) due to the oxidation states of Ti and Nb and the large number of edge-sharing MO 6 octahedra.The d 0 Ti 4+ and Nb 5+ cations are susceptible to second-order Jahn− Teller distortions when octahedrally coordinated by oxygen. 55,56This causes displacement of the cations away from the center of their coordinating octahedra.The edge-sharing MO 6 octahedra of TiNb 2 O 7 also undergo significant distortions due to the strong electrostatic repulsion between neighboring Ti 4+ and Nb 5+ cations. 43This repulsion increases the distance between edge-sharing cations, causing a further off-centering of each cation that simultaneously induces collateral distortions of their surrounding oxygen octahedron. 43he donation of electrons to the host upon Li insertion undoes many of the octahedral distortions that are initially present in TiNb 2 O 7 .The reduction of the Ti 4+ and Nb 5+ cations eliminates their susceptibility to second-order Jahn− Teller distortions.Furthermore, the redox mechanism described in the previous section, which leads to metal− metal dimer formation, has structural consequences that affect the lattice parameters of the host.Each edge-sharing pair of transition metal cations that hybridize to form metal−metal dimers undergo a contraction that pulls the cations back toward the centers of their octahedra.Especially notable is the abrupt contraction in a subset of the bond lengths that occurs between x = 1 and x = 2.In the lowest energy ground-state structures (gold points), this occurs first between Nb−Nb pairs, as is evident in Figure 8a, with the two gold points at x = 1.33 and x = 1.66, having values of approximately 2.7 Å. Edge-sharing Nb−Ti pairs (Figure 8c) also exhibit dimer formation, with a subset contracting to values close to 2.8 Å, but the contractions only set in appreciably after Nb−Nb pairs have started to contract.The contraction between edge-sharing Ti−Ti pairs (Figure 8b) is  less pronounced and begins to occur only at higher Li concentrations.
The onset of metal−metal dimer formation leads to sizable dimensional changes in the host.Figure 9a 57 Here as well, the agreement is very good.The b-lattice parameter is predicted to increase abruptly between x = 1 and x = 2.5 and then levels off at higher Li concentrations.The same trend is observed experimentally. 57 useful metric of the dimensional changes of the host is the strain order parameter , 19,58 which measures tetragonal distortions along the 3 × 3 block axis of Li x TiNb 2 O 7 .The Cartesian strains, E xx , etc., appearing in the expression of the strain order parameter, e 3 , are defined with respect to a Cartesian coordinate system whose ẑaxis is parallel to the block length of the TiNb 2 O 7 host structure.The Cartesian strains are calculated as Hencky strains 58 relative to the dimensions of the fully relaxed TiNb 2 O 7 host structure without any Li. Figure 9c collects the e 3 strain for all 937 fully relaxed structures of Li x TiNb 2 O 7 .The e 3 strains of the groundstate structures are connected by a light blue line.Similar to the variation in the b-lattice parameter in Figure 8b, the e 3 strain order parameter shows a rapid increase in a narrow Li composition interval between x = 1 and x = 2.5.A positive value of e 3 signifies an expansion along the block length and a contraction along the block waist.The abrupt increase in e 3 and in the b lattice parameter between x = 1 and x = 2.5 can be attributed to the onset of metal−metal dimer formation, which results in a reduction of the distance between edge-sharing transition-metal cations and a straightening of the highly distorted octahedra of the pristine TiNb 2 O 7 crystal structure.
The degree with which the MO 6 octahedra of TiNb 2 O 7 distort upon Li insertion can be analyzed by projecting the ionic displacements of each octahedron on symmetry adapted collective displacements as described in the Supporting Information.Figure 10a,b, for example, shows the average amplitude of the symmetry preserving breathing mode of the TiO 6 and NbO 6 octahedra as a function of Li concentration.Also shown is one standard deviation spread around the average.The averages were collected from the 937 Li-vacancy orderings in Li x TiNb 2 O 7 as relaxed with DFT-PBE. Figure 10 shows that the average volumes of the NbO 6 and TiO 6 octahedra increase steadily with the concentration of Li.Of particular interest in Figure 10b is the abrupt increase between x = 1 and x = 2 in the volume of the NbO 6 octahedra that share four edges with neighboring octahedra (purple curve).This concentration interval coincides with the start of the metal−metal dimer redox mechanism, with the first pairs that form dimers involving Nb cations in octahedra that share four edges with neighboring octahedra.Figure 10b also shows that the Nb in the corner-sharing octahedron at the center of the 3 × 3 block (orange curve) expands between x = 0 and x ≈ 1 but  then remains more or less constant until x ≈ 3.This is consistent with previous first-principles studies, 15,59,60 which showed that the transition metal cations of the corner-sharing octahedra at the center of the blocks of Wadsley−Roth phases play an important role in the redox processes at dilute Li concentrations.
There are a total of 15 symmetry adapted collective displacement modes for a perfect MO 6 reference octahedron.These naturally divide into six irreducible subspaces, as described in the Supporting Information.The breathing mode shown in the inset of Figure 10 forms a one-dimensional subspace.Another one is of dimension two and is spanned by the well-known first-order Jahn−Teller collective displacement modes.There are four additional irreducible subspaces, each of dimension three (see the Supporting Information for more details).Two of these are useful to analyze the distortion modes of Wadsley−Roth phases such as Li x TiNb 2 O 7 as they measure the extent of second-order Jahn−Teller distortions and of the octahedral shape changes that accommodate the changes in the distance between edge-sharing cations.
Figure 11a shows the three symmetry adapted collective displacement modes that characterize a second-order Jahn− Teller distortion of a d 0 transition-metal coordinated by an octahedron of oxygen ions.The three collective displacement modes of Figure 11a each describe an off-centering of the transition metal along one of the Cartesian axes and form a basis on which to describe an arbitrary off-centering.A measure of the degree of off-centering is the Euclidean length of the three amplitudes of the collective distortion modes of Figure 11a, as described in the Supporting Information.Figure 11b,c shows the average Euclidean length of the off-centering distortion mode along with a one standard deviation spread for the TiO 6 and NbO 6 octahedra as a function of Li concentration.The averages were again taken over the 937 fully relaxed Li x TiNb 2 O 7 structures.The large values at low Li concentrations indicate that the cations are displaced away from the center of their coordinating octahedra.The offcentering is more pronounced for the cations that share more edges with neighboring octahedra.The Nb of the cornersharing octahedron at the center of the 3 × 3 block (orange curve in Figure 11c) exhibits the smallest degree of offcentering, which decreases abruptly around x ≈ 1.As the central Nb reduces its oxidation state from its starting value of Nb 5+ , its susceptibility to a second-order Jahn−Teller distortion is lowered, and the degree to which it is offcentered decreases.The other transition-metal cations, which are more off-centered at x = 0 than the central Nb due to the electrostatic repulsion with neighboring edge-sharing transition-metal cations, also become less off-centered with an increasing Li concentration.The decrease in the degree of offcentering occurs at slightly higher Li concentrations than that of the central Nb and coincides with the composition at which metal−metal dimers start to form.The formation of bonding states between neighboring transition-metal cations leads to metal−metal dimers, as described above, and an overall centering of the transition-metal cations within their octahedra.
Figure 12a shows a second set of symmetry adapted collective displacement modes of MO 6 octahedra whose amplitudes in Li x TiNb 2 O 7 undergo large changes with Li concentrations.The three collective displacement modes of Figure 12a also form a basis to describe octahedral distortions that reside within a T 2u irreducible subspace, with each collective displacement involving four equatorial oxygen ions that distort perpendicular to their equatorial plane.These displacement modes measure the collateral distortions of the oxygen octahedra in response to the large relaxations that lead to an off-centering of edge-sharing transition-metal cations. 43igure 12b,c plots the average Euclidian distance of the amplitudes of the three orthogonal displacement modes for the TiO 6 and NbO 6 octahedra as a function of Li concentration.The octahedral distortions are large at dilute Li concentrations but decrease substantially over a small concentration interval between x = 1 and x = 2 upon the formation of metal−metal dimers.Upon formation of metal−metal dimers, the transitionmetal cations move to the center of their octahedra, thereby allowing the oxygen ions to adopt positions that are closer to those of an ideal octahedron.
3.5.Finite Temperature Electrochemical Properties.Room-temperature electrochemical properties were calculated by combining cluster expansions with Monte Carlo simulations. 35,37A cluster expansion 61,62 is a surrogate model that interpolates the energies of different Li-vacancy orderings as calculated with a computationally expensive first-principles method, such as DFT-PBE.The cluster expansion can then be used in Monte Carlo simulations to calculate the energies of microstates sampled in large unit cells according to the probability distribution of statistical mechanics. 37The cluster expansions used in this study were trained to the formation energies of the 937 Li-vacancy orderings in Li x TiNb 2 O 7 shown in Figure 2a.A Bayesian approach was followed to enable uncertainty quantification of calculated thermodynamic properties due to numerical noise on the training data and cluster expansion truncation.Ten different cluster expansions were sampled from a Bayesian posterior probability distribution as described in Ober et al. 41 Each cluster expansion was used in Monte Carlo simulations to calculate equilibrium voltage curves (related to the Li chemical potential according to the Nernst equation 37 ) and equilibrium Li site occupancies as a function of the overall Li concentration.
Figure 13a shows ten voltage curves as a function of Li concentration, each calculated with a different cluster expansion sampled from a Bayesian posterior distribution.The voltage curves were calculated with grand canonical Monte Carlo simulations, which generate the average Li concentration at each Li chemical potential and temperature.The smooth sloping voltage curves reflect a solid solution.The Monte Carlo simulations predict that the Li ions and vacancies are disordered at room temperature.
Similar to experimentally measured voltage profiles of Li x TiNb 2 O 7 , 12,25,63 the calculated voltage curve exhibits an initial steep decrease between x = 0 and x ≈ 0.5, which is followed by a flatter concentration dependence between x ≈ 0.5 and x ≈ 2. Beyond x ≈ 2, the decrease in voltage with the Li concentration is again steeper and exhibits several weak steps.We note that the middle portion, while having a shallow slope, is not as flat as that exhibited by experimental curves. 11igure 13b shows the Li site occupancy as a function of the Li concentration.Each Li site has ten curves, one for each cluster expansion sampled from the posterior distribution.It is clear in Figure 13b that differences in the predicted site occupancy as calculated with the different cluster expansions are small.The predicted trends in Figure 13b are consistent with those predicted at 0 K. Li primarily fills the pyramidal P s sites, which steadily become enriched with Li until they saturate around x = 4.The vertical window sites, W v , also accommodate Li ions early on, but do not saturate until approximately x = 5.The horizontal window sites, W h , start filling only around x = 2.The pyramidal P s ′ sites are overall the least favored sites and saturate only at the highest Li concentration.
It is of interest to analyze the Li site occupancy based on the surrounding Ti concentration.This is shown in Figure 14.Each Li site is distinguished by the number of Ti cations that share an edge with the Li site.The darker blue curves and uncertainty bounds track the concentration of Li in sites that are surrounded by more Ti, as indicated in the insets, while the lighter green curves track the Li concentration in sites surrounded by more Nb.Overall, Figure 14 shows that Li tends to fill sites differently, depending on the amount of coordinating Ti.This is very starkly evident for the P s ′ sites.

DISCUSSION
Our first-principles study of the Li x TiNb 2 O 7 Wadsley−Roth phase has shed light on the redox mechanisms accompanying the electrochemical lithiation of TiNb 2 O 7 .At least two redox mechanisms are identified based on an analysis of 937 fully relaxed Li x TiNb 2 O 7 structures.At dilute Li concentrations, DFT-PBE calculations predict that electrons donated by Li reduce Ti and Nb more or less uniformly.The calculations predict some degree of spin polarization for x < 1.At higher Li concentrations, DFT-PBE calculations predict that the redox mechanism changes qualitatively, shifting from the filling of cation-centric t 2g orbitals to the filling of the bonding states  that arise when the t 2g orbitals of edge-sharing transition-metal cations hybridize to form metal−metal dimers.The redox then occurs on extended molecular orbital-like states with enhanced charge density between pairs of edge-sharing transition-metal cations.The first metal−metal dimers to form involve Nb cations that occupy the sites with the highest number of edgesharing neighbors.The large electrostatic interactions between highly oxidized edge-sharing neighbors increase the driving force to undergo redox at those sites in order to lower their formal oxidation state.Nb−Nb dimer formation is followed by Nb−Ti dimer formation with less pronounced activity predicted to occur between Ti−Ti pairs.
The dimer formation between edge-sharing transition metals has structural consequences.The distance between metal− metal pairs that host the electrons donated by Li in bonding states undergoes a contraction, which in turn induces a straightening of the oxygen octahedra surrounding the affected transition-metal cations.This leads to an elongation of the block length of the TiNb 2 O 7 host that has macroscopic ramifications.The predicted variations in volume and b lattice parameter as a function of Li concentration are in good agreement with the experiment, 57 indicating that DFT-PBE is capable of accurately describing the redox mechanisms in this material.The structural distortions of the host induced by metal−metal dimer formation affect not only the macroscopic dimensions of the crystal but also those of the interstitial Li sites.The window sites, for example, are highly distorted in the pristine TiNb 2 O 7 structure and unfavorable for Li occupancy.Above x ≈ 2, however, when the MO 6 octahedral distortions become less extreme, the window sites become more square planar and more favorable for Li occupancy.
The predicted variation in the magnetic moment of Li x TiNb 2 O 7 as a function of Li concentration is also in very good agreement with the measurements of Griffith et al. 15 DFT-PBE predicts some degree of spin polarization that leads to a net magnetic moment at low Li concentrations.The net magnetization is predicted to drop to negligible values, however, once the metal−metal dimer redox mechanism commences.The DFT-PBE calculations predict that the bonding states associated with the metal−metal dimers are filled by an equal number of spin up and spin down electrons and do not contribute to a net magnetic moment.Griffith et al. 15 suggested that a Hubbard correction to DFT-PBE is necessary to describe the electronic structure of Li x TiNb 2 O 7 at dilute Li concentrations.The analysis of a large number of Livacancy orderings at dilute concentrations in the current study, however, has shown that DFT-PBE without a Hubbard U correction is already capable of predicting the observed magnetic behavior as a function of Li concentration.
We expect that a redox mechanism involving the bonding states of metal−metal dimers is not restricted to Li x TiNb 2 O 7 , but it is common in other Wadsley−Roth phases as well.In fact, similar metal−metal bonding has been predicted to occur in Li x PNb 9 O 25 . 19However, because Li x PNb 9 O 25 has only one type of transition metal and has a higher degree of cation ordering than Li x TiNb 2 O 7 , the metal−metal bonds are more extended, and the electronic states that emerge are more delocalized.Due to the presence of Ti and Nb disorder in Li x TiNb 2 O 7 , in contrast, the metal−metal dimer formation is more localized on individual edge-sharing pairs and therefore, more apparent as a mechanism of redox.Nb is known to form metal−metal bonds in other compounds, including NaNb 10 O 18 64 and NaNb 3 O 5 F, 65 while metal−metal bonds involving Mo have been characterized in NaMoO 2 . 66−69 Other compounds exhibiting molecular-orbital redox mechanisms include Na 2 Mn 3 O 7 32 and Li x ScMo 3 O 8 . 68,69In Na 2 Mn 3 O 7 , redox has been predicted to occur on antibonding states distributed over an extended ring of π-bonded Mn and oxygen orbitals surrounding a cation vacancy. 32In Li x ScMo 3 O 8 , charge donated by Li is accommodated on molecular orbital-like states derived from Mo metal trimer clusters formed by the hybridization of t 2g orbitals. 68,69he redox mechanism described here for Li x TiNb 2 O 7 induces structural changes due to the contraction in the distance between edge-sharing transition metal cations to form a favorable bonding state.−79 While significant structural changes due to redox processes are undesirable as they can lead to mechanical damage of the electrode material and hysteresis phenomena, 80,81 the redox mechanism involving metal dimer formation is less extreme than coordination changing redox mechanisms.The crystallographic diversity of Wadsley−Roth phases, with widely varying numbers and distributions of edge-sharing octahedra, opens up opportunities to tailor the sequence of redox processes and their structural consequences.
As a final comment, we have discussed predictions for only one particular Ti−Nb ordering within the Wadsley−Roth structure of TiNb 2 O 7 .The Supporting Information shows that similar variations in the b-lattice parameter and volume occur for a different Ti−Nb ordering over the cation sites of the TiNb 2 O 7 Wadsley−Roth structure.Many of the qualitative predictions of electrochemical properties for the Ti−Nb ground state ordering are also predicted for different Ti−Nb orderings, as summarized in the Supporting Information.

CONCLUSIONS
Our comprehensive first-principles study of Li insertion into the TiNb 2 O 7 Wadsley−Roth phase has revealed an unusual redox mechanism that has important consequences for the structure of the host and Li site preferences as a function of Li concentration.While electrons fill t 2g orbitals centered on transition-metal cations of TiNb 2 O 7 at dilute Li concentrations, they are accommodated by the bonding states of metal−metal dimers above x ≈ 1 in Li x TiNb 2 O 7 .The transition from cation centric redox to dimer bond redox centers results in significant structural changes of the host unit cell and oxygen octahedra coordinating transition-metal cations.The metal−metal dimer formation has consequences for the magnetic and electrochemical properties of the compound.The good agreement between predicted and experimentally measured magnetic moments and lattice parameter variations gives confidence in the validity of the predicted dimer redox mechanism.The insights of this work provide guidance as to how electrochemical properties can be tuned in early transition-metal oxide intercalation compounds by exploiting the rich structural and chemical diversity of Wadsley−Roth phases. 43ASSOCIATED CONTENT * sı Supporting Information The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.chemmater.3c02003.

Figure 1 .
Figure 1.Idealized Wadsley−Roth crystal structure of TiNb 2 O 7 consisting of corner-sharing and edge-sharing TiO 6 (light blue) and NbO 6 (dark blue) octahedra.The Ti and Nb of this model adopt their lowest energy arrangement over the octahedrally coordinated cation sites.The TiO 6 and NbO 6 octahedra are highly distorted in the fully relaxed TiNb 2 O 7 structure (not shown).Intercalated Li ions can occupy pyramidal and window sites.
plots the electronic charge density for the states between the bottom of the conduction band and the Fermi level of LiTiNb 2 O 7 .The electronic charge density concentrates around the transitionmetal cations and adopts the characteristic charge density distribution of t 2g orbitals (i.e., d xy , d xz , and d yz ).

Figure 2 .
Figure 2. (a) Formation energies of Li x TiNb 2 O 7 of 937 Li-vacancy orderings over the pyramidal and window sites of TiNb 2 O 7 having the Ti−Nb ordering of Figure 1.(b) 0 K voltage curve as a function of composition for Li x TiNb 2 O 7 .(c) Lithium occupancy of the pyramidal and window sites of Li x TiNb 2 O 7 in the ground-state orderings.

Figure 3 .
Figure 3. Electronic density of states (DOS) of (a) TiNb 2 O 7 and (b) LiTiNb 2 O 7 as calculated with DFT-PBE.The positive (negative) DOS corresponds to spin up (down) states.(c) Electronic charge density corresponding to the filled states in the conduction band of LiTiNb 2 O 7 .

Figure 4 .
Figure 4. (a) Electronic charge density corresponding to the peaks labeled δ in the DOS of Li 1.66 TiNb 2 O 7 shown in (b).(c) Electronic density of states for the remaining filled states below the Fermi level.

Figure 5 .
Figure 5. d xy orbitals of edge-sharing transition-metal cations can hybridize to form bonding and antibonding states.The bonding states have a lower energy than the unhybridized d xy states and can host two electrons of opposite spin.

Figure 6 .
Figure 6.Electronic DOS of Li 2.5 TiNb 2 O 7 and electronic charge density of states extending from the bottom of the conduction band up to the Fermi level.

Figure 8
plots the edge-sharing metal−metal bond lengths as a function of the Li concentration collected from the 937 fully relaxed Li x TiNb 2 O 7 structures.It is insightful to inspect the edge-sharing Nb−Nb, Nb−Ti, and Ti−Ti pair distances separately, as shown in Figure 8a−c, respectively.The nearest neighbor distances of edge-sharing metal−metal pairs in the ground-state configurations are shown in gold.The figures show a clear trend toward an overall contraction of the edgesharing metal−metal bonds with increasing Li concentration.

Figure 7 .
Figure 7. Magnetic moments normalized by the number of Li ions of 937 Li x TiNb 2 O 7 structures as calculated with DFT-PBE.The magnetic moments per Li of the ground-state structures are connected by the blue line.The red points are the magnetic moments per Li of the ground-state structures as calculated with DFT-SCAN.Orange points are magnetic moments per Li as measured by Griffith et al.15

Figure 8 .
Figure 8.Bond lengths in Li x TiNb 2 O 7 structures with formation energies within 50 meV/atom of the convex hull for edge-sharing (a) Nb−Nb pairs, (b) Ti−Ti pairs, and (c) Ti−Nb pairs.Gold points refer to the pair distances in ground state structures.
collects the change in volumes of the relaxed ground-state structures of Li x TiNb 2 O 7 relative to that of TiNb 2 O 7 and compares them to the experimentally measured 57 changes in volume.The agreement between the volume change, as calculated with DFT-PBE and the measured volume change, is very good.Figure 9b compares the change in the calculated lattice parameter b of the Li x TiNb 2 O 7 host, which is parallel to the 3 × 3 blocks, for the ground-state structures to the corresponding experimental values.

Figure 9 .
Figure 9. (a) Comparison of measured and calculated percent changes in the volume of the Li x TiNb 2 O 7 unit cell.Points labeled as triangles were measured experimentally by Guo et al. 57 Circles are for the ground-state structures of Li x TiNb 2 O 7 as calculated with DFT-PBE.(b) Comparison of the percent change in the measured (triangles) and calculated (circles) b lattice parameter.(c) Calculated e 3 strain relative to TiNb 2 O 7 for 937 Li x TiNb 2 O 7 structures.

Figure 10 .
Figure 10.Amplitude of the octahedral breathing mode for (a) the TiO 6 octahedra and (b) NbO 6 octahedra as a function of lithium composition.

Figure 11 .
Figure 11.(a) Symmetry adapted collective displacement modes that characterize a second-order Jahn−Teller distortion of a d 0 transition metal that is octahedrally coordinated by oxygen.The average amplitude of this type of displacement mode as a function of Li concentration for (b) TiO 6 and (c) NbO 6 octahedra.The amplitudes are calculated as a Euclidean distance within the space spanned by the three collective displacement modes of (a).

Figure 12 .
Figure 12.(a) Symmetry adapted the collective displacement modes of the oxygen octahedra that have large amplitudes in TiNb 2 O 7 .The average amplitude of this type of displacement mode as a function of Li concentration for (b) TiO 6 and (c) NbO 6 octahedra.The amplitudes are calculated as Euclidean distances within the space spanned by the three collective displacement modes of (a).

Figure 13 .
Figure 13.(a) Voltage profile at 300 K of Li x TiNb 2 O 7 as calculated with Monte Carlo simulations applied to cluster expansions of the formation energy.The Ti−Nb ordering of TiNb 2 O 7 is that of Figure 1.(b) Lithium site occupancy as a function of lithium concentration as calculated with Monte Carlo simulations applied to 10 different cluster expansions of the formation energy.

Figure 14 .
Figure 14.Li concentration of different sites coordinated by varying amounts of edge-sharing Ti for (a) P s , (b) P s ′, (c) W v , and (d) W h sites.
Description of the symmetry adapted collective displacement modes of an octahedron and methods to calculate their amplitudes in fully relaxed structures; formation energies of TiNb 2 O 7 orderings; Li-vacancy ordering ground states in Li x TiNb 2 O 7 ; density of states and charge densities for multiple Li-vacancy ground states; zero-temperature predicted energies, voltage, and Li site occupancy profiles for a high symmetry TiNb 2 O 7 ordering; and structural changes upon lithiation in a high symmetry TiNb 2 O 7 ordering (PDF) Anton Van der Ven − Materials Department, University of California, Santa Barbara, Santa Barbara, California 93106, United States; Email: avdv@ucsb.edu