Redox Mechanisms upon the Lithiation of Wadsley–Roth Phases

The Wadsley–Roth family of transition metal oxide phases are a promising class of anode materials for Li-ion batteries due to their open crystal structures and their ability to intercalate Li at high rates. Unfortunately, most early transition metal oxides that adopt a Wadsley–Roth crystal structure intercalate Li at voltages that are too high for most battery applications. First-principles electronic structure calculations are performed to elucidate redox mechanisms in Wadsley–Roth phases with the aim of determining how they depend on crystal structure. A comparative study of two very distinct polymorphs of Nb2O5 reveal two redox mechanisms: (i) an atom-centered redox mechanism at early stages of Li intercalation and (ii) a redox mechanism at intermediate to high Li concentrations involving the bonding orbitals of metal–metal dimers formed by edge-sharing Nb cations. Our study motivates several design principles to guide the development of new Wadsley–Roth phases with superior electrochemical properties.


I. INTRODUCTION
The high-power energy storage needs of automotive and aerospace applications require new electrode materials that can intercalate Li-ions at rates that exceed those of current carbonbased anodes.This has led to widespread interest in Wadsley− Roth phases as possible anode materials, as these phases are able to charge and discharge at exceptionally high rates within a voltage window of 2.5 and 0.75 V. 1−13 Wadsley−Roth phases exhibit a rich diversity of crystal structures, consisting of infinitely long, rectangular blocks of corner-sharing metal− oxygen octahedra (MO 6 ) that are joined together along "crystallographic shear planes" where the octahedra share edges. 14,15Their open structures allow for rapid Li insertion and removal, while the crystallographic shear planes give the blocks of corner-sharing octahedra some degree of rigidity.−23 A significant impediment to the widespread use of Wadsley−Roth phases, however, is their high open-circuit voltage relative to Li, which reduces the voltage of a battery when they are used as the anode.As a result, there is a need to understand mechanisms of redox in Wadsley−Roth phases and how they can be manipulated through modifications of chemistry and crystal structure.
Figure 1 shows several Wadsley−Roth crystal structures.The original naming scheme of Wadsley−Roth phases introduced by Cava et al. 14 was recently extended to be more descriptive and precise. 15The E iα [n × m] Wadsley−Roth phases, for example, consist of blocks of n × m corner-sharing octahedra joined together along shear boundaries that exclusively consist of edge-sharing octahedra.The subscript to E specifies the relative shift i in units of octahedral widths in the α direction (α = x or y). Figure 1a illustrates the E 1 [3 × 3]  structure of TiNb 2 O 7 , an important commercial anode material 1,16,24−29 (the shift direction α is redundant when n = m).The TiO 2 -B compound (Figure 1b), which is another promising high-rate anode material, 30,31 was recently identified as an E 1 [2 × 2] Wadsley−Roth phase. 15The T[n × m] Wadsley−Roth phases differ from the E iα [n × m] structures in that they have tetrahedrally coordinated cations at the corners of the blocks.Figure 1c illustrates the T[3 × 3] structure of PNb 9 O 25 , which has phosphorus cations (P 5+ ) residing in tetrahedral sites. 17,32Also shown in Figures 1d and e are two very different Wadsley−Roth structures that can be adopted by Nb 2 O 5 . 33The E[2 × ∞] structure shown in Figure 1d, known as R-Nb 2 O 5 , consists of 2 × ∞ corner-sharing octahedral blocks 34−37 and is the stable polymorph of γ-Ta 2 O 5 , 38,39 while the E 1 [4 × 4] structure shown in Figure 1e consists of 4 × 4 blocks shifted relative to each other by one octahedral spacing and is an important polymorph of Nb 2 O 5 . 15−43 The chemical and crystallographic diversity of Wadsley− Roth phases 14,15 should offer a variety of ways to modify the voltage profile and Li transport properties of anodes made of early transition metal oxides.The design space is further enlarged by the availability of two qualitatively distinct redox mechanisms predicted to occur in Wadsley−Roth phases.The first is an atom-centric redox mechanism involving t 2g orbitals of early transition metals residing in corner-sharing octahedra 44 that dominates at dilute Li concentrations.At less dilute Li concentrations, a different redox mechanism that relies on the formation of metal−metal dimers between edge-sharing MO 6 octahedra was recently predicted to dominate in Li x TiNb 2 O 7 . 45The t 2g orbitals (i.e., d xy , d yz , and d xz ) of early transition metal cations, having lobes that point between pairs of oxygen, can hybridize with neighboring t 2g orbitals to form bonding and antibonding states, as schematically illustrated in Figure 2. The bonding state, which has a lower energy than the unhybridized atom-centric t 2g orbitals, can host two electrons and thereby serve as a redox center to accommodate the electrons donated by Li.−51 In this paper, we analyze redox mechanisms in Wadsley− Roth phases from first principles.We focus on two distinct polymorphs of Nb 2 O 5 to identify crystallographic factors that affect redox mechanisms in Wadsley−Roth phases.Similar to TiNb 2 O 7 , 45 two redox mechanisms are found to accommodate Li insertion into Nb 2 O 5 : an atom-centered charge compensation mechanism dominating at dilute Li concentrations and a redox mechanism that relies on the formation of metal−metal dimers dominating at more concentrated Li contents.The metal−dimer redox mechanism induces structural distortions that result in sizable changes in lattice parameters and unit cell shape.The insights of this study taken together with those generated in previous work 15,18,44,45,52 suggest materials design principles with which the voltage of Wadsley−Roth phases can be tailored.

II. METHODS
Density functional theory (DFT) calculations were performed using the Vienna Ab initio Simulation Package (VASP) 53−56 within the approximation of Perdew, Burke, and Ernzerhof (PBE). 57Interactions between valence electrons and core electrons were treated with the projector augmented wave (PAW) method. 58,59All electronic structure calculations were performed spin-polarized and were initialized in the ferromagnetic state.

A. Formation Energies and Zero Kelvin Voltage
Profiles.To establish the role of structure on the redox mechanisms in Wadsley−Roth phases, we analyze the results of first-principles electronic structure calculations performed on two very distinct Wadsley−Roth crystal structures having the same chemical composition: Nb 2 O 5 in the E[2 × ∞] and E 1 [4 × 4] structures.Lithium ions in Wadsley−Roth phases can occupy pyramidal sites coordinated by five oxygen ions and square planar window sites coordinated by four oxygen ions. 18,44,45,63The pyramidal sites reside along the shear boundaries, at the edges of the n × m blocks of corner-sharing octahedra, while the window sites reside within the n × m blocks.The formation energies of a large number of Li arrangements over the pyramidal and window sites of the E[2 × ∞] and E 1 [4 × 4] forms of Nb 2 O 5 were calculated with DFT-PBE.
Figures 3a and b show the formation energies of Li x Nb 2 O 5 in the E[2 × ∞] and E 1 [4 × 4] structures, respectively, as a function of Li concentration.Figures 3c and d show the corresponding voltages at zero Kelvin relative to a metallic lithium reference electrode.At zero Kelvin, only the lowest energy Li-vacancy orderings that reside on the convex hull (i.e., common tangent construction applied to formation energies) are thermodynamically stable.These orderings are referred to as the ground states.The voltage relative to metallic Li is linearly related to the Li chemical potential of the electrode,  Inorganic Chemistry which at zero Kelvin is linearly related to the slope of the convex hull as a function of Li concentration. 64Each step in the voltage profile corresponds to the voltage window in which a particular ordered ground state is stable, while each plateau corresponds to the voltage at which one ground state transitions to an adjacent ground state. 64While the formation energies of a large number of Li arrangements within the two host structures were calculated (987 in E[2 × ∞] and 1265 in E 1 [4 × 4]), it is not possible to perform an exhaustive enumeration of all possible orderings.As a result, the true ground states may not have been determined for each host.
Figure 3c and d show experimental voltage curves for Li x Nb 2 O 5 having the E[2 × ∞] structure as measured by Parui et al. 37 and for Li x Nb 2 O 5 having the E 1 [4 × 4] structure as measured by Lian et al. 42 Li-ions in Wadsley−Roth phases tend to disorder at room temperature and exhibit sloping voltage profiles that are characteristic of solid solutions. 18,45,65s is evident in Figure 3c, the voltage curve for the E[2 × ∞] structure as measured by Parui et al. 37 overlaps the zero Kelvin voltage between x = 0 and 2, but is higher than the calculated voltage above x = 2.The measured voltage curve for the E 1 [4 × 4] structure 42 is substantially higher than the zero Kelvin voltage curve (Figure 3d).A possible explanation for the discrepancies between the measured and calculated voltage curves is that the true Li-vacancy ground state orderings have not been identified.For example, while the energies of 1265 different Li-vacancy arrangements over the interstitial sites of the E 1 [4 × 4] structure have been explicitly calculated, the large unit cell of this structure results in a combinatorial explosion of possible Li-vacancy orderings, even within the primitive unit cell of the host.In spite of these computational limitations, the very large number of configurations considered in this study for both the E[2 × ∞] and E 1 [4 × 4] forms of Li x Nb 2 O 5 should provide invaluable insights about redox mechanisms in these compounds.

B. Ion-Centered Redox at Dilute Li Concentrations.
The Nb cations of Nb 2 O 5 are octahedrally coordinated by oxygen and are in their maximum oxidation state of +5, having a d 0 electron configuration.A large band gap separates the valence bands, comprised primarily of oxygen p states, from empty conduction bands, made up of the Nb derived t 2g states (i.e., the Nb 4d xy , 4d yz , and 4d xz orbitals when octahedrally coordinated by oxygen).The electrons that accompany the inserted Li + ions reduce the Nb 5+ cations by filling the t 2g derived conduction bands.First-principles studies of Nb-and W-containing Wadsley−Roth phases by Kocȩr et al. 44,52 showed that the first transition metal cations of Wadsley−Roth phases to reduce are those that reside in corner-sharing octahedra at the center of the n × m blocks.This was subsequently also predicted in PNb 9 O 25 . 18Figure 4a and b show the DOS and electron charge density, respectively, of Li 0.25 Nb 2 O 5 in the E 1 [4 × 4] structure.Upon the insertion of a dilute concentration of Li, the Fermi level moves into the t 2g derived bands (Figures 4a).The electronic charge density of the filled t 2g states (Figures 4b) shows a high charge density on the Nb of the corner sharing octahedra at the center of the [4  × 4] block.Consistent with previous first-principles calculations of Wadsley−Roth phases with dilute Li concentrations, 44,45,52 charge density is also evident on the other Nb within the crystal.Note that the DOS of Figure 4a shows a spin up density that is slightly higher than that of the spin down density.This results in a net magnetic moment, which has been observed and predicted in Wadsley−Roth phases containing a dilute concentration of Li. 16,45,52 Figures 4c and d show similar DOS and electron charge density plots for dilute Li 0.5 Nb 2 O 5 in the E[2 × ∞] structure.Due to the very narrow block dimension in one direction, there are no Nb that reside in exclusively corner-sharing octahedra.The electrons donated to the host upon the insertion of a dilute concentration of Li are therefore uniformly distributed over all the Nb cations.In both the E 1 [4 × 4] and E[2 × ∞] structures, the increase in charge density is localized in atomcentered orbitals.
C. Structural Signatures of Metal-Dimer Redox.While the donation of electrons to the Nb 2 O 5 host upon the insertion of Li increases the charge density of atom-centered t 2g orbitals at dilute Li concentrations, the mechanism of charge accommodation changes at intermediate to high Li concentrations.Similar to predictions for TiNb 2 O 7 , 45 a metal-dimer redox mechanism involving the bonding states of hybridized t 2g orbitals of edge-sharing transition metal cations becomes evident at more concentrated Li concentrations within the E 1 [4 × 4] and the E[2 × ∞] structures of Nb 2 O 5 .
One signature of the onset of the metal-dimer redox mechanism is a shortening of the distance between pairs of edge-sharing metal cations.In pristine Wadsley−Roth phases (without Li) the electrostatic repulsion between edge-sharing transition metal cations in their fully oxidized state induces a significant off-centering and an increase in metal−metal distances. 15,52,66This is evident in Figure 5a, which shows a vertical cut through the Nb 2 O 5 E 1 [4 × 4] structure.As shown by the red arrow in Figure 5, the Nb in the edge-sharing octahedra along the shear boundaries are displaced away from each other, leading not only to an increase in the distance between pairs of edge-sharing transition metal cations, but also significant distortions of the coordinating octahedra of oxygen ions.Figure 5b shows similar increases in metal−metal The formation of bonding states between the t 2g orbitals of edge-sharing Nb to accommodate electrons donated by inserted Li leads to a contraction of the metal−metal pair distance.Figure 6 collects the pair distances between edgesharing Nb cations in the E[2 × ∞] and E 1 [4 × 4] structures of Nb 2 O 5 as a function of Li concentration.These pair distances were extracted from 987 and 1265 fully relaxed Livacancy configurations in the E[2 × ∞] and E 1 [4 × 4]  structures of Nb 2 O 5 , respectively.Figure 6 shows that the distances between a subset of edge-sharing Nb cations undergo a significant contraction (from approximately 3.4 Å to almost 2.6 Å) upon the insertion of 0.5 to 1.5 Li per transition metal.The gold points are metal−metal pair distances of edge-sharing Nb cations in the lowest energy configurations.Since there are multiple edge-sharing metal−metal pairs in each structure, there are multiple pair distances for each lowest energy configuration.
Figure 6 shows that there is a wide distribution of Nb−Nb pair distances, with a spread ranging between 2.6 and 3.7 Å.The Nb−Nb pair distances are sensitive to the Li concentration and the Li-vacancy ordering at a fixed composition.At room temperature, the Li ions are disordered, as manifested by the experimentally measured sloping voltage profiles. 37,42Lithium disorder will result in a distribution of Nb−Nb distances with little or no long-range order among the Nb−Nb pairs that form dimers. Hence, structural models obtained with diffraction experiments will reflect an average Nb−Nb bond length.A calculation of the average Nb−Nb bond length at room temperature would require an in-depth  The formation of metal-dimers has macroscopic consequences.As was shown for Nb-and W-containing Wadsley−Roth phases, 44 PNb 9 O 25 18 and TiNb 2 O 7 , 45 a contraction in the distances between a subset of edge-sharing cations along the crystallographic shear planes and corners leads to an elongation of the block length and a contraction along the block waist.A useful measure of this dimensional change is a strain order parameter defined as a symmetry-adapted linear combination strain according to 67 (2 ) The strains are measured within a Cartesian coordinate system with its z-axis aligned parallel to the Wadsley−Roth block length, while the xand y-axes are in the plane of the n × m blocks.
Figure 7 collects the e 3 strain order parameter for each of the fully relaxed Li-vacancy orderings in the E[2 × ∞] and E 1 [4 ×   4] structures of Nb 2 O 5 .There is an abrupt increase in e 3 for x between 0.5 and 1.5, which coincides with the onset of metaldimer formation.The transition is especially abrupt for the E[2 × ∞] structure, but is more smooth and gradual in the E 1 [4 ×  4] structure, presumably due to its large diversity in edgesharing Nb pairs.D. Electronic Signature of Metal-Dimer Redox.The bonding states of metal-dimer pairs are highly localized and host both an up and down spin electron.Figure 8a shows the electronic density of states (DOS) of the lowest energy Li ordering within Li x Nb 2 O 5 in the E 1 [4 × 4] structure at x = 0.875, the first E 1 [4 × 4] structure on the convex hull to undergo metal-dimer redox.The Fermi level in Figure 8a is denoted by the vertical dashed line.The DOS below ≈ −3 eV corresponds to oxygen-dominated levels, while the DOS above ≈ −1 eV is derived primarily from Nb t 2g orbitals.Especially striking in the DOS is a sharp peak below the Fermi level at −1 eV due to the formation of a metal−metal dimer.Figure 8b plots the electronic charge density corresponding to this peak.The charge density adopts the characteristic shape of two d xy orbitals centered on neighboring Nb cations with a clear enhancement of charge density between the pair of cations.The cations reside in octahedra of oxygen that share a common edge similar to the schematic of Figure 2. The DOS of Figure 8a shows that there is an equal number of up spin and down spin electrons occupying the states corresponding to the metal−metal dimer.The metal-dimer pair consists of Nb cations that share the largest number of edges with other Nb  cations.These cations are coordinated by four other cations in a high oxidation state and therefore experience a large driving force to undergo redox to a lower oxidation state.The formation of the metal-dimer bond leads to a contraction of the distance of the pair of Nb cations to approximately 2.6 Å, which is significantly smaller than the average pair distance of 3.4 Å between the other edge-sharing Nb pairs that do not form metal−metal bonds.The charge density due to the remaining t 2g derived states above the peak and extending up to the Fermi level is shown in Figure 8c.These states are more uniformly distributed over all the Nb cations within the crystal.Since these states have a sizable density at the Fermi level, they are expected to contribute to electron conduction.
Figure 9a shows the DOS for the next E 1 [4 × 4] convex hull structure at x = 1.9375.In this structure, the distances between several more edge-sharing Nb pairs have contracted from a mean of 3.36 Å at x = 0.875 to 2.97 Å at x = 1.9375 in Li x Nb 2 O 5 .The DOS of Figure 9a shows multiple localized peaks below the Fermi level.The charge densities associated with several of the well-localized peaks are shown in Figure 9b,  c, and d.The peaks correspond to localized charge densities between neighboring Nb cations and indicate the formation of metal−metal bonding states that are filled by spin up and spin down electrons.The Nb cations that form dimer bonds share 4, 3, and 2 edges with neighboring Nb. Figure 9e shows the electronic charge density of states with energies close to the Fermi level.The states close to the Fermi level are more centered on Nb cations and are also more uniformly distributed throughout the crystal.
The E[2 × ∞] structure has less diversity in the environments surrounding edge-sharing Nb pairs.As is clear from Figure 6a, the formation of shortened Nb−Nb distances occurs at a higher Li concentration in E[2 × ∞] than in E 1 [4 ×  4] (Figure 6b).Metal-dimer formation in the E[2 × ∞] structure only starts at x = 1.5 as is evident by the emergence of several peaks below the Fermi level in the DOS of Figure 10a.The charge density associated with these peaks, shown in Figure 10b and c, are more uniformly distributed over a large number of metal−metal pairs.In contrast to the E 1 [4 × 4]  structure, in which some Nb cations are coordinated by 4 cations and others by 3 or 2 Nb cations, all the Nb cations of E[2 × ∞] reside in an identical environment, being coordinated by 2 edge-sharing Nb cations.Hence, the onset of Nb−Nb bonding states is more uniform, but occurs at a higher Li concentration, as more electrons are required to fill a larger number of symmetrically equivalent bonding states.

E. Octahedral Distortions Due to Redox Mechanisms.
−73 There are two primary driving forces for these large octahedral distortions.The first is due to the d 0 electronic configuration of the metal cations of most Wadsley−Roth phases, which makes early transition metals susceptible to an off-centering second-order Jahn− Teller distortion. 44The second arises from the high density of edge-sharing octahedra along the shear boundaries of Wadsley−Roth phases.The transition metal cations residing in edge-sharing octahedra experience a strong electrostatic repulsion that results in a sizable off-centering that in turn causes collateral distortions of the coordinating octahedra of oxygen ions. 15he redox reactions that accompany Li insertion into Wadsley−Roth phases affect both driving forces of octahedral distortions.The atom-centric redox mechanism at dilute Li concentrations, which results in a filling of the empty d orbitals on early transition metal cations, removes the susceptibility for a second-order Jahn−Teller distortion.The formation of metal−metal bonds at higher Li concentrations leads to a contraction of the edge-sharing Nb−Nb bond distances.The cations forming metal−metal bonds are pulled to the center of their coordinating oxygen octahedra, thereby undoing the strong distortions of the surrounding oxygen ions.
There are two classes of symmetry-adapted collective displacement modes of a NbO 6 octahedron that capture the octahedral distortions of pristine Wadsley−Roth phases. 15,45he first class, described by the three orthogonal displacement modes shown in Figure 11a, corresponds to a cation offcentering and is a measure of the second-order Jahn−Teller distortion.The amplitudes of this distortion mode were collected for each NbO 6 octahedron in each fully relaxed Livacancy configuration in the E[2 × ∞] and E 1 [4 × 4] forms of Li x Nb 2 O 5 .The displacement vectors of the seven ions of an NbO 6 octahedron undergoing a symmetry-adapted collective distortion mode can be collected in a 7 × 3 = 21 dimensional vector that is normalized to have a length of 1.The amplitudes plotted in Figure 11b and c, with units of Å, are the coefficients of a decomposition of the fully relaxed displacement field of a NbO 6 octahedron onto the symmetry-adapted collective displacement mode of Figure 11a, as described by Saber et al. 45 Figures 11b and c show the average amplitude and a onestandard-deviation spread around the average of this offcentering distortion mode as a function of the Li concentration for the E[2 × ∞] and E 1 [4 × 4] structures, respectively.The off-centering amplitude is large in the pristine phases and at dilute Li concentrations.The amplitude of a particular octahedron is also very sensitive to the number of edges that it shares with neighboring octahedra.The corner-sharing octahedra at the center of the 4 × 4 blocks of E 1 [4 × 4]  (orange curve) has the smallest off-centering amplitude.The octahedra that share three or four edges in the E 1 [4 × 4]  structure have the largest amplitude.The E[2 × ∞] structure consists of only one type of NbO 6 octahedron, each sharing only two edges.The variation of the off-centering amplitude of octahedra that share only two edges with Li concentration is very similar in both the E 1 [4 × 4] and A second class of octahedral distortion modes captures the deformations of the oxygen ions surrounding edge-sharing Nb cations.The three orthogonal displacement modes that span this class of distortions are shown in Figure 12a.Figures 12b  and c show the average and one-standard-deviation spread for the E[2 × ∞] and E 1 [4 × 4] structures.Here as well, the amplitudes of this distortion mode are very large at dilute Li concentrations and very sensitive to the number of edges that a particular octahedron shares with neighboring octahedra.The octahedra that share edges with neighboring octahedra undergo an abrupt reduction in their distortion amplitude between x = 1 and 2, which coincides with the onset of metal− metal dimer formation.The behavior is very similar to that of other Wadsley−Roth phases 18,45 and early transition metal compounds such as anatase TiO 2 . 74

IV. DISCUSSION
The open crystal structures of Wadsley−Roth phases makes them attractive electrode materials for Li-ion batteries.−77 Unfortunately, the Li intercalation voltages of promising anode materials having a Wadsley−Roth crystal structure, such as TiNb 2 O 7 , 1 PNb 9 O 25 , 17 and ternary Nb−W oxides (e.g., Nb 16 W 5 O 55 and Nb 18 W 16 O 93 ), 2 are too high to be commercially competitive.A major objective in the design of new Wadsley−Roth phases for anode applications is, therefore, to identify chemistries and structures that have lower voltages than current candidates.This requires a detailed understanding of redox mechanisms and the chemical and structural factors that affect them.
Our first-principles study of the electronic properties of Li x Nb 2 O 5 having the E 1 [4 × 4] and E[2 × ∞] structures predict the occurrence of two distinct redox mechanisms and reinforce similar conclusions of previous studies on related materials. 18,44,45At dilute Li concentrations, electrons fill ioncentered transition metal t 2g orbitals. 15,18,44This occurs more or less uniformly; however, as shown by Kocȩr et al., 44,52 there is a slight preference for the transition metals in the cornersharing octahedra at the center of the blocks.At higher Li concentrations, the redox mechanism is predicted to change, 45 involving the low-energy bonding states of metal−metal dimers.The t 2g orbitals of edge-sharing transition metal cations hybridize to form bonding states that can accommodate electrons donated by Li (Figure 2).From our

Inorganic Chemistry
examination of redox mechanisms in two Nb 2 O 5 polymorphs, we have found that metal−metal bond formation, responsible for significant strain in TiNb 2 O 7 , 45 also occurs in Nb 2 O 5 , suggesting a universality of the metal−metal bond redox mechanism in Wadsley−Roth phases.8][49][50][51]78 The formation of metal−metal dimers has structural consequences, as it results in a shortening of the distance between edge-sharing cations. This  turn affects the lattice parameters and leads to shape changes of the crystal with changes in the Li concentration.The metal−metal dimers form first between transition metal cations that have the highest number of edge-sharing neighbors.These cations experience the strongest electrostatic interactions with other cations in high oxidation states, producing a large driving force to undergo redox and thereby reduce their oxidation state.The E 1 [4 × 4] structure has more flexibility in accommodating the formation of metal-dimers than the more ordered E[2 × ∞] structure.This is because the E 1 [4 × 4]  (where e refers to the charge of an electron and where the chemical potentials are in units of eV).64 The Li chemical potential in turn is related to the derivative of the Gibbs free energy of the electrode material with respect to concentration. Figure 13 schematically shows two ways in which the voltage of an electrode material can be reduced.One is to lower the free energy of the host at low Li concentrations, while the second is to increase the free energy at high Li concentrations.
In order to lower the voltage of Wadsley−Roth phases, new structures and chemistries must be identified that have a low free energy in the pristine state and/or upon the addition of a dilute concentration of Li-ions, but an unfavorable free energy at higher Li concentrations.This can be realized by using the most stable Wadsley−Roth polymorph for a given transition metal oxide composition.Low-energy Wadsley−Roth phases have structures that make them sufficiently flexible to accommodate octahedral distortions and that have cations arranged to minimize electrostatic interactions between edgesharing octahedra (i.e., the highest oxidation state cations should be segregated to the center of the blocks occupying corner-sharing octahedra 15,44 ).
It may not always be possible to use the most stable pristine Wadsley−Roth crystal structure for a particular transition metal oxide composition if the structure inhibits fast charge/ discharge kinetics.In that scenario, it is desirable to identify a structure that has a large number of favorable Li sites at dilute concentrations.This will ensure that the free energy decreases rapidly upon the insertion of a dilute concentration of Li-ions.While Li tends to first fill pyramidal sites along the shear boundaries, the window sites are closer to the transition metal cations that undergo redox at dilute Li concentrations, and thereby more favorable from electrostatic considerations.The window sites tend to be highly distorted in pristine Wadsley− Roth phases due the large distortions of the fully oxidized and edge-sharing MO 6 octahedra. 15,45Wadsley−Roth structures with larger block sizes, however, contain more square window sites that are less distorted and thereby more favorable for Li occupancy. 18,45This is evident in Figure 5a, which shows the distortions of the square planar sites of pristine E 1 [4 × 4]  Nb 2 O 5 .The square planar sites at the center of the block are less distorted than those along the shear-boundaries.The square planar sites of Wadsley−Roth phases with smaller blocks, such as TiNb 2 O 7 , which has the E 1 [3 × 3] structure, and PNb 9 O 25 , which has the T[3 × 3] structure, are highly distorted due to their close proximity to the shear boundaries and are therefore not favorable for Li occupancy at dilute Li concentrations (Figure 5b). 18,45Dilute Li in these compounds tend to fill the pyramidal sites along the shear boundaries, which are further away from the electron donated to the Nb at the center of the block. 18,45he voltage can be further lowered by suppressing metal− metal bond formation at intermediate to high Li concentrations.At nondilute Li concentrations, the results of this study and prior work 18,45 predict that the redox mechanism of Wadsley−Roth phases transitions from an atom-centered mechanism to one that relies on the formation of metal− metal bonds.Structures that are more resistant to shortened metal−metal bonds will have a higher free energy when this redox mechanism becomes active.A stiff crystalline backbone will require the conversion of a portion of the chemical free energy gain of forming metal−metal dimers into elastic energy, thereby lowering the voltage.This, however, may lead to undesirable hysteresis in the voltage profile. 79An alternative is to identify alloying elements that do not favor metal−metal dimer formation and also segregate to the edge-sharing octahedra at the shear boundaries.Wadsley−Roth phases containing Nb, V, and Mo, which are known to readily form metal−metal bonds, 46−49 should then be alloyed with elements that have a lower oxidation state.Alloying elements having a low maximum oxidation state will tend to segregate to the edge-sharing octahedra along the crystallographic shear planes 15 and thereby relegate the metal-dimer formers such as Nb, V, and Mo to corner-sharing octahedra where they are unable to form metal−metal bonds.The addition of alloying elements with lower oxidation states than Nb or W will stabilize Wadsley−Roth crystal structures with smaller block sizes or Wadsley−Roth phases with partially reduced transition metals to preserve charge neutrality. 15The suppression of metal-dimer redox will lead to a reliance on less favorable, atom-centered, redox mechanisms, and thereby a lowering of the voltage at higher Li concentrations.

V. CONCLUSION
We have performed a first-principles investigation of redox mechanisms accompanying Li insertion in two very distinct Wadsley−Roth crystal structures of Nb 2 O 5 .Consistent with previous first-principles studies of other Wadsley−Roth phases, 18,44,45 we identify two redox mechanisms: (i) an atom-centered redox mechanism that dominates at dilute Li concentrations and (ii) a dimer redox mechanism at intermediate to high Li concentrations whereby the bonding states of edge-sharing metal-dimers accommodate the electrons donated by Li to the host.The dimer redox mechanism leads to the shortening of metal−metal pair distances.This makes the mechanism highly dependent on the elastic flexibility of the host crystal structure.The results of this study suggest design principles with which the electrochemical properties of Wadsley−Roth phases can be tailored through chemical and structural modifications of the host.

■ ASSOCIATED CONTENT
A plane-wave energy cutoff of 650 eV and k-point grid with a reciprocal space discretization of 25 Kpoints per Å −1 were used.Lithium vacancy orderings in the E[2 × ∞] and E 1 [4 × 4] Nb 2 O 5 Wadsley−Roth phases were enumerated using the Clusters Approach to Statistical Mechanics (CASM) software package. 60−62

Figure 2 .
Figure 2. The t 2g orbitals, such as d xy , of early transition metal cations that reside in edge-sharing octahedra made of oxygen anions can hybridize to form metal−metal bonds that can host two electrons.Reproduced from Saber et al.45

Figure 4 .
Figure 4. Density of states (a) and charge density (b) for E 1 [4 × 4] Li 0.25 Nb 2 O 5 .The charge density plot is in a plane perpendicular to the z-axis, which is parallel to the block length.Density of states (c) and charge density (d) for E[2 × ∞] Li 0.5 Nb 2 O 5 .The charge density plots are on a plane perpendicular to the x-axis.The z-axis is again parallel to the block length of the E[2 × ∞] structure.

Figure 5 .
Figure 5. (a) A slice of the E 1 [4 × 4] Wadsley−Roth crystal structure along the block length.The orange circles are oxygen, and the blue circles are transition metal cations.The edge-sharing MO 6 octahedra along the shear boundaries are highly distorted due to the strong electrostatic repulsion between edge-sharing metal cations.This leads to highly distorted window sites that can host Li-ions.(b) The T[3 × 3] structure of PNb 9 O 25 undergoes similar distortions along the shear boundaries.

Figure 6 .
Figure 6.Bond lengths between edge-sharing Nb cations of (a) E[2 × ∞] Nb 2 O 5 and (b) E 1 [4 × 4] Nb 2 O 5 as a function of Li concentration for all structures (blue) and ground state structures (gold) where the gold line indicates the mean bond length for Nb cations in Nb sites in ground state structures.The bond lengths were collected from the fully relaxed DFT calculations of 987 (1265) Li configurations in E[2 × ∞] (E 1 [4 × 4]) Nb 2 O 5 .

Figure 8 .
Figure 8.(a) Density of states for the lowest energy E 1 [4 × 4] Li 0.875 Nb 2 O 5 ordering.(b) The electronic charge density for states with energies highlighted in orange in (a).(c) The electronic charge density for states with energies highlighted in blue in (a).

Figure 9 .
Figure 9. (a) The total density of states of Li 1.9375 Nb 2 O 5 in the E 1 [4 × 4] structure.The charge densities due to electron states within different energy intervals are shown in (b), (c), and (d).All charge densities are shown in a plane perpendicular to the z-axis (which is parallel to the block axis).

Figure 10 .
Figure 10.(a) The total density of states of E[2 × ∞] Li 1.5 Nb 2 O 5 .Charge densities for electronic states with energies in different intervals are shown in (b), (c), and (d).The charge density plots are shown in a plane that is perpendicular to the x-axis.The block axis is parallel to the z-axis.

Figure 11 .
Figure 11.(a) T 1u collective octahedral distortion modes that measure the degree of cation off-centering relative to the coordinating oxygen ions.The T 1u distortion modes for NbO 6 octahedra in the of E[2 × ∞] (b) and E 1 [4 × 4] (c) forms of Li x Nb 2 O 5 as a function of lithium concentration.

Figure 12 .
Figure 12.(a) T 2u collective octahedral distortion modes that measure symmetry-breaking deformations of the coordinating oxygen ions of Nb cations.T 2u distortion modes for NbO 6 octahedra in the E[2 × ∞] (b) and E 1 [4 × 4] (c) forms of Li x Nb 2 O 5 as a function of lithium concentration.

Figure 13 .
Figure 13.Voltage of a Wadsley−Roth transition metal oxide can be tailored through crystallographic and chemical modifications that alter the free energy of the host as a function of Li concentration (a).The voltage (b) is related to the slope of the free energy curve with respect to Li concentration.The voltage can be decreased by decreasing the free energy at dilute Li concentrations and increasing the free energy at higher Li concentrations.
structure has a wider diversity of local environments, containing NbO 6 octahedra that share zero, two, three, and four edges with neighboring octahedra.The E[2 × ∞] structure, in contrast, has only one octahedral environment, with each NbO 6 octahedron sharing exactly two edges with neighboring octahedra.As a result, E 1 [4 × 4] Nb 2 O 5 forms metal−metal bonding at a lower lithium concentration than E[2 × ∞] Nb 2 O 5 .The results of this study and those of prior studies suggest guidelines with which the structural and chemical diversity of Wadsley−Roth phases can be exploited to tailor the voltage of early transition metal oxides.The voltage of an electrode, as measured relative to a metallic Li reference electrode, is related to the difference in the Li chemical potential of the electrode,