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## Creator

[JALEM Randy](https://orcid.org/0000-0001-9505-771X), [TATEYAMA Yoshitaka](https://orcid.org/0000-0002-5532-6134), [TAKADA Kazunori](https://orcid.org/0000-0001-7568-1806), JANG Seonghoon

## Rights

This document is the Accepted Manuscript version of a Published Work that appeared in final form in The Journal of Physical Chemistry C, copyright © 2023 American Chemical Society after peer review and technical editing by the publisher. To access the final edited and published work see https://doi.org/10.1021/acs.jpcc.3c02801[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Multiobjective Solid Electrolyte Design of Tetragonal and Cubic Inverse-Perovskites for All-Solid-State Lithium-Ion Batteries by High-Throughput Density Functional Theory Calculations and AI-Driven Methods](https://mdr.nims.go.jp/datasets/31cd0c96-f1ea-400a-af53-1e35cf5cdf7b)

## Fulltext

Multi-objective solid electrolyte design of tetragonal and cubic inverse-perovskites for all-solid-state lithium-ion batteries by high-throughput DFT calculations and AI-driven methods Randy Jalem1,*, Yoshitaka Tateyama1, Kazunori Takada1, and Seong-Hoon Jang1 1Center for Green Research on Energy and Environmental Materials (GREEN), National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan  Abstract Solid electrolytes (SEs) are crucial materials to realize highly safe and practical all-solid-state Li+ ion batteries. Here, we performed a large-scale computational SE screening on a chemical space of >10,000 Li-rich inverse-perovskite (ip) compounds with tetragonal and cubic structures by high-throughput DFT and AI-driven methods. A total of 1,413 novel candidate compounds were predicted to be synthesizable based on thermodynamic decomposition energy ( 𝐸𝑑 ) and machine-learned experimental synthesis likelihood (𝐿𝑠). These compounds were further screened using a Pareto-front approximation set of a multi-objective Bayesian optimization task for 𝑘  = 3 DFT-calculated SE properties (𝑓𝑘, with 𝑘 = 1, 2, and 3): i) electrochemical window from electronic band gap energy (𝑓1: 𝐸𝑔), ii) chemical stability by reaction with moisture (𝑓2: 𝐸ℎ), and iii) 400-K bulk Li+ ion conductivity (𝑓3: Λ). As a result, the compound list was reduced down to 24 candidate ip SEs, examples include Cm Li8O2Cl3Br (𝐸𝑑 = 0, 𝐿𝑠 > 0.5, 𝐸𝑔 = 4.74 eV, 𝐸ℎ = -33.22 kJ/mol, Λ = 9.0 x 10-4 S/cm), Amm2 Li8OSCl4 (𝐸𝑑 = 0.070 eV/atom, 𝐿𝑠 > 0.5, 𝐸𝑔 = 4.14 eV, 𝐸ℎ = -40.70 kJ/mol, Λ = 9.2 x 10-2 S/cm), and Cmcm Li12O3SeClBr3 (𝐸𝑑 = 0.097 eV/atom, 𝐿𝑠 > 0.5, 𝐸𝑔 = 3.36 eV, 𝐸ℎ = -86.88 kJ/mol, Λ = 7.8 x 10-1 S/cm). Possible solid-state synthesis routes for the screened SE candidates were also explored using thermodynamic phase competition analysis and classical-nucleation-theory reaction barrier. Aside from providing a well-informed list of potentially novel ip-type SEs, our work also reports on an effective calculation methodology for tiered large-scale material screening which, at the same time, incorporates ‘small data’ learning on target property datasets that are computationally expensive to obtain. The generated datasets are expected as well to be of great utility for future data-driven material design efforts.          Introduction A low-carbon society has been for some time now a major goal of many countries that desire to further elevate the quality of life of people and be more responsive in the protection of the environment by limiting on fossil fuel dependence for energy generation. One specific technology that is expected to play a key role in this area is the all-solid-state rechargeable Li+ ion battery (ASSB).1-3 A high-performance ASSB is expected to meet several industry requirements, particularly on energy density, safety, and reliability. It can become widespread in applications such as electric vehicles (EVs), backup power, and household use.1 To achieve high energy density, an ASSB with a Li metal anode (3,860 mAh/g in theoretical capacity) is strongly sought and many R&D efforts have since been made so that it can become viable technology-wise.4 On the other hand, to realize high practical capacity, high-voltage cathodes are needed, one example is a recently designed LiNi0.8Mn0.1Co0.1O2 (NMC811) compound which is capable of 81.3% capacity retention after 2,000 cycles (at 1.5C rate) and a specific energy of up 631.1 Wh/kg.5 The combination of Li metal anode and high-capacity cathode is expected to meet the driving mileage demand for EVs.6-7 In addition, another critical component towards unlocking a high-performance ASSB is the solid electrolyte (SE) which has a role of shuttling the charge carrier (Li+) ions between the anode and the cathode, and vice versa, during battery discharge/charge operations. Several SE design goals include: i) Li-superionic conductivity, ii) mechanical robustness in a sufficiently thin configuration, iii) chemical stability during battery work operation, and iv) (electro)chemical compatibility with other solid-state cell components.8-11 So far, a number of promising SEs were reported to satisfy the ionic conductivity requirement (i.e., 10-3 S/cm or better), but finding a material that also meets other sought properties at the same time has still remained a longstanding challenge.10,12-15  To speed up the search of novel SEs, high-throughput density functional theory calculations (HT-DFT) and methods based on artificial intelligence (AI) have been employed recently in combination to provide experimentalists with a list of candidate materials that should be highly prioritized for actual synthesis and characterization. Several examples work on computational SE material screening have been reported, but they only focus on a single or a few SE properties at a time. Examples of these properties are ionic conductivity, thermodynamic stability, (electro)chemical stability, and suppression ability vs. Li dendrite growth.16-22  Among the class of already reported SEs, Li-rich inverse-perovskite (ip) compounds show great promise in terms of ionic conductivity and electrochemical property.23-25 These compounds can also demonstrate mechanical softness to enable for an intimate contact with electrodes and other battery materials in the cell for reduced interfacial resistance, show exceptional stability against Li metal via formation of electronic-insulating but ionic-conducting decomposition interphase layer, and be rapidly synthesized with high scalability.26-27 Their general formula can be expressed as Li3n+1XnZn+1 which, by symmetry, can be characterized as tetragonal (t-ip) for n ∈ [1, 2, 3, …] and cubic (c-ip) for n = ∞ (i.e., Li3XZ as reduced formula); the value of n is indicative of the thickness of the X/Z-Li (anion-centered) 2D slabs that are stacked in the crystallographic long-axis direction.28 Given the long development history of normal perovskites, it naturally follows that there is also a wide variety of structure derivatives (i.e., based on index n) and configurations (i.e., site alloying + symmetry group-subgroup relations) for c-ip and t-ip compounds. Consequently, there is a large chemical search space that is yet to be explored for undiscovered novel compounds. We have partly demonstrated this in our recent work which reported on several DFT-predicted novel t-ip (n = 1) SE candidates on a chemical space containing about 500 compounds.29 Here, using a combination of HT-DFT calculations and AI-driven methods, we explored the chemical space of >10,000 in-silico t-ip and c-ip compounds to find promising novel SEs. Multiple DFT-calculated SE properties were used as screening criteria: thermodynamic stability (by convex hull method), electrochemical window upper bound based by electronic band gap energy, air-exposed chemical stability by thermodynamic bulk reaction energy vs. H2O, and bulk Li+ ionic conductivity from multiple-temperature ab initio molecular dynamics (AIMD) approach. Additionally, the in-silico compounds were evaluated for their synthesizability and possible rational synthesis routes by DFT and recent state-of-the-art machine learning techniques. Aside from the useful atomistic-level insights on SE properties as well as practical information on ip compound synthesis, this study also provides a useful methodology template for multi-objective material SE design (i.e., Pareto-based optimization) and a large-scale DFT dataset of potentially novel compounds that can be exploited for future data-driven material design studies.                Computational details Structure geometry optimization DFT calculations were performed using the VASP software30,31 which is based on the projector augmented wave (PAW) approach32-34. The electron exchange-correlation part of the Hamiltonian was described using the generalized gradient approximation (GGA) method by Perdew, Burke, and Ernzerhof (PBE).35,36 Three parent crystal structure aristotypes based on the ip block stacking were used for chemical space generation: Li4XZ2 (n = 1), Li7X2Z3 (n = 2) and Li3XZ (n = ∞); the atomic coordinate data were taken from the Inorganic Crystal Structure Database (ICSD),37 see Figure 1. Except for the Li pseudopotential which explicitly includes the semi-core s states as valence states, standard pseudopotentials were employed for H, X (O, S, Se, Te) and Z (F, Cl, Br, I) elements. A 520-eV kinetic cutoff energy was set with a k-points resolution of ≥ 1000 in a Monkhorst-Pack grid scheme.38 All calculations were carried out with spin polarization. Energy and residual forces were ensured to be less than 1 meV/atom and 0.01 eV/Å, respectively.  Figure 1. Schematic illustration of different inverse perovskite Li3n+1XnZn+1 crystal structures classified according to the slab layer thickness expressed as integer n > 0 relative to the XLi6 unit along c-direction, such as in the case of I4/mmm tetragonal symmetry (e.g., n = 1, n = 2). As the XLi6 slab layer becomes infinitely thick (i.e., n = ∞), the structure assumes the Pm3̅m cubic symmetry. Green, red, blue spheres represent Li+ cation, X2- anion, and Z- halide anion, respectively.   Thermodynamic phase stability evaluation  In-silico t-ip and c-ip compounds were evaluated for their thermodynamic stability by DFT decomposition energy (𝐸𝑑) calculation.39 Briefly, a convex hull of tie lines formed by ground-state phases was constructed. In this formulation, ground-state phases lie on the hull surface as vertices and, by definition, have a hull distance of zero (i.e., 𝐸𝑑 = 0). Any in-silico compounds that lie above the hull surface of ground-state phases are predicted as metastable or as compounds that are thermodynamically driven to decompose. Formally, 𝐸𝑑 (in eV/atom) was calculated according to the following equation: 𝐸𝑑 = ∆𝐻𝑓 − ∆𝐻𝑐,     (1)  where ∆𝐻𝑓 and ∆𝐻𝑐 are the convex hull energies (using formation enthalpies, pressure and entropy effects were considered negligible at ambient conditions), relative to the phase equilibria at composition 𝑐 (i.e., related to stable competing phases). Calculations were performed using the total energies of relevant DFT-calculated ICSD compounds and compounds from the Materials Project database, using the latter’s application programming interface (i.e., Pymatgen library).40,41  Multi-objective optimization of electrolyte properties  Three DFT-calculated SE-related properties were used as objective functions (𝑓𝑘, where 𝑘 ∈ {1,2,3}) for finding optimal SE candidates. These are as follows: i) electrochemical window upper bound based on electronic band gap energy (𝑓1: 𝐸𝑔), ii) chemical stability measure based on hydration energy (𝑓2: 𝐸ℎ), and iii) bulk Li+ ionic conductivity at 400 K (𝑓3: Λ).  Previous works have already used 𝐸𝑔 as a filter for SE electrochemical stability window,42 following the general observation that a large 𝐸𝑔  often correlates with high electrochemical stability and negligible electronic conductivity. The value of 𝐸𝑔 was estimated from the energy difference between the valence band maximum (VBM) and conduction band minimum (CBM) of a compound’s DFT-PBE electronic density of states.  For 𝐸ℎ , the Li+/H+ exchange process leading to LiOH formation was considered as the relevant reaction for chemical stability evaluation in air:43-44 𝐿𝑖𝑎𝑋𝑏𝑍𝑐 + 𝐻2𝑂 → 𝐿𝑖𝑎−1𝐻𝑋𝑏𝑍𝑐 + 𝐿𝑖𝑂𝐻,  (2) where 𝐿𝑖𝑎𝑋𝑏𝑍𝑐  is the ip compound (e.g., {𝑎, 𝑏, 𝑐} = {81,27,27}  for Li81X27Z27 supercell). A supercell size of at least10 Å in cell edge lengths was imposed to remove any unphysical interaction between the atoms in the main cell and those from the mirror-image cells due to periodic boundary condition. For the Li-H arrangements, 10 random configurations were generated and then geometry-optimized by DFT. Using the lowest total-energy structure, the reaction free energy with H2O (ΔG𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝐻2𝑂) was determined as follows: Δ𝐺𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝐻2𝑂  =  Δ𝐻 −  𝑇Δ𝑆 (𝑇 = 298 K),  (3) where ΔG𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝐻2𝑂 < 0 (in kJ/mol) when the reaction is exothermic (i.e., spontaneous reaction with H2O). Otherwise, the reaction is endothermic ( ΔG𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝐻2𝑂  > 0) and would require external energy/heat source to drive the protonation process. A compound with 𝑓2: 𝐸ℎ > 0 is thus desired from the viewpoint of SE chemical stability. The entropic term (𝑇Δ𝑆) was computed following the scheme implemented for the Materials Project database and with the pymatgen library.45 Specifically, the entropy contribution term is added to the DFT energy of any compounds that are liquid or gaseous at room temperature (298 K), such as H2O in the present work; the entropy data that was used was derived from an experimental dataset source.46 For 𝑓3: Λ, AIMD calculation was performed for two types of defect-driven ion transport processes: Li dumbbell (interstitial) and vacancy-driven mechanism.47 The MD step size was fixed to 1 fs and the target temperature was set to 400 K. For the estimation of bulk Li+ ion activation energy (𝐸𝑎), the Arrhenius slope was determined from diffusivity data at several MD temperatures (700, 800, 900, 1000, and 1200 K). In the initial stage of the MD run, structures were equilibrated at the target temperature under NPT ensemble condition for 10 ps.48 This step was then followed by the actual production run under the NVT ensemble condition for 50 ps.49 The time-averaged mean square displacement (𝑀𝑆𝐷) of Li+ ions (in Å2) was calculated by the following equation:50 𝑀𝑆𝐷 = 〈[𝒓(𝑡 + 𝜏) − 𝒓(𝑡)]2〉,    (4) where 𝒓(𝑡)  is the position Li at time 𝑡 , and 𝜏  is the lag time between positions. The diffusion coefficient (cm2/s) was estimated based on the Einstein-Smoluchowski equation:51 𝐷 = lim𝑡→∞[(1/2𝑑𝑡)〈[𝒓(𝑡 + 𝜏) − 𝒓(𝑡)]2〉],  (5) where 𝑑 is the dimensionality of the lattice for Li+ ion diffusion. It was previously determined that t-ip and c-ip compounds have 𝑑 = 2 and 𝑑 = 3, respectively. It is noteworthy to mention that 𝐷 was derived from the slope at the diffusive regime of the 𝑀𝑆𝐷 plot; additional details are provided in the Supporting Information section (see Fig. S1). The Nernst-Einstein equation was then used to calculate the AIMD ionic conductivity (Λ):52 Λ = 𝜌𝐹𝑐2𝑧𝑐2𝐷 𝑅𝑇⁄ ,     (6) where 𝜌 is the mass density of Li+ ions, 𝑧𝑐 is the charge of Li+ ion, 𝐹𝑐 is the Faraday constant, 𝑅 is the gas constant, and 𝑇 is the temperature.   Machine learning methods Machine-learning descriptors of a given compound 𝑖  (𝒙𝑖 ) were extracted from its chemical and structure-/geometric-based information. Three natural descriptor groupings were considered: atom electronegativity (𝐸𝑁), atom-centered Voronoi real-feature values (𝑉𝑅), and atom-atom partial radial distribution functions (𝑃𝐹). Features were encoded by binning approach, essentially converting the compound information into histogram vectors.53  A surrogate-assisted multi-objective optimization algorithm based on Bayesian optimization (MOBO) with a batch-compound sampling scheme was employed, see Algorithm 1.54 The MOBO goal here was to iteratively improve a Pareto-front approximation set of SE-property objective functions. The non-dominated points of the Pareto front are referred to as the optimal solution sets (i.e., best ip compounds). Without loss of generality, objective functions 𝑦(𝑘) (i.e., 𝑘 = 3, {𝑦(1): 𝑓1, 𝑦(2): 𝑓2, 𝑦(3): 𝑓3}) were treated as mutually independent from each other. The 𝑦 function was approximated by a Gaussian process (𝐺𝑃 ) model that is characterized by a mean function 𝑚(𝒙): 𝜲 → ℝ  and a kernel (or covariance) function: 𝜲 × 𝑿 → ℝ . The 𝑦  values ( 𝑦1:𝑛 ≔ [𝑦(𝒙1), … , 𝑦(𝒙𝑛)]⊤ ) are assumed to be drawn from a distribution with a Gaussian nature at arbitrary finite 𝑛  data points 𝒙1:𝑛:55 𝑦1:𝑛|𝒙1:𝑛~𝒩(𝑚1:𝑛, 𝜥),    (7)  where 𝛫𝑖,𝑗 = 𝑘(𝒙𝑖 , 𝒙𝑗) (𝑖, 𝑗 ∈ {1, … , 𝑛}). The base functional form of 𝑘(𝒙𝑖 , 𝒙𝑗) was formulated with a radial basis function: 𝑘(𝒙𝑖, 𝒙𝑗) = 𝜎𝑓2 exp (−‖𝒙𝑖 − 𝒙𝑗‖22𝑙2⁄ ),  (8) where 𝜎𝑓2 is for the positive-valued signal variance (kernel amplitude), ‖𝒙𝑖 − 𝒙𝑗‖2 is for the squared Euclidean distance between two sample compounds (represented as descriptor vectors), and 𝑙 is for the positive-valued lengthscale. The variation of two hyperparameters, 𝜎𝑓2  and 𝑙 , results in the variation in the a priori correlation between sample compounds and consequently, the prediction variability for 𝑦. The latter is also assumed to have an associated noise (𝜖): 𝑦𝑖 = 𝑓(𝒙𝑖) + 𝜖𝑖      (9) with 𝜖𝑖~ 𝒩(0, 𝜎𝑛𝑜𝑖𝑠𝑒2 ). Given 𝑛 training samples 𝒟𝑡 ∶= {𝒙𝑖 , 𝑦𝑖}𝑖=1𝑡 , a posterior for 𝑓𝑘: 𝑓(𝒙) can be expressed into the following: 𝑓(𝒙)|𝑦1:𝑡~𝒩(𝜇𝑡(𝒙), 𝜎𝑡2(𝒙)),    (10) 𝜇𝑡(𝒙) = 𝑚(𝒙) + 𝒌(𝒙)⊤(𝜥 + 𝜎𝑛𝑜𝑖𝑠𝑒2 𝜤)−1(𝑦1:𝑡 − 𝑚1:𝑡), (11) 𝜎𝑡2(𝒙) = 𝑘(𝒙, 𝒙) − 𝒌(𝒙)⊤(𝜥 + 𝜎𝑛𝑜𝑖𝑠𝑒2 𝜤)−1𝒌(𝒙),  (12) where 𝒌(𝒙) , 𝑚1:𝑡 , 𝜇𝑡(𝒙) , and 𝜎𝑡2(𝒙)  represents the covariance among 𝒙  and 𝒙1:𝑡 , the mean function of 𝒙1:𝑡, the posterior mean, and the posterior variance, respectively. The 𝐺𝑃 models were improved by tuning the hyperparameters by marginal likelihood maximization approach.56  To avoid convergence issues related to the high-dimensionality of 𝑓(𝒙), a descriptor reduction step was performed as follows:57,58 i) projecting each of the grouped descriptor vectors (𝐸𝑁, 𝑉𝑅, 𝑃𝐹) down to 30 latent variables by kernel PCA,59 and ii) decomposing 𝑓(𝒙) into a linear combination relative to the descriptor groups: 𝑓(𝒙) = ∑ 𝑓(𝒙𝑖)𝐹=6𝐹=1 + 𝝐,    (13) where 𝐹 =  6 denotes 6 decomposition terms for 𝑓(𝒙)  and 𝝐~𝒩(0, 𝜎𝜖2)  is the imposed residual that is a zero-mean Gaussian noise. Moreover, each 𝑓(𝒙𝑖) was formulated with sub-kernels 𝑘𝐹 as follows: 𝑘1 = 𝑘(𝒙, 𝒙′)𝐸𝑁,     (14) 𝑘2 = 𝑘(𝒙, 𝒙′)𝑉𝑅,     (15) 𝑘3 = 𝑘(𝒙, 𝒙′)𝑃𝐹,     (16) 𝑘4 = 𝑘(𝒙, 𝒙′)𝐸𝑁𝑘(𝑥, 𝑥′)𝑉𝑅,    (17) 𝑘5 = 𝑘(𝒙, 𝒙′)𝐸𝑁𝑘(𝑥, 𝑥′)𝑃𝐹,    (18) 𝑘6 = 𝑘(𝒙, 𝒙′)𝑉𝑅𝑘(𝑥, 𝑥′)𝑃𝐹,    (19) where 𝑘1, 𝑘2, and 𝑘3 are sub-kernels for the original (primary) descriptor groups (𝐸𝑁, 𝑉𝑅, 𝑃𝐹) while 𝑘4 , 𝑘5 , and 𝑘6  are secondary product sub-kernels derived from 𝑘1 , 𝑘2 , and 𝑘3 . The abovementioned algorithm and the machine-learning models were coded using the python libraries scikit-learn and GPy.60,61 At each MOBO iteration step, a batch of query compounds were selected by an infill criterion based on hypervolume improvement with upper confidence bound (𝑢𝐻𝑉𝐼):54 𝑢𝐻𝑉𝐼 = 𝑉1 − 𝑉0 = 𝑉 (𝒚(𝑑0) ∪ 𝑔(𝒙𝑡+1,𝑚)) − 𝑉(𝒚(𝑑0)), (20) where 𝑔(𝒙𝑡+1,𝑚) = 𝜇𝑡+1.𝑚(𝒙) + 𝜎𝑡+1.𝑚(𝒙) is the upper confidence bound of a sample compound 𝑚  related to the DFT-calculated (𝑓1 , 𝑓2 , 𝑓3 ) values. Each sample compound 𝑚  in the list of 𝑀 unevaluated candidates (𝑚 ∈ 𝑀) was added to 𝑑0 at a time. All the 𝑢𝐻𝑉𝐼 scores of 𝑀 candidates were then sorted in decreasing order, with the top 𝑧 = 10 compounds in the ordered list being picked by a greedy selection rule. This batch of selected compounds were subsequently evaluated for their (𝑓1, 𝑓2, 𝑓3) values by DFT. The sequential nature of MOBO sampling makes it highly suitable for ‘small data’ learning on target properties that are expensive to obtain, such in the case of AIMD ionic conductivity.58 Since the optimal direction of the present MOBO sampling strategy was formulated towards large positive values of 𝑓1 , 𝑓2 , and 𝑓3 , the overall optimization setting can then be regarded as a maximization problem. Additionally, (𝑓1, 𝑓2, 𝑓3) values were weighted equally with the following scaling transformation: 𝑓1 → −𝑓1,     (21) 𝑓2 → −𝑓2/100, and     (22) 𝑓3 → −10𝑓3.     (23) The negative signs were applied for convenience in the hypervolume calculation step for the 𝑢𝐻𝑉𝐼 score. The nadir point was fixed to a scaled coordinate of (0, -10, 0) which was determined as the physically meaningful bound for ( 𝑓1 , 𝑓2 , 𝑓3 ). The batch compound sampling procedure was terminated when all the best (𝑓1, 𝑓2, 𝑓3) values do not change anymore for at least 3 MOBO iteration steps. The numerical calculation of hypervolume was performed using the python package PyGMO.62 Table 1 shows the MOBO algorithm in detail.  Table 1. Algorithm description of the compound sampling workflow that was developed in this work which is based on multi-objective Bayesian optimization (MOBO) with upper confidence bound (UCB) and greedy-selection batch sampling (𝑘 = 3 objective functions) 1. Build initial dataset 𝑑0 = (𝒙𝑡, (𝑦𝑡1, 𝑦𝑡2, 𝑦𝑡3)). 2. Calculate initial hypervolume 𝑉0 = 𝑉(𝑑0). 3. Build 𝐺𝑃 models for 𝑘 objective functions using 𝑑0. 4. Use 𝐺𝑃 models to estimate (𝑔𝑡+1,𝑚(1) (𝒙), 𝑔𝑡+1,𝑚(2) (𝒙), 𝑔𝑡+1,𝑚(3) (𝒙)) of all sampling point 𝑚 in the list of unevaluated M samples (i.e., 𝑔𝑡+1,𝑚(𝒙) = 𝜇𝑡+1,𝑚(𝒙) + 𝜎𝑡+1,𝑚(𝒙)). 5. For each 𝑚 unevaluated point (𝑥𝑡+1), calculate separately for hypervolume 𝑉1 ← 𝑉(𝒚(𝑑0) ∪𝑔𝑡+1,𝑚(𝒙)) and expected hypervolume improvement with UCB (𝑢𝐻𝑉𝐼 = 𝑉1 − 𝑉0). 6. Select top 𝑧 candidates based on 𝑎𝑟𝑔𝑚𝑎𝑥(𝑢𝐻𝑉𝐼) from 𝒙𝑡+1,𝑚 samples. 7. Evaluate 𝒚 of top 𝑧 candidates (z < M) by DFT calculations.   𝑑1 = {(𝒙𝑡+1,1, (𝑦𝑡+1,11 , 𝑦𝑡+1,12 , 𝑦𝑡+1,13 )), (𝒙𝑡+1,2, (𝑦𝑡+1,21 , 𝑦𝑡+1,22 , 𝑦𝑡+1,23 )), (𝒙𝑡+1,3, (𝑦𝑡+1,31 , 𝑦𝑡+1,32 , 𝑦𝑡+1,33 )), … , (𝒙𝑡+1,𝑧, (𝑦𝑡+1,𝑧1 , 𝑦𝑡+1,𝑧2 , 𝑦𝑡+1,𝑧3 )). 8. Stopping criterion met?   Yes: Output Pareto-front approximation set   No: Perform update (𝑑0 ← 𝑑0 ∪ 𝑑1, 𝑉0 ← 𝑉0 ∪ 𝑉1), repeat steps 3-7.   Fig. 2 shows the schematic workflow of the MOBO-based SE screening task. In step 1, derivative structures and series compounds from parent lattices (I4/mmm for n > 1, Pm3̅ m for n = ∞) were generated by: i) supercell operation with up to a maximal Miller index of 3 and ii) exclusive anion-site isovalent substitution with symmetry ordering constrained by maximal group-subgroup relations63. The sub-concentration ratios of the alloying elements at the X (O, S, Se, Te) and Z (F, Cl, Br, I) sites were restricted to 1:0, 0.75:0.25, 0.5:0.5, 0.67:0.33, and vice versa. Using heuristic guidelines on metastability,64,65 the different anionic configuration (microstates) in a given composition were each treated as distinct compounds that potentially can be accessible during actual experimental synthesis. As a result, the generated initial chemical space contains multiple entries at each unique composition. In step 2, the initial chemical space was reduced using a phase stability filter of 𝐸𝑑 < 0.1 eV/atom.65 The reduced chemical space was then used for the proposed MOBO-based SE screening task in step 3, with batch compound sampling in an iterative manner that involves the explicit DFT calculations of 𝑓1: 𝐸𝑔, 𝑓2: 𝐸ℎ, and 𝑓3: Λ. In step 4, the non-dominated points (i.e., optimal compounds) were determined from the constructed the Pareto-front approximation set (i.e., SE-property frontier surface).    Figure 2. Schematic workflow of the multi-objective Bayesian-optimization compound sampling that was employed in this work: (1) in-silico compound generation by anion site alloying and symmetry constraints, (2) thermodynamic (meta)stability filtering by convex hull method for compound space reduction, (3) compound sampling by multi-objective Bayesian optimization (see Table 1 for the algorithm in detail) based on DFT-calculated electrolyte-related properties (𝑓1: DFT electronic band gap energy in eV, 𝑓2: DFT hydration energy in kJ/mol, 𝑓3: DFT 400-K Li+ ionic conductivity in S/cm), and (4) Pareto-based analysis for the selection of optimal compounds.               Results and discussion Structure and thermodynamic stability  We previously proposed a modified formulation of the Goldschmidt tolerance factor (𝑡𝐺,𝑖𝑝) for t-ip compounds, the equation is given by:29,66  𝑡𝐺,𝑖𝑝 =𝑟𝐿𝑖+∑ 𝑤𝑍,𝑖𝑟𝑍,𝑖2𝑖=1√2(∑ 𝑤𝑋,𝑖𝑟𝑋,𝑖2𝑖=1 +∑ 𝑤𝑍,𝑖𝑟𝑍,𝑖2𝑖=1 ),   (24) where 𝑟𝐿𝑖, 𝑟𝑋,𝑖, and 𝑟𝑍,𝑖 are the Shannon ionic radii of Li+, X2- anion, and Z-, respectively, with their site-occupancy weights (i.e., 1 for Li+, 𝑤𝑋,𝑖 for X2-, and 𝑤𝑍,𝑖 for Z-). Here, we apply equation (24) to also evaluate the structure stability of c-ip compounds. Fig. 3a shows the plot on 𝐸𝑑 vs. 𝑡𝐺,𝑖𝑝 of 10,173 in-silico t-ip and c-ip compounds, highlighting a negative correlation and an increasingly narrower 𝐸𝑑 range as 𝑡𝐺,𝑖𝑝 increases. The physical meaning of 𝑡𝐺,𝑖𝑝 can be further analyzed with respect to chemistry, as displayed in Fig. 3b which is the 𝐸𝑑 heatmap as a function of weighted anion radii at the X and Z sites (see equation 24 denominator). There is a clear underlying structure of positive and negative correlations between 𝐸𝑑  and ∑ 𝑤𝑋,𝑖𝑟𝑋,𝑖2𝑖=1   and ∑ 𝑤𝑍,𝑖𝑟𝑍,𝑖2𝑖=1  , respectively, hinted by an overall shape characterized by low-lying 𝐸𝑑 contour lines in the upper left corner and high-lying contour lines in the lower right corner of the energy-stability surface. Consistent with our previous result,29 ip compounds tend to become less stable with less electronegative X anion (i.e., generally larger X anion radius) and/or more electronegative the Z halide anion (i.e., smaller Z anion radius), and vice versa. Given this strong correlation, 𝑡𝐺,𝑖𝑝 can thus be employed as a general stability screening parameter that encompasses both t-ip and c-ip compounds.29,67 By 𝐸𝑑 criterion, 1,501 compounds are predicted to be (meta)stable. Some of these compounds have already been experimentally confirmed previously, thus providing a quantitative validation of our calculation results: Li3OCl (0 eV/atom), Li3OBr (0.017 eV/atom), Li3OCl0.5Br0.5 (0.007 eV/atom), Li7O2Br3 (0.006 eV/atom), and Li6OSI2 (0.052 eV/atom). Meanwhile, several unreported and potentially novel compounds are predicted as stable or ground-state phases (i.e., DFT-𝐸𝑑 = 0), they belong to O- and Br/Cl-bearing compositions: I4/mmm Li7O2Cl3 (I4/mmm), Li7O2ClBr2 (I4mm, P4/mmm, P4/nmm), Li8O2ClBr3 (P4/nmm, P4/mmm, P4mm, Pmm2, P4̅m2, Cm, P1), Li4OClBr (P4mm, P4/mmm, I4mm, P4mm, Pmm2, Pma2, Cm), Li8O2Cl3Br (P4/mmm, P4/nmm, P4mm, P4̅m2, Pmm2, Cm), Li4OCl2 (I4/mmm), and Li4OBr2 (I4/mmm). Majority of compounds that are predicted to be unstable (i.e., 𝐸𝑑 > 0.1 eV/atom) belong to anion substitutions in the first and last half of {F, Cl, Br, I} and {O, S, Se, Te} series, respectively. Examples include I4/mmm Li7S2Cl3 (0.147 eV/atom), Pm3̅m Li3TeF (0.783 eV/atom), I4/mmm Li7Te2F3 (0.640 eV/atom), Pmmm Li8Se2Br3I (0.125 eV/atom), P4mm Li7SeTeFCl2 (0.295 eV/atom) and Cmcm Li12SeTe3F3Br (0.635 eV/atom).  The 𝐸𝑔 trend is also analyzed vs. phase stability. In Fig. 3a, ip compounds with large 𝐸𝑔 (> 4 eV) are noted to be highly stable as well. Three selected in-silico compounds are shown in Fig. 4, they contain O-Cl/I anions: n = 1 (P4mm Li8O2Cl3I), n = 2 (I4mm Li7O2Cl2I), and n = ∞ (P4/mmm Li12O4Cl3I). Given the large 𝐸𝑔 values, these ip compounds can be categorized as good insulators, as compared relative to other known inorganic-type SEs (see Table S1). Upon inspection of the electronic band structure, the VBM is found to be dominated by O-2p states and the VBM region has highly localized peak features that corresponds well to strong ionic bonding.   Figure 3. (a) Plot on DFT-calculated decomposition energy (𝐸𝑑) vs. modified Goldschmidt tolerance factor (𝑡𝐺,𝑖𝑝)29 of 10,173 in-silico inverse perovskite (ip) structures in the general formulae Li3XZ (cubic, n = ∞), Li4XZ2 (tetragonal, n = 1), and Li7X2Z3 (tetragonal, n = 2). Structures with PBE electronic band gap energy (𝐸𝑔) > 4 eV are highlighted in large symbols. (b) Contour map of lower-bound 𝐸𝑑  values (per unique composition) as a function of weighted-average anion radius per formula unit (rX,w, rZ,w) at the X and Z site. Contour line labels indicate the 𝐸𝑑 value.   Figure 4. Representative Cl-/I-bearing in-silico inverse-perovskite compounds, geometry-optimized by DFT and with their (meta)stability evaluated by decomposition energy (𝐸𝑑): (a) P4mm Li8O2Cl3I (n = 1 structure), (b) I4mm Li7O2Cl2 (n = 2 structure), and (c) P4/mmm Li12O4Cl3I (n = ∞ structure). The corresponding GGA-PBE electronic density of states are shown in (d), (e), and (f), respectively.   Multiple SE-property optimization Fig. 5a shows the MOBO-based SE screening results for in-silico t-ip and c-ip compounds, the best combination values for 𝑓1: 𝐸𝑔, 𝑓2: 𝐸ℎ, and 𝑓3: Λ are plotted for each MOBO iteration step. The termination condition is achieved at the 7th step, with 𝑓1 and 𝑓3 determined as the fastest and slowest SE-property function, respectively. The distribution ranges are 𝑓1 = [2.52, 4.99] eV, 𝑓2 = [-158.27, 56.54] kJ/mol, and 𝑓3 = [8.0 x 10-6, 7.8 x 10-1] S/cm. Fig. 5b depicts the Pareto-front approximation set (red-colored surface) at the 7th sampling step, containing a total of 24 optimal compounds (circle symbol) out of all the batch-sampled compounds (circle + x symbols). Table 2 summarizes Pareto-front (optimal) compounds, including their corresponding 𝑓1 , 𝑓2 , and 𝑓3  values. The best compounds with respect to each SE-property function are P4/nmm Li8O2Cl3Br (𝑓1: 𝐸𝑔 = 4.99 eV), I4/mmm Li7O2FBr2 (𝑓2: 𝐸ℎ  = 56.54 kJ/mol), and Cmcm Li12O3SeClBr3 (𝑓3: Λ  = 7.8 x 10-1 S/cm). Oppositely, worst compounds are Pmm2 Li12Se3TeBrI3 (𝑓1: 𝐸𝑔 = 2.52 eV), Fm3̅m Li6O2FCl (𝑓2: 𝐸ℎ = -158.27 kJ/mol) and P4mm Li8O2ClBr3 ( 𝑓3: Λ  = 8.0 x 10-6 S/cm). The small 𝐸𝑔  of Pmm2 Li12Se3TeBrI3 can be explained by the relatively energy-shallow anion density of states (DOS) of Se2- and Te2- contributions to the VB region, in converse to P4/nmm Li8O2Cl3Br (i.e., largest- 𝐸𝑔 compound) which has O anion DOS states located in the deeper VB-region energy level. Notably, most of 𝑓2 values are negative which indicate that H2O reactivity in ip compounds is rather difficult to avoid via anion-site alloying.   Figure 5. (a) SE-property maximization results for the in-silico inverse-perovskite chemical space (general formulae Li3XZ, Li4XZ2, Li7X2Z3) by multi-objective Bayesian optimization (MOBO) algorithm. SE properties include 𝑓1: electronic band gap energy (𝐸𝑔 in eV), 𝑓2: hydration energy (𝐸ℎ in kJ/mol), and 𝑓3: 400-K AIMD Li+ ionic conductivity (Λ in S/cm). MOBO iteration is stopped when the best values of all 3 SE-property functions do not change simultaneously for 3 consecutive iterations. (b) Evolved Pareto-front approximation set (red surface) at the 7th iteration step. Arrows indicate the direction for maximization. All sampled points (compounds) are represented as x marks while non-dominated points (i.e., optimal compounds) therein are separately marked as yellow circles.       Table 2. Approximate Pareto-front (optimal) compounds derived from multiple SE-property optimization (see Figure 5). The thermodynamic (meta)stability of sampled compounds is evaluated DFT-calculated decomposition energy (𝐸𝑑). Target SE properties that were simultaneously optimized are DFT-PBE electronic band gap energies (𝐸𝑔), DFT reaction energy with H2O (𝐸ℎ), and AIMD bulk Li+ ionic conductivity at 400 K (Λ). Composition Space group Base structure formula 𝐸𝑑 / eV/atom 𝑓1: 𝐸𝑔 / eV 𝑓2: 𝐸ℎ / kJ/mol 𝑓3: Λ / S/cm Li8Te2Br3I P1 Li4XZ2 (n = 1) 0.079 3.10 +13.36 5.3 x 10-1 Li8Se2BrI3 Cm Li4XZ2 (n = 1) 0.060 3.48 +15.45 2.2 x 10-1 Li8O2ClBr3 P4mm Li4XZ2 (n = 1) 0 4.77 -21.69 8.0 x 10-6 Li8OSCl4 Amm2 Li4XZ2 (n = 1) 0.070 4.14 -40.70 9.2 x 10-2 Li8OSCl4 Cmmm Li4XZ2 (n = 1) 0.077 4.12 -157.83 1.4 x 10-1 Li8Se2ClI3 Cm Li4XZ2 (n = 1) 0.067 3.49 +15.45 4.2 x 10-1 Li8O2ClBr3 Cm Li4XZ2 (n = 1) 0 4.53 -37.88 7.4 x 10-2 Li8O2Cl3Br Cm Li4XZ2 (n = 1) 0 4.74 -33.22 9.0 x 10-4 Li8O2Cl3Br P4/nmm Li4XZ2 (n = 1) 0 4.99 -39.39 1.0 x 10-5 Li8S2Cl3I Cm Li4XZ2 (n = 1) 0.059 4.11 +7.06 3.3 x 10-1 Li7O2FBr2 I4/mmm Li7X2Z3 (n = 2) 0.074 4.24 +56.54 5.0 x 10-5 Li7O2Cl2Br P4/nmm Li7X2Z3 (n = 2) 0 4.62 -22.49 3.0 x 10-3 Li3OCl Pm3̅m Li3XZ (n = ∞) 0 4.56 -22.43 4.8 x 10-2 Li6O2FCl Fm3̅m Li3XZ (n = ∞) 0.091 4.65 -158.27 1.3 x 10-1 Li6O2ClBr Fm3̅m Li3XZ (n = ∞) 0.009 4.36 -31.99 8.2 x 10-2 Li6SSeClI Pmm2 Li3XZ (n = ∞) 0.076 3.12 +16.12 1.4 x 10-1 Li9O3ClBr2 P4/mmm Li3XZ (n = ∞) 0.011 4.96 -34.66 9.0 x 10-3 Li9O2SBr2I P4mm Li3XZ (n = ∞) 0.086 3.90 -51.46 3.6 x 10-1 Li12OS3Cl2Br2 Amm2 Li3XZ (n = ∞) 0.077 3.74 +8.35 1.2 x 10-1 Li12OS3Cl2Br2 Pm Li3XZ (n = ∞) 0.072 3.54 -66.74 5.2 x 10-1 Li12SSe3ClI3 Cmmm Li3XZ (n = ∞) 0.096 3.28 -1.73 5.0 x 10-1 Li12Se3TeBrI3 Pmm2 Li3XZ (n = ∞) 0.094 2.52 +18.99 1.3 x 10-1 Li12O3SeClBr3 Cmcm Li3XZ (n = ∞) 0.097 3.36 -86.88 7.8 x 10-1 Li6OSeI2 Fm3̅m Li3XZ (n = ∞) 0.059 3.10 +38.42 3.8 x 10-2   Electrochemical window analysis Fig. 6 shows the DFT-calculated Li grand potential phase stability plots for various Li-rich inverse-perovskite (ip) solid electrolyte (SE) candidates with various anion-site substitutions as shown below; decomposition product phases in different voltage ranges are indicated.68 It is noted that the most of the SE candidates are predicted to be metastable (i.e., 𝐸𝑑 < 0.1 eV/atom). Given this, when in contact with the Li metal anode, the thermodynamic decomposition phases are expected to be the relevant phases to constitute the interphase region between the anode and the SE component. Consequently, the electrochemical window of the SE candidates may be considered as interphase-controlled. Meanwhile, a crucial consideration of the interphase region is that it should be Li-ion conducting but electronically insulating at the same time, so that further decomposition of the (kinetically stabilized) main-bulk SE component, as well as Li dendrite growth during the Li plating process (i.e., during battery charging), can be prevented.  For example, the relevant decomposition phases for P1 Li8Te2Br3I SE are LiI, LiBr, and Li2Te (see Figure (a)) in the electrochemical window of [0 V, 1.56 V]. Experimentally, LiI has been reported as a good Li ion conductor, while LiBr was shown to have a beneficial Li dendrite suppression effect.69,70 On the other hand, Li2Te has been predicted by DFT to have a low bulk Li ion activation energy.71 Overall, the interphase region formed by the 3 phases is predicted to provide a favorable Li ion transport between Li metal anode and Li8Te2Br3I SE. A similar analysis can be performed as well for the other SE compounds, such as Cm Li8S2Cl3I (LiI, LiCl, and Li2S in the range of [0 V, 2.34 V]), Fm3̅m Li6O2ClBr (Li2O, LiBr, and LiCl in the range of [0 V, 2.79 V]), Pmm2 Li6SSeClI (LiI, Li2Se, Li2S, and LiCl in the range of [0 V, 1.89 V]), Pmm2 Li12Se3TeBrI3 (Li2Te, LiI, LiBr, and Li2Se in the range of [0 V, 1.56 V]), and Fm3̅m Li6OSeI2 (Li2O, Li2Se, and LiI in the range of [0 V, 1.89 V]). Among the decomposition phases of these ip SEs, Li2S poses a detrimental effect for the interphase region because of its poor Li ionic conductivity of ~10-13 S/cm at room temperature.72 As a consequence, S-bearing ip compositions such as Cm Li8S2Cl3I and Pmm2 Li6SSeClI may not form an interphase region with high Li ion diffusivity across the anode - SE interface.  A notable observation for the oxidative stability (i.e., at low Li chemical potential condition) of ip SE compounds is that the interphase-controlled voltage window for O-bearing compositions are generally larger than for non-O-bearing ones, such as in the case of Fm3̅ m Li6O2ClBr ([0 V, 2.79 V]) vs. Li12Se3TeBrI3 ([0 V, 1.56 V]). Collectively, an oxidative stability trend down the periodic table of elements can be deduced: O2- > S2- > Se2- > Te2-. We emphasize that this trend is also consistent with the observed trend for the DFT electronic band gap energy (𝐸𝑔) which was used as an electrochemical window upper bound for ip SEs (see Figure 3a). Based on above thermodynamic analysis, ip SEs generally exhibit narrow electrochemical stability windows. However, they can be effectively extended by the decomposition phases that form the interphase.73 Thus, the wide electrochemical window that was demonstrated experimentally in a prior work using an ip SE may have stemmed from the interphase compounds at the SE-electrode interface.23  Figure 6. Li grand potential phase stability plots for selected Li-rich inverse-perovskite (ip) solid electrolyte (SE) candidates from Table 2: (a) P1 Li8Te2Br3I, (b) Cm Li8S2Cl3I, (c) Fm3̅m Li6O2ClBr, (d) Pmm2 Li6SSeClI, (e) Pmm2 Li12Se3TeBrI3, and (f) Fm3̅m Li6OSeI2. In the anodic or low voltage region, the decomposition binary phases determine the reductive stability, whereas in the cathodic or high voltage region (i.e., after the first step in the Li uptake axis), they are predicted to undergo oxidation and loses Li. Compounds listed in the plots are DFT-predicted phase equilibria (decomposition phases) at corresponding voltage regions (V vs Li/Li+).  The stability of ip SEs when in contact with known cathode materials were evaluated by calculating the interfacial mutual reaction energy. Specifically, the reaction energy minimum (∆𝐸𝑚𝑢𝑡𝑢𝑎𝑙,𝑚𝑖𝑛) with respect to varying reaction ratio of the cathode and the SE was calculated using an approach employed in a previous work:68 ∆𝐸𝑚𝑢𝑡𝑢𝑎𝑙,𝑚𝑖𝑛 = min𝑥∈[0,1]{1𝑁[𝐸𝑒𝑞(𝑥𝑐𝐴 + (1 − 𝑥)𝑐𝑆𝐸) − 𝑥𝐸(𝑐𝐴) − (1 − 𝑥)𝐸(𝑐𝑆𝐸)]}, (25) where 𝑥 is the ratio of cathode 𝐴, 𝐸(𝑐𝐴) and 𝐸(𝑐𝑆𝐸) are the DFT-calculated total energies at the convex hull at the composition of cathode A and SE, respectively, 𝑁 is the total number of atoms involved in the reaction, and 𝐸𝑒𝑞(𝑥𝑐𝐴 + (1 − 𝑥)𝑐𝑆𝐸) is the phase equilibria energy at composition 𝑥𝑐𝐴 + (1 − 𝑥)𝑐𝑆𝐸. Fig. 7a shows the pseudobinary phase diagram on ∆𝐸𝑚𝑢𝑡𝑢𝑎𝑙,𝑚𝑖𝑛 of several ip SE candidates when in contact with the NMC811 cathode (LiNi10/12Mn1/12Co1/12O2). Among the ip SEs, S-, Se- and Te-bearing compositions are predicted to have the highest reactivity. For instance, the reaction energy between Pm Li12OS3Cl2Br2 SE and NMC811 is -270 meV/atom. This result is in good agreement with our previous DFT-based study on the possible reactions and structures at the solid hetero-interface formed between an oxide-type cathode (LiCoO2) and a sulfide-type SE (β-Li3PS4), in which the O-S exchange reaction across the interface was predicted to be thermodynamically favorable.72 On the other hand, O-bearing compositions show the lowest reactivity, such as in the case of P4mm Li9O3ClBr2 which is only -7 meV/atom. Extending to other Ni-Mn-Co compositions (see Fig. 7b), it is evident that the interfacial reactivity of ip SEs are positively correlated with Ni content, with LiNiO2 (i.e., 100% Ni at the transition metal site of the layered structure) showing the most exothermic reactions (i.e., darkest-colored heatmap cells). The LiFePO4 cathode generally shows a relatively moderate reactivity vs. ip SEs, the reaction energy range is between -120 and -70 meV/atom. For the LiCoO2 cathode, O-bearing ip SEs show low reactivity (above -10 meV/atom), while the rest have moderate reactivity (-90 to -10 meV/atom). The least reactive cathode (i.e., with the lightest-colored heatmap cells) vs. ip SEs is LiMnO2, with reaction energies that are very close to 0. This predicted interfacial stability is consistent with the findings of a previous DFT work,75 it was suggested to have stemmed from the relative stability of the oxidation state of Mn as compared to Ni and Co.   Figure 7. (a) DFT-calculated pseudobinary phase diagram of NCM811 cathode (LiNi10/12Mn1/12Co1/12O2) that is in contact with Li-rich inverse-perovskite (ip) solid electrolyte (SE) candidates that are listed in Table 2 (the same compound ordering). The circle markers indicate the most exothermic reactions (i.e., most negative reaction energy) at a given cathode-SE reaction ratio. (b) Mutual reaction energy heatmap between known cathode materials and ip SEs that are listed in Table 2 (again, the same compound ordering).  Li+ ion transport analysis Bulk Li+ ionic conductivity (𝜎𝑏𝑢𝑙𝑘) can be largely affected by defects in the as-prepared SE material. Here, we investigate the 𝜎𝑏𝑢𝑙𝑘  of t-ip and c-ip compounds by deriving the Arrhenius plot from multiple AIMD temperatures. Two defect-driven Li+ ion transport processes are separately investigated: Li-interstitial and Li-vacancy mechanism. Fig. 8a shows the 𝜎𝑏𝑢𝑙𝑘  plot for selected compounds from Table 1: Cm Li8S2Cl3I (n = 1 structure), Cmcm Li12O3SeClBr3 (n = ∞ structure), and P4/nmm Li7O2Cl2Br (n = 2 structure). Moreover, the nature of the diffusion channels is displayed in Fig. 8b and 8d for the tetragonal symmetry (Cm Li8S2Cl3I and P4/nmm Li7O2Cl2Br) and in Fig. 8c for the cubic symmetry (Cmcm Li12O3SeClBr3). It is noted that the Li ion diffusion of ip compounds with the tetragonal structure has a low Li ion migration barrier within the intra-slab-layer, while the inter-slab-layer migration has a large energy barrier, as demonstrated in our previous work by DFT climbing-image nudged elastic band method.29 The large energy barrier for the inter-slab Li ion migration is characterized by a transition state in which Li ions need to pass between two halide anions. On the other hand, the energetically favorable intra-slab layer migration has a transition state which can be described by a triangular plane with vertices formed by one O2-/S2- anion and two halide anions. In the case of cubic Cmcm Li12O3SeClBr3 (Figure 8c), the slab layer size in the vertical direction is effectively infinite (n = ∞) and all the local transition states are similar (i.e., a triangular plane with vertices formed by one O2-/Se2- anion and two halide anions). Thus, the Li ion diffusion is relatively more percolated in 3D in the cubic structure than in the tetragonal structure. Based on the 𝜎𝑏𝑢𝑙𝑘T vs. 1000/T slope, it is evident that the Li-interstitial mechanism generally results in a relatively higher conductivity than the Li-vacancy case. This can be explained by the increased Li-Li repulsion effect due to shorter effective Li-Li interatomic distance (Fig. S2) that leads to Li site-potential shallowing, such as in the case of garnet-type Li7La3Zr2O12 SE.76,77 Therefore, it would be beneficial to stabilize this defect type, such as through stoichiometry control during synthesis (e.g., by excess X2- anion into the halide Z- anion site). Some optimization efforts have been made in related compositions, but they were focused on introducing Li vacancies, such as in Ba-doped Li3OCl and Li-Cl Frenkel defects in Li3OCl.23,78 Estimates on 𝐸𝑎  in Fig. 6a with Li-interstitial (Li-vacancy) mechanism are 0.15 (0.27), 0.37 (0.42) and 0.33 (0.50) eV, respectively. The lowest 𝐸𝑎 of the three compounds, which is for Cm Li8S2Cl3I, can be traced to the highly polarizable S2- anions which promote noticeable tilting of the linked anion-center SLi6 octahedral units in the geometry-optimized structure (see Fig. 8b). This degree of tilting correlates with the extent of the displacement amplitude of Li+ ions within their site cages and around the S2- anion, as qualitatively exhibited by the locally wider Li trajectory densities, as compared to the other selected compounds which have the less-polarizable O2- anions (Fig. 8c, 8d).  While the local Li pathway characteristics are shown to depend on anion chemistry, the structure type determines the diffusion dimensionality in ip structures. Specifically, Li+ ion diffusion in the t-ip structure (n = 1, 2) proceeds in a 2D manner, confined within each ip slab layer units. Although interstitial sites exist in the interslab region, these sites are apparently inaccessible for a migrating Li+ ion because of a large energy barrier.29 On the other hand, c-ip compounds generally have 3D diffusion, with contiguous Li site connectivity via corner-linking of anion-center XLi6 octahedral units.  Figure 8. (a) Conductivity plot of 3 novel inverse-perovkite (ip) compounds (Cm Li8S2Cl3I, Cmcm Li12O3SeClBr3, P4/nmm Li7O2Cl2Br) based on Li-interstitial and Li-vacancy mechanism. The corresponding Li+ ion trajectory densities (in yellow, with cross-section in blue) as derived by 400-K NVT-AIMD runs are shown in (b)-(d), respectively; bondings within the ip structures are displayed in anion-center view. The ip slab size are indicated by parameter 𝑛 which is 1, 2, ∞ for the Li4XZ2, Li7X2Z3 and Li3XZ parent structure, respectively.     Evaluation of synthesis likelihood Although 𝐸𝑑  has been shown in the past to be valid when predicting the phase stability of experimentally confirmed compounds, it cannot fully ascertain beforehand whether theoretically designed compounds are really synthesizable or not. This is expected, given the complexity and variability involved when actually synthesizing compounds in the lab. To supplement 𝐸𝑑  in evaluating the synthesis likelihood of in-silico ip compounds in this work, an additional stability parameter is considered. This parameter is determined from a deep-learning (DL) model that is trained on a structure dataset that is composed of positive (synthesizable) and negative (virtual or unsynthesized) compound entries from the Materials Project (MP) database; there are 46,781 entries that are originally from the ICSD database and 77,734 entries that are virtually generated ones.79 Essentially, the DL model performs a positive-unlabeled (semi-supervised) classification task, using descriptors that are derived from crystal graph network representation. The stability parameter is then expressed as a probability “crystal-likeness” score (𝐿𝑠). If 𝐿𝑠 > 0.50, a given compound is considered as sufficiently alike in terms of crystal structure as with known experimental compounds, otherwise 𝐿𝑠 is too low and the compound tag is set to ‘unlabeled’ (i.e., ambiguous).  Fig. 9a displays the 𝐿𝑠 vs. 𝐸𝑑 plot for 10,173 in-silico t-ip and c-ip compounds. Four regions of interest are shown based on the (dis)agreement between the two parameters: (1) 𝐿𝑠 > 0.5 and 𝐸𝑑 < 0.1 eV/atom, (2) 𝐿𝑠 > 0.5 and 𝐸𝑑 > 0.1 eV/atom, (3) 𝐿𝑠 < 0.5 and 𝐸𝑑 < 0.1 eV/atom, and (4) 𝐿𝑠 < 0.5 and 𝐸𝑑 > 0.1 eV/atom. A continuous 𝐿𝑠 distribution is also found, described with statistical range, mean, and skewness of [0.025, 0.909], 0.542 and -0.395, respectively; the negative skew implies that the median 𝐿𝑠 is larger than the mean 𝐿𝑠 which, is in turn, lies in the ‘synthesizable’ regime. About 61.25% (i.e., 6,231/10,173) are labeled as ‘synthesizable’ (regions 1 and 2), with 13.89% (1,413/10,173) labeled as both ‘synthesizable’ and thermodynamically (meta)stable (region 1). Meanwhile, 47.43% (4,825/10,173) are labeled as ‘synthesizable’ but are predicted to be thermodynamically unstable 𝐸𝑑 criterion (region 2). In contrast, only 0.86% (88/10,173) are labeled as ‘ambiguous’ compounds, even though they are DFT-predicted to be (meta)stable (region 3). Lastly, 37.70% (3,835/10,173) are predicted as ‘ambiguous’ and unstable (region 4). Out of the four regions, region 1 (i.e., 𝐿𝑠 > 0.5 and 𝐸𝑑 < 0.1 eV/atom) is of particular interest, it further excludes 1,501 (by 𝐸𝑑) – 1,413 (by 𝐿𝑠) = 88 compounds in terms of priority towards actual synthesis. In Fig. 9b, the breakdown of counted compounds and composition-stability relationship for region 1 is shown. The stability trend is found to be consistent with trends based on 𝐸𝑑 and 𝑡𝐺,𝑖𝑝. Specifically, the order of decreasing stability is reflected by the compound counts: O2-  S2-  Se2-  Te2- and I-  Br-  Cl-  F-. Interestingly, 𝐿𝑠 is shown to have a moderately negative correlation vs. 𝐸𝑑, indicating that it partly captured the information on thermodynamic stability despite the DL model not being explicitly trained on the 𝐸𝑑 dataset. The list of region-1 compounds is provided in the Supporting information (Table S2), including their 𝐿𝑠 and 𝐸𝑑 values. Overall, the combined synthesizability parameters 𝐿𝑠 and 𝐸𝑑 should provide experimentalists with a well-informed list of compounds for actual synthesis.     Figure 9. (a) Synthesis likelihood (𝐿𝑠 ) map by crystal-likeness (CL) score and thermodynamic decomposition energy (𝐸𝑑) plot for 10,173 in-silico inverse-perovskite (ip) compounds. The CL score was determined by a trained deep learning model on experimental and virtual compounds from Materials Project.79 The CL score indicates the label probability of a compound as either ‘synthesizable’ or ‘ambiguous’ (i.e., either synthesizable or not). The criterion for synthesis likelihood and thermodynamic stability were set to 𝐿𝑠 > 0.50 (horizontal line) and 𝐸𝑑 < 0.1 eV/atom (vertical line), respectively. Four regions of interest are shown, grouping compounds with: (1) 𝐿𝑠 > 0.5 and 𝐸𝑑 < 0.1 eV/atom, (2) 𝐿𝑠 > 0.5 and 𝐸𝑑 > 0.1 eV/atom, (3) 𝐿𝑠 < 0.5 and 𝐸𝑑 < 0.1 eV/atom, and (4) 𝐿𝑠 < 0.5 and 𝐸𝑑 > 0.1 eV/atom. (b) Anion-specific composition counts for region (1) in (a).   Analysis on solid-state synthesis pathways Synthesis reaction pathways can vary largely depending on experimental conditions (e.g., reaction energies, competing phases, nucleation barriers, reactant mixing homogeneity, heating protocol, and choice of starting reactants/precursors). To this end, the solid-state synthesis routes of in-silico t-ip and c-ip compounds are studied using a data-driven approach that is based on classical nucleation theory (CNT).80 The method starts with the description of steady-state CNT for a given reaction with solid reactants 𝛼𝑖 (where 𝑖 = 1, 2, 3, …). The heterogeneous nucleation rate (𝐽) of a target phase β (i.e., the in-silico ip compound in this work) on the surface of a reactant 𝛼𝑖 at temperature T is given by: 𝐽𝛼𝑖→𝛽 = 𝐽0𝑒[−∆𝐺𝛼𝑖→𝛽∗ 𝑘𝐵𝑇⁄ ],   (26) where 𝐽0 is the pre-exponential factor and ∆𝐺𝛼𝑖→𝛽∗  is the critical thermodynamic nucleation barrier. For a spherical nucleus, the latter can be expressed as follows: ∆𝐺𝛼𝑖→𝛽∗ =16𝜋𝛾𝛽3𝑓(𝑆𝛼𝑖→𝛽)3(∆𝐺)2 ,   (27) where 𝛾𝛽 is the surface energy of phase 𝛽, ∆𝐺 is the reaction free energy per volume of the formed cluster phase 𝛽, and 𝑓(𝑆𝛼𝑖→𝛽) is the factor for barrier reduction (in the range of [0, 1]) from the homogeneous or uncatalyzed limit on the surface of reactant 𝛼𝑖 as a function of 𝑆𝛼𝑖→𝛽 which is the cosine of the contact angle (𝜃) between reactant substrate 𝛼𝑖 and target phase 𝛽: 𝑆𝛼𝑖→𝛽 = cos 𝜃𝛽𝛼𝑖=𝛾𝛼𝑖−𝛾𝛽𝛼𝑖𝛾𝛽,   (28) where 𝛾𝛼𝑖  and 𝛾𝛽𝛼𝑖  are the surface energy of substrate 𝛼𝑖  and the interface energy between the substrate 𝛼𝑖 and nucleus 𝛽. The expression for 𝑓(𝑆𝛼𝑖→𝛽) is formulated as a monotonic function: 𝑓(𝑆𝛼𝑖→𝛽) =2−3𝑆𝛼𝑖→𝛽+𝑆𝛼𝑖→𝛽34.   (29) For the purpose of simply distinguishing the favorable reactions from non-favorable ones, 𝑆𝛼1→𝛽 is approximated based on crystal structure similarity and epitaxial matching arguments follows: 𝑆𝛼1→𝛽 = 1 − 𝑞𝛼𝑖,𝛽𝑠𝑖𝑚 − 𝑞𝛼𝑖,𝛽𝑒𝑝𝑖,   (30) where 𝑞𝛼𝑖,𝛽𝑠𝑖𝑚   and 𝑞𝛼𝑖,𝛽𝑒𝑝𝑖  are normalized distance metrics (scaled to [0, 1]). Meanwhile, the overall reaction barrier (∆𝐺𝛼𝑖→𝛽∗∗ ) was formulated with the inclusion of the reaction activation barrier (∆𝐸𝛼𝑖→𝛽, which is contained in 𝐽0): ∆𝐺𝛼𝑖→𝛽∗∗ = ∆𝐺𝛼𝑖→𝛽∗ + ∆𝐸𝛼𝑖→𝛽.   (31) Here, the structure similarity parameter 𝑞𝛼𝑖,𝛽𝑠𝑖𝑚   was used to approximate ∆𝐸𝛼𝑖→𝛽  (i.e., ∆𝐸𝛼𝑖→𝛽 ∝𝑞𝛼𝑖,𝛽𝑠𝑖𝑚). Finally, the minimum heterogeneous nucleation barrier (∆𝐺∗∗) for a given candidate reaction to form 𝛽 was determined among 𝑚 reactants: ∆𝐺∗∗ = min{∆𝐺𝛼1→𝛽∗∗ , ∆𝐺𝛼2→𝛽∗∗ , … , ∆𝐺𝛼𝑚→𝛽∗∗ }. (32)     From above formulations, candidate solid-state reactions for the in-silico ip compounds can be searched based on: i) heterogeneous nucleation barriers (that can be estimated using high-throughput thermochemical data) and ii) the number of competing phases aside from the target compound to be synthesized. Using these two parameters, optimal reaction pathways can be identified by Pareto analysis, with the goal of finding reactions that both have the lowest barrier and the fewest number of competing phases. For this analysis, the reaction temperature and pressure are set to 1000 K and 1 atm, respectively. Included in the reactant list are binary powders (e.g., Li2O, LiCl, LiBr, Li2O2), carbon-containing precursors (e.g., Li2CO3, C), elemental solids (e.g., Li, I), and O2 gas. Reactants with known extreme toxicity (e.g., SeO2, CCl4, SeBr4) are excluded. Fig. 10 displays the 2D Pareto plots of possible synthesis pathways of representative t-ip and c-ip compounds. It includes 2 experimentally confirmed compounds (Pm3̅m Li3OCl and I4/mmm Li4OBr2) and yet-to-be synthesized metastable candidates (P4/nmm Li8O2Cl3Br, I4/mmm Li7O2FBr2, Fm3̅ m Li6OSeI2, and Cmcm Li12O3SeClBr3). Each data point in the subplots represents a possible reaction route (Table S3-S8). The dashed lines (in red) show the Pareto-frontier of non-dominated data points (i.e., optimal reaction routes for a given target compound). Table 3 shows the detailed reactions for the compounds shown in Fig. 10. For Pm3̅m Li3OCl, the route with the fewest number of competing phases is the reaction using Li2O2 and LiCl (i.e., ∆1 with 2 competing phases). Meanwhile, for a combustion-type reaction (i.e., with a C source), the nucleation barrier is predicted to decrease down to a quarter (i.e., 3.72  0.78 a.u.), but this route can increase the number of competing phases (e.g., ∆2 with 3 competing phases). Replacing LiCl with ClOx precursors can also further reduce the nucleation barrier, but also at the expense of further increased number of competing phases. For I4/mmm Li4OBr2, one reaction route (∆3) can completely avoid the formation of competing phases, but it has a very large nucleation barrier (44.2 x 103 a.u.). Using a catalytic-type reaction (e.g., using C, Li2O2, and LiBr as reactants), the barrier is largely reduced by 4 orders of magnitude (∆4, 44.2 x 103 1.22 a.u.) but it can facilitate the formation of 2 competing phases. Replacing the Br source from LiBr to Br2O3, the barrier further reduces by another factor but the formation of more competing phases is predicted. For P4/nmm Li8O2Cl3Br, Pareto-front reactions all involve a C source and CO2 release. A relatively moderate kinetic barrier magnitude is predicted (12.06 a.u.) for a reaction involving Li2CO3, LiBr, and LiCl; this reaction results to 2 competing phases (∆5). A reduction of the barrier by 2 orders of magnitude is predicted (12.06  0.30 a.u.) with using 4 reactants (Li2O, LiBr, LiC12, and LiClO4), though the number of competing phases is increased to 3 (∆6). For I4/mmm Li7O2FBr2, the reaction route which assisted by Li2CO3 (with LiBr and LiF) only leads to 1 competing phase (∆7) and the nucleation barrier is relatively moderate (16.31 a.u.). If Li2CO3 is replaced with C and Li2O2, the barrier is reduced by 2 orders of magnitude (16.31  0.84 a.u.), but the number of competing phases increases to 2 (∆8). Other reaction routes, such as those involving with reactants Li2O and Br2O3, slightly reduce the kinetic barrier but they are accompanied with the further increase in the number of competing phases.  For Fm3̅m Li6OSeI2, one route does not result to the appearance of competing phases (∆8) and with a relatively low nucleation barrier (0.11 a.u.). This reaction is involved with 4 reactants such as Li2CO3, Li2Se, LiC12 and LiIO3. Changing the reactant types and their combination only increases the number of competing phases. For Cmcm Li12O3SeClBr3, all Pareto-based reactions (e.g., ∆10 and ∆11) are involved with 5 reactants and with relatively low nucleation barrier (0.10 a.u.), though the number of competing phases is at least 3. These reaction outcomes are rather intuitive, given the complex composition of the target phase (i.e., 5-element system).   Figure 10. Analysis on solid-state synthesis reaction pathways (at 1000 K, 1 atm) of screened inverse-perovskite (ip) compounds (see Table 2): (a) Pm 3̅ m Li3OCl, (b) I4/mmm Li4OBr2, (c) P4/nmm Li8O2Cl3Br, (d) I4/mmm Li7O2FBr2, (e) Fm3̅m Li6OSeI2, and (f) Cmcm Li12O3SeClBr3. The analysis was performed using a heuristic approach based on classical nucleation theory and temperature-dependent Gibbs reaction energy (as described in ref. 80). Red dashed lines indicate Pareto frontiers for the 2 objective functions: nucleation barrier and number of competing phases. Competing phases are compounds which form using the same precursor sets as the target ip compound. Competing phases determined from Gibbs free energy calculations for ternary, quaternary, and quinary ip compounds were searched to include {binary}, {binary, ternary}, and {binary, ternary, quaternary} phases, respectively. Selected (Pareto-frontier) reaction routes are labeled (∆1, ∆2, etc.). Nucleation barrier in (b) is normalized by a factor of 1000 for visual clarity.  Table 3. Summary of possible synthesis reaction pathways for selected inverse perovskite (ip) compounds based on Pareto analysis with 2 objective functions: number (#) of competing phases and approximated nucleation barrier (method according to ref. 80). The list includes 2 experimentally confirmed compounds (Pm3̅m Li3OCl, I4/mmm Li4OBr2) and 4 yet-to-be synthesized ones (P4/nmm Li8O2Cl3Br, Cm Li8S2Cl3I, Fm3̅m Li6OSeI2, Cmcm Li12O3SeClBr3). Reaction routes (labeled as ∆i) correspond to the same labels in Fig. 8 insets. Target compound #competing phases Nucleation barrier / a.u. Pareto-front reaction routes Pm3̅m Li3OCl 2 3.72 Li2O2 + LiCl  Li3OCl + 0.5 O2 (∆1) 3 0.78 0.5 C + Li2O2 + LiCl  Li3OCl + 0.5 CO2 (∆2) 4 0.68 ClO2 + 1.5 Li2O  Li3OCl + 1.25 O2 5 0.32 1.0 ClO3+ 1.4286 Li2O + 0.1429 LiC12  Li3OCl + 1.7143 CO2 6 0.31 0.5 Cl2O7 + 1.4184 Li2O + 0.1633 LiC12  Li3OCl + 1.9592 CO2 I4/mmm Li4OBr2 0 44.2 x 103 Li2O + 2.0 LiBr  Li4OBr2 (∆3) 2 1.22 0.5 C + Li2O2 + 2.0 LiBr  Li4OBr2 + 0.5 CO2 (∆4) 4 0.81 Br2O3 + 2.0 Li2O  Li4OBr2 + 2.0 O2 5 0.34 Br2O3 + 1.9184 Li2O + 0.1633 LiC12  Li4OBr2 + 1.9592 CO2 P4/nmm Li8O2Cl3Br 2 12.06 2.0 Li2CO3 + LiBr + 3.0 LiCl  Li8O2Cl3Br + 2.0 CO2 (∆5) 3 0.30 1.7551 Li2O + LiBr + 0.4898 LiC12 + 3.0 LiClO4   Li8O2Cl3Br + 5.8776 CO2 (∆6) 8 0.30 0.5 Br2O3+ 2.2143 Li2O + 0.5714 LiC12 + 3.0 LiClO4  Li8O2Cl3Br + 6.8571 CO2 I4/mmm Li7O2FBr2  1 16.31 2.0 Li2CO3 + 2.0 LiBr + LiF  Li7O2FBr2 + 2.0 CO2 (∆7) 2 0.84 C + 2.0 Li2O2 + 2.0 LiBr + LiF  Li7O2FBr2 + CO2 (∆8) 4 0.84 Br2O3 + 3.0 Li2O + LiF  Li7O2FBr2 + 2.0 O2 5 0.33 Br2O3 + 2.9184 Li2O + 0.1633 LiC12 + LiF  Li7O2FBr2 + 1.9592 CO2 Fm3̅m Li6OSeI2 0 0.12 0.8776 Li2CO3 + Li2Se + 0.2449 LiC12 + 2.0 LiIO3  Li6OSeI2 + 3.8163 CO2 (∆8) 1 0.11 3.5 C + Li2O2 + Li2Se + 2.0 LiIO3  Li6OSeI2 + 3.5 CO2 (∆9) 5 0.11 I2O5 + 1.8776 Li2CO3 + Li2Se + 0.2449 LiC12  Li6OSeI2 + 4.8163 CO2 6 0.11 4.0 C + I2O5 + 2.0 Li2O2 + Li2Se  Li6OSeI2 + 4.0 CO2 7 0.10 Li2Se + 2.0 LiC12 + 2.0 LiI + 24.5 O2  Li6OSeI2 + 24.0 CO2 13 0.10 I2O5 + Li2Se + 4.0 LiC12 + 46.0 O2  Li6OSeI2 + 48.0 CO2 Cmcm Li12O3SeClBr3 3 0.10 1.5 Br2O3 + ClO3 + 4.71 Li2O2 + Li2Se + 0.58 LiC12  Li12O3SeClBr3 + 6.96 CO2 (∆10) 6 0.10 1.5 Br2O3 + Li2Se + 9.0 LiC12 + LiClO4 + 105.25 O2  Li12O3SeClBr3 + 108.0 CO2 (∆11) 8 0.10 1.5 Br2O3 + 0.5 Cl2O7 + Li2Se + 10.0 LiC12 + 117.5 O2  Li12O3SeClBr3 + 120.0 CO2  Discussion The present multi-objective optimization task has successfully shortlisted over 10,000 in-silico ip compounds down to a few candidates for SE use. Instead of simply relying on recorded compounds from established structure databases, the target chemical space was largely extended through a systematic approach of generating derivative structures based on anion-site alloying and symmetry group-subgroup relations. This approach allows for potentially novel but often missed-out compounds to be uncovered in the SE screening procedure.  Out of the 3 SE-property objective functions, 𝑓1: 𝐸𝑔 is the least-costly to obtain as it can be readily derived with no extra computing cost after DFT geometry optimization. Since there is a systematic offset of the PBE-level 𝐸𝑔values relative to more accurate but more expensive calculation approaches (e.g., GW method),81 a meaningful ranking of compounds is still achieved. Aside from t-ip structures, correlations are also successfully established for 𝐸𝑔, 𝐸𝑑, and 𝑡𝐺,𝑖𝑝 to include for c-ip structures as well, further confirming the general utility of 𝑡𝐺,𝑖𝑝 as a substitute parameter for 𝐸𝑔 and 𝐸𝑑. Based on 𝑓2: 𝐸ℎ, most of screened ip compounds are predicted be reactive to moisture. This reactivity is consistent with the hard-soft acid-base theory, considering Li is a hard acid (in Li-rich ip compounds) and H2O molecule is a hard base. It is noted that some of the compounds have 𝐸ℎ > 0, but their values are close to the endothermic-exothermic reaction crossover point (𝐸ℎ = 0). This means that in the presence of other exothermic side reactions, proton incorporation is still conceivable for the said compounds. One likely reaction is related to LiOH (a possible secondary or pre-existing phase) and CO2 (as a contaminant from air):  𝐿𝑖𝑂𝐻 +12𝐶𝑂2 →12𝐿𝑖2𝐶𝑂3 +12𝐻2𝑂.   (33) The calculated DFT reaction energy for equation (32) is -60.5 kJ/mol. With this additional heat source, all the screened compounds are predicted to likely react with H2O under air-exposed condition. Experimental observations have demonstrated on the ease of protonation of some known ip compounds.80-82  Based on 𝑓3: Λ, Cmcm Li12O3SeClBr can be considered as the best compound (0.78 S/cm). It has a cubic Li3XZ base structure with a continuous bcc-type anion X-Z8 and Z-X8 sublattices in 3 cell directions. This anion framework has been suggested previously to facilitate for a fast 3D diffusion pathway for Li+ ions.85 However, its composition complexity indicates a difficulty in actually synthesizing it in high phase purity because of competing phases that can likely form with it. Meanwhile, for t-ip compounds, the linkage of X-Z8 and Z-X8 bcc-type units are interrupted in the long-axis direction. As a result, fast ion conduction is constrained in 2D within each of the X-Z-Li slab layer units. This suggests a potential morphology-dependent conductivity in this structure type, such as if the exposed particle facet can facilitate for a fast inter-grain transport or not. Also, it is cautioned that although the bulk conductivities of the screened compounds are promising, they can have significantly different total conductivities, depending on the type of bulk defects present, the extent of proton incorporation, and the activation energy at the grain boundary region.29, 82-84,86-89  Several candidate synthesis routes are predicted for the screened ip compounds in the present work. For example, Fm3̅m Li6OSeI2 can be synthesized under typical solid state reaction conditions, without forming extra phases and with minimal nucleation barrier. Thus, this material may serve as a promising candidate for actual synthesis. Meanwhile, other candidate compounds are shown to have reactions that involve phase competition. This may reflect the general difficulty of stoichiometry control at elevated temperatures for ip compounds, since the typical reactants are highly susceptible to undergo sublimation (e.g., Li2Se, Li2S, LiCl, and LiBr).90 Some synthesis routes are predicted to have no secondary phase formation, such as in the case of I4/mmm Li4OBr2, but it requires a high nucleation barrier for the reaction to proceed. An attempt to synthesize Li4OBr2 by high-pressure synthesis has been reported, but it resulted in a slightly different structure.91    Conclusions Using high-throughput DFT and machine learning techniques, a large-scale material screening and evaluation was performed on a ten-thousand-scale ip-compound space, with a goal to find novel SEs that can satisfy multiple property criteria. Some of the key findings are as follows: i) Out of 10,173 in-silico compounds, 1,413 compounds are found to have a high likelihood of synthesizability based on DFT-calculated thermodynamic decomposition energy ( 𝐸𝑑 ) and experimental crystal-likeness score (𝐿𝑠). ii) The modified formulation of the Goldschmidt tolerance factor (𝑡𝐺,𝑖𝑝) can serve as good descriptor for thermodynamic phase stability and 𝐸𝑔. iii) There are 24 promising ip SEs that are determined by the present material screening methodology, examples include Cm Li8O2Cl3Br (𝐸𝑑 = 0, 𝐿𝑠 > 0.5, 𝐸𝑔 = 4.74 eV, 𝐸ℎ = -33.22 kJ/mol, Λ = 9.0 x 10-4 S/cm), Amm2 Li8OSCl4 (𝐸𝑑 = 0.070 eV/atom, 𝐿𝑠 > 0.5, 𝐸𝑔 = 4.14 eV, 𝐸ℎ = -40.70 kJ/mol, Λ = 9.2 x 10-2 S/cm), and Cmcm Li12O3SeClBr3 (𝐸𝑑 = 0.097 eV/atom, 𝐿𝑠 > 0.5, 𝐸𝑔 = 3.36 eV, 𝐸ℎ = -86.88 kJ/mol, Λ = 7.8 x 10-1 S/cm). iii) The bulk Li+ ion conductivity of t-ip and c-ip compounds is strongly dependent on the type of stabilized defects, with the defects related to Li interstitials resulting to higher conductivity and lower diffusion barrier, as compared to defects that promote Li vacancies. iv) The synthesis of t-ip and c-ip compounds are predicted to be accompanied with the formation of competing phases with low nucleation barriers.  Our comprehensive work provides useful insights into the design of ip-type SEs.   Acknowledgments Authors are thankful for the support in part by JST through ALCA-SPRING grant number JPMJAL1301, and COI-NEXT grant number JPMJPF2016; by JSPS KAKENHI grant numbers JP19H05815 and JP21K14729; and by MEXT as Materials Processing Science project (“Materealize”) grant number JPMXP0219207397, and “Program for Promoting Research on the Supercomputer Fugaku” grant number JPMXP1020200301. The calculations were performed on the supercomputers at NIMS (Numerical Materials Simulator) and the supercomputer Fugaku at the RIKEN through the HPCI System Research Project (project ID: hp210105). Crystal structures were visualized using the VESTA software.92 The python library Matplotlib was used for making the plots.93 .  Supporting Information Table on DFT-calculated electronic band gap energy values of known inorganic solid electrolytes; details on MSD slope fitting procedure; figure on DFT-geometry-optimized structure of I4/mmm Li4OBr2 antiperovskite compound with an added Li interstitial in one of the OLi6 octahedral basal plane; table on summary list on region-1 compounds from Figure 9 of the main text; table on candidate solid state reactions based on a data-driven approach in Reference 80 of the main text for Pm3̅m Li3OCl, I4/mmm Li4OBr2, P4/nmm Li8O2Cl3Br, I4/mmm Li7O2FBr2, Fm3̅ m Li6OSeI2, and Cmcm Li12O3SeClBr3 compounds.   References 1. Janek, J.; Zeier, W. G. A solid future for battery development. Nat. Energy 2016, 1, 16141. 2. Takada, K. Progress in solid electrolytes toward realizing solid-state lithium batteries. J. Power Sources 2018, 394, 74-85. 3. Manthiram, A.; Yu, X. W.; Wang, S. F. Lithium battery chemistries enabled by solid-state electrolytes. Nat. Rev. Mater. 2017, 2, 16103. 4. 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