Ovarian clear cell carcinomas (OCCs) arise from endometriotic cysts that many women develop. Biomarkers for early OCC detection need to be identified. Extracellular vesicles have attracted attention as biomarker carriers. This study aims to identify cancer-specific miRNAs as novel OCC biomarkers using tissue-exudative extracellular vesicles (Te-EVs). Te-EVs were collected from four patients with OCC on one side and a normal ovary on the other side. Microarray analysis was performed to identify cancer-specific miRNAs in Te-EVs. Serum samples obtained before and after surgery from patients with OCC and atypical endometrial hyperplasia (AEH) (controls) were compared using real-time PCR to examine changes in the detected EV miRNA levels. Thirty-seven miRNAs were >2-fold upregulated on the OCC side compared with the normal ovarian side. We selected 17 miRNAs and created specific primers for 12 of these miRNAs. The levels of six EV miRNAs were significantly decreased in postoperative OCC serum compared to those in preoperative OCC serum. In contrast, no significant change was observed between the pre and postoperative values in the control group. We identified OCC tissue-specific miRNAs in the EVs secreted by OCC tissues. These EV miRNAs have potential for use as biomarkers for the early diagnosis and detection of OCC. According to GLOBOCAN 2020, 314,000 women are diagnosed with, and 207,000 women die from ovarian cancer every year. The incidence and mortality per 100,000 women are 6.6 and 4.2, respectively . OCCs arWestern blot analysis and electron microscopy were performed to confirm that EVs were obtained in our procedures. Electron microscopy was performed to Te-EVs obtained from the cancer specimens in case 4. Western blot analysis was performed to all Te-EVs obtained from cancer and normal specimens in each case . HoweverHomo sapiens. In each case, the levels of several miRNAs were two-fold higher in the EVs released from cancer cells compared with those in normal ovaries: An amount of 255 miRNAs in case 1  in EVs secreted from OCC. These EV-derived miRNAs may serve as potential biomarkers for the early diagnosis and detection of ovarian cancer.For the past two decades, miRNAs have been investigated for their roles in many cancers. They modulate gene expression and play significant roles in biological pathways, influencing proliferation, apoptosis, cell cycle, invasion, and migration. Interest is focused on their use as biomarkers for the early detection, diagnosis, and prognosis of cancers. They are released by cancer cells and detectable in all human biofluids, including plasma, serum, and urine. Many cancer-specific circulating miRNAs have been reported and are expected to serve as viable biomarkers for the early detection of various cancers ,19,20,21Several authors have reported miRNAs in EVs extracted from the serum or plasma of patients with ovarian cancer ,23,24. TSeveral studies have reported the presence of cancer-specific miRNAs in OCC. In these studies, miRNAs were extracted directly from cell lines and tissues, or from EVs secreted by cell lines ,26,27. WHowever, the functions of these miRNAs are not yet fully understood. Among the 37 miRNAs detected in this study, some are considered to belong to the miR−200 family, including miR−200a, miR−200b, miR−200c, and miR−141. The miR−200 family comprises five members: miR−200a, miR−200b, miR−200c, miR−141, and miR−429 . The miRSome authors have reported that miR−30a−5p functions as an anti-oncogene in several malignancies ,37,38. AThis study had several limitations. First, the control blood samples were obtained from AEH patients; AEH may have an influence on miRNAs. Second, the blood samples and OCC tissue specimens analyzed were obtained from two different groups of patients due to a low RNA quality in blood samples obtained from the patients who provided tissue specimens. Third, the sample size was too small to prove that miRNA in EVs is a useful biomarker. It is not clear whether other differences between patients  would not result in changes in these miRNAs. Therefore, an analysis of a larger sample of OCC patients and age matched individuals without tumors is needed.Specimens were obtained from four patients with OCC who underwent surgery between 2018 and 2019 at Osaka Medical and Pharmaceutical University. All four patients had an OCC on one side and a normal ovary on the opposite side. The surgically removed specimens were washed thoroughly with 500 mL of saline solution. Then, tissue fragments  were excised including the OCC and the normal ovary. The obtained fragments were individually incubated. 2. The tissue-immersed medium was filtered using a 0.22 μm filter  to remove cell debris and centrifuged at 2000× g for 30 min, 4 °C to remove cells and dead cells, and centrifuged 16,000× g for 30 min, 4 °C to remove cell debris. Blood samples were centrifuged at 2500× g for 10 min with 7 mL of blood within 1 h of collection. This centrifugation was performed to remove as many impurities as possible, while using a centrifugal speed and centrifugal time that does not destroy the blood cells. The supernatant was centrifuged at 20,000× g for 5 min. This centrifugation removed the leukocytes. Processes were completed within 2 h of blood collection to prevent elution of blood components into plasma and serum and stored at −80 °C until the next process. To pellet EVs from the collected supernatant, ultracentrifugation was performed using an Optima XE-100 , SW41 T1 (Beckman Coulter), and Ultra-Clear tube (Beckman Coulter) at 100,000× g for 70 min, 4 °C. EVs were collected in phosphate-buffered saline to make a total of 150 μL, placed in a 1.5 mL collection tube, and stored at −80 °C. The protein concentration content of EVs was measured using the PierceTM BCA protein assay kit . Western blotting and electron microscopy confirmed the EV extraction.EVs samples were dissolved in radioimmunoprecipitation assay buffer . An amount of 5.0 µg of EV samples was placed in each lane. The lysates were separated using 5–20% sodium dodecyl sulfate–polyacrylamide gel electrophoresis and transferred to polyvinylidene difluoride membranes. The membranes were blocked in 10% bovine serum albumin in 1X Tris-buffered saline and incubated overnight with specific primary antibodies against β-actin , CD9 , CD63 , and CD81  at 4 °C. Then, the membranes were incubated with horse radish peroxidase-conjugated secondary antibodies for 1 h. Bands were visualized using an enhanced chemiluminescence agent .l-lysine solution-coated beads . After drying, the beads were washed and fixed in 1.25% glutaraldehyde in 0.1 M phosphate buffer . Then, they were washed with PB and fixed in 1% osmium tetroxide for 40 min. After washing with PB, the beads were gradually dehydrated using a graded series of ethanol washes. The platinum–palladium was evaporated and vapor deposition was performed. Next, scanning electron microscopy analysis  was performed.EVs (1 μg) were incubated with poly-® miRNA 4.0  using the protocol provided by Affymetrix. The GeneChip miRNA 4.0 array contains over 30,000 probes from Sanger miRBase v.20. The array was analyzed using the Transcriptome Viewer software . The miRNA signal values were standardized by global normalization . Relative expression levels for each miRNA were validated by one-way analysis of variance or t-test (p < 0.05). MicroRNAs whose expression levels showed at least a 2-fold difference (|log2-fold change| > 1 and p < 0.05) between the test and control samples were analyzed.Initially, two total RNA extractions from EVs were performed using the miRNeasy Mini Kit according to the manufacturer’s protocol . The concentration and purity of RNA were assessed using spectrophotometry , and RNA integrity was checked using a 2100 Bioanalyzer  before microarray hybridization. The samples were hybridized, washed, and stained using a GeneChip Hybridization Oven 640, GeneChip Fluidics Station 450 , and Affymetrix GeneChipTotal miRNAs were extracted from the suspended EVs using the miRVana™ miRNA Isolation Kit according to the manufacturer’s recommendations .−ΔΔCq method.qRT-PCR was performed using a StepOnePlus Real-Time PCR System . Pre and postoperative OCC and AEH blood samples were used. The RNA concentration was adjusted to 2.0 ng/μL. The miRNAs were reverse transcribed using a microRNA reverse transcription kit (Thermo Fisher Scientific Inc.) and qPCR was performed according to the manufacturer’s instructions (Thermo Fisher Scientific Inc.). The following primers were used for PCR: miR−21−5p, miR−30a−5p, miR−30c−5p, miR−141−3p, miR−182−5p, miR−183−5p, miR−200a−5p, miR−200a−3p, miR−200b−5p, miR−200b−3p, miR−200c−3p, miR−210−3p, miR−652−3p, and RNU48 . RNU48 was used as the normalization control. qPCR of miRNA was performed three times in triplicate wells. The relative expression of miRNAs was calculated using the 2p < 0.05.Statistical analyses were performed using JMP Pro version 14.2.0 . The Wilcoxon test was used to compare continuous variables. Statistical significance was set at We identified the following OCC tissue-specific miRNAs in EVs secreted by tissues: miR−200a−3p, miR−200b−3p, miR−200c−3p, miR−200a−5p, miR−200b−5p, and miR−30a−5p. These miRNAs may serve as potential OCC-specific biomarkers.


C6, C8, C9, C10 dispersion coefficients using the MCLF method. The second algorithm computes the AIM polarizability tensors and C6 coefficients using the TS-SCS method. Linear-scaling computational cost is achieved using a dipole interaction cutoff length function combined with iterative methods that avoid large dense matrix multiplies and large matrix inversions. For MCLF, Richardson extrapolation of the screening increments is used. For TS-SCS, a failproof conjugate residual (FCR) algorithm is introduced that solves any linear equation system having Hermitian coefficients matrix. These algorithms have mathematically provable stable convergence that resists round-off errors. We parallelized these methods to provide rapid computation on multi-core computers. Excellent parallelization efficiencies were obtained, and adding parallel processors does not significantly increase memory requirements. This enables system-specific polarizabilities and dispersion coefficients to be readily computed for materials containing millions of atoms in the unit cell. The largest example studied herein is an ice crystal containing >2 million atoms in the unit cell. For this material, the FCR algorithm solved a linear equation system containing >6 million rows, 7.57 billion interacting atom pairs, 45.4 billion stored non-negligible matrix components used in each large matrix-vector multiplication, and ∼19 million unknowns per frequency point .We present two algorithms to compute system-specific polarizabilities and dispersion coefficients such that required memory and computational time scale linearly with increasing number of atoms in the unit cell for large systems. The first algorithm computes the atom-in-material (AIM) static polarizability tensors, force-field polarizabilities, and  Linear-scaling algorithms were developed to compute atom-in-material polarizabilities and dispersion coefficients using MCLF and TS-SCS methods. For sufficiently large systems, both the required memory and computational time scale linearly with increasing number of atoms in the unit cell. Our methods can easily be applied to materials containing millions of atoms in the unit cell. This is orders of magnitude larger than unit cells for materials previously studied with MCLF and TS-SCS methods. For small unit cells, our methods are still faster and require less memory than direct matrix inversion with negligible difference in computational precision.2 TS-SCS can be combined with multibody dispersion (MBD), a damping function, and density functional theory (DFT) to give a DFT + dispersion method.2 Ambrosetti et al. introduced range-separated dipole interaction tensors to avoid (or minimize) double-counting dispersion interactions in the combined MBD@rsSCS framework.3 The TS-SCS method requires atom-in-material 〈r3〉 radial moments as inputs.2,4 Several TS-SCS variants used different partitioning methods to compute the 〈r3〉 moments: TS-SCS/Hirshfeld,2,4,5 TS-SCS/iterative Hirshfeld,6,7 and TS-SCS/DDEC6.1 Normally, the TS method assumes a constant polarizability to 〈r3〉 moment ratio for a specific chemical element irrespective of its charge states in a material.2,4 Recently, Gould et al. introduced a Fractionally Ionic (FI) method that adjusts the polarizability to 〈r3〉 moment ratio using partial atomic charges.8 This requires a library of reference ion polarizabilities and C6 coefficients for free atoms in different charge states.8,9 However, these reference polarizabilities and C6 coefficients are hard to compute for anions such as O2− that contain unbound electrons10 in their free state.The TS-SCS method was introduced in 2012.CA6, dispersion coefficient is computed using the Casimir–Polder integral  is the dipole–dipole interaction tensor, s and t are spatial indices , δst is the Kronecker delta, αunscreenedA(u) is the isotropic AIM unscreened polarizability as a function of u, and  is the self-consistent screened AIM polarizability tensor as a function of u.2 A dipole interaction cutoff length function turns off dipole interactions between atoms separated by a distance larger than this cutoff length . This makes the coefficients matrix sparse when the material's unit cell is much larger than this cutoff length. Because the dipole interaction tensor is symmetric in both atomic indices and spatial indices, developing a computationally efficient TS-SCS algorithm is functionally equivalent to developing a computationally efficient algorithm to solve a set of real-valued symmetric sparse linear equations.In the TS-SCS method, the integral , which rMy = W having a sparse real-valued symmetric coefficient matrix M. A naïve approach would invert the matrix M to compute y = (inverse(M))W. However, this is computationally infeasible for matrices containing a large number of rows (Nrows). First, direct matrix inversion algorithms  and the more advanced Strassen algorithm) have computational costs scaling between (Nrows)2 and (Nrows)3.11 Second, inverse(M) may be dense (or at least considerably less sparse than M) leading to high matrix multiplication costs. The cost of multiplying two dense square matrices is between (Nrows)2 and (Nrows)3,11 and multiplying a dense square matrix times a dense vector would cost (Nrows)2 if using conventional multiplication. Therefore, the primary goal is to solve this linear equation system without any large dense matrix multiplies or large matrix inversions. Here, a new conjugate residual algorithm is introduced to achieve this with a computational cost proportional to Nrows. This new conjugate residual algorithm resists round-off errors.At its heart, therefore, this is a fundamental linear algebra problem. Specifically, to solve a linear equation system 1 the MCLF method includes numerous important innovations. It uses a conduction limit upper bound to ensure assigned AIM polarizabilities do not exceed those of a perfect conductor. It uses m-scaling to smoothly transition between the scaling behaviors of surface and buried atoms. It uses new scaling relationships to describe changes in the polarizability-to-〈r3〉 moment ratio as a function of the atomic charge state without requiring quantum mechanical (QM) computed reference polarizabilities and C6 values for charged atoms. This is an important advantage, because some isolated charged atoms are unstable . To compute several different kinds of AIM polarizabilities, MCLF separates directional from non-directional dipole interaction tensor contributions. This includes: (a) non-directionally screened polarizabilities to be used as force-field input parameters, (b) fluctuating polarizabilities to compute the C6 dispersion coefficients, and (c) static polarizabilities containing long-range dipole alignment due to a constant externally applied electric field. A new polarizability partition and iterative polarizability screening ensure assigned AIM polarizabilities are non-negative. A proof that MCLF αforce-field, αnon-dir(u), αlow_freq, and αscreened(u) are ≥0 is provided in the ESI.C6 coefficients. Quantum Drude oscillator (QDO) parameters yield higher-order AIM dispersion coefficients  and associated mixing rules.As described in the first article of this series,1 While TS-SCS solves a linear equation system  Because it has mathematically provable stable convergence that resists round-off errors, this conjugate residual algorithm should find widespread applications to solve a plethora of scientific computing problems involving large sparse linear equation systems. Potential applications for such a method are vast.The remainder of this article is organized as follows. Section 2 presents the linear-scaling MCLF algorithm. The failsafe conjugate residual and linear-scaling TS-SCS algorithms are introduced in Section 3. Section 4 contains the performance results: required computational times, required memory, and parallelization efficiencies. All computational timing and memory results reported in this paper are for the MCLF or TS-SCS analysis; they do not include the prior quantum chemistry calculation (if any) and prior atomic population analysis used to provide the input data. Section 5 concludes. The ESI12–18 All DDEC6 calculations were performed using the Chargemol program.12Finally, we comment on the choice of atomic population analysis method (APAM) used to provide input data to the MCLF and TS-SCS methods. The memory requirements for MCLF and TS-SCS analysis do not depend on which APAM provides the input data. The required computational time for MCLF analysis does not depend on which APAM provides the input data. The required computational time for TS-SCS analysis using the FCR algorithm only depends on the number of FCR iterations, which could potentially be weakly affected by the choice of APAM used to provide the input data. In this article, we used DDEC6  atomic p12 The electron and spin densities assigned to an atom in the material are zero outside this cutoff radius. This yields linear-scaling computational time and memory, because the number of integration points per atom does not increase as Natoms increases.12Natoms is the number of atoms in the unit cell.The overall sequence is: QM calculation → APAM → MCLF or TS-SCS analysis. DDEC6 and many other stockholder partitioning methods can be made strictly linear-scaling by using a cutoff radius around each atom in a material.62–69 Orbital-free DFT calculation  was performed for a million atom system.66 Further improvements are needed to enable DFT calculations on millions and billions of atoms. Even without DFT calculations of such large systems, the MCLF (or TS-SCS) method could still find applications to materials containing millions of atoms. For example, by using DFT calculations and DDEC6 analysis on smaller clusters to parameterize information for atom types, followed by MCLF (or TS-SCS) analysis for the full material. This strategy would make sense, because DDEC6 analysis is more localized  compared to MCLF (or TS-SCS) analysis .1,13QM computation of the material's electron and spin densities is the rate-limiting step in this overall sequence. Significant progress has been made developing linear-scaling DFT codes that have applications to studying unit cells containing ∼10 000 atoms.2.2.1Natoms increases when Natoms is sufficiently large. First, no allocatable arrays having multiple dimensions of size proportional to Natoms ) are allocated for Natoms greater than a threshold. Second, no nested DO loops having two indices of ranges proportional to Natoms  iterations) are executed for Natoms greater than a threshold. This is achieved by making sure the program is written such that all sets of nested DO loops and allocatable arrays  have at most one index whose range is proportional to Natoms.Strict linear-scaling means that each and every part of the MCLF program scales no worse than linear in computational time and memory as Natoms the total number of symmetry unique interacting atom pairs is proportional to Natoms. This is true even if the material is periodic and extends forever. This is conceptually equivalent to a sparse matrix algebra.This is physically enabled by using a dipole interaction cutoff length such that any two atoms farther apart have no direct interactions. As explained in Section 2.2 below, the program constructs lists of interacting atom pairs and uses these in subsequent calculations. Because each atom in the material only directly interacts with a limited number of other atoms, for sufficiently large Order(Nrows). As explained in Section 2.5, a linear-scaling algorithm was developed that avoids large matrix inversion. This algorithm is linear-scaling, because it requires a fixed number of large matrix-vector multiplications in which the matrix is sparse. This matrix is the dipole interaction tensor, whose non-zero components correspond to the interacting atom pairs. Hence, each large matrix-vector multiplication corresponds to looping over the lists of symmetry unique interacting atom pairs.Other mathematical innovations are also employed. First, a special lookup table method (see Section 2.4) is used to compute the total dispersion coefficientAs explained in Section 2.6, all of these linear-scaling innovations were parallelized to enable fast multi-core computing. Moreover, all of these innovations were programmed in a general way that handles 0, 1, 2, or 3 periodic boundary conditions (PBC). For input files of materials having periodic unit cells, these MCLF and TS-SCS programs include the option to use or ignore the input file PBC. For example, a molecule can be simulated in a planewave code (such as VASP) by placing it near the center of a large cube with PBC. In this case, to yield AIM polarizabilities and dispersion coefficients for the isolated molecule, the MCLF or TS-SCS program should be set to ignore these PBC; this turns off all interactions between periodic images.2.2e.g., A, B) to represent an atom in the reference unit cell. A small letter represents a translated atomic image that is not necessarily located in the reference unit cell. For example, b =  represents the image of atom B that is translated L1 times the first lattice vector v⃑1, L2 times the second lattice vector v⃑2, and L3 times the third lattice vector v⃑3.R⃑B is the nuclear position of the parent atom in the reference unit cell.We use a capital letter , (ii) L1 > 0, (iii) L1 = 0 and L2 > 0, or (iv) L1 = L2 = 0 and L3 > 0.18During MCLF analysis, two separate lists of interacting atom pairs are prepared after the unscreened calculation and before the screened calculation. For convenience, we refer to these as the ‘small’ and ‘large’ lists, where ‘small’ and ‘large’ refer to the interaction cutoff distance. Without loss of generality, the first atom in each pair can be considered to reside within the reference unit cell. The second atom is located somewhere within the dipole interaction cutoff length of the first atom. Only translation symmetry unique atom pairs need to be included in the loops over atom pairs. As explained in the earlier bond order article, an atom image pair is translation symmetry unique if and only if at least one of the following criteria is satisfied: (i) the index numbers of the atoms are different  is the static attenuation length. This criterion ensures erfc(υAb) ≈ exp(−(υAb2)) ≈ 0 for any atom pair not included in the list. The value υcutoff = 54/3 was chosen such that even if the  screened polarizability is larger than the unscreened polarizability by up to a factor of five, then erfc(υscreenedAb) ≤ erfc(5) = 1.5 × 10−12 and exp(−(υscreenedAb)2) ≤ exp(−25) = 1.4 × 10−11. The following information was saved in the small list array for each included atom pair: the index numbers of the first and second atoms, the translation integers for the second atom, the atomic number of each atom, dAb, the value of the smooth cutoff function, the value of the multibody screening function times the smooth cutoff function, and the following six tensor components: η1,1 = 3(Δx)2/dAb2 − 1, η1,2 = η2,1 = 3ΔxΔy/dAb2, η1,3 = η3,1 = 3ΔxΔz/dAb2, η22 = 3(Δy)2/dAb2 − 1, η2,3 = η3,2 = 3ΔyΔz/dAb2, η3,3 = 3(Δz)2/dAb2 − 1. Here, Δx, Δy, and Δz are the Cartesian components of R⃑A − R⃑b.The small lists all atom pairs having ‘overlapping’ Gaussian dipole model densities as defined by the cutoff criterionA, B) in the reference unit cell for which atom A (first atom) is located a distance less than dipole interaction cutoff length to at least one image of atom B (second atom). The self-pair  is included if and only if one of the non-trivially translated images of atom A is located a distance less than dipole interaction cutoff length to atom A in the reference unit cell. The number of atom pairs in the large list is always ≤Natoms(Natoms + 1)/2. For example, a NaCl crystal containing two atoms in the reference unit cell would contain three pairs in the large list: Na–Na, Na–Cl, and Cl–Cl. For each pair  in the large list, a loop is performed over all atom b images located within dipole interaction cutoff length of atom A, and the following sums are accumulated and stored:fcutoff and fMBS are the smooth cutoff function and multi-body screening function, respectively, described in the prior article.1 Since  and  are symmetric with respect to exchange of the spatial indices, only six components need to be computed and stored for each.The large lists all atom pairs , but each non-periodic basis vector is chosen to be perpendicular to the other basis vectors.1. Divide the unit cell parallelepiped into spatial regions: This unit cell parallelepiped is divided into a whole number of spatial regions along each basis vector. A periodic direction produces an infinite number of periodic images of each region, while a non-periodic direction has only the reference image. Atoms in the reference unit cell are classified by region, and a sorted list is prepared such that atoms of the same region are adjacent in this sorted list. Because each spatial region is defined such that its volume is less than that of a sphere of dipole interaction cutoff length radius, the number of atoms in each region is always below a threshold. The code ignores regions that do not contain any atoms. Because empty regions are skipped, having a few atoms in the center of an enormous unit cell would execute quickly.2. Construct arrays listing interacting region pair images: Two spatial region images interact if the minimum distance between inter-region points is ≤the dipole interaction cutoff length. Because the spatial regions and their images are coordinate system indexed, a list of interacting region pair images is constructed without having to construct a double summation over all region pairs. Thus, even for an extremely large unit cell  divided into many regions , the list of interacting region pair images is constructed in time and memory scaling linearly with increasing unit cell size. Different regions can interact with different periodic images. For example, in an extremely large unit cell, a region near the center would interact only with nearby regions in the reference unit cell, while a region not too far from the left edge would interact with some regions in the reference unit cell and some other region images in the left-translated unit cell. Thus, a first array is constructed listing pairs of regions having any interacting images, and a second array is prepared that lists which specific images of each particular region pair interact. Region pairs that do not interact are not included in these two arrays.3. Construct two lists of interacting atom pairs: Because interacting atom pairs must be contained in interacting region pair images, the code identifies the interacting atom pairs by executing an outer loop over the interacting region pair images and inner loops over the atoms in these regions . Using the list of atoms sorted by region makes this process cache access friendly. For each such atom pair, tests are performed to determine if it meets the small and large list inclusion criteria. If so, its information is added to the small and/or large lists. Because the number of interacting region pair images scales linearly with large Natoms and the number of atoms in each region is below a threshold, these small and large lists are constructed in linear-scaling computational time and memory for large Natoms.4. Having separate small and large lists provides the following computational efficiencies during the subsequent MCLF polarizability screening. First, non-directional screening only needs to be performed over the ‘overlapping’ atoms contained in the small list. Second, the computationally expensive erfc function only needs to be evaluated for atom pairs in the small list. This provides computational savings during directional screening that first loops over all pairs in the small list and then over all pairs in the large list. Third, the large list contains pre-computed sums over all periodic images of interacting atom pairs. For non-overlapping atoms, this avoids re-computing any sums over periodic images at each screening increment and each frequency point.These two lists of interacting atom pairs are used as follows. For each screening increment of each frequency point, non-directional screening loops over all atom pairs in the small list to compile the necessary dipole–dipole interaction terms. For each screening increment of each frequency point, directional screening first loops over all atom pairs in the small list and then over all atom pairs in the large list to compile the associated dipole–dipole interaction terms. More details are provided in Sections 2.5 and 2.6 below.αstatic was the most sensitive to the dipole interaction cutoff length, and αforce-field was the least sensitive. For 2-dimensional sheets such as graphene, doubling the dipole interaction cutoff length increases the computational cost by approximately four-fold. For dense materials with large unit cells, doubling the dipole interaction cutoff length increases the computational cost by approximately eight-fold. We selected dipole interaction cutoff length = 50 bohr as a good compromise between computational cost and precision. This value was used for all other results in this article.19–21 using the PBE22 functional, a 400 eV planewave cutoff, and the projector augmented wave  method. The k-point mesh and grid spacing followed previous recommendations.12 This yielded a nearest neighbor C–C distance of 1.42 Å. Our MCLF and TS-SCS calculations of graphene were performed using 2-D not 3-D periodic boundary conditions .The graphene primitive unit cell was optimized in VASP2.3CAB6, dispersion coefficient between two subsystems A and B to the product of their fluctuating polarizabilities at imfreq  integrated over all imfreqs:25The Casimir–Polder integral relates the A = B givesu = 0 corresponds to infinite imfreq. Near infinite imfreq, the polarizability becomes inversely proportional to ω2.26 This is incorporated into the TS-SCS2,4 and MCLF1 methods using a Padé approximation27 for the dynamic unscreened polarizability:u = 0 point contributes nothing to the integral.As shown in i.e., Romberg integration).28–30 Dividing the  interval into 2G segments and performing Romberg integration of order  yields an integration error of the order 2G(2G+2)−.28 Normally, Romberg integration of 2G segments corresponds to 2G + 1 integration points. Since the u = 0 point contributes nothing to the integral, this leaves only Nimfreqs = 2G nontrivial integration points. The ESIcRombergG,u of these 2G integration points for G = 1 to 5. The integral is computed using the following sum:We numerically integrated using Richardson extrapolation  in the C6 coefficient between 16 and 32 integration points. All other results were virtually identical for 16 and 32 integration points. Moreover, all results for 8 integration points differed by <1% from the higher integration points. This shows the results are highly converged for 16 integration points, which is the value we chose.2.4CAB6, dispersion coefficients could in principle be computed from the Casimir–Polder integral using αA(u) and αB(u), this would involve many integrations for unit cells containing thousands or more atoms. Therefore, we used the following mixing formula which is consistent with both Padé approximation27 and QDO31 models:Although the mixed αlow_freq, because these are the polarizabilities associated with the dispersion interaction. Of note, the TS and TS-SCS methods use similar mixing formula, except the polarizabilities appearing in the mixing formula are αTS and αTS-SCS,2,4 because those methods do not yield αlow_freq.In our method, the polarizabilities appearing in C6 dispersion coefficient per unit cell requires a total of (Natoms)(Natoms − 1)/2 individual CAB6, evaluations. This itemized approach to computing Ctotal6 would be computationally expensive for unit cells containing a billion or more atoms. Atomistic  simulations employing an interaction cutoff distance require only a small subset of {CAB6,} for large unit cells, because atoms outside the cutoff distance do not interact. Therefore, it would be extremely wasteful to compute and print all the individual CAB6, values for large unit cells. It is preferred to compute and print {αlow_freqA} and {CA6,} that can be inserted into CAB6, whenever needed. For a billion atoms, storing {αlow_freqA} and {CA6,} requires only 16 gigabytes (GB), while storing {CAB6,} would require 4 exabytes. Therefore, it is unnecessary for the program to compute, store, or print the full {CAB6,}.When using the mixing rule of Ctotal6 is now introduced to address this issue. DefiningA linear-scaling algorithm to compute −2 in j < 1 or > Num_lookup even in the presence of fixed precision floating point round-off. An array alpha_table having Num_lookup components is initialized to zero. A loop is then performed over all atoms in the unit cell. Each atom A's polarizability αA contributes to two adjacent array values:Linear-scaling computation is achieved by constructing a lookup table that uniformly spaces ln:Ctotal6 is computed by the following double summation (that involves an outer loop over lookup table entries and an inner loop over atoms in the unit cell):i = 0.After this loop completes, Natoms2), (Natoms × Num_lookup)) )  + Natoms . This function parallelizes Cunscreened6 or Cscreened6.The ESICtotal6 is always less than interval2/16, which is ≤∼4 × Num_lookup−2. Thus, with Num_lookup = 105, the URE is usually ≤∼4 × 10−10. This method has astonishingly high precision. As proved in Section S3 of the ESI,Ctotal6  − 12 digits  = ∼3 remaining digits of precision for individual CAB6, addition. 128-Bit reals have approximately 34 significant base-ten digits, which is adequate. Although using 128-bit reals for alpha_table and the temporary variable are not required, this helps preserve quadruple precision during Ctotal6 computation.All real variables were 64-bit, except al6 both , alpha_tal6 both  were 128Ctotal6 function  method described in Section 3 below. The same Num_lookup = 105 and threshold_wp_lookup = 2 × Num_lookup are used there. The same function described above is used, but called with the alpha_values and wp_values appropriate for computing CTS6 or CTS-SCS6 for the TS-SCS(FCR) method.This same wp lookup table method for computing 2.532,33 GEPP, QR factorization, Cholesky and LDL decomposition (for positive definite Hermitian matrices), and other common matrix inversion algorithms have computational costs scaling proportional to the number of rows (Nrows) cubed.11,33 The more complicated Strassen algorithm has scaling proportional to Nrowslog(7)/log(2) = Nrows2.807….11Direct inversion of large matrices is extremely computationally expensive. Gaussian elimination with partial pivoting (GEPP) is a common matrix inversion algorithm that exhibits numerical instability for some matrices but is usually stable in practice.34 Consider a material containing 1 million atoms in the unit cell. For this material, the dipole interaction tensor has 3 million rows and an equal number of columns. The computational cost of the matrix inversion step would be on the order of Nrows3 = (3 × 106)3 = ∼2.7 × 1019 floating point operations . This corresponds to ∼7500 computational hours on a teraflop computer or ∼7.5 computational hours on a petaflop computer for each imfreq point. Since the dipole interaction tensor would need to be inverted at several  imfreq points, a TS-SCS or MCLF calculation on this material would be computationally prohibitive when using direct matrix inversion. Moreover, storing the full multi-body polarizability matrix in double-precision arithmetic would take  × (3 × 106 rows) × (3 × 106 columns) = 7.2 × 1013 bytes = 72 terabytes of random access memory (RAM). This is a huge amount of memory.A direct matrix inversion algorithm would first construct the dipole interaction tensor, then invert it to get the multibody polarizability matrix, and then contract the multibody polarizability matrix to get the AIM polarizability tensors.Pi−1) to linear in Natoms as the unit cell becomes sufficiently large.To address this problem, we developed a new computational algorithm that converges to the same solution without requiring any matrix inversions. This inverse-free algorithm is conceptually related to the iterative Schulz method for matrix inversion. In the Schulz method, an estimate Qj+1 = Dj + Δjτj (see companion article1 for specific definitions of Dj and τj) into i = 1 gives the first Schulz iteration asQj+1). The subscript j + 1 is the screening iteration and should not be confused with the Schulz iteration (superscript i). Δj is the screening increment. Substituting Substituting τj includes non-zero blocks only for interacting atom pairs , (2) Dj and inv(Dj) are block-diagonal matrices, (3) the non-zero blocks of inv(Dj) are the partially screened atomic polarizabilities {αjA} , and (4) the multi-body polarizability matrix P is only needed in its contracted form as partially screened atomic polarizabilities {αjA}.Note that: (1) 1 atoms with larger pre-screened polarizability get a proportionally larger piece of the screening mixed polarizability contribution:s and t represent spatial directions ∈{x, y, z}. Three different types of atomic polarizabilities are computed using this inverse-free algorithm. As explained in the companion article,αnon-dirA(u), τjnon-dir,Ab(u) is defined in the companion article,1 where the attenuation length for the pair of atoms A and Bj + 1 using αjA(u) and αjB(u) to compute the spherical Gaussian dipole widthαnon-dirA(u) as the value on the left-side of For non-directional screening to compute αscreenedA(u), M1 and M2 are square matrices with 3 rows andFor directional screening to compute σAB,j(u) is updated at the start of screening iteration j + 1 using αjA(u) and αjB(u) from αscreenedA(u) as one-third the trace of the tensor on the left-side of In , φjAb and αjA are defined analogous to  from σAB,j(u = Nimfreqs) is updated at the start of screening iteration j + 1 using αjA and αjB inserted into  as the tensor on the left-side of 1 the following anisotropic polarizability correction is then applied to get the static polarizability tensors over all atoms in the unit cell gives the unit cell (or molecular) polarizability tensor. Taking one-third the trace of the AIM or molecular static polarizability tensors yields the corresponding isotropic static polarizabilities.For directional screening to compute Pj+1(1) is a calculated estimate of inv(Qj+1), we begin by defining the error residual asI is the identity matrix. Multiplying Qj+1 = Dj + Δjτj givesI, and taking the norm gives36 For convenience, we used fixed size screening increments: Δj = Δ. Δ2, where Δ is the screening increment. Since the total number of screening iterations is 1/Δ, the total difference between this inverse-free algorithm and direct matrix inversion will be on the order of Δ.The order of the error residual is now derived. Since Δ → 0, the inverse-free and direct matrix inversion algorithms converge to the identical solution. We used Richardson extrapolation30,37 to evaluate the limit Δ → 0. As explained in the prior paragraph, without Richardson extrapolation the overall error using screening increment Δ is of Order(Δ). Each Richardson extrapolation step removes a successive power of Δ in the error. After K Richardson extrapolation steps (RES), the remaining error will thus be of Order(ΔK+1). We extrapolated using screening increments of 1, 2−1, 2−2, …2K−. During each such screening process, the screening increments sum to 12, …2K screening points. The final Δ = 2K− undergoes K RES leading to a residual error of Order(2K(K+1)−). Note that 7 RES is approximately twice as expensive as 6 RES and four times as expensive as 5 RES.By taking the limit Richardson extrapolation was applied to the following atomic polarizabilities:Non-directional screening:Frequency-dependent directional screening:αξforce-field,A, αξscreened,A(u),  are the values computed using 2ξ−1 screening points . The coefficients cextrapK+1),ξ, five RES for the fluctuating polarizabilities used to compute C6 coefficients, and seven RES for the static polarizabilities. The static polarizability requires more RES than the force-field and fluctuating polarizabilities, because of the longer range dipole–dipole interactions contributing to the static polarizability. The inverse-free and GEPP algorithms converged to the same results in the limit Δ → 0. The inverse-free algorithm is preferable, because it exhibits better computational cost scaling than GEPP. As an additional test, the  RES calculation was repeated using a 20 000 atom supercell (which is a 100 × 100 replication of the 2 atom primitive unit cell) which shows the MCLF results for a large supercell are identical to those computed for a primitive unit cell. The 20 000 atom graphene supercell was constructed using DDEC6 AIM properties from the primitive cell; no DFT calculation on the large supercell was needed.2.638,39 The Fortran code was parallelized by adding OpenMP compiler directives that parallelized the most computationally intense loops. OpenMP codes can be compiled in serial mode by simply instructing the compiler to ignore the OpenMP directives. We used several techniques for efficient OpenMP parallelization discussed in one of our previous articles.12 Specifically, the array indices were ordered in a cache friendly manner, large arrays were declared as shared variables to avoid large memory increases when adding more parallel threads, and when needed reductions were used with the parallel directive rather than with the do schedule directive (see Section 3.3 for example).12 Directives such as single, atomic, and critical that require one thread to wait on another should be kept to a minimum.12,38,39 To maximize parallelization efficiency, each thread should be given enough work such that the overhead time to set up the threads is a small percentage of the parallel region time.12,38,39 For example, when parallel work is done over the ‘small’ and ‘large’ atom pair lists }, {αscreenedA(u)}, and , respectively. The array indices in these figures correspond to what is actually needed in the computer program: (1) the atom number appears as an array index rather than as a subscript, (2) the screening increment does not appear as an array index because temporary results for each screening increment are computed in-place, (3) the imfreq integration point u appears as an array index only where needed, etc.Paradoxically, the computer code to perform MCLF analysis is actually simpler than the mathematical equations that define MCLF analysis. The reason is that many array indices that appear in the mathematical equations are actually not required in the computer code, because these quantities can be computed in-place. For this reason, pseudocodes for the main parts of MCLF analysis are illustrated in this section. 3.3.14 to compute αTS and CTS6 for each atom, (b) setting up the ‘small’ and ‘large’ atom pair lists, (c) self-consistent screening to computed the TS-SCS polarizability at each imfreq point, and (d) Casimir–Polder integration to compute the TS-SCS C6 coefficients. Just as for MCLF, we used Romberg integration with 16 imfreq points to compute C6 for TS-SCS.The TS-SCS calculation consists of four main parts: (a) input file reading and unscreened calculation using the TS scaling lawfMBSA,b does not need to be stored for TS-SCS. For the ‘large’ list, only  needs to be stored and  does not need to be stored for TS-SCS. This allowed each pair in the ‘large’ list for TS-SCS to be stored in a single cache line , while two cache lines were used to store data for each pair in the ‘large’ list of MCLF analysis.  Using whole cache lines provides a slight simplification for tasks that parallelize over atom pairs in the ‘large’ list.)Step (b) follows a procedure analogous to that described in Section 2.2 with the following differences. For the ‘small’ list, similar information was stored for TS-SCS as for MCLF, except 1 This same smooth cutoff function could also be used for TS-SCS. However, for consistency with prior literature,40 we used the sharp cutoff functiondAb = rAB,L is the distance between atom A and the image b in bohr. dcutoff is the dipole interaction cutoff length. H is the Heaviside step function. In this article as well as the companion one,1 we set dcutoff to 50 bohr.For MCLF analysis, the smooth cutoff function described in the companion article was used.E⃑0A is the electric field acting on site A due to permanent charges, permanent multipoles, and externally applied electric field.  is the dipole moment induced on site B. αforce-fieldA is the polarizability of site A. Specifically, setting αforce-fieldA to αunscreenedA(u) and E⃑0A to  yields  that gives the first column of . Solving again with E⃑0A set to  yields  that gives the second column of . Finally, solving with E⃑0A set to  yields  that gives the third column of . Hence, a mathematical procedure that is optimized for solving the polarizability et al., the coefficients matrix in the polarizability equation can be expressed as a two-dimensional array  by combining spatial and atomic indices.34 Since there are three spatial components  in the dipole moment for each atom, the coefficients matrix for the polarizability equation  played a key role in the history of electronic computing.41–44 (The typeface is used to distinguish A as a matrix from A as an atom.) Here, we are particularly interested in solving large systems of sparse linear equations, and many different algorithms have been described in the prior literature for doing this.42,43,45,46 Krylov subspace techniques solve a linear equation system using iterations in which some error norm is minimized in a Krylov subspace whose order increases with iteration number.47,48 A Krylov subspace of order n defined by matrix M and vector W is48Developing algorithms to solve large linear equation systems .44,47,48 By definition, a Hermitian matrix equals its Hermitian conjugate . Conjugate gradient algorithms are a widespread class of Krylov subspace techniques.43,47,48,51A key consideration is how to generate an orthonormal vector basis that spans the Krylov subspace sequence without having to store vectors from all prior iterations.42,46,47,51,52 Conditioning multiplies the original coefficients matrix by one or more matrices to transform the original linear equation Ax = B into a new linear equation My = W that is easier to solve.42,46,47,51,52 Either left multiplication or right multiplication, or a combination of both, can be used during conditioning.42,47,51,52 Conditioning can accomplish one of more of the following:42,47,51,52Conditioning plays a key role in conjugate gradient type methods.A is not Hermitian, conditioning may make the new coefficients matrix M Hermitian., or(i) If the original coefficients matrix A is not positive (semi)definite, conditioning may make the new coefficients matrix M positive (semi)definite., or(ii) If the original coefficients matrix A is not square, conditioning may make the new coefficients matrix M square. This is important for solving least-squares problems., or(iii) If the original coefficients matrix M more clustered than that of A. After conditioning, conjugate gradient methods minimize some error norm within the Krylov subspace Kn.(iv) To improve convergence speed, conditioning may make the eigenspectrum of 53,54 Aviat et al. used the Orthomin47 conjugate gradient algorithm  truncated at fixed order to solve the multibody polarization equations efficiently without introducing spurious energy drifts during molecular dynamics simulations using polarizable force fields.53,54 The key limitation of the Orthomin conjugate gradient algorithm is the coefficients matrix M in the linear equation My = W must be Hermitian positive definite.47 Although the polarization 56 For many common circumstances the coefficients matrix in this polarization equation is likely to be positive definite, but we do not have any information at hand about exceptions that could potentially give an indefinite coefficients matrix in the polarization equation. If the coefficients matrix M is indefinite, the Orthomin conjugate gradient algorithm will fail if 〈zi)(|Mzi)( and ϑi)(. Because the choice of direction pi) and CGNR.42,43,47,55 Unfortunately, both of these algorithms often converge slowly.42,43,47 For example, we programmed CGNE and CGNR for TS-SCS analysis and tested them on the C50H24 polyacene, but convergence was not reached within 100 iterations for at least one direction at a single imfreq point. For these tests, we used the same convergence threshold as for the FCR algorithm; namely, <10−5 for the maximum absolute value of each conditioned residual component.Conjugate gradient of the normal equations rearranges the linear equation system so that the coefficients matrix is 3.2x and B may contain a single column or more than one column. Matrix A is non-singular if and only if its determinant is nonzero . In this case, matrix A is invertible and the equation Ax = B has the unique solution x = A−1B. If matrix A is singular, then Ax = B has either no solution or an infinite number of solutions. If the linear equation system is consistent , this FCR algorithm returns one of its solutions. When matrix B has more than one column, the FCR algorithm is applied separately to each column. When B = I (identity matrix), the method solves for x = A−1, the inverse of matrix A.This provided motivation to develop a failproof conjugate residual (FCR) algorithm that converges robustly and resists round-off errors. FCR solves any linear equation system with Hermitian coefficients matrixA to improve convergence speed:x0 is any initial guess for x. This shift always makes y0 = 0 as the initial guess for y. The conditioning matrix C must be non-singular. Since A is Hermitian, the matrix M will automatically be Hermitian for any conditioning matrix C:The goal of conditioning is to rotate and scale matrix v and w is defined asM = MH can be freely moved between sides of the dot product. For example,In this article, the dot product of two vectors i.e., i.e., has no exact solution), this FCR algorithm returns a statement that the linear equation system has no exact solution along with a value y = yFCR that minimizes the least-squares problemy irrespective of whether an exact solution to My = W is consistent arises as the special case of If the linear equation system i.e.,  is inconM is the set of all y values that solve My = 0,M is non-singular, y = 0 is the only vector in the kernel. When M is singular, the kernel includes y = 0 along with an infinite number of non-zero vectors. Manifestly, any vector from the kernel of M can be added to yFCR without changing the conditioned residual value. Therefore, the set of all equally good (aka ‘best’) solutions to M) is known, this allows calculating the entire set of {yall} that give the same minimum value of F. If My = W is consistent, this is the set of y values yielding F = 0 ( and qi):The FCR method minimizes the conditioned residual normz-search directions Mpi)( and Mqi)( and 〈Mzi)(|Mzi) if the absolute value of every conditioned residual component is less than a chosen CR_convergence_tol, the algorithm exits and returns a message that the linear equation system is consistent and returns the current value of y (which is a nearly exact solution). (2) Else if  is less than a chosen Mz_length_tolerance, the algorithm exits and returns a statement that the linear equation system is inconsistent and returns the current value of y . (4) Else the algorithm continues to the next iteration.After computing the new values of e.g., 100–1000) and exit via condition (3) represents a convergence failure. In case of convergence failure, the algorithm could be restarted using a different (and hopefully better) conditioning matrix. In case (B), the goal is to reach a specified number of FCR iterations rather than a specified precision. In classical atomistic simulations via polarizable force fields, using a fixed number of conjugate gradient iterations leads to continuous forces with enhanced energy conservation; therefore, exiting after a small constant number of iterations is preferred.53,54 This behavior can be achieved by setting max_FCR_iterations to a small whole number  and setting FCR_convergence_tol and Mz_length_tolerance to very small values.In some circumstances (A) an exit due to condition (3) is considered an error, but in other circumstances (B) it is the preferred exit condition. In this case (A), the goal is to reach a specified precision rather than a specified number of FCR iterations. In this case, max_FCR_iterations is set to a large value (pi)( is chosen to ensurepi)( is non-zero (see ESIThe direction qi)(|Mpi)(〉 or 〈Mqi)(|Mqi)(|Mpi)(〉 and 〈Mqi)(|Mqi)(|Mpi)(〉 = 0 can only occur if 〈Mzi−1)(|Mzi−1). 〈Mqi)(|Mqi)(〉 = 0 may arise from (a) round-off errors or (b) on the last iteration if all components of zi)( not in kernel(M) are already made zero by the Mpi)( (τi)( (βi)( (ξi)( (κ1i)( (κ2i)( (χ2i)( (σ2i)( (χ1i)( (σ1i)( (χ0i)( (All denominators in the method have the form 〈ter γ(i) , τ(i) (ei) (τ(i) , β(i) (ei) (β(i) , ξ(i) (ei) (ξ(i) , κ1(i)  (κ1(i) , κ2(i)  (κ2(i) , χ2(i)  (χ2(i) , σ2(i)  (σ2(i) , χ1(i)  (χ1(i) , σ1(i)  (σ1(i) , or χ0(i) (χ0(i)  is set ti.e., along Mpi)( and Mqi). In exact arithmetic, these directions are orthogonal to all previous directions Mpj<i)( and Mqj<i) iterations. In finite arithmetic, the FCR algorithm resists round-off errors as described above.In exact arithmetic, the algorithm converges to the exact solution in a finite number of iterations. The conditioned residual is eliminated along two directions in each iteration  iterations. For example, if matrix M contains Nrows = 1 million, but matrix M has only 12 distinct eigenvalues, then FCR will converge to the solution in at most 6 iterations in exact arithmetic.Further convergence analysis can be obtained by expanding yi)((λ). When the eigenvalues are close to one, this is trivial becausei)((λ) ≈ 1/λ for all of the present eigenvalues. Thus, the primary goal of conditioning  as shown in W contains any nonzero contribution  from an eigenvector V⃑j having associated zero eigenvalue , then the linear equation system is inconsistent. Since only the basis eigenvectors appearing in W need to be represented by My  of W have non-zero eigenvalues . Therefore, even My = W systems for singular M can be solved exactly by FCR as long as the zero eigenvalues  have zero contribution  to W. In this case, the returned solution yFCR has no contribution from any eigenvector whose associated eigenvalue is zero ).For singular y My see , the linMy cannot contain any eigenvectors from the kernel(M), because each term contributing to My contains the factor (λj)k+1≥1 which is zero for any Vj in kernel(M). Consequently, any Vj in kernel(M) having non-zero cj cannot be removed from the conditioned residual , while conditioned residual components in kernel(M) remain at their initial levels .As shown in dual see . TherefoThe ESI3.3u with E⃑0A set to . The FCR algorithm was then used to solve the linear equation My = W for y which was solved for  by multiplying by : gives the first column of . In this scheme,  corresponds to ‘x’ in the linear equation system ‘Ax = B’, and B corresponds to the externally applied electric field, {E⃑0A}. The conditioning matrix C is diagonal with  along the diagonal. Solving again with E⃑0A set to  yields  that gives the second column of . Finally, solving with E⃑0A set to  yields  that gives the third column of . This entire process is repeated at each imfreq point u.The following procedure was used to solve the TS-SCS equations using the FCR algorithm. First, the following quantities were defined14 the C50H24 polyacene geometry taken from the companion article,1 and a metal–organic framework whose geometry was taken from the Computation Ready Experimental (CoRE) metal–organic framework (MOF) database.58 We used the MOF having Cambridge Structural Database59 code KUCDIW. Electron densities for these three materials were generated in VASP using the PBE functional, a 400 (B-DNA and MOF) and 750 (C50H24) eV planewave cutoff, and the PAW method. The k-point mesh and grid spacing followed previous recommendations.12 FCR convergence for these three materials is summarized in We now discuss a few computational examples. For the examples we studied, the FCR algorithm indicated the linear equations were consistent and converged to a nearly exact solution. 1 were performed using 64-bit real numbers. For comparison, computations performed using 128-bit reals are also displayed in Except where otherwise specified, all computations in this article and the companion articleC6 for the unit cell  = M × Mp(:) and MMq = M × Mq(:).We used OpenMP to parallelize the most computationally intense parts of this TS-SCS algorithm. The loops parallelized included calculation of unscreened and screened total Natoms when the number of FCR iterations does not increase appreciably with increasing system size. To date, we have not observed any materials for which TS-SCS(FCR) converges slowly. However, the fixed number of required iterations for MCLF is a clear advantage compared to the variable number of iterations to converge TS-SCS. First, it makes MCLF convergence highly repeatable, which will become important for applications that involve differentiation with respect to atomic positions . Second, it makes MCLF computational times highly predictable, because the number of required iterations is known up front.The TS-SCS(FCR) code achieves linear-scaling computational cost for sufficiently large 4.4.1via Gaussian elimination with partial pivoting (GEPP). GEPP is a widespread algorithm described in many numerical methods textbooks.60 This allowed us to compare both the computational time and precision / × ).  i.e., one processor) TS-SCS(FCR) and MCLF calculations could not complete in less than one week; therefore, only parallel calculations were run. Secondly, the source code had to be compiled using 64-bit integers to accommodate the large range of array index values. This 64-bit integer source code and a large memory node were also used for the 1 053 696 atom system reported in To quantify the performance of these algorithms for larger systems, an ice supercell containing >2 million atoms in the unit cell was prepared from the DDEC6 AIM properties. Specifically, a 2 × 2 × 2 supercell of the 263 424 atom unit cell gave a periodic supercell containing 2 107 392 atoms. Due to the large size, serial  classification of atoms into spatial regions followed by constructing two lists of interacting atom pairs and (b) a lookup table method to compute Ctotal6.We developed computationally efficient algorithms to compute atom-in-material polarizabilities and dispersion coefficients using MCLF and TS-SCS analysis. Our MCLF algorithm uses Richardson extrapolation of the screening increments. Our TS-SCS algorithm uses a special conjugate residual algorithm that resists round-off errors. Both our algorithms have computational time and memory requirements scaling linearly with the number of atoms in the unit cell when the unit cell is much larger than the dipole interaction cutoff distance. For both small and large systems, our algorithms require less computational time and memory than direct matrix inversion, with negligible change in computational precision. Our algorithms achieved this by avoiding both large matrix inversions and large dense matrix multiplies. This is an important achievement, because direct matrix inversion and dense square matrix multiplication have computational costs scaling between Our algorithms were easily parallelized to take advantage of multiple computing cores. To minimize false sharing, our algorithms access data in cache line friendly order. Excellent parallelization efficiencies were obtained. Moreover, adding parallel computing cores did not significantly increase memory requirements.This made it possible to apply the MCLF and TS-SCS methods to materials containing orders of magnitude more atoms per unit cell than previously feasible. Our algorithms can be readily applied to materials containing millions of atoms in the unit cell. The largest example studied herein was an ice crystal containing >2 million atoms in the unit cell. To perform TS-SCS on this material, the FCR algorithm solved a linear equation system containing >6 million rows, 7.57 billion interacting atom pairs in the large list and 87 million in the small list, 45.4 billion stored non-negligible matrix components used in each large matrix-vector multiplication, and ∼19 million unknowns per frequency point . This problem was solved in 5.6 hours by 48 parallel computing cores with 500 GB RAM. The MCLF calculation for this material took 21.7 hours on 48 parallel computing cores with 950 GB RAM.xyz file containing the list of atoms , unit cell information , and AIM 〈r3〉 moments and (ii) a calculation_parameters.txt file listing calculation parameter values. Required inputs for the MCLF program are: (i) separate xyz files for AIM volumes, 〈r3〉 moments, 〈r4〉 moments, weighted 〈r4〉 moments, and net atomic charges, and (ii) a calculation_parameters.txt file listing calculation parameter values. Settings in the calculation_parameters.txt file choose whether to use an inverse-free method or direct matrix inversion, the Romberg integration order, the dipole interaction cutoff length, whether to ignore PBC, the number of Richardson extrapolation steps (for MCLF), the convergence threshold (for TS-SCS(FCR)), etc. The default values are reliable and almost never need to be changed.The MCLF and TS-SCS software programs described here will be distributed through the same code repository as the Chargemol program. Required inputs for the TS-SCS program are: (i) an Finally, the new failsafe conjugate residual algorithm should find widespread applications to a plethora of scientific computing problems, because it resists round-off errors and solves any linear equation system with Hermitian coefficients matrix. This conjugate residual algorithm has many desirable mathematical properties, because it minimizes the norm of the conditioned residual within a Krylov subspace of increasing order with each successive iteration.T. A. M. supervised the study, obtained funding, developed all computational and mathematical methods, generated and wrote all mathematical derivations and proofs , wrote the entire ESI,There are no conflicts of interest to declare.RA-009-C9RA01983A-s001RA-009-C9RA01983A-s002RA-009-C9RA01983A-s003RA-009-C9RA01983A-s004RA-009-C9RA01983A-s005RA-009-C9RA01983A-s006RA-009-C9RA01983A-s007RA-009-C9RA01983A-s008


Nowadays, the growing popularity of distance running has been accompanied by the increasing prevalence of vegan and vegetarian diets, especially among endurance athletes. The present study aimed to examine the association between diet type and dietary intake of distance runners competing at distances longer than 10 km. From a total of 317 participants, 211 endurance runners  were considered the final sample after applying the exclusion criteria. Runners were assigned to three groups based on the self-reported diet types: 95 omnivores, 40 vegetarians, and 76 vegans. Data collection was conducted using an online survey with questions about sociodemographic information, dietary intake, and dietary-associated motives. A comprehensive food frequency questionnaire with 53 food groups (categorized in 14 basic—plus three umbrella—food clusters) was used to assess dietary intake. Vegan runners had a higher intake of “beans and seeds”, “fruit and vegetables”, and “dairy alternatives”, as well as lower intakes of “oils” than other two groups. Vegetarian runners had a lower intake of “dairy products” and “eggs” than omnivores. A greater intake of “alcohol” and a lower intake of “meat alternatives” was observed in omnivorous runners compared to vegans and vegetarians. Despite the existence of a tendency toward the consumption of health-related food clusters by vegan runners, further investigations are needed to verify the predominance of vegans in health-oriented dietary patterns. Diet type can significantly impact the nutritional status of endurance athletes, thus influencing their health and performance ,2. Two mA crucial part of sports nutrition practice is to improve, optimize, and personalize exercise-related dietary strategies. With this, dietary assessment helps identify exercise-induced nutritional undersupply that may result in deficiencies . The nutritional requirements of distance runners are more serious than other runners due to the ridged training routine (high mileage and intensity) and racing activities . Evidenc12) in plant vs. animal-based foods [This concern seems to be more critical for vegan and vegetarian athletes if their diet is not planned appropriately . In thised foods ,18, inveed foods ,19,20,21ed foods ,23 may red foods ,24,25.Surveys on endurance runners have also reported that vegans/vegetarians had a higher reliance on carbohydrates and a lower reliance on fat and protein when compared to omnivorous runners ,27. RegaTo date, there are limited data on nutritional intakes of athletes based on dietary subgroups ,28. ThisThe present cross-sectional investigation is part of the Nutrition and Running High Mileage (NURMI) Study Step 2. The study protocol  was apprhttps://www.nurmi-study.com/en; accessed on 8 July 2022). A written description of the procedures was provided for participants, and they agreed to the informed consent before filling in the questionnaires. There were four initial inclusion criteria for participation in the NURMI Study Step 2, including (1) written informed consent, (2) ≥18 years of age, (3) completion of the questionnaire, and (4) successful participation in a distance running event  within the past two years.Endurance runners were mostly from Germany, Austria, and Switzerland, and were recruited through personal contacts via social media, online running communities, websites of organizers for marathon events, email lists of running magazines, as well as other multi-channel sources. In the NURMI Study Step 2, endurance runners were asked to complete an online survey available in English and German , vegetarian , and vegan diet  ,4. The i2 was excluded from the sample. Further, based on the exclusion criterion of consumption of ≤50% carbohydrates in their total daily energy intake  of the recreational runners adequately assessing their diet type. As a final sample, 211 runners  with complete data sets were included for data analysis. From the initial number of 317 participants, 106 runners were excluded from the sample (including 46 runners who did not meet the inclusion criteria). The Body Mass Index (BMI) approach, following the World Health Organization (WHO) guidelines ,37, was ociation ,39, we eociation . FinallyThe validated food frequency questionnaire (FFQ) of the “German Health Interview and Examination Survey for Adults (DEGS)”  ,41 was uBased on the 53 food groups of the DEGS-FFQ and according to the NOVA classification system developed by the Food and Agriculture Organization (FAO) of the United Nations (UN) ,43,44,45The statistical software R version 4.1.1 Core Team 2018  was used to conduct all statistical analyses. Exploratory analysis was performed by descriptive statistics, including mean values, standard deviation (SD), median, and interquartile range (IQR). 2, nominal scale) was used to examine the association of diet type with sex, nationality, academic qualification, marital status, race distance, and dietary motives. Kruskal–Wallis tests  were conducted (by using the F distributions) to examine the association of diet type with age, body weight, height, and BMI. Univariate tests were used to investigate the differences between diet type. Chi-square test (χFood clusters were defined by 53 manifest parameters (assessing how often and level of consumption of specific dietary items). In order to scale dietary intake displayed by measures, items, and clusters, a heuristic index (as a new compound variable) ranging between 0 and 100 was defined , and FFQ was calculated by multiplying the two questions and dividing by the maximum. Examination of significant differences in the intake of specific food clusters by diet type and age was conducted by a linear-regression model. The regression analysis assumptions have been confirmed by inspection of graphs of residuals. Differences in respective food clusters between dietary subgroups are also displayed by effect plots with the standardized regression coefficient (β) and 95% confidence interval (CI).p ≤ 0.05.The statistical significance level was set at 2 (IQR 3.4) were considered as the final sample for statistical analysis. Germany, Austria, and Switzerland  had the most participants , and the remaining runners were from other countries across the globe. Significant differences were found between dietary subgroups in age (p = 0.040), sex (p = 0.013), weight (p = 0.002), and BMI (p = 0.001). There were no significant differences in height, nationality, marital status, academic qualification, and race distance between omnivorous, vegetarian, and vegan runners. A total number of 211 runners  with a median age of 38 years (IQR 18) and a median BMI of 21.7 kg/mp = 0.028), “food scandals” (p = 0.010), “animal ethics” (p < 0.001), “ecological aspects” (p = 0.005), and “custom/tradition” (p = 0.007).p < 0.05). While vegans reported a more frequent intake of “beans and seeds” (p < 0.001), “fruit and vegetables” (p < 0.001), dairy alternatives” (p < 0.001), “meat alternatives” (p < 0.001), and “protein” (p < 0.001), consumption of “oils” were lower (p < 0.001) in vegans compared to omnivores and vegetarians. The consumption frequency of “dairy” and “eggs” were higher in omnivores than vegetarians (p < 0.001). Omnivores also reported a greater intake of “alcohol” compared to vegetarians and vegans (p = 0.004). No significant difference between diet groups was observed in “grains”, “snacks”, “water”, “beverages”, “processed foods”, or “free/added sugar” (p > 0.05). Significant differences were found between omnivores, vegetarians, and vegans in the consumption frequency of 11 food clusters , while the vegan diet was a significant predictor of 15 food groups (p < 0.05). Results indicate that age was a significant predictor of two clusters, including “fruit and vegetables” (p = 0.024) and “meat alternatives” (p = 0.011).The present investigation examined dietary intake of distance runners (based on 14 basic and three umbrella food clusters) differentiated by vegan, vegetarian, and omnivorous groups. Runners’ motives to follow diet types and the associations between dietary intake and sociodemographic information were also investigated. The essential results showing significant differences within dietary subgroups were that (1) vegan runners had a higher intake of “beans and seeds”, “fruit and vegetables”, and “dairy alternatives”, as well as a lower intake of “oils” compared to vegetarians and omnivores; (2) vegetarian runners had a lower intake of “dairy products” and “eggs” than omnivores; (3) omnivorous runners had greater intake of “alcohol” and a lower intake of “meat alternatives” compared to vegans and vegetarians. In addition, further important findings show that, independent of diet type, (4) “health and wellbeing”, “animal ethics”, and “ecological aspects” were identified as the most popular motives to follow the self-reported kind of diet; and (5) “diet type” has been found to be a predictor for consumption of the majority of food groups, but “age” has been shown to be a predictor of only two food clusters, including “fruit and vegetables” and “meat alternatives”. The existence of some health-related orientations in consumption of some food clusters by vegan runners seems to partially verify the present hypothesis on the positive trend of more frequent consumption of the food groups characterized by beneficially contributing to a health-promoting dietary intake. To the best of the authors' knowledge, this is the first investigation performed on endurance runner dietary intake–food consumption based on the frequency of specific food items–focusing on the strict examination of and comparison between vegan and non-vegan groups.As a central method to identify nutritional inadequacy, dietary assessment can help to develop personalized dietary strategies for improving health and performance levels in athletes and optimizing health-related insights in general populations ,47. DietIn line with the increasing prevalence of athletes who follow vegan/vegetarian diets, it has been suggested that well-designed plant-based diets can fulfill the nutritional requirements considering health and a successful endurance performance ,54,55,56Results on hydration habits showed that diet type were identified as a non-influencing variable in the consumption of water and beverages. However, there were significant between-group differences in the consumption of milk (predominantly omnivores) and vegetable juices (predominantly vegans) in the present study. While little is known about the hydration status of vegan and vegetarian athletes, it seems that training and racing behaviors play a stronger role in the hydration-related dietary behaviors of endurance athletes . ConcernIn the present study, no difference in grain consumption was observed between vegan, vegetarian, and omnivorous runners; however, differentiated results showed a significant association between diet type and refined (but not whole) grains, where vegan runners had a lower consumption of refined grains than vegetarians and omnivores. Vegan runners were also found to have a higher intake of two other carbohydrate sources, including “beans and seeds” and “fruit and vegetables”. Available literature shows that vegans and vegetarians generally consume a higher amount of carbohydrates than omnivores ,21. For Interesting findings were observed concerning the protein-based clusters, which are not associated with the characteristics of dietary subgroups. Vegans, followed by vegetarians, reported a higher intake of both “dairy alternatives” and “meat alternatives” compared to omnivores. While these findings may be considered compensating behaviors to reach the required amount of protein intake, evidence indicates that the proportion of protein and fat in the daily energy intake of vegans/vegetarians is generally lower than omnivores ,21,67. TRecent evidence indicates that distance runners have acceptable health-related behaviors regarding food choices since most of them prefer to intake healthy dietary ingredients . The preDespite the general fact that vegetarians only focus on specific food items among animal sources , the present findings show that they consume both “eggs” and “dairy products” considerably less than omnivores. This result may suggest that they are in the transition process of becoming vegan and may potentially follow a 100% plant-based diet in the future. Another finding to support this assumption was that vegetarians reported consuming dairy alternatives 77% higher than omnivores but 28% less than vegans, and meat alternatives 3.8 times (380%) higher than omnivores but only 17% less than vegans. While a possible explanation for these findings can be found in living conditions (particularly living with a partner or family member who follows a plant-based diet), it should be considered that the lack of discrimination between vegan and vegetarian diets in most comparable studies has made it difficult to compare and interpret the present findings with the similar investigations. As a well-established fact, personalized nutritional requirements of athletes, which is connected to their different performance-related challenges, is considered a remarkable factor for explaining most dietary contradictions [Educational background and specific knowledge regarding sports nutrition especially may also be associated with improved health behaviors, including a healthier diet . ConcernThe present investigation has some limitations necessary to mention. The study was set up as a cross-sectional design with self-reported data, thus, under- or over-reporting might have occurred on individual data sets. Misreporting of dietary intake is somewhat prevalent in athletic populations . HoweverAnalysis of endurance runner dietary intake showed that there are differences between vegan, vegetarian, and omnivorous runners in their food choices . Vegan runners had a lower consumption of “refined grains” and “oils” along with a higher consumption of “beans and seeds”, “fruit and vegetables”, and “dairy alternatives” compared to vegetarians and omnivores. These findings suggest that vegans have a higher tendency toward consumption of frequently-recommended food groups in favor of health . However, there are some considerations  that may potentially obscure any health-related interpretation regarding this result, but it appears that the health-oriented lifestyle of runners who follow plant-based diets can play a remarkable role in many dietary patterns. Considering the growing popularity of distance running and plant-based diets alike, the present findings may be of help for sports dietitians, nutritionists, and coaches to provide more precise dietary recommendations. These results may also produce new aspects for sports nutrition specialists and researchers regarding the hidden health-related dietary patterns of vegan and vegetarian runners, independent of the food items associated with animal- versus plant-based characteristics. However, more detailed interventions using further analyses of interacting factors are necessary to expand this area of knowledge, particularly to find the association between diet type and health-related dietary choices in endurance runners.


To make thesematerials more useful practically, the mechanisms controlling thetemperature over which this transition occurs and the magnitude ofcontraction need to be controlled. In this work, ball-milling wasused to decrease particles and crystallite sizes and increase thestrain in LaFe11.9Mn0.27Si1.29Hx alloys. Such size and strain tuning effectivelybroadened the temperature over which this transition occurs. The material’sNTE operational temperature window was expanded, and its peak wassuppressed by up to 85%. This work demonstrates that induced strainis the key mechanism controlling these materials’ phase transitions.This allows the optimization of their thermal expansion toward room-temperatureZTE applications.Zero thermal expansion (ZTE) composites are typicallydesignedby combining positive thermal expansion (PTE) with negative thermalexpansion (NTE) materials acting as compensators and have many diverseapplications, including in high-precision instrumentation and biomedicaldevices. La(Fe This issue is particularlyproblematic for devices with several constituents (such as multilayereddevices) where there is likely to be a thermal expansion mismatchbetween their constituent materials.9 Such thermal expansion mismatchbetween materials can cause mechanical cracks and disrupted electricalconnections. In such cases, controllable thermal expansion can enhancedevice reliability and prolong their lifetime.8 In order to tackle this technological problem, materialsand/or composites with near-zero thermal expansion behavior [zerothermal expansion (ZTE) or invar materials] are required.9Currently, there is anurge to mechanically control materials atthe micro- and nanoscale in a wide set of technological industries,for example, micro-devices, large telescopes, dentistry, sensors,and fuel cells.10 Functionalcomposites’ thermal expansion can be tailored by combiningPTE with NTE constituent materials. In particular, ZTE compositescan be achieved by combining materials with isotropic thermal expansions,where the NTE material will act as a thermal expansion compensator.13 Therefore,it is extremely important to understand the underlying intrinsic chemical–physicalbehavior of these NTE materials and provide fundamental insights,so that they are controllable for a wide range of applications.28While most materials expandwhen heated, exhibiting a positivethermal expansion (PTE), some possess an unusual property of volumecontraction—these are said to have a negative thermal expansion(NTE).10 oneis the occurrence of a magnetovolume coupling.32 This mechanism typically arises from a simultaneous magnetic andatomic lattice transition, where the large magnetization variationat the Curie temperature (TC) is accompanied by a large change in the crystal structure.The simultaneous variation of two order parameters (magnetizationand unit cell volume) typically occurs in materials with first-orderphase transitions such as the LaFeSi, HfNbFe, and MnZnN material families.5 This coupling provides the ability to tune the thermal expansionof these materials by adjusting their magnetic properties.Among the different mechanisms that lead to NTE behavior,16 In particular, several materials that present bulk PTE have beenfound to exhibit NTE when presenting at least one dimension confinedat the nanoscale.33 These materials’properties, such as magnetization, superficial/interfacial coordination,local lattice symmetry, and elemental distribution, can be changedthrough size reduction, enabling the possibility of tailoring associatedfeatures such as their thermal expansion.16Morerecently, another mechanism controlling the thermal expansionhas been attracting research attention: the size confinement of anygiven material . Such microstructural downsizing can lead to the appearanceof novel physical properties due to size-induced effects.x1–,Six)13-derived compounds are very well knownfor their giant magnetocaloric effect and NTE behavior.46 Their crystal structure is cubic with Fm-3c space group, and theydisplay a ferromagnetic-to-paramagnetic (FM-PM) isostructural transition,corresponding to a low-temperature high-magnetization (ordered) statethat evolves into a high-temperature low-magnetization (disordered)state as the temperature is increased across their TC.26 Along these transitions,the crystal structure is retained, but the unit cell volume contractsby 3000–30,000 ppm typically over a <50 K temperature interval.37 To act as compensators, NTE materials should havea more gradual change of volume over a larger temperature window,akin to the typical changes on standard PTE materials. Fortunately,it is possible to tune and shape the profile of the magnetizationversus temperature curves  by tuning bothextrinsic (pressure and magnetic field)46 and intrinsic (stoichiometry) properties.41 Recently, there have been remarkable examples ofthermal expansion control via chemical substitutions, where the typicallyabrupt NTE behavior is smoothed across a wider temperature window.41 For example, by replacing Feby Co and Mn, or by adjusting the Si/Fe ratio, or even by electrolytichydrogenating La–Fe–Si, Li’s group has successfullybeen able to tune the operational temperature and smooth out La–Fe–Sisharp volume transition.26The Lahas a strong impact on the magnetic transition profile: increasingmagnetic disorder leads to a broadening of the magnetic transitionthat is associated with a wider temperature distribution of Tc’s.59 Hence, due to La–Fe–Si strong magnetovolume coupling,in this work, we propose to make use of the microstructural strainto control the magnetic transition profile and consequently tune itsstructural transition and thermal expansion coefficient. Moreover,the size reduction effects on LaFe11.9Mn0.27Si1.29Hx morphological, microstructural,magnetic, and thermal expansion properties were thoroughly characterizedand correlated, providing a comprehensive physical model for the observedbehaviors.This work proposes an alternative mechanismto control the thermalexpansion of strongly magnetovolume coupled materials—microstructuralstrain induced by ball-milling (BM).211.9Mn0.27Si1.29Hx sample was prepared by arc-melting followedby hydrogenation.49 After melting, to ensurehomogeneity of the as-cast alloy, the ingot was encapsulated in quartztubes under an Ar atmosphere and then annealed at 1050 °C for7 days and subsequently quenched in water. Hydrogenation was carriedout on 1–2 mm fragments of the parent bulk sample in a furnacefilled with 0.9 bar H2 atmosphere, at 723 K for 1 hour,to saturate the H concentration. The as-prepared hydrogenated piecesof LaFe11.9Mn0.27Si1.29Hx alloy were hand-milled for a short period of time,resulting in an initial set of particles with a 100 μm averagesize and a size distribution width (FWHM) of 126 μm  and energy dispersive spectroscopy (EDS)using FEI Quanta 400FEG ESEM/EDAX Genesis X4M equipment. SEM imagestatistical analysis was performed using the ImageJ open source software.50 Several hundreds of particles from the SEM imageswere randomly chosen for each sample to determine the particle sizedistribution, namely, the average particle size and the size standarddeviation. Magnetic characterizations of the powder samples were performedin a commercial Superconducting Quantum Interference Device (SQUID)magnetometer  by measuring their magnetizationas a function of temperature between 350 and 5 K while cooling witha magnetic field of 1000 Oe and by measuring isothermally their magnetizationas a function of magnetic field at 20 K. All magnetic measurementswere taken under identical geometries and were properly correctedfor inevitable deviations to the dipole approximation in accordancewith the recent report by Amorim and co-authors.62The LaFef 126 μm 1a. This 51 a conversionof any given unit cell volume to temperature can be made, which resultsin Cu working as a local sample thermometer.The room-temperature crystal lattice parameterswere obtained byRietveld refinements against X-ray diffractograms measured in a RigakuSmartLab using a Cu X-ray source (Kα1 = 1.5406 Å). Thesamples’ thermal expansion was evaluated through Rietveld refinementsof X-ray diffractograms obtained while varying the temperature between250 and 350 K at 1 K/min under heating and cooling, with each diffractogramcorresponding to a 1 K step. These measurements were performed withsynchrotron radiation (λ = 0.1173 Å) at the Advanced PhotonSource (APS) of the Argonne National Laboratory (beamline 11-ID-C).Each powder sample was wrapped in a Cu thin foil such that the diffractionpattern includes reflections linked to the cubic Cu phase. As Cu thermalexpansion is very well known,33.1tBM, of 180 min, together witha 98% reduction of the standard deviation.3.211.9Mn0.27Si1.29Hx has the NaZn13-type crystal structure (space groupFm-3c), where the Fe atoms, 8b (Fe(I)) and 96i (Fe(II)), occupy twononequivalent 8b (Fe(I)) and 96i (Fe(II)) sites; Si atoms preferentiallyoccupy Fe(II) sites;35 Mn atoms tend to occupy equallyboth Fe(I) and Fe(II) sites;39 and the hydrogen atomslie at interstitial sites41 . TheRietveld refinements against all X-ray diffractograms at 273.6 K revealthe expected NaZn13-type crystal structure and a minorityphase fraction of α-Fe (space group Im-3m),which is commonly observed in samples prepared by this route.40 An X-ray diffractogram with a Rietveld refinementof sample BM15 is presented in Figure S2 (Supporting Information). In addition, peaks due to the Cu cubicphase (space group Fm-3m) used for sample wrappingare identified and fitted.LaFetBM.The average particle size was obtained by SEM statistical image analysis.In contrast, the crystallite size (D) and strain(ε) were obtained through a linear regression following theWilliamson–Hall relationship [see Figure S3 (Supporting Information)]w is the FWHM of theXRD peaks corresponding to the La13 structure, λ is the incident X-ray wavelength, and θ isthe peak position.55tBM leadsto a decreasein average particle and crystallite sizes following a decreasing trend to 0.26%(for the BM180 sample). A linear fit to this behavior yields a 1.22× 10–5 slope, meaning that a strain of ∼1.22× 10–3% is induced per every minute of BM.Thus, by relative comparison, it is possible to conclude that BM timehas a greater impact on the strain than on the crystallite size.Increasing ng trend2. Compar3.3x1–,Six)13-based compound family is stronglycorrelated with its magnetic transition, magnetization vs temperaturemeasurements were performed in all samples. M(T = 250 K)), M/M250K measured on cooling and with an appliedmagnetic field of 1 kOe, in the temperature range from 250 to 350K for all samples. M(T) curves, dM/dT(T). From these curves, one can extract the full width at half maximum(FWHM = ΔTM/dTd) and the absolute value of their global minimumpeaks, dM/dTpeak. TM/dTd as afunction of tBM and also the slope ofthe M(T) curves, M/MT = 250 K, evaluatedin the respective temperature range corresponding to ΔTM/dTd.Because the NTEbehavior of La profiles of the BM samples present an increasinglysmoother transition with increasing tBM. In fact, the M(T350K)/M(T = 250 K)value increases monotonously as the tBM increases, indicating that the FM–PM transition becomes moregradual with tBM.The magnetization increases on cooling, revealingthe typical PM–FMphase transition at M/dTpeak quantifiesthe “abruptness” of the phase transition  and indicates the temperature atwhich the magnetization is changing most drastically, correspondingto T ∼ TC. Itsdependence on tBM is represented in Figure S4 (Supporting Information) where a reductionfrom ∼0.23 K–1 (as-prepared)to ∼0.01 K–1 (BM180) isobserved.The dTM/dTd quantifies how broad the magnetic transitionsareacross temperatures. This quantity varies approximately linearly with tBM—increasing from ∼3 K (as-prepared)to ∼24 K (BM180), which shows how the broadening of the magnetictransition evolves with tBM.The ΔM(T) curvesdecreases rapidly with tBM—decreasingfrom ∼0.15 K–1 (as-prepared) to ∼0.01K–1 (BM180). This means that the temperature-inducedmagnetization variation is strongly suppressed as tBM increases. In addition, the profile of the slope ofthe M(T) curves vs tBM  via a β coefficient. In addition, this model also introduced a new term η,which is dependent on β and on the material compressibility(K), allowing us to quantitatively distinguish materialswith strong (η > 1) and weak (η < 1) magnetovolumecoupling. The M(T) curves were fittedusing the Bean–Rodbell model, assuming that the transitioncan be described not by a single TC, butinstead by a lognormal distribution of TC’s (associated with the induced microstrain), with a givenwidth, Figure S6 in the Supporting Information, tBM from ∼3 K (as-prepared) up to ∼110 K(BM180). Similar to the strain and ΔTM/dTd, tBM.Froma theoretical point of view, the Bean–Rodbell model has beenparticularly successful in explaining and predicting the magnetictransitions of materials with strong magnetovolume coupling.ith refs 64),. As,. As59 T3.4V(T)/V(T = 250 K)= [V(T) − V(T = 250K)]/V(T = 250 K). V(T)/V(T = 250K) curves, which is the equivalent to thecoefficient of thermal expansion as a function of temperature, αV(T). From this curve, oneobtains the FWHM, V(T)peak. tBM.V(T)/V(T = 250K) curves show that the volumedecreaseswith the increasing temperature, that is, αV < 0, demonstrating NTE behavior across the transition.In addition, the evolution of the ΔV(T)/V(T = 250K) curves’profiles with tBM is very similar to theevolution of the M(T) curves’profiles: a gradual broadening of the transitions, as a result ofthe strong magnetovolume coupling. In addition, the overall ΔV/V variation tends to diminish with tBM: the as-prepared sample has an overall ΔV/V of ∼11.5 × 10–3, whereas the BM180 has ∼9.8 × 10–3.ΔV(T)peak dependency with tBM isshownin Figure S7 (Supporting Information),where a value of ∼2210 × 10–6 K–1 is observed for the as-prepared sample and a valueof ∼260 × 10–6 K–1 is obtained for the BM180 sample.The αtBM increasing from ∼1 K (as-prepared) to ∼15K (BM180). Similar to what was observed with the magnetic transition,the tBM from ∼40 × 10–4 K–1 (as-prepared) to ∼3 × 10–4 K–1 (BM180). Additionally, the tBM, reaching then a plateau region. As withthe magnetization slope, this behavior suggests that further BM willnot induce significant tBM is linearly expanding the temperature intervalwhere the transition occurs and, concomitantly, making it less sharp.V(T) wasalso analyzed in the whole temperature interval, from 250 to 350 K, Figure S8 (SupportingInformation) shows the relation between tBM, wherea linear relation is observed. The as-prepared sample shows a –6 K–1 and the BM180 a –6 K–1.The αtBM, the relation between tBM, as shown in I–FeII interatomic distances (known as the interatomic distanceinfluencing Tc the most),60 and the Tc were plotted against ball-milling time as can be seen in Figures S13–15 in the Supporting Information.The absence of significant or monotonous changes observed in the centerpeak position, unit cell volume, FeI–FeII interatomic distances, or Tc as a functionof milling time, coupled with the clear, symmetric, and linear broadeningof the diffraction and magnetization derivative peaks, strongly suggeststhat the linearly increasing microstrain has the main role on tuningthe magnetic and structural transitions. This physical picture isfurther corroborated by the Bean–Rodbell analysis, where alinearly increasing broadening of the Tc distribution is required to fit the measured M(T) curves as a function of milling time .In order to try to disentanglethe roles of crystallite size and induced microstrain on the observedchanges of the magnetic and structural transitions with milling time,the raw patterns, the unit cell volume, the FeFigure S12. As can be seen, the saturation magnetization remainsnearly unchanged after the total 180 min of milling. This behaviorcontrasts with the behavior found for La–Fe–Si nanoparticles,where a 30% decrease in the saturation magnetization was observed.61 In fact, the suppression of the magnetizationvalues, together with shift of the transition temperatures, is typicallyobserved in a wide range of nanostructured magnetocaloric materials.47 Hence, these results demonstrate that the millingis not changing the individual magnetic moments but is instead inducingmagnetic disorder that manifests itself in broadened magnetic and structural phase transitions.Additionally, in order to further untanglethe effect of size confinementfrom that of induced microstrain, we have measured low-temperature(20 K), isothermal magnetization versus magnetic field curves forthe as-prepared and BM180 samples, as plotted in Figure S10, we can see thatthe relative variation in magnetization between 250 and 350 K is morepronounced than the relative change in unit cell volume. This suggeststhat the magnetic correlations are more sensitive to strain than theunit cell volume of the sample.Interestingly, comparing the magnetic and structural transitionsas seen in V(T) of these materials while providing pathwaysfor further thermal expansion coefficient tailoring. A remarkablysimilar color pattern is observed in Using this interpolation method, we are able to createan estimationfor new values of 411.9Mn0.27Si1.29Hx. A severe broadeningof the magnetization versus temperature profiles across TC is observed, quantified by a 93% suppression of itsM(T)/M(T = 250 K) slope and an 88% enhancement of the transition width(ΔTM/dTd). In the framework of the Bean–Rodbell model, suchtransition widening can be explained by the presence of an ever-wider TC distribution as a consequence of the increasedinduced microstrain (via tBM). Due tostrong magnetovolume coupling, both unit cell volume and magnetizationdependence on temperature show a remarkably similar evolution withBM time. In particular, the αV(T)peak decreases from ∼2210 (as-prepared)down to ∼ 260 ppm K–1 (after 180 min of BM),and the averaged αV in the temperaturerange from 250 to 350 K, –6 K–1 (as-prepared) to −98× 10–6 K–1 (BM180).In summary, BM leads toa significant reduction of the averageparticle and crystallite size and to a drastic linear enhancementof the induced microstrain in LaFex1–,Six)13-based compounds. Combining the method presented herewith the usual chemical control method, a synergistic effect is expectedtoward the development of a near-zero thermal expansion material.In addition, this La13 size-control study triggersan opening research window for other NTE materials with strong magnetovolumecoupling.This work demonstrates that the crystallite size and the ball-milled-inducedmicrostrain (in particular the latter) are remarkably efficient toolsto tune and customize the thermal expansion coefficient of La, smaller NTE particles are more effectivein tuning (decreasing) a composite thermal expansion than the largerones,