Understanding and predicting properties of these complex systems present an ongoing and increasingly important challenge. Over the last few decades computer simulation of interfaces has become vastly more powerful and sophisticated. However, theoretical interface screening remains based on largely heuristic methods and is strongly biased to systems that are amenable to modelling within constrained periodic cell approaches. Here we present an unconstrained and generally applicable non-periodic screening approach for systematic exploration of material’s interfaces based on extracting and aligning disks from periodic reference slabs. Our disk interface method directly and accurately describes how interface structure and energetic stability depends on arbitrary relative displacements and twist angles of two interacting surfaces. The resultant detailed and comprehensive energetic stability maps provide a global perspective for understanding and designing interfaces. We confirm the power and utility of our method with respect to the catalytically important TiO Predicting structures and stabilities of solid-solid interfaces presents an ongoing and increasingly important challenge for development of new technologies. Here authors report an unconstrained and generally applicable non-periodic screening method for systematic exploration of material´s interfaces. Interfaces between different materials are also key to designing heterostructures for use in a range of modern applications . Here, the formation of well-ordered interfaces is achieved by controlled deposition of one semiconductor on the surface of another which is largely constrained by epitaxial matching5. Recently, top-down manipulation of two-dimensional (2D) materials has created a new class of layered heterostructures in which epitaxial constraints are less pronounced due to the relatively weak van der Waals interfacial interactions7. The resulting freedom to carefully tune the interfaces in such systems (e.g. relative in-plane twist angles of layers8) is highly promising for developing the next generation of 2D nanodevices and has already yielded spectacular new phenomena9. Although clearly playing a huge role in established and emergent technologies, interfaces are highly complex systems whose properties are typically difficult to predict and/or rationalise. Computational modelling is playing an increasingly important role in helping to analyse and understand interfaces. Recent methodological advances have tended to focus on approaches for searching for detailed low-energy atomic/electronic structures of selected interfaces13. However, given the huge number of possible ways in which two surfaces can interact, efficient and accurate screening of the energetic/structural landscape of viable interfaces is a pre-requisite for more in-depth investigations. Machine learning has been used to screen the structures and energies of metallic tilt grain boundaries, but required prior training with 10,000’s of calculated examples14. Here we address the screening challenge with a simple, powerful and direct modelling approach which, in principle, allows for rapid, unconstrained and systematic exploration of energies and structures of interfaces between arbitrary solid surfaces.From Bronze Age alloys to advanced ceramics, technological progress heavily relies on the development and use of polycrystalline materials whose performance is largely dictated by crystallite interfaces (i.e. grain boundaries)16. The use of CSLT-based methods can often provide a useful first step towards developing possible interface models. However, such heuristic geometric approaches have been criticised for not providing a realistic description of the complex array of chemical interactions at interfaces17. For the latter, computational modelling employing periodic atomistic and/or electronic structure calculations is often used. Periodic density functional theory (DFT) based calculations, for example, have proven to be extremely useful in providing a deeper understanding of some specific interfaces24. A drawback of such an approach is that one must define a common crystallographic cell containing both sides of the interface, which necessarily introduces interfacial strain due to the mismatch between the in-plane lattice parameters of each surface25. Such calculations are only physically reasonable when the cell-induced strain is relatively small (e.g. <3%), and methods have been developed to find supercells which minimise interfacial strain28. Interface stabilities can then be estimated by subsequently subtracting strain energies. Structural and physicochemical properties of such inherently mismatched interfaces are, however, less amenable to such post-correction. This overall approach is further limited to a subset of supercells with sizes small enough to comfortably compute using periodic DFT calculations (i.e. less than ∼1000 atoms). Constraints due to model-induced strain and practical tractability can thus lead to potentially relevant interfaces being overlooked in a CSLT/strain-dictated screened periodic DFT approach.Theoretical approaches to studying materials interfaces are typically based on first finding low-energy alignments of two infinitely repeating model surfaces. Historically, coincidence site lattice theory (CSLT) and related methods which aim to maximise favourable interfacial atomic overlap have been widely used for this taskHerein, we demonstrate a general non-periodic method to screen the structure and stability of interfaces without the need of heuristically constructing interfaces within the constraints of a common supercell approach. Generally, once two surfaces on either side of an interface plane are defined, four degrees of interfacial freedom remain: the in-plane relative twist angle and in-plane and out-of-plane relative displacements of the two surfaces. From a grain boundary perspective, the definition of the surfaces relative to an interfacial plane (often reduced to a single out-of-plane tilt angle in symmetric systems) together with the in-plane twist angle contribute to the overall misorientation between grains. In-plane and out-of-plane displacements are linked to grain boundary sliding and grain separation, respectively. Within a periodic approach, varying the surfaces constituting a symmetric grain boundary interface is relatively straightforward as the repeat cell is similar for a range of tilt angles. Within our non-periodic disk interface method, the analogous set of interfaces can be modelled, and moreover the tilt angle can be varied more continuously. For sampling in-plane twist angles a periodic approach is relatively more limited due to the above-noted constraints, while our disk-based approach can be applied for arbitrary twist angles. Unlike periodic models, our method can be applied to any two defined surfaces possessing arbitrary lattice parameters and can then provide a systematic survey of atomically accurate interfaces and their relative energetic stabilities for all possible combinations of relative in-plane twists and displacements.2). The interaction between surfaces of anatase TiO2 has recently been highlighted in the photoreduction of CO230. Here, we focus on the anatase (001)/(101) interface which has been used to help rationalise these experimental findings31. By systematically scanning structure versus relative energetic stability for this system, we were able to find a number of new stable interfaces for this system. We also highlight the increased accuracy and detail of our approach with respect to a geometric CSLT-type approach based on atom site overlap. Finally, we apply our method to a more challenging heterojunction between stable surfaces of the distinct anatase and rutile crystal polymorphs of titania. The anatase-rutile interface has attracted huge interest due to its potential importance in helping to explain the efficiency of titania based photocatalysts34.Generally, the disk interface method thus promises to help open the door to more comprehensive and systematic studies of interfaces in many systems . Specifically, the enabled global overview of interfacial possibilities could help in screening for design/optimisation of new interfaces by tuning twist angle/displacement. We confirm the predictive power of our method by applying it to interfaces between surfaces of titania  which are each terminated by one of the two surfaces in the z-direction. At this point, the structures of the extended surface slabs (i.e. lattice parameters and atomic positions) can be obtained from experimental data or accurate theoretical models. Note that the definition of the reference system does not imply that the surf1 and surf2 have a common periodic cell or the same symmetry. A disk is first cut from each surface slab such that the circular area of the respective surfaces on each disk is the same. Generally, each disk can be cut at an arbitrarily chosen displacement from the origin in the x–y plane with respect to in-plane 2D vectors csurf1 and csurf2, respectively. Formally, this is equivalent cutting a disk from surf1 at the origin and cutting the disk from surf2 at s = , where s = csurf1 – csurf2. To model interfaces corresponding to any in-plane shift vector s, we move the s-centred disk by −s =  so that the disks have a shared radial axis. Maintaining this common axis, the two disks are then oriented such that the surfaces are parallel and facing one another. In addition, for any choice of s, the disks can be each rotated separately about the axis through their radially aligned centres while keeping the interfacial area constant. As with the displacement, without loss of generality we can assume that one disk is fixed while varying the twist angle, α, of the second disk. To avoid spurious structural relaxation on the radially terminated sides of each disk we fix an outer ring of atoms to their positions in the original reference slabs. Although, for any specific calculation, this limits the degree of radial relaxation of each disk as a whole, one can cut disks from reference periodic slabs in which arbitrary constraints have been applied (e.g. isotropic/anisotropic in-plane strain) to take into account long-range structural adaptions of one surface to the other. In the present work we use reference systems based on fully relaxed periodic slabs. A schematic representation of how we form our disk interface models is shown in Fig. The aim of our method is to generate a set of accurate and comparable interface models for any two chosen surfaces in which the twist angle and relative displacement can be arbitrarily and systematically varied. To achieve this, we create models based on two equally sized disk-shaped slabs, each which exhibit one of the two surfaces to form the interface on a circular face. The disks are then brought close together so that the two surfaces on the circular faces directly oppose one another. An interface is then formed by relaxation of the interacting disk system. To construct our disk models, we take a reference system consisting of two independent z-direction and wide enough in the x–y plane to minimise finite-size effects. The calculated relative energetic stabilities of interfaces formed by any two interacting disks is calculated using the work of separation (Esep) – see “Methods”. A suitable disk thickness for each surface can be determined from periodic calculations of converged surface energies. It is important that any finite representation of the surfaces maintains at least these thicknesses throughout the whole interface. Hemispherical cuts from surfaces have been proposed to model energies of grain boundaries in metals with different misorientation angles35. The lack of uniform thickness and edge effects in such an approach would likely lead to significant finite effects if applied to more complex inorganic interfaces as studied herein. To avoid such problems very large hemispherical cuts would be required leading to significantly larger and more computationally expensive models relative to a disk approach.To obtain reasonably accurate and converged structures and relative energetic stabilities of the interfaces, the disks should be both thick enough in the In addition to scanning twist angles, one of the main features of the present disk-based method is the ability to scan relative energetic stabilities for relative displacements at any selected twist angle for two interacting surfaces free from concerns of variable thickness-based finite-size effects. For an interface between two infinite surfaces, displacing one surface with respect to a fixed second surface and the complementary case are equivalent. For finite systems, this symmetrical relation is not strictly true due to the truncated representation of the infinite translational periodicity of each surface. This finite-size effect can be minimised by using wider disks and its extent for any diameter can be estimated by performing two scans where each disk is displaced using the other disk as a fixed reference. Such complementary scans are used to refine all reported relative energetic stability data in both twist angle and displacement scans (see more details in Supplementary note s) and the relative radial twist angle (α) of the disks, we can systematically scan arbitrary surf1/surf2 interfaces without constraints of periodicity or symmetry. We note that out-of-plane displacement can also be freely varied in this approach but, for simplicity, in our example case studies the disk-disk separation is fixed in the constrained interfacial regions for each in-plane displacement and twist angle sampled.By varying the relative in-plane displacement (2 anatase (101)/(001) interface. To obtain the relaxed structures and relative energetic stabilities of our model interfaces, each corresponding disk interface system is evaluated using computational chemical modelling. Following previously reported periodic DFT modelling studies of the anatase (001)/(101) interface31 we use reference slabs and disks with ∼1.2 nm thicknesses (i.e. six atomic layers). To further ensure reasonably converged energies in this system, we also found that disks of 5 nm diameter were adequate . The usefulness and accuracy of computationally efficient empirically parameterised IP for modelling inorganic oxide interface was established in early IP-based periodic calculations38. Nowadays, as in our study, such IPs are more often used to obtain fairly accurate and efficient results prior to more detailed calculations10.To provide a concrete example of how our method works we first consider the TiO39 could also be used for potentially increased accuracy and would also allow general application of our approach to interfaces between materials for which parameterised IPs are not currently available. Herein, we use empirically parameterised IPs that have been confirmed to well describe relative energetics and structures of the bulk, surfaces and nanostructures of TiO243. For disks of infinite diameter, the interface would be perfectly periodic and relative shifts of the type s =  would yield symmetrically equivalent interface configurations with the same energy, where a1, a2, b1, b2 are the in-plane lattice parameters of surf1 and surf2 and A–D are integers  in non-periodic IP-based calculations.We note that machine-learned IPsEsep) of a different TiO2 anatase (101)/(001) interfaces with respect to systematically varying the shifting vector s over a grid of 625 points in the range 0 Å ≤ sx ≤ 5.6 Å and 0 Å ≤ sy ≤ 5.6 Å, for a fixed twist angle of α = 0° (i.e. no relative rotation). The map reveals a complex energy landscape with numerous distinct minima and maxima spanning an energy range of the order ∼0.8 J·m−2. As expected, the translation symmetries of the (001) and (101) surfaces are recovered within the range of shift vectors sampled (see Supplementary note y-direction with a distance close to 3.7 Å. This distance corresponds to the a lattice parameter of both the (001) and (101) anatase surfaces  within the resolution of our energy sampling (∼0.2 Å grid spacing). In the y-direction the energy map has a y repeat distance of 3.78 Å corresponding to the b lattice parameter of the anatase (001) surface. The b lattice parameter of the (101) surface is 10.24 Å which is outside of the displacement range sampled. Within this translationally symmetric quasi-square region  disk, with respect to the (101) disk about their common radial axis for each displacement. Figure α = 30°, 45°, and 90°, each based on 625 sampled interfaces. The main overall effect of the twist angle is to cause a global rotation of the Esep displacement map for α = 0°. From Fig. α = 0° case. This cursory analysis suggests that varying the shifting vector has a primary effect on determining the relative interface stability, while the contribution from varying the twist angle has a secondary, but still significant, effect.We now investigate the relative energetic stability of interfaces for Esep for the four interfaces corresponding to the four lowest energy minima reported in Fig. α. These data, shown in Fig. −2 with respect to changes in twist angle, which is approximately one quarter of the maximum energy variation found with respect to displacements. We note that the angular increment of 5° used in these twist angle scans is of a sufficiently high resolution for capturing the main angular dependent tendencies in relative stability /(100) interfaces predicted by the disk interface method with those reported in the literature.Our comprehensive relative energetic stability maps (relative to arbitrary combinations of displacement and twist angle) provide a detailed tool for further interface exploration. As generated, our maps of relative interface energies correspond to locally optimised structures  coming from bringing two surfaces in contact. For interfaces at points of interest on these maps one could then use constrained molecular dynamics or global optimisation methods to search for more stable reconstructed interface structures47 Comparing the DFT-optimised reference slabs with the corresponding ones obtained from our IP-based periodic calculations we find a very reasonable agreement between the optimised lattice parameters (see Supplementary note a and b vectors defining the (101) surface plane. This interface was derived by inspection to ensure good cation-anion matching. We note that the optimised periodic supercell containing this interface has lattice vectors a = 10.923 Å and b = 10.527 Å, which induce moderate lattice strains of ∼3%. To assess the consistency of this prediction with our non-periodic IP-based approach, we extracted one of the four near-degenerate lowest energy interface structures  and (001) surfaces using the hybrid HSE06 density functional . Previous studies have shown that DFT calculations can provide reasonably reliable electronic properties for both anatase surfaces and more demanding extended interface calculationsα = 90°  showing a good agreement between the two approaches. From the IP-based periodic model of our newly predicted α = 90° interface we also calculate its interface energy with respect to bulk anatase (see “Methods”) to be 1.02 J m−2. These results demonstrate that our method can be used to systematically screen for new stable interfaces. The analysis of the α = 90° interface structure suggests that its relative higher stability is due to a better matching of ions at the interface where nine Ti–O bonds form, as compared with the formation of eight Ti–O bonds when α = 45° (where we take the presence of bond to be indicated by a Ti-O distance in the range 1.9 - 2.2Å).To further asses the accuracy of our method with respect to providing accurate relative interfacial energetics, we take the interface structure corresponding to the M1 energy minimum for see Fig. . For thiα = 45° and α = 90° interface structures. In particular, we focus on the band edge alignment as this is an important quantity for photocatalytic applications49. The photoactivity of this system has been linked to a type-II alignment between at this interface, with both valence and conduction band edges in the (001) surface being ∼0.5 eV higher in energy than those of the (101) surface31. This alignment tends to promote the migration of photo-generated electrons to the (101) side of the interface and holes to the (001) side29. We calculated the band offsets of the fully optimised DFT interfaces for both α = 45° and α = 90°, by means of the potential line-up method51. For both 90° and 45° the TiO2 (101) band edges are lower in energy than those for the (001) surface, leading to a type-II junction /(001) surface junction.Next, we analyse the electronic properties of the DFT-optimised α = 0°. Our atom-based CS approach identifies potential interfaces based on maximising the degree of coincidence (or overlap) of atom sites in the two respective surfaces which are treated as rigid geometric lattices /(001) interface are even more pronounced for other angles  and rutile (110) surfaces. Modelling this interface would be very challenging for periodic DFT calculations due to the very high lattice mismatch involved. For the simplest aligned (i.e. α = 0°) system, the a and b lattice vectors of anatase (101) are approximately 25% and 60% larger than the corresponding values for rutile (110), implying that large supercells should be needed to minimise the mismatch. A (4 × 5) anatase (101) supercell could then be used to reasonably match a (5 × 8) rutile (110) supercell, corresponding to ∼1600 atom periodic interface model. This demanding calculation would provide a single interface structure for one displacement and one twist angle. To systematically scan both displacements and twist angles of this system we use a disk interface model based on a rutile disk exhibiting the (110) surface interacting with an anatase disk exhibiting the (101) surface for α = 0° (see “Methods”). Figure Esep displacement map for this system. Clearly the resulting displacement map is more complex than those for the anatase (101)/(001) interface . We first tested our method with respect to the anatase (101)/(001) interface which has relevance to photocatalysis. Detailed systematic scans of thousands of interface structures with respect to displacements and twist angle reveals highly detailed and complex stability maps, which cannot simply be rationalised by geometric atom-based CS considerations. DFT calculations on energetically stable interfaces extracted from these maps, confirms that the disk interface method can both reproduce previously reported interface models and predict new interface structures with relatively higher stabilities. Lastly, we applied our method to the challenging and scientifically/commercially important anatase/rutile TiOThe unconstrained and general nature of the disk interface method means that, in principle, it could be applied to a wide range of disparate interfacial systems. Due to its non-periodic character, the method is particularly well suited to complex non-symmetric interfaces between dissimilar materials and for the inclusion of interfacial trapped charges and defects. Future applications of the disk interface method have the potential to systematically screen as yet unexplored interfaces and provide a fuller and more detailed understanding of the interfacial energy landscapes of important existing systems.2 anatase (101)/(001) and anatase (101)/rutile (110) disk interface models each containing >4000 atoms. Interface models were constructed from two face-to-face nanodisks, each with a 5 nm diameter and a 1.2 nm thickness (anatase) or 1.8 nm thickness (rutile) and cut from respective optimised periodic slab calculations. For the present work all surfaces were assumed to be defect-free although we note that the method allows the inclusion of arbitrary defects. Scanning of different interfaces was achieved by choosing disks with differing relative in-plane displacements (i.e. sx and sy) and by rotating one disk with respect to the other with an in-plane twist angle (α)—see also Fig. 40 partially constrained relaxations employing the GULP54 code. Specifically, the atoms in the inner part of each disk (3.4 nm diameter) were optimised, while the coordinates of the atoms in outer region were fixed to their original bulk positions in the corresponding reference slabs. The constrained regions of the disks were kept at separated by a fixed interfacial vertical distance of about 2.2 Å during all the scans. Rigid body relaxations of both disks  after the scans resulted in negligible changes in the calculated energies, confirming the adequacy of this choice. Convergence and finite-size testing for the overall nanodisk set-up for the anatase (101)/(001) system can be found in Supplementary notes We created 1000 s of TiOEsep):A is the interfacial area. In this way it is easily possible to identify regions characterised by a favourable interfacing (negative Esep < 0) and regions for which interfacing is unlikely (Esep > 0). All reported Esep values have been refined by complementary scans :Eint should then be subtracted55. For our reported Eint values, we extracted large periodic slab models from our disk interface scans for a low-energy anatase (101)/(001) interface and a low-energy anatase (101)/rutile (110) interface. Due to the large size of the models and we optimised the structures and obtained all energies using our adopted IPs and the GULP code. The calculated lattice vectors and number of atoms for each periodic interface slab model are reported in Table In those cases where we extracted a periodic slab model of selected energetically stable interfaces, we also could evaluate the interface energy with respect to the corresponding bulk phases  range-separated hybrid density functional61, which has been found to provide a reliable description of the structural and electronic properties of semiconducting metal oxides63. Bulk anatase was modelled using a single unit cell and a 4 × 4 × 4 Monkhorst−Pack (MP)64k-point grid. 2D slab models were used to study the (101) and (001) anatase surfaces (1.2 nm thickness) and the (101)/(001) interfaces (2.6 nm thickness). A 4 × 4 × 1 MP k-point grid was adopted for the slabs, and for the interface models the MP grid was reduced to 1 × 1 × 1 (i.e. gamma point only) because of the relatively large cell dimensions. All slabs were separated from their periodic images in the out-of-plane z-direction by a vacuum space of ∼2 nm. All models were structurally optimised until forces on all atoms were less than 10−2 eV/Å. The periodic models were generated from suitable disk interfaces by: (1) inspecting the nanodisk interface structure, (2) identifying regions of space where atoms are characterised by a near-periodicity, and (3) applying periodic boundary conditions.Periodic DFT-based calculations using the VASP 6.1.0 codeFurther information on research design is available in the Supplementary InfoPeer Review FileReporting SummaryDescription of Additional Supplementary FilesSupplementary Data 1


Electrically conductive two-dimensional metal–organic frameworks have emerged as promising materials for electronic and energy storage devices, but their stacked nature offers limited accessibility to the framework pores. Now, pillaring a conductive 2D MOF is shown to enhance gravimetric capacitance by more than double. Most electrically conductive MOFs reported so far are however two-dimensional in nature owing to favourable stacking interactions between conjugated ligands. This stacking between sheets limits ion accessibility to the internal pores, leading to inefficient diffusion. Now, Jihye Park and colleagues from the University of Colorado Boulder in the US demonstrate that post-synthetic pillaring of 2D MOFs is an effective strategy to produce conductive 3D frameworks with accessible pores (10.1021/acsnano.1c10838)1.Electrically conductive metal2, but can post-synthetically accommodate 4,4’-bipyridyl pillar ligands in an octahedral coordination sphere  ligands−1 at 10 mV s−1, more than double that of pristine Cu–THQ. The researchers attributed this to the increased electrochemical surface area that results from enhanced ion accessibility to the framework pores.“We hypothesized that if the 2D sheet-stacked structure could be interrupted, we would gain insight into the structure–transport relationship”, comments Park. “Unlike post-synthetic pillar insertion demonstrated in conventional MOFs, we expected pillar insertion in conductive MOFs to serve as a chemical tool to manipulate charge transport pathways.” The team found that an increased interlayer distance between the sheets in the pillared MOF as compared with its 2D counterpart led to a decrease in bulk electrical conductivity. More encouragingly, however, electrochemical studies found the 3D MOF to display a gravimetric capacitance of 66.1 F g“Beyond our chosen platform, we expect pillar insertion to be a facile means to structurally diversify and chemically functionalize other existing 2D electrically conducting MOFs”, concludes Park. Indeed, when one considers the large pool of possible pillaring ligands and the increasing number of known conductive 2D MOFs, it is easy to imagine that opportunities to exploit this strategy will be numerous.


High−temperature wear failure has been a major challenge to die parts. This work provides a comprehensive study on the high−temperature wear performance of a TiC/H13 composite coating prepared by laser metal deposition (LMD). The microstructures of wrought H13 samples, LMD−processed H13 and TiC/H13 samples were systematically investigated. The refined martensite size, the uniform distribution of TiC ceramic particles, as well as their bonding with the matrix endowed the fabricated composite coating with superior hardness. The LMD−prepared TiC/H13 composite coating material demonstrated outstanding wear resistance when compared with other counterparts, mainly due to the high thermal stability and the load-transferring effect triggered by the introduced TiC ceramic particles. The dominated wear mechanism transition from severe ploughing in the wrought H13 material to mild delamination in the TiC/H13 composite coating was confirmed. The present study is expected to shed light on high-temperature wear-resistant coating material design and applications within the highly demanding mould industry. H13 steel has been widely applied in the mould industry for the fabrication of extrusion dies, casting dies, and hot forming dies, mainly due to its excellent impact toughness, good thermal fatigue properties, and superior high-temperature strength ,3. HowevTo improve the die surface wear resistance at elevated temperatures, extensive efforts have been put to coat a hardened layer by means of chemical vapour deposition (CVD), physical vapour deposition (PVD) and laser surface treatment ,6,7,8,9.Ceramic particles−reinforced metal matrix composites (MMCs) provide multiple benefits of high strength−to−weight ratio, good wear resistance, and superior stiffness, which have received enormous attention in surface treatment industries ,15,16. TIn this study, to evaluate the applicability of TiC ceramic particles for enhancing the high-temperature wear resistance of H13 steel, the TiC/H13 composite coating material was prepared by LMD. The role of the incorporated TiC particles on the microstructures and microhardness of H13 steel after the LMD process was systematically investigated. In comparison with the standard wrought H13 and the LMD−prepared H13 materials, the elevated temperature wear performance of the TiC/H13 composite material was studied, and the underlying wear mechanisms were elucidated in detail. The current work is expected to provide important scientific guidance for the application of LMD−prepared high−temperature wear-resistant coatings, and more importantly, shed light on future high−temperature wear-resistant coating material development for the highly demanding die industry.The applied H13 powder materials with a typical size range of 53–150 μm for the LMD process were prepared by argon gas atomization. To make the TiC/H13 composite powders, the TiC and H13 powders with a respective volume fraction of 10:90 were mixed in a planetary ball milling machine. As can be seen in 2. Good surface finishes were obtained on both the single−track and multi−track samples  was conducted on a Bruker D8 Advance diffractometer with Cu−Kα (λ = 0.15046 nm) radiation, and operated at 40 kV and 40 mA with a scanning 2θ range of 20–100°and a step size of 0.02°. The samples for microstructural observations were all cut along the vertical sections of the deposited materials and then prepared with standard metallography procedures. A Zeiss Gemini 300 scanning electron microscope (SEM) equipped with a Bruker EDS setup was applied for grain structure and carbide investigations. Electron backscattered diffraction (EBSD) images of the sample cross−sectional areas were revealed by an OXFORD Symmetry S detector installed in the SEM equipment. Moreover, the worn track surface areas of the sample after high−temperature wear tests were also analyzed using SEM equipment. The worn track surface was further reconstructed using a Keyence VHX−7000 3D optical profilometer.3N4 ceramic balls, and the friction contact form was then determined as ball−on−disk. Three sets of wear experiments were conducted for each polished sample to ensure the repeatability of the tested data. The high−temperature wear tests were all conducted at 600 °C with a normal load of 50 N and a sliding speed of 5 mm/s, and the total testing time was 15 min. Before wear testing, the samples and the wear counterparts were first heated to the target temperature and then maintained at this temperature for at least 10 min to ensure the whole sample reached the target temperature. The coefficient of friction (COF) was automatically recorded from the tribometer during testing. The wear depth and width were determined based on the analysis of the 3D worn-surface track morphology, while the wear volume was then calculated by integrating the cross−sectional areas of the worn tracks.The microhardness values along the sample depth direction were recorded on a Vickers 1000EG hardness tester under a loading force of 100 g and a loading time of 15 s. The high−temperature wear behaviours of the studied materials were evaluated on a Bruker UMT−TL REC−1000 °C tribometer in a reciprocal manner. The corresponding wear counterpart was made of SiThe XRD analyses were performed to reveal the presented phases in the prepared samples . It can 23C6, VC, and Mo2C) with sizes measuring from dozens to hundreds of nanometers, and such particles can be regarded as main strengthening contributors at elevated temperatures [SEM images further demonstrated the above gradual refinement trend of the martensite microstructures. The typically tempered martensite was observed in the H13−W sample, and higher−magnification images revealed the granular carbides and the retained austenite along the martensite structure boundaries a,d. In feratures ,30. On teratures . InteresThe microhardness value distributions along the cross-sections of the LMD−processed alloys are shown in Clearly, the markedly refined martensite lath resulting from the high cooling rate of the laser process contributed to the hardness improvements in the LMD-fabricated coatings to a great extent. On the other hand, the in situ−formed carbide particles with sizes below a few hundred nanometers of H13 steel would have been expected to provide a strong strengthening effect by inhibiting dislocation motion. Moreover, due to the good bonding behaviour between the TiC particles and the alloy matrix, the load−transferring effect offered by the ceramic particles further elevated the global strength of the composite coating material .The LMD−fabricated coating materials and the wrought H13 alloy were subjected to wear testing at an elevated temperature of 600 °C. −2 mm3, 1.7 × 10−2 mm3 and 0.87 × 10−2 mm3, respectively The LMD-prepared TiC/H13 composite coating possessed refined martensite microstructures compared to the H13−W and H13−L samples, which could be a result of the high-temperature gradient, high undercooling, as well as the introduced TiC particles. The TiC ceramic particles demonstrated a finer size, uniform distribution, and good bonding with the H13 steel matrix.(2)0.1 to 648 HV0.1, while its wear track depth/width (from 11 μm/556 μm to 4.8 μm/360 μm) and wear volume (from 2.08 × 10−2 mm3 to 0.87 × 10−2 mm3) decreased significantly compared to its wrought counterpart.The TiC/H13 composite coating exhibited a superior microhardness and high-temperature wear performance than its LMD- and wrought−processed H13 counterparts. The microhardness value for the composite TiC/H13 coating increased by nearly 30% more than the H13-W sample, e.g., from 508 HV(3)Compared with the wrought H13 sample, the high microhardness and the high thermal stability of the LMD−prepared TiC/H13 composite coating promoted the dominant wear mechanism transformation from ploughing in the H13−W sample to mild delamination in the TiC/H13−L sample.In this study, the high-temperature wear behaviours and wear mechanisms of LMD−prepared TiC/H13 coatings were comprehensively studied. The following conclusions can be drawn:Overall, this study demonstrated the effective high-temperature wear performance enhancement of LMD−prepared TiC/H13 composite coatings, and we expect the methodology applied in this study can provide some useful guidance in future wear−resistant die surface repair or remanufacture.


The spin Hamiltonian of this dimer allows a complete set of operations to act as a 64-dimensional all-electron spin qudit, or, equivalently, as six addressable qubits. Electron paramagnetic resonance experiments show that resonant transitions between different spin states can be coherently controlled, with coherence times TM of the order of 1 µs limited by hyperfine interactions. Coordination complexes with embedded quantum functionalities are promising building blocks for quantum computation and simulation hybrid platforms.Artificial magnetic molecules can host several spin qubits, which could then implement small-scale algorithms. In order to become of practical use, such molecular spin processors need to increase the available computational space and warrant universal operations. Here, we design, synthesize and fully characterize dissymetric molecular dimers hosting either one or two Gadolinium(III) ions. The strong sensitivity of Gadolinium magnetic anisotropy to its local coordination gives rise to different zero-field splittings at each metal site. As a result, the [LaGd] and [GdLu] complexes provide realizations of distinct spin qudits with eight unequally spaced levels. In the [Gd Molecular spin processors are promising for quantum computing, but for universal applicability the available computational space needs to be expanded beyond three qubits while retaining the ability to perform universal quantum operations. Here, the authors report dissymetric molecular Gadolinium(III) dimers acting as 6-qubit quantum processors. Quantized spin projections give rise to a discrete set of states that can be coherently manipulated by external magnetic field pulses. In addition, they are insensitive to electric noise and can therefore show longer coherence times than qubits based on electric (e.g. superconducting) circuits. Outstanding examples are magnetic defects in semiconductor hosts, such as NV− centres in diamond2 or P+ donors in silicon3. These systems are however not easy to tune, and wiring them up into a scalable architecture remains very challenging. A promising alternative to address these questions relies on the use of electron spins hosted by artificial molecules, i.e., organic radicals or transition-metal complexes6. This chemistry-based approach affords the synthesis of macroscopic numbers of identical qubits. The design of the metal coordination provides some control over the spin levels, which determine the qubit frequencies8. Furthermore, the spin coherence can be enhanced by either reducing the number of neighbouring nuclear spins or by engineering qubit states that are insensitive to magnetic field fluctuations. As a result, molecular spin qubit coherence times TM have shown very significant improvements over the past years13.Electronic spins in solids are natural candidates to encode qubits, the basic units of future quantum computers6. At the synthetic level, this implies a necessary dissymmetry between the different magnetic centres. Early successful examples in either homometallic15 or heterometallic17 lanthanide dimers have shown that conditions to realize both CNOT and √SWAP two-qubit gates can be fulfilled18, and allowed measuring the quantum coherence of a CNOT gate with TM of about 400 ns21. Also, modular supramolecular approaches able to incorporate multiple weakly coupled Cr7Ni qubits have been developed22 and used to implement C-PHASE two-qubit gates23. A second strategy is to profit from the internal spin levels of each magnetic centre to create d-dimensional (d > 2) quantum systems, that is, qudits. Examples include the coherent manipulation of the multiple nuclear spin states of 159TbPc2 25, Et4N[163DyPc2] 26 and 173Yb 27 mononuclear complexes. The former of these enabled the first realization of a three-level Grover quantum search algorithm in a single molecule, using tools of molecular electronics to read-out the qudit state28. Electronic spin qudits can also be realized in some cases, for instance by using the S = 7/2 spin manifold of a Gd(III) ion provided that it has a sufficiently low magnetic anisotropy to make all levels experimentally accessible29.But arguably the most appealing trait of this approach is that it offers the possibility of scaling up computational resources within each molecular unit. The most straightforward strategy is to incorporate several (two or more) magnetic centres, each of them realizing a qubit. Several spin carriers can be assembled in a rational way, such that their nature, disposition and interaction result in a set of addressable transitions that allow coherently manipulating all qubit states32, at the molecular scale. These applications demand increasing the dimension of the computational space beyond three qubits while retaining the ability to perform universal quantum operations33. Meeting both conditions in a molecule still represents a daunting challenge.The integration of multiple levels in well-defined complexes increases the density of quantum information handled by these systems and could allow embedding specific functionalities, say quantum error correction or some simple quantum algorithms and simulations2] molecule in which the two Gd(III) ions have a slightly different coordination15. The isolated properties of each Gd(III) d = 8 qudit are first determined through the isolation and study of the [LaGd] and [GdLu] analogue molecules, in which the Gd(III) ions occupy either of the two coordination sites present in [Gd2]. The results show that the [Gd2] molecule holds the promise to act as a six-qubit quantum processor or as a d = 64 electronic spin qudit.Here we show that it is possible to design and synthesize such building blocks via the combination of the two strategies outlined above. The rational design of a coordination complex with multiple and magnetically inequivalent Gd(III) ions would allow to scale up to six or more addressable qubits within one single molecule, with the potential of realizing different quantum gate operations. We focus here on the realization of this scaling-up in a dissymmetric [Gd3L forms a series of dissymmetric homometallic dilanthanide complexes of formula (Hpy)[Ln2(HL)3(NO3)(py)(H2O)], where H3L = 6-(3-oxo-3-(2-hydroxyphenyl)propionyl)-pyridine-2-carboxylic acid and py = pyridine, among which the [Gd2] compound studied here15. The two different Ln environments resulting from the different coordination pockets created by the three HL2− ligands (HL2− = double deprotonated H3L) and their disposition give rise to two metal sites with markedly different Ln–O and Ln–N bond lengths, one position being systematically larger than the other. This allows forming in a controlled manner heterometallic dilanthanide molecules, provided their ionic radii are sufficiently different34. This unique synthetic strategy was first used to isolate the [CeEr], [LaEr] and [CeY] analogues19. Here, in order to study each Gd(III) ion present in [Gd2], albeit isolated from the other, the [LaGd] and [GdLu] analogues are prepared in a similar manner. Note that the differences in ionic radii Δri, respectively of 9.4 and 7.7 pm34, are sufficient to expect the formation of homogeneous heterometallic single phases in both cases17. Thus, equimolar amounts of Gd(NO3)3 and either La(NO3)3 or Lu(NO3)3 were made to react with H3L in pyridine. The layering of the resulting clear yellow solutions with Et2O resulted in homogeneous phases of yellow crystals that were found by single-crystal X-ray diffraction to consist of (Hpy)[LaGd(HL)3(NO3)(py)(H2O)]·5py ([LaGd]) and (Hpy)[GdLu(HL)3(NO3)(py)(H2O)]·5py ([GdLu]).We previously showed that the ligand HP21/n, with the asymmetric unit coinciding with their formula and Z = 4  pockets and one/two  chelates respectively for site 1/2. The coordination sphere is completed by a nitrate ion on site 1 (La in [LaGd] and Gd in [GdLu]), and by one pyridine molecule and a water molecule on site 2 (Gd in [LaGd] and Lu in [GdLu]). In [LaGd], the coordinated nitrate is bidentate, resulting in a coordination number CN of 10 for the largest La(III) ion, while the rest have CN of 9. The intermetallic separations are respectively 3.8941(3) and 3.761(1) Å for La···Gd and Gd···Lu  and 2.432 Å (Gd in [GdLu]), than at site 2, 2.405 Å (Gd in [LaGd]) and 2.342 Å (Lu in [GdLu]). The differences in average bond distances ΔO between both sites are thus respectively 0.139 and 0.090 Å, larger than those observed in the homometallic analogues [La2] (ΔO = 0.045 Å) and [Gd2] (ΔO = 0.026 Å)14, an indication that the lanthanide ions have taken selectively the optimal relative positions following their respective ionic radii17.Compounds [LaGd] and [GdLu] crystallize in the monoclinic space group 36. Thus, the Gd2 site in [LaGd] is closest to an ideal capped antiprism (C4v symmetry) and to an ideal tricapped trigonal prism (D3h symmetry). The Gd1 site in [GdLu] is more irregular as indicated by distances to any ideal polyhedron 3–4 times larger. These differences are identical to those found between the two Gd sites in [Gd2]. The metal site composition is confirmed for both compounds by the unreasonable relative displacement parameters and worse final refinement obtained for any other distribution of metals within the molecular structure.The coordination sphere at both Gd sites also differs in shape, as indicated by continuous shape measures (HL)2]+ fragment. These fragments correspond to the molecular moiety in the absence of the terminal ligands, which are the only species detected by the technique of the dissociation equilibrium. In the case of [GdLu], a small impurity of [Gd2] is observed indicating a process of partial scrambling. Indeed, the dissociation of the terminal ligands may conduce to a relaxation of the structure and therefore to a decrease of the selectivity as it was shown by previous ESI-MS and density functional theory studies with compounds of the same family17. Nevertheless, the selectivity remains important and follows the relative Δri, which for [LaGd] and [GdLu], is relatively high. While the ESI-MS technique does not allow a quantitative analysis, since the [Gd2] impurity arises from the dissociated fraction  and the selectivity of the full molecule is that of the solid state, we are confident the amount of impurity in solution remains negligible. Therefore, the heterometallic molecular composition seen in the solid-state structure is maintained in solution, which is relevant for the studies of coherent spin dynamics done on diluted solutions. These studies confirm that any fraction of [Gd2] in the solution of [GdLu] is not detectable since this would have an effect on the quantum coherence in the opposite direction as that observed (see below).The molecular metal composition observed in the solid-state is confirmed by Electrospray Ionization Mass Spectrometry (ESI-MS), which also allows ensuring the integrity of the molecular structure in solution. Thus, the spectra obtained from solutions of [LaGd],  and [GdLu] are shown in Fig. d 2 Fig. .Fig. 2Ma40. Taking into account the low symmetry of the Gd coordination sites in these molecules, we have considered the simplest lowest-order spin HamiltonianD and E are zero-field splitting parameters and g is the electron spin g-factor. The results of these fits are compared to the experimental results in Fig. D = 2 GHz, ǀEǀ = 0.67 GHz and g = 1.99 for [LaGd] and D = 2.4 GHz, ǀEǀ = 0.8 GHz and g = 1.99 for [GdLu]. The fact that the orthorombicity ǀEǀ/D is close to the maximum value (1/3) probably arises from the low symmetry of both coordination sites. The line broadenings suggest that there are sizeable distributions in D and E, of about 60% for both compounds.In order to render these arguments quantitative, we have performed fits of the experimental spectra using the EPR simulation package EasySpinentclass1pt{minimacP/R is dominated by excitations of vibrational modes. This lattice contribution had been measured directly on the diamagnetic derivative [YLa]19, and agrees very well with the high-temperature behaviour. The additional anomaly observed, at H = 0, for both [LaGd] and [GdLu] below 2 K must then be associated with the finite splitting of the Gd(III) electronic spin levels. A nice feature of these results is that the position of the specific heat maximum provides a direct measure of the overall zero-field splitting41.Another experimental technique that provides information on the structure of electronic energy levels is heat capacity. Results obtained for the two molecular “monomers” are shown in Fig. D and E parameters derived from EPR experiments (and with the same D- and E- strains). A good agreement is also found for data measured under non-zero magnetic fields. Results for the two compounds become progressively closer to each other as H increases, due to the relatively weak magnetic anisotropy of Gd(III) and to the predominance of the Zeeman term in Eq.  ions coordinated in the molecular structures of [LaGd] and [GdLu] have low-lying electronic energy level schemes that provide a basis for two different spin qudits.2]. This complex hosts the two magnetic ions, in different coordination sites, whose properties in isolation have been discussed in the previous section. This discussion suggests that this molecule, with (2S + 1) × (2S+1) = 64 = 26 unequally spaced levels, provides a proper implementation of a d = 64 qudit or of six qubits. However, an additional necessary ingredient, related to the condition of universality that we discuss below, is the existence of a net coupling between the two spins.In this section, we turn our attention to the molecular dimer [Gd2] complex with those measured on [LaGd] and [GdLu]. Figure χT plots of [LaGd] and [GdLu] agree with the predictions for isolated Gd(III) ions , the data for [Gd2] show a decrease below ~4 K. This behaviour is compatible with the existence of a weak isotropic coupling described by the following spin Hamiltonian1 and H2 are the spin Hamiltonians of each isolated Gd(III) ion, given by Eq.  or in terms of the states of six qubits . However, this condition, i.e., that the space dimension is large enough, is not sufficient to ensure that such a small “processor” could perform universal quantum operations. The condition of universality implies that any quantum superposition of the basis states (the 64 mentioned above) can be generated starting from any initial state, e.g., starting from the system initialized in its ground state. It implies that there exist non-forbidden transitions between different levels, which can be univocally addressed by setting the frequency of a microwave pulse or the external magnetic field, and which form a complete set, in the sense, defined above, of allowing “visiting” all possible spin states.The results described in previous sections show that Eq.  provides−1, which link the ground state with any other basis state. In order to perform a more rigorous demonstration, we have adapted the mathematical proof for universality42, based on the formalism of Lie algebras, to this particular situation. Figure 2]. Applying the same mathematical method, we find that the Hamiltonian given by Eq. . The reason for this condition, which we anticipated above, is that conditional operations between states of the two Gd(III) ions can be implemented with simple resonant pulses if and only if each of them has some anisotropy and the two are magnetically coupled.Figure 2] provides a suitable platform for a six-qubit quantum processor, in the sense of having the right Hilbert space and a sufficient number of allowed transitions. However, it also sets a final stringent condition, namely that transitions between these levels can be implemented coherently. The latter depends on the spin-coherence and spin-relaxation times, which we discuss in this section. The spin dynamics has been experimentally studied, at 6 K, on diluted solutions of [LaGd], [GdLu] and [Gd2] in MeOH-d4:EtOH-d6, with concentrations in the range 0.38–0.61 mmol L−1. As is well known9, the use of deuterated solvents reduces the hyperfine couplings with the solvent nuclear spins and enhances the electronic spin coherence. The comparison with results measured for a sample of [GdLu] diluted in a non-deuterated solvent mixture, shown in Supplementary Fig. The previous section shows that [Gd2] present measurable quantum coherence over the full magnetic field range, thus supporting the picture discussed in the previous section. The echo intensity varies with H, which allows obtaining echo-induced EPR spectra. These are in good agreement with the cw spectra, although the presence of a strong modulation of the echo decay (see below) results in variations with the time interval between pulses  separated by a varying interval τ. For all three compounds, the ESE intensity decreases in a similar manner with increasing τ, the decay being slightly more rapid in the case of [Gd2]. The ESE decay exhibits a strong modulation with a frequency that increases with H, independently of the compound. To quantify these effects, the ESE decays were modelled  ions feel decoherence sources affecting either site 1 or site 2. As it happens with other properties  refocused by a π pulse, with τ intervals (100 ≤ τ ≤ 200 ns) between the two pulses and between the refocusing pulse and ESE detection. Representative results are shown in Fig. 2], and in the supplementary material .In this work, we have designed a molecular structure that can host two Gd(III) ions in distinct coordination environments. Combined with the intrinsic properties of this ion, in particular, its 46, thus implementing simple algorithms. Of special relevance would be the correction of errors, e.g. phase errors that are likely to be most disturbing in the case of spin qubits, at the molecular scale30. For this, it is not even necessary that the dimension of the Hilbert space defined by the available spin states be a power of two. These molecular NISQS can also serve as a suitable basis where to map quantum simulations of simple systems, such as small molecules47. An advantage of using a single physical unit to perform these simple computations is that it reduces the number of non-local operations, which require switching on and off interactions between different parts of a quantum circuit, and that are often more sensitive to decoherence and, therefore, error32.The integration of a high number of addressable quantum states in well-defined molecular units represents one of the distinctive traits of the chemical approach to quantum technologies and can provide a number of competitive advantages in this field. Such molecules can locally act as tiny quantum processors themselves, the equivalent of NISQS (Noisy Intermediate Size Quantum Systems) in other schemes50. In the end, the ideal situation would be to explore the response of individual molecules, either via the application of single-molecule electronics28 or by enhancing the current sensitivity of magnetic spectroscopic techniques54.The fulfilment of this potential defines also a series of important challenges. A prominent one is how to individually address each of the relevant transitions. This requires working on well-oriented molecules, to avoid the difficulties discussed above in connection with nutation experiments and decreasing significantly the linewidth of each resonance. Even though coherence times are reasonably long in these systems, the resonance lines show a relatively large inhomogeneous broadening, which mainly arises from the random orientation of molecules in frozen solutions and from the existence of some distributions in the parameters that define the spin Hamiltonian (exchange and anisotropy constants). These difficulties can possibly be overcome by synthesizing well-organized molecular frameworks, in which all molecules need to be oriented in the same manner and, at the same time, magnetically diluted or sufficiently far away from each otherd = 64 qudit, the resonant transitions that provide the basic set of operations suffer already from a “frequency crowding”, which would seriously hinder addressing individual resonances for qudits of increasing dimension. Therefore, scaling beyond this point must proceed by linking different units via coherent mediators. Implementing such switchable couplings within molecules  remains a challenging goal56. The alternative is to combine a chemical, bottom up, approach with the integration of functional molecules into solid-state circuits. A recent proposal for a scalable architecture considers the possibility of applying superconducting circuits, namely on-chip resonators, to control, read-out and communicate magnetic molecules58. A further advantage of these devices, and of dealing with electronic spins having sizeable energy splittings, is that they are compatible with working at very low temperatures, which are necessary to initialize the qudit to its ground state (its population becomes larger than 99.99% for T < 0.1 K and a magnetic field of 0.5 T). Within this scheme, medium size molecular qudits, able to embed some critical functionalities such as phase error correction, would provide very attractive building blocks to reach computational performances difficult to match by other platforms.A second challenge is how to move forward beyond the level of a few qubits. As it becomes clear by inspecting the energy level scheme of a 6-(3-oxo-3-(2-hydroxyphenyl)propionyl)pyridine-2-carboxylic acid (H3L). The ligand H3L was synthesized as described previously14.(Hpy)[Gd2(HL)3(NO3)(py)(H2O)]·5py ([Gd2]). Compound [Gd2] was synthesized as described previously15. Purity was checked by elemental analysis, mass spectrometry and multiple cell determinations on single crystals, including full structure determination.(Hpy)[LaGd(HL)3(NO3)(py)(H2O)]·5py ([LaGd]). A 10 mL pyridine solution of ligand H3L  is added slowly to a 10 mL pyridine solution of La(NO3)3·6H2O  and Gd(NO3)3·6H2O . The mixture is stirred at room temperature for one hour, yielding a clear shiny yellow solution that is layered with diethylether. Crystals of [LaGd] form slowly and are recovered after one weak in a 63 % yield. ESI-MS: [LaGd(HL)2(H2L)]+m/z = 1146.98. Elemental analysis found  for [LaGd(H2L)(HL)2(NO3)(py)(H2O)]·3H2O: C 44.11 (44.16); H 2.77 (3.04); N 4.87 (5.15). IR : 3410 mb, 1618s, 1583 s, 1560 m, 1528 s, 1463 m, 1400 s, 1384 s, 1298 m, 1239w, 1204w, 1148w, 1120w, 1059 m, 949 m, 891w, 760 m, 706 m, 663 m, 635w, 568w.(Hpy)[GdLu(HL)3(NO3)(py)(H2O)]·5py ([GdLu]). A 10 mL pyridine solution of ligand H3L  is added slowly to a 10 mL pyridine solution of Gd(NO3)3·6H2O  and Lu(NO3)3·6H2O . The mixture is stirred at room temperature for one hour, yielding a clear yellow solution to which 200 μL conc. HNO3 is added. After 10 more min stirring, the solution is layered with diethylether. Crystals of [GdLu] are recovered after 2 weeks in a 53% yield. ESI-MS: [GdLu(HL)2(H2L)]+m/z = 1183.02. Elemental analysis found  for [GdLu(H2L)(HL)2(NO3)(py)(H2O)]·6.7H2O: C 40.33 (41.06); H 2.56 (3.34); N 5.50 (4.79). IR : 3422 mb, 1619s, 1585 s, 1560 m, 1528 s, 1465 m, 1401 s, 1384 s, 1325 m, 1299 m, 1239w, 1208w, 1147w, 1121w, 1058 m, 950 m, 892w, 756w, 708 m, 666w, 637w, 569w.Magnetic measurements were performed using a Quantum Design SQUID MPMS-XL magnetometer through the Physical Measurements unit of the Servicio de Apoyo a la Investigación-SAI, Universidad de Zaragoza. All data were corrected for the sample holders and grease contributions, determined empirically as well as for the intrinsic diamagnetism of the sample, estimated using Pascal constants.T = 0.35 K, with a commercial physical property measurement system  that makes use of the relaxation method. The samples, in powder form, were pressed into pellets and placed onto the calorimeter on top of a thin layer of Apiezon N grease that fixes the sample and improves the thermal contact. The raw data were corrected from the known contributions arising from the empty calorimeter and the grease.Heat capacity data were measured, down to Continuous-wave (cw) EPR measurements were performed with a Bruker Biospin ELEXSYS E-580 spectrometer operating in the X-band and Q-band. Solid-state cw-EPR measurements were performed at RT on polycrystalline samples placed in quartz tubes.2]) mmol L−1 solutions. Measurements on 0.13 and 0.08 mmol L−1 solutions of [GdLu] were also performed to discard any variation of T1 and TM in the range of concentrations used. Additional TD-EPR measurements were performed on a 0.38 mmol/L solution of [GdLu] in a 1:1 mixture of non-deuterated methanol and ethanol, in order to show the effect of the solvent nuclear spins on the spin coherence times and on the modulation of the ESE signals measured after 2-pulse and 3-pulse sequences. Finally, cw-EPR measurements were also done on the same frozen solutions, albeit in the range 20–80 K due to signs of saturation at 6 K.In addition, pulsed time domain (TD) measurements were performed at X-band frequencies. In these experiments, the typical widths of the π/2 and π pulses were 16 and 32 ns, respectively. In order to avoid unwanted echoes, a 2-step (4-step) phase cycle was used in the 2-pulse (inversion recovery and three-pulse) experiments. The high power microwave excitation was obtained by using a TWT amplifier. A dielectric low Q cavity from Bruker was used as resonator. TD-EPR measurements were done on frozen solutions, for which gas-flow Helium cryostats were used. The polycrystalline solids were dissolved in a 1:1 mixture of fully deuterated methanol and ethanol. The use of deuterated solvents is intended to limit decoherence due to protons. The TD-EPR experiments were performed at 6 K on 0.61 ([LaGd]), 0.38 ([GdLu]) and 0.44 (, remaining voids in the structure with no significant electron density peaks were analysed and taken into account with PLATON SQUEEZE62 that recovered a total of 224 electrons per cell, over four equivalent voids of 193 Å3. These figures being reasonable for one diffuse pyridine solvent molecule per void, i.e., one per formula unit, this has been reflected in the reported formula. For both structures, the metal site composition was confirmed by refining the structure with the two homometallic situations as well as with the Ln sites inverted. These resulted in relatively poorer agreement factors and most importantly in unrealistic combinations of Ueq values at the metal sites.Data for compounds [LaGd] and [GdLu] were collected at 100 K on a Bruker APEX II QUAZAR diffractometer equipped with a microfocus multilayer monochromator with MoKα radiation  or not, we write the spin Hamiltonian as,B1(t) is a time-dependent external magnetic field that induces resonant transitions between different eigenstates of the spin Hamiltonian d of the Hilbert spaces of all spin qudits considered in this work is d = 2N, with N = 3 for [LaGd] and [GdLu], and N = 6 for [Gd2], these eigenstates may be denoted as either |n〉  or as |00…0〉 to |11…1〉. If all of these states are accessible from any other one, the qudit realizes a N-qubit processor.Here n〉 and |m〉, by making the frequency ω of the driving magnetic field B1(t) resonant with the transition frequency One way to check universality is as follows. We use the fact that the energy spectrum has some nonlinearity, i.e., that the levels are not simply equidistant as those of a harmonic oscillator. This arises from the combination of single-ion anisotropy, the dissymmetry between the two Gd(III) coordination sites and the mutual interaction between the two Gd(III) spins. This means that we can address a transition between any two states, say |L will not cover all the necessary transitions. The natural question is then which extra transitions can be implemented by concatenating the different elements of L. The formal answer42 to this question is that the accessible transitions are those belonging to the Lie algebra L. This is a natural consequence of the Lie-Trotter formulas, i.e.It is however expected that L are also elements of the corresponding Lie group, Hnm by expressing gμBB1(t)S in the basis of eigenstates of The different commutators of the elements of We finally notice that the proof says nothing about how to perform operations. It only checks if they are possible.n〉 S, the system is presumably not universal. For example, a parity selection rule can impose that only transitionsn〉, the odd states cannot be accessed, and viceversa. This condition might introduce difficulties in situations that are closer to that described by the spin Hamiltonian of [Gd2]. The isotropic Heisenberg model, S. Therefore, a spin dimer described by this model is definitely not universal for an external driving gμBB1(t)S, with S = S1 + S2. Transitions between different total spin states are forbidden. Fortunately, in [Gd2] the different anisotropy terms acting on each spin break this symmetry, besides providing a non-equidistant level spectrum. However, contrary to what happens in the single qudit case, the dc magnetic field H cannot be much higher that J, D and E. The reason is twofold. If H is strong enough, the two Gd(III) ions are effectively decoupled. At the same time, the anisotropy terms become less important and the system is closer to an isotropic Heisenberg system. Our numerical calculations, whose results are shown in Fig. −1, we construct contour plots (Fig. J = 0 or increasing the dc-external field. The conclusion is that [Gd2] can, under the appropriate conditions, simulate either a six-qubit processor or two decoupled three-qubit systems.Putting together several qudits, they must interact as a prerequisite to be universal. The Gd dimer isots Fig.  of thoseSupplementary InformationDescription of Additional Supplementary FilesSupplementary Data 1Supplementary Data 2Peer Review File


In our study, we tested whether spontaneous gene deletion and gene duplication rates are also high in Dictyostelium discoideum, a eukaryote with among the lowest point mutation rates (2.5 × 10–11 per site per generation) and an AT-rich genome (GC content of 22%). We calculated mutation rates of gene deletions and duplications using whole-genome sequencing data originating from a mutation accumulation experiment and determined the association between the copy number mutations and GC content. Overall, we estimated an average of 3.93 × 10–8 gene deletions and 1.18 × 10–8 gene duplications per gene per generation. While orders of magnitude greater than their point mutation rate, these rates are much lower compared to gene deletion and duplication rates estimated from mutation accumulation lines in other organisms (that are on the order of ~ 10–6 per gene/generation). The deletions and duplications were enriched in regions that were AT-rich even compared to the genomic background, in contrast to our expectations if low GC content was contributing to low mutation rates. The low deletion and duplication mutation rates in D. discoideum compared to other eukaryotes mirror their low point mutation rates, supporting previous work suggesting that this organism has high replication fidelity and effective molecular machinery to avoid the accumulation of mutations in their genome.The study of spontaneous mutation rates has revealed a wide range of heritable point mutation rates across species, but there are comparatively few estimates for large-scale deletion and duplication rates. The handful of studies that have directly calculated spontaneous rates of deletion and duplication using mutation accumulation lines have estimated that genes are duplicated and deleted at orders of magnitude greater rates than the spontaneous point mutation rateThe online version contains supplementary material available at 10.1007/s00239-022-10081-1. Mutations induce genetic variation that underlies the evolution of all organisms. The mutational rate and fitness effects of spontaneous mutations are therefore important to understand evolution and heritable diseases , P. tetraurelia (28%) and D. discoideum (22%) are substantially lower than C. reinhardtii (64%), P. tricornutum (49%), D. pulex (40%), D. melanogaster (42%), S. cerevisiae (38%), and C. elegans (35%), with the exception of P. falciparum (20%). While it is possible that eukaryotes with AT-rich genomes have more efficient DNA repair or encounter fewer DSBs, there is not enough data on how GC content relates to large mutations, such as deletions and duplications.Amongst eukaryotes, the single nucleotide mutation rates calculated from MA experiments range between ~ 8 × 10S. cerevisiae at 5.5 × 10–6  or non-homologous end joining (NHEJ), which is more error-prone. The DSB repair pathway choice is therefore essential for maintaining genome integrity  that is also rich in simple sequence repeats (SSRs), making up over 14% of their 34 Mb genome  experiment  or greater than 1.5x (putative duplication) of the mean read depth. The resulting CNVs from all CNV callers were visually inspected with Integrative Genome Viewer (IGV) , where m is the number of observed mutations, n is the total number of genes in the genome, and T is the number of generations. Rates were calculated separately for complete gene CNVs and partial gene CNVs. The rate of deletions and duplications per genome per generation were also calculated.To determine genes that were within the putative deletions and duplications, annotated gene coordinates were assessed for overlap with CNVs using BEDTools. A complete gene deletion or duplication was assigned when an entire gene was encompassed in a CNV, while a partial gene CNV was assigned when only a portion of a gene overlapped with a CNV. Only annotated protein-coding genes were kept for calculating mutation rates. The gene deletion and gene duplication mutation rates (μ) were calculated for each MA line using the equation μ = https://cci-git.uncc.edu/wsung/ssrsearch) from a previous analysis of D. discoideum SSRs of the same MA samples  was calculated using BEDTools to determine whether GC content differed from the genomic background. The occurrence of simple sequence repeats (SSRs) among CNVs and their breakpoints was also analyzed to test whether CNVs overlapped SSRs more frequently than expected by chance. SSRs were determined using perl scripts . Most MA lines did not have any deletion or duplication , while 4 MA lines had more than one deletion and/or duplication. The maximum number of CNVs that occurred in a single MA line was four (MA line L54 had four deletions) and one MA line (QS50) had two duplications. The average deletion length was 3,305 bp, and the average duplication length was 4,191 bp. There was no effect of sequencing coverage on the number of deletions and duplications detected among MA lines , and no CNVs were detected in the region on chromosome 2 where duplications are frequently found among strains .Across all 37 MA lines and an average of ~ 1,500 generations, we identified a total of 18 deletions and 3 duplications (Table D. discoideum is gene-dense with about 68% covered by protein-coding genes. All but one of the CNVs overlapped at least one gene (Suppl. Table 2). There was a total of 40 different genes encompassed among the CNVs (including 2 non-coding tRNA genes). Based on random permutations of CNV regions across the genome, the observed number of CNVs overlapping genes (percentile = 0.648) and the number of protein-coding genes in CNVs (percentile = 0.079) were not significantly different than chance alone. Of the 38 protein-coding genes in CNV regions, there were 32 gene deletions , and 6 gene duplications . When considering only complete gene CNVs, the gene deletion and duplication rates were 2.24 × 10–8 and 5.91 × 10–9 per gene/generation, respectively. The partial gene deletion and duplication rates were similar to complete gene deletion and duplication rates, at 1.69 × 10–8 and 5.91 × 10–9 per gene/generation, respectively (Suppl. Table 1). Given the low rates recovered compared to other species, we investigated whether these were influenced by conservative CNV filtering (see Methods), in case we filtered out true positives along with our true negatives. We recalculated gene deletion and duplication rates prior to manual filtering, taking all deletions and duplications detected by the three CNV detection tools that were within the same length distribution as our verified deletions and duplications (below 20 kb). This analysis recovered a total of 72 complete gene deletions and 46 complete gene duplications, which resulted in an average of 9.23 × 10–8 gene deletions and 5.90 × 10–8 gene duplications per gene/generation. We also recalculated gene deletion and duplication rates after excluding reads with low mapping quality (MAPQ < 10) and only retaining genes that had > 20 × average coverage, but this had negligeable effects on our estimates resulting in a slightly higher combined rate of 5.40 × 10–8 gene deletions and duplications per gene per generation. We found neither a correlation between the CNV rate in a MA line and its base substitution rate  nor its indel rate , based on published small-scale mutation rates from the original MA study  and compared to genes (p = 2.3e-5), despite the already AT-rich genome , if we only analyzed 50 bp upstream and downstream of the CNV breakpoints (p = 0.006 compared to the genomic background), or if we only included gene CNVs (p = 0.016). We also found that CNVs and CNV breakpoints both overlap fewer simple sequence repeats (SSRs) than expected by chance (both p < 0.001), in contrast to point mutations that were found to preferentially occur in SSRs , raising the possibility that this high AT content contributes to a lower mutation rate. The GC content has been shown to have a notable impact on the probability of mutations  deletion and (B) duplication called by CNVnator, with the CNV region highlighted in the red box. Supplementary file1 (PDF 1907 KB)Example view in IGV of a deletion called by Delly, with the CNV region highlighted in the red box. Supplementary file2 (PDF 2594 KB)Example view in IGV of a (A) deletion and (B) duplication called by Manta, with the CNV region highlighted in the red box. Supplementary file3 (PDF 1247 KB)Supplementary file4 (XLSX 23 KB)Below is the link to the electronic supplementary material.