[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-122247-en":3,"doc-seo-122247-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},122247,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Cartesian atomic cluster expansion for machine learning interatomic potentials","Machine learning interatomic potentials enable rapid prediction of energies and forces from quantum-mechanical calculations, supporting large-scale exploration of materials and molecules. Atomic representations in Cartesian coordinates contain complete structural information but cannot be learned directly without enforcing symmetries such as translation, rotation, inversion, and atom permutations. Building on atomic cluster expansion and equivariant message passing, this work introduces a mathematically equivalent Cartesian alternative that avoids spherical harmonics while preserving rotational symmetry through Cartesian-coordinate operations. The resulting Cartesian Atomic Cluster Expansion (CACE) combines low-dimensional element embeddings, trainable radial channel coupling, and inter-atomic message passing, achieving strong accuracy, stability, and generalizability across bulk water, small molecules, and 25-element high-entropy alloys.","Cartesian atomic cluster expansion for machine learning interatomic potentials  \narXiv :2402 .07472v3 [physics .comp-ph] 30 Jul 2024  \nBingqing Cheng 1, 2, ∗  \n1 Department of Chemistry, University of California, Berkeley, CA, USA  \n2 The Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria (Dated: July 31, 2024)  \nMachine learning interatomic potentials are revolutionizing large-scale, accurate atomistic modelling in material science and chemistry. Many potentials use atomic cluster expansion or equivariant message passing frameworks. Such frameworks typically use spherical harmonics as angular basis functions, followed by Clebsch-Gordan contraction to maintain rotational symmetry. We propose a mathematically equivalent and simple alternative that performs all operations in the Cartesian coordinates . This approach provides a complete set of polynormially independent features of atomic environments while maintaining interaction body orders. Additionally, we integrate low-dimensional embeddings of various chemical elements, trainable radial channel coupling, and inter-atomic message passing. The resulting potential, named Cartesian Atomic Cluster Expansion (CACE), exhibits good accuracy, stability, and generalizability. We validate its performance in diverse systems, including bulk water, small molecules, and 25-element high-entropy alloys.  \nMachine learning interatomic potentials (MLIPs) can learn from quantum-mechanical calculations and predict the energy and forces of atomic configurations speedily, thus enabling more precise and comprehensive exploration of material and molecular properties at scale [1, 2] . Crucially, while atomic Cartesian coordinates encode all the essential information of a structure, they cannot be directly used in learning tasks, due to the lack of symmetries like translation, rotation, inversion, and atom permutations. Consequently, numerous methods have been developed to enforce these symmetries [2, 3] .  \nIn particular, atomic cluster expansion (ACE) [4] represents atomic environments based on a body-order expansion, using a complete set of bases of spherical harmonics and radial components. ACE can be viewed as a general framework of many other representations of the atomic environment, such as the Atom Centered Symmetry Functions (ACSF) [5], Smooth Overlap of Atomic Positions (SOAP) descriptor [6], Moment Tensor Potentials (MTPs) [7], and bispectrums [8] .  \nAnother important approach to learning atomic interactions uses message passing neural networks (MPNNs) . They represent structures as graphs with atoms as nodes, and apply message passing operations to learn atomic environment representations. Earlier MPNN models, such as SchNet [9], PhysNet [10], SphereNet [11] and GemNet [12], use internal features that are invariant under rotations. In models such as NewtonNet [13], EGNN [14], and PaiNN [15], vector features built using relative atomic positions are also used on top of invariant features. In later equivariant MPNNs, such as Cormorant [16], NequIP [17], MACE [18] exploit internal features based on irreducible representations of the E(3) symmetry group [19, 20] . E(3) equivariant MPNNs were able to achieve an unprecedented accuracy compared to  \n∗ [bingqingcheng@berkeley.edu](bingqingcheng@berkeley.edu)  \nthe previous invariant architectures [17, 18] .  \nUnder the hood, in both ACE and E(3) equivariant MPNNs, the key process is the following: Equivariant features or messages of body order ν are encoded in spherical harmonics with degrees l. The invariant features are subsequently created by contracting the equivariant ones via the Clebsch-Gordan coefficients. These invariants are used to predict energies or other invariant physical quantities. The use of spherical harmonics is motivated by that they forms a natural basis for the irreducible representations of the SO(3) group and thus is useful in operations involving rotational symmetries. Since the origi","cbCaif9lFNsh1Hlx","https://ap.wps.com/l/cbCaif9lFNsh1Hlx","pdf",1661592,1,13,"English","en",105,"# Background\n## Symmetry challenges in Cartesian learning\n## Atomic cluster expansion (ACE) and related descriptors\n## Message passing neural networks and E(3) equivariance\n# Proposed method\n## Cartesian Atomic Cluster Expansion (CACE) design\n## Feature construction in Cartesian space\n# Results\n## Benchmarks and evaluation focus","[{\"question\":\"Why can Cartesian atomic coordinates not be used directly for learning?\",\"answer\":\"Cartesian coordinates encode structural information, but learning needs invariance to symmetries like translation, rotation, inversion, and atom permutations. Without enforcing these symmetries, the model cannot generalize properly.\"},{\"question\":\"How does CACE relate to atomic cluster expansion (ACE)?\",\"answer\":\"CACE uses the same mathematical foundation as ACE, keeping the body-order expansion and completeness properties of atomic environment descriptions, while performing expansion and contraction directly in Cartesian coordinates.\"},{\"question\":\"What components are integrated to build the CACE potential?\",\"answer\":\"CACE combines low-dimensional embeddings of chemical elements, trainable radial channel coupling, and inter-atomic message passing to produce an accurate and stable potential.\"}]","Cartesian atomic cluster expansion for machine learning interatomic potentials | PDF",1785809624,33,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"cartesian-atomic-cluster-expansion-for-machine-learning-interatomic-potentials","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/cartesian-atomic-cluster-expansion-for-machine-learning-interatomic-potentials/122247/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why can Cartesian atomic coordinates not be used directly for learning?","Question",{"text":75,"@type":76},"Cartesian coordinates encode structural information, but learning needs invariance to symmetries like translation, rotation, inversion, and atom permutations. Without enforcing these symmetries, the model cannot generalize properly.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does CACE relate to atomic cluster expansion (ACE)?",{"text":80,"@type":76},"CACE uses the same mathematical foundation as ACE, keeping the body-order expansion and completeness properties of atomic environment descriptions, while performing expansion and contraction directly in Cartesian coordinates.",{"name":82,"@type":73,"acceptedAnswer":83},"What components are integrated to build the CACE potential?",{"text":84,"@type":76},"CACE combines low-dimensional embeddings of chemical elements, trainable radial channel coupling, and inter-atomic message passing to produce an accurate and stable potential.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]