{"id":"https://openalex.org/W7216209447","doi":"https://doi.org/10.26434/chemrxiv.15006158/v2","title":"Symmetry Numbers: A Flexible Approach for Molecules and Transition States","display_name":"Symmetry Numbers: A Flexible Approach for Molecules and Transition States","publication_year":2026,"publication_date":"2026-10-02","ids":{"openalex":"https://openalex.org/W7216209447","doi":"https://doi.org/10.26434/chemrxiv.15006158/v2"},"language":"en","primary_location":{"id":"doi:10.26434/chemrxiv.15006158/v2","doi":"https://doi.org/10.26434/chemrxiv.15006158/v2","is_oa":true,"landing_page_url":"https://doi.org/10.26434/chemrxiv.15006158/v2","pdf_url":"https://chemrxiv.org/doi/pdf/10.26434/chemrxiv.15006158/v2","source":{"id":"https://openalex.org/S4393918830","display_name":"ChemRxiv","issn_l":"2573-2293","issn":["2573-2293"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://chemrxiv.org/doi/pdf/10.26434/chemrxiv.15006158/v2","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5018194960","display_name":"Maxim Papusha","orcid":"https://orcid.org/0009-0000-8580-4777","observed_orcids":["https://orcid.org/0009-0000-8580-4777"]},"institutions":[{"id":"https://openalex.org/I887968799","display_name":"RWTH Aachen University","ror":"https://ror.org/04xfq0f34","country_code":"DE","type":"education","lineage":["https://openalex.org/I887968799"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Maxim Papusha","raw_affiliation_strings":["Institute of Technical Thermodynamics","RWTH Aachen University"],"raw_orcid":"https://orcid.org/0009-0000-8580-4777","affiliations":[{"raw_affiliation_string":"Institute of Technical Thermodynamics","institution_ids":[]},{"raw_affiliation_string":"RWTH Aachen University","institution_ids":["https://openalex.org/I887968799"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033642875","display_name":"Andreas V. Copan","orcid":"https://orcid.org/0000-0002-8682-0813","observed_orcids":["https://orcid.org/0000-0002-8682-0813"]},"institutions":[{"id":"https://openalex.org/I165733156","display_name":"University of Georgia","ror":"https://ror.org/00te3t702","country_code":"US","type":"education","lineage":["https://openalex.org/I165733156"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Andreas V. Copan","raw_affiliation_strings":["College of Engineering","Department of Chemistry","University of Georgia"],"raw_orcid":"https://orcid.org/0000-0002-8682-0813","affiliations":[{"raw_affiliation_string":"College of Engineering","institution_ids":[]},{"raw_affiliation_string":"Department of Chemistry","institution_ids":[]},{"raw_affiliation_string":"University of Georgia","institution_ids":["https://openalex.org/I165733156"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071330264","display_name":"Brandon M. Rotavera","orcid":"https://orcid.org/0000-0003-0533-7131","observed_orcids":["https://orcid.org/0000-0003-0533-7131"]},"institutions":[{"id":"https://openalex.org/I165733156","display_name":"University of Georgia","ror":"https://ror.org/00te3t702","country_code":"US","type":"education","lineage":["https://openalex.org/I165733156"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Brandon Rotavera","raw_affiliation_strings":["College of Engineering","Department of Chemistry","University of Georgia"],"raw_orcid":"https://orcid.org/0000-0003-0533-7131","affiliations":[{"raw_affiliation_string":"College of Engineering","institution_ids":[]},{"raw_affiliation_string":"Department of Chemistry","institution_ids":[]},{"raw_affiliation_string":"University of Georgia","institution_ids":["https://openalex.org/I165733156"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5050082199","display_name":"Kai O. Leonhard","orcid":"https://orcid.org/0000-0001-6231-6957","observed_orcids":["https://orcid.org/0000-0001-6231-6957"]},"institutions":[{"id":"https://openalex.org/I887968799","display_name":"RWTH Aachen University","ror":"https://ror.org/04xfq0f34","country_code":"DE","type":"education","lineage":["https://openalex.org/I887968799"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Kai Leonhard","raw_affiliation_strings":["Institute of Technical Thermodynamics","RWTH Aachen University"],"raw_orcid":"https://orcid.org/0000-0001-6231-6957","affiliations":[{"raw_affiliation_string":"Institute of Technical Thermodynamics","institution_ids":[]},{"raw_affiliation_string":"RWTH Aachen University","institution_ids":["https://openalex.org/I887968799"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10002","display_name":"Advanced Chemical Physics Studies","score":0.6801999807357788,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10002","display_name":"Advanced Chemical Physics Studies","score":0.6801999807357788,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.08889999985694885,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12527","display_name":"Chemistry and Stereochemistry Studies","score":0.061400000005960464,"subfield":{"id":"https://openalex.org/subfields/1605","display_name":"Organic Chemistry"},"field":{"id":"https://openalex.org/fields/16","display_name":"Chemistry"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/symmetry-number","display_name":"symmetry number","score":0.9829999804496765},{"id":"https://openalex.org/keywords/automorphism-groups","display_name":"automorphism groups","score":0.6510000228881836},{"id":"https://openalex.org/keywords/molecular-graphs","display_name":"molecular graphs","score":0.5590000152587891},{"id":"https://openalex.org/keywords/stereochemistry","display_name":"stereochemistry","score":0.6000000238418579},{"id":"https://openalex.org/keywords/transition-state","display_name":"transition state","score":0.7490000128746033},{"id":"https://openalex.org/keywords/fluxional-molecules","display_name":"fluxional molecules","score":0.8500000238418579},{"id":"https://openalex.org/keywords/pseudorotation","display_name":"pseudorotation","score":0.8270000219345093},{"id":"https://openalex.org/keywords/thermochemistry","display_name":"thermochemistry","score":0.48100000619888306},{"id":"https://openalex.org/keywords/reaction-kinetics","display_name":"reaction kinetics","score":0.7549999952316284}],"concepts":[{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.6500999927520752},{"id":"https://openalex.org/C5534955","wikidata":"https://www.wikidata.org/wiki/Q7661343","display_name":"Symmetry operation","level":3,"score":0.5187000036239624},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5126000046730042},{"id":"https://openalex.org/C33026886","wikidata":"https://www.wikidata.org/wiki/Q593950","display_name":"Rotational symmetry","level":2,"score":0.4690999984741211},{"id":"https://openalex.org/C32909587","wikidata":"https://www.wikidata.org/wiki/Q11369","display_name":"Molecule","level":2,"score":0.43529999256134033},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.43309998512268066},{"id":"https://openalex.org/C22511879","wikidata":"https://www.wikidata.org/wiki/Q7661339","display_name":"Symmetry number","level":5,"score":0.42559999227523804},{"id":"https://openalex.org/C37197862","wikidata":"https://www.wikidata.org/wiki/Q2066775","display_name":"Molecular symmetry","level":3,"score":0.4156000018119812},{"id":"https://openalex.org/C38976095","wikidata":"https://www.wikidata.org/wiki/Q752641","display_name":"Asymmetry","level":2,"score":0.4138000011444092},{"id":"https://openalex.org/C44306375","wikidata":"https://www.wikidata.org/wiki/Q902019","display_name":"Symmetry group","level":2,"score":0.3806000053882599},{"id":"https://openalex.org/C58721731","wikidata":"https://www.wikidata.org/wiki/Q1105337","display_name":"Explicit symmetry breaking","level":4,"score":0.36739999055862427},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3630000054836273},{"id":"https://openalex.org/C2779291638","wikidata":"https://www.wikidata.org/wiki/Q1455736","display_name":"Local symmetry","level":2,"score":0.3447999954223633},{"id":"https://openalex.org/C33332235","wikidata":"https://www.wikidata.org/wiki/Q18362","display_name":"Theoretical physics","level":1,"score":0.3386000096797943},{"id":"https://openalex.org/C2779413573","wikidata":"https://www.wikidata.org/wiki/Q556992","display_name":"Pseudorotation","level":3,"score":0.33640000224113464},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.32690000534057617},{"id":"https://openalex.org/C164772539","wikidata":"https://www.wikidata.org/wiki/Q7092304","display_name":"One-dimensional symmetry group","level":3,"score":0.313400000333786},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.31279999017715454},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.30649998784065247},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.2953999936580658},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.29429998993873596},{"id":"https://openalex.org/C33559828","wikidata":"https://www.wikidata.org/wiki/Q397226","display_name":"Coordination number","level":3,"score":0.2906000018119812},{"id":"https://openalex.org/C192423722","wikidata":"https://www.wikidata.org/wiki/Q1455736","display_name":"Global symmetry","level":4,"score":0.2892000079154968},{"id":"https://openalex.org/C2778242569","wikidata":"https://www.wikidata.org/wiki/Q1996344","display_name":"Continuous symmetry","level":2,"score":0.2865000069141388},{"id":"https://openalex.org/C3168419","wikidata":"https://www.wikidata.org/wiki/Q280222","display_name":"Tetrahedral symmetry","level":3,"score":0.2676999866962433},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.259799987077713},{"id":"https://openalex.org/C147597530","wikidata":"https://www.wikidata.org/wiki/Q369472","display_name":"Computational chemistry","level":1,"score":0.25850000977516174}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.26434/chemrxiv.15006158/v2","doi":"https://doi.org/10.26434/chemrxiv.15006158/v2","is_oa":true,"landing_page_url":"https://doi.org/10.26434/chemrxiv.15006158/v2","pdf_url":"https://chemrxiv.org/doi/pdf/10.26434/chemrxiv.15006158/v2","source":{"id":"https://openalex.org/S4393918830","display_name":"ChemRxiv","issn_l":"2573-2293","issn":["2573-2293"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"}],"best_oa_location":{"id":"doi:10.26434/chemrxiv.15006158/v2","doi":"https://doi.org/10.26434/chemrxiv.15006158/v2","is_oa":true,"landing_page_url":"https://doi.org/10.26434/chemrxiv.15006158/v2","pdf_url":"https://chemrxiv.org/doi/pdf/10.26434/chemrxiv.15006158/v2","source":{"id":"https://openalex.org/S4393918830","display_name":"ChemRxiv","issn_l":"2573-2293","issn":["2573-2293"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320328924","display_name":"Covestro","ror":"https://ror.org/01b44s469"},{"id":"https://openalex.org/F4320332169","display_name":"Directorate for Computer and Information Science and Engineering","ror":"https://ror.org/025kzpk63"},{"id":"https://openalex.org/F4320337377","display_name":"Office of Advanced Cyberinfrastructure","ror":"https://ror.org/04nh1dc89"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7216209447.pdf","grobid_xml":"https://content.openalex.org/works/W7216209447.grobid-xml"},"referenced_works_count":35,"referenced_works":["https://openalex.org/W1974136124","https://openalex.org/W1990194134","https://openalex.org/W2011600249","https://openalex.org/W2030307268","https://openalex.org/W2038312514","https://openalex.org/W2046557827","https://openalex.org/W2054118021","https://openalex.org/W2095401846","https://openalex.org/W2140871833","https://openalex.org/W2151995023","https://openalex.org/W2169745180","https://openalex.org/W2321326197","https://openalex.org/W2330868039","https://openalex.org/W2795196889","https://openalex.org/W2886777349","https://openalex.org/W2977953554","https://openalex.org/W3014445245","https://openalex.org/W3037545945","https://openalex.org/W3048777150","https://openalex.org/W3165308978","https://openalex.org/W4293052120","https://openalex.org/W4313650965","https://openalex.org/W4362588628","https://openalex.org/W4386968510","https://openalex.org/W4389438498","https://openalex.org/W4390265352","https://openalex.org/W4395064433","https://openalex.org/W4396217750","https://openalex.org/W4403859849","https://openalex.org/W4413951804","https://openalex.org/W4416354678","https://openalex.org/W4416364499","https://openalex.org/W7130852483","https://openalex.org/W7141430845","https://openalex.org/W7159589150"],"related_works":["https://openalex.org/W2169745180","https://openalex.org/W2011600249","https://openalex.org/W7098050071","https://openalex.org/W2140871833","https://openalex.org/W1974136124","https://openalex.org/W7172415908","https://openalex.org/W7141430845","https://openalex.org/W2095401846","https://openalex.org/W183560356","https://openalex.org/W2020896544"],"abstract_inverted_index":{"Symmetry":[0],"numbers":[1,37,142,167],"are":[2,109],"essential":[3],"corrections":[4],"in":[5,16,53],"thermochemistry":[6],"and":[7,24,74,88,124,130,139,173,202,207],"chemical":[8],"kinetics,":[9],"but":[10],"their":[11],"reliable":[12],"assignment":[13],"remains":[14],"difficult":[15],"automated":[17,200],"workflows,":[18],"especially":[19],"for":[20,34,81,143,152,168,199],"stereochemical,":[21],"fluxional,":[22,114],"cyclic,":[23],"transition-state":[25,131],"structures.":[26],"We":[27],"present":[28],"a":[29,82,90],"stereochemistry-aware":[30],"automorphism":[31,68],"(SAA)":[32],"approach":[33,192],"computing":[35],"symmetry":[36,59,96,132,141,150,157,166],"using":[38],"the":[39,54,58,64,67,79,134,155,164],"StereoMolGraph":[40],"(SMG)":[41],"framework.":[42],"In":[43],"this":[44],"representation,":[45],"local":[46,50],"stereodescriptors":[47],"encode":[48],"rigid":[49],"geometries":[51],"directly":[52],"molecular":[55,107,129],"graph,":[56],"so":[57],"number":[60,97,151,158],"is":[61,101,159,193],"obtained":[62],"as":[63,111],"order":[65],"of":[66,94,118,205],"group":[69],"that":[70],"preserves":[71],"both":[72],"connectivity":[73],"encoded":[75],"stereochemistry.":[76],"This":[77],"avoids":[78],"need":[80],"post":[83],"hoc":[84],"\u201clabel":[85],"stereocenter\u201d":[86],"correction":[87],"provides":[89],"direct":[91],"graph-theoretical":[92],"analogue":[93],"permutation-based":[95],"definitions.":[98],"Model":[99],"dependence":[100],"handled":[102],"explicitly":[103],"by":[104],"choosing":[105],"which":[106],"features":[108],"treated":[110],"fixed":[112],"or":[113,179],"enabling":[115],"consistent":[116],"treatment":[117],"rigidity,":[119],"free":[120],"rotation,":[121],"umbrella":[122],"inversion,":[123],"pseudorotation":[125],"assumptions.":[126],"Beyond":[127],"overall":[128,156],"numbers,":[133],"same":[135],"framework":[136],"yields":[137],"external":[138,149],"internal":[140],"hindered-rotor":[144],"treatments,":[145],"including":[146,183],"an":[147],"effective":[148],"cases":[153,175],"where":[154,176],"non-separable.":[160],"Benchmark":[161],"calculations":[162],"reproduce":[163],"expected":[165],"all":[169],"supported":[170],"benchmark":[171],"molecules":[172,206],"resolve":[174],"existing":[177],"rule-based":[178],"corrected-connectivity":[180],"approaches":[181],"fail,":[182],"stereochemically":[184],"distinct":[185],"structures":[186],"with":[187],"identical":[188],"connectivity.":[189],"The":[190],"SAA":[191],"highly":[194],"robust,":[195],"making":[196],"it":[197],"suitable":[198],"thermochemical":[201],"kinetic":[203],"modeling":[204],"reactions.":[208]},"counts_by_year":[],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2026-10-03T00:00:00"}
