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PreprintCited by 5Crystal structures of chemical compoundsAdvanced Chemical Physics StudiesAdvanced Physical and Chemical Molecular Interactions

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Determining Molecular Structure, Coordination Geometry, and Molecular Symmetry

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Sørensen, Thomas Just, Nielsen, Villads·2025·Nature Communications
DOI 10.21203/rs.3.rs-6860832/v1PMC PMC12706060

Preprint — published as 10.1038/s41467-025-66054-7 · Paper page

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Abstract

Chemists have a strong language describing and defining idealized polyhedra P and symmetry point groups G , but no efficient measure to correlate these to real molecular structures Q . Symmetry and structure have been ascribed through experience, but this approach is error-prone and provides no measure that can correlate molecular structure to molecular properties. Quantification of molecular structure and symmetry is crucial, and this requires two tools: a tool that orients a molecular structure to a symmetry axis, and a tool that measures the deviation of the molecular structure from symmetry. The first approach developed was the Continuous Symmetry Measure (CSM), which does both tasks and is exact across all symmetries as well as mathematically elegant. Unfortunately as a tool the CSM is close to impossible to implement. This led to the development of the Continuous Shape Measure (CShM) that measures how close a molecular structure Q is to a set of selected structures P ’. The CShM tool have severe limitations, and we suggest that the Continuous Symmetry operation Measure (CSoM) should be the preferred tool. The CSoM tool can quantify the symmetry of any structure that can be described as a list of points in space. Here, we describe the CSM, CShM, and CSoM approaches and demonstrate how these can be used as tools to determine molecular structure, coordination geometry, and symmetry of water, organic molecules, transition metal complexes, and lanthanide compounds.

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Fig. 1 The definition of the molecular entity using the molecular structure Q with the experimental measurable Ø, and the steps to determine coordination geometry P and molecular symmetry i.e., point group G of methane. a Defining the molecular coordinate system (x,y,z). b Identifying the molecular structure Q. c The structure Q can be determined to be identical to a polyhedron P = T-4 (a tetrahedron). d The structure Q is highly symmetric with symmetry elements like: a C3-rotational axis and three mirror planes. e The symmetry of methane is described by G = Td and Q is reproduced by all 24 symmetry operations in G = Td. Note that the symmetry axis z depends on the symmetry operation, while the molecular Z axis remain constant. Atom legend. carbon = brown, hydrogen = pink, chlorine = green.
Fig. 1 The definition of the molecular entity using the molecular structure Q with the experimental measurable Ø, and the steps to determine coordination geometry P and molecular symmetry i.e., point group G of methane. a Defining the molecular coordinate system (x,y,z). b Identifying the molecular structure Q. c The structure Q can be determined to be identical to a polyhedron P = T-4 (a tetrahedron). d The structure Q is highly symmetric with symmetry elements like: a C3-rotational axis and three mirror planes. e The symmetry of methane is described by G = Td and Q is reproduced by all 24 symmetry operations in G = Td. Note that the symmetry axis z depends on the symmetry operation, while the molecular Z axis remain constant. Atom legend. carbon = brown, hydrogen = pink, chlorine = green.CC BY-NC-ND · PubMed Central
Fig. 2 Three approached for determining and describing molecular structure. Determining molecular structure Q and symmetry point group G of methane and chloroform either using polyhedral P with the (a). CSM, b CShM or symmetry with c CSoM. CSM provides S-values, CShM provides σideal-values and CSoM provides σsym-values as deviations from ideal fit to P or the symmetry G. Atom legend: carbon = brown, hydrogen = pink, chlorine = green.
Fig. 2 Three approached for determining and describing molecular structure. Determining molecular structure Q and symmetry point group G of methane and chloroform either using polyhedral P with the (a). CSM, b CShM or symmetry with c CSoM. CSM provides S-values, CShM provides σideal-values and CSoM provides σsym-values as deviations from ideal fit to P or the symmetry G. Atom legend: carbon = brown, hydrogen = pink, chlorine = green.CC BY-NC-ND · PubMed Central
Fig. 3 Molecules and complexes investigated with the CSM, CShM, and CSoM method. Atom legend: oxygen = red phosphorous = blue, curium coordination polyhedra = gray, tungsten coordination polyhedra = deep purple.
Fig. 3 Molecules and complexes investigated with the CSM, CShM, and CSoM method. Atom legend: oxygen = red phosphorous = blue, curium coordination polyhedra = gray, tungsten coordination polyhedra = deep purple.CC BY-NC-ND · PubMed Central
Fig. 4 Water and aromatic molecules evaluated with CSoM for point group G providing deviations from the symmetry as σsym-values. a Ideal geometry of H2O with C2v symmetry that can only be broken by the asymmetric stretch. b 2000 H2O molecules from a molecular dynamic simulation34 evaluated with the C2, Cσ, and C2v point groups, illustrating the small effect of the asymmetric stretch on symmetry. c The symmetry of water molecules and clusters in ICE III and d the symmetry of water molecules and clusters in ICE IV130,131. e CSoM analysis of a single in silico optimized benzene as well as three crystal structures showing the visual numerical CSoM output. f CSoM analysis of 11 benzene molecular structures172–174,192–195. g CSoM output for the in silico and crystal structure of 1,3,5-trimethyl-benzene. h CSoM analysis for in silico optimized structures and experimental crystal structures of naphthalene, triphenylene, and hexabenzocoronene. The symmetry aligned structures from in silico and the crystal structures are overlaid for visual comparison132,133,180 Atom legend: carbon = brown, hydrogen = pink, oxygen = red.
Fig. 4 Water and aromatic molecules evaluated with CSoM for point group G providing deviations from the symmetry as σsym-values. a Ideal geometry of H2O with C2v symmetry that can only be broken by the asymmetric stretch. b 2000 H2O molecules from a molecular dynamic simulation34 evaluated with the C2, Cσ, and C2v point groups, illustrating the small effect of the asymmetric stretch on symmetry. c The symmetry of water molecules and clusters in ICE III and d the symmetry of water molecules and clusters in ICE IV130,131. e CSoM analysis of a single in silico optimized benzene as well as three crystal structures showing the visual numerical CSoM output. f CSoM analysis of 11 benzene molecular structures172–174,192–195. g CSoM output for the in silico and crystal structure of 1,3,5-trimethyl-benzene. h CSoM analysis for in silico optimized structures and experimental crystal structures of naphthalene, triphenylene, and hexabenzocoronene. The symmetry aligned structures from in silico and the crystal structures are overlaid for visual comparison132,133,180 Atom legend: carbon = brown, hydrogen = pink, oxygen = red.CC BY-NC-ND · PubMed Central
Fig. 5 Determination of the symmetry of the coordination polyhedra and molecules with CSoM for complexes and materials. Comparing coordination polyhedra P determined with CShM using σideal-values to the symmetry expressed with point groups G determined with CSoM via σsym-values for: a FeIICl4135, b [Ni(L)]+134, c FeIIICl6136, d CdCl215-crown-5139, e CuMn-MFU-4140, f Fe(btz)33+(CN = 6)137. g The four phases observed for the BaTiO3 perovskite analyzed with CSoM for the of BaO12 h and TiO4 i centers in the BaTiO3 perovskite as a function temperature. Structural data from reference142 and 141. Atom legend: carbon = brown, hydrogen = pink, oxygen = red, nitrogen = light blue, chlorine = green, titanium = ice blue, manganese = blue, iron = gold, copper = purple, nickel = light gray, cadmium = light purple, barium = bright green.
Fig. 5 Determination of the symmetry of the coordination polyhedra and molecules with CSoM for complexes and materials. Comparing coordination polyhedra P determined with CShM using σideal-values to the symmetry expressed with point groups G determined with CSoM via σsym-values for: a FeIICl4135, b [Ni(L)]+134, c FeIIICl6136, d CdCl215-crown-5139, e CuMn-MFU-4140, f Fe(btz)33+(CN = 6)137. g The four phases observed for the BaTiO3 perovskite analyzed with CSoM for the of BaO12 h and TiO4 i centers in the BaTiO3 perovskite as a function temperature. Structural data from reference142 and 141. Atom legend: carbon = brown, hydrogen = pink, oxygen = red, nitrogen = light blue, chlorine = green, titanium = ice blue, manganese = blue, iron = gold, copper = purple, nickel = light gray, cadmium = light purple, barium = bright green.CC BY-NC-ND · PubMed Central
Fig. 6 Lanthanide complexes analysed with SHAPE v2.1 (CShM) and CSoM providing σideal-values for a selected polyhedra P and σsym-values for selected point groups G. a Dy(bbpen)X145, b Nd(pc)2146, c Nd(oda)3147, d Dy(tp-py)2148. e CSoM investigation of the structure of Ln(sst)3THF2 series. f CSoM investigation of the structure of Tm.DOTA and Dy.DOTA(H2O) and the electronic structure of Dy.DOTA(H2O) as a function of symmetry163. Atom legend: carbon ligand backbone = pale gray, silicon ligand backbone = blue, hydrogen = pink, oxygen = red, aromatic nitrogen = light blue, aliphatic nitrogen = dark blue, lanthanum = green, neodymium = deep purple, dysprosium = bright green, thulium= aquamarine, lutetium = pale green. Images in panel f used with permission of Royal Society of Chemistry, from Chemical Science, M. Briganti et al, 10, 7233-7245, 2019; permission conveyed through Copyright Clearance Center, Inc.
Fig. 6 Lanthanide complexes analysed with SHAPE v2.1 (CShM) and CSoM providing σideal-values for a selected polyhedra P and σsym-values for selected point groups G. a Dy(bbpen)X145, b Nd(pc)2146, c Nd(oda)3147, d Dy(tp-py)2148. e CSoM investigation of the structure of Ln(sst)3THF2 series. f CSoM investigation of the structure of Tm.DOTA and Dy.DOTA(H2O) and the electronic structure of Dy.DOTA(H2O) as a function of symmetry163. Atom legend: carbon ligand backbone = pale gray, silicon ligand backbone = blue, hydrogen = pink, oxygen = red, aromatic nitrogen = light blue, aliphatic nitrogen = dark blue, lanthanum = green, neodymium = deep purple, dysprosium = bright green, thulium= aquamarine, lutetium = pale green. Images in panel f used with permission of Royal Society of Chemistry, from Chemical Science, M. Briganti et al, 10, 7233-7245, 2019; permission conveyed through Copyright Clearance Center, Inc.CC BY-NC-ND · PubMed Central
Fig. 7 Emission spectra and a CSoM analysis of relevant point groups for 10 Eu(III) complexes, σsym-values for selected point groups G are reported for the molecular structure Q and for the coordinating atoms Qinner. The structures and spectra are redrawn from literature data: a. Eu:Ba2MgWO6196, b. Eu2(SO4)6111, c. EuOCl197, d.Eu(pic)3·3(aza)198, e. Eu.DOTA(H2O)199, f. Eu.L’(H2O)199, g. Eu(oda)3200, h. Eu(dpa)3201, i. EuTp3202, and j. Eu.edta(H2O)3201. Atom legend: europium = purple, carbon = brown, oxygen = red, nitrogen = light blue, chlorine = green, sulfur = yellow, barium = orange, tungsten = dark orange. Hydrogen atoms are omitted for clarity.
Fig. 7 Emission spectra and a CSoM analysis of relevant point groups for 10 Eu(III) complexes, σsym-values for selected point groups G are reported for the molecular structure Q and for the coordinating atoms Qinner. The structures and spectra are redrawn from literature data: a. Eu:Ba2MgWO6196, b. Eu2(SO4)6111, c. EuOCl197, d.Eu(pic)3·3(aza)198, e. Eu.DOTA(H2O)199, f. Eu.L’(H2O)199, g. Eu(oda)3200, h. Eu(dpa)3201, i. EuTp3202, and j. Eu.edta(H2O)3201. Atom legend: europium = purple, carbon = brown, oxygen = red, nitrogen = light blue, chlorine = green, sulfur = yellow, barium = orange, tungsten = dark orange. Hydrogen atoms are omitted for clarity.CC BY-NC-ND · PubMed Central
Fig. 8 The effect of symmetry on the Nd(III) electronic structure and emission spectra. a Crystal field splitting in the lowest energy term 4I9/2 of Nd(III) in 10 different environments: Nd:LaCl3203, Nd(H2O)9(EtSO4)3204, Nd(H2O)9(BrO3)3205, Nd2(N3)8206, Nd(oda)385, Nd(dpa)3 (Note: CSoM values have been taken from the Eu(dpa)3 structure as the crystal field splitting has only been determined in solution)167, Nd:La2O3207, Nd(btmsm)3208, Nd:LiLuF4168, and Nd:YAG169. The energy levels were determined with emission spectroscopy for each molecular structure Q and the point group G was determined with a CSoM analysis. The relevant σsym(Q,G)-values are shown on the plot. The emission spectra of the 4F3/2 → 4I9/2 band for selected molecular structures Q with their CSoM determined point group G, are shown for b Nd2(N3)8, c Nd(oda)3, d Nd:YAG, e Nd(H2O)9, f Nd(dpa)3, and g Nd:LiLuF4 complexes, the spectra are redrawn from earlier work85,137,167–169,206. Atom legend: neodymium = deep purple, carbon = brown, oxygen = red, nitrogen = light blue. Hydrogen atoms are omitted for clarity.
Fig. 8 The effect of symmetry on the Nd(III) electronic structure and emission spectra. a Crystal field splitting in the lowest energy term 4I9/2 of Nd(III) in 10 different environments: Nd:LaCl3203, Nd(H2O)9(EtSO4)3204, Nd(H2O)9(BrO3)3205, Nd2(N3)8206, Nd(oda)385, Nd(dpa)3 (Note: CSoM values have been taken from the Eu(dpa)3 structure as the crystal field splitting has only been determined in solution)167, Nd:La2O3207, Nd(btmsm)3208, Nd:LiLuF4168, and Nd:YAG169. The energy levels were determined with emission spectroscopy for each molecular structure Q and the point group G was determined with a CSoM analysis. The relevant σsym(Q,G)-values are shown on the plot. The emission spectra of the 4F3/2 → 4I9/2 band for selected molecular structures Q with their CSoM determined point group G, are shown for b Nd2(N3)8, c Nd(oda)3, d Nd:YAG, e Nd(H2O)9, f Nd(dpa)3, and g Nd:LiLuF4 complexes, the spectra are redrawn from earlier work85,137,167–169,206. Atom legend: neodymium = deep purple, carbon = brown, oxygen = red, nitrogen = light blue. Hydrogen atoms are omitted for clarity.CC BY-NC-ND · PubMed Central

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Metadata

Authors
Sørensen, T. J., & Nielsen, V.
Affiliations
University of Copenhagen (DK)
Corresponding author
Sørensen, Thomas Just
ORCID
Venue
Nature Communications
Publisher
Springer Science and Business Media LLC
Published
2025-06-19
Volume
16
Issue
1
Pages
11122-11122
ISSN
2041-1723 (ISSN-L)
DOI
10.21203/rs.3.rs-6860832/v1
PMC
PMC12706060
OpenAlex
W4416506368
Cited by
5
References
1
Citations by year
2026: 5
FWCI
1.09
Type
journal article
Language
en
Open access
Licence
Locations
  • Nature Communications (publishedVersion, cc-by-nc-nd)
  • Research Square (acceptedVersion, cc-by)
  • Europe PMC (PubMed Central) (publishedVersion, cc-by-nc-nd)
  • PubMed (publishedVersion)
  • PubMed Central (publishedVersion, cc-by-nc-nd)
  • Research at the University of Copenhagen (University of Copenhagen) (publishedVersion, cc-by-nc-nd)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
Funders
  • Teknologi og Produktion, Det Frie Forskningsråd
  • Villum Fonden
  • Carlsbergfondet
  • Novo Nordisk
  • Novo Nordisk Fonden
Topics
  • Physical Sciences › Chemistry › Inorganic Chemistry › Crystal structures of chemical compounds
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Advanced Chemical Physics Studies
  • Physical Sciences › Chemistry › Physical and Theoretical Chemistry › Advanced Physical and Chemical Molecular Interactions
Keywords
molecular symmetry, coordination geometry, molecular structure determination, polyhedra, transition metal complexes, lanthanide compounds, water, phase transitions, luminescence

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CoChat. (2026). Citation check: Determining Molecular Structure, Coordination Geometry, and Molecular Symmetry. https://app.cochat.ai/papers/10.21203/rs.3.rs-6860832/v1
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Sørensen, T. J., & Nielsen, V. (2025). Determining Molecular Structure, Coordination Geometry, and Molecular Symmetry. Nature Communications. https://doi.org/10.21203/rs.3.rs-6860832/v1