# Neodymium(III) Aqua Ion as a Model System in Ab Initio Crystal Field Analysis Beyond Point Charges and Crystal Field Theory

Villads R. M. Nielsen, Maxime Grasser, Boris Le Guennic, Thomas Just Sorensen

► To cite this version:

Villads R. M. Nielsen, Maxime Grasser, Boris Le Guennic, Thomas Just Sorensen. Neodymium(III) Aqua Ion as a Model System in Ab Initio Crystal Field Analysis Beyond Point Charges and Crystal Field Theory. Inorganic Chemistry, 2025, 64 (7), pp.3463-3475. 10.1021/acs.inorgchem.4c05044 .
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Distributed under a Creative Commons Attribution 4.0 International License
The Neodymium(III) Aqua Ion as a Model System

in *Ab Initio* Crystal Field Analysis Beyond Point

Charges and Crystal Field Theory

Villads R. M. Nielsen†‡, Maxime Grasser‡, Boris Le Guennic‡, Thomas Just Sørensen†*

† Department of Chemistry and NanoScience Centre, University of Copenhagen,
Universitetsparken 5, 2100 Copenhagen, Denmark, TJS@chem.ku.dk

‡ Univ Rennes, CNRS, ISCR (Institut des Sciences Chimiques de Rennes), UMR 6226, F-35000 Rennes, France

ABSTRACT. Correlating the molecular structure with the electronic structure of lanthanide(III) solvates is a challenging task. Neodymium(III) in aqueous solution serves as an appealing and straightforward model to address this issue. Herein, the experimentally determined electronic structure of neodymium(III) in water is compared with *ab initio* calculated electronic structures based on various models of its molecular structure. This comparison enables the determination of the most reliable molecular structure. The findings reveal that the molecular structure of the neodymium(III) aqua ion that best aligns with its electronic structure corresponds to a nine coordinated neodymium(III) complex, surrounded by 17 water molecules in the second coordination sphere. The role of second-sphere water molecules was investigated by calculating the crystal field splitting of the five Kramers doublets within the <sup>4</sup>I<sub>9/2</sub> low-energy multiplet for
several calculated molecular structures with coordination numbers of eight, nine, and ten. The
results demonstrated that the shape of the donor molecular orbitals plays a critical role in the crystal
field splitting of the neodymium(III) ion. Furthermore, the findings confirmed that the orientation
of the donating orbitals, specifically the orientation of the O-H bonds in water, is essential for
accurately describing the electronic structure. Finally, manual alteration of the Nd-O bond lengths
revealed that CAS(3,7)CF calculations tend to underestimate the crystal field strength.

Introduction

Understanding the solution chemistry of trivalent lanthanides is fundamental for improving
the separation and extraction processes necessary for their purification. ¹, ² With the growing
demand for these elements in a wide range of high-tech applications, such as magnets,³, ⁴ lasers,⁵,
⁶ and optical materials,⁷⁻¹⁰ the development of efficient and environmentally friendly separation
processes is essential to ensure a sustainable supply for these technologies.¹¹ In this regard,
neodymium(III) ion is of particular interest given its extensive use in permanent magnets and
lasers.³⁻⁶ However, the purification of lanthanides is challenging due to the similar coordination
chemistry shared across the series.¹² This similarity arises because the inner xenon-core electrons
are radially extended beyond the 4f valence electrons, thereby shielding the valence orbitals from
interactions with the chemical environment. Consequently, the chemistry of lanthanide(III) ions is
governed primarily by electrostatics and the steric hindrance imposed by the ligands.¹³, ¹⁴

To increase our understanding, we have investigated neodymium(III) in the simplest
relevant system: water, without any additional ligands. The molecular structures of lanthanide(III)
ions in water are largely dictated by the ionic radius of the central ion. As a monodentate ligand,
water coordinates in arrangements that can be approximated as solutions to the Thomson
problem.¹⁵ For a coordination number (CN) of 8, the donor atoms adopts a square antiprism (SAP), while for CN = 9, the donor atoms forms a tricapped trigonal prism (TTP). However, for CN = 9, the capped square antiprism (cSAP) is energetically comparable to the TTP, and the minimal distortion pathway between these geometries suggests that both need consideration.¹⁶ For the neodymium(III) aqua ion, a CN of 9 is widely accepted.¹⁷⁻³⁸ The TTP is the most frequently reported structure,¹⁷, 25-27, 29, 30, 34, 37 though results from DFT calculations²⁷, 29, 30, 36 and molecular dynamics simulations¹⁷, 25, 35 indicate that the molecular structure is dynamic and often deviates from ideal cSAP and TTP geometries. Precise molecular structures, however, are challenging to deduce from experimental data. As a result, both experimental¹⁹, 22, 24, 26, 39, 40 and computational²³, 25-27, 29-31, 36, 40 studies have largely focused on determining coordination numbers. Key methods include extended X-ray absorption fine structure and molecular dynamics, but x-ray diffraction,¹⁸ neutron diffraction,²⁰, ²¹ Raman spectroscopy,³⁹, ⁴¹ and luminescence spectroscopy,⁴²⁻⁴⁴ have also been able to provide valuable insights on the structural trends. Optical spectroscopy, particularly high-resolution spectroscopy, has proven invaluable for structural analysis of lanthanide(III) ions in solution, as it can provide insights into the electronic structure and crystal field splitting.¹⁴, ⁴⁵, ⁴⁶ The experimentally extracted electronic structures can then be compared with computational results derived from structural models.⁴⁴, ⁴⁷⁻⁵⁴ For neodymium(III), resolving the crystal field splitting in solution has historically been challenging, with early studies focusing solely on the positions of the optical bands and emission lifetimes,⁵⁵⁻⁵⁷ but we recently were able to resolve the crystal field splitting in solution.⁴⁷, ⁵⁸

The levels in ⁴I<sub>J</sub> multiplets of neodymium(III) have previously been studied using a combination of optical spectroscopy and semi-empirical parameterized crystal field theory in various single crystals.⁵⁹⁻⁶³ While these parameterized models have proven useful they lack the predictive power
offered by accurate *ab initio* models. Over the past decade, *ab initio* multiconfigurational methods, such as CASSCF (complete active space self-consistent field) combined with spin-orbit (SO) coupling, have provided a robust framework for calculating the electronic structure of low-energy multiplets in lanthanide(III) complexes.<sup>49-51, 53, 64-68</sup> These methods have been extensively applied in molecular magnetism to model the magnetic properties, particularly for complexes based on dysprosium(III),<sup>65, 66, 69-71</sup> terbium(III),<sup>72, 73</sup> and ytterbium(III).<sup>49, 51, 52, 67, 74-76</sup> For neodymium(III), *ab initio* CASSCF calculations have been successful in reproducing data from inelastic neutron scattering,<sup>83</sup> electron paramagnetic resonance spectroscopy,<sup>84</sup> and magnetic measurements.<sup>78-82</sup> Recently, we demonstrated that CASSCF calculations yield a highly accurate description of the energy levels in the <sup>4</sup>I<sub>J</sub> multiplets of neodymium(III), showing excellent agreement with experimental data from luminescence measurements.<sup>54</sup>

In this study, we investigate the crystal field splitting of the neodymium(III) aqua complex, extracted from high-resolution luminescence measurements on neodymium(III) triflate in acidified D<sub>2</sub>O, and computed using different molecular models at the CASSCF level. The crystal field is determined by the molecular structure, where the donor atoms create an electrostatic potential that perturbs the energy states of the central neodymium(III) ion. The analysis of the computed electronic structures focuses on the crystal field splitting within different multiplets of neodymium(III). The concept of how the crystal field splitting affects the neodymium ion and 4f-elements in general are illustrated in Figure 1. In a simple atomic orbital model, a hydrogen-like 4f<sup>1</sup> system is considered without spin-orbit coupling. In the free ion, all seven 4f orbitals forms fully degenerate energy levels. Placing the ion in a crystal field with C<sub>∞</sub> symmetry partially lifts the degeneracy, splitting the orbitals into four distinct energy levels. In a linear molecular structure, this can be modeled by two negative charges in axial positions. The resulting electrostatic
interactions repel the 4f electrons differently, creating the observed energy levels For neodymium(III), the inclusion of spin-orbit coupling causes the three 4f valence electrons to form 41 multiplets, yielding a total of 182 Kramers doublets, and resulting in a significantly more complex electronic structure. Figure 1 illustrates this for the lowest energy multiplet, ⁴I₉/₂, in cubic and other symmetries. Despite this increased complexity, the fundamental concept of crystal field splitting remains the same: the electronic states of the metal ion are perturbed by the electric field generated by the donor atoms.

Here, we evaluate the electronic structure of neodymium(III) based on the five Kramers doublets in the lowest energy multiplet, ⁴I₉/₂, and the two Kramers doublets in the emitting multiplet, ⁴F₃/₂. The influence of the crystal field on the electronic structure is examined using point charges, fluoride ions, and water molecules to represent the donor atoms. From a symmetry perspective, two options are possible for the neodymium(III) ion: it may either retain some degeneracy (cubic symmetry) or exhibit a lifting of degeneracy into doubly degenerate Kramers doublets (Figure 1). Non-cubic symmetry results in observable splitting into two and five energy levels for ⁴F₃/₂ and ⁴I₉/₂ multiplets, respectively. Based on these considerations, we model the crystal field of neodymium(III) aqua ions with molecular structures based on ideal polyhedra to determine how the donor atoms must be described and oriented for the computed electronic structure to align with experimental results. First, we evaluate the electronic structure of neodymium(III) ion in 12 different ideal molecular geometries spanning CN of 8, 9, and 10. We analyze the most stable molecular structures for each CN, as determined through geometry optimizations. Finally, we incorporate and optimize second-sphere solvent molecules to refine the molecular structure and accurately reproduce the experimentally determined crystal field splitting.
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Top: Generic electronic structure of the lowest energy multiplet, $^4I_{9/2}$, for neodymium(III) after spin-orbit coupling in the free ion and in cubic and other symmetries. Bottom: Electronic structure of a $4f^1$ ion where spin-orbit coupling is not considered. The electronic structure is shown for the free ion and in a $C_\infty$ symmetric crystal field generated by two axial point charges. The low energy and high energy orbitals show how the crystal field effect is defined by electron repulsion. Orbitals are drawn using the Wolfram demonstration project.⁸⁶</figcaption></figure>

**Experimental and Computational Section**

No uncommon hazards are noted.

**Chemicals.**

Two samples have been made in this study: Neodymium in D₂O and H₂O. All chemicals were used as received. 35.5 ± 0.2 mg Nd(CF₃SO₃)₃ (> 97.0 %, TCI Chemicals) were dissolved in 3 mL H₂O (demineralized) or D₂O (Eurisotop 99.90%D) up to a total of 20 mM neodymium(III).
**Optical Spectroscopy**

All experiments have been done in 10 mm quartz cuvettes (23/Q/10) from Starna Scientific.

**Emission spectra.** Emission spectra of samples at room temperature (293 K) and liquid nitrogen temperature (77 K) were performed on our custom-build spectrometer. The samples were excited by a supercontinuum laser (NKT SuperK Fianium FIU-15) that was coupled to a tunable band pass filter (NKT LLTF Contrast VIS/SWIR HP8). The laser power was set to 90% with maximum repetition rate (78MHz). A 750B grating and the PyLoN detector was used for measurements below the 950 nm point, and a 1200B grating and the NIRvana detector was used for measurements above 950 nm. The emission slit was set to 10 µm for all measurements. The excitation wavelength was set to 523 nm, long pass filter of 850 nm was used. Measurements are done with an exposure time of 2.5 s below 950 nm and 60 s above 950 nm. A background has been created for each measurement with identical acquisition and has been used for background subtraction. Details are shown in Figure S1–S3.

**Excitation spectra.** Excitation spectra were recorded by changing the wavelength of the excitation source through our own Python code that connects to the tunable band pass filter. At each excitation wavelength, an emission spectrum was recorded and integrated by summing all data points spanning the region of interest. Two different acquisitions are used for two different ranges of the spectrum. In the range 450 to 835 nm the emission from 850 to 900 nm is integrated with an integration time of 0.2 s. In the range 720 to 925 the emission from 1040 to 1070 nm is integrated with an integration time of 0.5 s. The two spectra are stitched together by normalizing with respect to the area under the peak in the range 725 to 750 nm.

**Polyhedron models**
The 12 Polyhedron models were constructed based on the model polyhedron structures provided in previous studies.²⁵, ⁸⁷, ⁸⁸ The mean Nd-O distance in each polyhedron was scaled to match the experimental results and set to 2.525 Å, as previously determined experimentally.²⁴

CN = 8: Square antiprism (SAP), Square-face bicapped trigonal prism or bicapped trigonal prism (BTP), Cube (Cube), and triangular-face dodecahedron (DD).

CN = 9: Square-face monocapped antiprism or capped square antiprism (cSAP), tricapped trigonal prism (TTP), Square-face monocapped prism or capped cube (cCube), and heptagonal bipyramid (HBP)

CN = 10: Square-face bicapped square antiprism or Bicapped square antiprism (bcSAP), Bicapped dodecahedron (bcDD), Staggered dodecahedron (SDD), and Pentagonal antiprism (PAP). The reader could note that bcDD and SDD are not IUPAC polyhedra, but correspond to structures as reported in ref 87.

## Ab initio calculations

All coordinates are available in individual files in the Supporting Information. The electronic structure of neodymium(III) was calculated with the software OpenMOLCAS (version 22.10).⁸⁹,⁹⁰ State average (SA) complete active space (CAS), and self-consistent field (SCF) approaches were used.⁹¹⁻⁹³ Relativistic corrections at the scalar level have been taken into account by the use of the Douglas-Kroll-Hess Hamiltonian⁹⁴⁻⁹⁶ and the all-electron atomic natural orbital relativistic contracted (ANO-RCC) basis-set from the Molcas library.⁹⁷, ⁹⁸ The following contractions were used: [8s7p4d3f2g1h] for the Neodymium, [3s2p1d] for the oxygen, and [2s1p] for hydrogen. Spin-orbit (SO) coupling has been added within the restricted active space state interaction (RASSI) method on the number of previously calculated roots with the use of the atomic mean field integral (AMFI) theory. SA-CAS(3,7)SCF/RASSI-SO calculations were carried out
including three electrons spread in the seven 4f orbitals of the neodymium(III) ion. All spin-free
states were calculated, that is 35 quadruplets and 112 doublets for neodymium(III) ion. Additional
calculations with larger basis sets and with the inclusion of dynamic correlation through second
order perturbation theory (PT2) were attempted as shown in Figures S26-27. None improve the
results and only calculations at the SA-CAS(3,7)SCF/RASSI-SO level are discussed in the main
text.

The geometry optimizations were performed using density functional theory (DFT),¹⁰¹ with the
hybrid meta-GGA M06-2X exchange-correlation functional.¹⁰² Initial optimizations using GGA-
PBE did not converge to the same geometry when starting with different geometries of Nd(H₂O)CN
(CN = 8 and 9). The hybrid meta-GGA M06-2X exchange-correlation functional overcame this
issue, and has also been shown to provide accurate bond lengths in lanthanide(III) complexes in a
previous study.¹⁰³

Calculations have been performed with the ADF suite (version 2022.103)¹⁰⁴, ¹⁰⁵ with basis sets
at the TZP level for all atoms. The ZORA formalism with the MAPA potential was used to describe
scalar relativistic effects as implemented in ADF.¹⁰⁶, ¹⁰⁷ The COSMO model has been used for the
screening of a water-based electrostatic potential environment. Geometry optimizations and
frequency analysis were performed with the AMS 2023.1 program package.¹⁰⁹ All calculations
were performed without any symmetry constraint.

The *ab initio* calculations were performed on multiple different models for the molecular
structure of Nd(H₂O)CN (CN = 8, 9, and 10):

**Ideal polyhedron with point charges at the vertices.** For the point charge models, *ab initio*
calculations were performed on neodymium(III) ion where point charges (-1 e) where placed at
the vertices of the polyhedra.
**Ideal polyhedra with fluoride ions at the vertices.** Performed with fluoride ions (F⁻) at the vertices of the polyhedron.

**Ideal polyhedra with Water at the vertices.** Performed with the oxygen of water molecules (H₂O) at the vertices. The hydrogen atoms were inserted manually into the structures in meaningful positions. Then the geometry was optimized with fixed oxygen positions.

**Optimized neodymium(III) aqua ion complexes.** Each geometry of the ideal polyhedra with water at the vertices was optimized resulting in three converged molecular structures: Nd(H₂O)₈³⁺, Nd(H₂O)₉³⁺, and Nd(H₂O)₁₀³⁺. The Nd(H₂O)₈³⁺ and Nd(H₂O)₉³⁺ resulted in the minimum energy structure as obtained from frequency calculations, while for the Nd(H₂O)₁₀³⁺ this could not be achieved. See Figures S20 – S22 and Tables S46 – S48 in the supporting information for details. All of the final structures were used as input structures for CASSCF calculations.

**Molecules with varied distances of the Nd-O bond length.** For each of the coordination numbers, the fully optimized neodymium(III) aqua ion complex with lowest calculated Gibbs free energy was used to generate complexes with varied Nd-O bond distances. This was done by manually changing the Nd-O bond length followed by an additional optimization of the hydrogen atoms with fixed oxygen atoms to the new geometries with changed bond distances.

**Optimized neodymium(III) aqua ion complexes including explicit solvent molecules in the second sphere.** For the Nd(H₂O)₈ and Nd(H₂O)₉ structures, a second sphere of coordination was constructed. This was achieved by manually adding water molecules to the second coordination sphere until each first-sphere water molecule formed a hydrogen bond with a second-sphere oxygen. Then, their geometries were optimized resulting in two molecular structures: [Nd(H₂O)₈](H₂O)₁₅³⁺ and [Nd(H₂O)₉](H₂O)₁₇³⁺.

**Wavefunction component analysis**
To assess the evolution of the nature of the wavefunctions in $^4I_{9/2}$ as the molecular structure is changed, we used a model that expresses a set of wavefunctions as a linear combination of another set of wavefunctions, both spanned by the same basis set. The five Kramers levels of $^4I_{9/2}$ split differently in one crystal field A compared to another crystal field B, but the individual states are still constructed from the same basis set, that is the 10 m$_J$ states, which in zero field define a basis of five degenerate Kramers levels (k): $|\pm1/2\rangle, |\pm3/2\rangle, |\pm5/2\rangle, |\pm7/2\rangle, |\pm9/2\rangle$. In crystal field A, the Kramers levels, (i), can be described by a linear combination of the basis (k):

$$ \text{Eq. 1} \qquad (i) = \sum_{k=0}^{4} c_{k,i} |k\rangle $$

Where k is the Kramers levels 0 to 4 of the basis, $c_{k,i}$ is the linear coefficient of each component in the basis, and i goes from 0 to 4 representing the 5 Kramers levels in $^4I_{9/2}$. In a different crystal field B, the new Kramers levels, (f), can similarly be described by a linear combination of the 5 degenerate Kramers levels in zero field from the basis (k), but also by a linear combination of the five levels of (i) with different coefficients:

$$ \text{Eq. 2} \qquad (f) = \sum_{i=0}^{4} d_{i,f} (i) $$

Where $d_{i,f}$ is a 5x5 matrix with the linear coefficients of each (i) used to construct (f). The five Kramers levels (k) and their linear combinations creating (i) and (f) are the outputs from the CASSCF calculations given as percentages.

All Kramers levels in (i) and (f) contain contributions from each of the basis vectors (k). Both (i) and (f) can therefore be expressed as linear combination of the (k) levels using the percentage contribution as weights. From a randomly initialized coefficient matrix, $d_{i,f}$, (f) can be calculated by Eq. 1 and Eq. 2, from which the vector distance, sigma, between the two can be constructed:

$$ \text{Eq. 3} \qquad \sigma_{ma_f} = \left| \sum_{k=1}^{5} c_{k,f} |k\rangle - \sum_{i=1}^{5} d_{i,f} (i) \right| $$
The coefficient matrix, $d_{i,f}$, is found numerically by minimizing the sum of the vector distances of the (f) calculated by both Eq. 1 and Eq. 2:

$$ \text{Eq. 4} \qquad |\sigma\sigma\sigma|^2 = \sum_{f=1}^{5} \sigma\sigma\sigma^2_f $$

Equation 4 is used to track the evolution that occurs between crystal fields. If (k) describes the free ion, then each of the (k) levels can be followed when lifting the degeneracy in different molecular structures.

## Results and discussion

Multireference *ab initio* calculations (see computational details) were performed to investigate the crystal field splitting and the electronic structure of the $^4I_{9/2}$ multiplet of neodymium(III). The crystal field splitting in Nd(H$_2$O)$_8$ with cubic symmetry was calculated using different models, as shown in Figure 2. As expected, modelling the free ion results in all levels being energetically separated by less than 2 cm$^{-1}$, making them fully degenerate. The expected separation is 0 cm$^{-1}$, and the deviation from this value is uncertain. Discrepancies of a similar magnitude are observed for higher-energy multiplets, which are most likely due to computational inaccuracies, but could also reflect underlying physical phenomena. When the crystal field is modelled with point charges at the vertices of the polyhedron, the five Kramers doublets, referred to as levels, in $^4I_{9/2}$ group are distributed in a (2, 2, 1) pattern. The two lowest energy levels are doubly degenerate (in addition to the inherent Kramers doublet degeneracy), while the highest energy level is singly degenerate. The total magnitude of the splitting is 130 cm$^{-1}$. This partial lifting of degeneracy is expected, as the coordination symmetry is cubic and belongs to the O$_h$ point group. Moving beyond the point charge model, two chemically meaningful systems are included in Figure 2, where fluoride ions and water molecules are placed at the vertices of the polyhedron. For the water model, the positions of the hydrogen atoms were optimized using DFT, with fixed oxygen positions at the vertices (see
Computational Details). We observe that replacing the point charges with fluoride ions does not reduce the symmetry from $O_h$ symmetry. The splitting of the energy levels in $^4I_{9/2}$ still follows a (2, 2, 1) distribution, identical to that of the point charge model. However, the total splitting is greater compared to the point charge model, indicating that the atomic orbitals of fluoride ions directly perturb the energy levels of the neodymium(III) ion more significantly than the point charges. Replacing the fluoride ions with water molecules reduces the symmetry from $O_h$ to $D_4$. Although the oxygen atoms are located in $O_h$ symmetry, the position of the hydrogen atoms alters the oxygen atomic orbitals, reducing the symmetry. Thus, the degeneracy is lifted, and all five energy levels in $^4I_{9/2}$ have different energies. Despite this loss of degeneracy, the qualitative ordering of the levels can still be meaningfully described by a (2, 2, 1) distribution. It is important to note that the stronger donor strength of the negative fluoride ions leads to a smaller overall splitting (~350 cm⁻¹), compared to the asymmetry induced by the water molecules (~450 cm⁻¹).

Figure 2 includes dotted lines that track the evolution of the five Kramers doublets in the $^4I_{9/2}$ multiplet. The three crystal fields – defined by point charges, fluoride ions and water molecules – perturb both the energy and the composition of the wavefunctions. The resulting compositions of the $m_J$ wavefunctions for each Kramers doublet can be traced by examining how much each contains of the initial free ion basis, as summarized in Table S2. A line is drawn between two levels of the different models if they share at least 25 % of the same $m_J$ components as determined from Eq. 1-4. The energy levels, wavefunctions, and the linear coefficient matrices derived from these calculations are shown in Tables S1, S2, and S3 respectively, with the methodology detailed in the Method Section. The analysis reveals that the highest energy level (4) has a similar composition across all three crystal fields, consisting of approximately 40 % $|±9/2⟩$ and 55 % $|±1/2⟩$. In contrast, the lower energy levels exhibit different mixing in each crystal field. The four
lowest energy levels form two degenerate levels (which are each doubly degenerate Kramers doublets) for both the point charge and the flouride ion donors. These split into four levels with water molecule. The nature of the wave composition is different in all cases. One example is the ground state that changes from 50 % $|±7/2⟩$ and 32 % $|±5/2⟩$ with the point charge at the vertices to 39 % $|±9/2⟩$ and 43 % $|±1/2⟩$ with fluoride ions and to 50 % $|±9/2⟩$ and 43 % $|±1/2⟩$ with water molecules as donors.

Although the three molecular structures are constructed with the same geometry and coordination symmetry, it is concluded that the electronic structure and the relative energy splitting differ significantly between the crystal fields. This result emphasizes the importance of accounting for the molecular structure of the donor ligands when modelling the crystal field splitting of neodymium(III), a principle that can likely be extended to the entire lanthanide(III) series.
<figure><img src="image_3.png" /><figcaption>**Figure 2.** The electronic configuration of the $^4I_{9/2}$ of neodymium(III) ions calculated in four different molecular structures of increasing complexity: as a free ion, and with point charges, fluoride ions, and water molecules at the vertices of the square polyhedron with neodymium(III) in the centre. The energy levels have been normalized to the barycentre of the multiplet.</figcaption></figure>
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 3.** Optical transitions of neodymium(III) in D₂O. a) Excitation collected by integration of the emission in the range 1040 to 1070 nm. b) Emission collected after excitation at 465 nm. Each band for the 77 K spectra is normalized to the same maximum as the 293 K spectra. c) Fit of the ⁴I₉/₂ → ⁴F₃/₂ transitions. d) Fit of the ⁴F₃/₂ → ⁴I₉/₂ transitions. e) Fit of the ⁴F₃/₂ → ⁴I₁₁/₂ transitions. f) Fit of the ⁴F₃/₂ → ⁴I₁₃/₂ transitions.</figcaption></figure>

Optical spectroscopy

After establishing the computational approach to model the electronic structure of neodymium(III), we now turn to the experimentally determined electronic structure. The electronic structure of neodymium(III) can be experimentally resolved in solution using optical spectroscopy,
following the methodology from previous work.<sup>47, 54</sup> Figure 3 shows the optical spectra of neodymium(III) in D<sub>2</sub>O, along with the 15 Kramers levels of the lowest energy multiplet at room temperature (~293K) and liquid nitrogen temperature (77K). Each spectral band is provided in high resolution in the supporting information (Figures S4–S13). Both the lowest energy multiplet (<sup>4</sup>I<sub>9/2</sub>) and emitting multiplet (<sup>4</sup>F<sub>3/2</sub>) are well separated from other multiplets. Therefore, excitation and emission are predominantly observed from <sup>4</sup>I<sub>9/2</sub> and <sup>4</sup>F<sub>3/2</sub>, respectively, with only minor contributions of thermally activated emission at 293 K from <sup>4</sup>F<sub>5/2</sub>, and <sup>2</sup>H<sub>9/2</sub> at 800 nm and 1000 nm (as shown in Figure 3b). Given that the neodymium(III) aqua ion is believed to be highly dynamic,<sup>17, 25, 27, 29, 30, 35, 36</sup> we investigated whether multiple emissive species could be observed in frozen solution (77 K). Emission spectra were collected at varied excitation wavelengths across the 580 nm excitation band (see Figures S14–S16). While minor variations were observed in the emission spectra, these were attributed to stochastic variations around an average spectrum. Therefore, we conclude that, despite the possibility of high lability, only a single coordination number of neodymium can be observed.

Analyses of individual bands provide the experimental crystal field splitting. The bands from <sup>4</sup>I<sub>9/2</sub> → <sup>4</sup>F<sub>3/2</sub> and <sup>4</sup>F<sub>3/2</sub> → <sup>4</sup>I<sub>9/2</sub> are particularly useful, as the combined information from the transition intensities in both directions allows for a complete resolution of the electronic structure of these multiplets.<sup>47</sup> Spectral deconvolutions of the bands <sup>4</sup>I<sub>9/2</sub> → <sup>4</sup>F<sub>3/2</sub>, <sup>4</sup>F<sub>3/2</sub> → <sup>4</sup>I<sub>9/2</sub>, <sup>4</sup>F<sub>3/2</sub> → <sup>4</sup>I<sub>11/2</sub>, and <sup>4</sup>F<sub>3/2</sub> → <sup>4</sup>I<sub>13/2</sub> are shown in Figures 3c-3f. Starting with <sup>4</sup>I<sub>9/2</sub> → <sup>4</sup>F<sub>3/2</sub> and <sup>4</sup>F<sub>3/2</sub> → <sup>4</sup>I<sub>9/2</sub> (Figures 3c and 3d), a maximum of 10 lines can be observed in both the excitation and emission spectra, as there are up to two levels in the emitting multiplet <sup>4</sup>F<sub>3/2</sub>, and up to five levels in the lowest energy multiplet <sup>4</sup>I<sub>9/2</sub>. In the excitation band <sup>4</sup>I<sub>9/2</sub> → <sup>4</sup>F<sub>3/2</sub> at 77 K (Figure 3c), two lines separated by 44 cm<sup>-1</sup> are observed at ~11600 cm<sup>-1</sup>, both with near-equal intensity. Consulting the emission band
$^4F_{3/2} \rightarrow ^4I_{9/2}$ in Figure 3e, the same pair of two lines are observed, along with two additional sets of new lines. However, in this case, the low-energy transition exhibits significantly higher intensity. This can be rationalized by considering the relative Boltzmann population. At 77 K, two levels separated by 45 cm$^{-1}$ have relative populations of 70 % and 30 %, respectively, while the relative populations are near 50 % and 50 % at 293 K. Based on this, the two Kramers doublets in $^4F_{3/2}$ and the five Kramers levels in $^4I_{9/2}$ can be assigned. From this point, the three emission bands in Figures 3d-3f were fitted with a sum of sets of Voigt functions, following a similar approach as reported previously,$^{47,54}$ where each set describes the two transitions from $^4F_{3/2}$. The crystal field splitting of $^4F_{3/2}$ is optimized as a global parameter, and the result gives the crystal field splitting of the $^4I_J$ multiplets. For the $^4I_{11/2}$ multiplet, the splitting is not large enough for all lines to be fully resolved at 77K. The relative energies in each multiplet are provided in Figure 3, and the details of the fit parameters are supplied in Tables S4 – S7.

Ab initio calculations: ideal polyhedra

With the experimental crystal field splitting resolved from optical spectroscopy, we can now model the molecular structure to obtain an *ab initio* crystal field splitting that aligns with the measurements. To explore the diversity of crystal field splitting that can be obtained for the neodymium(III) aqua ion, 36 molecular structures was constructed. These were based on 12 ideal polyhedra with CN of 8, 9, and 10. For each polyhedron, the three crystal fields discussed above are explored: A point charge electrostatic crystal field, an *ab initio* crystal field with fluoride ions, and an *ab initio* crystal field with water molecules, where the positions of the hydrogen atoms have been geometry optimized at the DFT level (see Computational Details). The point group symmetries were evaluated for both point charges and water molecules at the vertices with the continuous symmetry operation measure (CSoM), and the assigned point groups are provided
along the molecular structures in Figure 4 (see Tables S8 – S10 for details). The magnitude of the
point charges was fixed to 1 *e* to enable comparison with the fluoride ions. Additionally,
calculations were performed using a charge of -0.7477 *e*, which corresponds to the average
Mulliken charge found on the oxygen atoms of water molecules. The crystal field splittings under
this adjustment were generally unchanged, with a slight decrease in magnitude observed (See
Figures S17-S19).

The computed electronic structure of the <sup>4</sup>I<sub>9/2</sub> multiplet in the three crystal fields is shown in Figure 4. The wavefunction component analysis is included in the plots (see tabular data in Tables S1–S3 and S13–S45). Before evaluating the crystal field splitting obtained for the 12 molecular structures, let us first consider the general results. As previously reported,<sup>53</sup> a description of the crystal field based on point charges alone is insufficient, and full *ab initio* evaluation of the crystal field is necessary. Furthermore, it is evident that charge density and the orbital symmetry of the donor atoms are also crucial factors, as discussed above.

To analyze the individual crystal fields, we establish a nomenclature for describing the <sup>4</sup>I<sub>9/2</sub>
multiplet. This multiplet comprises five Kramers doublets that are split by the crystal field. The
total splitting is reported in cm<sup>-1</sup> and corresponds to the energy difference between the lowest and
the highest energy levels in the multiplet. When energy levels are close (a threshold of 50 cm<sup>-1</sup> is
used), they are grouped. For instance, in the O<sub>h</sub> symmetry case shown in Figure 2, the splitting
distribution is characterized as (2, 2, 1). For the *ab initio* crystal field splitting, several molecular
structures – including SAP, BTP, Cube, cSAP, TTP, cCube, bcSAP, and PAP – yield similar
distributions of energy levels and wavefunction components with both fluoride ions or water
molecules. However, for other molecular structures, such as DD, HbP, bcDD, and SDD, the crystal
fields generated by fluoride ions and water molecules produce significantly different electronic
structures. The most pronounced differences in splitting and wavefunction composition were observed for the molecular structure defined by the HbP polyhedron. Here, the distribution shifts from (1, 1, 1, 2,) to (1, 1, 1, 1, 1) when water molecules replace the fluoride ions. When fluorides occupy the vertices, the structure exhibits $D_{7h}$ symmetry, while the inclusion of the hydrogen atoms from water molecules reduces the symmetry to $C_2$. In contrast, for the Cube structure, the splitting and wavefunction compositions remain nearly identical whether fluoride ions or water molecules are used. In this case, a (2, 2, 1) energy level distribution is observed across all crystal fields, and the hydrogen atoms maintain the $D_4$ symmetry derived from the $O_h$ symmetric cube. Similarly, only minor changes in electronic structure are found for the SAP and PAP geometries when substituting fluoride ions with water molecules. This indicates that the shape of the oxygen atomic orbitals, influenced by the position of the hydrogen atoms, plays a significant role. When the reduction in symmetry between fluoride and oxygen orbitals is minimal and the overall symmetry remains high, the distribution of energy levels and wavefunction composition are consistent between the aqua ions and fluoride complexes.

In the cSAP, cCube, and bcSAP structures, the hydrogen atoms of the capping water molecules reduce the symmetry from $C_4$ to $C_2$. This reduction does not significantly alter the distribution of energy levels. We speculate that the capping water molecules are positioned far enough away that changes in atomic orbital symmetry have a negligible impact. For instance, in the cSAP molecular structure, the Nd-O distance are 3.33 Å for the capping water molecules and 2.43 Å for the other water molecules. Consequently, the effective symmetry remains close to $C_4$ (see Tables S9 and S10).

The crystal field splitting of the $^4I_{9/2}$ multiplet was calculated to range between 200 cm$^{-1}$ and 700 cm$^{-1}$ across the twelve different fluoride complexes and aqua ions. The energy level distributions
vary significantly and cannot be predicted *a priori*. For example, a (2, 3) distribution is observed for the SAP aqua ion, while a (1, 1, 1, 1, 1) distribution is found for the BTP aqua ion. Notably, none of the constructed molecular structures accurately reproduce the experimentally determined (1, 2, 2) distribution. The closest match is the TTP geometry with fluoride ions. We conclude that while these molecular structures provide valuable insights into the nature of the crystal field splitting in neodymium(III) ions, the idealized polyhedral geometries do not fully capture the properties of the neodymium(III) aqua ion as observed experimentally.
<figure><img src="image_10.png" /><figcaption>**Figure 4.** Computed energies for the Kramers doublets in the $^4I_{9/2}$ multiplet for neodymium(III) ions in molecular structures constructed from ideal polyhedra with CN = 8 (top), 9 (middle), and 10 (bottom). Three different ligands are used for each polyhedra: Point charges (black), fluoride ions (orange), and water molecules (blue). The experimental values for the Kramers doublets in the $^4I_{9/2}$ multiplet are included for reference (red). The evolution of the wavefunctions is shown with dotted lines. The molecular structure of the neodymium(III) aqua ion is shown for each</figcaption></figure>
polyhedron, and the point group symmetry for the donor atoms are indicated (* indicates point group if the hydrogens are considered). The color code of the aqua complexes is: Orange = neodymium, red = oxygen, and white = hydrogen.

CASSCF calculations on DFT optimized structures

The crystal fields depicted in Figure 4 results from fixed molecular structures. When these structures are relaxed and optimized using DFT, they all converge into a single molecular geometry for each coordination number: [Nd(H₂O)₈]³⁺, [Nd(H₂O)₉]³⁺, and [Nd(H₂O)₁₀] (see the Methodology Section). Figure 5 illustrates these optimized structures and Table 1 presents the results of a continuous symmetry operation measure (CSoM) analysis for each. The oxygen atoms in these complexes exhibit high symmetries: D₄d, D₃h, and D₄d for [Nd(H₂O)₈]³⁺, [Nd(H₂O)₉]³⁺, and [Nd(H₂O)₁₀]³⁺, respectively. However, when the positions of the hydrogen atoms are taken into account, the symmetry is reduced to S₈, D₃, and C₂. This reduction is significant, as the hydrogen atoms define the molecular orbital shape of water, which in turn generate the ab initio crystal field. Thus, the lower symmetry must be considered in the analysis. It is also worth noting that the symmetries correspond to those of SAP, TTP, and bcSAP polyhedra, with slight deviations in bond lengths and angles from the idealized polyhedra shown in Figure 4. The optimized molecular structures can be regarded as solutions to the Thomson problem,¹⁵ aligning with previous studies on the geometry of lanthanide(III) aqua ions.²⁷, ²⁹, ³⁰, ³⁶ Four independent reports suggest that the neodymium-oxygen distance in water is 2.51 ± 0.01 Å,¹⁹, ²², ²⁴, ¹¹⁰ However, none of the molecular structures within the first coordination sphere reproduce this exact bond length (see Table 1). A comparison of the Gibbs free energies of the [Nd(H₂O)₉]³⁺ structure and a DFT-optimized CN = 8 structure with a single explicit solvent molecule [Nd(H₂O)₈](H₂O)³⁺ revealed
that $[Nd(H_2O)_9]^{3+}$ is more stable by -46 kJ/mol. For details, refer to equation S3 and S4, as well as Table S40.

Table 1. Average distances between neodymium(III) and oxygen in the five optimized water structures. The point group G and the symmetry measure $\sigma_{Sym}(Q, G)$ for relevant point groups as determined by CSoM analysis considering either the oxygen atom positions or the molecular structure, when hydrogen is included in the analysis.

<table><thead><tr><th>Structure Q</th><th>[Nd(H<sub>2</sub>O)<sub>8</sub>]<sup>3+</sup></th><th>[Nd(H<sub>2</sub>O)<sub>9</sub>]<sup>3+</sup></th><th>[Nd(H<sub>2</sub>O)<sub>10</sub>]<sup>3+</sup></th><th>[Nd(H<sub>2</sub>O)<sub>8</sub>](H<sub>2</sub>O)<sub>15</sub>]<sup>3+</sup></th><th>[Nd(H<sub>2</sub>O)<sub>9</sub>](H<sub>2</sub>O)<sub>17</sub>]<sup>3+</sup></th></tr></thead><tbody><tr><td>Nd–O bond distance (Å)</td><td>2.44</td><td>2.48</td><td>2.53</td><td>2.46</td><td>2.51</td></tr><tr><td>Point group G (oxygen atoms)</td><td>D<sub>4d</sub></td><td>D<sub>3h</sub></td><td>D<sub>4d</sub></td><td>D<sub>4d</sub> / C<sub>2v</sub> / D<sub>2</sub></td><td>C<sub>4v</sub> / D<sub>3h</sub></td></tr><tr><td>σ<sub>Sym</sub>(Q<sub>O</sub>, G)</td><td>0.1</td><td>0</td><td>0.2</td><td>2.2 / 0.7 / 0.8</td><td>0.6 / 10.4</td></tr><tr><td>Point group (molecular structure)</td><td>S8</td><td>D<sub>3</sub></td><td>C<sub>2</sub></td><td>C<sub>1</sub></td><td>C<sub>1</sub></td></tr><tr><td>σ<sub>Sym</sub>(Q<sub>H2O</sub>, G)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr></tbody></table>

Figure 5 presents a comparison of the CASSCF-calculated crystal field splitting of the $^4I_{9/2}$ multiplet for the optimized neodymium(III) aqua complexes with experimentally determined energy levels (see Table S50 for detailed values). While none of the molecular structures achieve a perfect match, the $[Nd(H_2O)_9]^{3+}$ structure correctly reproduces the (1, 2, 2) distribution of levels. For all structures, the calculated absolute crystal field splitting is smaller than the experimental value, indicating that the crystal field strength is underestimated. This degree of
underrepresentation in the *ab initio* crystal field splitting is a common limitation at this level of theory for lanthanide(III) complexes.<sup>49, 53, 54</sup> To further investigate this discrepancy, we manually varied the Nd-O distance and recalculated the electronic energy levels, as shown in Figure 5 (and Figures S23 – S25). The Nd-O distance was manually varied between 2.4 and 2.55 Å, maintaining the relative geometry of the optimized structures. Following this adjustment, the oxygen atoms positions were fixed, and the hydrogen atoms were re-optimized. The wavefunction characteristics remained consistent throughout the process. The slope of the energy as a function of bond length provides insights into how the crystal field strength affects each energy level. This manipulation revealed that the splitting pattern of [Nd(H<sub>2</sub>O)<sub>9</sub>]<sup>3+</sup> consistently retains the (1, 2, 2) distribution. Moreover, decreasing the bond distance by 2 %, improved the agreement between the *ab initio* crystal field splitting and the experimentally determined values. In contrast, the energy splitting for [Nd(H<sub>2</sub>O)<sub>8</sub>]<sup>3+</sup> and [Nd(H<sub>2</sub>O)<sub>10</sub>]<sup>3+</sup> did not align with experimental results at any Nd-O distance.

To enhance the agreement between computational and experimental results, higher-level calculations were performed, incorporating larger basis sets and dynamic correlations effects. However, these adjustments did not lead to any significant improvements in the results (see Figures S26 and S27 for details).
<figure><img src="image_11.png" /><figcaption>Figure 5. Top: DFT Optimized neodymium(III) aqua complexes for CN = 8, 9, and 10. Color code: Orange = neodymium, red = oxygen, and white = hydrogen. Bottom: CASSCF computed electronic energy levels in the $^4I_{9/2}$ multiplet of neodymium(III) of three geometry-optimized structures for CN = 8, 9, and 10: $[Nd(H_2O)_8]^{3+}$, $[Nd(H_2O)_9]^{3+}$, and $[Nd(H_2O)_{10}]$. The experimentally obtained Kramers levels are also provided as reference.</figcaption></figure>

### CASSCF calculations on DFT optimized structures including the second coordination sphere

Finally, we modelled the molecular structure of the neodymium(III) aqua complex, including explicit solvent molecules in the second sphere. These structures are not true minima, as the complexity of the hydrogen-bonding network allows for a large family of structures with similar energies. Nevertheless, the converged structures selected here are considered representative of the
configurations typically found in solution. The neodymium(III) aqua complexes for CN = 8 with 15 water molecules [Nd(H₂O)₈](H₂O)₁₅³⁺ and CN = 9 with 17 water molecules in the second sphere [Nd(H₂O)₉](H₂O)₁₇³⁺ are shown in Figure 6. For [Nd(H₂O)₉](H₂O)₁₇³⁺, the Nd-O distance is 2.51 Å, matching the reported literature value of 2.51 ± 0.01 Å. ¹⁹, ²², ²⁴, ¹¹⁰ In contrast, the Nd-O distance for [Nd(H₂O)₈](H₂O)₁₅³⁺ is significantly shorter, at 2.46 Å (see Table 1). Table 1 also summarizes the average Nd-O bond lengths and the results of the CSoM analysis. Notably, the inclusion of the second sphere reduces the symmetry of the molecular structure to C₁ (see Table S12 for details). Considering only the positions of the coordinating oxygen atoms, the addition of solvent molecules to the CN = 8 structure alters the near-perfect D₄d symmetry to a distorted D₄d, which is likely better described as either C₂v or D₂d symmetry, as the bond lengths become asymmetrically distorted. For the CN = 9 structure, the solvent shell induces a change in symmetry from D₃h to C₄v. While the [Nd(H₂O)₉](H₂O)₁₇³⁺ molecular structure with cSAP symmetry (C₄v) provides a good match to the experimental data, we note that the [Nd(H₂O)₉]³⁺ structure with TTP symmetry also offers an acceptable description. Both are static models. However, in aqueous solution, the molecular structure of neodymium(III) is not static and is generally considered to be highly labile.¹⁷, ²⁵⁻²⁷, ²⁹, ³⁰, ³⁴, ³⁷ The TTP and cSAP are challenging to distinguish, as the geometrical distortion path between these two structures is minimal.¹⁶ Thus, a mixture of molecular structures is likely necessary to describe the experimentally obtained data.

Figure 6, shows the energy splitting of the ⁴I₉/₂, ⁴I₁₁/₂, and ⁴I₁₃/₂ multiplets, calculated both with and without the second sphere. The inclusion of the second sphere alters the energy levels for both complexes, with the difference being more pronounced for the CN = 8 structure. Comparing the calculated energy diagrams with the experimentally resolved energy levels, it is found that the CN = 9 structure, with the second sphere included, provides a near-perfect match, assuming the crystal
field strength is somewhat underestimated (refer to the bond length modifications in Figure 5). As in Figure 5, we adjusted the bond distances in the coordination sphere geometry optimized with the second sphere, followed by re-optimization of the hydrogen positions. However, this re-optimization did not include the second sphere, and the hydrogen positions did not converge to the same local minima when optimized without it. Despite these changes in molecular structure, it was also found that slightly decreasing the bond distance improved the match with experimental data (see Figure S28).

In summary, the optimized [Nd(H₂O)₉](H₂O)₁₇³⁺ molecular structure exhibits an *ab initio* crystal field splitting that closely matches the experimentally determined values. Additionally, CASSCF calculations on the [Nd(H₂O)₉](H₂O)₁₇³⁺ complex, both with and without the inclusion of the second-sphere water molecules, were performed (see Table S50). Minor differences were observed and it is evident that the second sphere primarily influences the positions of the water molecules in the inner sphere by orientating the hydrogen atoms through hydrogen bonding. Similar results were found in calculations of ytterbium(III) solvates.⁴⁴ In dysprosium(III) complexes with the DOTA (1,4,7,10-tetraazadodecane-1,4,7,10-tetraacedic acid) ligand, the orientation of a capping water molecule was also found to significantly affect both the electronic structure and the magnetic properties.⁶⁹, ⁷¹ Therefore, we can conclude that the orientation of the hydrogen atoms plays a crucial role.

While the [Nd(H₂O)₉](H₂O)₁₇³⁺ complex with cSAP symmetry (C₄v) provides a good match to the experimental data, we note that [Nd(H₂O)₉]³⁺ with TTP symmetry also offers an acceptable description. Both are static models. However, in aqueous solution, the molecular structure of neodymium(III) is not static and is generally considered to be highly labile.¹⁷, ²⁵, ²⁷, ²⁹, ³⁰, ³⁵, ³⁶ Thus, a mixture of molecular structures is likely necessary to describe the experimentally obtained data.
Our findings show that both the crystal field strength (see Figure 5), and the modification of the
crystal field symmetry—here engineered by adding the second-sphere solvent—improve the fit to
the experimental data. This suggests that molecular structures resembling TTP-like and cSAP-
like geometries can generate variations in the crystal field that are consistent with the experimental
observations. Additionally, since no distinct structures were observed experimentally (see Figures
S14–S16), a distribution of structures with similar electronic characteristics could account for the
broadening of the emission lines. In solution, the lines in the optical spectra are generally broader
than those observed in the solid state.⁴⁶,⁵⁴

<figure><img src="image_12.png" /></figure>
Figure 6. Left: ab initio computed and experimentally observed electronic energy levels of the
⁴I₁₉/₂, ⁴I₁₁/₂, and ⁴I₁₃/₂ multiplets of the neodymium(III) aqua ion. Calculations are provided for
complexes of CN = 8 and 9, geometry optimized with and without second-sphere water
molecules. Right: Molecular structure of complexes optimised with second-sphere water
molecules, [Nd(H₂O)₈](H₂O)₁₅³⁺ (CN = 8, top) and [Nd(H₂O)₉](H₂O)₁₇³⁺ (CN = 9, bottom).
Color code: Orange = neodymium, red = first sphere oxygen, purple = second sphere oxygen,
and white = hydrogen.

**Conclusions**

The electronic structure of neodymium(III) in water was spectroscopically resolved at 77 K. To
reproduce the experimentally determined crystal field splitting, a computational approach at the
CASSCF/RASSI-SO level was used to explore various models for the neodymium aqua complex.
Initially, 12 molecular structures constructed from ideal polyhedra were used to examine possible
variations in the crystal field splitting. Based on this investigation, we conclude that point charges
are insufficient for modeling the crystal field, and that the electron density of the donor atoms
plays a crucial role in defining the crystal field.

To obtain an *ab initio* crystal field that matches the experimental data, a second coordination
sphere had to be included, and the crystal field strength had to be increased. This was simulated
by shortening the Nd-O bond length (an unrealistic approximation). As anticipated, the calculated
molecular structure has a coordination number of nine. Surprisingly, the coordination geometry
resembles a capped square antiprism (cSAP), rather than the commonly assumed tricapped trigonal
prism (TTP). Although these two structures are difficult to differentiate without detailed analysis,
the Continuous Symmetry operation Measure (CSOM) clearly identifies the molecular structure as
having C₄ᵥ symmetry. It is likely that the molecular structure is highly dynamic, a factor that cannot
be captured by the static models used here. The next step is to explore the impact of structural
fluctuations on the electronic structure through dynamic studies.

ASSOCIATED CONTENT

(Supporting Information. Additional experimental data, fitting of high-resolution emission spectra, extended computational data, symmetry determination, computational details and molecular coordinates are available as Supporting Information.

The following files are available free of charge.

Supporting information (.PDF)

Molecular coordinates as xyz files (.zip)

AUTHOR INFORMATION

**Corresponding Author**

* TJS@chem.ku.dk

Funding sources

The Carlsberg Foundation, Villum Fonden, Novo Nordisk Fonden, Independent Research Fund
Denmark, Tranes Foundation, and William Demand Foundation.

ACKNOWLEDGMENT

We thank the Carlsberg Foundation, Villum Fonden, Novo Nordisk Fonden, Independent Research Fund Denmark, Tranes Foundation, William Demand Foundation, and the University of Copenhagen for generous support.
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