# An Expedited Route to Optical and Electronic Properties at Finite Temperature via Unsupervised Learning

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{2}(NCS)

_{2}]

^{4−}(dcbpy = 4,4′-dicarboxy-2,2′-bipyridine) in water solution, also called “N3

^{4−}” (Figure 1, right panel), which is a popular example of Ru-based dye sensitizers for solar cells and light-harvesting applications [95,96,97]. Light absorption by N3

^{4−}in the visible region induced excitation to a dense manifold of metal-to-ligand charge-transfer (

^{1}MLCT) states. N3

^{4−}photo-physical behavior is characterized by an ultrafast relaxation pathway among the singlet and triplet manifolds, induced by a complex interplay between closely spaced coupled electronic states, nuclear motion and solvent rearrangement [98,99,100,101,102,103,104,105], potentially influencing the dynamics and efficiency of the electron injection into a semiconductor substrate [106,107,108,109,110]. The N3

^{4−}complex, both for its dense

^{1}MLCT manifold and its conformational dynamics in the solution [111], represents therefore another ideal model system for testing an efficient MD/ML clustering approach for the simulation of electronic spectra including finite-temperature effects.

## 2. Results and Discussion

#### 2.1. The TCNE:$\pi $:1ClN Case Study

_{1}and S

_{2}excited states in Table 2. We recall that the weaker electronic transitions below 4.00 eV have a charge transfer (CT) nature (for S

_{1}and S

_{2}see ${\omega}_{\mathrm{CT}}$ charge transfer descriptor parameter in Table 2). Conversely, the very bright ones are characterized by electronic density reorganization occurring in the same molecular unit, hence they are of a local excitation (LE) character. For medoid 1, the TCNE is located on an edge of the 1ClN ring and it mainly contributes to the absorption bands above 2.50 eV (see light green curve in Figure 5), while for the S

_{0}–S

_{1}electronic transition at 1.807 eV characterized by a strong CT nature (${\omega}_{\mathrm{CT}}=0.968$) the probability is negligible, $f=0.002$. For medoids 2 and 4, the TCNE lies on the ring bearing the chlorine atom and they share roughly the same electronic properties in terms of transition probability and energy range (see orange and magenta curves in Figure 5, respectively). Both show absorption bands in all regions of the spectrum. In this case, the first two states S

_{1}and S

_{2}of both medoids contribute, respectively, to the bands at ∼2.00 and 2.80 eV. Also in these cases, the S

_{1}and S

_{2}states are characterized by a strong charge transfer nature as can be easily deduced from the values of the ${\omega}_{\mathrm{CT}}$ descriptor close to unity, reported in Table 2. In medoids 3 and 5, the TCNE is placed on the unfunctionalized six-membered ring of the 1ClN and we observe that the electronic properties show considerable differences. The electronic features of medoid 3 (violet curve in Figure 5) cover the entire spectral range considered, 1.50–5.00 eV, while only high energy electronic transitions (>3.50 eV) are bright for medoid 5 (dark green curve in Figure 5).

_{1}states of representative frames 4, 2 and 3, each having, in turn, a clear 1ClN → TCNE charge-transfer character. Analogously, the second band at ∼2.80 eV appears constituted by the S

_{2}CT states of medoids 1, 4, 3 and 2.

#### 2.2. The N3^{4−} Case Study

^{4−}dynamics at room temperature in water solution is characterized, on the one hand, by the rigidity of the dcbpy ligands, due to the chelation to the Ru center and, on the other hand, by the flexibility of the NCS

^{−}ligands, exploring conical-shaped regions (please see Figure 1, right panel, to recall the system under investigation). The vibrational dynamics induce therefore instantaneous deviations from the ideal ${C}_{2}$ symmetry, which could improve the transition probability of otherwise dark excited states [111]. The clustering procedure applied to the collected N3

^{4−}trajectory suggested a partition into seven distinct clusters. Projection into the two-dimensional principal component subspace (actually accounting for 56.9% of the total variance) shows indeed a quite clear separation between the clusters and the medoids representing them (Figure 7). Again, the observed partial superposition could be a spurious effect of data visualization through a low-dimensional PCA. According to the feature values shown by the cluster medoids, these representative structures (reported in Figure 8) actually seem to capture both the conformational (torsional) freedom of the coordinated NCS

^{−}ligands (${\varphi}_{1}$ and ${\varphi}_{2}$ torsional angles) and the different degrees of asymmetry sampled by the N3

^{4−}dynamics (Table 3, please refer also to Section 3.2 for N3

^{4−}features definitions). In particular, the values of continuous symmetry measure of deviation from ${C}_{2}$ symmetry (${C}_{2}$-CSM, Section 3.2) most sampled by the MD trajectory (distribution maxima at 0.09, 0.17, 0.22, 0.31, Figure 9) are close to the values by the cluster medoids, further confirming the representation capabilities of the latter.

^{4−}electronic spectrum was simulated up to ∼3.7 eV, comprising the two experimentally characterized bands at ∼2.50 and ∼3.36 eV [114]. The spectra calculated for each cluster medoid actually slightly differ in the absorption band positions (energies) and intensities, since the medoids represent different regions of the accessible conformational space (Figure 10). In particular, the spectrum obtained from the only seven representative frames can actually quite well reproduce that from the complete MD sampling at $T=298\phantom{\rule{3.33333pt}{0ex}}K$ in water solution (Figure 11), although with an increased sub-structure, due to the lower number of frames involved in the spectrum calculation. The selection of representative frames through a clustering analysis allowed one therefore to achieve a remarkable ∼70-fold decrease in the total computational cost for N3

^{4−}electronic spectrum simulation.

^{4−}spectral characterization, which could be otherwise difficult to perform. In particular, the calculated band at 2.07 eV (Figure 11) results from medoid 1 S

_{2}, medoid 2 S

_{2}, medoid 7 S

_{1}, medoid 6 S

_{1}and medoid 5 S

_{2}states, which are mainly Ru → (dcbpy)

_{2}(${\Omega}_{RP}\approx 0.55$, ${\Omega}_{SP}\approx 0.25$, Table 4) CT states. Analogously, medoid 6 S

_{5}, medoid 7 S

_{5}, medoid 2 S

_{6}, medoid 1 S

_{5}and medoid 3 S

_{5}, with similar metal-to-ligand charge-transfer (MLCT) spatial features, contribute to the more intense calculated band at 2.39 eV. The higher-energy bands are characterized instead by a less homogeneous set of excited states. In fact, the calculated band at 3.23 eV results from medoid 2 S

_{8}, medoid 7 S

_{13}Ru → (dcbpy)

_{2}states (${\Omega}_{RP}\approx 0.55$, ${\Omega}_{SP}\approx 0.25$, Table 4), medoid 2 S

_{18}Ru → (dcbpy)

_{2}state, but with an increased dcbpy localized-excitation character (${\Omega}_{RP}\approx 0.40$, ${\Omega}_{PP}\approx 0.30$), medoid 4 S

_{15}Ru → (dcbpy)

_{2}state, with increased (NCS)

_{2}donor contribution (${\Omega}_{RP}\approx 0.50$, ${\Omega}_{SP}\approx 0.40$) and medoid 7 S

_{19}state, which is mainly an (NCS)

_{2}→ (dcbpy)

_{2}CT state (${\Omega}_{SP}\approx 0.40$, ${\Omega}_{RP}\approx 0.30$). The close calculated 3.38 eV band has instead a quite different average character. In fact, the contributing medoid 1 S

_{40}state is mostly an (NCS)

_{2}→ (dcbpy)

_{2}CT state (${\Omega}_{SP}\approx 0.60$), medoid 2 S

_{37}and medoid 5 S

_{33}states have an increased localized character (${\Omega}_{SP}\approx 0.40$, ${\Omega}_{PP}\approx 0.30$), while medoid 3 S

_{21}and medoid 6 S

_{34}are localized excitations on dcbpy ligands (${\Omega}_{PP}\approx 0.60$ and $\approx 0.50$, respectively).

## 3. Materials and Methods

#### 3.1. Ab Initio Molecular Dynamics

^{4−}model systems were sampled through ab initio molecular dynamics simulations. In particular, the Atom-centered Density Matrix Propagation (ADMP) method was employed: the density matrix in an orthonormalized atomic basis is included in an extended Lagrangian as an additional degree of freedom and propagated together with the nuclear degrees, avoiding a self-consistent procedure at each step [115,116,117,118,119].

^{4−}system was simulated instead for 8.6 ps with a 0.1 fs time step [111]. A velocity rescaling every 1 ps allowed to keep a 298 K temperature. Explicit water solvation was included in the N3

^{4−}ground state sampling, in order to better model the specific solute–solvent interactions at the several solvation sites. A 22 Å-radius spherical solvent box (∼1500 molecules) was extracted from a pre-equilibrated cubic one and placed around N3

^{4−}. A hybrid quantum mechanics/molecular mechanics potential was employed: B3LYP/def2-SVP [135] for the QM portion (the N3

^{4−}molecule) with associated electronic core potential for the Ru atom [136] and the TIP3P water model [137] for the MM part (the water spherical box), re-parametrized to allow a bending motion [11]. The QM and MM potentials were combined through the ONIOM QM/MM scheme [138,139,140], including the MM charges into the QM hamiltonian (i.e., an “electronic embedding”). General AMBER Force Field [141] atom types (and so van der Waals non-bonding parameters) were assigned, moreover, to N3

^{4−}atoms. Non-periodic boundary conditions were introduced through a hybrid explicit/implicit solvent model. Long-range electrostatic effects and short-range dispersion–repulsion forces between the explicit and the bulk solvent were, respectively, modeled through C-PCM self-reaction field and an empirical confining potential, which has to be parametrized for the specific solvent model [10,124,142,143,144]. We refer the reader to previous works for more details about the ab initio molecular dynamics simulations of the model systems and the employed potentials [34,94,111].

#### 3.2. Feature Selection and Clustering of Molecular Dynamics Trajectories

^{4−}, instead, a continuous symmetry measure of deviation from the ideal ${C}_{2}$ symmetry [145,146,147], calculated as the minimized root-mean-square deviation from the images generated through the ${C}_{2}$ symmetry operations, was considered. In particular, ${C}_{2}$-CSM was evaluated on the smallest subset of N3

^{4−}atoms showing a symmetry not higher than ${C}_{2}$, as the complete molecule. Since the non-linearity of the NCS

^{−}coordination in the water solution (C(NCS)-N(NCS)-Ru angle less than 180°) and their torsional mobility were previously recognized [111], the C(NCS)-N(NCS)-Ru-N(dcbpy) dihedrals describing the NCS

^{−}orientations were also included. In this regard, to avoid problems due to the periodicity around ±180°, each dihedral $\varphi $ was included as a $(cos(\varphi ),sin(\varphi \left)\right)$ pair to keep a metric feature space [148]. The MD datasets in the feature space $\mathit{X}$ were standardized (i.e., shifted to zero mean and scaled to unit variance) before following analyses.

^{4−}model systems in their respective feature spaces were clusterized with the K-Medoids algorithm [152], since the cluster medoids, which are representative of the corresponding clusters, are trajectory frames themselves and, compared to K-Means centroids, should be less sensitive to possible outliers [151].

#### 3.3. Dimensionality Reduction for MD Data Visualization

^{4−}trajectories was performed only for data visualization purposes on two-dimensional plots and not as a pre-processing step for clustering analysis. In fact, the dimensionalities of their respective feature spaces (Section 3.2) are actually quite small, likely not involving any “curse of dimensionality” issues.

#### 3.4. Excited State Characterization and Spectra Simulations

^{4−}excited states were computed with the linear-response TD-DFT approach at CAM-B3LYP/GD3/C-PCM(DCM)/6-31+G(d,p) and B3LYP/C-PCM(water)/def2-SVP/SDD(Ru) levels of theory, respectively. Electronic spectra in the solution at $T=298$ K were simulated on 500 frame subsets of the collected MD samplings (i.e., every 20 fs and 17.2 fs, respectively). The first 8 and 40 singlet excited states were calculated for TCNE:$\pi $:1ClN and N3

^{4−}, respectively. The complete spectra were obtained by summation of Gaussian-shaped contributions over each frame and each calculated excited state:

_{0}excitation.

## 4. Conclusions

_{2}(NCS)

_{2}]

^{4−}complex in solution at room temperature. Cluster medoids, taken as representative structures, were found and analyzed in terms of main structural parameters, principal component dynamics, electronic excitations and charge transfer indices, showing how such medoids can satisfactorily cover the system dynamics and optical properties with a very good agreement with experiments.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

1ClN | 1-chloronaphthalene |

ADMP | Atom-centered density matrix propagation |

C-PCM | Conductor-like polarizable continuum model |

CT | Charge transfer |

dcbpy | 4,4′-dicarboxy-2,2′-bipyridine |

DCM | dichloromethane |

DFT | Density functional theory |

LMCT | Ligand-to-metal charge-transfer |

MD | Molecular dynamics |

ML | Machine learning |

MLCT | Metal-to-ligand charge-transfer |

MM | Molecular mechanics |

N3^{4−} | [Ru(dcbpy)_{2}(NCS)_{2}]^{4−} |

PC | Principal component |

PCA | Principal component analysis |

PES | Potential energy surface |

QM | Quantum mechanics |

SDF | Spatial distribution function |

TCNE | Tetracyanoethylene |

TD-DFT | Time dependent density functional theory |

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**Figure 1.**Case studies investigated in the present work. The TCNE:$\pi $:1ClN non-covalent dimer and Ru(II) complex ([Ru(dcbpy)

_{2}(NCS)

_{2}]

^{4−}or “N3

^{4−}”, dcbpy = 4,4′-dicarboxy-2,2′-bipyridine) are presented from left to right, respectively (Carbon is in gray, Hydrogen in white, Chlorine in green, Sulphur in yellow, Oxygen in red, Nitrogen in blue, Ruthenium in pink).

**Figure 2.**Side, front and top views of the spatial distribution function of the center-of-mass of the TCNE acceptor monomer around the 1ClN subunit.

**Figure 3.**Structures of the five cluster medoids in top (

**left**panel) and side (

**right**panel) views. The TCNE and 1ClN are represented as ball and stick in blue and red, respectively. The color code is uniform with that of Figure 4.

**Figure 4.**TCNE:$\pi $:1ClN trajectory in the features’ first two principal components space. Cluster partition is represented through different colors. Cluster medoids are also highlighted (as star symbols). The color scheme adopted is kept fixed throughout this section.

**Figure 5.**TCNE:$\pi $:1ClN absorption spectrum (in eV) calculated at TD-CAM-B3LYP/6-31G(d,p)/GD3/C-PCM(DCM) level of theory from each medoid and as the sum spectrum of the structures representative of the conformational equilibrium in the ground state. The color code is presented in the graph legend. The sum spectrum (red dashed curve) was obtained as the sum of individual medoid contributions (presented in the figure as well, see color legend), each one already multiplied by the k-th cluster population. See Equation (5) and the procedure explained in Section 3.4 for more details.

**Figure 6.**

**Top**panel: comparison of TCNE:$\pi $:1ClN simulated absorption spectra in the 1.50–3.50 eV range.

**Bottom**panel: experimental UV-Vis spectrum, retrieved from Ref. [93], of the TCNE:$\pi $:1ClN complex measured in DCM solution (molar absorptivity, $\epsilon $). The color code is presented in the graph legend.

**Figure 7.**N3

^{4−}trajectory in the features’ first two principal components subspace. Cluster partition is represented through different colors. Cluster medoids are also highlighted (as star symbols).

**Figure 8.**Structures of the N3

^{4−}seven cluster medoids. The atoms determining the features employed for clustering analysis are highlighted as ball and stick. The color code is uniform with that of Figure 7.

**Figure 9.**Distribution of ${C}_{2}$-CSM symmetry deviation parameter from N3

^{4−}trajectory in water solution. Values of the medoid structures from trajectory clustering analysis are also shown as vertical bars (with arbitrary heights). The color code is uniform with that of Figure 7.

**Figure 10.**N3

^{4−}absorption spectra (in eV) calculated at TD-B3LYP/C-PCM/def2-SVP/SDD(Ru) level of theory from each medoid, weighted by the population of the corresponding cluster and the spectrum resulting from the sum over the medoids (red dashed curve). The color code is presented in the graph legend. See Equation (5) and the procedure explained in Section 3.4 for more details.

**Figure 11.**

**Top**panel: comparison of N3

^{4−}simulated absorption spectra in the 1.50–4.00 eV range.

**Bottom**panel: experimental N3

^{4−}UV-Vis spectrum, retrieved from Ref. [114], measured in water solution. The color code is presented in the graph legend.

**Table 1.**Clustering feature values of the five cluster medoids from TCNE:$\pi $:1ClN trajectory. ${\theta}_{r}$: rotation angle (angle between versors normal to the two molecular planes, degrees), ${\widehat{n}}_{r}$: rotation axis (versor normal to the former ones), ${\overrightarrow{r}}_{N-E}$: relative position vector (between 1ClN and TCNE geometric centers). Vector quantities are given as cartesian components (Å) in a fixed frame of reference.

Medoid | ${\mathit{\theta}}_{\mathit{r}}$ | ${\mathit{n}}_{\mathit{r},\mathit{x}}$ | ${\mathit{n}}_{\mathit{r},\mathit{y}}$ | ${\mathit{n}}_{\mathit{r},\mathit{z}}$ | ${\mathit{r}}_{\mathit{N}-\mathit{E},\mathit{x}}$ | ${\mathit{r}}_{\mathit{N}-\mathit{E},\mathit{y}}$ | ${\mathit{r}}_{\mathit{N}-\mathit{E},\mathit{z}}$ |
---|---|---|---|---|---|---|---|

1 | 16.68 | −0.660 | 0.398 | 0.638 | 1.513 | 2.847 | 2.190 |

2 | 14.60 | −0.816 | 0.141 | 0.561 | 2.604 | 1.673 | 1.852 |

3 | 19.36 | 0.723 | −0.407 | −0.558 | 1.543 | 2.444 | 2.276 |

4 | 15.19 | 0.714 | 0.021 | −0.699 | 2.666 | 1.773 | 1.736 |

5 | 16.27 | 0.112 | −0.736 | 0.668 | 1.833 | 2.401 | 2.436 |

**Table 2.**Characterization of S

_{1}and S

_{2}excited states of TCNE:$\pi $:1ClN cluster medoids. ${\nu}_{i}$ (eV): vertical excitation energy, ${f}_{i}$: oscillator strength (arb. units), ${\Omega}_{AB}$: transition density population analysis for A (hole) and B (electron) fragments, ${\omega}_{\mathrm{CT}}$: charge transfer descriptor (please refer to Section 3.4 for definitions). Fragment labels: E: TCNE, N: 1ClN.

Medoid | ${\mathit{\nu}}_{\mathit{i}}$ | ${\mathit{f}}_{\mathit{i}}$ | ${\Omega}_{\mathit{EE}}$ | ${\Omega}_{\mathit{EN}}$ | ${\Omega}_{\mathit{NE}}$ | ${\Omega}_{\mathit{NN}}$ | ${\mathit{\omega}}_{\mathit{CT}}$ | |
---|---|---|---|---|---|---|---|---|

1 | ${S}_{1}$ | 1.807 | 0.002 | 0.014 | 0.000 | 0.968 | 0.018 | 0.968 |

${S}_{2}$ | 2.973 | 0.035 | 0.016 | 0.000 | 0.965 | 0.018 | 0.966 | |

2 | ${S}_{1}$ | 2.003 | 0.052 | 0.031 | 0.001 | 0.943 | 0.025 | 0.944 |

${S}_{2}$ | 2.835 | 0.014 | 0.018 | 0.001 | 0.955 | 0.027 | 0.955 | |

3 | ${S}_{1}$ | 1.928 | 0.026 | 0.019 | 0.000 | 0.963 | 0.017 | 0.963 |

${S}_{2}$ | 2.713 | 0.009 | 0.018 | 0.000 | 0.963 | 0.018 | 0.964 | |

4 | ${S}_{1}$ | 2.063 | 0.067 | 0.031 | 0.001 | 0.938 | 0.030 | 0.939 |

${S}_{2}$ | 2.863 | 0.013 | 0.014 | 0.000 | 0.953 | 0.032 | 0.954 | |

5 | ${S}_{1}$ | 2.084 | 0.005 | 0.009 | 0.000 | 0.980 | 0.012 | 0.980 |

${S}_{2}$ | 2.994 | 0.005 | 0.017 | 0.000 | 0.971 | 0.012 | 0.971 |

**Table 3.**Clustering feature values of the seven cluster medoids from N3

^{4−}trajectory. ${\varphi}_{1}$: C(NCS1)-N(NCS1)-Ru-N(dcbpy) dihedral angle (degrees), ${\varphi}_{2}$: C(NCS2)-N(NCS2)-Ru-N(dcbpy) dihedral angle (degrees), ${C}_{2}$-CSM: continuous symmetry measure for deviation from ${C}_{2}$ symmetry.

Medoid | ${\mathit{\varphi}}_{1}$ | ${\mathit{\varphi}}_{2}$ | ${\mathit{C}}_{2}$-CSM |
---|---|---|---|

1 | −30.77 | 5.57 | 0.176 |

2 | −54.10 | 107.01 | 0.172 |

3 | −143.43 | 140.61 | 0.219 |

4 | 52.26 | −131.41 | 0.174 |

5 | 69.09 | 83.05 | 0.215 |

6 | −127.10 | 40.02 | 0.116 |

7 | 107.85 | −137.07 | 0.352 |

**Table 4.**Characterization of the excited states of N3

^{4−}cluster medoids most contributing to the calculated absorption bands. ${\nu}_{i}$ (eV): vertical excitation energy, ${f}_{i}$: oscillator strength, ${\Omega}_{AB}$: transition density population analysis for A (hole) and B (electron) fragments, ${\omega}_{\mathrm{CT}}$: charge transfer descriptor (please refer to Section 3.4 for definitions). Fragment labels: S: (NCS)

_{2}, R: Ru, P: (dcbpy)

_{2}.

Medoid | ${\mathit{\nu}}_{\mathit{i}}$ | ${\mathit{f}}_{\mathit{i}}$ | ${\Omega}_{\mathit{SP}}$ | ${\Omega}_{\mathit{RP}}$ | ${\Omega}_{\mathit{PP}}$ | ${\mathit{\omega}}_{\mathit{CT}}$ | |
---|---|---|---|---|---|---|---|

1 | ${S}_{2}$ | 2.097 | 0.029 | 0.269 | 0.566 | 0.113 | 0.858 |

${S}_{5}$ | 2.342 | 0.086 | 0.284 | 0.542 | 0.119 | 0.851 | |

${S}_{40}$ | 3.556 | 0.056 | 0.578 | 0.235 | 0.127 | 0.851 | |

2 | ${S}_{2}$ | 2.083 | 0.026 | 0.235 | 0.580 | 0.111 | 0.846 |

${S}_{6}$ | 2.569 | 0.079 | 0.323 | 0.507 | 0.103 | 0.862 | |

${S}_{8}$ | 2.840 | 0.055 | 0.275 | 0.536 | 0.114 | 0.843 | |

${S}_{18}$ | 3.237 | 0.046 | 0.265 | 0.422 | 0.271 | 0.713 | |

${S}_{37}$ | 3.498 | 0.063 | 0.401 | 0.219 | 0.310 | 0.662 | |

3 | ${S}_{5}$ | 2.536 | 0.118 | 0.294 | 0.545 | 0.108 | 0.862 |

${S}_{21}$ | 3.389 | 0.043 | 0.133 | 0.215 | 0.583 | 0.391 | |

4 | ${S}_{15}$ | 2.983 | 0.056 | 0.366 | 0.511 | 0.087 | 0.894 |

5 | ${S}_{2}$ | 2.114 | 0.022 | 0.276 | 0.572 | 0.086 | 0.876 |

${S}_{33}$ | 3.454 | 0.057 | 0.373 | 0.277 | 0.289 | 0.693 | |

6 | ${S}_{1}$ | 2.041 | 0.025 | 0.242 | 0.627 | 0.093 | 0.884 |

${S}_{5}$ | 2.433 | 0.141 | 0.267 | 0.570 | 0.109 | 0.859 | |

${S}_{34}$ | 3.472 | 0.030 | 0.335 | 0.099 | 0.521 | 0.469 | |

7 | ${S}_{1}$ | 1.935 | 0.029 | 0.249 | 0.575 | 0.136 | 0.842 |

${S}_{5}$ | 2.316 | 0.101 | 0.267 | 0.551 | 0.121 | 0.844 | |

${S}_{13}$ | 3.022 | 0.056 | 0.278 | 0.553 | 0.148 | 0.840 | |

${S}_{19}$ | 3.165 | 0.059 | 0.427 | 0.312 | 0.212 | 0.773 |

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## Share and Cite

**MDPI and ACS Style**

Perrella, F.; Coppola, F.; Rega, N.; Petrone, A.
An Expedited Route to Optical and Electronic Properties at Finite Temperature via Unsupervised Learning. *Molecules* **2023**, *28*, 3411.
https://doi.org/10.3390/molecules28083411

**AMA Style**

Perrella F, Coppola F, Rega N, Petrone A.
An Expedited Route to Optical and Electronic Properties at Finite Temperature via Unsupervised Learning. *Molecules*. 2023; 28(8):3411.
https://doi.org/10.3390/molecules28083411

**Chicago/Turabian Style**

Perrella, Fulvio, Federico Coppola, Nadia Rega, and Alessio Petrone.
2023. "An Expedited Route to Optical and Electronic Properties at Finite Temperature via Unsupervised Learning" *Molecules* 28, no. 8: 3411.
https://doi.org/10.3390/molecules28083411