Electronic and Optical Properties of Small Metal Fluoride Clusters.

We report a systematic investigation on the electronic and optical properties of the smallest stable clusters of alkaline-earth metal fluorides, namely, MgF2, CaF2, SrF2, and BaF2. For these clusters, we perform density functional theory (DFT) and time-dependent DFT (TDDFT) calculations with a localized Gaussian basis set. For each molecule ((MF2) n , n = 1-3, M = Mg, Ca, Sr, Ba), we determine a series of molecular properties, namely, ground-state energies, fragmentation energies, electron affinities, ionization energies, fundamental energy gaps, optical absorption spectra, and exciton binding energies. We compare electronic and optical properties between clusters of different sizes with the same metal atom and between clusters of the same size with different metal atoms. From this analysis, it turns out that MgF2 clusters have distinguished ground-state and excited-state properties with respect to the other fluoride molecules. Sizeable reductions of the optical onset energies and a consistent increase of excitonic effects are observed for all clusters under study with respect to the corresponding bulk systems. Possible consequences of the present results are discussed with respect to applied and fundamental research.

Introduction

For years, fluorides and fluorite-type crystals have attracted much interest for their intrinsic optical properties and their potential applications in optoelectronic devices, in particular for those operating in the ultraviolet (UV). CaF2, e.g., has a direct band gap at the Γ point of 12.1 eV and an indirect gap estimated around 11.8 eV.[1] Calcium fluoride, as well as other difluorides, is a highly ionic system that adopts the cubic Fm3̅m crystal structure with three atoms per unit cell.[2]

Due to their importance for applied and basic research, experimental studies on CaF2 and other difluorides have been carried out for decades. Different experimental techniques were applied to study the optical, structural, and electronic properties of these compounds, such as discharge tube experiments,[3] reflectance,[1] dielectric loss techniques,[4] photoelectron spectrometry measurements,[5] light absorption[6] and spectrophotometry techniques,[7] neutron diffraction,[8] and polarized-light methods.[9,10]

From the theoretical and computational point of view, the most recent results concern optical and electronic excitations. Linear and non-linear optical properties of insulators with a cubic structure (CaF2, SrF2, CdF2, BaF2) were calculated within the first-principles orthogonalized linear combination of atomic orbitals (OLCAO) method.[11] Other studies, focused on point defects in CdF2, were performed within the plane wave-pseudopotential (PW-PP) method.[12] Electronic excitation and energy bands of CaF2 and other fluorides were determined by the quasiparticle DFT-GW approach, using a plane wave basis set and ionic pseudopotentials, i.e., a PW-PP scheme.[13] Using the hybrid B3PW functional, the electronic structures of defected fluorides, namely, CaF2 and BaF2, were evaluated.[14−16] After an iterative procedure using an effective Hamiltonian, the imaginary part of the dielectric function ε2 was calculated for CaF2,[17] within a PW-PP scheme and considering a screened interaction for electron–hole (e–h) parts. Native and rare-earth-doped defect complexes in β-PbF2 were studied by atomistic simulations.[18]

Bulk cubic fluorides have been investigated yet by some researchers of the present collaboration. Some of us investigated the cubic fluorides in detail by means of DFT within the local density approximation (LDA) for the exchange-correlation energy.[19] The ground-state electronic properties were calculated for the bulk cubic structures of CaF2, SrF2, BaF2, CdF2, HgF2, and β-PbF2, using a plane wave expansion of the wave functions. The results showed good agreement with existing experiments and previous theoretical predictions. General trends of the ground-state parameters, the electronic energy bands, and transition energies for all fluorides considered have been given and discussed in detail. Moreover, for the first time, results for HgF2 were presented. In following works, the same authors studied the electronic and optical properties of two of the above bulk compounds, namely, CdF2 and BaF2, using state-of-the-art computational techniques for the one- and two-particle effects.[20,21] Also, in these cases, the obtained results were in good agreement with existing experimental data.

In the present paper, we face the problem of the calculation of the electronic and optical properties of the three smallest clusters of alkaline-earth metal fluorides, namely, MF2 with M = Mg, Ca, Sr, Ba. The interest in these finite fluoride systems rises from the fact that clusters are the smallest pieces of matter that can exist in a stable form. Due to the small volume/surface ratio and the high number of unsaturated bonds, small clusters usually show different structural, electronic, and optical and reactive features with respect to their bulk parent compounds. They can be considered as prototypical examples of fragments of larger fluoride samples.

On the experimental side, it is known that monomers, dimers, and trimers (namely, MX2 , (MX2)2 , (MX2)3 with M as the metal atom and X as the halide atom) are contained in high-temperature vapors of alkaline- earth dihalides.[22] While the monomers have already been studied either experimentally and theoretically, less information is available for dimers and trimers. For example, for the dimers, information has been obtained from infrared (IR) and Raman spectra of their vapors trapped in solid matrices.[23] For MgF2 dimers and trimers, an extended comparison between Hartree–Fock/Moeller–Plesset calculations and experimental data has been performed by Francisco et al.[24] Studies on other families of fluoride clusters have been also performed: e.g., the lowest-energy isomers of coinage metal fluoride and chloride clusters have been analyzed systematically within a DFT scheme.[25]

The present study is further motivated by the recent interest on the electronic and optical properties of alkaline-earth metal fluoride clusters emerging in the literature.[26−29] Here, we focus on (MX2) systems with n = 1, 2, 3. Therefore, our attention is dedicated to subnanometric fluoride shards for the first smallest three members of each family for which larger deviations from the corresponding bulk solids in the structural, electronic, and optical properties are expected.

Therefore, the electronic and optical properties of the clusters will then be directly compared with their bulk counterparts.

Results and Discussion

Ground State and Morphological Properties

In Figure , the geometrical structures of the clusters studied here are reported. According to the present calculations, only MgF2 shows a linear structure with D∞ point group symmetry. This finding is in accordance with the results of a recent paper by Pandey and coworkers.[28] On the other hand, for Ca, Sr, and Ba fluoride monomers, we find a bent configuration, with point symmetry group C2, in accordance with Levy and Hargittai,[26] Koput and Roszczak[27] (for CaF2), and Calder et al.[29] In Table , our results for the M–F bond angles and distances of the monomers are in agreement with the results of Levy and Hargittai[26] with deviations within 2% (angles) and 0.5% (distances). The comparison of angles with experiments is also good.[29] On the contrary, Pandey and coworkers present linear geometries for all the monomers. A linear structure for Ca, Sr, and Ba fluoride monomers is neither confirmed here nor by other previous theoretical and experimental results.[26] Their conclusions on the structure of the monomers could be ascribed to the use of a less sophisticated basis set. This fact was addressed for CaF2 by Koput and Roszczak[27] and for all the systems by Levy and Hargittai.[26]

Figure 1

Structure of (MF2) clusters with n being the number of MF2 units. The point group symmetry for each cluster is also reported. (a) M = Mg. (b) M = Ca, Sr. (c)M = Ba. Fluorine atoms are in cyan color, metal atoms in other colors.

Table 1

M–Fa Angles and Distances for the MF2 Monomers after Present Calculations and after Other Calculations and Experimentsb

results	angle (°)	distance (Å)
CaF2
present	143.7	1.997
other-1	142.4	1.990
other-2	140
SrF2
present	131.0	2.130
other-1	129.0	2.120
other-2	108
BaF2
present	119.9	2.245
other-1	117.8	2.236
other-2	100

M represents the metal atom and F the fluorine one.

Other-1 corresponds to results of calculations by Levy and coworkers with B3LYP XC functionals.[26] Other-2 corresponds to the M–F angles measured with the matrix isolation infrared spectroscopy technique (MI-IR).[29]

Besides the case of the MgF2 monomer, we also found differences with previous results in the case of (BaF2)3. In fact, (BaF2)3 clusters show a different geometry with respect to (CaF2)3 and (SrF2)3. According to our findings, the n = 3 Ba cluster belongs to the C point group symmetry, while this cluster is assigned to the C2 group by Pandey and coworkers.[28]

It is well known from the literature[24,28] that (MF2), for n = 2, 3, shows many different local minima at energies as low as few tenths of eV over the ground state (global minimum), so we have performed a search for the lower-energy isomers, as outlined in the Supporting Information. The results show that the ground-state structures previously discussed are in fact the lowest-energy isomers.

It is interesting to analyze also the electronic properties of the ground state and their chemical trends. We refer to Table for these calculated observables. The average distance between the fluorine and metal atom within the clusters behaves differently in the case of the Mg fluoride clusters. In this case, the average distance is at a maximum for n = 2, while for all the other clusters (Ca, Sr, and Ba), that observable is larger for n = 3. Our results compare well with those of Pandey and coworkers.[28] From these results, it seems that the larger is the number of MF2 units present in the cluster, the larger is the average distance between the metal and the fluorine atoms.

Table 2

Ground-State Properties of the Fluoride Clustersa

cluster	RM–F [Å]	Efrag [eV]	IEV [eV]	EAV [eV]
MgF2	1.747 (1.72)		12.94 (12.1)	0.45 (−0.4)
(MgF2)2	1.997 (2.06)	2.47 (3.6)	12.19 (11.5)	0.71 (−0.4)
(MgF2)3	1.867 (1.86)	2.56 (3.5)	11.91(11.3)	0.76 (−0.4)
CaF2	1.997 (2.06)		11.52 (9.3)	0.69 (0.0)
(CaF2)2	2.166 (2.2)	2.67 (4.0)	11.08 (8.9)	1.03 (1.6)
(CaF2)3	2.187 (2.24)	2.83 (4.8)	10.68 (8.9)	0.84 (1.1)
SrF2	2.130 (2.2)		10.94 (9.0)	0.78 (0.1)
(SrF2)2	2.316 (2.31)	2.56 (3.9)	10.55(10.2)	0.96 (1.4)
(SrF2)3	2.332 (2.38)	2.83 (4.7)	10.13 (8.8)	0.80 (0.3)
BaF2	2.245 (2.33)		10.58 (8.1)	0.61 (3.6)
(BaF2)2	2.29 (2.47)	2.31(3.7)	9.93 (7.7)	0.69 (2.1)
(BaF2)3	2.56 (2.52)	2.58 (4.5)	9.70 (7.4)	0.56 (2.4)

The average distance between the metal and the fluorine atom RM–F, the fragmentation energy Efrag, the vertical ionization energy IEV, and vertical electron affinity EAV are listed. In parentheses, the corresponding data from the work of Pandey et al.[28] are given.

Another important observable for the molecules is their fragmentation energy, i.e., the energy required to remove an MF2 unit from an (MF2) cluster[28]where E((MF2)) represents the total ground-state energy of the cluster made by n MF2 units. Efrag(n) is the cost in energy to extract an MF2 unit from an n-unit cluster. In contrast to the results of Pandey and coworkers, it seems that for n = 2, 3, the Mg fluoride clusters show a different behavior compared to the clusters with Ca, Sr, and Ba metal atoms. While in the latter cases, the fragmentation energy is larger at n = 3 by about 20%, in the case of the Mg fluoride clusters, this energy is almost the same for n = 2, 3.

The values for the fragmentation energies reported in Table are of the same order of magnitude as those reported by the group of Pandey, even if slightly smaller. Passing from n = 2 to n = 3, for all the clusters considered here, one obtains larger energies for the creation of an MF2 unit. For n = 2, the smallest fragmentation energy appears for the Ba fluoride cluster while the largest one occurs for the Ca clusters. For n = 3 the smallest fragmentation energy is reported for the Mg fluoride cluster, while the largest is found for the fluoride Ca and Sr clusters. The differences w.r.t. Pandey et al. could be a direct consequence of the different choices for the basis set, which led in some cases also to different ground-state geometries.[28]

The comparison of the vertical ionization energy and the vertical electron affinity is also interesting. These quantities are defined aswhere EN0, EA0, and EC0 are total energies defined in the section Computational Methods.

The MgF2 clusters exhibit the largest IEV, with calculated values around 12.3 eV, while for CaF2, SrF2, and BaF2, the results and trends compare well with those reported by Pandey et al.[28] The ionization energy for the MgF2 monomer, 12.94 eV, is also in fair agreement with experimental findings 13.3 ± 0.3 eV[24] and 13.6 ± 0.2 eV.[24] No clear trend is found for the absolute values of the vertical electron affinity EAV, with values around 0.6 eV for MgF2 and BaF2 clusters and close to 0.8 eV for the remaining ones. This is in contrast to the values reported by Pandey and coworkers, who found a negative electron affinity for MgF2 clusters. We remark that the evaluation of EA for clusters within DFT suffers per se of well-known problems. Even in the atomic case, electron affinities cannot be correctly obtained because the long-range behavior of the exchange-correlation potential is incorrect for negative ions.[30] These facts are related to the need to correctly consider self-interaction correction (SIC) terms of a system (molecule) to evaluate properly its corresponding EA.[31] Therefore, for this observable, the comparison with other results, obtained within different methods, could be less satisfactory.[32,33]

Electronic Excitations and Optical Properties

We turn here to the study of the electronic excitation properties, discussing the optical absorption spectra of the clusters. From total energy differences, it is possible to evaluate important electronic observables. In fact, the ΔSCF technique enables us to calculate the fundamental gap for each cluster. From the ΔSCF method, one obtains the quasiparticle gap (or fundamental gap) by the expression[34,35]

Related to this, from the knowledge of the optical gap energy Eopt, defined as the first optically active transition in the absorption spectrum, an estimate of the exciton binding energy can be obtained through the difference Eb = Egap– Eopt.

An exciton can be considered as an elementary excitation resulting from a bound state made of an electron and a hole. It is created as a consequence of the absorption of a photon; an electron and a hole are attracted to each other by the electrostatic Coulomb force. It can be considered as an electrically neutral quasiparticle that can exist in insulators and semiconductors that can transport energy without transporting net electric charge.[33,36]

The resulting qausiparticle (QP) energies from eq are reported in Table . The average values for each family are 10.7, 10.3, 9.7, and 9.4 eV, respectively, for MgF2, CaF2, SrF2, and BaF2 and exhibit a decreasing trend with increasing size of the metal atom; for a given metal atom, they decrease when going from n = 1 to n = 3 with a spread of about 1.0–0.75 eV.

Table 3

Excited and Optical Properties of the Clustersa

cluster	Egap [eV]	Eopt [eV]	Eb [eV]
MgF2	12.49	6.78 (0.14; H-3 → L, H-2 → L)	5.71
(MgF2)2	11.48	6.56 (0.03; H-1 → L, H → L+1)	4.92
(MgF2)3	11.15	6.66 (0.03; H-1 → L, H-1 → L+1, H → L)	4.49
CaF2	10.90	5.64 (5.9 × 10–5; H → L)	5.19
(CaF2)2	10.04	5.42 (1.9 × 10–4; H-1 → L, H → L)	4.62
(CaF2)3	9.84	5.39 (1.1 × 10–4; H → L, H → L+4)	4.45
SrF2	10.16	5.26 (7.0 × 10–4; H → L)	4.9
(SrF2)2	9.60	5.11 (1.6 × 10–3; H-2 → L)	4.49
(SrF2)3	9.33	5.10 (4.0 × 10–4; H-1 → L)	4.23
BaF2	9.98	5.40 (8.0 × 10–4; H → L)	4.58
(BaF2)2	9.24	5.25 (1.6 × 10–3; H → L)	3.99
(BaF2)3	9.15	5.40 (1.3 × 10–3; H → L)	3.75

The quasiparticle gap (Egap), the optical onset (Eopt in bold), and the binding energy of the exciton (Eb) are given. In the third column, for Eopt, the oscillator strength of the transition followed by the states involved in the transition in the form initial state → final state with H for HOMO and L for LUMO states is given in parentheses.

Table also reports additional information on the details and nature of the electronic transitions contributing to the onset. All clusters begin to absorb in the middle UV (MUV; 4.13–6.20 eV), and we study their spectra up to the far UV (FUV; 6.20–10.16 eV). Besides the Mg fluoride clusters, which show an average value of Eopt of about 6.7 eV, the other systems display average onset energies in the range of 5.1–5.5 eV. This observable shows very small variations with respect to n at a fixed cation. It is clear from Table that while for the MgF2 and CaF2 clusters, the onset energies are determined by different transitions around the HOMO–LUMO gap, for the heaviest metal clusters (Sr and Ba), typically, only one transition enters in the first absorption peak. In particular, for all Ba clusters, the HOMO to LUMO transition is responsible for absorption at the optical onset.

Optical Absorption Spectra

The absorption spectra of all clusters were determined through the TDDFT scheme and are displayed in Figures –5. The details related to the onset energies are reported in Table .

Figure 2

Optical absorption spectra for MgF2 clusters.

Figure 5

Optical absorption spectra for BaF2 clusters.

Optical absorption spectra for CaF2 clusters.

Optical absorption spectra for SrF2 clusters.

Clear chemical and structural trends are visible. While the absorption onset increases going from Mg to Ba, the average absorption strength decreases. The number n of MX2 units is related to the electronic confinement. In fact, the size of the cluster determines the importance of quantum confinement as it is clear from the trend of Egap that is larger for smaller clusters. The absorption edge slightly shifts toward lower energies with increasing n. In the second column of the table, the oscillator strength of the transition followed by the states involved in the transition in the form initial state → final state with H for HOMO and L for LUMO states is reported in parentheses.

All the Mg systems present sharp peaks of absorption in the range 6.5–10.5 eV with intensites that are almost double w.r.t. the other MF2 systems. Moreover, the absorption spectrum of the MgF2 monomer displays a region 2 eV wide, from about 7.4 eV to about 9.4 eV in which the material is transparent. There is no analogous behavior in the other systems studied here.

Another important fact distinguishes the MgF2 clusters: All the onsets have peak intensities comparable with the other structures of the spectrum. (CaF2), for n = 2, 3, and all SrF2 and BaF2 clusters only exhibit a tiny absorption structure at the onset in the range 5.1–5.6 eV, much weaker than the other absorption peaks at higher energy. In the case of CaF2 monomer, the onset peak is located at a slightly higher energy in proximity of a stronger structure.

The differences in the absorption onsets and quasiparticle gaps are also listed in Table . They characterize the electron–hole binding in the lowest-energy exciton. The values for Eb are huge, of the order of 5 eV. They decrease slightly going from Mg to Ba. The decrease of Eb with n for a given cation illustrates the influence of electronic confinement on the electron–hole interaction.

Bulk Versus Cluster: Ground- and Excited-State Properties

The comparison between the ground- and excited-state properties of the clusters in Tables and and those of the corresponding bulk systems in Tables and may help to understand the effect of nanostructuring on electronic and optical properties. In Table , the ground- and excited-state properties of bulk cubic CaF2 and BaF2 are reported.[20,38] The computational schemes used to tackle bulk properties are DFT methods based on ionic pseudopotentials and a plane wave expansion of the electronic wave functions.[20,37] The reported observables are the distance between metal and fluorine atoms, R′M–F, and the vertical ionization energy and electron affinity, IEV and EAV, respectively. IEV and EAV are calculated as differences of energy levels for the (111) surface within a DFT-GGA scheme as difference of energy levels defining the vacuum level by the average electrostatic potential.[37] The quasiparticle gap (Egap), the optical onset (Eopt), and the binding energy of the exciton (Eb) are reported as well. For these quantities, many-body techniques have been used. In Table , we summarize the excited and optical properties of bulk rutile (tetragonal) MgF2.[39] It is clear from Table that the average distance between the metal and the fluorine atom RM–F in the clusters is larger than R′M–F in the corresponding bulk systems. This can be ascribed to the fact that cluster systems are less constrained with respect to an infinite, translationally invariant bulk. IEV and EAV are smaller in Table for the clusters than those in Table 4 for the bulk systems, implying that it is easier to extract electrons from or add electrons to molecules. As far as the excitation properties are concerned, the quasiparticle band gap energies of the clusters calculated here are smaller than in bulk but follow the same chemical trend. On the other hand, if we compare the energies of the optical onset, then we observe dramatic changes going from bulk to clusters. For example, the onset energy of MgF2 (see Table ) jumps from 10.9 eV (EUV) for bulk to an average value of 6.7 eV (FUV) for the clusters studied here. In the case of bulk CaF2, the onset energy is 10.7 eV (EUV), and the average value for the corresponding clusters is 5.4 eV (MUV). Therefore, the onset energies in the case of the clusters belong to a different UV domain with respect to their corresponding bulk systems. This fact can be ascribed to the presence of molecular transitions in clusters at energies for which the bulk system does not allow electronic transitions.

Table 4

Ground-State, Excited-State, and Optical Properties for Bulk Cubic CaF2 and BaF2 Crystalsa

solid	R′M–F	IEV [eV]	EAV [eV]	Egap [eV]	Eopt [eV]	Eb [eV]
CaF2	2.35 (2.36)	11.84 (11.96)	1.04 (−0.15)	11.8 (12.1)	10.7 (11.2)	1.1 (0.9)
BaF2	2.68 (2.68)	10.88 (10.7)	0.87 (0.21)	11.58 (11.0)	10.0 (10.0)	1.5 (1.0)

In the left part of the panel—the distance between metal and fluorine atoms, R′M–F, and the vertical ionization energy and electron affinity, IEV and EAV, respectively. In parentheses, the experimental values for each observable are listed. IEV and EAV are calculated for the (111) surface. These data are taken from a work of Matusalem and coworkers.[37] The quasiparticle energy gap (Egap), the optical onset (Eopt), and the binding energy of the exciton (Eb) are reported in the right side of the panel. In parentheses, the experimental values for each observable are reported. These quantities are referred to the fundamental direct transition for each bulk material. The data for CaF2 are from a paper of Ma and Rohlfing[38] while those for BaF2 are from a more recent work.[20]

Table 5

Excited and Optical Properties for Bulk MgF2 Crystalsa

solid	Egap [eV]	Eopt [eV]	Ev [eV]
MgF2	12.17 (12.4)	10.90 (11.2)	1.127 (1.2)

This crystal has the rutile structure. These data are taken a work of Yi and Jia.[39] The quasiparticle gap (Egap), the optical onset (Eopt), and the binding energy of the exciton (Eb) are reported. In parentheses, the experimental values for each observable as quoted in the same reference are reported. These quantities are referred to the fundamental direct transition of MgF2.

The large differences in Egap and Eopt between clusters and bulk have an important consequence on the exciton binding energy, which results strongly amplified in clusters. This fact can be ascribed to the interplay between confinement effects and reduced screening taking place in the finite systems. The results therefore suggest that there is a clear optical signature of the formation of clusters or other nanostructures or fragments compared to the bulk in the case of fluorides: the strongly reduced optical onset energy and the high exciton binding energy. The presence of these effects produces an optical signature of the formation of such molecular systems, e.g., in an optical absorption experiment for fluorides. Therefore, these two facts may have application-related consequences. The important message for possible UV applications is that the optical absorption of the fluoride clusters starts in an energy region, the MUV one, which is much below the optical onset regions of their corresponding solids, which absorb the FUV part of the spectrum. This means that the same material, a fluoride compound, if prepared as a (small) cluster or as a bulk crystal, shows very different UV properties with possibly very different application-related consequences for UV devices.

Here, we studied only small clusters and showed that their electronic and optical properties are very different in comparison to bulk crystals. It could be interesting to find out if the same scenario shows up for larger fluoride molecules, which are expected to resemble more the electronic and optical behavior of the solid phase. We postpone this point to a forthcoming study on these systems.

Conclusions

We presented a systematic investigation on the electronic and optical properties of the first three smallest stable clusters of fluoride compounds, namely, MgF2, CaF2, SrF2, and BaF2. We compared several electronic and optical properties for clusters of different sizes with the same metal atom and for clusters of the same size with different metal atoms. From this analysis, it turns out that MgF2 clusters are predicted to have distinguished ground- and excited-state properties with respect to the other fluoride systems studied here. Moreover, comparing the optical properties of the three cluster classes with those of the corresponding bulk fluorides, two important features are clearly visible: In the case of the clusters, a strong redshift of the onset energy and a corresponding rise of the exciton binding energy with respect to the bulk cases are observed. These effects, aside from their importance in the ongoing basic research on fluorides, turn out to be important for possible applications. Their presence/absence could be used as a discriminating/sorting criterion in optical measurements to check the formation and presence of such fragments in the target.

Computational Methods

For the case of (MX2) clusters treated here, with respect to geometry optimization, one has to consider an appropriate basis set due to the presence of fluorine and metal atoms of the second group. For all the clusters studied, different choices of the basis sets are reported in the literature.[26−29] Here, we employ def2-QZVPPD[40] for all atoms, i.e., the Karlsruhe quadruple-ζ valence basis set with polarization functions[41] and the addition of moderately diffuse basis functions.[42] The capability of this basis set to reproduce several properties (e.g., dipole polarizabilities, transition dipole moments, Raman intensities, optical rotations, and non-linear optical coefficients) of different clusters with high accuracy has been widely demonstrated.[41,42]

For Sr and Ba atoms, inner-shell electrons are modeled by effective core potentials (ECP), which reduce the required basis set size and account for relativistic effects.[41] ECPs deliver the accuracy of all-electron calculations with considerable reduction of the computational load, as shown in a work of Kaupp and coworkers.[43]

Our approach substantially differs from the recent study of Pandey and coworkers where the LanL2DZ and 6-31G* basis sets were used for alkaline-earth metal and fluorine atoms, respectively.[28]

With respect to the exchange-correlation (XC) potential, we chose the B3LYP one since with respect to other possibilities, e.g., the Perdew–Burke–Ernzerhof (PBE) functional, it reproduces better results for different clusters, e.g., for many PAH molecules, for both ground and excited states, as previously established.[33−36]

We perform all the DFT and time-dependent DFT (TDDFT) calculations using the Gaussian16 computational code, an all-electron Gaussian-based package.[44] This code enables us to characterize different clusters with respect to ground-state and excited-state properties.[34,35] In particular, the ground-state and the electronic properties, e.g., fragmentation energies, electron affinities, ionization energies, and quasiparticle (QP) gaps, are obtained here through the DFT method for all the fluoride molecules analyzed. Second, the time-dependent counterpart of this scheme allows the calculation of the optical properties (optical onsets, exciton binding energies) and to work out the absorption spectra from the visible up to the UV range covering the middle ultraviolet (MUV [4.13–6.20 eV]) and part of the far ultraviolet (FUV [6.20–10.16 eV]) spectral range. In particular, the Casida computational scheme is used for the calculations of the poles of the polarizability function in the frequency domain. These poles correspond to vertical excitation energies, whereas their strengths represent the oscillator strengths of the involved transitions.[45,46] Finally, we use the ΔSCF scheme[47,48] to calculate the vertical electron affinities (EAV) and ionization energies (IEV) as differences between the ground-state total energy of the neutral system, EN0, and the energies of the charged clusters (the anion EA0 and the cation EC0), at the neutral geometry.

All the spectra presented here are calculated within TDDFT, including exchange and correlation effects in the B3LYP approximation.[45,49] The comparison of TDDFT calculations with experimental data is typically very good for small clusters.[32,50,51] In particular, in the present work, we are mainly interested in studying trends in the optical absorption of families of fluoride clusters. Therefore, the TDDFT scheme turns out to be particularly advisable, efficient, and reliable.[33,36,52]