A Note on Computational Characterization of Dy@C₈₂: Dopant for Solar Cells

Abstract

Dy@C₈₂ is one of the metallofullerenes studied as dopants for improvements of stability and performance of solar cells. Calculations should help in formulating rules for selections of fullerene endohedrals for such new applications in photovoltaics. Structure, energetics, and relative equilibrium populations of two potential-energy-lowest IPR (isolated pentagon rule) isomers of Dy@C₈₂ under high synthetic temperatures are calculated using the Gibbs energy based on molecular characteristics at the B3LYP/6-31G*∼SDD level. Dy@$C_{2v}\left(9\right)$-C₈₂ and Dy@$C_s\left(6\right)$-C₈₂ are calculated as 58 and 42%, respectively, of their equilibrium mixture at a synthetic temperature of 1000 K, in agreement with observations. The Dy@$C_{2v}\left(9\right)$-C₈₂ species is found as lower in the potential energy by 1.77 kcal/mol compared to the Dy@$C_s\left(6\right)$-C₈₂ isomer.

1. Introduction

Fullerenes and metallofullerenes have recently been considered as dopants for improvements of stability and performance of solar cells [1,2,3,4,5,6,7], for example, Er@C₈₂ (the improvements could be related [5] to its hydrophobicity). Applications of various fullerene derivatives and tuning their optical and electronic characteristics in order to increase the light conversion efficiency in organic solar cells are surveyed in a recent review by Mumyatov and Troshin [7].

Dy@C₈₂ represents another metallofullerene species studied for augmentation of photovoltaic systems. Yang et al. [1] treated the Langmuir–Blodgett films with Dy@C₈₂ and reported enhanced quantum yields, up to nine-fold higher [2,3]. Actually, just two isomers of Dy@C₈₂ are characterized by observations, namely [8,9,10,11,12,13,14,15,16] the species with the cage symmetries $C_{2v}\left(9\right)$ (major isomer) and $C_s\left(6\right)$ (minor isomer). Iida et al. concluded [8] the $C_{2v}$ cage symmetry for the major species while Takabayashi et al. arrived [9] at the $C_s$ cage symmetry with the minor isomer (more specifically, the cage is coded as $C_s\left(c\right)$ or $C_s\left(6\right)$). It should be mentioned that there are two cage-labeling conventions [17] applied in the literature for the altogether nine C₈₂ IPR (isolated pentagon rule) fullerene cages. The older system refers to the symmetries of the carbon cages (actually to their highest topological symmetries, i.e., without possible symmetry reductions caused, for example, by the Jahn–Teller distortions): $C_{3v}\left(a\right)$, $C_{3v}\left(b\right)$, $C_{2v}$, $C_2\left(a\right)$, $C_2\left(b\right)$, $C_2\left(c\right)$, $C_s\left(a\right)$, $C_s\left(b\right)$, and $C_s\left(c\right)$. The newer labeling system employs serial isomer-enumeration numbers [17] produced by the so-called spiral algorithm: $C_{3v}\left(7\right)$, $C_{3v}\left(8\right)$, $C_{2v}\left(9\right)$, $C_2\left(3\right)$, $C_2\left(1\right)$, $C_2\left(5\right)$, $C_s\left(2\right)$, $C_s\left(4\right)$, and $C_s\left(6\right)$. Metal encapsulation into the C₈₂ cages frequently leads to at least two isomeric endohedral species as known from both observations and calculations [17,18,19,20,21,22,23]. The present work is focused on the two Dy@C₈₂ IPR isomers characterized in observations (the two isomers are the lowest structures in the potential energy). In particular, the temperature-dependent equilibrium relative isomeric populations are calculated for the two Dy@C₈₂ isomers: Dy@$C_{2v}\left(9\right)$-C₈₂ and Dy@$C_s\left(6\right)$-C₈₂. Such calculated relative equilibrium populations are useful in interpretations of experimental data obtained for conditions close to the inter-isomeric thermodynamic equilibrium. In particular, the calculated characteristics should be helpful in formulations of rules for future rational screening of fullerene endohedrals for their applications in photovoltaics. The first step towards such selection rules is accumulation of calculated molecular characteristics of metallofullerenes already known [1,2,3,4,5,6,7] as applicable dopants for improvements of stability and performance of solar cells.

Let us mention that the stabilities of nanocarbons, and in particular metallofullerenes, are usually treated using just potential-energy terms (i.e., without evaluation of temperature developments of their populations). Nevertheless, there are several calculations [24,25,26,27,28,29,30,31,32,33] showing that the Gibbs energy should be considered instead, as the entropic part becomes increasingly important when temperature raises. In particular, a structure that is not the lowest in potential energy can still become the most populated species at high synthetic temperatures (used in nanocarbon preparations, for example, with production of metallofullerenes). Clearly enough, it is not possible to obtain such more complex relative-stability interplay only from the mere potential energies. Thus, calculations are performed in this study on the equilibrium relative populations of the two observed (and the potential-energy-lowest) Dy@C₈₂ IPR isomers at higher temperatures, considering both the enthalpy and entropy parts of the Gibbs energy in order to predict their reliable equilibrium isomeric populations at high-temperature synthetic conditions.

2. Calculations

The molecular structures of the two Dy@C₈₂ isomers were optimized with the density functional theory (DFT) approach, first using a combined atomic basis set—the standard 3-21G basis [34] for C atoms and SDD basis [35] with the SDD effective core potential on Dy atom (denoted here as 3-21G∼SDD). The applied DFT treatment combines Becke’s three-parameter functional [36] with the non-local Lee–Yang–Parr correlation functional [37], i.e., the unrestricted B3LYP/3-21G∼SDD approach. Moreover, the B3LYP/3-21G∼SDD optimized molecular geometries were further re-optimized using the standard 6-31G* basis set [38] for carbon atoms, i.e., the more advanced B3LYP/6-31G*∼SDD approach. The calculations are carried out for the quintet electronic state as the multiplicity produces the lowest energy at the B3LYP/6-31G*∼SDD computational level. The geometry optimizations at the B3LYP/6-31G*∼SDD level point out that the two isomers known from observations, cages $C_{2v}\left(9\right)$ and $C_s\left(6\right)$, are indeed the lowest isomers in the potential energy (

Table 1. B3LYP/6-31G*∼SDD relative potential energies $\mathsf{\Delta }{E}_{pot,rel}$ and enthalpies at 0 K $\mathsf{\Delta }{H}_{0,rel}^o$ of the two energy-lowest Dy@C₈₂ isomers.

Species	$\mathbf{\Delta }{\mathit{E}}_{\mathit{pot},\mathit{rel}}$/kcal·mol−1	$\mathbf{\Delta }{\mathit{H}}_{0,\mathit{rel}}^{\mathit{o}}$/kcal·mol−1
$C_s\left(6\right)$ a	1.77	1.91
$C_{2v}\left(9\right)$ a	0.0	0.0

^a See Figure 1.

). Other isomers would have insignificant relative equilibrium populations owing to the enthalpy and entropy terms. In the optimized B3LYP/6-31G*∼SDD geometries, the harmonic vibrational analysis was carried out with the analytical force-constant matrix in order to generate the vibrational frequencies needed for the partition functions considered in the thermodynamic-stability description.

The SCF wavefunction stability [39,40,41] was checked throughout in order to prevent misleading unstable SCF solutions (that can happen rather frequently with metallofullerenes and could create a wrong stability picture if the wavefunction instability is not treated accordingly). The calculations were performed using the Gaussian 09 program [41] in parallel regime on computers operating with 8–24 processors (computational frequency up 3 GHz each and with the available operational memory up to 60 GB).

The relative equilibrium populations (expressed in the terms of mole fractions $x_i$) in our set of the two observed isomers can be expressed [42,43] with the help of their partition functions $q_i$ and the ground-state energies or enthalpies at absolute zero temperature $\mathsf{\Delta }{H}_{0,i}^o$ (i.e., the relative potential energies corrected for the vibrational zero-point energies) by a formula:

$$x_i={\displaystyle \frac{q_{i}exp[-\mathsf{\Delta }{H}_{0,i}^o/\left(RT\right)]}{{\sum }_{j=1}^2{q}_{j}exp[-\mathsf{\Delta }{H}_{0,j}^o/\left(RT\right)]}}.$$

(1)

The formula just presupposes the conditions of the inter-isomeric equilibrium (and thus—it cannot reflect a kinetic control if present in a reaction system). The rotational and vibrational partition functions are constructed [43] here with the conventional rigid-rotator and harmonic-oscillator (RRHO) approximation. No vibrational-frequency scaling is considered, as it is not significant [44,45] for the $x_i$ values at high temperatures. The temperature regions where metallofullerene electric-arc syntheses take place is not well known, however, the recent observations [46] suggest some arguments to expect it, for empty fullerenes, somewhere around 1300 K (though the region for metallofullerene syntheses could be somewhat lower owing to catalytic effects produced by the metals involved [47]).

For description of the metal motions in the cage, a modified [23,48] RRHO approach is however considered, respecting findings [49] that the encapsulated atoms can exhibit large amplitude vibrational motions, in particular at higher temperatures (interestingly, the motions can be restricted by cage derivatizations [50]). It can be expected that if the encapsulated atom is relatively free then, at high temperatures, its motions in different cages will yield about the same contribution to the partition functions. However, such comparable contributions should then cancel out in Equation (1). This approach is called the [23,48] free, fluctuating, or floating encapsulate model (FEM) and consists of two further simplifications. One is suppression of the three lowest vibrational frequencies (belonging to the metal motions in the cage), the other simplification takes symmetries of the cages as the highest (topologically) possible, which reflects averaging effects of the large-amplitude vibrational motions. For example, with the Dy@C₈₂ isomer based on the IPR $C_{2v}\left(9\right)$ cage (Table 1), the $C_{2v}$ symmetry is considered within the FEM scheme though its static [51] symmetry (i.e., just resulting from the geometry optimizations) is only simple $C_1$ (Figure 1). Generally speaking, the FEM treatment is known to yield a better agreement [23,48] with observed data than the conventional RRHO approach. Hence, the FEM approach is also applied in this study.

3. Results and Discussion

Table 1 reports the Dy@C₈₂ relative inter-isomeric energetics calculated at the B3LYP/6-31G*∼SDD level, i.e., either the difference in the potential energy, $\mathsf{\Delta }{E}_{pot,rel}$, or the difference in the enthalpy at absolute zero temperature, $\mathsf{\Delta }{H}_{0,rel}^o$. The two terms are related by the difference of zero-point vibrational energies $\mathsf{\Delta }ZPE$:

$$\mathsf{\Delta }{H}_{0,rel}^o=\mathsf{\Delta }{E}_{pot,rel}+\mathsf{\Delta }ZPE.$$

(2)

The Dy@$C_{2v}\left(9\right)$-C₈₂ isomer is lower in the B3LYP/6-31G*∼SDD potential energy by 1.77 kcal/mol compared to the other stabilomer Dy@$C_s\left(6\right)$-C₈₂ (Figure 1), or by 1.91 kcal in the enthalpy at 0 K scale. The two Dy@C₈₂ isomers from Table 1 are considered in the following evaluation of their thermodynamic-equilibrium populations.

Table 2. The selected characteristics of the two energy-lowest Dy@C₈₂ isomers—the closest Dy-C contact ^a $r_{Dy-C}$, the Mulliken charge ^b on Dy $q_{Dy}$, the dipole moment ^b p, the lowest vibrational frequency ^a ${\omega }_{low}$.

Species	${\mathit{r}}_{\mathbf{Dy}-\mathit{C}}$/Å	${\mathit{q}}_{\mathbf{Dy}}$	p/Debye	${\mathit{\omega }}_{\mathbf{low}}$/cm−1
$C_s\left(6\right)$c	2.480	2.018	0.801	27.5
$C_{2v}\left(9\right)$c	2.513	2.011	1.345	18.2

^a B3LYP/6-31G*∼SDD terms. ^b B3LYP/3-21G∼SDD terms. ^c See Figure 1.

presents selected calculated molecular characteristics of the two observed (and potential-energy-lowest) Dy@C₈₂ isomers. The B3LYP/6-31G*∼SDD calculated closest contacts $r_{Dy-C}$ between the Dy atom and the cage are around 2.5 Å and hence rather similar to the values reported previously for other C₈₂-based metallofullerenes [6,22,52]. The Mulliken atomic charge $q_{Dy}$ on Dy evaluated at the B3LYP/3-21G∼SDD level is about 2.01 (in the units of the electron elementary charge), similar as found [6] for Er@C₈₂. The charges on the cage carbons are both positive and negative (which is important for repulsions between the carbon atoms). The Dy@$C_{2v}\left(9\right)$-C₈₂ isomer exhibits carbon charges from −0.201 to 0.010, Dy@$C_s\left(6\right)$-C₈₂ from −0.205 to 0.012. The B3LYP/3-21G∼SDD calculated dipole moments of Dy@C₈₂ are also similar to those found previously [6] for Er@C₈₂. Such charge-distribution characteristics should help in analysis of the metallofullerene dopants for improvements of solar cells [5]. The lowest vibrational frequencies ${\omega }_{low}$ presented in Table 2 are in agreement with the expected relatively free motion of metal atoms encapsulated in metallofullerenes. It should be mentioned that the charge distributions evaluated with the 3-21G∼SDD basis set (in contrast to, e.g., the 6-31G*∼SDD computational level) produce [32] for metallofullerenes a good agreement with the available observed charges [53]. Moreover, there are also general methodological reasons [54,55,56,57] why larger basis sets should not be used for evaluation of the charge distributions (larger basis sets can sometimes produce quite unphysical charge values [55]).

Figure 2 presents the key results of this study—temperature development of the relative equilibrium populations for the two studied Dy@C₈₂ isomers in a wide temperature region. The relative populations are calculated with the FEM treatment based on the B3LYP/6-31G*∼SDD energy and entropy contributions. Clearly enough, at very low temperatures, the structure lower in the $\mathsf{\Delta }{H}_{0,i}^o$ scale must prevail, i.e., the Dy@$C_{2v}\left(9\right)$-C₈₂ species. However, the population of the other observed species Dy@$C_s\left(6\right)$-C₈₂ increases rather fast so that both isomers have comparable concentrations at high temperatures around 1500 K (their precise equimolarity 50:50% is reached at 1570 K). At a supposed synthetic temperature [46,47] of 1000 K, the equilibrium populations are calculated as 58.0 and 42.0% for the Dy@$C_{2v}\left(9\right)$-C₈₂ and Dy@$C_s\left(6\right)$-C₈₂ isomer, respectively. The FEM B3LYP/6-31G*∼SDD calculations thus agree with the observations [8,9,10,11,12,13,14,15,16] of the two cage symmetries, $C_{2v}\left(9\right)$ (major isomer) and $C_s\left(6\right)$ (minor isomer). Interestingly, when the entropic contributions are completely neglected in the evaluation of the relative populations (i.e., if just the simple Boltzmann factors [42,43] are considered), the relative isomeric population at the same temperature of 1000 K is equal to 71.0 and 29.0% for the Dy@$C_{2v}\left(9\right)$-C₈₂ and Dy@$C_s\left(6\right)$-C₈₂ isomer, respectively (while in the conventional RRHO approximation the isomeric populations differ even more from the recommended FEM values—the RRHO terms are 82.0 and 18%).

It should be mentioned that although Equation (1) deals with just the two isomers we are interested in, the ratio itself of their relative populations would not be changed if they would be placed into the equilibrium mixture with more isomers (i.e., the two relative poulations would be changed, however, not their ratio). If more isomers would be considered, there would be a different denominator in Equation (1). Still, when a ratio of the two isomers is evaluated, the denominator would be just cancelled out. In particular, the two isomers even in presence of some other isomers would again exhibit their equimolarity at the temperature of 1570 K (as found for the two isomers alone—Figure 2)—hence, this equimolarity temperature represents a kind of transferable parameter.

The B3LYP/6-31G*∼SDD vibrational frequencies serve here primarily in the partitions functions in Equation (1), however, they can also be used for simulation of the IR vibrational spectra (Figure 3). The IR spectra could be used for identification of the isomers.

Some similarities are to be expected with the results previously obtained [18,30,58,59,60,61,62,63] in calculations of other C₈₂-based metallofullerenes with a comparable metal-to-cage charge transfer. Such similarities can be related to the aspect that metallofullerenes are not formed by a covalent but by ionic bond [64,65,66]. There are however some general problems with comparison of calculated and observed data. The observed isomeric populations can depend on applied metal form [67]. The feature can be related to catalytic and kinetic factors [47,68,69]. Moreover, it is generally difficult to check if the supposed inter-isomeric thermodynamic equilibrium is really achieved in observations. Yet, another experimental issue is possible with different solubility [70,71,72] of various endohedral isomers in the applied extraction solvent.

4. Conclusions

The calculations of the relative equilibrium populations for the two observed IPR isomers of Dy@C₈₂ in the temperature region used [46] in metallofullerene syntheses, employing the Gibbs energy (which is based on quantum-chemical molecular parameters), agree with available observed data. Still, it can be interesting from the methodological point of view to check the results at yet higher quantum-chemical levels as well as to improve Equation (1) with a higher description of the partition functions—when technically possible. Overall, the results encourage applications of the Gibbs-energy approach to the equilibrium populations of even more complex nanocarbon systems (e.g., [73,74,75,76,77,78,79,80,81,82,83,84,85]). In particular, further calculations are in progress in order to obtain a more detailed understanding of the functions of fullerene endohedrals as dopants for higher performance and stability of solar cells [6,86,87,88]. In addition to Dy@C₈₂ [1,2], also Er@C₈₂ [5,6] and Gd@C₈₂ [61,87,88,89] are studied, especially their molecular and electronic structures as well as their energetics, thermodynamics, and also hydrophobicity [5,90,91,92,93,94,95]. The theoretical treatments should further be used in the formulation of molecular rules for rational screening of fullerene endohedrals for their applications in photovoltaics.