Confinement Effects of a Noble Gas Dimer Inside a Fullerene Cage: Can It Be Used as an Acceptor in a DSSC?

A detailed density functional theory investigation of He2-encapsulated fullerene C36 and C40 has been presented here. When confinement takes place, He-He bond length shortens and a non-covalent type of interaction exists between two He atoms. Energy decomposition analysis shows that though an attractive interaction exists in free He2, when it is confined inside the fullerenes, repulsive interaction is observed due to the presence of dominant repulsive energy term. Fullerene C40, with greater size, makes the incorporation of He2 much easier than C36 as confirmed from the study of boundary crossing barrier. In addition, we have studied the possibility of using He2-incorporated fullerene as acceptor material in dye-sensitized solar cell (DSSC). Based on the highest energy gap, He2@C40 and bare C40 fullerenes are chosen for this purpose. Dye constructed with He2@C40 as an acceptor has the highest light-harvesting efficiency and correspondingly will possess the maximum short circuit current as compared to pure C40 acceptor.


INTRODUCTION
Ever since the discovery of fullerene (Kroto et al., 1985), it has been studied extensively due to its fascinating properties, leading to versatile applications in various fields of nano and optoelectronics (Bhusal et al., 2016;Paul et al., 2017Paul et al., , 2018aQu et al., 2019), besides being used as sensors (Jaroš et al., 2019;Parey et al., 2019) and hydrogen storage devices (Chandrakumar and Ghosh, 2008;Srinivasu and Ghosh, 2012). Fullerenes are also known to exist in different isomeric forms (Małolepsza et al., 2007;Zhao et al., 2018). Insertion of various atoms and molecules into the fullerene cage has attracted a lot of interest in the scientific community and is designated as the endohedral fullerenes. It is believed that the endohedrally trapped species can manipulate different properties and affect the reactivities of the fullerene cage as well Subramanian, 2011, 2012). Thus, this class of fullerenes is studied both theoretically and experimentally, having their extensive use in the field of electronics, medicine, and nanotechnology (Guha and Nakamoto, 2005;Martín, 2006;Yamada et al., 2010). They can also serve the purpose of being an acceptor in photovoltaic devices (Osuna et al., 2011). Reactivity pattern of a system (atoms and molecules) significantly changes due to confinement (Chattaraj and Sarkar, 2003;Sarkar et al., 2009Sarkar et al., , 2012Khatua et al., 2014c;Deb et al., 2016a,b); as a result, stability increases and reduces the activation barrier. The encapsulated atoms or molecules can be either metal, non-metal, or noble gas (Ng). Noble gases, due to their closed electronic shell configuration, are generally reluctant of forming any chemical compound. However, it is possible to successfully encapsulate them inside fullerenes using techniques, such as ion bombardment (Weiske et al., 1991), molecular surgery (Saunders et al., 1994), and involving high pressure and high-temperature methods. Synthesis of the first xenon-containing molecule opened a way and interest for the researchers that noble gases are eligible to take part in chemical reactions (Bartlett, 1962). Ng-containing compound HNgY (Ng = Ar, Kr, Xe; Y = electron-withdrawing group) has been successfully synthesized by Pettersson et al. (1998). Literature survey also reveals that Feldman et al. has prepared hydrides and other Ng-related compounds (Feldman and Sukhov, 1996;Feldman et al., 1997). A number of theoretical studies on Ng have also been done apart from experimental investigations (Pan et al., 2013a(Pan et al., ,b,c, 2014a(Pan et al., ,b,c, 2015aSaha et al., 2015;Ghara et al., 2016;Ayub, 2017). Inclusion and exclusion of Ng inside fullerene cages have been reported (Saunders et al., 1993;Hummelen et al., 1995), where an intermediate is formed by the rupture of one or more bonds, thus making an open window for the encapsulation of these noble gas atoms, although some disagreement is also set forth (Haaland et al., 2004). Experimental studies show that noble gases put inside the hollow cavity of C 60 acquire an activation barrier energy as high as 90 kcal/mol −1 during its dissociation (Becker et al., 1996). Ng atoms are not only encapsulated inside larger fullerene cages (C 60 , C 70 ), but they can also be successfully trapped inside much smaller cages such as C 10 H 10 , C 20 H 20 , C 32 , etc. (Darzynkiewicz and Scuseria, 1997;Jiménez-Vázquez et al., 2001;Zou et al., 2010;Chakraborty et al., 2016;Sekhar et al., 2017). Further studies on trapping two Ng atoms forming an Ng dimer (Ng 2 ) inside fullerene have been first theoretically predicted by Giblin et al. (1997) and just after a year, He and Ne dimers are experimentally spotted into C 70 fullerene Laskin et al., 1998). Confinement of Ng dimers in C 60 fullerene starting from He to Xe atoms have been studied theoretically by some groups (Krapp and Frenking, 2007;Fernández et al., 2014;Khatua et al., 2014b), where it was found that the Ng-Ng bond distances relatively get smaller when the Ng 2 molecule is trapped inside the cage as compared to the bond distances of the isolated dimer (Krapp and Frenking, 2007). However, as a whole, these systems are thermodynamically not stable owing to the exclusion of the Ng atoms. In a study as reported by Krapp and Frenking (2007) a genuine chemical bond is found to exist between the two Xe atoms when its dimer is confined inside the C 60 cage, while for its lighter analogs, i.e., He and Ne, the presence of weak van der Waals interaction is detected. Cerpa et al. (2009) have studied the effect of confinement on the electronic interaction between He-He bond in the host C 20 H 20 and their results followed that a shorter He-He bond does not always indicate a chemical bond. Not only are the carbon related compounds considered as the hosts for entrapping the Ng atoms and their dimers, but hosts constituted with other elements such as B 40 cage (Pan et al., 2018), B 12 N 12 (Khatua et al., 2014a;Paul et al., 2019), B 16 N 16 (Khatua et al., 2014a), Sn 2− 12 , and Pb 2− 12 (Sekhar et al., 2017) are also reported. Encapsulation of Ng and Ng 2 in B 12 N 12 and B 16 N 16 fullerenes show that the dimer He 2 undergoes not only vibration and translation, but also rotation inside the cage. These investigations on noble gases encourage further comprehensive study on their confinement with different host materials.
Fullerenes possessed with electron-deficient characteristics are used as electron acceptors in a solar cell device (Liu and Troisi, 2011;Eom et al., 2014;Leng et al., 2014;Shimata et al., 2016). Dye-sensitized solar cell (DSSC) (O'Regan and Grätzel, 1991) is one of the widely used organic solar cells because it offers the possibility to convert light photons to electrical energy at a low cost. A dye in a DSSC consists of three parts: an electron-rich donor, a spacer, and an electron-deficient acceptor. Among the acceptors, [6,6]-phenyl-C 61 -butyric acid methyl ester (PCBM) has been widely investigated (Liu and Troisi, 2011). Apart from fullerene C 60 in PCBM, C 70 cage is also used (Leng et al., 2014). Endohedral fullerenes can be used in designing solar cells (Ross et al., 2009) where the confined atoms are supposedly contributing to the efficiencies of the carrier transport (Yamada et al., 2010).
Inspired by all these works on noble gas, herein, we have investigated the influence of He 2 molecule when trapped inside C 36 and C 40 fullerenes by calculating the energetic stability, bonding analysis, energy decomposition analysis, distortion energy, and boundary-crossing barrier energy of He 2 @C X (X = 36, 40). A glimpse of the UV-visible absorption spectra of He 2 @C X is presented. Finally, considering the energetically most stable fullerene among the two (He 2 @C 40 ), various parameters associated with the study of a dye-sensitized solar cell have been investigated. To make a comparison, pristine C 40 fullerene is considered for this particular case study.

COMPUTATIONAL DETAILS
All the structures have been optimized using density functional theory (DFT) methodology as implemented in Gaussian 09 program package (Frisch et al., 2009), using M05-2X functional (Hohenstein et al., 2008). We have used 6-31+g(d,p) basis set for the C atoms and def2-TZVP basis set for the He atoms.
Dissociation energy (D e ) and distortion energy (E d ) are calculated using the relations: where E C X is the total energy of pristine C X fullerenes, E He 2 @C X is the total energy of the He 2 encapsulated fullerenes (i.e., He 2 @C 36 and He 2 @C 40 ), E C X (expanded) is the energy calculated by removing the He 2 dimer and evaluating single point energy of the expanded C X fullerenes, and E He 2 is the energy of He 2 dimer. The boundary-crossing barrier of the He 2 dimer through one hexagonal face of the fullerenes has been obtained by scanning the potential energy surface at different distances from the center of the fullerene cages. GAMESS software (Schmidt et al., 1993) has been implemented in calculation of energy decomposition analysis. Multiwfn software (Lu and Chen, 2011) is used to perform the topological analysis of electron density. Time-dependent density functional theory (TDDFT) has been employed to check the absorption spectra for the first 40 excited state transitions of the fullerenes and GaussSum software (O'Boyle et al., 2008) has been used for plotting. For calculating solar cell parameters, we have considered B3LYP functional and 6-31 g basis set for all the atoms.

RESULTS AND DISCUSSION
The optimized structure of He 2 dimer encapsulated fullerenes are shown in Figure 1 below.
Upon optimization, He 2 dimer orients itself in the middle of the fullerene cage in such a way that it can move through the midpoint of a six-membered ring easily when put inside C 36 and C 40 cages. In the beginning, empty C 36 and C 40 fullerenes ( Figure S1) possess D 2d and D 2 symmetry, respectively. On enclosing He 2 inside them, the symmetry of C 36 (i.e., He 2 @C 36 ) changes to C 2 but C 40 (i.e., He 2 @C 40 ) successfully retains its original symmetry D 2 . The highest dissociation energy is found when He 2 is encapsulated inside C 36 cage (−53.1 kcal/mol) and naturally, the least is found for C 40 fullerene, having a value of −45.6 kcal/mol. Negative values of dissociation energy reveal that He 2 encapsulation in them is not favorable. It should be noted that cage distortion and Pauli repulsion plays an important role in destabilization. However, with the increase in the size of the fullerene cage (C 36 < C 40 ), destabilization decreases due to enlargement of space inside the cavity. Hence, it may be inferred that larger cages may lead toward favoring He 2 encapsulation in them. This result is also supported by the values of energy gap of He 2 -incorporated C 36 and C 40 fullerenes. The larger fullerene C 40 records a value of 3.980 eV, whereas C 36 has 3.028 eV energy gap when they both hold the He 2 dimer inside them, indicating that He 2 @C 40 is more stable than He 2 @C 36 owing to its greater energy gap in comparison.

Distortion Energy
The encapsulation of the He 2 dimer inside the fullerene cages expands the surface area of the fullerene cage and this expansion may increase the binding energy of the system. The calculated distortion energy for He 2 @C 36 is 1.6 kcal/mol and for He 2 @C 40 is 1.7 kcal/mol. Thus, He 2 dimer has to expend this amount of energy barriers to get encapsulated inside the fullerene cages, as can be seen from the distortion energy values.

Structural Properties and Bonding Analysis
In order to get an idea about how the confinement of He 2 affects itself as well as the fullerene encaging it, we have calculated the bond lengths of the He 2 dimer both in unconfined and confined state. We have also performed electron density analysis, which will further strengthen the results. The equilibrium bond length for an isolated He 2 is calculated to be 2.666 Å, which is exactly same as that of Khatua et al. (2014a). As expected, confinement reduces the bond length of He 2 dimer, which is found from the investigation. More specifically, when He 2 is entrapped inside C 36 , its bond length diminishes to 1.520 Å, while with the larger C 40 cage, it has a value of 1.546 Å.
To understand the nature of interaction between the two He atoms both inside and outside fullerene C 36 and C 40 and surrounding C atoms when He 2 is placed inside the cages, the electron density analysis at the bond critical point (BCP) was performed (Deb et al., 2018) by employing the Bader's atoms-inmolecules theory (Bader, 1990). The corresponding topological descriptors are tabulated in Table 1 and the contour plot of the Laplacian of electron densities at the BCPs are shown in Figure 2. At the BCP, depletion and accumulation of electron density [ρ (r c )] can be well understood from the positive and negative values of the Laplacian of electron density (∇ 2 ρ BCP ), respectively. Thus, when the magnitude of ∇ 2 ρ BCP < 0, a covalent or shared bond is formed between two atoms while a non-covalent bond is expected when ∇ 2 ρ BCP > 0. This criterion is helpful in explaining the bonding nature in many systems, but for the systems involving heavier atoms or with 3d orbitals, considering only [∇ 2 ρ (r c )] is not adequate to describe the nature of bonding connecting the heavier atoms. Therefore, some more topological descriptors like the local kinetic energy density [G (r c )], local potential energy density [V (r c )], and local electron energy density [H (r c )], including two ratios −G (r c ) /V (r c ) and G (r c ) /ρ(r c ), become essential to knowing the bonding nature. Based on these parameters, Cremer and Kraka (1984) proposed that if ∇ 2 ρ(r c ) > 0 and H(r c ) < 0, then the nature of bonding is partially covalent. It is also stated that if the ratio −G (r c ) /V (r c ) falls in the range 0.5-1, then there exists some degree of covalent character (partial covalent type) and if −G (r c ) /V (r c ) > 1, a purely non-covalent type of interaction is formed. Further, the magnitude of G (r c ) /ρ(r c ) < 1 is also indicative of the presence of covalency in any bond.
Present case deals with two types of bonds, i.e., one between He-He (in He 2 and He 2 @C 36 , He 2 @C 40 ) and another between He-C (in He 2 @C 36 and He 2 @C 40 ); corresponding labeled figure is provided in Figure S2. From Table 1, it is seen that both ∇ 2 ρ(r c ) and H(r c ) have values > 0, which means that a noncovalent type of bonding exists between them. The ratios −G (r c ) /V (r c ) and G (r c ) /ρ(r c ) both have a value > 1, which again emphasizes the finding that a non-covalent type of interaction is present between both He-He and He-C bonds. It should be mentioned here that along with positive ∇ 2 ρ(r c ), the magnitude of ρ(r c ) at the BCP is also lower than 0.1 au, which is the threshold value for ρ(r c ) considered here, and again confirms that the studied bonds are not of covalent type. Now examining the He-He bonds in the two host fullerenes C 36 and C 40 , one can find that the ρ(r c ) value of He-He bond present in C 36 is the highest between the two hosts, recording a value of 0.045 au. However, it may be noted that for unconfined He 2 , the values of ρ(r c ), ∇ 2 ρ(r c ), G (r c ), V (r c ), and H (r c ) are very small and almost close to zero, but when it gets confined inside the cages, the values are considerably increased. This indicates that confinement of He 2 inside fullerene C 36 and C 40 has affected its various parameters.

Energy Decomposition Analysis
To examine the interdependence of He 2 inside C 36 and C 40 fullerene cages, we studied the interaction energy and the different contributing energy terms in connection with the total interaction obtained from all-electron calculations using localized molecular orbital energy decomposition analysis (LMO-EDA). To make a comparison, we also checked the FIGURE 1 | Optimized geometry of (A) He 2 @C 36 and (B) He 2 @C 40 .
TABLE 1 | Electron density descriptors (in au) at the bond critical points (BCPs) in between some selected bonds of He 2 , He 2 @C X (X = 36, 40).
He 2 He (1) interaction energy for an unconfined He 2 dimer. The interaction energy (E int ) can be decomposed into five energy components: electrostatic (E elec ), exchange (E exc ), repulsion (E rep ), polarization (E pol ), and dispersion (E disp ). This particular study will shed light into the attractive (negative E int ) or repulsive (positive E int ) nature of interaction between the He 2 molecule and the concerned fullerene cages. For the He 2 -entrapped C 36 and C 40 fullerenes, He 2 molecule is considered as one fragment and the associated fullerene as the other. Here, the electrostatic energy represents the classical Coulomb interaction, the exchange energy includes the Pauli repulsion. The repulsion energy deals with the contribution coming from the exchange of parallel spin electrons between the two fragments. The energy gained due to the orbital relaxation of one fragment because of the existence of other fragment having undistorted charge distribution in the former fragment's proximity is accounted for the polarization energy. Lastly, the dispersion energy comes from the instantaneous correlation of fluctuating electron density distribution between the two fragments. From Table 2, we may find that attractive nature of interaction is found when two He atoms form a He 2 dimer, which is clear from its negative interaction energy. However, the situation is changed as soon as the dimer gets confined inside the two fullerene cages and records a positive value of interaction energy. In case of free He 2 dimer, all the energy components are less in magnitude, with zero contribution coming from the polarization energy. Now, coming to the confined He 2 inside the cages, it can be seen that E elec , E exc , E pol , and E disp energies are attractive in nature. Of all the attractive terms, E exc contributes the most toward the total attraction energy, the second leading contributor being E disp for both the systems. The electrostatic energy makes the third-highest attractive contribution and the polarization term with a smaller value (as compared to other attractive terms) is put in the last of all. However, the Pauli repulsion energy component, E rep , is so repulsive with the highest magnitude (positive) among all other energy terms that it makes the net interaction energy term positive and thus overall repulsive interaction energy is found for He 2 incorporated fullerenes. The highest repulsive energy is found for He 2 @C 36 (149.05 kcal/mol), whereas He 2 @C 40 records much less repulsive energy than He 2 @C 36 , with a value 120.18 kcal/mol. It may be noted here that the steep rise in the destabilizing Pauli repulsive energy term in the case of He 2 -entrapped C 36 and C 40 fullerene and a positive interaction energy is due to the compression of bond distance in He 2 dimer confined inside the cages in comparison to the isolated He 2 dimer. One important finding is that when compared to He 2 entrapped in B 12 N 12 and B 16 N 16 cages (Khatua et al., 2014a), the total interaction energy is much lowered in the present study owing to the increase in the size of the fullerene cages. This may indicate that larger cages can accommodate He 2 dimer easily and may at some point stabilize the whole system.  All the values are in kcal/mol.

Barrier Crossing Energy
The barrier crossing energy of He 2 @C 36 and He 2 @C 40 fullerenes have been depicted in Figure 3. The potential of an atom or a molecule to enter/exit into/from a fullerene cage is measured by speculating the boundary-crossing barriers. The corresponding energies provide information about the permeability and kinetic barrier of the atom or the molecule for crossing the boundary of the fullerene cage. Thus, to observe the movement of He 2 through C 36 and C 40 fullerenes, the boundary-crossing barrier has been studied. Also, five pictures of He 2 encapsulation in fullerenes are shown in Figure 4. Now, since an easy translation of the dimer into the cage can be done through a six-membered ring (instead of five-membered ring), the scanning process for its barrier crossing is calculated through a hexagon only. From Figure 3, we may observe that there are two peaks for each of the curves, with the second one (2nd maxima) from left much steeper than the first one (1st maxima). The barrier crossing energy is calculated in such a way that at the beginning the He 2 dimer is put at a distance of 7 Å from the center and then it is moved toward the cage to finally get placed in the center of the fullerene cage. The energy along this path is noted and the resulting curve is shown in Figure 3. The presence of two peaks is due to the two He atoms, which enter through the cage. The first peak from right (2nd maxima) corresponds to the energy when the first He atom passes through the cage, then the energy comes down slightly corresponding to the valley between two peaks, where the two He atoms are equally placed, one inside and another outside the surface of the cage. After that, when the second He atom just reaches the surface of the fullerene cage, the corresponding energy again rises (1st maxima), but with a relatively lower magnitude than that of the former peak value. Thereafter the energy value gradually decreases as the He 2 dimer reaches the center of the cage, where it takes the minimum energy position. Among the two cages, namely fullerene C 36 and C 40 , it is clear from Figure 3 that He 2 dimer possesses lower barrier crossing energy when it enters the C 40 fullerene as compared to C 36 . Thus, it is obvious that with increase in the size of the cage, it becomes comparatively easier for the He 2 dimer to be encapsulated inside it. Moreover, high energy barrier for the encapsulation process of any atom/molecule to be confined inside any closed complex suggests their kinetic stability, and once these encapsulated clusters are formed, they cannot dissociate into fragments owing to higher kinetic barrier (Sekhar et al., 2017). Here in this study, both the He 2 encapsulated fullerenes, He 2 @C 36 (327. 3 kcal/mol at 1st maxima and 342.2 kcal/mol at 2nd maxima) and He 2 @C 40 (305.4 kcal/mol at 1st maxima and 320.4 kcal/mol at 2nd maxima), have a much higher energy barrier when the two He atoms of the constituting He 2 dimer cross the surface of the fullerene cages. Thus, they may be considered as kinetically stable systems. FIGURE 5 | UV-visible absorption spectra of C X and He 2 @C X (X = 36, 40).

Absorption Spectra
Table S1 provides the maximum absorption wavelength (λ max ), corresponding transition energy (E 0 ), highest oscillator strength (f max ), and the major electronic transitions of the He 2 encapsulated C 36 and C 40 fullerenes along with their empty counterparts. The associated absorption spectra are provided in Figure 5. Also, to get a clear view of the highest absorption peak and highest oscillator strength of both bare and He 2encapsulated fullerenes, a zoom plot is provided. We may see that both the He 2 -encaged fullerenes possess absorption maxima in the ultraviolet (UV) region of the spectra with their λ max values of 283.652 and 277.090 nm for He 2 @C 36 and He 2 @C 40 , respectively. One may observe that the highest oscillator strength of He 2 confined C 36 cage (0.1521) is much higher than that of C 40 cage (0.0324). In addition, the corresponding λ max of He 2 @C 36 is red-shifted toward a higher wavelength compared to He 2 @C 40 . Again, from the absorption spectra, it should be pointed out that the maximum absorption of the empty fullerenes, i.e., C 36 (282.572 nm) and C 40 (280.669 nm), also occur in the UV region, having slight displacement of the peaks with respect to their He 2 -confined analogs. C 36 , He 2 @C 36 has higher intensity of absorption compared to C 40 , He 2 @C 40 . This is because the former set (C 36 and He 2 @C 36 ) has recorded a much higher value of oscillator strength, which significantly increased its absorption maxima with respect to the later set (C 40 and He 2 @C 40 ). In addition, the increment in the oscillator strength of C 36 and He 2 @C 36 occurs may be due to higher value of transition dipole moment, which ultimately puts an impact on the intensity of the highest electronic transition (Wang et al., 2015). The shift in the absorption maxima (either blue or red) depends on various factors; for example, the electronic structure (Yanagisawa et al., 2013;Giri et al., 2014), implementation of gas or solvent phase (Giri et al., 2014), orientation of embedded molecule inside any closed cage (Wang et al., 2018), substitution of donor or acceptors (Ma et al., 2019), and sometimes dependency on the method of calculation is also observed (Cárdenas-Jirón et al., 2017). Here, in this case, may be the electronic structure (symmetry, orientation of He 2 inside the fullerenes) plays an important role in this shift. In addition, red shift of the absorption spectra is also associated with a smaller energy gap (HOMO-LUMO gap) (Giri et al., 2014). Thus, He 2 @C 36 (3.028 eV) with comparatively lower value of energy gap than He 2 @C 40 (3.980 eV), has its absorption maxima red shifted toward the greater wavelength region. These systems with their maximum absorption peak falling in the UV region are usable in designing UV light protection devices.

Solar Cell Parameters
The most stable He 2 encapsulated fullerene (He 2 @C 40 ) among the two hosts is chosen as the acceptor for designing a solar cell device. To make a comparison, its free counterpart, pristine C 40 , is also taken as another acceptor. For the study, anthracene as donor and benzene as spacer are being used. For simplicity, we name the dye with acceptor as pristine C 40 as D1 and that with He 2 @C 40 as D1@He 2 . The optimized dyes are shown in Figure 6.
The energy levels of a dye molecule, which primarily indicates its HOMO and LUMO, play a very important role in its performance as a solar cell and the corresponding energy level diagram of the dyes is shown in Figure 7 (HOMO level of the dyes is zoomed). Eventually, the HOMO level of the dye must lie below the redox potential of I − 3 /I − couple (µ redox = −4.80 eV), while the LUMO level must be placed above the conduction band (CB) edge of TiO 2 semiconductor (E CB = −4.00 eV) (Qin et al., 2007). We found that the HOMO level of both the dyes lie just below the redox potential of I − 3 /I − couple, with a magnitude of −4.821 eV (for D1) and −4.816 eV (for D1@He 2 ), which confirms that charge regeneration of the dyes is possible. The LUMO value of dye, D1, is calculated to be −3.736 eV, while that of D1@He 2 is −3.795 eV. This means the LUMO levels of both the dyes lie above the conduction band of TiO 2 , and thus electron injection from the excited dyes to the conduction band of the semiconductor (TiO 2 ) will become easier. However, on comparing the HOMO and LUMO values of both the dyes, one may observe that the LUMO of D1 acquires some appreciable changes when He 2 dimer is inserted in the acceptor (i.e., D1@He 2 dye), but there is very less change in the HOMO energy. Therefore, it can be inferred that changing the acceptor can only affect the LUMO energy level of the dye. Moreover, dye D1 with comparatively higher LUMO value than D1@He 2 will help to increase the open-circuit voltage, opening a path to improve the efficiency of the dye-sensitized solar cell. The energy gap of dyes D1 and D1@He 2 are calculated to be 1.085 and 1.021 eV, respectively. Thus, dye D1@He 2 with a relatively smaller energy gap and a comparatively higher light-harvesting efficiency will show a better result.

Absorption Properties
Here we have calculated the UV-visible absorption spectra of both the dyes (Figure 8) considering the lowest 40 transitions and results are provided in Table S2. One may observe that the maximum absorption wavelength corresponding to the highest oscillator strength (0.1456) of D1 dye falls at 547.126 nm with major transitions from HOMO → LUMO+3 (55%), HOMO → LUMO+5 (20%). For dye D1@He 2 , the highest oscillator strength increases to 0.1570 and there is a blue shift of the maximum absorption peak as compared to dye D1, located at 545.897 nm. There are two major transitions found for D1@He 2 dye that take place from HOMO → LUMO+3 (53%) and HOMO → LUMO+5 (20%). Both the dyes show absorption in the visible region occurring at around 550 nm. From this study, we can infer that He 2 dimer incorporation inside C 40 increases the oscillator strength of dye D1. Thus, based on the study of absorption spectra, it is found that dye D1@He 2 possesses a comparatively higher light-harvesting efficiency resulting a greater photocurrent response.

Short-Circuit Current and Open-Circuit Voltage
The short circuit current (J SC ) (Zhang et al., 2012) is given by the relation It greatly depends on the light-harvesting efficiency (LHE) and electron injection efficiency (φ inject ). For the systems differing only in the choice of dyes, η collect can be assumed constant.
LHE may be determined using LHE = 1 − 10 −f max (4) On the other hand, φ inject is related to the free energy of electron injection from the excited states of dye to the semiconductor surface ( G inject ), which can be expressed as G inject can be estimated using the following equation (Katoh et al., 2004), where E dye * ox represents the excited state oxidation potential of the dye, E dye ox represents the ground state oxidation potential of the dye, E 0 is the vertical transition energy corresponding to the maximum absorption wavelength of the dye, and E CB refers to the conduction band edge of the semiconductor (TiO 2 ).
The computed results of various quantities involved in the calculation of J SC are provided in Table 3 for both the dyes. With the increase in the magnitude of E dye * ox , G inject increases, which ultimately increases the φ inject . Out of the two dyes, dye with the He 2 dimer records the highest LHE value and thus will have a tendency to absorb more photons, which will lead to a higher magnitude of the short circuit current in comparison to dye D1. Also, according to a study (Islam et al., 2003), the electron injection efficiency of a dye is approximately equal to 1 when G inject > 0.2. Here, we can see that both the dyes have a value of G inject , which are much higher than 0.2. Thus, both of them possess sufficient driving force for electron injection to the semiconductor TiO 2 .
The open-circuit voltage (V OC ) of a dye molecule can be evaluated approximately by finding the difference in The eV OC values of the dyes are given in Table 3. The equation indicates that higher the magnitude of the LUMO level of the dye, the higher open-circuit voltage (V OC ) it will generate. From Table 3 we can see that dye D1 records greater value of eV OC than its He 2 dimer encapsulated counterpart, which has a comparatively higher value of G inject . This may be due to the fact that too high value of G inject leads to energy redundancy, making a fall in the value of V OC (Li et al., 2017). Hence though the dye D1@He 2 has a higher G inject , but it possesses a smaller value of V OC than dye D1.

Non-linear Optical Properties
To explore the relationship between the efficiency of the dye molecules and their non-linear optical (NLO) properties, the isotropic polarizability of the dyes has been calculated. The response of any system when it is subjected to an external electric field is characterized by the study of NLO properties of that system (Deb et al., 2020). In case of dye D1, the isotropic polarizability is found to be 574.045 au, while for the dye D1@He 2 the value increases to 577.945 au. This means that inclusion of a He 2 dimer inside C 40 acceptor improves the dye's polarizability. In addition, dyes having higher magnitudes of polarizability possibly will create strong interaction with its surroundings and will increase the local concentration of acceptor. Thus, the local concentration of the acceptor of the dye D1@He 2 , i.e., C 40 @He 2 is increased on the semiconductor surface, which in turn will increase the possibility of this acceptor to perforate into the dye adsorption layer.

CONCLUSION
Density functional theory has been implemented to assess the various properties of He 2 dimer when encaged inside two fullerene cages, C 36 and C 40 . Our study shows that when He 2 is confined in the cages, its bond length considerably decreases. The compression in bond length is more when C 36 holds the dimer as compared to that of its relatively larger counterpart, C 40 fullerene. Non-covalent type of interaction exists between the He-He bond, whether in isolated or confined inside the cages, which is confirmed from the electron density analysis. From energy decomposition analysis it is observed that attractive interaction is found for He 2 dimer, but after being encapsulated in the fullerenes, the interaction becomes repulsive. The presence of larger repulsive energy compared to the other attractive energy terms may be responsible for this change in the case of He 2 @C 36 and He 2 @C 40 . Fullerene C 40 bearing a larger cavity makes He 2 incorporation energetically much easier than that of C 36 as observed from the investigation of barrier crossing energy. Absorption spectra analysis of both the He 2 @C X shows that they can be potentially used as UV light protectors since they possess absorption maxima in the UV region. Next, we designed a DSSC with free C 40 and He 2 confined C 40 as acceptors.
Charge regeneration and electron injection, which are the two most important qualities, are being fulfilled here by the dyes. In addition, both the dyes show an absorption peak in the visible region, which is another criterion for a DSSC. He 2 @C 40 , when used as acceptor, records the highest LHE value and thus will have a higher magnitude of J SC . NLO properties of the dyes are also calculated, and we found that the dye with He 2confined C 40 acceptor has greater polarizability and thus will have higher possibility to perforate into the dye adsorption layer. This indicates that He 2 incorporation inside fullerene really has a good effect on different properties.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

AUTHOR CONTRIBUTIONS
DP has played the lead role in data collection, analysis, visualization, and writing original draft. HD has played a supporting role in data extraction and formal analysis. US has a lead role in supervision, funding and software, and supporting role in conceptualization. All authors contributed to the article and approved the submitted version.