An investigation into the structural, electronic, and non-linear optical properties in CN (N = 20, 24, 26, 28, 30, 32, 34, 36, and 38) fullerene cages

The present study attempts to investigate the structural, electronic, and non-linear optical properties of CN (N = 20, 24, 26, 28, 30, 32, 34, 36, and 38) fullerene cages based on Density Functional Theory (DFT). In the DFT calculations, the B3LYP/6-311G(d,p) and CAM-B3LYP/6–311 +  + G(d,p) level of theories were used. The isomers of each fullerene have been received from the Fullerene Structure Library. These isomers have optimized using the B3LYP/6-311G(d,p). The results included optimization of the neutral and ionic state structures according to their multiplicity. Geometries, optimization energies, relative energies, frequencies, HOMO, LUMO, and HOMO–LUMO gap of these stable fullerene cages have been predicted by B3LYP/6-311G(d,p). Afterwards, the most stable structures have been re-optimized using the CAM-B3LYP /6–311 +  + G(d,p). Finally, non-linear optical properties, Fukui functions, density of state, electron affinity, and ionization potential values of the most stable fullerene cages have been found out by the DFT/ CAM-B3LYP /6–311 +  + G(d,p) level of theory. All calculation results have been compared with both C60 fullerene and the relevant literature on corresponding fullerenes.


Introduction
Organic nanostructures such as fullerenes, nanotubes, and graphene generally receive much attention among the scholars. The main reason behind this is that organic nanostructures have properties suitable for practically various applications in different scientific fields. Among these organic nanostructures, fullerenes have outstanding properties in many different fields. With the discovery of the C 60 fullerene by Kroto [1], an array of theoretical and experimental studies has been carried out to determine the physical and chemical properties of fullerenes, thereby characterizing a wide range of fullerenes over the years [2][3][4].
Currently, fullerenes find wide development application for modern science such as physics, medicine, materials science, and biology. More specifically, they have wide applications in many fields such as organic solar cells, super-capacitors, catalyzers, and superconducting materials. In addition to these properties, nano-size and thermally stable fullerenes have a wide range of applications in electronics, photonics, and nonlinear optics due to their electronic, sensing, and optical properties. The fact that fullerenes have all these application areas is due to their properties. For instance, they have a high electron affinity, unique geometric structure, electronic, and physicochemical properties. Among these properties, high electron affinity is one of its most important properties. Thanks to this property, it can have the ability to attract extra electrons and form a bound mono or poly-anion. Besides, it is nanostructures that have applications in artificial photosynthesis and photovoltaic devices [5].
Fullerenes, especially C60, have been an interesting research area for researchers and have been extensively studied. Today, they are an important building block for preparing materials with potential applicability in research fields such as photovoltaics, nonlinear optics, optoelectronics, and medicine [6]. Especially after the developments in C60 chemistry, it has allowed the preparation of many fullerene derivatives covalently bonded to the donor moieties. Fullerenes are used extensively in intramolecular processes such as energy and electron transfer. The C60 fullerene is known to be a particularly interesting electron acceptor in photochemical molecular devices, due to its symmetrical shape, large size, and the properties of its π-electron system [7]. They are used in the most efficient "bulk heterojunction" devices. In these devices, the fullerene derivatives are blended with conjugated semiconducting polymers. The fullerenes act as an electron acceptor to separate excitons formed when the polymer absorbs light [8].
Another important issue is analogues. The BN pair has been proposed as boron nitride analogs of the fullerenes, especially B 30 N 30 as the analog of C 60 . Carbon and boron nitride, being isoelectronic, tend to form similar compounds or materials. As the isoelectronic counterparts to carbon nanostructures, the boron nitrides cages and nanotubes have been extensively investigated due their high-temperature stability, low dielectric constant, large thermal conductivity, ultra-violet light emission, and oxidation resistance. The bond length, the long-order parameters, and the lattice constants are very similar between BN and C. Thus, it has been expected that the similarities should also exist between the nanoscale structures. In the present work, the analogs of C 2X are the BN X (X: 10, 12, 13, 14, 15, 16, 17, 18, and 19). They could have similar structures, from one-dimensional nanotubes to three-dimensional diamond structures. Besides the similar structural features exhibited by the two species, there are also many differences between BN-made compounds and C-based compounds. For example, B and N form strong polarized covalent bonds because of the difference in their electronegativities. Therefore, BN structures have different mechanical and thermal properties. In addition to, BN nanostructures are expected to have higher reactivity than analogous carbon structures, due to the polar nature of B-N bonds. For instance, theoretical investigations predict that the BN clusters could store H 2 molecules more readily than the carbon clusters [9][10][11][12][13][14].
Despite all the theoretical and experimental studies, there are still many deficiencies in the characterization of fullerenes. So, theoretical and experimental characterization studies of fullerenes are still critical. The aim of this study is to determine the structural, electronic, and nonlinear optical properties of C N (N = 20, 24, 26, 28, 30, 32, 34, 36, and 38) fullerene cages. This study provided the theoretical calculation results of C N fullerene cages have been presented. To the best of our knowledge, a comparative study of these fullerenes is not reported in the literature yet. To fill such a void, structural, electronic, and non-linear optical properties analyses of these fullerenes are provided in the manuscript in detail. All isomers of C N fullerenes with various spin multiplicities were simulated by Density Functional Theory (DFT). Finally, a detailed study of the Mulliken charge, density of state, and Fukui function of these fullerenes has been presented.
Firstly, the isomers of all considered fullerene cages are fully optimized at B3LYP/6-311G(d,p) level of theory (abbreviated as DFT/B3LYP level) and the nature of the stationary points is checked by frequency analysis at the same computational level. The geometries of all fullerene cages were taken from the Fullerene Structure Library built by Mitsuho Yoshida [20]. Harmonic vibration frequencies at the same level were also calculated to check the stability of the geometries on the potential energy surface. Electrostatic potential maps, HOMO, and LUMO plots from results of the optimization calculations of the most stable structure of each fullerene cage were visualized with the same level.
Later, the CAM-B3LYP/6-311 + + G(d,p) (abbreviated as DFT/CAM-B3LYP level) was used on non-linear optical properties. The B3LYP method overestimated (hyper) polarizations for some large systems [18]. A new density functional Coulomb attenuated hybrid exchange correlation functional (CAM-B3LYP) has been developed and is suitable for predicting the molecular NLO properties of a large system [21][22][23][24][25]. This functional has been shown to provide good results for electronic excitation energies, first, second hyperpolarizabilities. Specifically, the CAM-B3LYP method was found to be suitable for calculating the hyperpolarizability of some nanotubes and nanoclusters [26]. Therefore, the CAM-B3LYP method may be a satisfactory choice to investigate the static hyperpolarizability of currently studied systems. Besides, some visualizations (Mulliken atomic charge, density of state, and Fukui function) and some calculations (ionization potentials (IP) and electron affinities (EA) and the vertical detachment energy (VDE) calculations) were performed by CAM-B3LYP methods.
Since IP and EA are essential properties that reflect the stability of fullerene cages, IP and EA were calculated from the optimization energies of the most stable geometries of neutral and charged fullerene cages [27]. The IP was obtained by considering the difference between the optimized energies of the neutral fullerene cages and their cations. In this study, the researchers evaluated the IP versus fullerene cage size.
EA is defined as the energy obtained when an electron is added to the isolated atom. Adiabatic electron affinity (EA ad ) values for the lowest energy isomers of fullerene cages were computed in this study. The EA ad is defined as the optimization energy difference in the most stable neutral and anion state [27].
The vertical electron affinity (EA vert ) is the energy difference between the optimized neutral state and the anionic state without changing the geometry in the neutral state [27].
The vertical detachment energy (VDE) is the energy required to remove an electron from the anionic fullerene cages without changing its geometry. As understood from the definition, it is regarded as the energy difference between the neutral state without changing the geometry in the optimized anionic state and the optimized anion state [27].
It is important to determine the nonlinear optical properties of fullerenes. Theoretically, making (hyper)polarizability calculations might provide data for future studies of fullerenes. The energy of a system in the weak and homogeneous electric field can be defined as: where E 0 is the total molecular energy in the absence of an electric field. F is the electric field component along the α direction. The , , and denote dipole moment, polarizability, and the first-order hyperpolarizability respectively [28]. The dipole moment ( ), the mean polarizability ( ), the anisotropy of the polarizability ( Δ ), and the first-order hyperpolarizability (β 0 ) are defined as: In general, the main purpose of molecular electrostatic potential (MEP) maps is to explain the charge distribution of the working system. In this calculation, a map was created due to the properties of the nucleus and the nature of the electrostatic potential energy. These visualizations were used to illustrate concepts such as polarity, electronegativity, and similar properties. These maps were sampled over the entire accessible surface of the studied structure. The three-dimensional isosurfaces of MEPs showed electrostatic potentials superimposed on a surface of electron density. The most negative electrostatic potential was shown in red while the most positive electrostatic potential was presented in blue [29].
Fukui function is an important concept in conceptual DFT, and it has been widely used in prediction of reactive site. It is also used as a descriptor in quantitative structure-activity relationships. It calculated with the help of the following equations. In these equations, q k is the atomic charge at the r th atomic site. The N, N + 1, and N − 1 impressions show neutral, anionic, and cationic states, respectively. [29].
Euler's theorem [30] provides a rule for pentagonal and hexagonal numbers for C 20 cage fullerene and fullerenes with an even number of atoms greater than C 20 cage. C nm (n ≥ 2; m: even numbers) fullerene cages have twelve pentagons. The number of hexagons was determined by the expression (n/2 − 10). Pentagonal and hexagonal numbers were given in the correspond table for each fullerene.

Results and discussions
The present study investigates the structural, electronic, and non-linear optical properties of neutral, cationic (single positive charge), and anionic (single negative charge) fullerenes. Geometry optimization of the neutral and charged states with different spin multiplicities was carried out for each of the fullerenes. The lowest-energy structures of these fullerenes were shown in Figs. 1-9. Pentagon/hexagon numbers, (10) y = yyy + yzz + yxx (11) z = zzz + zxx + zyy (12) 0 = xxx + xyy + xzz 2 + yyy + xxy + yzz 2 + zzz + xxz + yyz for neutral (radical) attack symmetry, the optimized energies, and relative energies of all isomer of the fullerene cages with various spin multiplicities were given in Tables S1-S9. The results were discussed in the following sections.

C 2N (N:0, 4, 6, and 8) fullerenes
Fullerenes are closed-cage carbon structures consisting of 12 pentagons and certain number of hexagons. For each fullerene, the possible hexagon numbers could be 0, 2, 3, 4, 5, and more. The structure consisting of 12 pentagons without hexagons is the smallest possible fullerene. Thus, the smallest theoretically possible fullerene is C 20 . The C 20 with a dodecahedral cage structure is regarded as the smallest fullerene existing. It only comprises 12 pentagons and 30 bonds. Kroto [1] expressed the pentagon isolation rule, which stipulates that the most stable fullerenes should have 12 pentagons and that these pentagons should be as far apart as possible. Considering this rule, the C 20 fullerene cage should be highly unstable. It is sometimes called "unconventional fullerene." The structure of the C 20 fullerene cage has I h symmetry. Besides, the C 20 fullerene cage also has eight structures with subgroup symmetry. Wang [31] et al. reported that a symmetry path from I h symmetry to C 1 symmetry found for C 20 fullerene cage. Based on this result, the structure with D 2h symmetry is the most stable in this path. The researchers took Wang's suggestion for the C 20 as a starting point. However, negative frequency was detected in the structure's calculation with D 2h symmetry. Therefore, the calculations were continued with the structure with D 2 symmetry. It is structurally very similar between the structure with D 2 symmetry and the structure with D 2h symmetry. The researchers have done all the following calculations for the C 20 fullerene cage with geometry D 2 . Table S1 presented all calculation results such as optimization energy for each multiplicity, symmetry, and geometry. The results showed that the most stable state of C 20 fullerene obtained the singlet spin multiplicity for neutral state and quartet spin multiplicity for ionic cases. Figure Figure 1 shows the gaps between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) plots. The researchers calculated at -5.442 eV (HOMO energy), -3.506 eV (LUMO energy), and 1.937 eV (HOMO-LUMO gap energy) by DFT/B3LYP level.
Prinzbach [32] et al. accomplished the first synthesis of C 20 . In their study, C 20 H 20 was firstly converted to C 20 Br 20 by substitution of H atoms with Br atoms. Then, C 20 Br 20 was debrominated to synthesize C 20 fullerene. Apart from this study, many studies have been carried out on the synthesis of C 20 [33]. Prinzbach [32] et al. reported that the C 20 fullerene had an EA of 2.25 eV by its photoelectron spectrum. In other studies on EA of the C 20 fullerene cage, this value was determined as 2.7 eV by Yang [34] et al., 2.65 eV by Wang [35] et al., and 2.689 eV by Wang [36] et al. According to the results from the present calculations, 2.391 eV was predicted as EA of C 20 fullerene cage by DFT/CAM-B3LYP level ( Table 1). The C 20 fullerene cage has a very symmetrical structure. As seen in Table 1, the and β 0 components of the C 20 fullerene cage are all zero. However, the C 20 fullerene has relatively small polarizability. The and the Δ were predicted at 27.59 × 10 −24 esu and 1.552 × 10 −24 esu by DFT/CAM-B3LYP level, respectively.  Table 2S provides all calculation results such as optimization energy for each multiplicity, symmetry, and geometry for C 24 fullerene cage. Considering the optimized geometry in Table S2, C 24 fullerene cage comprises two hexagons at the top and bottom, and 12 pentagons at the waist. Because the first member of the fullerene cage family having a magic number is the C 24 fullerene cage, Kroto suggested that fullerenes Cn with magic numbers (n: 24, 28, 32, 36, 50, 60, and 70) should have enhanced stability relative to those with similar numbers of atoms [1]. The fullerene possesses ideal D 6d symmetry [37]. As a starting point of the study, the C 24 fullerene cage with D 6d symmetry was optimized. However, negative frequency was obtained in the structure's calculation with D 6D symmetry. Jensen [37] et al. noted that this structure with D 6d symmetry is stable geometry. Nevertheless, it was found that C 24 fullerene cage with D 6d was in the transition structure in the calculations made on D H(D) and 6-31G (d) basis sets. Similarly, it was determined that C 24 fullerene cage with D 6d was in a transition structure in the present studies. Hence, the C 24 symmetry was reduced from D 6d to D 6 symmetry. The C 24 fullerene cage with D 6d and its D 6 symmetric structure were very similar structurally. In that symmetry, C 24 fullerene cage with D 6 was the most stable. The researchers noted that the most stable state of C 24 fullerene cage determined the singlet spin multiplicity for neutral state, doublet spin multiplicity for anionic case, and quartet spin multiplicity for cationic case. As seen in Fig. 2, the Mulliken charge of the atoms shown in red color is -0.148 a.u., and the charge of atoms shown in green color is 0.148 a.u. by DFT/CAM-B3LYP level. If the Mulliken charge distribution is examined, it could be seen that two different charges are distributed symmetrically. The researchers calculated at − 6.050 eV (HOMO energy), − 4.221 eV (LUMO energy), and 1.829 eV (HOMO-LUMO gap energy) by DFT/B3LYP level.
Janjanpour [38] et al. reported that the EA and IP of the C 24 fullerene cage were predicted at 7.47 eV and 2.98 eV by B3LYP/6-31 + G (d). In the present study, the computed IP, EA ad, and VDE values were obtained at 7.691 eV, 3.252 eV, and 2.699 eV by means of DFT/CAM-B3LYP, respectively ( Table 1). The results showed that C 24 fullerene cage obtained a great EA vert value at 3.091 eV by DFT/ CAM-B3LYP level. It indicated that C 24 fullerene cage was a great electron acceptor.
The most stable geometry of the C 24 fullerene cage possesses D 6 symmetry group. Given this structure, it could be stated that C 24 had a very symmetrical structure. So, and β 0 components of the C 24 fullerene cage were all zero. Kosar [39] et al. reported that of the C 24 fullerene cage theoretically obtained 31.863 × 10 −24 esu (215 a.u.) by B3LYP/6-31 + G(d) level. In the present study, and Δ were predicted at 32.04 × 10 −24 esu, and 8.242 × 10 −24 esu by DFT/CAM-B3LYP level ( Table 2). The following closed fullerene cage by size is C 26 , of which there is only one classical closed-cage isomer. The highest possible symmetry of C 26 is D 3h . The C 26 fullerene cage comprises three consecutively connected hexagons and 12 pentagons. A ring is formed by connecting hexagons. Pentagons have a structure that could form bridges to these rings. Table S3 provides all calculation results such as optimization energy for each multiplicity, symmetry, and geometry for C 26 fullerene cage. The researchers could not get a negative frequency in the frequency calculations of this fullerene cage. The results indicated that the most stable state of C 26 fullerene cage determined the quintet spin multiplicity for neutral state, quartet spin multiplicity for anionic case, and sextet spin multiplicity for cationic case. Figure 3 showed the optimized geometry (a), Mulliken charge distribution (b), MEP counter (c), and HOMO-LUMO plot (d) of the most stable isomer of the C 26 fullerene cage.
As seen in Fig. 3, the Mulliken atomic charges on C in the C 26  To gain insight into the electronic properties of the lowest energy isomer, we evaluated the IP, EA, and VDE of C 26 fullerene cage at the DFT/CAM-B3LYP level ( Table 1). In the present study, the researchers obtained at 7.443 eV (IP), 2.618 eV, 2.452 eV (adiabatic and vertical EA), and 2.815 eV (Vertical detachment energy). Janjanpour [38] et al. reported that the IP and EA in the C 26 fullerene cage were predicted at 2.95 eV and 3.34 eV by B3LYP/6-31 + G (d), respectively. An [41] et al. argued that the EA vert of the most stable geometry of C 26 fullerene cage were 2.72 eV.   The most stable geometry of the C 26 fullerene cage possesses D 3h symmetry group. Regarding this structure, it could be stated that C 26 had a very symmetrical structure. So, and β 0 components of the C 26 fullerene cage were all zero. The of C 26 fullerene cage was calculated as 34.14 × 10 −24 esu and the Δ of C 26 fullerene cage was calculated to be 5.588 × 10 − 24 esu by DFT/CAM-B3LYP level ( Table 2). C 28 fullerene plays an important role in theoretical studies on fullerenes smaller than C 30 . C28 are one of the magicnumber small fullerenes The researchers reported that the C 28 had two isomers with T d and D 2 symmetry [42]. The researchers optimized the two isomer with different spin states to achieve the most stable geometry. According to the results of the optimization calculations, the isomer of T d symmetry of the C 28 fullerene cage had the most stable isomer. The C 28 fullerene cage comprises 4 hexagons and 12 pentagons. The C 28 has been reported to have a tetrahedral structure, in which there are three isolated pentagons, one at each corner, which are not directly fused. In the isomer with T d symmetry, the researchers reported that the most stable state of C 28 fullerene cage determined the quintet spin multiplicity for neutral state and quartet spin multiplicities for ionic cases. The researchers could not get negative frequency in the frequency calculations of this structure. Table S4 presents all calculation results such as optimization energy for each multiplicity, symmetry, and geometry for C 28 fullerene cage. Figure 4 demonstrated the optimized geometry (a), Mulliken charge distribution (b), MEP counter (c), and HOMO-LUMO plot (d) of the most stable isomer of the C 28 fullerene cage.
As seen in Fig. 4, the Mulliken atomic charges on C in the C 28 fullerene cage were divided into three groups: 0.318 a.u. (twelve atoms in green color), − 0.109 a.u. (twelve atoms in dark red color), and − 0.627 a.u. (four atoms in red color). The HOMO-LUMO gap of the neutral C 28 fullerene cage with Td symmetry was predicted at 4.201 eV for alpha orbital and 2.321 eV for beta molecular orbital by DFT/ B3LYP level. These values were quite large when compared to the values of those of C 60 and C 70 fullerene cages.
Castro [43] et al. have put forward that the IP and EA for C 28 fullerene cage would contribute to the understanding of its behavior in electron detachment and electron affinity situations. The calculated IP and EA Ad were obtained as 7.69 eV and 3.39 eV. The results of the present study showed that 8.349 eV and 2.864 eV values were determined as IP and EA Ad by DFT/CAM-B3LYP level. EA vert were obtained at 2.733 eV and 3.007 eV for VDE by DFT/CAM-B3LYP level (Table 1).
In the present study, the researchers calculated the of C 28 fullerene cage as 35.99 × 10 -24 esu. The researchers note it is consistent with the value 40.44 × 10 −24 esu calculated by Sabirov et al. [44]. Since C 28 fullerene cage has a high symmetry (T d symmetry), all components of the and β 0 values are found as zero ( Table 2). C 3N (N:0, 2, 4, 6, and 8

) fullerenes
Based on the DFT calculation results, three C 30 isomers were theoretically differentiated by D 5h -I-C 30 isomer, C 2v -II-C 30 isomer, and C 2v -III-C 30 isomer. Table S5 provides all calculation results such as optimization energy for each multiplicity, symmetry, and geometry for C 30 fullerene cages. Accordingly, the ground state C 2v -II-C 30 isomer with singlet multiplicity was more stable than C 2v -I-C 30 and D 5h -I-C 30 isomers. The relative energy of C 2v -I-C 30 and D 5h -I-C 30 isomers determined as 28 kcal mol −1 and 55.81 kcal mol −1 . The C 30 cage comprises 5 hexagons and 12 pentagons. The researchers could not get negative frequency in the frequency calculations of this structure. The most stable state of the cationic and anionic C 2v -II-C 30 isomer obtained doublet spin multiplicity. Figure 5 showed the optimized geometry (a), Mulliken charge distribution (b), MEP counter (c), and HOMO-LUMO plot (d) of the most stable isomer of the C 30 fullerene cage.
As seen in Fig. 5, the Mulliken atomic charges on C in the C 30 fullerene cage were divided into ten groups. The three groups have positive Mulliken charges of 0.289 a.u., 0.188 a.u., and 0.183 a.u., and their colors on six atoms appear Fig. 4 Geometry, Mulliken charge, ESP cour map and HOMO, and LUMO plots of C28 fullerene cage as green color and its shades. The three positively charged groups have a charge of 0.029 a.u., 0.008 a.u., and 0.001 a.u., and their colors on eleven atoms appear in dark green color and its shades. The two groups have negative Mulliken charges of − 0.251 a.u. and − 0.195 a.u., and their colors on four atoms appear as red color and its shades. The three negatively charged groups have a charge of − 0.078 a.u. and − 0.068 a.u., and their colors on eight atoms appear in dark red color and its shades.
Paul [5] et al. argued that 7.352 eV and 2.761 eV values were obtained as IP and EA by ωB97XD functional with 6-311 + G (d, p) basis set, respectively. In the present study, these properties were calculated at 7.808 eV for IP and 3.559 eV for EA ad by DFT/CAM-B3LYP level. Moreover, 3.385 eV and 3.689 eV were determined as EA vert and VDE, respectively (Table 1). Paul et al. [5] reported that the of C30 fullerene cage was calculated at 0.12 Debye by B3LYP functional with 6-311 + G (d, p) basis set. In the present study, this value was predicted at 0.119 Debye by DFT/ CAM-B3LYP level. In the same study of Paul et al. [5], value of C30 fullerene cage was calculated as 43.79 × 10 −24 esu, and this value was determined as 40.151 × 10 −24 esu in the present study. Similarly, while the Δ value of C 30 fullerene cage was determined as 17.64 × 10 −24 esu by Paul et al. [5], it was determined as 5.438 × 10 −24 esu in the present study. The β 0 value of C 30 fullerene cage was obtained at 0.76 × 10 −30 esu (CAM-B3LYP/6-311 + G(d,p)) by Paul et al. [5] In the present study, this value was predicted as 2.062 × 10 -30 esu (DFT/CAM-B3LYP) ( Table 2).
As seen in Fig. 6, the Mulliken atomic charges on C in the C 32 fullerene cage were divided into two groups. The Mulliken charges of the first group were 0.280 a.u. and 0.210 a.u. The charges were on twenty atoms and their color was green and its shades. The Mulliken charges of the second group were − 0.049 a.u., − 0.097 a.u., − 0.176 a.u., and − 0.503 a.u. The charges were on twelve atoms and their color was red and its shades. Lin et al. [45] reported that HOMO-LUMO energy gap of the lowest-energy of C 32 fullerene cage with D 3 symmetry was obtained at 2.602 eV by means of B3LYP/6-31G(d,p) level. In the present study, the HOMO, LUMO, and HOMO-LUMO gap energies of the C 32 fullerene cage were determined at − 6.624 eV, − 4.014 eV, and 2.610 eV for DFT/B3LYP level. The C 70 and C 60 fullerene cages were much larger and much more stable than the C 32 fullerene cage. However, neutral C 32 seems to have a much larger HOMO-LUMO gap value than those of C 70 (1.3 eV) and C 60 (1.6 eV) [45].
The EA of C 32 was experimentally reported as ~ 2.8 eV [46]. In the present results (DFT/CAM-B3LYP level), EA ad of C 32 fullerene cage was obtained at 3.025 eV. Similarly, its EA vert was calculated as 2.769 eV. IP and VDE values of C 32 fullerene cage were found as 8.178 eV and 3.305 eV by DFT/CAM-B3LYP level ( Table 1). The and the Δ  Table 2). All components of and β 0 values were calculated as zero.
There were six isomers for C 34 fullerene cage. These isomers optimized by using the DFT/B3LYP level. Table S7 presents all calculation results such as optimization energy for each multiplicity, symmetry, and geometry of the C 34 fullerene cages. These six isomers were determined according to their symmetry as follows: three C 2 , two C S , and one C 3V . In the DFT calculations, the most stable C 34  The negative atomic charges on C in the C 34 fullerene cage were determined at − 0.146 a.u., − 0.068 a.u., − 0.047 a.u., − 0.117 a.u., − 0.021 a.u., − 0.066 a.u., − 0.436 a.u., − 0.034 a.u., and − 0.278 a.u. (each charge was on two atoms.). These charges were shown in red color and its shades as seen in Fig. 7. The positive atomic charges on C in the C 34 fullerene cage was found 0.222 a.u., 0.073 a.u., 0.261 a.u., 0.005 a.u., 0.163 a.u., 0.313 a.u., 0.064 a.u., and 0.112 a.u. (each charge is on two atoms.). These charges were shown in green color and its shades in the Fig. 7. S. A. Halim et al. [47] predicted the electronic structure and stability of C 34 and transition metal doped C 34 derivative by using B3PW91/6-31G(d) level. In their study, the HOMO, LUMO, and HOMO-LUMO gap energies of the C 34 fullerene cage were reported as-− 5.53 eV, 4.06 eV, and 1.47 eV, respectively. In the present study, the C 34 fullerene cage had alpha and beta molecular orbitals because the most stable structure of C 34 had triplet multiplicity. The HOMO energies of α and β orbitals were obtained as − 5.693 eV  S. A. Halim et al. [47] reported the calculation results of the IP, EA, chemical hardness, electronegativity, chemical potential, electrophilicity of pure C 34 , and transition metal-doped C 34 . In that study, these data were calculated according to Koopman's theorem. It was addressed that the estimated IP value was 5.530 eV and the estimated EA value was 4.057 eV by the B3PW91/6-31G(d) level. In the present computational work, the IP, EA ad , EA vert, and VDE values of C 34 fullerene cage were obtained as 7.450 eV, 2.955 eV, 3.062 eV, and 3.163 eV, respectively (Table 1).
In the study by S.A. Halim et al. [47], all components of the , , Δ , and β 0 values of C 34 fullerene cage were calculated and reported. In the same study, the , the Δ , , and β 0 values were also obtained by B3PW91/6-31G (d) level. In that study, 40.07 × 10 −24 esu, 8.613 × 10 −24 esu, 0.155 Debye, and 0.955 × 10 −30 esu values were determined as the , Δ , , and β 0 , respectively. In the present study, 46.062 × 10 −24 esu, 8.103 × 10 −−24 esu, 0.113 Debye, and 2.656 × 10 −30 esu values were determined as the , Δ , , and β 0 by DFT/CAM-B3LYP level, respectively ( Table 2). C 36 was one of the magic-number small fullerenes detected by mass spectroscopy in the very early days. C 36 had 15 conventional fullerene isomers. Table S8 presents all calculation results such as optimization energy for each multiplicity, symmetry, and geometry of the C 36 fullerene. D 6h and D 2d isomers contained a minimum number of adjacent pentagons. Therefore, they were candidates for the most stable structure. In the DFT calculations, it was revealed that the D 6h -XV-C 36 fullerene cage with triplet multiplicity case was the most stable structure.
As seen in Fig. 8, the Mulliken atomic charges on C in the C 36 fullerene cage were divided into three groups: − 0.057 a.u. (twelve atoms, red color), 0.052 a.u. (twelve atoms, green color), and 0.005 a.u. (twelve atoms, dark color). C 36 fullerene cage had HOMO-LUMO gaps of 1.718 eV (alpha molecular orbital) and 1.744 eV (beta molecular orbital) which were calculated theoretically by DFT/B3LYP level. The HOMO-LUMO gap value (~ 1.7 eV) of C 36 fullerene cage was determined close to those of C 60 (1.6 eV) which was very stable and larger than itself.
Using photoelectron spectroscopy, the EA of the C 36 − anion in the gas phase was measured as 3.0 eV [46]. In theoretical study, EA and IP values of C 36 fullerene cage with D 6h symmetry were calculated as 6.70 eV and 2.50 eV by B3LYP/6-31G(d) level [46]. In the present study, these properties were predicted as 7.335 eV (IP), 2.889 eV (EA ad ), 2.815 eV (EA vert ), and 2.962 eV (VDE) by DFT/CAM-B3LYP level (Table 1).
In the present study, the calculated the of C 36 fullerene cage was obtained at 47.561 × 10 −−24 esu by DFT/CAM-B3LYP level. Sabirov 29 et al. marked that the of C 36 fullerene cage was 52.41 × 10 −24 esu. The Δ of C 36 fullerene cage was calculated as 0.909 × 10 −24 esu by DFT/CAM-B3LYP level. Since C 36 fullerene cage had a D 6h symmetry, all components of and β 0 values were found zero ( Table 2).

Comparison of some properties
In conceptual DFT, Fukui functions could be employed as local descriptors to predict nucleophilic and electrophilic attacks. As seen in Figure S1, Fukui functions of C N (N = 20, 24, 26, 28, 30, 32, 34, 36, and 38) fullerene cages are exhibited. Since the results from the Hirshfeld charge were reliable, the researchers calculated the Fukui function using the Hirshfeld charge. The nucleophilic region covered by the negative isosurfaces were the regions colored in blue and the electrophilic region covered by the positive isosurfaces were colored in green. As understood from the figure, both nucleophilic (blue areas in the figures) and electrophilic (green areas in the figures) regions were determined using the Hirshfeld charge of the carbon atoms in the fullerene. The images of the Fukui functions were obtained through the Multiwfn 3.8.8 program [48].
C 60 was the most stable fullerene comprising 20 hexagons and 12 pentagons. Therefore, the focus of most of the experimental studies on fullerenes was C 60 fullerene. In the present study, the researchers compared the HOMO LUMO values of the calculated fullerenes with the data of Fig. 9 Geometry, Mulliken charge, ESP counter map and HOMO, and LUMO plots of C38 fullerene cage C 60 . The HOMO-LUMO gaps of fullerenes calculated at the DFT B3LYP/6-311G (d,p) level have been proved to be useful in determining their stabilities. Fullerenes with HOMO-LUMO gap value greater than 1.3 eV had high stability. However, HOMO-LUMO gap value less than this value showed low stability. The studied fullerenes had HOMO-LUMO gap values greater than 1.3 eV (Table 3).
While the C 26 and C 28 fullerene cages were the most stable in quartet multiplicities, the C 34 and C 36 fullerene cages were the most stable in triplet multiplicities. As seen in Fig. 10, as the multiplicity increased, especially the LUMO value of the α orbitals increased. Therefore, the researchers compared the C 60 values with fullerenes that were stable only at the singlet multiplicity. Since the C 26 , C 28 , C 34 , and C 36 fullerene cages were found to be stable at high multiplicities, the researchers did not make comparisons with the C 60 data.
Wang et al. [49] calculated the HOMO, LUMO, and HOMO-LUMO gap values of C 60 by B3LYP/6-311G(d,p) level. In their study, − 6.402 eV, − 3.658 eV, and 2.744 eV were predicted as HOMO, LUMO, and HOMO-LUMO gap of C 60 , respectively. In the present study, the C 32 fullerene cage had the highest HOMO-LUMO gap value (2.610 eV). C 32 is a magic number carbon cluster that always gives very intense signals in gas phase experiments. In PES studies, it is stated that the neutral C 32 molecule has a HOMO-LUMO gap comparable to C 70 and C 60 [46]. In the present study, the HOMO-LUMO value of the C 32 fullerene cage closest to the HOMO-LUMO value of C 60 was determined by DFT calculation.
Veries et al. reported that the IP of C 60 was experimentally determined to 7.58 eV by using single-photon ionization [50]. As seen in Fig. 11, the sequence of magnitude of the IP values was determined as C 28 > C 32 > C 30 > C 24 > C 26 > C 36 > C 38 > C 20 . When evaluated on IP data, the researchers slightly overestimated the values of C 24 , C 28 , C 30 , and C 32 fullerene cages from the experimental value of C 60 . The IP value of the other fullerene cage were calculated slightly smaller than the C 60 experimental data. The most accurate EA of C 60 was determined as 2.6835 eV by the high-resolution photoelectron imaging [51]. When compared to the previous studies on fullerenes with the EA value of C 60 , only the C 20 and C 26 fullerene cages values were calculated slightly small while the other fullerene cages were calculated slightly larger.
Overall, it could be stated that the IP and EA values of the studied fullerenes were compatible with the C 60 data. Nabil et al. reported that the of C 60 fullerene cage was calculated at 77.59 × 10 −24 esu by CAM-B3LYP/6-31 + G(d,p) level [52]. Among the fullerenes studied, the smallest value was determined as 27.59 × 10 −24 esu in C 20 fullerene cage by DFT/CAM-B3LYP level. This value increased continuously for all fullerenes up to C 38 fullerene. The value of C 38 fullerene cage was determined as 49.95 × 10 −24 esu by DFT/CAM-B3LYP level. Hyperpolarizability values of the C 20 , C 24 , C 26 , C 28 , C 32 , and C 36 fullerene cages with zero dipole moment were found zero. The other fullerenes had small dipole moments and small hyperpolarizability values.
Density of state (DOS) pictograms were obtained from Gaussian curves of unit height. DOS pictograms was obtained using the Mulliken population analysis results. DOS pictograms of fullerenes were obtained using the GaussSum 2.2 program [53]. The full width at half maximum (FWHM) at half height was used as 0.3 eV. The spectra of the state densities, which carry the orbital information of the molecules and emerge with the Gaussian type curves, were presented in Figure S2. The state density plots presented the population analysis per orbital. Figure S2 showed the DOS pictograms in the energy range − 20 to 0 eV. As seen in Figure S2, the DOS pictograms of the stable fullerenes in the singlet or high multiplicities were visualized. Total DOS pictograms in the singlet case and DOS pictograms of the α occupied and β virtual orbitals in the high multiplex case were created.

Conclusions
This study reported quantum chemical study of C N (N = 20, 24,26,28,30,32,34,36, and 38) fullerene cages. More specifically, detailed structural, electronic, non-linear optical properties, Mulliken charge, density of state, and Fukui function analysis of these fullerenes were provided. In conclusion, the symmetries and multiplicities of the global minimum structure of these fullerene cages were found D 2 with singlet (C 20 ), D 6 with singlet (C 24 ), D 3h with quintet (C 26 ), T d with quintet (C 28 ), C 2v with singlet (C 30 ), D 3 with singlet (C 32 ), C 2 with triplet (C 34 ), D 6h with triplet (C 36 ), and C 2 with singlet multiplicity (C 38 ). Hyperpolarizability values of the C 20 , C 24 , C 26 , C 28 , C 32 , and C 36 fullerene cages with zero dipole moment were found zero. The smallest mean polarizability value was determined in C 20 fullerene cage. This value increased continuously for all fullerenes up to C 38 fullerene. Succinctly, it could be stated that the ionization potential and electron affinity values of the studied fullerenes were compatible with the C 60 data. The HOMO-LUMO value of the C 32 fullerene cage closest to the HOMO-LUMO value of C 60 was determined by DFT calculation.