Fullerene related materials have attracted attention because of their unique physical and chemical properties.  Fullerenes as cage compounds built exclusively from carbon atoms and their metal-containing forms known as metallofullerenes were for the first time observed in the gas phase [2–6]. The unique spherically shaped cage of ‘’C60’’ fullerene contains empty space that is large enough to incorporate atoms or molecules inside it . Endohedral metallofullerenes having a single metal atom inside the spherical carbon cage are of interest as new molecules or materials with novel properties such as in designing electronic devices, super fibers, catalytic materials, etc., electrical conductance, magnetism, Ferro electricity, nonlinear optical properties, etc., which are expected to differ much from those of empty fullerenes. Moreover, these properties have also made them prominent species in nanotechnology and bio-medicine [2–6]. Experimentally, although carbon clusters of ‘’C70’’ are produced in the laser-vaporization process, except for ‘’C60’’ and ‘’C70’’, it is still difficult to isolate large quantities of the smaller fullerenes for structural determination. On the other hand, existing theoretical studies can be grouped into two categories are the in- and out-side of cage for TM. In the geometric approach, educated guesses based on mostly symmetry considerations were used to construct plausible topological networks, [7–12]. The topological networks of the fullerenes are constructed by symmetry considerations and by the leapfrog procedure [8–13]. Experiments suggest that fullerenes which incorporate alkali metals posses catalytic properties. Enyashin et al.  analyzed the electronic structures and stability of the crystalline modifications of ‘’C28’’ and Zn@C28 using a cluster model. From the point of view of theory, there are several abinitio calculations involving endohedral Li@C20  using the density functional nonequilibrium Green’s function method, and found that the equilibrium conductance of Li@C20 becomes larger than that of the empty ‘’C20’’ molecule. Chen et al..  investigated the stabilities and molecular structures of noble gas atoms (i.e. He, Ne, Ar, and Kr) inside the ‘’C32’’ cage. Also Garg et al.  studied the magnetic properties of 3d transition metals (TM) encapsulated in smaller fullerenes ranging from ‘’C20’’ to ‘’C36’’ by spin polarized density functional theory. In addition to Wu et al.  has been systematically investigated the structure, electronic property, and infrared spectroscopy of endohedral metallofullerenes TM@C20 (TM =Ce and Gd) with the aid of the hybrid DFT-B3LYP functional, and found that the endohedral Metallofullerene Gd@C20 has the high-thermodynamic stability. Moreover the large empty space inside fullerene molecule may be used as storage materials with high capacity and stability. The storage of hydrogen in fullerene has attracted much experimental and theoretical interest, to use in fuel cells for power generation . In the present work we have studied the stability, geometries, binding energies per atom of six fullerene isomers of ‘’C34’’ at DFT/B3PW91 and investigations of endohedral metallofullerenes TM@C34 (TM =Ti, V, Cr, Mn, Fe, Co, Ni, and Cu) to predict their stabilities and electronic properties for exploring their potential application. Some properties of these endohedral metallofullerenes have been investigated on the stable structures, including equilibrium structures, relative stabilities, energy gaps (highest occupied molecular orbital [HOMO]–lowest unoccupied molecular orbital [LUMO]), dissociation energy, charge transfers, Natural charges and natural electron configurations have been performed with the Natural Bond Orbital analysis (NBO),density of states (DOS), molecular electrostatic potentials (MEP), and non-linear optical (NLO)properties. Formation energy (ε), ionization potential (I), electron affinity (A), chemical hardness (η), electronegativity (χ), electrophilicity (ω) and thermodynamic properties were investigated on TM@C34 (TM =Ti).
All the DFT calculations have been performed with the Gaussian 09W (version A.02) software package . The definition for “B3PW91” method is Becke’s three-parameter (B3) hybrid exchanges functional  with the correlation functional of Perdew and Wang (PW91) . Su, K. H.et al. [23,24] shown that B3PW91 method can reproduce the equilibrium structures systematically better than the other methods, such as DFT at BPW91 and B3LYP levels of theory, second order Møller–Plesset perturbation theory (MP2), and quadratic configuration interaction at QCISD and QCISD(T) levels. We considered 3d24s2, 3d34s2, 3d54s1, 3d54s2, 3d64s2, 3d74s2, 3d84s2 and 3d104s1 electronic configurations for Ti, V, Cr, Mn, Fe, Co, Ni and Cu, respectively. Full geometry optimizations and energy calculations of endohedral metallofullerenes TM@C34(TM =Ti, V, Cr, Mn, Fe, Co, Ni, and Cu) were calculated at the B3PW91/6-31G(d) level of theory. All the stationary point geometries were analyzed by evaluating the harmonic vibrational frequencies of the same theoretical level. The figures were generated using the associated Gauss View software. The density of states (DOS) was calculated by using the Gauss Sum 2.2.5 program .
RESULTS AND DISCUSSION
Relative stability of C34 fullerenes
The total energies (Etot), relative energies (Erel), HOMO–LUMO energy gaps (E.G.), binding energies (BE) and dipole moment (DM) systems of the ‘’C34’’ six isomers are presented in Table 1. The optimized structures of the ‘’C34’’ six isomers are obtained using the B3PW91/6-31G(d) level, ‘’C34’’ has six fullerene isomers, i.e., three C2, two Cs, and one C3v , are shown in Fig. 1(a–f). Analysis of data onto Table 1 and Fig. 1(a–f) show that: the e(C2) isomer has the lowest energy among the six isomers we studied at the two methods HF/6-31G(d) level and B3PW91/6-31G(d) level. At the HF/6-31G(d) level, the stability order for ‘’C34’’ is e(C2)< d(C2) <c(CS) < f(C3V) < b(CS) < a(C2). The energy difference between e(C2) and d(C2) is 0.41 eV and that between d(C2) and c(CS) is 0.83 eV. The energy difference between c(CS) and f(C3V) is 0.15 eV and that between f(C3V) and b(CS) is 0.05 eV. The a(C2) isomer is the least stable. At the B3PW91/6-31G(d) level, the stability order for ‘’C34’’ is e(C2)< d(C2) <c(CS) < b(CS) < f(C3V) < a(C2). The relative energies are 0.00,0.68, 1.36, 1.40, 1.55, and 3.35 eV for e(C2), d(C2), c(CS), b(CS), f(C3V), a(C2) respectively. They are comparable with the values calculated at HF/6-31G(d)level. At the B3PW91/6-31G(d) level, the order of the binding energies (BE) for ‘’C34’’ is e(C2)< d(C2) <c(CS) < b(CS) < f(C3V) < a(C2). The binding energies values per atom are 8.85, 8.83, 8.81, 8.81, 8.80, and 8.57 eV for e(C2), d(C2), c(CS), b(CS), f(C3V) a(C2) respectively, this values indicating that the e(C2) isomer is more stable. Fig. 2, shows the isodensity surface of HOMO and LUMO of the ‘’C34’’ six isomers. The HOMO–LUMO energy gap has been used as an index of kinetic stability for fullerenes , which means that a fullerene molecule with a larger HOMO–LUMO gap will be chemically less reactive. The results show that the HOMO–LUMO gap between isomer d(C2) is the biggest one (1.73 eV) and shows its kinetic stability. The e(C2) isomer, predicted to have the lowest energy, has a large HOMO–LUMO gap between 1.47 eV in our calculation, indicating that it is kinetically stable.
Optimized equilibrium geometries of endohedral metallofullerenes TM@C34
In this study, we have considered the electronic state for free TM (TM =Ti, V, Cr, Mn, Fe, Co, Ni, and Cu) in the low-spin state at the B3PW91/6-31G(d) levels of theory. In Table 2, the theoretical electronic configuration of TM in the gas phase is given together with the spin multiplicity, spin state, electronic state and total electronic energy. The ‘’C34’’ cage with e(C2) symmetry Fig. 1 is considered the starting cage configuration to investigate the endohedral metallofullerenes TM@C34(TM =Ti, V, Cr, Mn, Fe, Co, Ni, and Cu). The average C–C and C=C bond lengths , distance between transition metal and center, the shortest transition metal-carbon bond length (in Å) are given in Table 3. For Mn@C34, the stable structure found with the metal atom located at off-center of the ‘’C34’’ cage not in the middle of the carbon cage according to the data in Table 3 is 0.20 Å, while Cu@C34, the stable structure found with the metal atom located at middle -center of the ‘’C34’’ cage and according to the data in Table 3 is 0.00 Å for Cu@C34. In the case of Ti@C34, V@C34, Cr@C34, Fe@C34, Co@C34, and Ni@C34, the stable structure found with the metal atom moved from the center of one pentagonal face and then linked to this face.
‘’C34’’ has an average diameter of 4.84 Å and the binding energy per atom is -300.8 eV with average C–C and C=C bond lengths 1.45-1.50 Å and 1.39-1.45 Å respectively. The optimized structures of the endohedral metallofullerenes TM@C34(TM =Ti, V, Cr, Mn, Fe, Co, Ni, and Cu) are shown in Fig. 3. As a result of TM substitution inside ‘’C34’’ cage, the average diameter shows a slight increase in a maximum value of 0.93 Å for Fe@C34. The average C–C and C=C bond lengths of TM@C34 complexes lie in the range 1.45-1.52 Å (see Table 3). The binding energy per atom is higher for Ti@C34 ,and V@C34, -7.117 eV and -6.089 eV for Ti and V respectively and decreasing value for other TMs. Further, the interaction between the fullerene cage and TMs along with the stabilities of TM@Cn complexes is described by evaluating their cohesive energies. We define the cohesive energy of TM@Cn complexes as follows:
Where ETotal (TM@Cn), E(Cn) and E(TM)) are the total energies of the endohedral complex, isolated fullerene and TM atoms, respectively. The cohesive energies ‘‘∆E’’ calculated using eqn (1) are presented in Table 3. The positive values of ∆E indicate the greater stability of TMs inside fullerene cage and are likely to be formed experimentally. The negative values of ∆E indicate that the corresponding TM encapsulated in the fullerene cage is unstable or less stable and have not been experimentally observed. The cohesive energies tabulated in Table 3 shows positive cohesive energies for all the TMs leading to the fact that they all form stable complexes about ‘’C34’’. The stability order for TM@C34 is (TM= Ti< Cr < V< Fe 34 is (TM= Ti< Cr < V< Fe
Frontier orbital analyses, energy gaps
The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are frequently referred to as frontier molecular orbital’s (FMOs). Fig. 4 shows the frontier orbital surfaces (0.02 eV/a.u.) of HOMO and LUMO of empty ‘’C34’’ and TM@C34(TM =Ti, V, Cr, Mn, Fe, Co, Ni, and Cu). From the frontier molecular orbitals illustrated, in Fig. 4 we could see that there is no hybridization between Mn and Cu atomic orbitals and the ‘’C34’’ cage orbitals. The HOMO and LUMO of TM@C34 (TM =Ti, V, Cr, Fe, Co, and Ni) shows obviously mixing between the orbitals of the ‘’C34’’ cage and 3d orbitals of the transition metal atoms. The orbital hybridization between the guest transition metal atoms and fullerene cage is a common characteristic in metallofullerenes [28 – 32].
The energy gaps between the empty ‘’C34’’ cage and TM@C34 (TM =Ti, V, Cr, Mn, Fe, Co, Ni, and Cu) are given in Table 4. The HOMO–LUMO gap between ‘’C34’’ is 1.47 eV which is calculated from HOMO (-5.53 eV) and LUMO (-4.06 eV). When the endohedral metallofullerenes TM@C34 (TM =Ti, V, Fe, Ni, and Cu), both of the HOMO and LUMO orbital energies are higher by (0.26,0.15,0.19,0.23,0.05) and (0.58,0.55,0.39,0.27,0.31) eV, respectively; this leads to the slightly large HOMO–LUMO gap (1.79,1.89,1.67,1.51 and 1.73 eV) respectively. The energy gaps between Mn@C34 , is the same as that of the empty ‘’C34’’ cage (1.47 eV). It was confirmed that the Mn and Cu atom and the ‘’C34’’ cage are not covalent. A large HOMO– LUMO gap can be used as a simple indicator of high kinetic stability and low chemical reactivity . Thus, it is indicated that the modified ‘’C34’’ derivatives by Ti, V, Mn, Fe, Ni, and Cu atoms increase their kinetic stabilities. The energy gaps between Cr@C34 and Co@C34 are smaller than that of the empty ‘’C34’’ cage, being 1.46, and 1.43 eV, respectively. Interestingly the Fe@C34 and Cr@C34 possesses the larger dipole moments, being 2.93 and 2.42 D, respectively.
Natural population analysis
Natural bond orbital (NBO) has an important role in interaction in complex, specially charge to transfer. The natural charges and natural electron configurations on TM atoms in the most stable TM@C34 (TM = Ti, V, Cr, Mn, Fe, Co, Ni, and Cu) are listed in Table 4. From this table, we can see that the charge transfer from the TM atoms to the ‘’C34’’ cage, so we can consider the fullerene ‘’C34’’ cage acts an electron acceptor. Also, from our calculation results Table 4 we show that no matter whether the valence electrons of the TM atoms are 4–11, the charge transfer is less than 2e. The isolated Ti atom has the following natural electron configurations 4s23d2, but in Ti @C34 the Ti atom has the following electron configurations: 4S0.013d2.664p0.374d0.016S0.26. In going from isolated Ti atom to Ti@C34, the occupation of the 4s orbitals is strongly reduced. The Ti atom forms an ionic bond between the ‘’C34’’ cage, and transfers only 0.70e to the cage. The charge is donated by the 4s orbitals of Ti atom and transfers 0.66e to its 3d orbitals. Similarly, the V, Mn, Fe, Co, and Ni atom transfer 0.76e, 1.06e, 0.69e, 0.90e, and 0.64e to the ‘’C34’’ cage, respectively. These results indicated that the 3d orbitals are the main valence orbitals. For Cr@C34 and Cu@C34 the situation is different, in Cr@C34 the electron configurations on Cr atom is 4S0.183d4.864p0.375S0.014d0.01. When compared with the 4s13d5 configurations of isolated Cr atom, it was found that the Cr atom transfers 0.58e to the ‘’C34’’ cage that charge is donated by the 4s and 3d orbitals of Cr atom. In Cu@C34, we found that the Cu atom transfers 0.38e to the ‘’C34’’ cage that charge is donated by the 4s and 3d orbitals of Cu atom.
Density of states and molecular electrostatic potentials
As shown in Table 4, the HOMO–LUMO energy gaps are reduced under the effect of introduced Cr@C34 and Co@C34 and implies that the density of states (DOS) become more abundant near Fermi levels. In solid state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. A high DOS at a specific energy level means that there are many states available for occupation. The calculated DOS are presented in Fig. 5 to clarify the effects of endohedral metallofullerenes TM@C34(TM =Ti, V, Cr, Mn, Fe, Co, Ni, and Cu). As shown, DOS become more abundant near Fermi levels in both Cr@C34 and Co@C34, where the overall features of DOS change, and deep vales develop on both sides nearby Fermi levels.
Molecular electrostatic potential (MEP)
The MEP contours or surfaces have been established extensively as a guide to the interpretation and prediction of molecular behavior . It has been shown to be a useful tool for studying both electrophilic and nucleophilic processes, in particular the ‘‘recognition’’ of one molecule by another . MEPs are either negative, low potentials that are characterized by an abundance of electrons and reactive with electrophiles, or positive, high potentials that are characterized by an absence of electrons and reactive with nucleophiles. We denote the former by a deep red color, and the later with a deep blue color. The molecular electrostatic potential surfaces (MEP) of empty ‘’C34’’ and endohedral metallofullerenes TM@C34(TM =Ti, V, Cr, Mn, Fe, Co, Ni, and Cu) are given in Fig. 6. The electron color scheme for the MEP surface is as follows: (deep red) low MEPs for electron rich, (the negative charge); (deep blue) high potentials for deficient, (the positive charge); respectively.
As shown, the intensities of the positive and negative MEP surfaces are more affected by endohedral metallofullerenes than the empty ‘’C34’’. For both Cr@C34 and Co@C34, the negative (deep red) low MEPs that are characterized by an abundance of electrons and reactive with electrophiles predominate the region of the ‘’C34’’ and extend to the Cr and Co. Simultaneously, the positive (deep blue) high potentials are significantly reduced in these regions. It is therefore concluded that the TM regions of both Cr@C34 and Co@C34 are more readily available for electrophilic processes, while the ‘’C34’’ regions are more readily available for nucleophilic processes. For both Mn@C34 and Cu@C34, we notice that Mn and Cu are located at off-center of the ‘’C34’’ cage and the MEP distributions for them are too similar to the empty ‘’C34’’ cage.
The non linear optical (NLO) properties, such as polarization and hyper- polarization characterize the response of a system in an applied electric field and determine the strength of molecular interactions as well as the cross-sections of different scattering and collision processes. The electronic dipole moment, molecular polarization, anisotropy of polarization and first hyper- polarization of the empty ‘’C34’’ and TM@C34(TM =Ti, V, Cr, Mn, Fe, Co, Ni, and Cu) are investigated. The higher values of molecular polarization and dipole moments are important for more active NLO performance. The NLO phenomena occur to sufficiently intense fields. As the applied field strength increases, the polarization response to the medium is no longer linear and the induced polarization (P) becomes a function of the applied field .
where the χ(2) and χ(2) coefficients represent the second and third-order susceptibilities of the medium, respectively. At the molecular level, Equation (2) is expressed as 
Where αij = polarization, βijK=first hyper- polarization (second-order effects), γijKl=second hyper- polarization (third-order effects), and i,j,K,l correspond to the molecular coordinates.
The average linear polarization (αtot) and the anisotropy of the polarization (∆α) are calculated as follows 
αtot = 1/3 (𝛼𝑥𝑥 + 𝛼𝑦𝑦 + 𝛼𝑧𝑧), (4)
Δ𝛼 = ((𝛼𝑥𝑥 − 𝛼𝑦𝑦) 2 + (𝛼𝑦𝑦 − 𝛼𝑧𝑧) 2 + (𝛼𝑧𝑧 − 𝛼𝑥𝑥) 2/2)1/2, (5)
The tensor of the static first-order hyper- polarization is a result of the third derivative of the energy with respect to the electric field components. The second order polarization or first hyper-polarization (βtot) is calculated as follows:
⟨𝛽⟩ = (𝛽2𝑥 + 𝛽2𝑦 + 𝛽2𝑧) 1/2, (6)
𝛽𝑥 = 𝛽𝑥𝑥𝑥 + 𝛽𝑥𝑦𝑦 + 𝛽𝑥𝑧𝑧,
𝛽𝑦 = 𝛽𝑦𝑦𝑦 + 𝛽𝑥𝑥𝑦+ 𝛽𝑦𝑧𝑧,
𝛽𝑧 = 𝛽𝑧𝑧𝑧 + 𝛽𝑥𝑥𝑧+ 𝛽𝑦𝑦𝑧.
The total dipole moment μtot can be calculated by using the following equation:
μtot = (𝜇2𝑥 + 𝜇2𝑦 + 𝜇2𝑧) 1/2 (7)
Where μi (i =x,y,z) is the electronic dipole moment.
The hyper- polarization, dipole moment and polarization for empty ‘’C34’’ cage and TM@C34 (TM =Ti, V, Fe, Ni, and Cu) are presented in Table 5. Several facts emerge from Table 5: (1) the larger polarization αtot is assigned to Mn@C34 and dominated by the term αxx. All of the polarization terms are significantly affected by incorporation of TM; (2) the larger resultant dipole moment μtot is assigned to Fe@C34, and dominated by the term μy. The dipole moment components μx, μy, and μz, are significantly affected by incorporation of TM; (3) the larger positive value of the total hyper- polarization term βtotal is assigned to Co@C34 and dominated by the positive termβxxx. All of the hyper- polarization terms are significantly affected by incorporation of TM. These results indicate that the polarization α, resultant dipole moment μ, and hyper-polarization β distinguish between TM@C34 complex and empty ‘’C34’’ cage, where larger values of polarization, hyper- polarization, and dipole moments are assigned to the former. For these systems, the relative NLO activities are in order: Cu@C34> Ni@C34> Ti@C34> Cr@C34> Fe@C34> Mn@C34> V@C34> Co@C34 , indicating that the structural modification by TM atom strengthens the second-order NLO response because the encapsulated atoms induce the localized bonding interactions in the inner surface of cage. There are no available experimental or theoretical values for comparison; however, we hope that all theoretical evidences would provide some chemical insight to characterize these complexes for the sake of their future experimental exploration.
The ionization potential (I) and electron affinity (A) can be calculated from relations 8 and 9:
I = E+1 – E0 (8)
A= E0 – E-1 (9)
Where I and A are obtained from the total electronic energies E+1, E0 and E-1 of the cation, neutral, and anion forms of TM@C34 at the neutral geometry, i.e. the optimized geometry of neutral complex TM@C34 E0 used for calculations of energies of both its cation E+1 and anion forms E-1 . In DFT, the natural (optimized geometry complex TM@C34) way to approximate electronegativity (χ)[39,40] and chemical hardness (η) [41,42] is to evaluate them directly from the calculated I and A as shown in the following equation (10,11):
𝜒 = - μ = (𝐼 + 𝐴)/2 (10)
𝜂 = (𝐼 − 𝐴)/2 (11)
The electrophilicity [43–45] (ω) is defined as
ω = μ2/ 2𝜂 (12)
The Ionization potential(I/eV), electron affinity (A/eV), chemical hardness (η/eV), electronegativity (χ / eV), chemical potential (μ / eV) and electrophilicity (ω/eV) of TM@C34 are collected in Table 6. Chemical hardness (η) can be used as complementary tool to describe the thermodynamic aspects of chemical reactivity. Our computational study shows that the hardness (η) of the complex Co@C34 is smaller than that of the complex V@C34 indicating a decrease in the rigidity of the Co@C34 complex thereby decreasing its stability. A good electrophile will be characterized by a high value of electrophilicity (ω). It is therefore evident that while the complexs V@C٣٤ are better electrophile than Co@C34, it is more stable and less reactive than Co@C34 complex.
Vibrational analysis and statistical thermodynamics, the standard thermodynamic functions, the heat capacity (C0 p,m), entropy (S0 m), and enthalpy (H0m ), for both TM@C34 and TM =Ti were obtained at the B3PW91/ 6−31G(d) level and are listed in Table 7. The table shows that the standard heat capacities, entropies, and enthalpies increase at any temperature from 200.00 to 600.00 K, because the intensities of the molecular vibration increase with the increasing temperature. The correlations between these thermodynamic properties and temperatures T are shown in Fig. 7. The correlation equations are as follows:
C0p,m,C34 = −9.98222 + 0.24615 T − 4.38172×10−4 T2; ( R2 = 0.99378 ) , (13)
C0p,m,Ti-C34 =−8.83333 + 0.25447 T − 4.59044×10−4 T2; ( R2 = 0.99361 ) , (14)
S0 m,C34 = 37.47622 + 0.22856 T + 0.84529×10−5 T2; (R2=0.99973) , (15)
S0m,Ti-C34 = 38.12089+ 0.24011 T – 0.80505×10−4 T2; (R2=0.99978) , (16)
H0m,C34 = 111.22422 - 0.00532 T + 2.2347 ×10−4 T2; (R2 = 0.98804 ) , (17)
H0m,Ti-C34 = 111.51378 - 0.00551 T + 2.3134 ×10−4 T2; (R2 = 0.98838 ) . (18)
These equations will be helpful for the further studies for both TM@C34 and TM =Ti compounds [46,47].
In summary, using the ab initio and density functional theory (DFT) methods, the author has studied the structure and stability of ‘’C34’’ fullerene isomers and predicted that the e(C2) isomer is the ground state of ‘’C34’’. In this work, we have performed the theoretical investigations into TM@C34 (TM =Ti, V, Cr, Mn, Fe, Co, Ni, and Cu) using the hybrid DFT-B3PW91 functional in conjunction with 6-31G(d) basis sets. For systematically exploring these complexes, some properties were considered, such as relative stabilities EHOMO, ELUMO, energy gap, dissociation energy , natural population analysis, the density of states (DOS), molecular electrostatic potential (MEP) , and nonlinear optical (NLO) properties. We found that the stable structure of Mn@C34 and Cu@C34 are situated in the center of ‘’C34’’ cage. While Ti@C34, V@C34, Cr@C34, Fe@C34, Co@C34, and Ni@C34 moved from the center towards one pentagonal face and then linked to this face. The cohesive energy of TM@C34 complexes suggests that Ti and V are most stable inside all cage structures, therefore indicating a strong possibility of their production. Cu is found to be least stable for encapsulation. All encapsulation processes are endothermic; this is likely due to swelling to the ‘’C34’’ cage upon encapsulation of TM. Here, all eight TM@C34 is at the local minima because they have all real frequencies. The HOMO and LUMO of Mn@C34 and Cu@C34 indicates that there is no hybridization between the ‘’C34’’ orbitals and the endohedral atom orbitals, while the HOMO and LUMO of Ti@C34, V@C34, Cr@C34, Fe@C34, Co@C34, and Ni@C34 indicate that there exists hybridization between the ‘’C34’’ orbitals and the end atom orbitals. It was also found that the energy gaps between HOMO and LUMO of TM@C34(TM= Cr, and Co) are smaller than that of the empty ‘’C34’’ cage. However, the energy gaps between HOMO and LUMO of TM@C34 (TM =Ti, V, Fe, Ni, and Cu) are larger than that of the empty ‘’C34’’ cage. Natural population analysis shows that the charges always transfer from the TM atoms to the ‘’C34’’ cage, so the fullerene ‘’C34’’ cage acts an electron acceptor. In going from isolated TM atom to TM@C34, the occupation of the 4s orbital is strongly reduced. The introduction to TM to the empty ‘’C34’’ leads to more active NLO performance. The correlations between the thermodynamic properties Cop,m, Som, Hom, and temperatures T are also obtained.
This research was supported by the facilities of Dep. Chemistry, Faculty of Education, Ain Shams University. My gratitude and deep thanks to Prof. Dr. A.M. El Mahdy for his interests, and useful discussions.
CONFLICT OF INTEREST
The authors declare that there are no conflicts of interest regarding the publication of this manuscript.