Computational method
The calculations to search low lying structures of M@Ge12 (M = Co, Pd, Tc, and Zr) in this work began with lot of previous geometries where transition metal atom sits at various different position i.e. (1) Substitution (2) Endohedral (3) Exohedral on the basis of optimized Ge12 and calculated at all possible spin states as reported in the literature[1-10]. All the initial geometries optimized without any symmetry constraint. For the optimization [to get total minimum energy], we used B3LYP [31-34] exchange correlation function with spin polarized generalized gradient approximation (GGA) as implemented in the Gaussian 03 computational code[35] which is based on linear combination of atomic orbitals density functional theory method. A very standard Gaussian basis (LANL2DZ) sets coupled with effective core potential to express molecular orbital as linear combination of atom-centered basis function is used on all atoms. This basis set can reduce the difficulties in two electron integrals caused by the doped transition metal atoms [36-39]. We have used 3d7 4s2, 4d105s0, 4d5 5s2, and 4d2 5s2 configuration for Co, Pd, Tc, and Zr and 4s2 4p2 configuration for Ge respectively. The accuracy of standard Gaussian basis (LANL2DZ) sets for different transition metal atoms doped in germanium cage clusters was validated by many recently publications[2-4, 15-17]. In present work, the minimum energy structure accepted as optimized when the maximum displacement of atoms, RMS displacement of atoms and the maximum force of atoms have very less magnitudes respectively. Furthermore, we have also corrected the zero point energy correction of the isomer however they are not expected to affect the relative binding energy [40]. The ground state structures were calculated at the same level of theory and found zero imaginary frequency to make sure that the optimized geometries corresponds to real local minima. To find the nature of materials and localized and delocalized electrons near the Fermi level, we also obtained the partial density of states (PDOS) using GaussSum software [41]. The natural bond orbital analysis (NBO) analysis [42-43] was also conducted to find out the charge analysis of valence orbitals on each atom and contribution near the Fermi level in the DOS.
Further to check the quality of our adopted method, test calculations were performed on the Ge-Ge and Ge-Co, Ge-Pd, Ge-Tc, and Ge-Zr dimmers. The bond length and frequencies of these dimmers are 2.54 Å (245 cm-1), 2.26 Å (273 cm-1), 2.25 Å (307 cm-1), 2.25 Å (291 cm-1), and 2.44 Å (307 cm-1) respectively. All the calculated structural parameter such as the bond length and frequency compared with other theoretical and experimental results shown in table 1. To further check the validity of our functional we determined bond length and frequency with different functional such as (B3LYP, MPW1PW91, and B3PW91). All the related parameters are shown in table 2. The outcomes gained by the B3LYP functional with LANL2DZ ECP basis set are acceptable as it is in good agreement with reported theoretical and experimental results [1-4, 44-49]