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]