## 3 Results and Discussion
### Transmission Characteristics
This can be seen in the MPSH for the device shown in Figure 5a, where the transport around the Fermi energy level after coupling the electrodes depends on the closer LUMO, where the PScCp has spin-up LUMO of \(-0.13\) eV and spin-down LUMO of \(-0.08\) eV, while PVCp has \(0.09\) eV and \(0.04\) eV, with the spin-down LUMO always being closer to the Fermi energy level. Moreover, the transmission spectrum (Figure 6a) also shows that there is an obvious spin-down peak near the Fermi energy level, indicating that the spin-down wave function matches the wave function of the electrode better. The device shown in Figure 5b appears semblable phenomenon, in which spin-up and spin-down channels interchange, resulting in perfect SFE. For the AP type, the transmission eigenvalues are very small, so the junctions will present tremendous GMR ratio. There are multiple transmission peaks around the Fermi level for Figure 5c,d devices with P and AP types, implying the GMR phenomenon disappears. Then, we attempt to explain it by scattering states at the Fermi level, where for each magnetic type, only the two eigenstates that contribute most are analyzed. For Figure 5a device, the d\({}_{xz}\) and d\({}_{yz}\) orbitals of Ni atom are well coupled with p\({}_{x}\) orbitals of N, d\({}_{xz}\) and d\({}_{yz}\) orbitals of V, and p\({}_{x}\) orbitals of C atoms on Cp ring. For the AP configuration, the wave functions of right-electrode Ni and C atoms are almost completely orthogonal, which makes it difficult for electrons hop from the left electrode to the right electrode through the intermediate molecule, resulting in a very low transmission eigenvalue. The LUMO is closest to the Fermi level in the spin-down energy level for PVCp, and the eigenstates are shown in Figure 7a. The highest energy (\(1.7\times 10^{-4}\) eV) of the spin-down band across the Fermi level lower than the LUMO, where we get from the band structure of 2 A Ni nanowires, as shown in Figure 7b, energy mismatch is the main reason why the spin-down electrons cannot pass through and the Figure 5b device presents \(100\%\) SFE. We also think that their wave functions are completely orthogonal as they are very close in energy and delocalized, but have no hybrid state. For AP, the band structure of the right electrode is the same as that of the image of the left but the spin is opposite, which inhibits the transmission of the spin-up electrons to the right electrode, so the total conductivity is zero. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**PScCp**} & \multicolumn{2}{c}{**PTiCp**} & \multicolumn{2}{c}{**PVCp**} & \multicolumn{2}{c}{**PFeCp**} \\  & **(Singlet)** & **(Doublet)** & **(Triplet)** & **(Sexet)** \\ \hline  & monoatomic & 27.05 & 52.85 & 58.5 & 54.94 \\ \(G_{P}(\mu S)\) & 2 Å & 2.16 & 6.14 & 35.31 & 0.7 \\  & 3.5 Å & 17.08 & 39.29 & 82.53 & 33.15 \\  & 4.5 Å & 6.18 & 16 & 15.83 & 65.72 \\ \hline  & monoatomic & 1.69 & 1.79 & 5.1 & 0.15 \\ \(G_{AP}(\mu S)\) & 2 Å & 0 & 0 & 0 & 0 \\  & 3.5 Å & 18.16 & 47.06 & 77.39 & 21.48 \\  & 4.5 Å & 4.64 & 11.49 & 15.98 & 52 \\ \hline  & monoatomic & 88.2\% & 93.4\% & 84\% & 99.5\% \\  & 2 Å & 100\% & 100\% & 100\% & 100\% \\  & 3.5 Å & \(-\)3\% & \(-\)9\% & 3.2\% & 21.4\% \\  & 4.5 Å & 14.3\% & 16.4\% & 0.5\% & 11.7\% \\ \hline  & monoatomic & 100\% & 100\% & 100\% & 100\% \\ SFE(P) & 2 Å & 100\% & 100\% & 100\% & 100\% \\  & 3.5 Å & 0.9\% & 32.8\% & 50.6\% & 100\% \\  & 4.5 Å & 70\% & 35.9\% & 55.5\% & 14.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Conductance (G), magnetoresistance (MR), and spin filtering efficiency (SFE) for P type of ground-state PTMCp (TM = Sc, Ti, V and Fe) with four kinds of electrodes. When the electrodes consist of 3.5 A and 4.5 A Ni nanowires, the transmission value at the Fermi level for the P is close to that for the AP, and the GMR effect disappears completely. The number of energy bands near the Fermi level increases due to the number increase of atoms in the cell, and more electronic states are coupled with the molecule.