## 3 Results and Discussion
There are three kinds of possible covalent hybridizations, i.e., between Ga and Mn(A/C), between Mn(A) and Mn(C) along c axis and between Mn(A/C) and Mn(B), which compete to determine the atomic magnetic moments. When c/a < 1, with the increase of the c/a ratio, the following occurs: (1) The distance between Ga and Mn(A/C) decreases, which leads to the strengthened p-d orbital covalent hybridization between Ga and Mn(A/C). Thus, the magnetic moment of Mn(A/C) decreases with the increase of c/a ratio. (2) The distance between Mn(A) and Mn(C) along the c axis increases, and the d-d orbital covalent hybridization between them was weakened. So, the magnetic moment of Mn(A/C) increases. (3) The distance between Mn(A/C) and Mn(B), namely, the distance between two nearest neighbor atoms, decreases and the d-d covalent hybridization is strengthened. Thus, the magnetic moments of Mn(A/C) and Mn(B) show a decrease trend. According to the above phenomena, it can be observed that when the volume is small (v = 17 nm\({}^{3}\)), the d-d orbital covalent hybridization between Mn(A) and Mn(C) atoms plays a dominant role to lead to a sharp decrease of the magnetic moment of Mn(A/C) with the increase of c/a. When the volume is greater than 21 nm\({}^{3}\), the p-d orbital covalent hybridization between Ga and Mn(A/C) and the d-d orbital covalent hybridization between Mn(A/C) and Mn(B) atoms are the main contributors. As for the cases of 18-20 nm\({}^{3}\), which have moderate distances among atoms, the p-d orbital covalent hybridization between Ga and Mn(A/C), the d-d orbital covalent hybridization between Mn(A) and Mn(C) atoms, and the d-d orbital covalent hybridization between Mn(A/C) and Mn(B) atoms counteract each other. Thus, the magnetic moment of Mn(A/C) remains almost unchanged with the increase of c/a. From Figure 4a, it can also be seen that when c/a > 1, the moment of Mn(A/C) first increases and then generates a downward trend. We might consider the case of c/a < 1 to understand the situation. Firstly, with the increase of c/a ratio, the distance between Mn(A/C) and Ga (also Mn(B)) increases. So, the p-d (d-d) covalent hybridization decreases and the moment of Mn(A/C) increases. Secondly, the distance between Mn(A) and Mn(C) atoms along c axis increases. Thus, the d-d covalent hybridization was weakened and the magnetic moments of Mn(A/C) increase. Thirdly, with the increase of c/a ratio, the distance between the Mn(A) and Mn(C) along the a or b axis decreases, which leads to the strengthened of d-d covalent hybridization between them and a decrease of the Mn(A/C) moment. So, we can know that the Mn(A/C) magnetic moment increases first, and then decreases with the increase  
Figure 5: The atomic magnetic moments of Mn(A/C) (**a**,**b**) and Mn(B) (**c**,**d**) as functions of c/a ratio for Mn\({}_{3}\)Z (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) alloys. of c/a, which may be attributed to the change of covalent hybridization originating from the change of interatomic distance. Furthermore, with the increase of volume, the position of the inflection point to go down gradually shifts to the right. This is because Mn\({}_{3}\)Ga alloy with larger volume needs a larger degree of distortion (a larger c/a ratio) to make the distance between Mn(A) and Mn(C) along the a or b axes sufficiently small to achieve the same strength of d-d orbital covalent hybridization. The magnetic moment of Mn(B) as a function of c/a ratio is plotted in Figure 4b for Mn\({}_{3}\)Ga alloys with different volumes. In the process of tetragonal distortion, the distance between Ga and Mn(B) atoms along the c axis gradually increases, and the p-d orbital covalent hybridization between these atoms gradually weakens, which makes the Mn(B) moment continue to increase. At the same time, the distance between Ga and Mn(B) atom along the a or b axis gradually decreases with the increase of c/a ratio. So, the p-d orbital covalent hybridization between these atoms gradually strengthens and the Mn(B) moment decreases.