## 3 Results and Discussion  
### Elastic Properties  
The structural parameters of metal compounds, such as lattice constant a, cell volume V, structure and space group, should be figured out before analyzing their elastic properties. In Table 1, detailed structural information and reference data (Ref. [18]) of Co\({}_{2}\)Zr and Co\({}_{2}\)Ti at zero pressure are given. We can find that the optimized lattice constant, crystal volume and so on are in accordance with the given experimental and other theoretical results. The errors of lattice parameters are 0.47% and 0.25% for Co\({}_{2}\)Zr and Co\({}_{2}\)Ti, respectively. In addition, the values of B\({}_{0}\), B\({}_{0}\)\({}^{\prime}\) and B\({}_{0}\)\({}^{\prime\prime}\) under 0 GPa pressure are provided, where B\({}_{0}\) is the bulk elastic modulus, B\({}_{0}\)\({}^{\prime}\) is the pressure derivative of the bulk modulus, and B\({}_{0}\)\({}^{\prime}\) is the second pressure derivative of the bulk modulus. It can be seen from the table that the values of B\({}_{0}\), B\({}_{0}\)\({}^{\prime}\) and B\({}_{0}\)\({}^{\prime\prime}\) of Co\({}_{2}\)Zr crystal derived from the quasi-harmonic Debye model were 192 GPa, 4.370 and \(-\)0.031, respectively, while those that are 192 GPa, 4.426 and \(-\)0.034 derived from the natural strain equation of state (EOS). For Co\({}_{2}\)Ti, we can get B\({}_{0}\) = 213 GPa, B\({}_{0}\)\({}^{\prime}\) = 4.533, and B\({}_{0}\)\({}^{\prime\prime}\) = \(-\)0.033 through the Debye model along with B\({}_{0}\) = 199 GPa, B\({}_{0}\)\({}^{\prime}\) = 5.247 and B\({}_{0}\)\({}^{\prime\prime}\) = \(-\)0.045 through natural strain equation of state. Although Co\({}_{2}\)Zr and Co\({}_{2}\)Ti have the same cubic structure, lattice constant \(a\) of Co\({}_{2}\)Zr is larger than that of Co\({}_{2}\)Ti while values of B\({}_{0}\) and B\({}_{0}\)\({}^{\prime}\) are smaller than the counterparts under the same condition. There are three independent elastic constants, C\({}_{11}\), C\({}_{12}\) and C\({}_{44}\), in cubic crystal, and their values at 0 GPa, 0 K are given in Table 2. We can calculate adiabatic bulk modulus B\({}_{\text{S}}\), shear modulus G and Debye temperature \(\Theta\) through Formulas (1), (2), and (5) with their results recorded in the Table 2. It was seen that the calculated values of Co\({}_{2}\)Ti basically coincide with the reference values from Ref. [23]. Meanwhile, the data of other structures, such as CoTi [26] and Co\({}_{3}\)Ti [24], are adopted to make comparison with these values. Unfortunately, there are no data available to compare with our results of Co\({}_{2}\)Zr, thus we only exhibit the data of CoZr [26]. The values of C\({}_{11}\), C\({}_{12}\) and C\({}_{44}\) of Co\({}_{2}\)Zr and Co\({}_{2}\)Ti at 0 K and 0-100 GPa are provided in Tables 3 and 4. It may be seen from this table that our values agree well with other theoretical and similar structural results, with the errors of C\({}_{11}\), C\({}_{12}\) and C\({}_{44}\) being, respectively, 10%, 14.9%, 3.9% for Co\({}_{2}\)Ti compared with Ref. [23]. According to the well-known Born stability criteria [35; 36; 39; 40], if the three independent elastic constants satisfy Formula (4), the crystals are stable. Therefore, we can find that Co\({}_{2}\)Zr is always stable under the pressure from 0 to 100 GPa, whereas Co\({}_{2}\)Ti can satisfy the Born stability criteria only under pressures smaller than 80 GPa, which indicates that Co\({}_{2}\)Ti is unstable when the pressure is more than 80 GPa. What is more, it is possible that a transition phrase exists under the pressure of 85 GPa.