Calculating the vibrational frequencies and infrared spectra of a methane CH4 molecule using quantum chemistry methods requires the use of computational chemistry software and a suitable level of theory. Here, I will provide a general outline of the process, but the actual values will depend on the specific method and basis set used in the calculations.1. Choose a computational chemistry software package: There are several software packages available for quantum chemistry calculations, such as Gaussian, ORCA, or NWChem. You will need to choose one that suits your needs and is available to you.2. Select a level of theory: The accuracy of the vibrational frequencies and infrared spectra depends on the level of theory used in the calculations. Commonly used methods include Hartree-Fock HF , Density Functional Theory DFT , or higher-level ab initio methods like Mller-Plesset perturbation theory MP2 or Coupled Cluster CC . Additionally, you will need to choose a basis set, which is a mathematical representation of the atomic orbitals. Popular basis sets include 6-31G d , 6-311++G 2d,2p , or cc-pVTZ.3. Optimize the geometry: Before calculating the vibrational frequencies, you need to optimize the geometry of the methane molecule. This means finding the lowest energy structure, which corresponds to the equilibrium bond lengths and angles.4. Calculate the vibrational frequencies: Once the geometry is optimized, you can perform a frequency calculation. This will provide you with the vibrational frequencies and their corresponding normal modes. Methane has 12 vibrational degrees of freedom, but since it is a non-linear molecule, it has 3N-6 = 9 vibrational modes where N is the number of atoms . However, due to its high symmetry T_d , only 4 unique vibrational modes are present: 2 A_1 modes symmetric stretch and bending , 1 E mode doubly degenerate bending , and 1 T_2 mode triply degenerate bending .5. Obtain the infrared spectra: The output of the frequency calculation will also include the infrared intensities of each vibrational mode. These intensities can be used to simulate the infrared spectra of methane. The symmetric stretch A_1 mode has the highest frequency, followed by the doubly degenerate bending E mode, the triply degenerate bending T_2 mode, and finally the symmetric bending A_1 mode.Please note that the actual vibrational frequencies and infrared spectra will depend on the specific method and basis set used in the calculations. To obtain accurate values, you will need to perform the calculations using a suitable computational chemistry software package and level of theory.