To calculate the vibrational frequencies and predict the infrared spectrum of methane CH4 using quantum chemical methods, we will follow these steps:1. Determine the molecular geometry and normal modes of vibration.2. Calculate the force constants.3. Calculate the vibrational frequencies.4. Predict the infrared spectrum.Step 1: Determine the molecular geometry and normal modes of vibrationMethane CH4 has a tetrahedral geometry, with the carbon atom at the center and four hydrogen atoms surrounding it. There are 12 vibrational degrees of freedom 3N - 6, where N is the number of atoms , which can be classified into normal modes of vibration: 3 stretching modes 1, 2, 3 and 9 bending modes 1, 2, 3, 4, 5, 6, 7, 8, 9 .Step 2: Calculate the force constantsTo calculate the force constants, we need to perform a quantum chemical calculation using a suitable method, such as Hartree-Fock HF or Density Functional Theory DFT . These calculations provide the potential energy surface PES of the molecule, which can be used to determine the force constants k for each normal mode of vibration.Step 3: Calculate the vibrational frequenciesOnce we have the force constants, we can calculate the vibrational frequencies using the following formula: = 1/2 * k/ where is the reduced mass of the vibrating atoms. For methane, the reduced mass for the stretching modes is approximately the mass of a hydrogen atom 1 amu , and for the bending modes, it is approximately half the mass of a hydrogen atom 0.5 amu .Step 4: Predict the infrared spectrumThe vibrational frequencies calculated in step 3 correspond to the peaks in the infrared spectrum. The stretching modes 1, 2, 3 will appear in the region of 2800-3200 cm^-1, while the bending modes 1, 2, 3, 4, 5, 6, 7, 8, 9 will appear in the region of 1200-1500 cm^-1. The intensities of these peaks depend on the change in the dipole moment during the vibration, which can also be calculated using quantum chemical methods.In summary, to predict the infrared spectrum of methane using quantum chemical methods, we need to perform a quantum chemical calculation to obtain the force constants, calculate the vibrational frequencies using these force constants, and then predict the peak positions and intensities in the infrared spectrum based on these frequencies.