To calculate the vibrational frequencies and corresponding infrared spectra of carbon dioxide CO2 using quantum chemical methods, we will follow these steps:1. Determine the molecular geometry and normal modes of vibration for CO2.2. Calculate the force constants for each normal mode.3. Calculate the vibrational frequencies using the force constants.4. Determine the infrared activity of each vibrational mode.Step 1: Molecular geometry and normal modes of vibrationCO2 has a linear molecular geometry with the carbon atom in the center and the two oxygen atoms on either side. There are three normal modes of vibration for CO2:- Symmetric stretching v1 : Both oxygen atoms move away from or towards the carbon atom simultaneously.- Bending v2 : One oxygen atom moves towards the carbon atom while the other moves away, causing the molecule to bend.- Asymmetric stretching v3 : One oxygen atom moves away from the carbon atom while the other moves towards it.Step 2: Calculate the force constantsTo calculate the force constants, we can use quantum chemical methods such as Hartree-Fock or Density Functional Theory DFT . These methods involve solving the Schrödinger equation for the molecule and determining the potential energy surface. The force constants can be obtained from the second derivative of the potential energy with respect to the normal mode coordinates.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 vibrational frequency, k is the force constant, and is the reduced mass of the vibrating atoms. The reduced mass can be calculated using the atomic masses of carbon and oxygen: = m_C * m_O / m_C + m_O Step 4: Determine the infrared activityTo determine the infrared activity of each vibrational mode, we need to consider the change in the molecular dipole moment during the vibration. In CO2, the symmetric stretching mode v1 does not result in a change in the dipole moment and is therefore not infrared active. The bending mode v2 and asymmetric stretching mode v3 both result in a change in the dipole moment and are therefore infrared active.In summary, using quantum chemical methods, we can calculate the vibrational frequencies and corresponding infrared spectra of CO2. The symmetric stretching mode is not infrared active, while the bending and asymmetric stretching modes are infrared active.