To calculate the vibrational frequencies and infrared spectra for a molecule of carbon dioxide CO2 using quantum chemistry methods, we can use the following steps:1. Determine the molecular geometry and normal modes of vibration:Carbon dioxide CO2 has a linear molecular geometry with the carbon atom in the center and the two oxygen atoms on either side. There are a total of 4 normal modes of vibration for CO2: symmetric stretching, asymmetric stretching, and two bending modes.2. Calculate the force constants:To calculate the vibrational frequencies, we need to determine the force constants for each normal mode. This can be done using quantum chemistry methods such as Hartree-Fock or Density Functional Theory DFT calculations. These calculations require a basis set, which is a set of mathematical functions used to approximate the molecular orbitals. Common basis sets include STO-3G, 6-31G, and cc-pVDZ.3. Calculate the vibrational frequencies:Once the force constants are obtained, we can calculate the vibrational frequencies using the following formula:v = 1/2 * k/ where v is the vibrational frequency, k is the force constant, and is the reduced mass of the vibrating atoms. The reduced mass can be calculated as: = m1 * m2 / m1 + m2 where m1 and m2 are the masses of the two atoms involved in the vibration.4. Determine the infrared spectra:The infrared spectra can be determined by calculating the intensities of the vibrational transitions. This can be done using the dipole moment derivatives with respect to the normal coordinates. The intensity of a vibrational transition is proportional to the square of the change in the dipole moment during the vibration.5. Analyze the results:The calculated vibrational frequencies and infrared spectra can be compared to experimental data to validate the accuracy of the quantum chemistry calculations. The vibrational frequencies of CO2 are typically found to be around 1330 cm^-1 for symmetric stretching, 2349 cm^-1 for asymmetric stretching, and 667 cm^-1 for bending modes.In summary, calculating the vibrational frequencies and infrared spectra for CO2 involves determining the molecular geometry, normal modes of vibration, force constants, and vibrational frequencies using quantum chemistry methods. The infrared spectra can then be determined by calculating the intensities of the vibrational transitions based on the dipole moment derivatives.