To calculate the vibrational frequency and the infrared spectrum of a water molecule H2O in its ground state using the harmonic oscillator model, we need to consider the three fundamental vibrational modes of the molecule: symmetric stretch, asymmetric stretch, and bending.1. Symmetric stretch: Both hydrogen atoms move away from or towards the oxygen atom simultaneously.2. Asymmetric stretch: One hydrogen atom moves away from the oxygen atom while the other moves towards it.3. Bending: The angle between the hydrogen-oxygen-hydrogen atoms changes.The vibrational frequency of a diatomic molecule can be calculated using the formula: = 1/2 * k/ where k is the force constant of the bond and is the reduced mass of the molecule. The reduced mass can be calculated using the formula: = m1 * m2 / m1 + m2 where m1 and m2 are the masses of the two atoms in the bond.For water, the reduced mass for the O-H bond can be calculated using the atomic masses of oxygen 16 amu and hydrogen 1 amu : = 16 * 1 / 16 + 1 = 16/17 amuThe force constant k for the O-H bond in water is approximately 1000 N/m.Now, we can calculate the vibrational frequency: = 1/2 * 1000 N/m / 16/17 amu To convert amu to kg, we multiply by 1.66054 x 10^-27 kg/amu: = 1/2 * 1000 N/m / 16/17 * 1.66054 x 10^-27 kg/amu 3.56 x 10^13 HzThe infrared spectrum of a water molecule can be obtained by calculating the wavenumber cm^-1 for each vibrational mode. The wavenumber is related to the frequency by the speed of light c : = / cFor the symmetric stretch, the wavenumber is approximately 3657 cm^-1.For the asymmetric stretch, the wavenumber is approximately 3756 cm^-1.For the bending mode, the wavenumber is approximately 1595 cm^-1.These values represent the main peaks in the infrared spectrum of a water molecule in its ground state.