To calculate the vibrational frequencies and corresponding infrared spectra of the hydrogen fluoride HF molecule using quantum mechanical methods, we can use the harmonic oscillator model. This model is a good approximation for the vibrational motion of diatomic molecules like HF.1. Determine the reduced mass of the HF molecule: = m * m / m + m where m and m are the masses of hydrogen H and fluorine F atoms, respectively. The mass of hydrogen is approximately 1 amu atomic mass unit , and the mass of fluorine is approximately 19 amu. = 1 * 19 / 1 + 19 = 19 / 20 = 0.95 amu2. Calculate the force constant k of the HF bond:The force constant can be determined experimentally or theoretically. For this example, let's assume the force constant k is 950 N/m, which is a reasonable value for the HF bond.3. Calculate the vibrational frequency using the following formula: = 1 / 2 * k / First, convert the reduced mass to kg:0.95 amu = 0.95 * 1.66054 x 10^-27 kg = 1.5775 x 10^-27 kgNow, calculate the vibrational frequency: = 1 / 2 * 950 / 1.5775 x 10^-27 = 3.289 x 10^13 Hz4. Convert the vibrational frequency to wavenumbers cm :Wavenumber = / cwhere c is the speed of light in cm/s approximately 3 x 10^10 cm/s . = 3.289 x 10^13 Hz / 3 x 10^10 cm/s = 1096 cm5. Infrared spectra:The infrared spectra of the HF molecule will show a single absorption peak at the calculated wavenumber 1096 cm , corresponding to the vibrational frequency of the molecule.In summary, the vibrational frequency of the HF molecule is approximately 3.289 x 10^13 Hz, and its corresponding infrared spectra will show a single absorption peak at 1096 cm. Keep in mind that these calculations are based on the harmonic oscillator model, which is an approximation, and more accurate results can be obtained using advanced quantum mechanical methods and computational chemistry software.