To calculate the fundamental vibrational frequency and corresponding infrared spectrum of the water molecule H2O in its ground state, we need to consider the vibrational modes of the molecule. Water has three normal modes of vibration: symmetric stretch, asymmetric stretch, and bending.The vibrational frequencies of these modes can be calculated 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 molecule.For water, the force constants and reduced masses for the three vibrational modes are:1. Symmetric stretch:Force constant k1 = 4.95 x 10^2 N/mReduced mass 1 = 1.63 x 10^-26 kg2. Asymmetric stretch:Force constant k2 = 2.00 x 10^3 N/mReduced mass 2 = 1.63 x 10^-26 kg3. Bending:Force constant k3 = 7.53 x 10^1 N/mReduced mass 3 = 9.17 x 10^-27 kgNow, we can calculate the vibrational frequencies for each mode:1. Symmetric stretch:v1 = 1/2 * k1/1 = 1/2 * 4.95 x 10^2 N/m / 1.63 x 10^-26 kg 3.65 x 10^13 Hz2. Asymmetric stretch:v2 = 1/2 * k2/2 = 1/2 * 2.00 x 10^3 N/m / 1.63 x 10^-26 kg 7.76 x 10^13 Hz3. Bending:v3 = 1/2 * k3/3 = 1/2 * 7.53 x 10^1 N/m / 9.17 x 10^-27 kg 1.60 x 10^13 HzThe corresponding infrared spectrum can be obtained by converting these frequencies to wavenumbers cm^-1 :1. Symmetric stretch:1 = v1 / c = 3.65 x 10^13 Hz / 3.00 x 10^10 cm/s 3650 cm^-12. Asymmetric stretch:2 = v2 / c = 7.76 x 10^13 Hz / 3.00 x 10^10 cm/s 7760 cm^-13. Bending:3 = v3 / c = 1.60 x 10^13 Hz / 3.00 x 10^10 cm/s 1600 cm^-1So, the fundamental vibrational frequencies and corresponding infrared spectrum of the water molecule H2O in its ground state are:1. Symmetric stretch: 3.65 x 10^13 Hz 3650 cm^-1 2. Asymmetric stretch: 7.76 x 10^13 Hz 7760 cm^-1 3. Bending: 1.60 x 10^13 Hz 1600 cm^-1