35Cl2 (560 cm-1), 39K35Cl (281 cm-1), 1H2 (4401 cm-1). Calculate the force constants for these molecules if treated as harmonic oscillators. 

Calculate the root-mean-square displacement of the nuclei of l2Cl6O in the v = 0 state and compare it with the equilibrium bond length of 112.832 pm. Later, in Table 13.4, we will find that the following molecules have the indicated vibrational frequencies: 35Cl2 (560 cm-1) 39K35C1(281 cm-1) 1H2(4401 cm-1) What are the force constants for these molecules if we treat them as harmonic oscillators? (b) Assuming that the force constant for 37Cl2 is the same as for 35Cl2, predict the fundamental vibrational frequency of 37Cl2. Check the normalization of psi0 and spi1 for the harmonic oscillator and show that they are orthogonal. Substitute the v = 1 eigenfunction for the harmonic oscillator into the Schrodinger equation for the harmonic oscillator, and obtain the expression for the eigenvalue (energy). In the vibrational motion of HI. the iodine atom essentially remains stationary because of its large mass. Assuming that the hydrogen atom undergoes harmonic motion and that the force constant k is 317 N m-1, what is the fundamental vibration frequency v0? What is v0 if H is replaced by D? What are the expectation values for x and x2 for a quantum mechanical harmonic oscillator in the v = 1 state? What is the standard deviation x? 12C16O is an example of a stiff diatomic molecule, and it Table 13.4 Constants of Diatomic Molecules Source: K. P Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV, Constants of Diatomic Molecules. New York: Van Nostrand, 1979.

2. Assuming that the vibrations of a 3C2 molecule are equivalent to those of a harmonic oscillator with a force constant k 329 N/m, what is the zero point energy of vibration of this molecule? (Mass 35C-34.9688 amu)