Consider the H2 molecule: a. The force constant of H2 is k = 510 N/m. Calculate the fundamental vibrational frequency (in units of cm-¹), the zero-point energy (i.e. energy of the ground vibrational state) in Joules, and the energies of the first two excited vibrational states, assuming that the H2 molecules behaves as a simple harmonic oscillator. b. When an IR spectrum of H2 is measured experimentally, no actual absorption of IR light is observed at the vibrational frequency calculated above (or at any other frequency for that matter). Why? c. An analytical expression that provides a good approximation to a potential energy curve for a real diatomic molecule as a function of the internuclear separation, x, is a so-called Morse potential given by: V(x) = D(1-e-B(x-R₂))2 where Re is the equilibrium bond distance, and parameters D and ẞ represent the dissociation energy of the molecule and the measure of curvature of V(x) at its minimum. Given that for H₂ we have k = 510 N/m, R = 74.1 pm, D = 7.61×10-19 J and ẞ = 0.0193 pm-¹, on the same graph: i. plot the harmonic oscillator potential V(x) = 1½ k(x - R₂)² for x between 0 and 400 pm ii. plot the Morse potential energy curve (expression above) for x between 0 and 400 pm iii. indicate as horizontal lines the energies of the ground vibrational state and the first two excited vibrational states calculated in part (a) in the harmonic oscillator approximation d. The solution of the Schrödinger equation for the Morse potential yields the following expression for the vibrational energy levels (in units of cm-1) of a diatomic molecule: En (n+1/2) (n + 1/2)² where n is the vibrational quantum number, v is the fundamental vibrational frequency (in cm-1) that you calculated in part a), and is the anharmonicity constant given by: x=hcv/4D. Assuming that the H2 molecule is accurately described by a Morse potential energy curve, calculate what is the highest vibrational energy level that the molecule can occupy without dissociating into two H atoms.