Question

1 Vibrational states of a diatomic molecule 1. Use Taylor expansion to get a harmonic approximation Vharmonic( 0.5k(r Ro2 of the following potential 2. Find the expressions for the equilibrium distance Ro and for the harmonic 3. Calculate the zero point energy in terms of the parameters of the given 4. Calculate the energy of a photon emitted upon a transition between ad- force constant k potential (a, ro and D jacent levels in terms of the parameters of the given potential (a, ro and

Answer 1

Given data use Taylor expansion to get a harmonic approximation V_harmonic (r) = 0.5 k(r - R_0)^2 of me following potential v(r) = D(1 - e^-a(r - r_0))^2 rightarrow 1 the expression for me equilibrium distance R_0 and for me harmonic force constant K The zero point energy in terms of me parameters of me given potential (a,r_0 and D) me energy of a photon emitted upon a transition between adjacent levels in terms of me parameters of me given potential (a,r_0 and D) v(r) = D(1 - e^-a(r - r_0))^2 \r\n = D(1-(1- a(r-r_0) + a^2(r - r_0)/2! )) = D(a(r - r_0) -a^2/2!(r - r_0)^2 + ) if number of terms of order 3 and above v(r) = Da^2(r - r_0)^2 comparing most is like a harmonic oscillation potential with equilibrium at r_0 E_0 = 1/2 kw = 1/2 1/n squareroot k/m = k/2 squareroot 2Da^2/m E_n rightarrow E_(n-1) = kw = k squareroot 2Da^2/m