In this and following questions, we develop a model for spontaneous emission of a photon by a diatomic molecule AB (a model molecule), which rotates and vibrates. In intermediate calculations, atomic units (a.u.) will be used: unit of mass = the mass of electron, unit of charge is the proton charge e, (e is a positive constant so that the charge of electron is -e). The initial state of the molecule is an excited rotational (1=1) and excited vibrational state (v=1). We consider a molecule with the reduced mass μ = 10,000 a.u. (it is similar to the mass of CO). After emitting a photon, the molecule will go to the I=0, v=0 state. The first question is about the model potential of the molecule. It is represented by a potential of the form: C12 C6 V(r) = Co p12 - where r is the distance between A and B in the molecule, C6 and C12 are positive constants (C6 =2 and C₁2-1). This potential has a well meaning that the molecule is bound. The first thing to do is find vibrational states of the molecule and the corresponding wave functions. We use the normal mode approximation, i.e. we consider that in the bottom of the well the potential is approximately quadratic. Therefore, the first step in your model is to find the distance ro of the minimum of the potential and the second derivative at the minimum. What is the second derivative?

In this and following questions, we develop a model for spontaneous emission of a photon by a diatomic molecule AB (a model molecule), which rotates and vibrates. In intermediate calculations, atomic units (a.u.) will be used: unit of mass = the mass of electron, unit of charge is the proton charge e, (e is a positive constant so that the charge of electron is -e). The initial state of the molecule is an excited rotational (1=1) and excited vibrational state (v=1). We consider a molecule with the reduced mass μ = 10,000 a.u. (it is similar to the mass of CO). After emitting a photon, the molecule will go to the I=0, v=0 state. The first question is about the model potential of the molecule. It is represented by a potential of the form: C12 C6 V(r) = Co p12 - where r is the distance between A and B in the molecule, C6 and C12 are positive constants (C6 =2 and C₁2-1). This potential has a well meaning that the molecule is bound. The first thing to do is find vibrational states of the molecule and the corresponding wave functions. We use the normal mode approximation, i.e. we consider that in the bottom of the well the potential is approximately quadratic. Therefore, the first step in your model is to find the distance ro of the minimum of the potential and the second derivative at the minimum. What is the second derivative?