The vibration of a diatomic molecule can be modelled using the quantum harmonic oscillator.
The Schrodinger equation can be written as:
+\frac{p_2^2}{2m_2}\psi(x_1,x_2)+\frac{1}{2}k(x_2-x_1-x_0)^2\psi(x_1,x_2)=E\psi(x_1,x_2)\;\;\;\;\;\;\;\;47)
To solve the equation, we need to make a change of coordinates to achieve a separation of variables. The new coordinate system consists of the variables 
and 
, where 
is the centre of mass of the molecule.
Question
What do the terms in eq47 mean? What is a separation of variables?
Answer
Eq47 is an eigenvalue equation, where the Hamiltonian ^2)
has two kinetic energy terms, one for each atom, and a potential energy term that describes the interaction between the atoms. According to the classical harmonic oscillator, the potential energy of the system is equal to 
, where 
is the displacement of the spring from its equilibrium position. For the diatomic molecule in the diagram above, 
.
To solve eq47, we need it to be of the form +\hat{H}_2(x_2)\]\psi(x_1,x_2)=(E_1+E_2)\psi(x_1,x_2))
, where the Hamiltonian is separated into two, each being a function of only one variable. Unless it is transformed, the potential energy term in eq47 does not allow the separation to occur.
The derivatives of 
and with respect to time are 
and 
respectively. Substituting 
and 
in 
and rearranging, we have 
and 
respectively. Using 
, the Hamiltonian of eq47 becomes
+\frac{1}{2}kr^2=\frac{p_s^2}{2M}+\frac{p_r^2}{2\mu}+\frac{1}{2}kr^2\;\;\;\;\;\;\;\;48)
where 
, 
, 
and 
.
Since 
and are the total mass of the molecule and the centre of mass coordinate of the molecule respectively, 
corresponds to the kinetic energy of the translational motion of the molecule. The other kinetic energy energy term 
must correspond to the internal motion (rotational and vibrational motion) of the molecule. However, there are only two degrees of freedom for a diatomic molecule in a one-dimensional space – a translational degree of freedom and a vibrational degree of freedom. Discarding the kinetic energy term for the translational motion, the Schrodinger equation describing the vibrational motion of the molecule is
\psi(r)=E\psi(r)\;\;\;\;\;\;\;\;49)
Eq49 is equivalent to eq4 except for the substitution of 
, which is called the reduced mass. Its solutions are given by eq45 (with 
replaced with ). With reference to eq21, the vibrational energies of the diatomic molecule are:
\hbar\omega\;\;\;\;n=0,1,2,\cdots\;\;\;\;\;\;\;\;50)
where 
.
The potential energy curve of 
in eq49 for a typical diatomic molecule is represented by the dashed curve in the diagram below. The minimum potential energy of the molecule occurs when the atoms are at their equilibrium positions. In the next article, we will show that the actual potential energy curve has the shape of the solid curve, which correctly shows that the potential energy of the molecule increases rapidly as the bond length shortens and approaches a constant value representing the dissociation energy of the molecule as increases. Despite that, the two curves coincide at low vibrational energy levels,. This implies that the harmonic oscillator model of a diatomic molecule only serves as a good approximation at low vibrational energy levels.
