Morse Potential - Vibrational States and Energies

Vibrational States and Energies

To write the stationary states on the Morse potential, i.e. solutions and of the following Schrödinger equation:

it is convenient to introduce the new variables:

$x=a r \text{; } x_e=a r_e \text{; } \lambda =\frac{\sqrt{2 m D_e}}{a \hbar } \text{; } \varepsilon _v=\frac{2 m }{a^2\hbar ^2}E(v).$

Then, the Schrödinger equation takes the simple form:

$\left(-\frac{\partial ^2}{\partial x^2}+V(x)\right)\Psi _n(x)=\varepsilon _n\Psi _n(x),$

$V(x)=\lambda ^2\left(e^{-2\left(x-x_e\right)}-2e^{-\left(x-x_e\right)}\right).$

Its eigenvalues and eigenstates can be written as:

$\varepsilon _n=-\left(\lambda -n-\frac{1}{2}\right)^2$

$\Psi _n(z)=N_nz^{\lambda -n-\frac{1}{2}}e^{-\frac{1}{2}z}L_n^{2\lambda -2n-1}(z),$

where $z=2\lambda e^{-\left(x-x_e\right)} \text{; } N_n=n!\left^{\frac{1}{2}}$ and is Laguerre polynomial:

$L_n^{\alpha }(z) = \frac{z^{-\alpha }e^z}{n!} \frac{d^n}{d z^n}\left(z^{n + \alpha } e^{-z}\right)=\frac{\Gamma (\alpha + n + 2)/\Gamma (\alpha +2)}{\Gamma (n+1)} \, _1F_1(-n,\alpha +1,z),$

There also exists the following important analytical expression for matrix elements of the coordinate operator (here it is assumed that and )

$\left\langle \Psi _m(x)|x|\Psi _n(x)\right\rangle =\frac{2(-1)^{m-n+1}}{(m-n)(2N-n-m)} \sqrt{\frac{(N-n)(N-m)\Gamma (2N-m+1)m!}{\Gamma (2N-n+1)n!}}.$

The eigenenergies in the initial variables have form:

where is the vibrational quantum number, and has units of frequency, and is mathematically related to the particle mass, and the Morse constants via

Whereas the energy spacing between vibrational levels in the quantum harmonic oscillator is constant at, the energy between adjacent levels decreases with increasing in the Morse oscillator. Mathematically, the spacing of Morse levels is

This trend matches the anharmonicity found in real molecules. However, this equation fails above some value of where is calculated to be zero or negative. Specifically,

This failure is due to the finite number of bound levels in the Morse potential, and some maximum that remains bound. For energies above, all the possible energy levels are allowed and the equation for is no longer valid.

Below, is a good approximation for the true vibrational structure in non-rotating diatomic molecules. In fact, the real molecular spectra are generally fit to the form1

in which the constants and can be directly related to the parameters for the Morse potential.

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