I am trying to represent the difference between the an anharmonic and harmonic treatment of vibrations. The easiest way to do this is to present an example of a harmonic and anharmonic potential energy surface (PES).

I would like to present a figure similar to the one attached below (from Herzberg),$^1$ where we have the morse oscillator function presented with the vibrational energy levels indicated.

Anharmonic PES of HCl with vibrational levels indicated by horizontal lines

I have some spectroscopic data from an undergraduate experiment I did about $\ce{I2}$, and have the data points (or the equation) to draw the PES, and the energies of the vibrations in order to plot their values.

Is there any software which is able to generate something like this for me, or are there any tricks people know where this can be done in other graphing software (preferably Excel or Igor Pro)? The difference between the spacing of each successive overtone in an experimental vs. harmonic spectrum is part of the explanation, so it's important to illustrate that in the figure, if at all possible.

EDIT I have since come across a Python script which will do the job (see answer below); however, I'm sure that other ways to generate these diagrams that people know would be appreciated (particularly for people who do not know much about Python, such as myself...)!

  1. G. Herzberg, Spectra of Diatomic Molecules, Krieger Publishing Company, Malabar, Florida, Reprint edn., 1989.
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    $\begingroup$ In general, this is most easy using software like R or python, but in Excel you can do it in the following way: Assume column A contains $r$ values and column B contains your potential values $V(r)$. Using =IF($B1>En0, NA(), En0) for the values in column C where En0 is your first energy. Use the other columns for the additional eigenvalues and plot with Scatter. $\endgroup$
    – Paul
    Commented Jul 10, 2023 at 10:59
  • $\begingroup$ @Paul what do you mean by my 'first energy'? Do you mean the first vibration (i.e. $\upsilon = 1$)? $\endgroup$ Commented Jul 10, 2023 at 13:56
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    $\begingroup$ Perhaps the question is suitable for mattermodelling.se (example). $\endgroup$
    – Buttonwood
    Commented Jul 10, 2023 at 20:22
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    $\begingroup$ @Buttonwood that did occur to me;I saw that exact post on this exchange, and a similar comment on that, but that comment also said it would be appropriate on either exchange (assume we call these different 'topics' exchanges?). I also thought it would be something relatively common in chemistry, which is why I'm so surprised there are no programs which do it automatically! $\endgroup$ Commented Jul 11, 2023 at 6:56
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    $\begingroup$ @Buttonwood I don't think it fits there. The question is about how to draw a Morse potential from experimental values. This should have all the information you need: chem.libretexts.org/Courses/Duke_University/… $\endgroup$ Commented Jul 12, 2023 at 0:30

2 Answers 2


I have made this basic python 3 script which runs in a Jupyter notebook (or in Visual Studio Code). With different molecules you will need to adjust the plot scale. The first 10 levels are plotted then every third one. You can adjust all this.

The code plots the potential, energy levels and wavefunctions.

# import all python add-ons etc that will be needed later on 
%matplotlib inline.  # for jupyter notebook only
import numpy as np
import matplotlib.pyplot as plt
from scipy.special import eval_genlaguerre   #associate Laguerre polynom
plt.rcParams.update({'font.size': 14})   # set font size for plots

# HCl  and Iodine
# Assoc Laguerre polys & Morse potential.  n is quantum number
# wavefunct not normalised

fig = plt.figure(figsize=(5,5))

hbar =   1.05449e-34  # Js/2pi
c    =   2.9979e8     # m/s
nm   =   1.0e-9       # nanometres
amu  =   1.66043e-27  # atomic mass unit

mol  = 1

if mol == 0:

    m1   = 127            # iodine
    m2   = 127
    we   = 128.0          # frequency cm^{-1}
    xewe = 0.834          # anharmionicity no units 
    reB  = 0.3016         # equilib bond length nm
    xm1  = 2              # used to limit plot length
    xm0  =0.8
    m1   = 1              # HCl
    m2   = 35
    we   = 2989.7         # frequency cm^{-1}
    xewe = 52.05          # anharmionicity no units 
    reB  = 0.123746       # equilib bond length nm
    xm1  = 3.5
    xm0  = 0.5

mass = m1*m2/(m1 + m2) # reduceD mass

fac3 = hbar/(amu*2*np.pi*100*c*nm**2)        #  factor to get constants sorted
DeqB = we**2/(4.0*xewe)                      # Morse dissn energy cm^{-1}

k    = 4.0*DeqB/we   
beta = we*np.sqrt(mass/(2.0*fac3*DeqB))      # 1/nm

d = k/2.0  
zeta = lambda x: np.exp(-beta*x)  
psi  = lambda n,x : np.exp(-zeta(x)*k/2.0)*\

En = lambda n:  we*(n+0.5)-xewe*(n+0.5)**2     # energy levels  cm^{-1}
nm = 0
while En(nm + 1) > En(nm):
    nm = nm + 1
print('max Q number',nm)

V = lambda x: DeqB*(1-np.exp(-beta*x) )**2     # Morse potential  cm^{-1}

xm = lambda E: -np.log(1-np.sqrt(E/DeqB) )/beta+reB  # find limits of potential
xp = lambda E: -np.log(1+np.sqrt(E/DeqB) )/beta+reB

x = np.linspace(xm0*reB, xm1*reB, 500)           # nm

for n in range(11):                              # plot first 10 levels
    E0 = En(n)
    n0 = max(abs(psi(n,x-reB)) )*3/we            # normalise on plot

for n in range(11, nm, 3):                       # plot some higher q number wavefunctions gaps of 3
    E0 = En(n)
    n0 = max(abs(psi(n,x-reB)) )*3/we            # normalise on plot
plt.xlabel(r'$r\; /\; nm$')
plt.ylabel(r'$Energy\; /\;cm^{-1}$')


An example plot is shown below

HCl morse

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    $\begingroup$ This is great! I'm totally adapting this for my class to show the effects of the anharmonic oscillator. $\endgroup$ Commented Jul 13, 2023 at 14:37
  • $\begingroup$ @GeoffHutchison the scripts I linked below do something similar (including plotting selected wave functions), so they might also be worth checking out? $\endgroup$ Commented Jul 14, 2023 at 6:23
  • $\begingroup$ Really like the look of this! Although I'm a little confused where the HCl comes into it? Or is that just another example of how the script can be used? $\endgroup$ Commented Jul 14, 2023 at 6:27
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    $\begingroup$ Just an example; one with a large force constant and one with a smaller one. $\endgroup$
    – porphyrin
    Commented Jul 14, 2023 at 9:18

I have come across two Python scripts which will do the job, written by Christian Hill. The links to them are:

Visualizing vibronic transitions in a diatomic molecule

The Morse oscillator

For example, here is a diagram for the $\ce{O2}$ PES generated by one of the codes linked above (I edited it slightly to remove the wave functions which you can overlay on the vibrational levels): PES of O2 molecule

They are not exactly what I was looking for (the curve could extend a little further towards the lower $r$ values to show how the energy keeps increasing beyond the dissociation energy), but it does plot the vibrational transitions the way I want.


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