Visualizing vibronic transitions in a diatomic molecule


The Morse oscillator code introduced in a previous blog post can be used to visualize the vibronic transitions in a diatomic molecule by creating two Morse objects (one for each electronic state) and plotting their potential energy curves and energy levels on the same Matplotlib Axes object.

enter image description here

import numpy as np
from matplotlib import rc
import matplotlib.pyplot as plt
from scipy.constants import h, c
from morse import Morse, FAC
rc('font', **{'family': 'serif', 'serif': ['Computer Modern'], 'size': 14})
rc('text', usetex=True)

# Figure resolution and size in pixels.
DPI = 100
w, h = 700, 525
# The colour to plot wavefunctions in.
COLOUR1 = (0.6196, 0.0039, 0.2588, 1.0)

# Atom masses and ground state parameters for (14N)2.
mA = mB = 14.003074004
X_re = 1.09768e-10
X_Te = 0
X_we, X_wexe = 2358.57, 14.324

X = Morse(mA, mB, X_we, X_wexe, X_re, X_Te)
X.V = X.Vmorse(X.r)

# First excited electronic state parameters for (14N)2.
A_re = 1.2866e-10
A_Te = 50203.6
A_we, A_wexe = 1460.64, 13.87
A = Morse(mA, mB, A_we, A_wexe, A_re, A_Te)

fig, ax = plt.subplots(figsize=(w/DPI, w/DPI), dpi=DPI)
# Plot the ground state potential energy curve and vibrational energy levels.
X.plot_V(ax, color='k')
X.draw_Elines(range(0, X.vmax, 3), ax, lw=0.5)
# Draw and label the ground state vibrational wavefunction.
X.plot_psi([0], ax, scaling=4, color=COLOUR1)
X.label_levels([0], ax, [r"$v''=0$"])

# Plot the excited state potential energy curve and vibrational energy levels.
A.plot_V(ax, color='k')
A.draw_Elines(range(0, A.vmax, 3), ax, lw=0.5, color='g')
# Draw and label the excited state vibrational wavefunctions for v=0 and v=15.
A.plot_psi([0, 15], ax, scaling=4, color=COLOUR1)
A.label_levels([0, 15], ax, [r"$v'=0$", r"$v'=15$"])

# Some figure labelling and tidying.
ax.set_xlabel(r'$r \;/ \mathrm{\AA}$')

# Indicate a couple of transitions.
Epp = X.Emorse(0, 'cm-1', include_Te=True)
Ep = A.Emorse(0, 'cm-1', include_Te=True)
ax.vlines(*1.e10, Epp, Ep, color='r')
Ep = A.Emorse(15, 'cm-1', include_Te=True)
ax.vlines(*1.e10*1.01, Epp, Ep, color='b')
ax.set_xlim(0.5, 3)

plt.savefig('morse-vibronic.png', dpi=DPI)

There are a couple of changes to the original code, which are included in the updated version of here:

import numpy as np
from scipy.constants import h, hbar, c, u
from scipy.special import factorial
from scipy.special import genlaguerre, gamma

# Factor for conversion from cm-1 to J
FAC = 100 * h * c

class Morse:
    """A class representing the Morse oscillator model of a diatomic."""

    def __init__(self, mA, mB, we, wexe, re, Te):
        """Initialize the Morse model for a diatomic molecule.

        mA, mB are the atom masses (atomic mass units).
        we, wexe are the Morse parameters (cm-1).
        re is the equilibrium bond length (m).
        Te is the electronic energy (minimum of the potential well; origin
            of the vibrational state energies).


        self.mA, self.mB = mA, mB = mA*mB/(mA+mB) * u
        self.we, self.wexe = we, wexe = re
        self.Te = Te

        self.De = we**2 / 4 / wexe * FAC = (2 * np.pi * c * 100 * we)**2 *
        #  Morse parameters, a and lambda.
        self.a = self.calc_a()
        self.lam = np.sqrt(2 * * self.De) / self.a / hbar
        # Maximum vibrational quantum number.
        self.vmax = int(np.floor(self.lam - 0.5))

        self.V = self.Vmorse(self.r)

    def make_rgrid(self, n=1000, rmin=None, rmax=None, retstep=False):
        """Make a suitable grid of internuclear separations."""

        self.rmin, self.rmax = rmin, rmax
        if rmin is None:
            # minimum r where V(r)=De on repulsive edge
            self.rmin = - np.log(2) / self.a
        if rmax is None:
            # maximum r where V(r)=f.De
            f = 0.999
            self.rmax = - np.log(1-f)/self.a
        self.r, self.dr = np.linspace(self.rmin, self.rmax, n,
        if retstep:
            return self.r, self.dr
        return self.r

    def calc_a(self):
        """Calculate the Morse parameter, a.

        Returns the Morse parameter, a, from the equilibrium
        vibrational wavenumber, we in cm-1, and the dissociation
        energy, De in J.


        return (self.we * np.sqrt(2 * * np.pi *
                c * 100)

    def Vmorse(self, r):
        """Calculate the Morse potential, V(r).

        Returns the Morse potential at r (in m) for parameters De
        (in J), a (in m-1) and re (in m).


        return self.De * (1 - np.exp(-self.a*(r -**2

    def Emorse(self, v, units='J', include_Te=False):
        """Calculate the energy of a Morse oscillator in state v.

        Returns the energy of a Morse oscillator parameterized by
        equilibrium vibrational frequency we and anharmonicity
        constant, wexe (both in cm-1).

        vphalf = v + 0.5
        E = self.we * vphalf - self.wexe * vphalf**2
        if include_Te:
            E += self.Te
        if units == 'cm-1':
            return E
        elif units == 'J':
            return E * FAC
        raise ValueError('unrecognised units "{}" in Emorse'.format(units))

    def calc_turning_pts(self, E):
        """Calculate the classical turning points at energy E.

        Returns rm and rp, the classical turning points of the Morse
        oscillator at energy E (provided in J). rm < rp.


        b = np.sqrt(E / self.De)
        return ( - np.log(1+b) / self.a,
       - np.log(1-b) / self.a)

    def calc_psi(self, v, r=None, normed=True, psi_max=1):
        """Calculates the Morse oscillator wavefunction, psi_v.

        Returns the Morse oscillator wavefunction at vibrational
        quantum number v. The returned function is "normalized" to
        give peak value psi_max.


        if r is None:
            r = self.r
        z = 2 * self.lam * np.exp(-self.a*(r -
        alpha = 2*(self.lam - v) - 1
        psi = (z**(self.lam-v-0.5) * np.exp(-z/2) *
               genlaguerre(v, alpha)(z))
        psi *= psi_max / np.max(psi)
        return psi

    def calc_psi_z(self, v, z):
        alpha = 2*(self.lam - v) - 1
        psi = (z**(self.lam-v-0.5) * np.exp(-z/2) *
               genlaguerre(v, alpha)(z))
        Nv = np.sqrt(factorial(v) * (2*self.lam - 2*v - 1) /
                     gamma(2*self.lam - v))
        return Nv * psi

    def plot_V(self, ax, **kwargs):
        """Plot the Morse potential on Axes ax."""

        ax.plot(self.r*1.e10, self.V / FAC + self.Te, **kwargs)

    def get_vmax(self):
        """Return the maximum vibrational quantum number."""

        return int(self.we / 2 / self.wexe - 0.5)

    def draw_Elines(self, vlist, ax, **kwargs):
        """Draw lines on Axes ax representing the energy level(s) in vlist."""

        if isinstance(vlist, int):
            vlist = [vlist]
        for v in vlist:
            E = self.Emorse(v)
            rm, rp = self.calc_turning_pts(E)
            ax.hlines(E / FAC + self.Te, rm*1.e10, rp*1e10, **kwargs)

    def label_levels(self, vlist, ax, labels=None):
        if isinstance(vlist, int):
            vlist = [vlist]
        if labels is None:
            labels = [r'$v={}$'.format(v) for v in vlist]

        for i, v in enumerate(vlist):
            E = self.Emorse(v)
            rm, rp = self.calc_turning_pts(E)
            ax.text(s=labels[i], x=rp*1e10*1.25,
                    y=E / FAC + self.Te, va='center')

    def plot_psi(self, vlist, ax, r_plot=None, scaling=1, **kwargs):
        """Plot the Morse wavefunction(s) in vlist on Axes ax."""
        if isinstance(vlist, int):
            vlist = [vlist]
        for v in vlist:
            E = self.Emorse(v)
            if r_plot is None:
                rm, rp = self.calc_turning_pts(E)
                x = self.r[self.r<rp*1.2]
                x = r_plot
            psi = self.calc_psi(v, r=x, psi_max=self.we/2)
            psi_plot = psi*scaling + self.Emorse(v)/FAC + self.Te
            ax.plot(x*1.e10, psi_plot, **kwargs)
Current rating: 4.8


Comments are pre-moderated. Please be patient and your comment will appear soon.

Yoo 2 years, 11 months ago

Gave me so much help thanks a lot !

Link | Reply
Currently unrated

Doug Brown 1 year, 11 months ago

from matplotlib had to import rcParams
and set rcParams['text', usetex'] = True
to get latex to work.

This what You did in the recent Cambridge Temp Visualization blog.

Link | Reply
Currently unrated

Trevor_LI 1 year, 10 months ago

Nice plots. I generated similar but less pretty ones for my advanced p-chem lab class at Stony Brook, but yours look much clearer. I'l l update mine a bit.

Link | Reply
Current rating: 5

Trevor_LI 1 year, 8 months ago

There's a Latex typo in
It should be '$r\; / \; \mathrm{\AA}$'
to get the angstrom symbol to print with the forward slash for the units spaced nicely

Link | Reply
Currently unrated

christian 1 year, 8 months ago

Thank you – fixed!

Link | Reply
Currently unrated

Dario 1 year, 6 months ago

It would be very nice to extend the Morse class to include the effects of centrifugal potential.

Link | Reply
Currently unrated

Dario 1 year, 6 months ago

The following reference shows the effects of the centrifugal potential on the eigenvalues but has not said anything about the eigenfunctions as far as I understand:

Link | Reply
Currently unrated

New Comment


required (not published)