## Vibrational spectra of diatomic molecules

October 25th, 2016

Let’s perform the interpretation of the vibrational spectra of diatomic molecules  with the use of the quantum harmonic oscillator model (fig.1):

Fig.1. The harmonic oscillator: A – in a static state, B – in a position of a motion

The equation of oscillation of particles with weights  $m_1$ and $m_2$ of the harmonic oscillator  can be reduced to equation oscillation of one material point with the weight  $\mu=\frac {m_1m_2}{m_1+m_2}$  and the frequency  $\nu=\sqrt \frac {k}{4 \pi^2 \mu}$:

$\frac {\mu d^2 (r-r_e)}{dr^2}=k(r-r_e)$       (1)

The equation of the quantum harmonic oscillator is given below:

$\bigtriangledown^2 \Psi + \frac{8 \pi_2 \mu}{h^2} (E - \frac {kx^2}{2}) \Psi =0$ ,           (2)

where $x=r-r_e$.

The solution of this equation is expressed by the energy of the oscillator from the vibrational quantum number:

$E(\upsilon) = h \nu (\upsilon + \frac{1}{2})$    (3)

Then the difference between the vibrational levels is equal to:

$\Delta E = h \nu (\upsilon_2 - \upsilon_1)$       (4)

Quantum mechanical selection rules for vibrational quantum numbers are equal to:

$\Delta \upsilon = \pm 1$      (5)

It leads to the fact that the vibrational energy levels are equidistant. As a result, the oscillating spectrum of the harmonic oscillator consists of a single line. For example, having fulfilled the calculation for FeO particle using HyperChem, we received only one line, as shown in Fig. 2.

Fig.2. Vibrational spectrum of FeO molecule which was calculated using HyperChem

(The information has been taken from article: Szwec W. Zastosowanie programu HyperChem w nauczaniu fizyki i chemii / W. Szwec // Fizyka w Szkole z Astronomią. – 2016. – № 4. – S. 16–18, URL: http://www.aspress.com.pl/fizyka.html. More information can be found at https://ipood-kiev.academia.edu/ValentynaShvets. )

Task 1. The cyclic frequency of the molecule HCl equals to
ω=5.63·1014s-1, the coefficient of the anharmonicity γ=0,0201. Define:

a) excitation energy of the molecule from zero vibrational level to the first;

b) number n of vibrational energy levels;

c) maximal vibrational energy;

d) dissociation energy.

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Task 2. Determine the number of vibrational energy levels, which has the molecule HBr, if the coefficient of anharmonicity is γ=0.0208.
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Task 3. How much differ the minimum and maximum differences between the two next power levels for the molecule H2 (γ=0.0277)?

Task 4. Define the maximal vibrational energy $E_{max}$ of the molecule O2, for which the net cyclic frequency $\omega=2.98 \cdot 10^{14} s^{-1}$ and the coefficient of anharmonicity $\gamma= 9.46 \cdot 10^{-3}$ are known.

Task 5. Find the excitation energy of the molecule CO from the first vibrational level to the second, and the dissociation energy if the net frequency of the molecule is equal to $\omega=4.08 \cdot 10^{14} s^{-1}$, and the coefficient of anharmonicity $\gamma=5.83 \cdot 10^{-3}$.
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(The tasks have been taken from the book «Elements of Quantum Mechanics” (author V.D. Shvets), part of which was published in the journal “Problems in Education” (article: “Programming of students’ educational activity in the training of elements of quantum mechanics”), 2006. – № 43. – P. 50–64. This book has Recommendation of Ministry Education of Ukraine № 44 from 14.01.1999. )