Estimating the vibrational frequency of Hydrogen molecule

• stevenytc
In summary, a diatomic molecule can vibrate when excited, with its relative distance between atoms following a periodic oscillation. The time dependence of this vibration is equivalent to a particle of reduced mass moving in a potential energy function, with the Morse potential being a commonly used approximation. The frequency of small classical vibrations about the minimum of the Morse potential can be calculated using the harmonic oscillator equation and the given parameters of B, r0, and β. For the specific case of H2, the calculated frequency is 4.17 x 10^12 Hz, which is two orders off from the observed vibrational frequency of 1.32 x 10^14 Hz. The discrepancy may be due to not accounting for the correct mass units
stevenytc

Homework Statement

A diatomic molecule, when excited, can vibrate in such a way that the relative distance
between the two atoms, r(t), executes periodic oscillations. If the potential energy of the
molecule as a function of r is given by V (r), the time dependence of r(t) is identical to that
of a particle of reduced mass

μ= m1 m2/ (m1 + m2)

moving in the potential V (r). An analytic expression which is sometimes used to approximate the actual interatomic interaction is the Morse potential,
V (r) = B[1 − exp(−β(r−r0 ))]^2 − B

B is the depth of the well, r0 is the equilibrium separation, and β is a parameter which
governs how rapidly the energy rises as one moves away from the equilibrium position.

(3.a) Find the frequency of small classical vibrations about the minimum of the Morse
potential in terms of B, r0 , and β.

(3.b) For the molecule H2 , β = 1.93 Angstrom-1 , r0 = 0.74 angstrom, and B = 4.8 eV. The mass of a hydrogen atom is 1.67 × 10^ -24 grams. What is the frequency of small vibrations? The observed vibrational frequency is 1.32 × 10^14 Hz.

The Attempt at a Solution

I used Taylor expansion to expand the exponent in V(r) and get, to first order,
V = Bβ^2(r-r0)^2 - B
Compare this to the energy for harmonic oscillator, I conclude that
ω = 2Bβ^2 / μ

Putting in the numbers give me a frequency of 4.17 x 10^12 Hz, which is two orders off.

Did you take care of the mass units? g or kg?

oh I got it! Thanks a lot

1. What is the significance of estimating the vibrational frequency of a Hydrogen molecule?

The vibrational frequency of a molecule is directly related to its chemical bonds and can provide important information about the molecule's structure and behavior. In the case of Hydrogen, which is the simplest molecule, studying its vibrational frequency can help us better understand the fundamental principles of molecular vibrations.

2. How is the vibrational frequency of a Hydrogen molecule estimated?

The vibrational frequency of a molecule can be estimated using quantum mechanical calculations, which take into account the mass and force constants of the atoms in the molecule. For Hydrogen, the calculations involve solving the Schrödinger equation to determine the energy levels of the molecule's vibrational modes.

3. What factors can affect the accuracy of the estimated vibrational frequency?

The accuracy of the estimated vibrational frequency can be affected by various factors such as the level of theory used in the calculations, the choice of basis set, and the inclusion of environmental effects. The accuracy can also be influenced by the assumption of a rigid molecule, as well as any anharmonic effects.

4. Why is it necessary to estimate the vibrational frequency of Hydrogen in different states?

Hydrogen can exist in different states, such as gas, liquid, and solid, and each state has a different molecular environment that can affect its vibrational frequency. Estimating the vibrational frequency in different states allows us to understand how the molecule behaves under different conditions and how its bonds are affected by the surrounding environment.

5. How is the estimated vibrational frequency of Hydrogen used in practical applications?

The estimated vibrational frequency of Hydrogen can be used in various practical applications, such as in spectroscopy, where it helps in identifying and characterizing different molecules. It is also useful in studying the dynamics of chemical reactions and in the design and development of new materials and compounds.

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