In quantum mechanics the vibrational energies of a diatomic molecule can be approximated by a harmonic spring (oscillator) potential
where n is an integer quantum number ranging from 0 to ∞, ν is the frequency of the spring and h is Planck's constant.
The lowest vibrational energy can be computed with the Quantum Chemistry package. Let us set the Digits to 15 and load the package
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| (3.1.1) |
Define the molecule hydrogen chloride at its equilibrium geometry
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| (3.1.2) |
We can compute the lowest vibrational mode in cm-1 with the command VibrationalModes (may take a minute)
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| (3.1.3) |
The vibration of hydrogen chloride in its ground vibrational state can be animated with the command VibrationalModeAnimation. After executing the command, click on the plot and click play to see one vibration (note when you click on the plot, a plot menu appears; by changing with the menu icon to the left of fps, you can make the vibrations run continuously until you press pause or stop!)
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We can convert cm-1 (cycles per cm) to Hz (cycles per s) by multiplying by the speed of light c
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| (3.1.4) |
| (3.1.5) |
or combining units with Maple
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| (3.1.6) |
To obtain the vibrational energies, we need Planck's constant
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| (3.1.7) |
Let us define the vibrational energies of hydrogen chloride as a function of the quantum number n
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With the function we can compute the first 10 vibrational energy states in Joules
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| (3.1.9) |
(a) What is the frequency of the hydrogen chloride bond in Hz?
(b) What are the energies of the three lowest vibrational energies?
In the next section we will use the Boltzmann distribution to determine the probability of a hydrogen chloride molecule being in a given vibrational energy state.