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Quantum Chemistry: Dr. Rohul Hayat Adnan Department of Chemistry UTM

1) The quantum harmonic oscillator model is used to describe the energy levels of a quantum particle confined in a potential well that follows a harmonic oscillator pattern. 2) The energy levels are quantized and take the form En = (n + 1/2)ħω, where n is an integer and ħω is the energy spacing between levels. 3) Examples are given of calculating energy levels and spacing for different molecular systems using the quantum harmonic oscillator model. Limitations of the model are also discussed.

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308 views7 pages

Quantum Chemistry: Dr. Rohul Hayat Adnan Department of Chemistry UTM

1) The quantum harmonic oscillator model is used to describe the energy levels of a quantum particle confined in a potential well that follows a harmonic oscillator pattern. 2) The energy levels are quantized and take the form En = (n + 1/2)ħω, where n is an integer and ħω is the energy spacing between levels. 3) Examples are given of calculating energy levels and spacing for different molecular systems using the quantum harmonic oscillator model. Limitations of the model are also discussed.

Uploaded by

Nazratul Najwa
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Quantum Chemistry

Lecture 6

Dr. Rohul Hayat Adnan


Department of Chemistry UTM
Physics is becoming
too difficult for
physicists
- David Hilbert
(mathematician
Classical Harmonic Oscillator

• Imagine a mass, m, attached to a spring


that oscillates up and down
• The force of oscillation, F = -kx
• The potential energy, V
1
𝑉 = − ‫ 𝑥𝑘 = 𝑥𝑑 𝐹 ׬‬2
2
• Angular frequency
𝑘
𝜔= = 2𝜋𝑓
𝜇
Quantum Harmonic Oscillator
ℏ2 𝑑 2 𝜓
• Write down the TISE: − + 𝑉𝜓 = 𝐸𝜓
2𝑚 𝑑𝑥 2
• The potential energy, V
1
𝑉 = 𝑘𝑥 2
2
ℏ2 𝑑 2 𝜓
• The TISE then becomes − + 𝑉𝜓 = 𝐸𝜓
2𝑚 𝑑𝑥 2
2
• The WF takes the form 𝜓 𝑥 = 𝐴𝑛 𝐻𝑛 𝑥 𝑒 −𝛼𝑥

Normalization Gaussian function


constant
Hermite polynomial
• Boundary conditions: 𝜓 𝑥 → 0 𝑎𝑠 𝑥 → ±∞
1 1
• Energy level, 𝐸𝑛 = 𝑛 + ℏ𝜔 = 𝑛 + ℎ𝑓
2 2
Summary - Quantum Harmonic Oscillator
1
• Energy level 𝐸𝑛 = 𝑛 + ℏ𝜔 (cf. for
2 2 2
2
𝑛 𝜋 ℏ
particle in a box 𝐸𝑛 =
2𝑚𝐿2
• Consistent with energy level of emission
from blackbody (∆𝐸 = ℏ𝜔)
• Energy level of ground state, n=0, is 𝐸 =
1
ℏ𝜔
2
• The energy levels are equally space
• The #nodes = quantum no.
Exercise
• Calculate the frequency of vibration and energy level spacing for 1H35Cl
molecule
Hint: Mass of H = 1.0078 amu, Mass of Cl = 34.9688 amu, k = 480 N/m
• A hydrogen atom is adsorbed on the surface of gold nanoparticles by a
bond force constant of 855 N/m. Calculate its zero-point vibrational energy
• Strongest IR band for 12C16O occurs at ν = 2143 cm-1. Find the force
constant.
Hint: mass of C = 12 amu, mass of O = 15.995 amu
Morse Potential
• HO cannot represent vibration of real
systems because bonds break at high
energy
• Only GS is well-represented by HO
• Better model is to use Morse Potential
that include anharmonicity of vibration
1 1
𝐸𝑣𝑖𝑏 = 𝑛+ ℎ𝑓𝑒 − 𝑛 + ℎ𝑓𝑒 𝑥𝑒
2 2

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