Ep 02 — Why Do Molecules "Eat" Light? The Nature of Molecular Vibrations and Infrared Absorption

Series: Encyclopedia of Infrared Spectroscopy: From Principles to Practice
Chapter: Part 1 · Introduction — The Code of Light
Audience: High school students, undergraduates, beginners in chemistry/materials/pharmacy
Prerequisite: Ep 01 (Basic Concepts of Infrared Light)
Reading Time: ~18 minutes

Introduction: Are Chemical Bonds Like Springs?

If you could magnify a molecule to a scale visible to the naked eye, you would discover a surprising fact: the atoms inside a molecule are not stationary but are constantly vibrating — like two balls connected by a spring, moving back and forth around an equilibrium position [1][2].

This is not just a simple analogy. In classical mechanics, the vibration of a spring follows Hooke's Law, and the vibrational behavior of a chemical bond at small displacements can indeed be described by the same mathematical model [3]. It is this "spring-ball model" that provides the most intuitive physical picture for understanding infrared absorption.

But molecules are not classical springs — they are residents of the quantum world. When we re-examine this model with the lens of quantum mechanics, a deeper and more precise picture will unfold before us.

I. The Spring-Ball Model: Hooke's Law

1.1 Basic Model

Imagine two atoms (masses m₁ and m₂) connected by a chemical bond (spring). When the atoms deviate from their equilibrium positions, the restoring force F produced by the bond is proportional to the displacement x [3][4]:

$$F = -k \cdot x$$

where:

• k is the force constant, representing the "stiffness" of the chemical bond, in N/m
• x is the displacement from the equilibrium bond length
• The negative sign indicates that the force always points toward the equilibrium position (restoring force)

1.2 Vibrational Frequency Formula

According to Newton's second law and Hooke's law, the vibrational frequency of this system is [3][4][5]:

$$\nu = \frac{1}{2\pi} \sqrt{\frac{k}{\mu}}$$

where μ is the reduced mass, defined as [3][4]:

$$\mu = \frac{m_1 \cdot m_2}{m_1 + m_2}$$

The reduced mass is introduced because in a diatomic molecule, both atoms move — we cannot simply treat one atom as a "wall." The reduced mass equivalently describes the vibration of an object of mass μ relative to a fixed reference point [3].

1.3 Two Decisive Factors

From the formula ν = (1/2π)√(k/μ), two core principles emerge [5][6]:

Factor 1: Stronger bonds have higher vibrational frequencies

The force constant k is positively correlated with bond order: triple bond > double bond > single bond [5][6]. Textbooks often give approximate values in mdyn/Å (1 mdyn/Å = 100 N/m); converted to the same N/m as in Table 1, they are approximately:

• C≡C (triple bond): k ≈ 1200–1800 N/m (about 12–18 mdyn/Å) → ~2100–2260 cm⁻¹
• C=C (double bond): k ≈ 900–1100 N/m (about 9–11 mdyn/Å) → ~1620–1680 cm⁻¹
• C-C (single bond): k ≈ 400–600 N/m (about 4–6 mdyn/Å) → ~800–1300 cm⁻¹

Factor 2: Lighter atoms give higher vibrational frequencies

Hydrogen (H) is the lightest element, so bonds containing hydrogen have very high vibrational frequencies [5][6]:

• O-H: ~3200–3600 cm⁻¹
• N-H: ~3300–3500 cm⁻¹
• C-H: ~2850–3300 cm⁻¹

Although C-H is a single bond (moderate force constant), the extremely small mass of hydrogen (small reduced mass) gives it a vibrational frequency even higher than many double and triple bonds. This is a classic "counterintuitive" point [5].

📷 Figure 1: Relationship between bond force constant and vibrational frequency
Source: OpenOChem IR Spectroscopy Tutorial [5]
https://learn.openochem.org/l…

1.4 Example Data

The following table shows experimental data for some diatomic molecules, confirming the above principles [4]:

Table 1: Vibrational frequencies and force constants of selected diatomic molecules (Data source: LibreTexts Physical Chemistry Textbook [4])

Two key points to note:

1. The force constants of H₂ and D₂ are nearly identical (510 vs 527 N/m), but the frequency of D₂ is significantly lower than that of H₂ — this is because deuterium (D) is twice as heavy as hydrogen (H), increasing the reduced mass and thus lowering the frequency [4][7]. This is the isotope effect.
2. The force constant of CO (1857 N/m) is much higher than that of Cl₂ (319 N/m) because CO has triple bond character, while Cl₂ is a pure single bond [4].

💡 ftir.fun Practical Links: Want to check the characteristic frequency ranges of specific functional groups?

• Carbonyl C=O: https://ftir.fun/ir/group/car…
• Hydroxyl O-H: https://ftir.fun/ir/group/hyd…
• Amine N-H: https://ftir.fun/ir/group/ami…
• Alkyl C-H: https://ftir.fun/ir/group/alk…

II. Quantized Vibrational Energy Levels: From Classical to Quantum

2.1 Limitations of the Classical Model

Classical mechanics tells us that the energy of a spring oscillator is continuous — you can give it any amount of energy, and the amplitude will change accordingly. But real molecules are not like that [7][8].

Quantum mechanics reveals a startling fact: the vibrational energy of a molecule is discontinuous (quantized), taking only specific discrete values [7][8][9].

2.2 Quantum Harmonic Oscillator Model

Treating the chemical bond as a harmonic oscillator and solving the Schrödinger equation yields quantized vibrational energy levels [7][8][9]:

$$E_v = \left(v + \frac{1}{2}\right) h\nu, \quad v = 0, 1, 2, 3, \ldots$$

where:

• v is the vibrational quantum number, which can only take non-negative integers
• h is Planck's constant (6.626 × 10⁻³⁴ J·s)
• ν is the classical vibrational frequency

This formula tells us three important things [7][8][9]:

1. Energy levels are equally spaced

The energy difference between adjacent levels is ΔE = hν, a constant. This means the transition from v=0 to v=1 requires the same energy as the transition from v=1 to v=2 [8].

2. There is a zero-point energy

Even when the molecule is in its lowest energy state (v=0), its energy is not zero but E₀ = ½hν. This means the molecule never completely stops vibrating [7][8]. This is a direct consequence of the Heisenberg uncertainty principle: if the molecule were completely stationary at the equilibrium position, both momentum and position would be determined, violating the uncertainty principle [8].

3. Selection rule: Δv = ±1

In the harmonic oscillator model, infrared absorption only allows transitions v → v±1, i.e., only one photon is absorbed to jump one level [7]. This is why we mainly see fundamental absorptions (v=0 → v=1) in IR spectra [7][8].

📷 Figure 2: Potential energy curve and quantized energy levels of a harmonic oscillator
Source: UCI Chemistry Handouts [3]
https://www.chem.uci.edu/~uni…

[1] Atkins, P. W., & De Paula, J. (2014). Atkins' Physical Chemistry (10th ed.). Oxford University Press.
[2] Herzberg, G. (1950). Molecular Spectra and Molecular Structure: Vol. I. Spectra of Diatomic Molecules (2nd ed.). Van Nostrand.
[3] UCI Chemistry. (n.d.). Lecture B3: Harmonic Oscillator. Retrieved from https://www.chem.uci.edu/~uni…
[4] LibreTexts. (n.d.). Diatomic Molecular Vibrations. Retrieved from https://chem.libretexts.org/B…)
[5] OpenOChem. (n.d.). IR Spectroscopy: The Position of Absorption Bands. Retrieved from https://learn.openochem.org/l…
[6] Silverstein, R. M., Webster, F. X., & Kiemle, D. J. (2005). Spectrometric Identification of Organic Compounds (7th ed.). Wiley.
[7] McQuarrie, D. A., & Simon, J. D. (1997). Physical Chemistry: A Molecular Approach. University Science Books.
[8] Griffiths, D. J. (2005). Introduction to Quantum Mechanics (2nd ed.). Pearson.
[9] Levine, I. N. (2014). Quantum Chemistry (7th ed.). Pearson.

2.3 The Nature of Resonant Absorption

When infrared light irradiates a molecule, if the photon energy exactly equals the energy difference between adjacent vibrational levels (E = hν), the molecule absorbs the photon and transitions from v=0 to v=1 [1][3].

$$\Delta E = E_1 - E_0 = h\nu$$

This is resonant absorption—the frequency must match precisely, just like tuning a radio to a specific frequency to receive a signal [1].

"The frequency of incident infrared light must exactly match the frequency of molecular vibration for absorption to occur."
—— University of Delaware Infrared Spectroscopy Lecture Notes [1]

3. Beyond the Harmonic Oscillator: Anharmonicity and the Morse Potential

3.1 Limitations of the Harmonic Oscillator Model

Although the harmonic oscillator model is simple and elegant, it has a fatal flaw: the potential energy curve is a perfect parabola, meaning the molecule never dissociates [7][10].

Real molecules behave differently. When a chemical bond is stretched sufficiently, the bond breaks—the molecule dissociates into atoms. When the bond is overly compressed, Coulomb repulsion between nuclei causes the potential energy to rise sharply [7][10].

3.2 The Morse Potential

In 1929, physicist Philip Morse proposed a more realistic molecular potential energy function [7][10]:

$$V(r) = D_e \left(1 - e^{-a(r - r_0)}\right)^2$$

where:

• Dₑ is the dissociation energy (energy required to break the bond from equilibrium to complete dissociation)
• a is a parameter controlling the width of the potential well
• r₀ is the equilibrium bond length

The Morse potential has the shape of an asymmetric well: steep on the left (inter-nuclear repulsion) and gradually leveling off on the right (dissociation limit) [10].

3.3 Three Consequences of Anharmonicity

Under the Morse potential, vibrational energy levels are no longer equally spaced [7][10]:

$$E_v = h\left[\left(v + \frac{1}{2}\right)\nu_e - \left(v + \frac{1}{2}\right)^2 \chi_e \nu_e\right]$$

where χₑ is the anharmonicity constant.

Consequence 1: Level spacing decreases with increasing v

As the vibrational quantum number v increases, the spacing between adjacent levels becomes smaller, eventually approaching zero at the dissociation limit [10]. This means highly excited molecules need very little energy to be further excited to dissociation [7][10].

Consequence 2: Overtone absorptions become possible

In the harmonic oscillator model, Δv = ±1 is the only selection rule. But in an anharmonic oscillator, Δv = ±2, ±3, etc., transitions, though weak, can occur [7][10]. These are overtone absorptions:

• v=0 → v=1: fundamental, strongest
• v=0 → v=2: first overtone, much weaker
• v=0 → v=3: second overtone, even weaker

Overtone absorptions are typically only 1–10% of the intensity of the fundamental, but they appear in the near-infrared region and form the basis of near-infrared spectroscopy [10].

Consequence 3: The fundamental frequency is slightly lower

Due to anharmonicity, the experimentally observed fundamental (v=0→1) is slightly lower than the frequency predicted by the harmonic oscillator model [7]. Using HCl as an example [7]:

Table 2: Comparison of harmonic predictions and experimental values for HCl vibrational transitions (Data source: Datafield Quantum Mechanics Case Study [7])

📷 Figure 3: Comparison of Morse potential and harmonic oscillator potential, with non-equally spaced energy levels
Source: Vectree Knowledge Graph [10]
https://vectree.io/pdf/c/quan…

4. Types of Molecular Vibrations: More Than Just "Stretching and Shrinking"

Diatomic molecules have only one vibrational mode—stretching along the bond axis. But polyatomic molecules are much more complex [1][2][11].

For a molecule containing N atoms, the number of vibrational modes is [1][11]:

• Nonlinear molecules: 3N − 6 vibrational modes
• Linear molecules: 3N − 5 vibrational modes

(The subtracted 3 are translational degrees of freedom; for nonlinear molecules, an additional 3 rotational degrees are subtracted, while for linear molecules, only 2 rotational degrees are subtracted [11])

For example:

• Water (H₂O, 3 atoms, nonlinear): 3×3−6 = 3 vibrational modes
• CO₂ (3 atoms, linear): 3×3−5 = 4 vibrational modes
• Benzene (C₆H₆, 12 atoms): 3×12−6 = 30 vibrational modes

4.1 Two Major Categories of Vibrations

Molecular vibrations are divided into two major categories [1][2][11]:

Stretching Vibration (ν)

Atoms move along the bond axis, causing periodic changes in bond length [1][11].

• Symmetric stretch (νₛ): both bonds lengthen or shorten simultaneously
• Asymmetric stretch (νₐₛ): one bond lengthens while the other shortens

Using methylene (-CH₂-) as an example [11]:

• νₛ (symmetric stretch): ~2850 cm⁻¹
• νₐₛ (asymmetric stretch): ~2930 cm⁻¹

Asymmetric stretching frequencies are generally higher than symmetric stretching frequencies [11].

Bending Vibration (δ)

Bond angles change periodically, with atomic motion not along the bond axis [1][11]. Bending vibrations are divided into in-plane and out-of-plane types [11]:

In-plane bending:
| Type | Description | CH₂ frequency |
|------|-------------|---------------|
| Scissoring (δ) | Two bond angles open and close like scissors | ~1450 cm⁻¹ |
| Rocking (ρ) | The entire group rocks side to side in the plane | ~750 cm⁻¹ |

Out-of-plane bending:
| Type | Description | CH₂ frequency |
|------|-------------|---------------|
| Wagging (ω) | Both atoms move out of the plane in the same direction | ~1250 cm⁻¹ |
| Twisting (τ) | Both atoms move out of the plane in opposite directions | ~1250 cm⁻¹ |

Table 3: Four types of bending vibrations (Data sources: Chongqing Medical University Micro-course [11]; baike.com Vibration Forms [2])

📷 Figure 4: Schematic of the six vibrational modes of methylene (-CH₂-)
Source: Chongqing Medical University Infrared Micro-course [11]
https://sooc.iclass.cn/wcapi/…

4.2 Energy Order

Generally, for the same type of bond, the vibrational energy order is [1][11]:

Asymmetric stretch > Symmetric stretch > Bending (scissoring > rocking/twisting/out-of-plane bending)

Because stretching a chemical bond requires more effort (overcoming a larger restoring force) than bending, stretching vibrations have higher frequencies than bending vibrations [1][11].

5. Classic Examples: H₂O and CO₂

5.1 Water Molecule (H₂O) — 3 Vibrational Modes

Water is a nonlinear molecule with 3N−6 = 3 vibrational modes [1][3][8]:

Next section: 5.2 CO₂ molecule (linear) — 4 vibrational modes (including degenerate modes)

Table 4: Three vibrational modes of water molecules (gas phase frequencies; data source: UCI lecture notes [3]; Fiveable [8]). Condensed phase/liquid water shows broad envelopes around ~3400 cm⁻¹ (O–H stretching) and ~1640 cm⁻¹ (bending); the common "water peak" in sample preparation refers to the latter.

All three vibrations involve changes in dipole moment, so they are all infrared active (Ep 03 will explain in detail).

📷 Figure 5: Animation schematic of the three vibrational modes of water
Source: Wikimedia Commons (public domain)
https://upload.wikimedia.org/…

5.2 Carbon Dioxide (CO₂) – 4 Vibrational Modes

CO₂ is a linear molecule with 3N−5 = 4 vibrational modes [1][11]:

Table 5: Vibrational modes of CO₂ (data source: University of Memphis lecture notes [11]; UDel lecture notes [1]). Bending mode is denoted as degenerate ν₂ following Herzberg convention; no separate ν₄ is assigned.

Key knowledge point: The symmetric stretching vibration (ν₁) of CO₂ does not change the dipole moment of the molecule (because the dipole moments of the two C=O bonds are equal in magnitude and opposite in direction, canceling each other during simultaneous stretching), so it is not IR active [1]. This is why we need Ep 03 to explain the dipole moment change rule in detail.

📷 Figure 6: Schematic of vibrational modes of CO₂
Source: Wikimedia Commons (public domain)
https://upload.wikimedia.org/…

Summary of This Episode

Questions for Thought

1. Why is the vibrational frequency of a C–H single bond (~2900 cm⁻¹) higher than that of a C=C double bond (~1650 cm⁻¹)?
2. H₂O has 3 vibrational modes; why does the infrared spectrum often show only two distinct broad peaks?
3. If H in HCl is replaced by D (deuterium), how will the vibrational frequency change? Estimate the change ratio.
4. Why is the symmetric stretching vibration of CO₂ not observed in the infrared spectrum? By what technique can this vibration be detected?

References

[1] University of Delaware. "IR Spectroscopy Lecture Notes." Department of Chemistry and Biochemistry.
https://www1.udel.edu/chem/fo…

[2] baike.com. "Vibrational mode (mode of vibration)." Baidu Encyclopedia.
https://m.baike.com/wiki/%E6%…

[3] University of California, Irvine. "Lecture B3: Vibrational Spectroscopy." Chemistry H2A Handouts.
https://www.chem.uci.edu/~uni…

[4] LibreTexts. "Group Work 5: The Quantum Harmonic Oscillator and Molecular Vibrations." Chemistry LibreTexts.
https://chem.libretexts.org/@…

[5] OpenOChem. "The Position of Absorption Bands." IR - Infrared Spectroscopy.
https://learn.openochem.org/l…

[6] Pearson. "Based on Hooke's law, choose the bond in each pair that you expect to vibrate at a higher wavenumber." Organic Chemistry Channel.
https://www.pearson.com/chann…

[7] Datafield. "Case Study 1: Molecular Vibrations — The Quantum Harmonic Oscillator in Chemistry." Quantum Mechanics, Chapter 4.
https://datafield.dev/quantum…

[8] Fiveable. "4.3 Harmonic Oscillator and Rigid Rotor." Physical Chemistry II Study Guide.
https://fiveable.me/physical-…

[9] LibreTexts. "3.1: Introduction to Vibrations." Physical Chemistry LibreTexts.
https://chem.libretexts.org/@…

[10] Vectree. "Quantum Mechanics of Anharmonic Oscillators and Molecular Overtones." Knowledge Map.
https://vectree.io/pdf/c/quan…

[11] Chongqing Medical University. "Molecular Vibrations." Micro-lecture, Deng Ping.

https://sooc.iclass.cn/wcapi/…

Next Episode Preview: Ep 03 — The Rule of Dipole Moment Change: Not All Vibrations Can Be \"Seen\"
We will delve into the criteria for IR active and IR inactive, understand why the symmetric stretching vibration of CO₂ is \"invisible\" in the infrared spectrum, and explore the complementary relationship between IR and Raman.

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