Vibrational Spectroscopy

Introduction

Vibrational spectroscopy probes the quantized vibrational motions of molecules and provides a direct window into molecular bonding, geometry, and dynamics. The two principal techniques — infrared (IR) absorption and Raman scattering — are complementary: IR spectroscopy measures the absorption of light when vibrational transitions change the molecule's electric dipole moment, while Raman spectroscopy detects inelastic scattering of light associated with changes in molecular polarizability.

Together, IR and Raman spectroscopy form the backbone of molecular characterization in chemistry, materials science, biology, and environmental monitoring. Every covalent bond vibrates at a characteristic frequency determined by the bond strength (force constant $k$) and the masses of the bonded atoms (reduced mass $\mu$). This classical picture, refined by quantum mechanics, yields the selection rules, overtone patterns, and rotational fine structure that make vibrational spectra so information-rich.

In this chapter we derive the quantum theory of vibrational spectroscopy from first principles: the harmonic and anharmonic (Morse) oscillator models, selection rules for IR and Raman transitions, rotation-vibration coupling that produces the characteristic P, Q, and R branches, and the normal-mode analysis of polyatomic molecules. We then explore applications ranging from functional group identification to atmospheric science.

Derivation 1: Quantum Harmonic Oscillator

The simplest model of a vibrating diatomic molecule treats the bond as a harmonic spring with potential energy $V(x) = \tfrac{1}{2}kx^2$, where $x = r - r_e$ is the displacement from equilibrium and $k$ is the force constant. The time-independent Schrödinger equation for this system is:

where $\mu$ is the reduced mass. We introduce the dimensionless coordinate $\xi = \alpha x$ with $\alpha = (\mu\omega/\hbar)^{1/2}$and $\omega = (k/\mu)^{1/2}$. The Schrödinger equation becomes:

Define $\lambda = 2E/(\hbar\omega)$. For large $|\xi|$, the equation is dominated by the $\xi^2$ term, so $\psi \sim e^{-\xi^2/2}$. We therefore write $\psi(\xi) = H(\xi)\,e^{-\xi^2/2}$ and substitute to obtain Hermite's differential equation:

A power-series solution $H(\xi) = \sum_{n=0}^{\infty} a_n \xi^n$ yields the recurrence relation:

For the wavefunction to be normalizable, the series must terminate. This requires$\lambda = 2v + 1$ for some non-negative integer $v = 0, 1, 2, \ldots$, giving the celebrated energy eigenvalues:

The corresponding wavefunctions are products of a Hermite polynomial and a Gaussian envelope:

The first several Hermite polynomials are $H_0 = 1$, $H_1 = 2\xi$,$H_2 = 4\xi^2 - 2$, $H_3 = 8\xi^3 - 12\xi$. Each successive wavefunction has one more node, with even/odd $v$ giving even/odd parity.

Derivation 2: Selection Rules for IR and Raman

IR Absorption Selection Rule

The probability of a vibrational transition induced by infrared radiation is proportional to the square of the transition dipole moment:

We expand the dipole moment function as a Taylor series about the equilibrium position:

The first term gives zero by orthogonality ($\langle v'|v\rangle = \delta_{v'v}$). The leading contribution comes from the linear term. Using the ladder operator result $x = \sqrt{\hbar/(2\mu\omega)}\,(\hat{a}^\dagger + \hat{a})$:

Raman Scattering Selection Rule

Raman scattering arises from the modulation of the molecular polarizability $\alpha$by the vibration. The induced dipole in an oscillating electric field is:

Expanding the polarizability about equilibrium:

Substituting $q(t) = q_0\cos(\omega_{\text{vib}}t)$ and using the product-to-sum identity:

The first term is Rayleigh (elastic) scattering. The second and third terms give Stokes ($\omega_0 - \omega_{\text{vib}}$) and anti-Stokes ($\omega_0 + \omega_{\text{vib}}$) Raman scattering.

Mutual Exclusion Rule

Derivation 3: Anharmonic Oscillator — Morse Potential

Real molecular potentials are anharmonic: they are asymmetric about the minimum, with the repulsive wall rising more steeply than the attractive tail. The Morse potential captures these features with the form:

Here $D_e$ is the well depth (measured from the minimum to the dissociation asymptote), $r_e$ is the equilibrium bond length, and $a$ controls the width of the well:

The remarkable property of the Morse potential is that the Schrödinger equation can be solvedexactly. Introducing the substitution $z = 2\lambda\,e^{-a(r-r_e)}$where $\lambda = \sqrt{2\mu D_e}/(\hbar a)$, the radial equation transforms into the associated Laguerre equation. The exact energy eigenvalues are:

where the anharmonicity constant is:

Key consequences of anharmonicity:

• Energy levels converge: The spacing decreases with increasing $v$, approaching zero near dissociation. The number of bound states is finite: $v_{\max} \approx 1/(2x_e) - 1/2$.
• Overtones become allowed: Since the wavefunctions of the anharmonic oscillator are not pure Hermite-Gaussian functions, the strict$\Delta v = \pm 1$ selection rule breaks down. Transitions with $\Delta v = \pm 2, \pm 3, \ldots$ (overtones) acquire non-zero intensity, though each successive overtone is typically an order of magnitude weaker.
• Combination bands: In polyatomic molecules, anharmonic coupling between modes allows combination bands where two or more modes are simultaneously excited ($\nu_i + \nu_j$).

The dissociation energy measured from the potential minimum is $D_e$, while the experimentally accessible quantity is $D_0 = D_e - E_0$, the dissociation energy from the ground vibrational state. Using the Morse model:

For HCl: $\omega_e = 2990.95$ cm$^{-1}$, $\omega_e x_e = 52.82$ cm$^{-1}$, giving $D_e \approx 42\,300$ cm$^{-1}$ ($\approx 5.24$ eV) and a maximum of about 23 bound vibrational states.

Derivation 4: Rotation-Vibration Coupling

A real diatomic molecule vibrates and rotates simultaneously. The total energy depends on both quantum numbers $v$ and $J$. We start from the full rovibrational Hamiltonian:

The rotational constant depends on the instantaneous bond length: $B(r) = \hbar^2/(2\mu r^2)$. As the molecule vibrates, the average value of $1/r^2$ differs from $1/r_e^2$. Taking the expectation value over the vibrational state $|v\rangle$ gives thevibrationally averaged rotational constant:

where $B_e = \hbar/(4\pi c\mu r_e^2)$ is the equilibrium rotational constant and $\alpha_e$ is the rotation-vibration coupling constant (always positive for a Morse oscillator because larger $v$ means larger average bond length and smaller $B$). The combined energy expression is:

P, Q, and R Branches

For a rovibrational transition $v'' \to v'$ with $\Delta J = 0, \pm 1$, the transition frequencies for the fundamental band ($v = 0 \to 1$) are:

where the band origin is $\tilde{\nu}_0 = \omega_e - 2\omega_e x_e$ (for the fundamental).

Since $B_1 < B_0$ (the bond lengthens upon vibrational excitation), the coefficient $(B_1 - B_0) < 0$. This means:

• The R-branch lines converge at high $J$ (spacing decreases), eventually forming a band head.
• The P-branch lines diverge (spacing increases with $J$).
• There is a gap at $\tilde{\nu}_0$ because $J = 0 \to J = 0$ is forbidden for molecules without electronic angular momentum.

Derivation 5: Normal Modes of Polyatomic Molecules

A molecule of $N$ atoms has $3N$ degrees of freedom. Of these, 3 are translational and either 3 (nonlinear) or 2 (linear) are rotational. The remaining internal degrees of freedom are vibrational:

The GF Matrix Method

Wilson's GF matrix method provides a systematic procedure for determining normal-mode frequencies from molecular geometry and force constants. The classical equations of motion for small vibrations are, in mass-weighted coordinates:

where $\mathbf{F}$ is the force constant matrix in internal coordinates and$\mathbf{G}$ is the kinetic energy matrix that encodes the molecular geometry and masses. The normal-mode frequencies are obtained by solving the secular equation:

The eigenvalues $\lambda_k$ give the squared frequencies, and the eigenvectors describe the atomic displacement patterns of each normal mode.

Application to H$_2$O

Water ($C_{2v}$ symmetry, $N = 3$) has $3(3) - 6 = 3$ normal modes. These can be classified by the irreducible representations of $C_{2v}$:

Since H$_2$O is noncentrosymmetric, the mutual exclusion rule does not apply, and all three modes are both IR and Raman active. The symmetric stretch and bend both belong to the totally symmetric representation A$_1$, while the asymmetric stretch transforms as B$_2$.

The force constant matrix in internal coordinates $(r_1, r_2, \theta)$ for the valence force field model takes the form:

where $f_r$ is the O–H stretching force constant ($\approx 8.45$ mdyn/Å),$f_\theta$ is the bending force constant ($\approx 0.76$ mdyn·Å),$f_{rr}$ is the stretch-stretch interaction constant, and $f_{r\theta}$ is the stretch-bend interaction constant. Solving $|\mathbf{GF} - \lambda\mathbf{I}| = 0$ yields frequencies in excellent agreement with experiment.

Applications

Historical Context

Simulation: Morse Potential & Rovibrational Spectrum

The Python simulation below plots the Morse and harmonic potentials for HCl with their energy levels, compares the harmonic and anharmonic level spacings, generates the rovibrational stick spectrum showing P and R branches with Boltzmann-weighted intensities at 300 K, and displays the quantum harmonic oscillator wavefunctions (Hermite polynomials times Gaussian).

Computation: Diatomic Vibrational Parameters

The Fortran program below computes vibrational frequencies, anharmonicity constants, dissociation energies, and rovibrational band parameters for eight diatomic molecules (HCl, HBr, HF, HI, CO, NO, N$_2$, O$_2$) using the Morse oscillator model. It also tabulates the detailed anharmonic energy levels for HCl up to the dissociation limit.