Molecular Degrees of Freedom and Classical Vibrations

Chem 3240 · Lecture 4.1

Davit Potoyan

A molecule is four motions at once

• Translation: the molecule flies
• Rotation: it tumbles
• Vibration: bonds stretch and bend
• Electronic: electrons rearrange

Each motion has its own energy ladder

\[E= \epsilon_{trans}+ \epsilon_{rot}+ \epsilon_{vib}+\epsilon_{elec}\]

• Level spacings differ by orders of magnitude
• That is why different spectroscopies exist: microwave for rotation, infrared for vibration, UV-Vis for electrons

Separating the Hamiltonian

• Born-Oppenheimer separates nuclei from electrons, leaving

\[\hat{H} = \hat{H}_{tr} + \hat{H}_{rot} + \hat{H}_{vib}\]

• Separation means the wavefunction factorizes:

\[\psi = \psi_{tr}\,\psi_{rot}\,\psi_{vib}\]

• One hard problem becomes three toy models: particle in a box, rigid rotor, harmonic oscillator

Counting the coordinates

• \(N\) nuclei carry \(3N\) coordinates
• Translation always takes 3
• Rotation takes 3, or only 2 for a linear molecule
• Vibrations get the rest: \(3N-6\) (nonlinear) or \(3N-5\) (linear)

molecule	\(3N\)	trans	rot	vib
H\(_2\)O	9	3	3	3
CO\(_2\)	9	3	2	4
benzene	36	3	3	30

The classical harmonic oscillator

• Bead on a spring: displacement \(x\) meets a restoring force

\[F=-kx\]

• Hooke’s law; \(k\) is the spring stiffness

Solving the motion

\[m \ddot x+kx = 0\]

Solving the motion

\[m \ddot x+kx = 0 \quad\Longrightarrow\quad \ddot{x}+\omega^2 x =0, \qquad \omega=\sqrt{\frac{k}{m}}\]

• Stiffer spring: faster oscillation; heavier mass: slower

\[x(t)= A \sin(\omega t+\phi)\]

Amplitude \(A\) and phase \(\phi\) come from initial conditions

Energy sloshes but is conserved

\[V(x) = \frac{kx^2}{2}, \qquad E=\frac{p^2}{2m} + \frac{kx^2}{2}\]

• Kinetic and potential energy interconvert at frequency \(\omega\)
• This total energy becomes the quantum Hamiltonian next lecture

A diatomic is one bead in disguise

• Two masses, one spring: the center of mass moves freely
• Relative coordinate obeys the one-bead equation with the reduced mass

\[\mu=\frac{m_1 m_2}{m_1+m_2}, \qquad \omega = \sqrt{\frac{k}{\mu}}\]

Why harmonic? Taylor says so

\[U(x) = \frac{1}{2}k(x-x_0)^2+\frac{\gamma}{3!}(x-x_0)^3+...\]

• Near any minimum the quadratic term dominates
• \(k = U''(x_0)\): curvature of the true potential
• Higher terms are the anharmonicity

Takeaway

Molecular motion separates into translation, rotation, vibration, and electronic parts, with \(3N-6\) (or \(3N-5\)) vibrations. Each vibration is classically a harmonic oscillator: \(\ddot{x} = -\omega^2 x\), \(\omega = \sqrt{k/\mu}\), \(E = \frac{p^2}{2\mu} + \frac{kx^2}{2}\), valid because every smooth potential is a parabola near its minimum.