Molecular Degrees of Freedom

Chem 3240 · Lecture 4.1

Davit Potoyan

Where we are headed

Bound systems have quantized energy.

Next: how each kind of motion is quantized, and how spectroscopy probes it.

Toy models map onto real motions:

  • Particle in a box → translation
  • Harmonic oscillator → vibration
  • Rigid rotor → rotation

Four kinds of molecular motion

A molecule’s energy splits into independent pieces:

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

Each kind of motion is quantized differently.

Different spacings, different spectroscopy

Boundary conditions set the level spacing.

Electronic ≫ vibrational ≫ rotational ≫ translational.

Each scale → its own spectroscopy.

Counting: 3N nuclear coordinates

\(N\) nuclei, each with \(x, y, z\):

\[3N \text{ total degrees of freedom}\]

Born–Oppenheimer: separate nuclei from electrons.

\[\hat{H} = \sum_{i=1}^{N} -\frac{\hbar^2}{2m_i}\nabla_{R_i}^2 + E(R_1,\dots,R_N)\]

Splitting the Hamiltonian

The nuclear motion separates into three parts:

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

  • \(\hat{H}_{tr}\), \(\hat{H}_{rot}\): no potential
  • \(\hat{H}_{vib}\): carries the potential \(E\) between nuclei

Separation lets the wavefunction factorize:

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

Three Schrödinger equations

\[\hat{H}_{tr}\psi_{tr} = E_{tr}\psi_{tr}\]

\[\hat{H}_{rot}\psi_{rot} = E_{rot}\psi_{rot}\]

\[\hat{H}_{vib}\psi_{vib} = E_{vib}\psi_{vib}\]

Translation: no boundary condition → continuous spectrum.

Rotation (cyclic boundary) and vibration (potential \(E\)) → quantized.

Bookkeeping the degrees of freedom

Start with \(3N\), then subtract:

  • Translation: always \(3\) → leaves \(3N - 3\)
  • Rotation: \(3\) nonlinear, \(2\) linear
  • Vibration: whatever remains
    • Nonlinear: \(3N - 6\)
    • Linear: \(3N - 5\)

At a glance

Translation Rotation Vibration
Linear 3 2 \(3N-5\)
Nonlinear 3 3 \(3N-6\)

Linear molecules have one fewer rotation, so one more vibration.

Takeaway

A molecule has \(3N\) degrees of freedom: \(3\) translation, \(2\) or \(3\) rotation, the rest (\(3N-5\) or \(3N-6\)) vibration. Born–Oppenheimer separation factorizes the motion, and boundary conditions decide which parts are quantized and how spectroscopy probes them.