Basics of Diatomic Molecules

Stationary quantum system = Some number of nuclei and electrons in a stable arrangement, e.g. atoms and molecules

Quantum state = EΨ = Solution of the time-independent Schrodinger Equation using the Hamiltonian of a given quantum system. A quantum (electronic) state has an energy and a wavefunction (describing the positions of the electrons and nuclei).

Quantum numbers are human labels that are used to distinguish between different quantum states. Useful quantum numbers are chosen so they tell you something about the energy and/or wavefunction of the quantum state.

Quantum constant = A constant that relates the energy of a quantum state and one of its quantum numbers.

Summary:

Quantum System: Diatomic Molecule

Quantum States: States with lowish energies

Quantum Numbers: Electronic, Vibrational, Rotational

Quantum Constants: The goal of this project!

Electronic States in Hydrogen Atom:

We only have one electron, so we just need to describe its properties. An electron can be in any ONE orbital around the hydrogen nucleus; some orbitals are shown to the right. They have labels like 1s, 2s, 2p_x, 3d_xy etc.

Electronic States in Multi-electron Atoms:

We have many electrons, but an electronic state describes their positions and momentums of all electrons. The wavefunction is the collective group of orbitals including the single and paired electrons.

Shape of wavefunction (probability space of the electrons is the square of the wavefunction)

• s → S
• p → P
• d → D etc.

Number of unpaired electrons

• Singlet (¹) (0 unpaired electrons)
• Doublet (²) (1 unpaired electrons)
• Triplet (³) (2 unpaired electrons)
• Quartet (⁴) (3 unpaired electrons)
• Quintet (⁵) (4 unpaired electrons)
• And so on...

Electronic States in Diatomic Molecules:

In diatomic molecules, we describe the electronic state using things like shape of overall wavefunction

• s→ Sigma (Σ)
• p → Pi (Π)
• d→ Delta (Δ)
• f → Phi (ϕ)
• g → Gamma (Γ)

Electronic Energy:

The electronic energy of a molecule is the largest component of its overall energy. This, unlike vibrational and rotational can only be calculated experimentally.

T_e (Term Energy): energy of electronic state relative to the lowest electronic state of the molecule.

El. State (Electronic State): Quantum number associated with electronic component of quantum state. Electronic State is a more complex label, e.g.

A set of electronic states with transitions between these states.

Molecular systems have nuclei that can move. The nuclei rotate and vibrate.

Vibration:

As the energy levels increase the amount of nodes in the wave function increases meaning that probability of being within the wavefunction decreases.

ν : Vibrational Quantum Number (begins at 0 and increases in integer steps)

Using harmonic oscillator approximation:

Rotation:

Using rigid-rotor approximation:

Exercises: