Questions: Schrödinger Equation for Molecular Systems

Question 1

A student claims to have 'measured the molecular wavefunction ψ directly' in their spectroscopy experiment. What is fundamentally wrong with this claim?

• A. Nothing — ψ can be measured directly by X-ray diffraction
• B. The wavefunction ψ is not directly observable; only |ψ|² (the probability density) corresponds to measurable quantities
• C. Only ground-state wavefunctions can be measured; excited states are always inaccessible
• D. Wavefunctions can be directly measured for atoms but the math becomes intractable for molecules

Answer: (B)

This is the core misconception listed in the topic. The wavefunction ψ is a mathematical object — it can be complex-valued and has no direct physical meaning on its own. What is physically observable is |ψ|², the probability density, which gives the likelihood of finding electrons at particular positions. This is what electron density maps (from X-ray crystallography) actually measure — a probability distribution, not ψ itself.

Question 2

The Born-Oppenheimer approximation makes the molecular Schrödinger equation tractable. What physical insight does it exploit?

• A. Nuclear kinetic energy is negligible compared to electron kinetic energy, so nuclei can be ignored entirely
• B. Electron-electron repulsion and nuclear-nuclear repulsion cancel each other out at equilibrium
• C. Nuclei are thousands of times more massive than electrons, so electrons adjust almost instantaneously to any nuclear arrangement — allowing nuclear positions to be treated as fixed parameters
• D. Molecules can be decomposed into non-interacting atom-sized subunits that each solve independently

Answer: (C)

The Born-Oppenheimer approximation exploits the enormous mass difference: protons and neutrons are ~1800 times heavier than electrons. Nuclei therefore move far more slowly, and from the electron's perspective, nuclei are essentially stationary. This allows us to 'clamp' nuclei at fixed positions, solve the electronic Schrödinger equation at that geometry, then repeat at many geometries to map out the potential energy surface. Option A is wrong — nuclei aren't ignored, their positions are just treated as parameters rather than dynamic variables.

Question 3

In the Born-Oppenheimer approximation, nuclear coordinates are treated as variables in the electronic Schrödinger equation.

• T. True
• F. False

Answer: False

Nuclear coordinates are treated as *parameters* — fixed values — not variables. The approximation 'clamps' the nuclei at a specific geometry and solves for the electronic wavefunction at that fixed configuration. This is then repeated at many different nuclear geometries to map out the potential energy surface. Treating nuclei as variables would mean solving for electronic and nuclear motion simultaneously, which is the full molecular Schrödinger equation — exactly what the approximation is designed to avoid.

Question 4

The potential energy surface obtained from the Born-Oppenheimer approximation has minima corresponding to stable molecular geometries and saddle points corresponding to transition states.

• T. True
• F. False

Answer: True

The potential energy surface (PES) maps the electronic energy of the molecule as a function of nuclear geometry. Minima on this surface are stable (or metastable) molecular structures where the energy is locally minimized — small distortions in any direction increase the energy. Saddle points are geometries where the energy is a maximum along the reaction coordinate but a minimum in all perpendicular directions — these correspond to transition states in chemical reactions. The curvature around minima also determines vibrational frequencies.

Question 5

Why can't the molecular Schrödinger equation be solved exactly for any system beyond H₂⁺, and what strategy does the Born-Oppenheimer approximation use to make it tractable?

Model answer: For systems with multiple electrons, the Hamiltonian contains electron-electron repulsion terms that couple all electrons together. This makes the equation non-separable — you cannot factor it into independent one-electron problems and solve each separately. Every electron's behavior depends on the positions of all other electrons simultaneously, creating an intractable many-body problem. The Born-Oppenheimer approximation exploits the ~1800:1 mass ratio between nuclei and electrons: nuclei move so slowly that electrons effectively adjust instantaneously. By treating nuclear positions as fixed parameters rather than dynamic variables, the full problem reduces to an electronic Schrödinger equation at each nuclear geometry. This is still a many-electron problem, but it is tractable with variational and perturbation methods, and it must be solved only once per geometry point on the potential energy surface.