Theoretical and computational chemistry
Schrödinger equation and Born–Oppenheimer approximation
How the quantum equation describes molecular electrons and nuclei, and why separating their motions makes molecular structure calculable.
IntuitionA quantum picture of a molecule
A molecule is not a miniature solar system: its electrons are described by a wavefunction, which predicts probabilities and energies. The nuclei are much heavier, so they usually move more slowly than the electrons.
This separation of time scales motivates the Born–Oppenheimer approximation: first solve the electronic problem for fixed nuclear positions, then let the nuclei move on the resulting energy surface.
SchoolFrom atoms to an equation
Definition: Wavefunction and probability
For a normalized state, |Ψ|² gives the probability density for a measurement of particle positions. The wavefunction also encodes the state’s energy and other observables.
The time-independent Schrödinger equation is an eigenvalue problem: the Hamiltonian operator Ĥ contains kinetic and potential energy, and allowed stationary states have energies E.
Example: Particle on a line
An electron is confined to a one-dimensional box of length L with infinitely high walls. What happens to its allowed energies when L is halved?
Solution
For a box, Eₙ ∝ n²/L². At fixed quantum number n, halving L makes each energy four times larger.
UndergraduateThe molecular Hamiltonian
The terms are electron and nuclear kinetic energy, electron–nucleus attraction, electron–electron repulsion, and nucleus–nucleus repulsion. The coupled coordinates make the exact many-particle equation difficult for all but the simplest systems.
Definition: Born–Oppenheimer electronic problem
At fixed nuclear geometry R, solve Ĥₑ(R)ψₑ(r;R)=Eₑ(R)ψₑ(r;R). The electronic eigenvalue, plus nuclear repulsion, defines a potential-energy surface on which the nuclei move.
| Quantity | Role |
|---|---|
| Electronic state | Fast response to nuclear geometry |
| Nuclear motion | Slower motion on an energy surface |
AdvancedLimits and deeper theory
The approximation neglects derivative couplings between electronic states. It can fail near conical intersections, avoided crossings, or when electronic and nuclear motions become strongly coupled; nonadiabatic dynamics then requires coupled electronic–nuclear treatment.
References
- Zur Quantentheorie der Molekeln · M. Born, J. R. Oppenheimer, 1927
- Molecular Quantum Mechanics · P. W. Atkins, R. S. Friedman, 2011