The Born–Oppenheimer Approximation
On the Quantum Theory of Molecules (1927)
The examples of wave functions we have seen up till now have typically involved single particles. The physics of more complex systems like molecules becomes increasingly difficult as more particles are added to the picture.
The Born-Oppenheimer approximation was one of the first attempts to solve this dilemma and has become a cornerstone of quantum chemistry and molecular physics. Originally introduced by Max Born and J. Robert Oppenheimer in 1927, this approximation allows for the separation of nuclear and electron motion in a molecule, vastly simplifying quantum mechanical calculations.
Though quite esoteric from a popular perspective, any well-read physicist will assure you that this approach is invaluable to the field.
The Molecular Quantum Problem
The Schrödinger equation, which governs the dynamics of a quantum system over time, must account for the motion of all particles in a molecule including their interactions and their wave-like behavior. This is embedded in a mathematical structure called the Hamiltonian which describes the total energy of a system:
Here, H^ is the Hamiltonian operator, Ψ(r,R) is the wave function describing the molecule that depends on both the electron coordinates r and the nuclear coordinates R. E is the total energy of the system.
This wave function lives in an abstract high-dimensional space because every coordinate of every particle must be considered. For just a small molecule like water (H₂O), the full quantum description involves 10 electrons and 3 nuclei—already a model containing 39 dimensions! Solving this full equation exactly is practically impossible for anything larger than the simplest systems.