Born Oppenheimer

Born-Oppenheimer approximation The complete Hamiltonian for a system of interacting nuclei and electrons is:

h2 H=− 2me

h2 ∑i ∇ − 2 2 i

Zα Z β e 2 1 2 1 ⎛1⎞ ∇α + ∑ ⎜ ⎟ ∑∑ 4πε 0 ⎝ 2 ⎠ α ≠ β Rαβ α mα

1 ⎛1⎞ e2 1 + − ∑∑ ⎜ ⎟ 4πε 0 ⎝ 2 ⎠ i ≠ j rij 4πε 0

Zα e 2 ∑∑ riα i α

Hψ ( Rα , ri ) = εψ ( Rα , ri ) Here the first two terms of the first equation describe the kinetic energy of the electrons and nuclei respectively, the third term describes the Coulomb repulsion between the positively charged nuclei, the fourth describes the Coulomb repulsion between the electrons, while the final term describes the Coulomb attraction between electrons and nuclei. What would be the error if the wavefunction of the electron-nuclear system, ψ(Rα,ri), were expressed as a product of an electronic wavefunction u(Rα,ri) times a nuclear wavefunction v(Rα)? The electronic Schrödinger equation may be written as: 2 2 α β 2 i α≠ β e i αβ 0 α i 2 2

⎡ h Z Z e 1 ⎛1⎞ ∇ + ⎢− ∑ ⎜ ⎟ ∑∑ m R πε 2 4 ⎝2⎠ ⎢ ⎢ ⎢ + 1 ⎛⎜ 1 ⎞⎟ ∑∑ e − 1 ∑∑ Zα e ⎢⎣ 4πε 0 ⎝ 2 ⎠ i ≠ j rij 4πε 0 i α riα

= E ( Rα ) u ( Rα , ri )

⎤ ⎥ ⎥u R ,r ) ⎥ ( ⎥ ⎥⎦

Here the electronic wavefunction is a function of the electron co-ordinates, while it depends parametrically on the nuclear co-ordinates. The Schrödinger equation for nuclear motion on the other hand may be written as:

⎡ h2 ⎢− ⎣ 2

⎤ 1 2 ∇α + E ( Rα ) ⎥ v ( Rα ) = ε v ( Rα ) ∑ α mα ⎦

Here the electronic energy serves as a potential for the nuclear motion. Substituting the product wavefunction in the complete Schrödinger equation, we find:

⎡ h2 h2 1 2 2 − ∇ − ∇α ⎢ ∑ ∑ i 2 α mα ⎢ 2me i ⎢ Zα Z β e 2 1 1 ⎛ ⎞ ⎢+ ⎜ ⎟ ∑∑ ⎢ 4πε 0 ⎝ 2 ⎠ α ≠ β Rαβ ⎢ e2 1 ⎢+ 1 ⎛ 1 ⎞ − ⎢ 4πε ⎜⎝ 2 ⎟⎠ ∑∑ 4πε 0 i≠ j rij 0 ⎣ h2 = ε u ( Rα , ri ) v ( Rα ) − 2

⎤ ⎥ ⎥ ⎥ ⎥ u ( Rα , ri ) v ( Rα ) ⎥ ⎥ 2 Zα e ⎥ ∑∑ riα ⎥⎦ i α

1 ∑ α mα

⎛ ∂u ∂v ∂ 2u ⎞ +v 2 ⎟ ⎜2 ∂Rα ⎠ ⎝ ∂Rα ∂Rα

This equation differs by virtue of the last summation on the RHS from the desired result which would provide the justification for a product wavefunction involving separation of the nuclear and electron variables in the sense indicated. However, these extra summation terms are small.