It’s good that nucleus is heavy

Electronic Structure Theory Quantum Chemistry

To solve Schroedinger Equation for even the smallest possible molecule can be complicated as the simplest molecule contains two nuclei and one electron. And hence all the molecular theories start with the follwing precious assumption. Without this molecular theories would have had a real bad time.

Born-Oppenheimer Approximation

The assumption starts with a simple fact:

Note

Nucleus is heavier in mass than an Electron

This simplifies the mathematics to a big extent. How? Let’s see together.

Time Independent Schroedinger Equation

Let’s put all the pieces together of a time independent Schroedinger Equation, so the equation looks like the following

\[ \hat{H} \psi = E \psi \tag{1}\]

Now we will write the Hamiltonian explicitly from Equation 1

\[ \hat{H} = \hat{H}_e + \hat{H}_N=-\frac{\nabla^2_e}{2} -\frac{\nabla^2_N}{2} + V_{ee} - V_{Ne} + V_{NN} \]

Here, the operators subscripted with \(e\) represents electronic operators and \(N\) represents nuclear operators.