# The Schrödinger Equation

Typically referring to the time-independent Schrodinger Equation:

$\hat{H}\Psi = E \Psi$

which relates the total energy of the system, $E$, its wavefunction $\Psi$, a function of particle coordinates (spatial or momentum, spin), and the Hamiltonian operator $\hat{H}$.

The Hamiltonian for a typical chemical system will be something like:

$\hat{H} = \hat{T}_e + \hat{V}_{ee} + \hat{V}_{en} + \hat{T}_n + \hat{V}_{nn}$

where $\hat{T}_e$ and $\hat{T}_n$ are kinetic energy terms for the electrons $e$ and nuclei $n$ and $\hat{V}_{ee}$, $\hat{V}_{en}$ and $\hat{V}_{nn}$ are potential energy terms for electron-electron, electron-nuclear and nuclear-nuclear interactions.

$\hat{T}_e = - \frac{\nabla^2 i}{2}$
$\hat{T}_n = - \frac{\nabla^2 A}{2 M_A}$
$\hat{V}_{ee} = \sum_{i
$\hat{V}_{en} = -\sum_{A,i} \frac{Z_A}{r_{Ai}}$
$\hat{V}_{nn} = \sum_{A

where:

• $i$ and $j$ indicate electrons,
• $A$ and $B$ indicate nuclei with nuclear charge $Z$ and mass $M$,
• $\nabla$ is the Laplacian