Variational Principle

Quantum Chemistry

Solving for the Hydrogen Ground State

To solve for the ground state of hydrogen we can use the variational principle. This states that:

E_{0} \leq \frac{⟨ ψ ∣ H ^ ∣ ψ ⟩}{⟨ ψ ∣ ψ ⟩}

Which, in non-equation form means: We can construct a guess wavefunction $ψ$ , take the expectation of hamiltonian, and then normalize it and that guess will be greater than or equal to the ground state. A very powerful statement indeed, so we can make a terrible guess and find an upper bound for the ground state. From a practical perspective, this almost always means we make a guess which depends on some parameter, and then we optimize that parameter to minimize the energy.

For the hydrogen atom our guess could look like this if we assume the atom is centered in the origin. This sets $R_{A} = (0, 0, 0)$ and we are left with:

1 s (r) = e^{- ζ ∣ r ∣}

$ζ$ will be our parameter which means this will be a nonlinear variational problem.