The Variational Theorem and Molecular Quantum Mechanics

In quantum chemistry, solving the Schrödinger equation for a molecule means finding the allowed energy levels and wave functions that describe the behavior of its electrons. However, for systems with more than one electron, solving this equation exactly is generally impossible. This is where the variational theorem becomes essential.

What is the Variational Theorem?

The variational theorem states that for any normalized trial wave function, the expected energy calculated using the Schrödinger Hamiltonian is always greater than or equal to the true ground state energy:

$$ E[\psi_{\text{trial}}] = \langle \psi_{\text{trial}} | \hat{H} | \psi_{\text{trial}} \rangle \geq E_0 $$

Here, $ \hat{H} $ is the Hamiltonian (energy operator), $ \psi_{\text{trial}} $ is any normalized trial wave function, and $ E_0 $ is the exact ground state energy. This means that the lowest possible energy you can compute with a trial wave function gives an upper bound to the true ground state energy.

Variational Principle (Concise Statement)

For any normalized trial wavefunction $\Psi$, the energy functional satisfies

$$ E[\Psi] = \frac{\langle \Psi | \hat H | \Psi \rangle}{\langle \Psi | \Psi \rangle} \ge E_0 \ . $$

Equality holds iff $\Psi$ is the exact ground state (up to phase).
Enlarging the trial space (basis / ansätze) monotonically lowers the upper bound.

Why is This Useful?

Since the exact wave function is unknown and too complex to calculate, we construct an approximate (trial) wave function with adjustable parameters. By tuning these parameters to minimize the energy, we get closer to the true ground state. This process turns the difficult problem of solving the Schrödinger equation into a variational optimization problem.

Applications in Molecular Calculations

The variational principle forms the basis of many computational chemistry methods:

Hartree-Fock theory: Uses a single Slater determinant and variationally optimizes orbitals.
Configuration Interaction (CI) and Møller-Plesset (MP) theory: Extend Hartree-Fock using excited determinants or perturbative corrections.
Density Functional Theory (DFT): Reformulates the variational principle using the electron density instead of the wave function.

Summary

The variational theorem is a foundational tool in molecular quantum mechanics. It provides a practical and rigorous way to estimate the ground state energy of molecules by optimizing trial wave functions—making the intractable solvable.