Brillouin's theorem

Within quantum chemistry, Brillouin's theorem, proposed by the French physicist Léon Brillouin in 1934, runs as follows:

Suppose D₁ is an optimized single determinant function and D_j is a determinant corresponding to any single excitation out of an orbital φ_j occupied in D₁ and into the virtual subspace (orthogonal complement) of D₁, then no improvement in energy is possible taking ψ = c₁D₁+c₂D_j.

The proof of this theorem is as follows: begin with a basis set that spans a function space. A self-consistent field (SCF) calculation is performed, which produces the best single-determinant wavefunction we can possibly get within this function space. Call this D₁. D_j differs from D₁ in only one atomic orbital, which means they differ in only one row. A general property of determinants is that, if two of them differ in only one row or column, any linear combination of the two can be expressed as one determinant. Thus, any combination c₁D₁ + c₂D_j can still be written as a single determinant. Since D_j makes no use of functions outside the original basis set, c₁D₁ + c₂D_j is a single determinant within the original function space. However, D₁ is already known to be the single determinant within this function space that gives the lowest energy, and therefore c₁D₁ + c₂D_j cannot do better.