natural orbital

The orbitals defined (P. Lowdin) as the eigenfunctions of the spinless one-particle electron density matrix. For a configuration interaction wave-function constructed from orbitals $Φ$ , the electron density function, $ρ$ , is of the form:

$ρ = ∑ i ∑ j a i j Φ i * Φ j$

where the coefficients $a i j$ are a set of numbers which form the density matrix. The NOs reduce the density matrix $ρ$ to a diagonal form:

$ρ = ∑ k b k Φ k * Φ k$

where the coefficients $b k$ are occupation numbers of each orbital. The importance of natural orbitals is in the fact that CI expansions based on these orbitals have generally the fastest convergence. If a CI calculation was carried out in terms of an arbitrary basis set and the subsequent diagonalisation of the density matrix $a i j$ gave the natural orbitals, the same calculation repeated in terms of the natural orbitals thus obtained would lead to the wave-function for which only those configurations built up from natural orbitals with large occupation numbers were important.

Source:

PAC, 1999, 71, 1919 (Glossary of terms used in theoretical organic chemistry) on page 1954