https://doi.org/10.1351/goldbook.NT07079

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: \[\unicode[Times]{x3C1} = \sum_{i}\sum _{j}a_{ij}\,\varPhi_{i}^{*}\,\varPhi_{j}\] where the coefficients ai j are a set of numbers which form the density matrix. The NOs reduce the density matrix ρ to a diagonal form: \[\unicode[Times]{x3C1} = \sum _{k}b_{k}\mathit{\Phi}_{k}^{*}\mathit{\Phi}_{k}\] where the coefficients bk 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 ai 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 (https://doi.org/10.1351/pac199971101919)

PAC, 1999,