natural orbital

https://doi.org/10.1351/goldbook.NT07079
The orbitals defined (P. Lowdin) as the eigenfunctions of the spinless one-particle @E01986@ matrix. For a configuration interaction wave-function constructed from orbitals \(\varPhi \), the @ET07024@, \(\unicode[Times]{x3C1}\), is of the form: \[\unicode[Times]{x3C1} = \sum_{i}\sum _{j}a_{ij}\,\varPhi_{i}^{*}\,\varPhi_{j}\] where the coefficients \(a_{ij}\) are a set of numbers which form the density matrix. The NOs reduce the density matrix \(\unicode[Times]{x3C1}\) to a diagonal form: \[\unicode[Times]{x3C1} = \sum _{k}b_{k}\mathit{\Phi}_{k}^{*}\mathit{\Phi}_{k}\] 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 @BT06999@ and the subsequent diagonalisation of the density matrix \(\text{a}_{ij}\) 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 [Terms] [Paper]