An approximate exchange-correlation hole density as a functional of the natural orbitals

The Fermi and Coulomb holes that can be used to describe the physics of electron correlation are calculated and analysed for a number of typical cases, ranging from prototype dynamical correlation to purely nondynamical correlation. Their behaviour as a function of the position of the reference electron and of the nuclear positions is exhibited. The notion that the hole can be written as the square of a hole amplitude, which is exactly true for the exchange hole, is generalized to the total holes, including the correlation part. An Ansatz is made for an approximate yet accurate expression for the hole amplitude in terms of the natural orbitals, employing the local (at the reference position) values of the natural orbitals and the density. This expression for the hole amplitude leads to an approximate two-electron density matrix that: (a) obeys correct permutation symmetry in the electron coordinates; (b) integrates to the exact one-matrix; and (c) yields exact correlation energies in the limiting cases of predominant dynamical correlation (high Z two-electron ions) and pure nondynamical correlation (dissociated H