MOLECULAR PHYSICS, 2002, VOL. 100, NO. 4, 401 ± 421 An approximate exchange-correlation hole density as a functional of the natural orbitals M. A. BUIJSE and E. J. BAERENDS* Afdeling Theoretische Chemie, Chemistry Department, FEW, Vrije Universiteit, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands (Received 8 March 2001; accepted 5 June 2001) 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 (disso- ciated H 2 ). 1. Introduction Density matrices received much attention in the early days of quantum chemistry, from the mid ®fties, say, until the mid seventies. An excellent exposition of reduced density matrices is to be found in the 1976 monograph by Davidson [1], which is still a source of much of our current knowledge concerning analytical properties, symmetry aspects and physical meaning of the one-electron and two-electron density matrices. Density matrices have not had the expected impact on practical ab initio methods for the calculation of elec- tronic wave functions and expectation values. By now, such calculations, going beyond Hartree±Fock (HF) in accuracy and including Coulomb correlation at some level, have become routine. The Coulomb correlation modi®es the electron pair density (with respect to HF). The concept of electron pairs plays a central role in chemistry and it would therefore be natural to suppose that two-electron density matrices and the correlated pair density would have been studied extensively. Also this, however, is not the case. The number of interpret- ative studies of pair densities (both HF and correlated) is very small in comparison to, for example, studies of the electron density. Nevertheless, the description of electron correlation in terms of Fermi holesÐdescribing exchange eectsÐand Coulomb holesÐdescribing pre- dominantly correlation between electrons of unlike spinÐyields a clear picture of the physics of electron correlation [2]. We will use the insights obtained from such pictures to make an Ansatz for a simple yet rather accurate approximation of the two-electron density matrix in terms of the natural orbitals. The present paper is based on chapter 5 of [2]. The description of the physics of electron correlation in terms of exchange and Coulomb holes has played a more signi®cant role in density functional theory (DFT). The practical success of DFT depends on accurate mod- elling of these holes. The initial success of the local density approximation (LDA) has been explained by the favourable properties of the implicitly used hole, and generalized gradient approximations (GGA) have often been proposed using arguments invoking known properties of the exact holes. Further improvement, beyond the GGAs, might come from improved model- ling of the holes. We will argue in this paper that improved modelling of holes is possible when invoking, apart from the occupied orbitals, also the virtual orbitals. First, we will analyse some eects of electron correlation using the two-electron density matrix and the related holes, in order to understand some features of electron correlation in a physical and visual manner. We will next make a con- tribution to accurate hole modelling by introducing the concept of a hole amplitude in which the correlated hole density is approximated by the square of an amplitude, Molecular Physics ISSN 0026±8976 print/ISSN 1362±3028 online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/0026897011007024 3 * Author for correspondence. e-mail: baerends@chem.vu.nl