Four-Component Relativistic Kohn–Sham Theory TROND SAUE, 1 TRYGVE HELGAKER 2 1 UMR 7551 CNRS/Universite ´ Louis Pasteur, Laboratoire de Chimie Quantique et Mode ´lisation Mole ´culaire, 4 rue Blaise Pascal, F-67000 Strasbourg, France 2 Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway Received 20 November 2001; Accepted 14 December 2001 Published online 11 April ience.wiley.com). DOI 10.1002/jcc.10066 Abstract: A four-component relativistic implementation of Kohn–Sham theory for molecular systems is presented. The implementation is based on a nonredundant exponential parametrization of the Kohn–Sham energy, well suited to studies of molecular static and dynamic properties as well as of total electronic energies. Calculations are presented of the bond lengths and the harmonic and anharmonic vibrational frequencies of Au 2 , Hg 2 2+ , HgAu + , HgPt, and AuH. All calculations are based on the full four-component Dirac–Coulomb Hamiltonian, employing nonrelativistic local, gradient-corrected, and hybrid density functionals. The relevance of the Coulomb and Breit operators for the construc- tion of relativistic functionals is discussed; it is argued that, at the relativistic level of density-functional theory and in the absence of a vector potential, the neglect of current functionals follows from the neglect of the Breit operator. © 2002 Wiley Periodicals, Inc. J Comput Chem 23: 814 – 823, 2002 Key words: Kohn–Sham theory; four-component relativistic implementation Introduction The universal acceptance of the periodic table towards the end of the 19th century was in large part due to Mendeleevs successful prediction 1 of the properties of gallium, scandium, and germani- um— elements unknown in 1871. However, if the vacant slots had been located in the lower part of the periodic table, the predictions of Mendeleev are likely to have been less convincing. The prime origin of the deviation from the periodic behavior for the heavier elements is relativity. 2,3 Indeed, over the past 30 years, the inclu- sion of the effects of relativity has become an essential ingredient in theoretical studies of heavy-element chemistry. However, in the high-Z regime, the effects of electron corre- lation can be just as important as those of relativity, making it important to develop efficient methods for the simultaneous treat- ment of correlation and relativity. The density-functional theory (DFT) has shown itself very capable of treating electron correla- tion at a reasonable computational cost. 4 Relativistic DFT has been formulated within the framework of quantum electrodynamics, where the renormalization procedure provides a minimum princi- ple that has made possible the relativistic extension 5 of the Ho- henberg–Kohn theorem. 6 Subsequently, relativistic versions of the Kohn–Sham equations have been developed independently by Rajagopal 7 and by MacDonald and Vosko. 8 Recent reviews of relativistic DFT have been given by Engel and Dreizler 9 and by Engel et al. 10 The combination of DFT and relativistic treatments has now found its way into many quantum-chemistry software packages, mostly within the framework of one- and two-component schemes, 11 although some four-component implementations have also been reported (see refs. 12, 13 , and 14 , and references therein). In this article, we report the implementation of four-component relativistic DFT based on nonrelativistic functionals in the molec- ular code DIRAC. 15 Among the current four-component relativistic molecular programs, DIRAC has the greatest functionality. It is hoped that the inclusion of DFT in DIRAC can facilitate the exten- sion of the four-component relativistic Kohn–Sham method to properties beyond spectroscopic constants, such as excitation en- ergies and various electric and magnetic properties. Theory The theory part consists of four sections. We first discuss the structure of the molecular relativistic Hamiltonian in the next subsection, emphasizing those aspects that are relevant for DFT, in particular the significance of the Coulomb and Gaunt operators. Next, we consider the Kohn–Sham energy and its parameterization in terms of a set of nonredundant variational parameters. A general technique for expanding the Kohn–Sham energy in the variational parameters is then presented and applied to first order. Finally, we discuss some aspects related to the optimization of the Kohn– Sham energy. Correspondence to: T. Helgaker; e-mail: trygve.helgaker@kjemi.uio.no © 2002 Wiley Periodicals, Inc.