A dual-level approach to four-component relativistic density-functional theory Wataru Mizukami a,⇑ , Takahito Nakajima b , Kimihiko Hirao c,e , Takeshi Yanai a,d a The Graduate University for Advanced Studies, Myodaiji, Okazaki, Aichi 444-8585, Japan b Computational Molecular Science Research Team, Advanced Institute for Computational Science, RIKEN, 7-1-26, Minatojima-minami, Cyuo, Kobe, Hyogo 650-0047, Japan c Next-generation Molecular Theory Unit, Advanced Science Institute, RIKEN, 2-1, Hirosawa, Wako, Saitama 351-0198, Japan d Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Okazaki, Aichi 444-8585, Japan e CREST, JST (Japan Science and Technology Agency), Saitama 332-0012, Japan article info Article history: Received 18 January 2011 In final form 6 April 2011 Available online 9 April 2011 abstract An efficient approach to the fully relativistic density-functional theory (DFT) is proposed to accelerate Dirac–Kohn–Sham calculation that uses high-quality basis sets and hybrid exchange–correlation func- tional. The dual-level approach proposed by Nakajima and Hirao underlies the present method, estimat- ing high-level four-component DFT energy perturbatively from reference density matrix, which is determined by a relatively inexpensive self-consistent calculation using low-quality basis sets and low-cost functional. A further approximation based on Infinite-Order Two-Component relativistic Ham- iltonian is incorporated into the low-level treatment. Accuracy and efficiency were examined by bench- mark calculation of spectroscopic values for MH, M 2 (M = Cu, Ag, and Au), and AtH. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction A direct approach to describing relativity in motion of electrons for quantum chemical system is to handle the molecular electronic structure with the four-component wave function as a solution of the Dirac-Coulomb equation. A great deal of effort has been directed toward developing an efficient implementation of the four-component formalism for molecules with fully relativistic many-electron theory, which accurately accounts for scalar relativ- istic effects, spin–orbit coupling, and electron correlation in a straightforward fashion. We have developed several four-compo- nent electronic structure methods, Dirac–Hartree–Fock (DHF) [1], Dirac–Kohn–Sham (DKS) [2], and the relativistic electron correla- tion methods practiced with a sophisticated four-index integral transformation algorithm [3]. They were efficiently implemented by introducing the four-component one-electron basis scheme [1] built upon the generally contracted spherically harmonic Gaussian spinors as the facile basis. For the basis, the highly optimized inte- gral algorithm was developed for the orbital calculations with the DHF and DKS methods [1,2,4]. This study presents an improvement that enhances the effi- ciency of the four-component density-functional theory (DFT) cal- culations by incorporating the dual-level approach proposed by Nakajima and Hirao [5,6] into the DKS method [2]. At the heart of the dual-level approach is that high-cost hybrid DFT calculation using high-quality basis can be reproduced from a simple pertur- bation to much lower-cost DFT calculation that uses small basis and low-level pure exchange–correlation functional. The accuracy of this approach hinges on the insight that the description of total density is insensitive to the level of calculation. Significant compu- tational saving arises in avoiding self-consistent field (SCF) proce- dure of iterating the time-consuming evaluation of Fock matrix in large basis representation as well as its large-dimensional diago- nalization. Several other developments along this direction have been reported, such as dual-basis scheme [7–21], dual-grid scheme [22], dual-functional perturbation correction [23], etc. In the dual-level approach, a pair of the basis sets and exchange– correlation functionals is defined for specifying each level of the low- and high-cost calculations. We apply this scheme to four- component DFT calculations. Although this can be approached in a rather straightforward way, we attempt to mix an extra ingredi- ent associated with the relativistic extension. In the present study, the duality of relativistic treatment is further invoked to achieve a further cost saving in the low-level calculations, for which the two-component relativistic approximation is employed. There has been considerable progress in the development of a computationally less demanding two-component Hamiltonian [24–42], which effectively neglects negative states. As the most suitable two-component method that allows for transparently bridging a gap between low- and high-level (i.e. four-component) descriptions, we employ a quasi-relativistic scheme proposed by Iliaš and Saue [41] for implementing the Infinite-Order Two- Component (IOTC) relativistic Hamiltonian derived by Barysz and Sadlej [34]. It is generated from a one-step decoupling transforma- tion starting from the Dirac operator in the finite basis representa- tion. The key reason for employing the IOTC Hamiltonian is that the decoupling matrix for the transformation is obtained as unitary. 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.04.031 ⇑ Corresponding author. E-mail address: mizukami@ims.ac.jp (W. Mizukami). Chemical Physics Letters 508 (2011) 177–181 Contents lists available at ScienceDirect Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett