University of Groningen An ab initio two-component relativistic method including spin- orbit coupling using the regular approximation Faas, S.; van Lenthe, J. H.; Hennum, A. C.; Snijders, J. G. Published in: Journal of Chemical Physics DOI: 10.1063/1.1288387 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2000 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Faas, S., van Lenthe, J. H., Hennum, A. C., & Snijders, J. G. (2000). An ab initio two-component relativistic method including spin- orbit coupling using the regular approximation. Journal of Chemical Physics, 113(10), 4052 - 4059. 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For technical reasons the number of authors shown on this cover page is limited to 10 maximum. An ab initio two-component relativistic method including spin–orbit coupling using the regular approximation S. Faas, J. H. van Lenthe, A. C. Hennum, and J. G. Snijders Citation: J. Chem. Phys. 113, 4052 (2000); doi: 10.1063/1.1288387 View online: https://doi.org/10.1063/1.1288387 View Table of Contents: http://aip.scitation.org/toc/jcp/113/10 Published by the American Institute of Physics Articles you may be interested in Relativistic total energy using regular approximations The Journal of Chemical Physics 101, 9783 (1994); 10.1063/1.467943 Molecular density functional calculations in the regular relativistic approximation: Method, application to coinage metal diatomics, hydrides, fluorides and chlorides, and comparison with first-order relativistic calculations The Journal of Chemical Physics 109, 392 (1998); 10.1063/1.476576 SPOCK.CI: A multireference spin-orbit configuration interaction method for large molecules The Journal of Chemical Physics 124, 124101 (2006); 10.1063/1.2173246 Quasirelativistic theory equivalent to fully relativistic theory The Journal of Chemical Physics 123, 241102 (2005); 10.1063/1.2137315 An improved ab initio relativistic zeroth-order regular approximation correct to order The Journal of Chemical Physics 113, 9957 (2000); 10.1063/1.1323266 Density-functional thermochemistry. III. The role of exact exchange The Journal of Chemical Physics 98, 5648 (1993); 10.1063/1.464913 JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 10 8 SEPTEMBER 2000 An ab initio two-component relativistic method including spin–orbit coupling using the regular approximation S. Faas, J. H. van Lenthe,a) and A. C. Hennumb) Theoretical Chemistry Group, Debye Institute, Utrecht University, Padualaan 14, 3584 CH Utrecht, The Netherlands J. G. Snijders Theoretical Chemistry, Materials Science Centre, Rijksuniversiteit Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands ͑Received 30 March 2000; accepted 16 June 2000͒ In this paper we present the implementation of the two-component scaled zeroth-order regular approximation ͑ZORA͒ method in the molecular electronic structure package GAMESS-UK.Itisthe first application of this method, which was earlier investigated in the context of density functional theory, in molecular ab initio basis set calculations. The performance of the method is tested in atomic calculations, which we can compare with numerical results, on xenon and radon and in molecular calculations on the molecules AgH, HI, I2, AuH, TlH, and Bi2. In calculations on the I2 molecule we investigated the effect of the different approaches regarding the internal Coulomb matrix used in the ZORA method. For the remaining molecules we computed harmonic frequencies and bond lengths. It is shown that the scaled ZORA approach is a cost-effective alternative to the Dirac–Fock method. © 2000 American Institute of Physics. ͓S0021-9606͑00͒32034-7͔ I. INTRODUCTION basis sets optimized originally for Dirac–Fock calculations, see, e.g., Ref. 15. The inclusion of relativistic effects is essential in the In the present paper we present the first implementation quantum chemical description of compounds containing one and applications of the two-component ZORA Hamiltonian or more heavier elements, see, e.g., Ref. 1. Highly accurate, within an ab initio basis set framework. Unlike the earlier but unfortunately extremely costly, calculations can be per- described scalar ZORA Hamiltonian,7 this Hamiltonian in- formed using the full theory of quantum electrodynamics. cludes spin–orbit coupling. We give a short review of the Due to the computational requirements, these calculations theory followed by details of the implementation of the can only be performed on small atomic systems. To be able method in the GAMESS-UK package.9 Within this implemen- to treat systems of chemical interest it is therefore necessary tation there is room for several approaches regarding the so- to make approximations. called internal Coulomb matrix. An explanation of this con- An accurate method is based on the solution of the four- cept and the different approximations used will be given. component Dirac–Fock equations, the relativistic analog of These approximations are tested in calculations on the I2 Hartree–Fock. This method is still very costly mainly be- molecule. cause of the use of large basis sets needed for a proper de- We test our implementation by calculations on the xenon scription of the small component. An attractive alternative is and radon atoms. These atomic calculations are compared to transform the four-component Dirac equation to a two-, or with numerical ZORA calculations using our ZORA imple- even one-, component formalism. The ZORA ͑zeroth-order mentation in the GRASP2 program.10 regular approximation͒ or CPD method, originally developed In molecular calculations on the diatomics I2, HI, AgH, 2,3 by Chang, Pellisier, and Durand and by Heully et al. and AuH, TlH, and Bi2 we investigate the performance of the later generalized to molecular density functional theory by ZORA Hamiltonian. The quality of the wave functions is 4,5 van Lenthe, Baerends, and Snijders and in its scalar form asserted by comparison of orbital energies (I2) with four- to ab initio methods by the present authors,6,7 performs such component Dirac–Fock results, obtained using the MOLFDIR 11 a transformation. A calculation based on the ZORA method program package. The calculations on the equilibrium ge- can therefore, in principle, be performed using only a large ometries and harmonic frequencies of the molecules HI, component basis, thereby circumventing the use of kineti- AgH, AuH, TlH and Bi2 are used to test the validity of the cally balanced basis sets. Since the ZORA orbitals resemble one-center approximation to the internal Coulomb matrix. the Dirac large components spinors,8 we can make use of Furthermore the results are compared with experiment and full Dirac–Fock results obtained with the DIRAC program.12 The comparison with scalar ZORA calculations unveils the a͒Electronic mail: [email protected] b͒Permanent address: Department of Chemistry, University of Oslo, P.O. effect of the inclusion of spin–orbit coupling in the self- Box 1033 Blindern, N-0315 Oslo, Norway. consistent field ͑SCF͒ procedure. 0021-9606/2000/113(10)/4052/8/$17.004052 © 2000 American Institute of Physics J. Chem. Phys., Vol. 113, No. 10, 8 September 2000 Two-component relativistic method 4053 II. THEORY ␳͑ Ј͒ϭ ␺†͑ ͒␺ ͑ Ј͒ ͑ ͒ r,r ͚ i r i r . 7 Using an expansion in E/(2c2ϪV) on the Foldy– iocc Wouthuysen transformed Dirac–Fock equation one can de- It implies rive the ab initio ZORA and scaled ZORA equations.6 The ϩ Ϸ ͑ ͒ equations, which still contain the large and small compo- J␾␾ J␹␹ J␺␺ . 8 nents of the Dirac spinor, read In the same spirit we assume 1 K␾␾ϷK␺␺ , FZORA␺ ϭ⑀ZORA␺ , FZORA␺ ϭ⑀ZORA ␺ , ͑1͒ i i i S i scaled,i i ͑9͒ K␾␹ ,K␹␾Ϸ0. where ␺ is the two-component ZORA orbital. The ZORA i This leads to the removal of the off-diagonal and small Fock operator is given by component Coulomb and exchange operators. The resulting 1 ZORA Fock operator is given by ZORAϭ͑ ␴ Ϫ ͒ ͑ ␴ Ϫ ͒ F c p K␾␹ 2 ␾␹ c p K␹␾ • ͑2c ϪV ϩK␹␹͒ • c2 c ZORA ␺ F ϭ␴ p ␺ ␴ pϩV ϪK␺␺ ͑10͒ ϩ ␾␹Ϫ ͑ ͒ • 2c2ϪV • c Vc K␾␾ 2 c and the scaling factor by with ␺ϭ ϩ ͑ ͒ Sϭ ϩ͗␺ ͉͑ ␴ Ϫ ͒ Vc Vnuc J␺␺ 11 1 i c •p K␾␹ and the scaling factor 1 ϫ ͑ ␴ Ϫ ͉͒␺ ͘ ͑ ͒ ͑ 2Ϫ ␾␹ϩ ͒2 c •p K␹␾ i . 3 c2 2c Vc K␹␹ Sϭ1ϩ͗␺ ͉͑␴ p͒ ␺ ͑␴ p͉͒␺ ͘. ͑12͒ i • ͑2c2ϪV ͒2 • i We use the abbreviation c ␾␹ We can now perform two-component ZORA calcula- V ϭV ϩJ␾␾ϩJ␹␹ . ͑4͒ c nuc tions using only a large component basis set. The large and small components of the original Dirac ␾ ␹ ␺ spinor are denoted by i and i . The relation between i , ␾ ␹ III. IMPLEMENTATION i , and i is determinated by the Foldy–Wouthuysen transformation6 and has become energy independent by the The two-component ZORA method is implemented in a use of the expansion in E/(2c2ϪV).