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Using JCP Format View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by DSpace at VU JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 19 15 MAY 2000 Density functional calculations of nuclear quadrupole coupling constants in the zero-order regular approximation for relativistic effects Erik van Lenthe and Evert Jan Baerends Afdeling Theoretische Chemie, Vrije Universiteit, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands ͑Received 29 November 1999; accepted 10 February 2000͒ The zeroth-order regular approximation ͑ZORA͒ is used for the evaluation of the electric field gradient, and hence nuclear quadrupole coupling constants, in some closed shell molecules. It is shown that for valence orbitals the ZORA-4 electron density, which includes a small component density ͑‘‘picture-change correction’’͒, very accurately agrees with the Dirac electron density. For hydrogen-like atoms exact relations between the ZORA-4 and Dirac formalism are given for the calculation of the electric field gradient. Density functional ͑DFT͒ calculations of the electric field gradients for a number of diatomic halides at the halogen nuclei Cl, Br, and I and at the metallic nuclei Al, Ga, In, Th, Cu, and Ag are presented. Scalar relativistic effects, spin–orbit effects, and the effects of picture-change correction, which introduces the small component density, are discussed. The results for the thallium halides show a large effect of spin–orbit coupling. Our ZORA-4 DFT calculations suggest adjustment of some of the nuclear quadrupole moments to Q(79Br)ϭ0.30(1) barn, Q(127I)ϭϪ0.69(3) barn, and Q(115In)ϭ0.74(3) barn, which should be checked by future highly correlated ab initio relativistic calculations. In the copper and silver halides the results with the used gradient corrected density functional are not in good agreement with experiment. © 2000 American Institute of Physics. ͓S0021-9606͑00͒30517-7͔ I. INTRODUCTION An alternative to ab initio calculations is the use of den- sity functional theory ͑DFT͒, since for many properties it can 1 In a recent article the zeroth-order regular approxima- provide accurate results at a low computational cost. In Ref. 2–5 tion ͑ZORA͒ to the Dirac equation was used for the cal- 17, for example, results of scalar relativistic DFT calcula- culation of the magnetic dipole hyperfine interaction, which tions of the EFG at iron in various solids were compared ͑ ͒ is the interaction between the effective electronic spin of a with Mo¨ssbauer spectroscopic data for the NQCC to obtain paramagnetic molecule of interest and a magnetic nucleus in the NQM of 57Fe. On the other hand, recently Schwerdtfeger the molecule. In this article we will consider the electric et al.18 questioned the use of DFT for the calculation of quadrupole hyperfine interaction, which is the electrostatic EFGs in transition metal compounds, since they found a poor interaction between an electric quadrupole of a nucleus and performance of many of the presently used functionals for all other charges in the compound. This interaction can lead the Cu EFG in CuCl. We will test one of those functionals in to splitting of lines in spectroscopic studies and the measured our calculations on a number of closed shell diatomic mol- splittings are often reported as the nuclear quadrupole cou- ecules. We use the ͑nonrelativistic͒ local density functional pling constants ͑NQCC͒. Such a NQCC, which can be mea- ͑LDA͒ with gradient correction ͑GGC͒ terms added, namely sured for example with microwave and nuclear quadrupole the Becke correction for exchange ͑Becke88͒,19 and the Per- resonance spectroscopy, is proportional to the electric field dew correction for correlation.20 gradient ͑EFG͒ at the nucleus and the electric nuclear quad- rupole moment ͑NQM͒ of that nucleus. The EFG, which Recently DFT was also used for the calculation of the gives valuable information about the electron distribution EFG at iron in some iron porphyrins and other molecules in surrounding the nucleus, is the property that we will calcu- Refs. 21, 22, for a comparison with Mo¨ssbauer spectroscopic late in this article. data, and in Ref. 23, nonrelativistic DFT calculations were One of the most accurate ways to determine the NQM of performed of the EFG at iodine in some iodine compounds. a certain nucleus is to combine the calculation of the EFG at In fact there are many articles with results of nonrelativistic that nucleus with the measured NQCC.6 Highly correlated ab DFT and nonrelativistic ab initio calculations of EFGs in initio calculations can give accurate EFGs for open shell at- molecules, but only few with results from fully relativistic oms or small closed shell molecules, see for example Refs. calculations. 7–12. In these references relativistic effects in the molecules An alternative to such fully relativistic calculations in were often approximated with the Douglas–Kroll–Hess molecules can be the use of approximate relativistic meth- Hamiltonian.13,14 Fully relativistic all-electron ͑correlated͒ ods, like the mentioned Douglas–Kroll–Hess method or the ab initio calculations of EFGs start to appear,15,16 but they ZORA method, which is used in this article. These are both are still computationally demanding even for small mol- two-component relativistic methods, for which it is impor- ecules if they contain heavy elements. tant to include picture-change effects when evaluating expec- 0021-9606/2000/112(19)/8279/14/$17.008279 © 2000 American Institute of Physics Downloaded 13 Mar 2011 to 130.37.129.78. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 8280 J. Chem. Phys., Vol. 112, No. 19, 15 May 2000 E. van Lenthe and E. J. Baerends 24,25 tation values. We will discuss such effects in the ZORA c2 ␳SRϭ͑␾SR͒†␾SRϩ ͑p␾SR͒† p␾SR . ͑5͒ case. It will be demonstrated that the proper evaluation of the i i i ͑2c2ϩ⑀SRϪV͒2 i • i EFG in the ZORA method requires the introduction of the i density from the small components, leading to what we call This is not the only possible scalar relativistic equation, as the ZORA-4 density. The precise relation of the use of the was discussed in Ref. 31, where several scalar relativistic ZORA-4 density with the picture-change correction to order equations are compared. cϪ2 will be explicitly discussed. The ͑SR͒ ZORA equation is the zeroth-order of the regu- Several other technical aspects of the calculation of lar expansion of the ͑SR͒ relativistic equation, EFGs will also be considered. In view of the heavy weight- HZORA⌿ ϭ͑VϩT͓V͔͒⌿ ϭ⑀ ⌿ , ͑6͒ ing of the near-nuclear region by the EFG operator, due to its SR i i i i 1/r3 behavior, polarization of the core, even if only slightly, with may lead to nonnegligible effects. The possibility of a frozen c2 core treatment therefore needs to be investigated. A related ZORA͓ ͔ϭ␴ ␴ T V p 2Ϫ p issue is the quality of the basis sets that are required, in • 2c V • particular in the core region. It should be possible to describe c2 c2 the core-orthogonality wiggles of the valence orbitals accu- ϭp pϩ ␴ ͑ Vϫp͒, ͑7a͒ • 2c2ϪV ͑2c2ϪV͒2 • “ rately with the basis set used. We will also discuss separately the scalar relativistic and spin–orbit effects on the calculated c2 TZORA͓V͔ϭp p. ͑7b͒ EFGs. SR • 2c2ϪV We calculate the EFG at the position of the halogen nuclei in the hydrogen halides, the interhalogens and in some The effective molecular Kohn–Sham potential V used in metal halides, where the metals are aluminum, gallium, in- our calculations is the sum of the nuclear potential, the Cou- dium, thallium, copper, and silver. For these diatomic halides lomb potential due to the total electron density and the we also calculated the EFG at aluminum, gallium, and in- exchange-correlation potential, for which we will use nonrel- dium. Some of these molecules were also discussed in recent ativistic approximations. The ZORA kinetic energy operator reviews by Palmer et al.26–28 on experimentally observed TZORA depends on the molecular Kohn–Sham potential. The ͑ ͒ ZORA halogen nuclear quadrupole coupling constants and ab initio scalar relativistic SR ZORA kinetic energy operator TSR calculations on a whole range of molecules. is the ZORA kinetic energy operator without spin–orbit cou- pling. For convenience we will refer to the ͑SR͒ ZORA ki- netic energy as T͑SR͓͒V͔. II. THE ZORA EQUATION AND ELECTROSTATIC An improved one-electron energy can be obtained by 32 PERTURBATION using the scaled ZORA energy expression ⑀ If only a time-independent electric field is present, the ⑀scaledϭ i ͑ ͒ i ϩ ⌿ ͉ ͉⌿ , 8 one-electron Dirac Kohn–Sham equations can be written in 1 ͗ i Q͓V͔ i͘ ϭϪ atomic units (p i“), as an equation for the large compo- with nent which after elimination of the small component ͑esc͒, reads c2 ZORA͓ ͔ϭ␴ ␴ ͑ ͒ Q V •p ͑ 2Ϫ ͒2 •p, 9a c2 2c V Hesc␾Dϭͩ Vϩ␴ p ␴ pͪ ␾Dϭ⑀D␾D , ͑1͒ i • 2c2ϩ⑀DϪV • i i i c2 i ZORA͓ ͔ϭ ͑ ͒ QSR V p 2 2 p. 9b and a companion equation which generates the small com- • ͑2c ϪV͒ ponent from the large component The scaled ZORA method is the basis of ͑bond͒ energy evaluations, since it remedies the gauge dependency problem D c D ␹ ϭ ␴ p␾ . ͑2͒ of the unscaled ZORA method, see the discussion in Ref. 32. i 2c2ϩ⑀DϪV • i i Let us now consider the effect of a small electric field The normalization is such that the four-component Dirac described by the perturbing potential VЈ(r), such as the po- ␳D electron density i , tential due to a nuclear quadrupole.
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