An indirect approach to the determination of the nuclear quadrupole moment by four-component relativistic DFT in molecular calculations

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Abstract

Commonly used exchange-correlations functionals are known to produce inaccurate electric field gradient (EFG) values at the nuclei of transition metals and heavy atoms in molecular calculations. This makes density functional theory (DFT) essentially inapplicable for the determination of nuclear quadrupole moments (NQM) from absolute EFG estimates. However, in a recently proposed indirect approach, the NQM is determined from the changes in the EFG along a series of molecules. We investigate this indirect approach within four-component relativistic DFT, showing that, at least in a series of chemically strictly related molecules, EFG variations can be computed quite accurately. This leads to surprisingly stable and reliable estimates of the NQM, even in notoriously ‘difficult’ cases such as 197Au.

Graphical abstract

Commonly used exchange-correlations functionals are known to produce inaccurate electric field gradient (EFG) values at the nuclei of transition metals and heavy atoms in molecular calculations. This makes density functional theory (DFT) essentially inapplicable for the determination of nuclear quadrupole moments (NQM) from absolute EFG estimates. However, in a recently proposed indirect approach, the NQM is determined from the changes in the EFG along a series of molecules. We investigate this indirect approach within four-component relativistic DFT, showing that, at least in a series of chemically strictly related molecules, EFG variations can be computed quite accurately. This leads to surprisingly stable and reliable estimates of the NQM, even in notoriously ‘difficult’ cases such as 197Au.

Introduction

The most popular method to determine the nuclear quadrupole moment (NQM) Q(X) of an atom X is based on the combination of a theoretical prediction of the electric field gradient (EFG) at the nucleus of interest with the experimental determination of the nuclear quadrupole coupling constant (NQCC) obtained by high resolution spectroscopy [1], [2], [3]. The relevant formula isQ(X)=kνQ(X)q(X),where νQ(X) is the NQCC in MHz, q(X) is the EFG at the nucleus X in au and k = 1/0.2349647 is the conversion factor to obtain Q(X) in millibarns (mb) (1 barn = 10−28 m2). If accurate NQCCs are available, Eq. (1) shows that the accuracy of the NQM obtained is determined entirely by the quality of the calculation of the EFG. Vice versa, if accurate NQMs are available for a nucleus, the NQCCs may be predicted on the basis of EFG estimates.

Eq. (1) has been applied extensively to light and moderately heavy atoms and small molecules where the EFG can be calculated to high precision using correlated ab initio methods that may include also some estimate of relativistic effects. The computation of accurate EFGs for heavy elements is by far not as straightforward, because of the substantial increase of electron correlation and relativistic effects. One of the most effective methods to introduce relativistic effects, often essential also for the EFG of relatively light atoms, and avoiding picture change effects, is the four-component Dirac–Coulomb (DC) method, followed by coupled-cluster (CC) calculations to introduce dynamic correlation effects. In all cases, obtaining accurate EFGs requires special care in the selection of appropriate atomic basis sets, affording, in particular, a good description of the atomic core region. Convergence of the results with respect to the basis should be assessed, but this task rapidly goes beyond feasibility for many-electron systems.

The possibility of using computationally less expensive procedures to compute the EFG, like density functional theory (DFT) would be highly desirable since, for large molecules, wavefunction based methods are computationally too demanding if electron correlation has to be taken into account. The performance of DFT in the determination of the EFG has been investigated in several papers. DFT performs well for EFG calculations of compounds containing main group elements, but commonly used density functionals contain systematic errors for transition metal complexes like for instance copper [4], [5] and silver [6] compounds. Similar and arguably related problems are encountered in the DFT determination of electric dipole moments of molecules containing transition metals [4]. Recent four-component relativistic DFT calculations [7] for the series of AuX molecules (X = F, Cl, Br, I) have indeed yielded unreliable results for the NQM of the 197Au nucleus.

Very recently, we proposed [8] a simple and very useful extension of Eq. (1) for the determination of the NQM via EFG calculations, which one can adopt in molecular calculations. When computing the nuclear EFG by calculations on several different molecules, it is more effective, rather than using the absolute value of the EFG q(X) in Eq. (1) (which we shall refer to as the ‘direct’ method), to concentrate only on its variation along the molecular series, due to changes in the chemical environment of the atom of interest. This leads immediately to the use of the alternative ‘indirect’ formulaQ(X)=kΔνQ(X)Δq(X)employing the shifts ΔνQ(X) and Δq(X) in the NQCC and EFG, respectively, with respect to a reference molecule. This procedure resembles the calculation of chemical shifts instead of absolute shieldings in the analysis of Nuclear Magnetic Resonance spectroscopy.

On the basis of the indirect formula, in combination with highly correlated DC-CCSD-T relativistic molecular calculations, including also an estimate of the Breit interaction and an extensive basis set study, we have been able to arrive at a new reliable estimate of the NQM of 197Au [8] of 510 ± 15 mb. We have shown that the indirect approach has the useful advantage of minimizing the impact of systematic errors in the evaluation of the EFG, because only differences in the EFG of various molecules enter the equation, thus speeding up the convergence of the results with respect to the basis set.

In the light of these features of the indirect formula, and of the mentioned advantages and shortcomings of DFT calculations of the EFG, it appears of great interest to study the performance of Eq. (2) also in the context of DFT. This is the subject of the present letter and we report the results of extensive four-component relativistic DFT calculations of the NQM of 197Au in a series of molecular systems, using several commonly available exchange-correlation functionals. We performed calculations on six linear gold molecules, namely AuF and its complexes XeAuF, KrAuF, ArAuF, (OC)AuF, plus AuH. The mentioned recent completion of our extensive ab initio study on the same molecules, using highly correlated methods and large basis sets, [8] gives us a timely and appropriate context for comparison and assessment of the present results.

Section snippets

Computational details

The calculations have been carried out within the framework of relativistic four-component Kramers-restricted DFT using the Dirac–Coulomb–Hamiltonian within the electronic structure code DIRAC [9], [10]. Relativistic corrections to the exchange-correlation functionals have not been considered here. We used two generalized gradient approximations (GGAs) of Becke–Lee–Yang–Parr (BLYP) [11], [12] and Perdew–Wang (PW91) [13] and the hybrid functionals B3LYP [14] and BHHLYP [12], [14]. Validation

Results and discussion

In Table 1, we report the EFG values obtained in this work and compare them with the recently presented values determined at the DC-CCSD-T level [8]. For the sake of completeness we include also the DC-HF and DC-MP2 values from the same previous work. Contrary to other transition metal compounds reported in the literature [5], for all molecules studied here, electron correlation gives a significant contribution, positive in all cases, to the EFGs. This makes these systems particularly

Concluding remarks

In conclusion, the present work has confirmed that the electric field gradient at heavy nuclei is a very sensitive probe of the accuracy of electron density calculations in molecules and the results obtained with standard DFT functionals for absolute values of the EFG are extremely variable and unreliable. However, the shift in the EFG of gold in AuF upon polarization by noble gas atoms and CO is captured very well by all functionals considered in this work. On the other hand, DFT fails in

Acknowledgements

L. B. thanks the Project HPC-EUROPA (RII3-CT-2003-506079), with the support of the European Community-Research Infrastructure Action under the FP6 ‘Structuring the European research Area’ Program. This work has been partially supported by FIRB 2003 ‘Molecular compounds and hybrid nanostructured materials with resonant and non-resonant optical properties for photonic devices’. A.W.G. and L.V. thank The Netherlands Organization for Scientific Research for financial support via the ‘Vici

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