^{1}, Giampaolo Barone

^{2}, Roland Lindh

^{3,a),b)}and Markus Reiher

^{4,a),c)}

### Abstract

In this work we present a comprehensive study of analytical electric field gradients in hydrogen halides calculated within the high-order Douglas-Kroll-Hess (DKH) scalar-relativistic approach taking picture-change effects analytically into account. We demonstrate the technical feasibility and reliability of a high-order DKH unitary transformation for the property integrals. The convergence behavior of the DKH property expansion is discussed close to the basis set limit and conditions ensuring picture-change-corrected results are determined. Numerical results are presented, which show that the DKH property expansion converges rapidly toward the reference values provided by four-component methods. This shows that in closed-shell cases, the scalar-relativistic DKH(2,2) approach which is of second order in the external potential for both orbitals and property operator yields a remarkable accuracy. As a parameter-dependence-free high-order DKH model, we recommend DKH(4,3). Moreover, the effect of a finite-nucleus model, different parametrization schemes for the unitary matrices, and the reliability of standard basis sets are investigated.

The authors thank the Schweizer National Fonds (Project no. 200021-113479/1) and the Swedish Research Council (VR) for financial support.

I. INTRODUCTION

II. THEORETICAL BACKGROUND

III. COMPUTATIONAL DETAILS

IV. BASIS SET CONSTRUCTION AND BASIS SET DEPENDENCE OF THE EFG

V. RESULTS AND DISCUSSION

A. Convergence of the DKH electric field gradient

1. Light homologues: HF, HCl

2. Heavy homologues: HBr, HI, HAt

B. Effects of a finite-nucleus model on the EFG

C. EFG calculated with standard basis sets

D. Dependence of the EFG on the parametrization of the unitary transformation

E. Comparison to data from the literature

VI. CONCLUSION

### Key Topics

- Basis sets
- 16.0
- Wave functions
- 14.0
- Electric fields
- 11.0
- General molecular properties
- 6.0
- Chemical properties
- 5.0

## Tables

and exponents of the decontracted basis sets used in all nonrelativistic, DKH, and four-component calculations.

and exponents of the decontracted basis sets used in all nonrelativistic, DKH, and four-component calculations.

, , and exponents of the decontracted basis sets used in all nonrelativistic, DKH, and four-component calculations.

, , and exponents of the decontracted basis sets used in all nonrelativistic, DKH, and four-component calculations.

Electronic energy and the principal component of the diagonalized electric field gradient tensor in HF (in a.u.) calculated at the F nucleus.

Electronic energy and the principal component of the diagonalized electric field gradient tensor in HF (in a.u.) calculated at the F nucleus.

Electronic energy and principal component of the diagonalized electric field gradient tensor in HCl (in a.u.) calculated at the Cl nucleus.

Electronic energy and principal component of the diagonalized electric field gradient tensor in HCl (in a.u.) calculated at the Cl nucleus.

Electronic energy and principal component of the diagonalized electric field gradient tensor in HBr (in a.u.) calculated at the Br nucleus.

Electronic energy and principal component of the diagonalized electric field gradient tensor in HBr (in a.u.) calculated at the Br nucleus.

Electronic energy and principal component of the diagonalized electric field gradient tensor in HI (in a.u.) calculated at the I nucleus.

Electronic energy and principal component of the diagonalized electric field gradient tensor in HI (in a.u.) calculated at the I nucleus.

Electronic energy and principal component of the diagonalized electric field gradient tensor in HAt (in a.u.) calculated at the At nucleus.

Electronic energy and principal component of the diagonalized electric field gradient tensor in HAt (in a.u.) calculated at the At nucleus.

Percental scalar-relativistic effect on and relative picture-change error as defined by Eq. (14).

Percental scalar-relativistic effect on and relative picture-change error as defined by Eq. (14).

Electronic energy and principal component of the diagonalized electric field gradient tensor in the halogen halide series (in a.u.) calculated with a DKH(2,2) and DKH(4,3) formalism at the halogen nucleus with standard ANO-RCC basis sets (Ref. 37). Relative differences (in %) to the DKH(2,2) and DKH(4,3) values obtained with large decontracted basis sets are given for comparison.

Electronic energy and principal component of the diagonalized electric field gradient tensor in the halogen halide series (in a.u.) calculated with a DKH(2,2) and DKH(4,3) formalism at the halogen nucleus with standard ANO-RCC basis sets (Ref. 37). Relative differences (in %) to the DKH(2,2) and DKH(4,3) values obtained with large decontracted basis sets are given for comparison.

Electronic energy and principal component of the diagonalized electric field gradient tensor in HI and HAt (in a.u.) calculated with the DKH(6,6) and DKH(5,4) protocols at the I and At nucleus, using different parametrization schemes. These schemes provide different sets of expansion coefficients given in Ref. 5. Note that the energy is classified by and hence either of sixth or fifth order. The fifth order is below the sixth order following the oscillatory convergence in Ref. 10.

Electronic energy and principal component of the diagonalized electric field gradient tensor in HI and HAt (in a.u.) calculated with the DKH(6,6) and DKH(5,4) protocols at the I and At nucleus, using different parametrization schemes. These schemes provide different sets of expansion coefficients given in Ref. 5. Note that the energy is classified by and hence either of sixth or fifth order. The fifth order is below the sixth order following the oscillatory convergence in Ref. 10.

Principal component of the diagonalized electric field gradient tensor in the hydrogen halides series HX (in a.u.) at the halogen nucleus compared to EFG values calculated by Neese *et al.* (Ref. 21), Malkin *et al.* (Ref. 20), and Visscher *et al.* (Ref. 30). Bond distances and basis sets are given for comparison. (FT)/(BT) refer to “forward transformation” and “back transformation,” respectively. The “basis” is the uncontracted basis set at the halogen atom and the bond distance (in bohrs) is denoted as . All four-component DHF and CCSD(T) [denoted as 4-c-CCSD(T)] calculations were performed with a Gaussian nuclear charge distribution, while the DKH(2,2) results were obtained with a point-charge nucleus model.

Principal component of the diagonalized electric field gradient tensor in the hydrogen halides series HX (in a.u.) at the halogen nucleus compared to EFG values calculated by Neese *et al.* (Ref. 21), Malkin *et al.* (Ref. 20), and Visscher *et al.* (Ref. 30). Bond distances and basis sets are given for comparison. (FT)/(BT) refer to “forward transformation” and “back transformation,” respectively. The “basis” is the uncontracted basis set at the halogen atom and the bond distance (in bohrs) is denoted as . All four-component DHF and CCSD(T) [denoted as 4-c-CCSD(T)] calculations were performed with a Gaussian nuclear charge distribution, while the DKH(2,2) results were obtained with a point-charge nucleus model.

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