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Rusakova IL, Rusakov YY. On the Utmost Importance of the Basis Set Choice for the Calculations of the Relativistic Corrections to NMR Shielding Constants. Int J Mol Sci 2023; 24:6231. [PMID: 37047204 PMCID: PMC10094374 DOI: 10.3390/ijms24076231] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2023] [Revised: 03/22/2023] [Accepted: 03/23/2023] [Indexed: 03/29/2023] Open
Abstract
The investigation of the sensitivity of the relativistic corrections to the NMR shielding constants (σ) to the configuration of angular spaces of the basis sets used on the atoms of interest was carried out within the four-component density functional theory (DFT). Both types of relativistic effects were considered, namely the so-called heavy atom on light atom and heavy atom on heavy atom effects, though the main attention was paid to the former. As a main result, it was found that the dependence of the relativistic corrections to σ of light nuclei (exemplified here by 1H and 13C) located in close vicinity to a heavy atom (exemplified here by In, Sn, Sb, Te, and I) on the basis set used on the light spectator atom was very much in common with that of the Fermi-contact contribution to the corresponding nonrelativistic spin-spin coupling constant (J). In general, it has been shown that the nonrelativistic J-oriented and σ-oriented basis sets, artificially saturated in the tight s-region, provided much better accuracy than the standard nonrelativistic σ-oriented basis sets when calculating the relativistic corrections to the NMR shielding constants of light nuclei at the relativistic four-component level of the DFT theory.
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Affiliation(s)
- Irina L. Rusakova
- A. E. Favorsky Irkutsk Institute of Chemistry, Siberian Branch of the Russian Academy of Sciences, Favorsky St. 1, 664033 Irkutsk, Russia;
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Rusakova IL, Rusakov YY, Krivdin LB. Computational 199 Hg NMR. MAGNETIC RESONANCE IN CHEMISTRY : MRC 2022; 60:929-953. [PMID: 35737297 DOI: 10.1002/mrc.5296] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/14/2022] [Revised: 06/20/2022] [Accepted: 06/21/2022] [Indexed: 06/15/2023]
Abstract
Theoretical background and fundamental results dealing with the computation of mercury chemical shifts and spin-spin coupling constants are reviewed with a special emphasis on their stereochemical behavior and applications.
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Affiliation(s)
- Irina L Rusakova
- A.E. Favorsky Irkutsk Institute of Chemistry, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia
| | - Yuriy Yu Rusakov
- A.E. Favorsky Irkutsk Institute of Chemistry, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia
| | - Leonid B Krivdin
- A.E. Favorsky Irkutsk Institute of Chemistry, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia
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Quantum Chemical Approaches to the Calculation of NMR Parameters: From Fundamentals to Recent Advances. MAGNETOCHEMISTRY 2022. [DOI: 10.3390/magnetochemistry8050050] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
Abstract
Quantum chemical methods for the calculation of indirect NMR spin–spin coupling constants and chemical shifts are always in progress. They never stay the same due to permanently developing computational facilities, which open new perspectives and create new challenges every now and then. This review starts from the fundamentals of the nonrelativistic and relativistic theory of nuclear magnetic resonance parameters, and gradually moves towards the discussion of the most popular common and newly developed methodologies for quantum chemical modeling of NMR spectra.
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Rusakova IL, Rusakov YY. Quantum chemical calculations of 77 Se and 125 Te nuclear magnetic resonance spectral parameters and their structural applications. MAGNETIC RESONANCE IN CHEMISTRY : MRC 2021; 59:359-407. [PMID: 33095923 DOI: 10.1002/mrc.5111] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2020] [Revised: 09/01/2020] [Accepted: 10/07/2020] [Indexed: 06/11/2023]
Abstract
An accurate quantum chemical (QC) modeling of 77 Se and 125 Te nuclear magnetic resonance (NMR) spectra is deeply involved in the NMR structural assignment for selenium and tellurium compounds that are of utmost importance both in organic and inorganic chemistry nowadays due to their huge application potential in many fields, like biology, medicine, and metallurgy. The main interest of this review is focused on the progress in QC computations of 77 Se and 125 Te NMR chemical shifts and indirect spin-spin coupling constants involving these nuclei. Different computational methodologies that have been used to simulate the NMR spectra of selenium and tellurium compounds since the middle of the 1990s are discussed with a strong emphasis on their accuracy. A special accent is placed on the calculations resorting to the relativistic methodologies, because taking into account the relativistic effects appreciably influences the precision of NMR calculations of selenium and, especially, tellurium compounds. Stereochemical applications of quantum chemical calculations of 77 Se and 125 Te NMR parameters are discussed so as to exemplify the importance of integrated approach of experimental and computational NMR techniques.
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Affiliation(s)
- Irina L Rusakova
- A. E. Favorsky Irkutsk Institute of Chemistry, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russian Federation
| | - Yuriy Yu Rusakov
- A. E. Favorsky Irkutsk Institute of Chemistry, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russian Federation
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Liu W. Erratum: “Essentials of relativistic quantum chemistry” [J. Chem. Phys. 152, 180901 (2020)]. J Chem Phys 2020; 152:249901. [DOI: 10.1063/5.0015698] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Wenjian Liu
- Qingdao Institute for Theoretical and Computational Sciences, Shandong University, Qingdao, Shandong 266237, People’s Republic of China
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Zhang T, Kasper JM, Li X. Localized relativistic two-component methods for ground and excited state calculations. ACTA ACUST UNITED AC 2020. [DOI: 10.1016/bs.arcc.2020.07.005] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/19/2023]
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Mühlbach AH, Reiher M. Quantum system partitioning at the single-particle level. J Chem Phys 2018; 149:184104. [DOI: 10.1063/1.5055942] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Adrian H. Mühlbach
- Laboratorium für Physikalische Chemie, ETH Zürich, Vladimir-Prelog-Weg 2, CH-8093 Zürich, Switzerland
| | - Markus Reiher
- Laboratorium für Physikalische Chemie, ETH Zürich, Vladimir-Prelog-Weg 2, CH-8093 Zürich, Switzerland
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Yoshizawa T, Zou W, Cremer D. Calculations of atomic magnetic nuclear shielding constants based on the two-component normalized elimination of the small component method. J Chem Phys 2017; 146:134109. [DOI: 10.1063/1.4979499] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Terutaka Yoshizawa
- Computational and Theoretical Chemistry Group (CATCO), Department of Chemistry, Southern Methodist University, 3215 Daniel Ave, Dallas, Texas 75275-0314, USA
| | - Wenli Zou
- Computational and Theoretical Chemistry Group (CATCO), Department of Chemistry, Southern Methodist University, 3215 Daniel Ave, Dallas, Texas 75275-0314, USA
| | - Dieter Cremer
- Computational and Theoretical Chemistry Group (CATCO), Department of Chemistry, Southern Methodist University, 3215 Daniel Ave, Dallas, Texas 75275-0314, USA
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Autschbach J. Relativistic calculations of magnetic resonance parameters: background and some recent developments. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2014; 372:20120489. [PMID: 24516182 DOI: 10.1098/rsta.2012.0489] [Citation(s) in RCA: 51] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
This article outlines some basic concepts of relativistic quantum chemistry and recent developments of relativistic methods for the calculation of the molecular properties that define the basic parameters of magnetic resonance spectroscopic techniques, i.e. nuclear magnetic resonance shielding, indirect nuclear spin-spin coupling and electric field gradients (nuclear quadrupole coupling), as well as with electron paramagnetic resonance g-factors and electron-nucleus hyperfine coupling. Density functional theory (DFT) has been very successful in molecular property calculations, despite a number of problems related to approximations in the functionals. In particular, for heavy-element systems, the large electron count and the need for a relativistic treatment often render the application of correlated wave function ab initio methods impracticable. Selected applications of DFT in relativistic calculation of magnetic resonance parameters are reviewed.
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Affiliation(s)
- Jochen Autschbach
- Department of Chemistry, State University of New York at Buffalo, , Buffalo, NY 14260-3000, USA
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Cremer D, Zou W, Filatov M. Dirac‐exact relativistic methods: the normalized elimination of the small component method. WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL MOLECULAR SCIENCE 2014. [DOI: 10.1002/wcms.1181] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Dieter Cremer
- Computational and Theoretical Chemistry Group (CATCO), Department of ChemistrySouthern Methodist UniversityDallasTXUSA
| | - Wenli Zou
- Computational and Theoretical Chemistry Group (CATCO), Department of ChemistrySouthern Methodist UniversityDallasTXUSA
| | - Michael Filatov
- Computational and Theoretical Chemistry Group (CATCO), Department of ChemistrySouthern Methodist UniversityDallasTXUSA
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Peng D, Middendorf N, Weigend F, Reiher M. An efficient implementation of two-component relativistic exact-decoupling methods for large molecules. J Chem Phys 2013; 138:184105. [PMID: 23676027 DOI: 10.1063/1.4803693] [Citation(s) in RCA: 141] [Impact Index Per Article: 12.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022] Open
Abstract
We present an efficient algorithm for one- and two-component relativistic exact-decoupling calculations. Spin-orbit coupling is thus taken into account for the evaluation of relativistically transformed (one-electron) Hamiltonian. As the relativistic decoupling transformation has to be evaluated with primitive functions, the construction of the relativistic one-electron Hamiltonian becomes the bottleneck of the whole calculation for large molecules. For the established exact-decoupling protocols, a minimal matrix operation count is established and discussed in detail. Furthermore, we apply our recently developed local DLU scheme [D. Peng and M. Reiher, J. Chem. Phys. 136, 244108 (2012)] to accelerate this step. With our new implementation two-component relativistic density functional calculations can be performed invoking the resolution-of-identity density-fitting approximation and (Abelian as well as non-Abelian) point group symmetry to accelerate both the exact-decoupling and the two-electron part. The capability of our implementation is illustrated at the example of silver clusters with up to 309 atoms, for which the cohesive energy is calculated and extrapolated to the bulk.
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Affiliation(s)
- Daoling Peng
- ETH Zurich, Laboratorium für Physikalische Chemie, Wolfgang-Pauli-Str. 10, CH-8093 Zurich, Switzerland
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Malček M, Bučinský L, Biskupič S, Jayatilaka D. The quasirelativistic contact interaction and effective electron and spin densities at the nucleus: A model based on weighting the electron density with the finite Gaussian nucleus model. Chem Phys Lett 2013. [DOI: 10.1016/j.cplett.2013.06.039] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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Autschbach J, Peng D, Reiher M. Two-Component Relativistic Calculations of Electric-Field Gradients Using Exact Decoupling Methods: Spin–orbit and Picture-Change Effects. J Chem Theory Comput 2012; 8:4239-48. [DOI: 10.1021/ct300623j] [Citation(s) in RCA: 53] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/04/2023]
Affiliation(s)
- Jochen Autschbach
- Department of Chemistry, University
at Buffalo, State University of New York, Buffalo, New York 14260-3000, United States
| | - Daoling Peng
- ETH Zürich, Laboratorium für Physikalische Chemie, Wolfgang-Pauli-Strasse
10, CH-8093 Zürich, Switzerland
| | - Markus Reiher
- ETH Zürich, Laboratorium für Physikalische Chemie, Wolfgang-Pauli-Strasse
10, CH-8093 Zürich, Switzerland
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Zou W, Filatov M, Cremer D. An improved algorithm for the normalized elimination of the small-component method. Theor Chem Acc 2011. [DOI: 10.1007/s00214-011-1007-8] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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17
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Reiher M. Relativistic Douglas–Kroll–Hess theory. WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL MOLECULAR SCIENCE 2011. [DOI: 10.1002/wcms.67] [Citation(s) in RCA: 97] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/01/2023]
Affiliation(s)
- Markus Reiher
- Laboratorium für Physikalische Chemie, ETH Zurich, Zurich, Switzerland
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Nataraj HS, Kállay M, Visscher L. General implementation of the relativistic coupled-cluster method. J Chem Phys 2010; 133:234109. [DOI: 10.1063/1.3518712] [Citation(s) in RCA: 43] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Sikkema J, Visscher L, Saue T, Iliaš M. The molecular mean-field approach for correlated relativistic calculations. J Chem Phys 2009; 131:124116. [DOI: 10.1063/1.3239505] [Citation(s) in RCA: 160] [Impact Index Per Article: 10.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Kutzelnigg W, Liu W. Relativistic theory of nuclear magnetic resonance parameters in a Gaussian basis representation. J Chem Phys 2009; 131:044129. [DOI: 10.1063/1.3185400] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023] Open
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Abstract
The Dirac operator in a matrix representation in a kinetically balanced basis is transformed to the matrix representation of a quasirelativistic Hamiltonian that has the same electronic eigenstates as the original Dirac matrix (but no positronic eigenstates). This transformation involves a matrix X, for which an exact identity is derived and which can be constructed either in a noniterative way or by various iteration schemes, not requiring an expansion parameter. Both linearly convergent and quadratically convergent iteration schemes are discussed and compared numerically. The authors present three rather different schemes, for each of which even in unfavorable cases convergence is reached within three or four iterations, for all electronic eigenstates of the Dirac operator. The authors present the theory both in terms of a non-Hermitian and a Hermitian quasirelativistic Hamiltonian. Quasirelativistic approaches at the matrix level known from the literature are critically analyzed in the frame of the general theory.
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Affiliation(s)
- Wenjian Liu
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, Peking University, Beijing 100871, People's Republic of China
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Liu W, Peng D. Infinite-order quasirelativistic density functional method based on the exact matrix quasirelativistic theory. J Chem Phys 2006; 125:44102. [PMID: 16942129 DOI: 10.1063/1.2222365] [Citation(s) in RCA: 182] [Impact Index Per Article: 10.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The exact one-electron matrix quasirelativistic theory [Kutzelnigg and Liu, J. Chem. Phys. 123, 241102 (2005)] is extended to the effective one-particle Kohn-Sham scheme of density functional theory. Several variants of the resultant theory are discussed. Although they are in principle equivalent, consideration of computational efficiency strongly favors the one (F(+)) in which the effective potential remains untransformed. Further combined with the atomic approximation for the matrix X relating the small and large components of the Dirac spinors as well as a simple ansatz for correcting the two-electron picture change errors, a very elegant, accurate, and efficient infinite-order quasirelativistic approach is obtained, which is far simpler than all existing quasirelativistic theories and must hence be regarded as a breakthrough in relativistic quantum chemistry. In passing, it is also shown that the Dirac-Kohn-Sham scheme can be made as efficient as two-component approaches without compromising the accuracy. To demonstrate the performance of the new methods, atomic calculations on Hg and E117 are first carried out. The spectroscopic constants (bond length, vibrational frequency, and dissociation energy) of E117(2) are then reported. All the results are in excellent agreement with those of the Dirac-Kohn-Sham calculations.
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Affiliation(s)
- Wenjian Liu
- Beijing National Laboratory for Molecular Sciences, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, People's Republic of China.
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Kutzelnigg W, Liu* W. Quasirelativistic theory I. Theory in terms of a quasi-relativistic operator. Mol Phys 2006. [DOI: 10.1080/00268970600662481] [Citation(s) in RCA: 78] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Hu XQ, Wang F, Li LM. An approach to study the relativistic effects of large systems with some parts containing heavy elements. Chem Phys 2005. [DOI: 10.1016/j.chemphys.2004.09.024] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Peralta JE, Uddin J, Scuseria GE. Scalar relativistic all-electron density functional calculations on periodic systems. J Chem Phys 2005; 122:84108. [PMID: 15836021 DOI: 10.1063/1.1851973] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Scalar relativistic effects are included in periodic boundary conditions calculations with Gaussian orbitals. This approach is based on the third-order Douglas-Kroll-Hess approximation, allowing the treatment of all electrons on an equal footing. With this methodology, we are able to perform relativistic all-electron density functional calculations using the traditional local spin-density and generalized gradient approximations (GGA), as well as meta-GGA and hybrid density functionals. We present benchmark results for the bulk metals Pd, Ag, Pt, and Au, and the large band gap semiconductors AgF and AgCl.
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Affiliation(s)
- Juan E Peralta
- Department of Chemistry, Rice University, Houston, TX 77005-1892, USA
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Filatov M, Cremer D. Connection between the regular approximation and the normalized elimination of the small component in relativistic quantum theory. J Chem Phys 2005; 122:064104. [PMID: 15740364 DOI: 10.1063/1.1844298] [Citation(s) in RCA: 45] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The regular approximation to the normalized elimination of the small component (NESC) in the modified Dirac equation has been developed and presented in matrix form. The matrix form of the infinite-order regular approximation (IORA) expressions, obtained in [Filatov and Cremer, J. Chem. Phys. 118, 6741 (2003)] using the resolution of the identity, is the exact matrix representation and corresponds to the zeroth-order regular approximation to NESC (NESC-ZORA). Because IORA (=NESC-ZORA) is a variationally stable method, it was used as a suitable starting point for the development of the second-order regular approximation to NESC (NESC-SORA). As shown for hydrogenlike ions, NESC-SORA energies are closer to the exact Dirac energies than the energies from the fifth-order Douglas-Kroll approximation, which is much more computationally demanding than NESC-SORA. For the application of IORA (=NESC-ZORA) and NESC-SORA to many-electron systems, the number of the two-electron integrals that need to be evaluated (identical to the number of the two-electron integrals of a full Dirac-Hartree-Fock calculation) was drastically reduced by using the resolution of the identity technique. An approximation was derived, which requires only the two-electron integrals of a nonrelativistic calculation. The accuracy of this approach was demonstrated for heliumlike ions. The total energy based on the approximate integrals deviates from the energy calculated with the exact integrals by less than 5 x 10(-9) hartree units. NESC-ZORA and NESC-SORA can easily be implemented in any nonrelativistic quantum chemical program. Their application is comparable in cost with that of nonrelativistic methods. The methods can be run with density functional theory and any wave function method. NESC-SORA has the advantage that it does not imply a picture change.
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Affiliation(s)
- Michael Filatov
- Department of Theoretical Chemistry, Göteborg University, Kemigården 3, S-41296 Göteborg, Sweden.
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Filatov M, Cremer D. A gauge-independent zeroth-order regular approximation to the exact relativistic Hamiltonian—Formulation and applications. J Chem Phys 2005; 122:44104. [PMID: 15740232 DOI: 10.1063/1.1839856] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A simple modification of the zeroth-order regular approximation (ZORA) in relativistic theory is suggested to suppress its erroneous gauge dependence to a high level of approximation. The method, coined gauge-independent ZORA (ZORA-GI), can be easily installed in any existing nonrelativistic quantum chemical package by programming simple one-electron matrix elements for the quasirelativistic Hamiltonian. Results of benchmark calculations obtained with ZORA-GI at the Hartree-Fock (HF) and second-order Moller-Plesset perturbation theory (MP2) level for dihalogens X(2) (X=F,Cl,Br,I,At) are in good agreement with the results of four-component relativistic calculations (HF level) and experimental data (MP2 level). ZORA-GI calculations based on MP2 or coupled-cluster theory with single and double perturbations and a perturbative inclusion of triple excitations [CCSD(T)] lead to accurate atomization energies and molecular geometries for the tetroxides of group VIII elements. With ZORA-GI/CCSD(T), an improved estimate for the atomization energy of hassium (Z=108) tetroxide is obtained.
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Affiliation(s)
- Michael Filatov
- Department of Theoretical Chemistry, Göteborg University, Kemigården 3, S-41296 Göteborg, Sweden.
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