The hydration of all trivalent lanthanoid (Ln) ions is studied theoretically from two aspects: energy and wave function. With the help of the incremental scheme, for the first time the lanthanoid(III) aqua complexes are computed at the CCSD(T) level using large basis sets. These computations prove that SCS-MP2 is nearly as accurate as CCSD, thus enabling us to give the most accurate first principle hydration Gibbs free energies and reliable preferred coordination numbers (CNs) of lanthanoid(III) aqua complexes: 9, 8, and both, for light, heavy, and intermediate lanthanoids, respectively. Then a series of wave function analyses were performed to explore the deeper reasons for the preference of specific CNs. An unexpected observation is that as Ln goes from samarium to lutetium, the capping Ln–O bonds in nona-aqua lanthanoid complexes become weaker while they get shorter. Therefore, as the capping Ln–O bonds are getting easier to disrupt, heavier lanthanoids will prefer a low CN, i.e., 8. On the basis of this and previous work of other groups, a model for the water exchange kinetics of lanthanoid(III) ions is proposed. This model suggests that the capping Ln–O bonds of moderate strength, which occur for intermediate lanthanoids, are advantageous for the formation of a bicapped trigonal prism intermediate during water exchange. This explains some NMR experiments and, more importantly, an observation which puzzled investigators for a long time, i.e., that the exchange rate reaches a maximum for the middle region but is low at the beginning and end of the lanthanoid series. This nontrivial behavior of capping Ln–O bonds is interpreted and is believed to determine the hydration behavior of lanthanoid(III) ions.

AbstractDie Energietransferpfade in Lanthanoid-Antennensonden können im Rahmen der gegenwärtig verfügbaren Modelle nicht vollständig berechnet werden; ihre Aufklärung bleibt daher eine Herausforderung. Auf Grundlage quantenchemischer Ab-initio-Berechnungen typischer Eu-Antennenkomplexe wird ein neuartiges Energieresonanzmodell vorgeschlagen, das von einer umfassenden Nonett-Quintett-Intersystemkreuzung aufgrund von Spin-Bahn-Kopplung in den Unterniveaus der beteiligten Zustände gesteuert wird.

AbstractThe energy transfer pathways in lanthanide antenna probes cannot be comprehensively rationalized by the currently available models, and their elucidation remains to be a challenging task. On the basis of quantum-chemical ab initio calculations of representative europium antenna complexes, an innovative energy resonance model is proposed, which is controlled by an overall nonet–quintet intersystem crossing on the basis of spin–orbit coupling among the sublevels of the involved states.

The global optimization of molecular clusters is an important topic encountered in many fields of chemistry. In our previous work (Phys. Chem. Chem. Phys., 2015, 17, 24173), we successfully applied the recently introduced artificial bee colony (ABC) algorithm to the global optimization of atomic clusters and introduced the corresponding software “ABCluster”. In the present work, ABCluster was extended to the optimization of clusters of rigid molecules. Here “rigid” means that all internal degrees of freedom of the constituent molecules are frozen. The algorithm was benchmarked by TIP4P water clusters (H2O)N (N ≤ 20), for which all global minima were successfully located. It was further applied to various clusters of different chemical nature: 10 microhydration clusters, 4 methanol microsolvation clusters, 4 nonpolar clusters and 2 ion–aromatic clusters. In all the cases we obtained results consistent with previous experimental or theoretical studies.

The geometric and electronic structure of the recently experimentally studied molecules ZCeF2 (Z = CH2, O) was investigated by density functional theory (DFT) and wave function-based ab initio methods. Special attention was paid to the Ce–Z metal–ligand bonding, especially to the nature of the interaction between the Ce 4f and the Z 2p orbitals and the possible multiconfigurational character arising from it, as well as to the assignment of an oxidation state of Ce reflecting the electronic structure. Complete active space self-consistent field (CASSCF) calculations were performed, followed by orbital rotations in the active orbital space. The methylene compound CH2CeF2 has an open-shell singlet ground state, which is characterized by a two-configurational wave function in the basis of the strongly mixed natural CASSCF orbitals. The system can also be described in a very compact way by the dominant Ce 4f1 C 2p1 configuration, if nearly pure Ce 4f and C 2p orbitals are used. In the basis of these localized orbitals, the molecule is almost monoconfigurational and should be best described as a Ce(III) system. The singlet ground state of the oxygen OCeF2 complex is of closed-shell character when a monoconfigurational wave function with very strongly mixed Ce 4f and O 2p CASSCF natural orbitals is used for the description. The transformation to orbitals localized on the cerium and oxygen atoms leads to a multiconfigurational wave function and reveals characteristics of a mixed valent Ce(IV)/Ce(III) compound. Additionally, the interactions of the localized active orbitals were analyzed by evaluating the expectation values of the charge fluctuation operator and the local spin operator. The Ce 4f and C 2p orbital interaction of the CH2CeF2 compound is weakly covalent and resembles the interaction of the H 1s orbitals in a stretched hydrogen dimer. In contrast, the interaction of the localized active orbitals for OCeF2 shows ionic character. Calculated vibrational Ce–C and Ce–O stretching frequencies at the DFT, CASSCF, second-order Rayleigh–Schrödinger perturbation theory (RS2C), multireference configuration interaction (MRCI), as well as single, doubles, and perturbative triples coupled cluster (CCSD(T)) level are reported and compared to experimental infrared absorption data in a Ne and Ar matrix.

The third-order incremental dual-basis set zero-buffer approach (inc3-db-B0) is an efficient, accurate, and black-box quantum chemical method for obtaining correlation energies of large systems, and it has been successfully applied to many real chemical problems. In this work, we extend this approach to high-spin open-shell systems. In the open-shell approach, we will first decompose the occupied orbitals of a system into several domains by a K-means clustering algorithm. The essential part is that we preserve the active (singly occupied) orbitals in all the calculations of the domain correlation energies. The duplicated contributions of the active orbitals to the correlation energy are subtracted from the incremental expansion. All techniques of truncating the virtual space such as the B0 approximation can be applied. This open-shell inc3-db-B0 approach is combined with the CCSD and CCSD(T) methods and applied to the computations of a singlet–triplet gap and an electron detachment process. Our approach exhibits an accuracy better than 0.6 kcal/mol or 0.3 eV compared with the standard implementation, while it saves a large amount of the computational time and can be efficiently parallelized.

Global optimization of cluster geometries is of fundamental importance in chemistry and an interesting problem in applied mathematics. In this work, we introduce a relatively new swarm intelligence algorithm, i.e. the artificial bee colony (ABC) algorithm proposed in 2005, to this field. It is inspired by the foraging behavior of a bee colony, and only three parameters are needed to control it. We applied it to several potential functions of quite different nature, i.e., the Coulomb–Born–Mayer, Lennard-Jones, Morse, Z and Gupta potentials. The benchmarks reveal that for long-ranged potentials the ABC algorithm is very efficient in locating the global minimum, while for short-ranged ones it is sometimes trapped into a local minimum funnel on a potential energy surface of large clusters. We have released an efficient, user-friendly, and free program “ABCluster” to realize the ABC algorithm. It is a black-box program for non-experts as well as experts and might become a useful tool for chemists to study clusters.

For a wide range of trivalent lanthanide ion coordination complexes of tricapped trigonal prism or monocapped square antiprism configurations, the bonds between the central lanthanide ions and the capping ligands are found to violate Badger’s rule: they can get weaker as they get shorter. We demonstrate that this observation originates from the screening and repulsion effect of the prism ligands. Both effects enhance as the electric field of the central ion or the softness of the prism ligands increases. Thus, for heavier lanthanides, despite the fact that the capping bond could be shorter, it is more efficient to be weakened by the prism ligands, being inherently labile. This concept of “labile capping bonds phenomenon” is then successfully used to interpret many problems in lanthanide(III) hydration, e.g., why the water exchange rate of a lanthanide(III) complex is much higher in a twisted square antiprism than in square antiprism configuration. Thus, the theory proposed in this paper offers new insights in understanding chemical problems.

The Gibbs energies of hydration of actinoid(III) ions are evaluated for density functional optimized geometries of [An(H2O)h]3+ complexes (h = 8, 9) at the coupled cluster singles, doubles, and perturbative triples level by means of the incremental scheme. Scalar-relativistic 5f-in-core pseudopotentials for actinoids and basis sets of polarized triple-ζ quality were applied. The calculated Gibbs energies for the octa- and nona-aquo complexes agree within 1% with the experimental values which are available only for uranium and plutonium. Compared to the hydrate complexes of the lanthanoid(III) ions those of the actinoid(III) series are slightly less stable.

•The ground state wavefunction of cerocene is analyzed.•At least two configurations are needed to describe cerocene.•Cerocene can be interpreted as a Ce(IV) or Ce(III) compound.Two alternative interpretations are given for the electronic ground state of bis-η8-annulene[8] cerium, cerocene, based on the same relativistic Douglas–Kroll–Hess complete active space all-electron wavefunction. Rotations in the spaces of the one- and many-electron wavefunctions, leaving the total energy invariant, show that the system can be viewed as a complex of a closed-shell Ce(IV) ion sandwiched by two aromatic annulene[8] dianions and bonded with a significant Ce 4f – ring ππ covalency, or as a Ce(III) ion with an almost atomic-like 4f1 subconfiguration, coupled to the unpaired electron in the rings highest energy occupied ππ orbitals in a Kondo-type fashion.

An efficient way to obtain accurate CCSD and CCSD(T) energies for large systems, i.e., the third-order incremental dual-basis set zero-buffer approach (inc3-db-B0), has been developed and tested. This approach combines the powerful incremental scheme with the dual-basis set method, and along with the new proposed K-means clustering (KM) method and zero-buffer (B0) approximation, can obtain very accurate absolute and relative energies efficiently. We tested the approach for 10 systems of different chemical nature, i.e., intermolecular interactions including hydrogen bonding, dispersion interaction, and halogen bonding; an intramolecular rearrangement reaction; aliphatic and conjugated hydrocarbon chains; three compact covalent molecules; and a water cluster. The results show that the errors for relative energies are <1.94 kJ/mol (or 0.46 kcal/mol), for absolute energies of <0.0026 hartree. By parallelization, our approach can be applied to molecules of more than 30 atoms and more than 100 correlated electrons with high-quality basis set such as cc-pVDZ or cc-pVTZ, saving computational cost by a factor of more than 10–20, compared to traditional implementation. The physical reasons of the success of the inc3-db-B0 approach are also analyzed.

Segmented contracted scalar-relativistic (23s16p12d6f)/[18s12p9d3f] all-electron basis sets for lanthanides La–Lu primarily for use in second-order Douglas–Kroll–Hess density functional calculations are presented. Atomic test calculations at the scalar-relativistic Hartree–Fock level reveal an accurate description of the first to fourth ionization potentials as well as low-energy d–f and d–p excitation energies; i.e., reference data obtained with optimized (34s28p22d16f) even-tempered basis sets are reproduced with mean absolute errors of 0.003 (IP1), 0.013 (IP2), 0.030 (IP3), 0.098 (IP4), 0.070 (d–f), and 0.018 (d–p) eV. Results of molecular test calculations are presented for the lanthanide trihalides LnX3 (Ln = La–Lu, X = F, Cl, Br, I) at the PBE0 hybrid density functional theory level. Compared to recently published basis sets of identical size, the sets proposed here show substantially smaller errors in the atomic test calculations as well as lower total energies and produce results of similar accuracy in the molecular calibration study.

Improved energy-optimized (6s5p4d) and (7s6p5d) primitive valence basis sets have been derived for energy-consistent scalar-relativistic 4f-in-core pseudopotentials of the Stuttgart-Cologne variety modeling divalent lanthanides with a \(4\hbox{f}^{n+1}\) occupation (n = 0–13 for La–Yb). Segmented contracted basis sets covering the range of polarized double-, triple-, and quadruple-zeta quality, augmented by 2f1g correlation sets, were created for use in molecular calculations. The basis sets contain smaller (4s4p3d) and (5s5p4d) primitive subsets, which are designed in particular for solid state calculations of crystals containing divalent lanthanide ions. Hartree–Fock, density functional theory and coupled cluster results obtained with the new basis sets for lanthanide atomic ionization potentials as well as of geometry optimizations of various test molecules, i.e. selected lanthanide mono- and dihydrides, mono- and difluorides, and monooxides, show a satisfactory agreement with experimental data as well as with corresponding scalar-relativistic all-electron results. Core-polarization potentials are found to improve the results, especially for the atomic first and second ionization potentials.

The bis(salicylhydroxamato) and bis(benzohydroxamato) complexes of UO22+ in aqueous solution have been investigated in a combined experimental and computational effort using extended X-ray absorption fine structure and UV−vis spectroscopy and density functional theory (DFT) techniques, respectively. The experimentally unknown bis(benzoate) complex of UO22+ was investigated computationally for comparison. Experimental data indicate 5-fold UO22+ coordination with mean equatorial U−O distances of 2.42 and 2.40 Å for the salicyl- and benzohydroxamate systems, respectively. DFT calculations on microsolvated model systems [UO2L2OH2] indicate UO22+ η2-chelation via the hydroxamate oxygen atoms in excellent agreement with experimental data; calculated complex stabilities support that UO22+ prefers hydroxamate over carboxylate coordination. The 414 nm absorption band of UO22+ in aqueous solution is blue-shifted to 390 and 386 nm upon complexation by salicyl- and benzohydroxamate, respectively. Calculated time-dependent DFT excitation energies of [UO2L2OH2], however, occasionally fail to reproduce accurately experimental UV−vis spectra, which are dominated by UO22+ ← L− charge-transfer contributions. We additionally show that the UVI large-core pseudopotential approximation recently developed by some of the authors can routinely be applied for electronic structure calculations not involving uranium 5f occupations significantly different from UVI.

The experimentally observed extraction complexes of trivalent lanthanide EuIII and actinide AmIII/CmIII cations with purified Cyanex301 [bis(2,4,4-trimethylpentyl)dithiophosphinic acid, HBTMPDTP denoted as HL], i.e., ML3 (M = Eu, Am, Cm) as well as the postulated complexes HAmL4 and HEuL4(H2O) have been studied by using energy-consistent 4f- and 5f-in-core pseudopotentials for trivalent f elements, combined with density functional theory and second-order Møller−Plesset perturbation theory. Special attention was paid to explaining the high selectivity of Cyanex301 for AmIII/CmIII over EuIII. It is shown that the neutral complexes ML3, where L acts as a bidentate ligand and the metal cation is coordinated by six S atoms, are most likely the most stable extraction complexes. The calculated metal−sulfur bond distances for ML3 do reflect the cation employed; i.e., the larger the cation, the longer the metal−sulfur bond distances. The calculated M−S and M−P bond lengths agree very well with the available experimental data. The obtained changes of the Gibbs free energies in the extraction reactions M3+ + 3HL → ML3 + 3H+ agree with the thermodynamical priority for Am3+ and Cm3+. Moreover, the ionic metal−ligand dissociation energies of the extraction complexes ML3 show that, although EuL3 is the most stable complex in the gas phase, it is the least stable in aqueous solution.

Lanthanide(III) hydration was studied by utilizing density-functional theory and second-order Møller–Plesset perturbation theory combined with scalar-relativistic 4f-in-core pseudopotentials and valence-only basis sets for the LnIII ions. For [LnIII(H2O)h]3+ (h = 7, 8, 9) and [LnIII(H2O)h−1·H2O]3+ (h = 8, 9) molecular structures, binding energies, entropies and energies of hydration as well as Gibbs free energies of hydration were calculated using (8s7p6d3f2g)/[6s5p5d3f2g] basis sets for LnIII and aug-cc-pV(D,T)Z basis sets for O and H in combination with the COSMO solvation model. At the generalized gradient approximation level of density-functional theory a preferred hydration number of 8 is found for LaIII–TmIII and 7 for YbIII–LuIII, whereas hybrid density-functional theory predicts a hydration number 8 for all LnIII. At the SCS-MP2 level of theory the preferred hydration number is found to be 9 for LaIII–SmIII and 8 for EuIII–LuIII in good agreement with experimental evidence.

Calibration studies of actinide and lanthanide trifluorides are reported for actinide and lanthanide scalar-relativistic energy-consistent f-in-core pseudopotentials, respectively, accompanying valence basis sets as well as core-polarization potentials. Results from Hartree–Fock and coupled-cluster singles, doubles, and perturbative triples f-in-core pseudopotential calculations are compared to corresponding data from f-in-valence pseudopotential and all-electron calculations as well as to experimental data. In general, good agreement is observed between the f-in-core and f-in-valence pseudopotential results, whereas due to the lack of experimental data for the actinides only a good agreement of the calculated and experimentally determined bond lengths of the lanthanide systems can be established. Nevertheless, the results indicate that the core-polarization potentials devised here for actinides improve the f-in-core results.

In order to assess the accuracy of a recently adjusted relativistic energy-consistent small-core pseudopotential for uranium, the U5+ (5f1 subconfiguration) spin−orbit splitting as well as the fine structure of the U4+ (5f2 subconfiguration) spectrum have been calculated. The pseudopotential has been adjusted to four-component all-electron data, i.e., at the multiconfiguration Dirac−Hartree−Fock level using the Dirac−Coulomb Hamiltonian with a Fermi nucleus charge distribution and perturbatively including the Breit interaction. Its performance in a dressed effective Hamiltonian spin−orbit configuration interaction framework is compared to that of an older scalar-relativistic Wood−Boring adjusted pseudopotential, supplemented by a valence spin−orbit term, as well as to all-electron calculations using the Douglas−Kroll−Hess Hamiltonian. Electron correlation is accounted for by the multireference configuration interaction method with and without the Davidson correction and with different frozen-orbital spaces. Our best calculations show satisfactory agreement with experimental data; i.e., the mean absolute (relative) deviations amount to 183 (2.4%) and 948 cm−1 (5.1%) for the U5+ and the U4+ fine-structure energy levels, respectively. Even better agreement, comparable to the one for rigorous highly correlated four-component all-electron data, is obtained in intermediate Hamiltonian Fock-space coupled-cluster calculations applying the new pseudopotential.

The options to adjust accurate relativistic energy-consistent pseudopotentials for actinides are explored using uranium as an example. The choice of the reference data and the core−valence separation is discussed in view of a targeted accuracy of 0.04 eV or better in atomic energy differences such as excitation energies and ionization potentials. A new small-core pseudopotential attributing 60 electrons to the core has been generated by an energy adjustment to state-averaged multiconfiguration Dirac−Hartree−Fock/Dirac−Coulomb−Breit Fermi nucleus reference data of 100 nonrelativistic configurations of U to U7+ corresponding to 30190 reference J levels. At the finite-difference multiconfiguration Hartree−Fock level the mean absolute errors are 0.002 and 0.024 eV for the configurations and J levels, respectively. A first molecular application to uranium monohydride UH yields very satisfactory agreement with results from all-electron calculations based on the Douglas−Kroll−Hess Hamiltonian.

A systematic computational approach to AnIII hydration on a density-functional level of theory, using quasi-relativistic 5f-in-core pseudopotentials and valence-only basis sets for the AnIII subsystems, is presented. Molecular structures, binding energies, hydration energies, and Gibbs free energies of hydration have been calculated for [AnIII(OH2)h]3+ (h = 7, 8, 9) and [AnIII(OH2)h−1·OH2]3+ (h = 8, 9), using large (7s6p5d2f1g)/[6s5p4d2f1g] AnIII and cc-pVQZ O and H basis sets within the COSMO implicit solvation model. AnIII preferred primary hydration numbers are found to be 8 for all AnIII at the gradient-corrected density-functional level of theory. Second-order Møller–Plesset perturbation theory predicts preferred primary hydration numbers of 9 and 8 for AcIII–MdIII and NoIII–LrIII, respectively.

Quasirelativistic energy-consistent 5f-in-core pseudopotentials modeling divalent (5fn+1 occupation with n =  5–13 for Pu–No) respectively tetravalent (5fn-1 occupation with n =  1–9 for Th–Cf) actinides together with corresponding core-polarization potentials have been generated. Energy-optimized (6s5p4d) and (7s6p5d) valence basis sets as well as 2f1g correlation functions have been derived and contracted to polarized double, triple, and quadruple zeta quality. Corresponding smaller (4s4p) and (5s5p) respectively (4s4p3d) and (5s5p4d) basis sets suitable for calculations on actinide(II) respectively actinide(IV) ions in crystalline solids form subsets of these basis sets designed for calculations on molecules. Results of Hartree–Fock test calculations for actinide di- and tetrafluorides show a satisfactory agreement with calculations using 5f-in-valence pseudopotentials.

Quasirelativistic energy-consistent 5f-in-core pseudopotentials modelling trivalent actinides, corresponding to a near-integral 5fn occupation (n = 0–14 for Ac–Lr), have been generated. Energy-optimized (6s5p4d), (7s6p5d), and (8s7p6d) primitive valence basis sets contracted to polarized double to quadruple zeta quality as well as 2f1g correlation functions have been derived. Corresponding smaller basis sets (4s4p3d), (5s5p4d), and (6s6p5d) suitable for calculations on actinide(III) ions in crystalline solids form subsets of these basis sets designed for calculations on neutral molecules. Results of Hartree–Fock test calculations for actinide(III) monohydrates and actinide trifluorides show a satisfactory agreement with corresponding calculations using 5f-in-valence pseudopotentials. Even in the beginning of the actinide series, where the 5f shell is relatively diffuse, only quite acceptable small deviations occur as long as the 5f-shell does not participate significantly in covalent bonding.

Lanthanide(III) hydration was studied by utilizing density-functional theory and second-order Møller–Plesset perturbation theory combined with scalar-relativistic 4f-in-core pseudopotentials and valence-only basis sets for the LnIII ions. For [LnIII(H2O)h]3+ (h = 7, 8, 9) and [LnIII(H2O)h−1·H2O]3+ (h = 8, 9) molecular structures, binding energies, entropies and energies of hydration as well as Gibbs free energies of hydration were calculated using (8s7p6d3f2g)/[6s5p5d3f2g] basis sets for LnIII and aug-cc-pV(D,T)Z basis sets for O and H in combination with the COSMO solvation model. At the generalized gradient approximation level of density-functional theory a preferred hydration number of 8 is found for LaIII–TmIII and 7 for YbIII–LuIII, whereas hybrid density-functional theory predicts a hydration number 8 for all LnIII. At the SCS-MP2 level of theory the preferred hydration number is found to be 9 for LaIII–SmIII and 8 for EuIII–LuIII in good agreement with experimental evidence.

Global optimization of cluster geometries is of fundamental importance in chemistry and an interesting problem in applied mathematics. In this work, we introduce a relatively new swarm intelligence algorithm, i.e. the artificial bee colony (ABC) algorithm proposed in 2005, to this field. It is inspired by the foraging behavior of a bee colony, and only three parameters are needed to control it. We applied it to several potential functions of quite different nature, i.e., the Coulomb–Born–Mayer, Lennard-Jones, Morse, Z and Gupta potentials. The benchmarks reveal that for long-ranged potentials the ABC algorithm is very efficient in locating the global minimum, while for short-ranged ones it is sometimes trapped into a local minimum funnel on a potential energy surface of large clusters. We have released an efficient, user-friendly, and free program “ABCluster” to realize the ABC algorithm. It is a black-box program for non-experts as well as experts and might become a useful tool for chemists to study clusters.

The global optimization of molecular clusters is an important topic encountered in many fields of chemistry. In our previous work (Phys. Chem. Chem. Phys., 2015, 17, 24173), we successfully applied the recently introduced artificial bee colony (ABC) algorithm to the global optimization of atomic clusters and introduced the corresponding software “ABCluster”. In the present work, ABCluster was extended to the optimization of clusters of rigid molecules. Here “rigid” means that all internal degrees of freedom of the constituent molecules are frozen. The algorithm was benchmarked by TIP4P water clusters (H2O)N (N ≤ 20), for which all global minima were successfully located. It was further applied to various clusters of different chemical nature: 10 microhydration clusters, 4 methanol microsolvation clusters, 4 nonpolar clusters and 2 ion–aromatic clusters. In all the cases we obtained results consistent with previous experimental or theoretical studies.

A systematic computational approach to AnIII hydration on a density-functional level of theory, using quasi-relativistic 5f-in-core pseudopotentials and valence-only basis sets for the AnIII subsystems, is presented. Molecular structures, binding energies, hydration energies, and Gibbs free energies of hydration have been calculated for [AnIII(OH2)h]3+ (h = 7, 8, 9) and [AnIII(OH2)h−1·OH2]3+ (h = 8, 9), using large (7s6p5d2f1g)/[6s5p4d2f1g] AnIII and cc-pVQZ O and H basis sets within the COSMO implicit solvation model. AnIII preferred primary hydration numbers are found to be 8 for all AnIII at the gradient-corrected density-functional level of theory. Second-order Møller–Plesset perturbation theory predicts preferred primary hydration numbers of 9 and 8 for AcIII–MdIII and NoIII–LrIII, respectively.