Ab Initio Ligand-Field Theory Analysis and Covalency Trends in

Article pubs.acs.org/IC

Ab Initio Ligand-Field Theory Analysis and Covalency Trends in Actinide and Lanthanide Free Ions and Octahedral Complexes Julie Jung,† Mihail Atanasov,*,†,‡ and Frank Neese*,† †

Max Planck Institut für Chemische Energiekonversion, Stifstrasse 34−36, D-45470 Mülheim an der Ruhr, Germany Institute of General and Inorganic Chemistry, Bulgarian Academy of Sciences, Akad. Georgi Bontchev Street 11, 1113 Sofia, Bulgaria

‡

S Supporting Information *

ABSTRACT: Actinide chemistry is gaining increased focus in modern research, particularly in the fields of energy research and molecular magnetism. However, the structure−function and structure−property relationships of actinides have still not been studied as intensely as those for transition metals. In this work, we report a detailed ab initio study of the spectroscopic, magnetic, and bonding properties of the trivalent actinide free ions and their associated hexachloride complexes in octahedral symmetry. The electronic structures of these systems are examined using complete active-space self-consistent-field calculations followed by second-order N-electron valence perturbation theory, including both scalar relativistic and spin− orbit-coupling effects. The computed energies and wave functions are further analyzed by means of ab initio ligand-field theory (AILFT) and finally chemically interpreted by means of the angular overlap model (AOM). The derived Slater−Condon and spin−orbit parameters have allowed us to systematically rationalize the spectroscopic and magnetic properties of the investigated free ions and complexes along the entire actinide series. Overall, the AILFT- and AOM-derived parameters accurately reproduce the multireference electronic structure calculations. The small observed discrepancies with respect to experimentally derived ligand-field parameters are essentially due to an underestimation of the electronic correlation, which arises from both the constrained size of the active space (restricted to the f orbitals) and the limit of the perturbation approach to account for dynamical correlation. Our analysis also provides insight into the metal−ligand covalency trends along the series. Consistent with natural population analysis, the nephelauxetic (Slater−Condon parameters) and relativistic nephelauxetic (spin−orbit-coupling) reductions determined for these complexes indicate a decrease in the covalency along the series. These trends also hold, to varying extents, for the corresponding tetravalent derivatives, as well as the lanthanide analogues.

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INTRODUCTION Actinide complexes play a central role in a variety of industrial and technological applications.1,2 For instance, they are of utmost importance in the field of nuclear fuel reprocessing.3,4 They may also be used as catalysts to activate small and inert molecules5−8 such as H2,9,10 CO2,11,12 or N2.13,14 More recently, they have arisen as promising candidates for the next generation of single-molecule magnets (SMMs).15−31 To make these systems more competitive in the socioeconomic context of the 21st century, a more fundamental understanding of how the structural and chemical features of actinide complexes correlate with their reactivity, and, in turn, their spectroscopic, bonding, and magnetic properties, is needed. While experimental techniques such as optical spectroscopy (e.g., absorption, luminescence, TRLFS),32−35 X-ray spectroscopy (e.g., XPS, XANES, EXAFS),36−41 and magnetic measurements (e.g., SQUID, EPR, NMR)42−44 can give us insight into the electronic structure of actinide complexes, ab initio calculations undoubtly provide a fuller and more accurate, but also more complicated, picture. In this context, combined experimental and computational investigations represent the most promising approach to reaching the most detailed level of © 2017 American Chemical Society

understanding of the electronic structure of actinide complexes.45−48 In such kinds of studies, the electronic structure calculations are usually translated into quantities that make more sense to chemists and spectroscopists and may be compared to the basic parameters extracted from the experiment. In the present work, we combine correlated electronic structure calculations with ab initio ligand-field theory (AILFT) to investigate the spectroscopic and magnetic properties, ligand-field effects, and bonding trends in several series of actinide complexes. In particular, we focus on the actinide hexachloride complexes, which have already been investigated both experimentally39,40,48−67 and computationally40,48,68−71 for various oxidation states. Yet, to analyze the covalency of the An−Cl bonds in the context of ligand-field theory, we must also consider the free ions. First, we will consider the trivalent free ions (An3+) and associated hexachloride complexes ([AnIIICl6]3−). Because the trivalent oxidation state is stable along the whole series (from Received: March 13, 2017 Published: July 14, 2017 8802

DOI: 10.1021/acs.inorgchem.7b00642 Inorg. Chem. 2017, 56, 8802−8816

Article

Inorganic Chemistry

Figure 1. Slater−Condon parameters F2 (a), F4 (b), and F6 (c) and SOC parameter ζ (d) for the whole trivalent actinide free-ion series (cm−1). Open blue circles and filled red circles correspond to the CASSCF and NEVPT2 values, respectively, and dashed lines correspond to the experimental data76 from refs 78 (black), 79 (blue), and 80 (red).

ThIII of configuration 5f1 to NoIII of configuration 5f13), these two series allow us to investigate the bonding and interaction trends for a complete series of trivalent actinide complexes. Experimentally, this is hardly achievable because of lifetime and toxicity issues. Special attention is paid to the free-ion case not only because the free-ion AILFT parameters are indispensable to assess the level of f covalency in the An−Cl bonds of the complexes but also because these parameters will help us to evaluate the accuracy of our approach. Next, we will turn our attention to the corresponding tetravalent series. Because this oxidation state is only stable for the first-half of the actinide series, we only consider the PaIV−BkIV derivatives. Finally, we will compare the trivalent actinide and trivalent lanthanide series, which were recently investigated by Aravena et al. using AILFT to rationalize the regularities of the structural, thermodynamical, and spectroscopic properties evidenced from previous experimental studies.36,72−74

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added through NEVPT2, while SOC and scalar relativistic effects are accounted for by means of quasi-degenerate perturbation theory (QDPT), using the second-order scalar relativistic Douglas−Kroll− Hess Hamiltonian formalism. On top of that, AILFT is used to extract ligand-field parameters such as the Slater−Condon parameters for interelectronic repulsion, F2, F4, and F6, the one-electron effective parameter for SOC, ζ, and the 28 parameters for ligand-field interactions. The latter parameters are further considered in the context of the angular overlap model (AOM), giving rise to the eσ and eπ parameters only. Further details on both the electronic structure calculations and AILFT are provided in the Supporting Information (sections S1 and S2).

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RESULTS AND DISCUSSION Trivalent Actinide (AnIII with An = Th−No). Free Ions An3+. AILFT Analysis. The Slater−Condon parameters for interelectronic repulsion, F2, F4, and F6, computed for the trivalent actinide free ions (Pa3+−Md3+) are shown in Figure 1a−c and section S3.1.1 (together with their standard deviations). For each series, the computed parameters increase with increasing atomic number, following the trend drawn by the experimental data76 (dashed lines in Figure 1). Because the number of 5f electrons increases along the series while the ionic radii and radial expectation values (section S3.1.2) decrease, it is expected that the interelectronic repulsion associated with the 5f shell becomes larger across the series. This contraction of

COMPUTATIONAL DETAILS

All of the electronic structure calculations discussed in the present paper are performed at the complete active-space self-consistent-field (CASSCF)/second-order N-electron valence perturbation theory (NEVPT2) + spin−orbit coupling (SOC) level of theory, using the ORCA 4.0 quantum chemistry package.75 For all investigated systems, the f−f excitation spectrum is computed using the CASSCF method. Dynamical correlation is 8803


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Table 1. Lower Part of the Excitation Spectrum of U3+ (cm−1)a

the 5f shell from left to right of the actinide series is induced by the more and more incomplete screening of the nuclear charge. The F2 parameters computed at the CASSCF level are about 70% (Pa3+) to 40% (Md3+) larger than the experimental ones.77 The F4 and F6 parameters, on the other hand, are overestimated by 40% (Pa3+) to 15% (Md3+). This overestimation is partially corrected when accounting for the dynamical correlation, dropping to 15−20% for F2, 0−20% for F4, and 10−20% for F6. The remaining difference between the computed and experimental ligand-field parameters can have different origins. First, from a computational perspective, one may consider the basis-set incompleteness, which may influence the description of the metal center. Second, there is the limit of the active space. Restriction of the active space to the 5f shell when excited-state configurations of the same g−u symmetry as the 5fn configurations are very close in energy may indeed lead to an incomplete treatment of the electronic correlation (see section S3.1.3 for further details). Last, there is the limit of the NEVPT2 approach (and of all PT2 approaches in general), which can still miss about 15−30% of the dynamical correlation effects.81,82 Overestimation of the Slater−Condon parameters likely results from a combination of all of the aforementioned effects. On a separate note, there is also the question of the relevance of the experiment versus computation comparison. Indeed, because there are no experimental data available for the actinide trivalent free ions, experimental data are taken from the AnCl3 series, the largest trivalent series for which experimental data are available. Yet, upon complexation, the associate Slater− Condon parameters are expected to be smaller than those for the free ions. We have computed this reduction to be less than 5% (vide supra). Hence, this alone should not account for the overestimation of the ligand-field parameters. In the same context, another source of discrepancy could arise from the very way the parameters are extracted. Not only does the model Hamiltonian used experimentally contain more parameters than the one used here but the experimental fit itself makes use of multiple approximations that we do not.80,83−87 Also, while we account for the whole fn electronic spectrum, the experimental fit is only performed for the energy window determined by the spectroscopic techniques. All of these issues are further discussed in section S3.1.4. Finally, similar to the interelectronic repulsion parameters, the effective one-electron spin−orbit parameters appear to increase along the free-ion series (Figure 1d). The computed values follow the experimental trend, which is directly correlated with the increase of the effective nuclear charge with increasing atomic number. The computed values are slightly overestimated, from 2% for Md3+ to 13% for U3+. This level of accuracy is satisfactory for our purposes and suggests that the multiplet splitting induced by SOC is properly described. The same kind of agreement was obtained for the trivalent lanthanide ions by Aravena et al.88 The combination of these results with the already documented accuracy of the spin−orbit mean-field treatment for transition metals and lighter main-group elements89 lends credence to this approach. Excitation Spectra. To get an idea of how the AILFT parameters relate to the electronic structure, the complete spectra of all free ions were computed at both the CASSCF and NEVPT2 levels and compared to the experimental data76 available from the literature. Hereafter, instead of considering all particular features of all spectra, we only describe the excitation spectrum of U3+ below 15000 cm−1 (Table 1). The

Experiment E 0 4293 7403 7927 9436 10114 11069 11284

J

w.f.

/2 11 /2 3 /2 13 /2 9 /2 5 /2 15 /2 3 /2

83% 4I + 15% 2H 95% 4I 64% 4F + 25% 2D 92% 4I 38% 2H + 33% 2G + 12% 4I + 11% 4F 72% 4F + 13% 4G + 5% 2F 80% 4I + 18% 2K 63% 4S + 22% 2P + 14% 4F CASSCF

9

E

J /2 /2 13 /2 3 /2 15 /2 9 /2 5 /2 3 /2 NEVPT2

0 4652 8635 11760 12526 13128 14667 15902 J 9

/2 11 /2 3 /2 13 /2 9 /2 5 /2 15 /2 3 /2

E 0 4990 7292 9415 9802 11003 11279 13144 13794

79% 85% 88% 72% 83% 44% 87% 74%

11

E 0 4652 8239 8822 10930 11184 12568 12983

w.f.

9

J 9

/2 11 /2 3 /2 13 /2 9 /2 5 /2 3 /2 15 /2 5 /2

4

I + 8% 2H I 4 I 4 F + 14% 2D 4 I 2 H + 26% 2G 4 F + 6% 2D 4 S + 16% 2P 4

w.f. 75% 85% 67% 85% 33% 76% 77% 67% Ruipérez et al.90

4

I + 10% 2H 4 I 4 F + 16% 2D 4 I 2 H + 29% 2G + 6% 4F 4 F + 6% 2D 4 I + 7% 2K 4 S + 21% 2P

w.f. (major contribution only) 4

I I 4 F 4 I 2 H, 2G 4 F 4 S 4 I 4 G 4

a Energies (E) are given with respect to the ground state. Each state is identified with its associate quantum number J, and its wave function (w.f.) is expressed in terms of spectroscopic terms (2S+1L).92 Experimental data76 are taken from ref 78. The complete spectrum is provided in section S3.1.5.

complete spectrum of U3+, as well as the lower part of the excitation spectra of Np3+−Fm3+, are reported in sections S3.1.5 and S3.1.6, respectively. The quantum number J and the wave function associated with the relativistic ground state of all ions from Th3+ to No3+ are shown in Table 2, including also the energy of the lowest excited state. Upon investigation of the excitation spectrum of U3+ (Table 1), it is observed that the CASSCF spin−orbit states are misordered and that their energies are largely overestimated. This is in line with the Fk Slater−Condon and ζ spin−orbit 8804


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Inorganic Chemistry Table 2. Quantum Number J and Wave Function (in Terms of 2S+1L Spectroscopic Terms92) Associated to the Ground State of U3+−Fm3+a

configurations91) and thus outside our active space, which would explain the deterioration of the energy of these higher excited states. The quantum number J associated with the ground state is well reproduced in both the CASSCF and NEVPT2 calculations for all ions of the U3+−Fm3+ series. Similarly, very good agreement is found for the ground-state wave function, with an average difference of less than 10% in the coefficients of the relativistic wave functions. Also, the wellknown deviation from Hund’s rules95 is nicely reproduced, in agreement with previous observations.42,96 For example, in the ground-state wavefuntion of Am3+, the weight of the 7F term is less than 50%. This is due to strong SOC that mixes electronic states stemming from different 2S+1L spectroscopic terms but with the same quantum number J. This explains why the intermediate coupling scheme is needed to describe the electronic structure of actinide systems (Tables 2 and S10). Magnetic Susceptibility. The temperature dependence of the magnetic susceptibility (χM) is another physical property, besides absorption spectroscopy, that can be used to glimpse at the electronic structure of the actinide trivalent free ions because it indirectly probes the ground-state magnetic sublevels in the temperature range of the experimental data.76 For this property, however, a comparison of the measured and computed data does not require the use of AILFT, thus avoiding any discrepancies connected to the definition of the model Hamiltonian (see section S3.1.4). Also, in particular cases where the total splitting of the ground manifold (Δ) is smaller than the thermal energy (kBT;60 which is the case here), the magnetic susceptibility reaches a linear regime that can be determined from the Curie law shown below (eq 1), with T the temperature (K), C the Curie constant, and χTIP the temperature-independent paramagnetic term.

a

For the other ions, the data are provided in section S3.1.7. Experimental data76 are always given on the first line of each row (in black) and were taken from refs 78 and 93 for U3+−Cm3+ and from ref 94 for Bk3+−Fm3+. The second (blue) and third (red) lines correspond to the CASSCF and NEVPT2 results, respectively. The energy differences (ΔE) to the first excited configuration are provided in the last column (cm−1).

χM =

parameters being overestimated. After accounting for the dynamical correlation, the state ordering is completely corrected, the energies are in better agreement with the experimental values, and the wave functions are closer to those derived by Carnall and Wybourne in 1964.78 On average, the energy differences are less than 1000 cm−1 and the coefficients of the wave functions (when expressed in terms of 2S+1L spectroscopic terms) change by less than 15%. This level of agreement is comparable to that obtained by Ruipérez et al.90 for the same kinds of calculations on U3+ (Table 1). The same level of accuracy is obtained for all other ions with respect to both the energies and wave functions (Table 2). For instance, for Np3+−Am3+, the first excited state, which is underestimated by a few hundred wavenumbers at the CASSCF level, becomes much closer to the experimental value when accounting for the dynamical correlation. On the other hand, for Bk3+−Fm3+, this energy is rather overestimated at the CASSCF level and does not drastically improve with the NEVPT2 correction. From these considerations, it appears that the average error of the CASSCF/NEVPT2 + SOC approach to compute the energy of the first excited spin−orbit-free multiplet in actinide systems is around a few hundred wavenumbers. The same goes for all states within the first 15000−20000 cm−1 of the many-particle spectrum (see section S3.1.6); however, these estimates worsen for higher excited states. Presumably, these excited states interact with excited configurations that are beyond the 5f shell (e.g., 5fn−26d2

C + χ TIP T

(1)

Hence, for all ions of the series, the temperature dependence of the magnetic susceptibility (χMT vs T) has been computed with both the CASSCF and NEVPT2 approaches. The data are fitted to eq 1 in the 50−300 K temperature range, i.e., where Δ < kBT, in order to extract C and χTIP. The results are shown in Table 3 for the ions for which experimental data76 are available, i.e., U3+−Cm3+, and in section S3.1.8 for the other ions. For U3+, Pu3+, and Cm3+, the computed values of C are in good agreement with the experiment. The observed differences (less than 10% of the experimental value) are directly connected to the differences found in the ground-state wave function (Table 2): the more the computed C value deviates from the experimental one, the more the coefficients of the computed wave functions deviate from the experiment. The same holds true for Pu3+ but with larger differences (up to about 30% of the experimental value). Indeed, with C being one order of magnitude smaller for Pu3+ than for the other ions, the margin of error in the experimental value is probably higher, resulting in a larger experimental/theoretical discrepancy. For some ions, the computed Curie constants are found to deviate from the values expected from Hund’s rules. For instance, in Pu3+, the computed values of C (0.142 cm3 K mol−1 with CASSCF and 0.170 cm3 K mol−1 with NEVPT2) are about twice CHund for the 6H term (i.e., 0.089 cm3 K mol−1). 1 This feature is particularly obvious toward the center of the series (Np3+−Bk3+) and seems to correlate with that observed in the ground-state wave functions (Table 2 and section 8805


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Inorganic Chemistry Table 3. Curie Constant C (cm3 K mol−1) and the TIP term χTIP (1 × 10−4 cm3 mol−1) for U3+−Cm3+a

Table 4. Computed An−Cl Bond Lengths (Å; ab Initio) for All Complexes of the Trivalent Hexachloride Seriesa

a

For the other ions, the data are shown in section S3.1.8. Experimental data76 are always given on the first line of each row (in black) and were obtained by converting the Landé g factor, gJ, provided in ref 60, to C NAβ 2gJ 2J(J + 1) using the following expression: C = with NA the 3kB Avogadro number, β the Bohr magneton, and kB the Boltzmann constant.98 The second (blue) and third (red) lines correspond to the and CHund are the CASSCF and NEVPT2 results, respectively. CHund 1 2 Curie constants expected from Hund’s rules for the two 2S+1L spectroscopic terms having the largest and second-largest contributions to the ground-state wave functions (given in parentheses). They are obtained from the expression above, such as [S(S + 1) − L(L + 1)] 3 gJ = + . 2 2J(J + 1)

AnIII

ab initio

ThIII PaIII UIII NpIII PuIII AmIII CmIII BkIII CfIII EsIII FmIII MdIII NoIII

2.884 2.844 2.824 2.799 2.776 2.757 2.749 2.723 2.711 2.695 2.693 2.682 2.667

expt

ionic limit

2.723 2.707 2.698 2.681 2.669 2.656 2.647

2.85 2.835 2.82 2.81 2.785 2.78 2.77 2.76

a

Experimental An−Cl distances (expt) are taken from the Cs2NaAnIIICl6 elpasolite derivatives.57 The ionic limit is obtained from the sum of Shannon’s ionic radii for AnIII (for CN = 6) and ClI (for CN = 1).

Despite the small discrepancy, the computed An−Cl distances nicely reproduce the actinide contraction brought about by the incomplete screening of the nuclear charge.101 Interestingly, from [ThIIICl6]3− to [NoIIICl6]3−, the computed An−Cl distances decrease faster than predicted by both Shannon’s ionic radii and atomic radial expectation values (see section S3.1.2). AILFT Analysis. Instead of considering the AILFT parameters of the [AnIIICl6]3− complexes directly, we will rather look at how they differ from the free-ion case (Figure 2). Herein, we only consider the CASSCF results because of the potentially large errors encountered in the NEVPT2 AILFT parameters (see section S3.2.3). Also, as discussed in detail in the next session, the NEVPT2 results are actually not relevant to the bonding analysis.102 From Figure 2a, it is seen that the Slater− Condon parameters are reduced by 1−6% at the CASSCF level and by 2−10% at the NEVPT2 level. Similarly, the effective one-electron spin−orbit parameter is reduced from 9% (for ThIII) to 1% (NoIII) at the CASSCF level. This indicates that the 5f orbitals accommodate the presence of the chloride ions in order to reduce the interelectronic repulsion among the and that there is a small quenching of the angular momentum of the 5f orbitals due to bond formation with the chloride ions. These effects are known as the nephelauxetic and relativistic nephelauxetic reductions, respectively. The fact that these reductions are stronger in the first half of the series suggests that the chloride ions have a stronger interaction with the 5f orbitals of the early actinide ions, i.e., for AnIII = ThIII−AmIII, likely due to actinide contraction. The AOM parameters, eσ and eπ, are shown in Figure 2b and section S3.2.2. Both parameters decrease monotonically along the [AnIIICl6]3− series. This is consistent with the idea that chloride ions interact more strongly with actinide ions from the beginning of the series. This finding is also in line with the observed reduction of the interelectronic repulsion parameters (Figure 2a). Interestingly, eσ is more than twice as large as eπ, implying that the interaction between the 5f orbitals of the actinide ions and the Cl(3p) orbitals of the chloride ions has a much stronger σ than π antibonding character. Excitation Spectra. Of all existing Cs2NaAnIIICl6 elpasolite compounds, detailed absorption spectra are only available for

S3.1.7). Indeed, the ground-state wave function of Pu3+ appears to only hold 65% (NEVPT2) to 75% (CASSCF) of the 6H character. Even further, this deviation appears to be driven by the spectroscopic term having the second-most-significant contribution to the ground-state wave function, meaning that C tends to deviate from CHund in order to get closer to CHund . 1 2 Additionally, the second-largest contribution to the groundstate wave function comes from the 4G term, for which CHund is 2 0.357 cm3 K mol−1, remarkably about twice the computed values. This trend holds along the whole series, which suggests that the deviation from Hund’s rules evidenced in the groundstate wave functions (Table 2) also is reflected in C. This is a consequence of the Landé formula strictly holding only within the Russell−Saunders (or LS) coupling scheme. These results are fully consistent with the recent work by Autillo et al.97 [AnIIICl6]3− Complexes. Geometry Optimization. The structure of each [AnIIICl6]3− complex is optimized at the NEVPT2 level of theory (see section S1.3) while constraining the overall symmetry to octahedral. The resulting An−Cl bond lengths are reported in Table 4. Even though all computed distances are 2−5% larger than that reported by Shoebrechts et al.57 for the Cs2NaAnIIICl6 elpasolite compounds, they are in the same range as the CASPT299 and RASPT2100 distances recently reported by Beekmeyer and Kerridge for similar complexes.69 This tendency to overestimate the bond distances is actually inherent to our approach. Indeed, if optimization is performed in the presence of an embedding potential to mimic the crystal environment (see sections S1.4 and S3.2.1), the computed An− Cl distances approach the experimental one. For instance, with such periodic conditions, the computed U−Cl equilibirium distance is 2.697 Å, which is less than 1% below the experimental value (i.e., 2.723 Å). 8806


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Figure 2. (a) Nephelauxetic and relativistic nephelauxetic reductions (%) and (b) AOM parameters (cm−1) associated with all [AnIIICl6]3− complexes. In part a, the blue, red, and green colors are for the reduction of F2, F4, and F6, respectively, and the black color is for the reduction of ζ. All reductions are defined as (1 − αcomplex/αfree ion) × 100, with α one of the aforementioned parameters. In part b, blue open circles correspond to eσ and red filled circles to eπ.

the AmIII derivative.53−56 This is mostly because of the complexes being so highly symmetric that Laporte’s rule for f−f transitions is hardly violated.103 Both the experimental and computed excitation spectra are shown in Table 5 and section S3.2.4 (for further details). Overall, there is good qualitative agreement between the computed and experimental spectra in that the energies and state ordering are well reproduced. Provided that dynamical correlation is accounted for, this agreement is nearly quantitative for the states below 10000 cm−1. Indeed, the average error in the excitation energies drops from 650 cm−1 at the CASSCF level to only 120 cm−1 at the NEVPT2 level. Also, while the overall energy splitting for CASSCF is much smaller than the measured one, the NEVPT2 splitting is typically slightly larger. For the higher excited states (above 15000− 20000 cm−1), both the CASSCF and NEVPT2 excitation energies are significantly larger than those of the Cs2NaAm III Cl6 compound, with errors exceeding 2000 cm−1. Some discrepancies also start to appear in the state ordering, with some being resolved with the addition of dynamical correlation and others not. Yet, it is difficult to say whether these result from experimental constraints (upon recording of the spectrum) or computational limitations. It is probably both. Despite these small discrepancies, the computed spectrum provides valuable help to further analyze the excitation spectrum of [AmIIICl6]3−. For instance, while Barbanel et al. could not unambiguously attribute the state measured at 18950 cm−1 to either the Γ4 or Γ5 irreducible representation (irrep), the present calculations clearly evidence that it belongs to the Γ5 irrep (i.e., the CASSCF/NEVPT2 state at 22116/21418 cm−1). In addition, from Table 5, it appears that most states can be described by a single quantum number J. Interestingly, this suggests that J mixing, i.e., the mixing of electronic states with different J values that belong to the same irrep through the ligand field, is very small for [AmIIICl6]3−. This is most likely due to the fact that the energy difference between states belonging to the same irrep is relatively large (typically, a few thousand wavenumbers). There is, however, one case in which J mixing appears to be nonnegligible (boldface in Table 5): Around 13000 cm−1, two states belonging to the Γ1 irrep with J = 0 and 6 mix with each other in an 80/20 ratio. In this very

case, the energy difference between both states is only 125 cm−1, which explains why J mixing is so large. Magnetic Behavior of the [AnIIICl6]3− Complexes. Similar to the free-ion case, we have computed and analyzed the magnetic behavior of the hexachloride complexes (Table 6). Because the magnetic susceptibility of some Cs2NaAnIIICl6 compounds were thoroughly investigated by Hendricks et al.60 in the 1970s, we have decided to focus on the same derivatives, i.e., AnIII = UIII−PuIII and CmIII. However, because the magnetic behavior of these Cs2NaAnIIICl6 compounds is weakly paramagnetic above 50 K, we only consider the low-temperature regime, i.e., from 3 to 77 K. For the UIII and CmIII derivatives, the computed effective magnetic moments and Curie constants are in very good agreement with the values obtained by Hendricks et al., with deviations of less than 10%. For the NpIII derivative, this difference rises to about 20%, and all computed quantities are found to be larger than their experimental counterparts. Finally, for the PuIII derivative, the computed effective magnetic moments are about 40−45% smaller than the experimental ones, while the Curie constants are about 80−85% smaller. This large difference is most likely due to the fact that all levels of the ground-state manifold are thermally populated under the experimental conditions (i.e., saturation is reached), while this is not the case for the computed data. Indeed, when fitted in the 150−300 K temperature range (where all states are populated), the CASSCF (NEVPT2) effective moment is 0.95 ± 0.06 μB (0.98 ± 0.06 μB) and the Curie constant 0.112 cm3 K mol−1 (0.121 cm3 K mol−1). These values are in much better agreement with those reported by Hendricks et al. and actually indicate that the splitting of the ground-state manifold is probably overestimated in the calculations. Surprisingly, the Curie constants computed for the UIII and NpIII derivatives are less than half of those of the corresponding free ions (Table 6). This is most likely due to the octahedral splitting of the free ion J states combined with the lack of thermal saturation in the ground-state multiplet. Indeed, while the splitting of the ground-state manifold spans about 500 cm−1, the magnetic susceptibility has not been measured beyond 100 K (i.e., 69.5 cm−1). Also, for both complexes, J mixing is found to be negligible in the ground-state manifold (Table 7), most likely because of the large energy gap to the 8807


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Inorganic Chemistry Table 5. Energies of the f−f Transitions of [AmIIICl6]3− (cm−1)a expt E 0 2725 4992 5408

9050 9619 9652

12050 12075 12210

12782

16850

18950 19365 19488 19503 20809 21530

irrep

CASSCF 2S+1

LJ

E

NEVPT2 J

irrep

Γ1 Γ4 Γ3 Γ5 Γ2 Γ5 Γ4 Γ5 Γ3 Γ4 Γ1 Γ5 Γ4 Γ3 Γ4 Γ3 Γ5 Γ2 Γ1

7

F0 F1 7 F2 7 F2 7 F3 7 F3 7 F3 7 F4 7 F4 7 F4 7 F4 7 F5 7 F5 7 F5 7 F5 7 F6 7 F6 7 F6 5 D0

0 2140 4331 4612 6679 6722 6818 8540 8902 8928 8950 10461 10493 10517 10853 11854 11888 11966 12204

(1) (3) (2) (3) (1) (3) (3) (3) (1) (3) (2) (3) (3) (2) (3) (2) (3) (1) (3)


0 1 2 2 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6

Γ5 Γ4 Γ1

7

F6 F6 7 F6

12242 (3) 12273 (1) 13982 (1)

Γ4 Γ1 Γ1

Γ4 Γ1 Γ4 Γ4?Γ5 Γ5 Γ2 Γ3 Γ3 Γ5

5

18779 21953 22029 22116 22434 22606 22663 23432 23819

Γ4 Γ1 Γ4 Γ5 Γ2 Γ5 Γ3 Γ5 Γ3

7

7

D1 L6 5 L6 5 L6 5 L6 5 L6 5 L6 5 G2 5 G2 5

(3) (1) (3) (3) (1) (3) (2) (3) (2)

E (1) (3) (2) (3) (1) (3) (3) (3) (1) (3) (2) (3) (3) (2) (3) (2) (3) (1) (1)


6 (98.1%) 6 (99.4%) 0 (99.8%)

12969 (3) 13000 (3) 13049 (1)

Γ5 Γ4 Γ1

1 6 6 6 6 6 6 2 2

17928 21418 21485 21499 21604 21622 21634 22461 22826

Γ4 Γ5 Γ4 Γ1 Γ2 Γ3 Γ5 Γ5 Γ3

(99.8%) (99.6%) (99.7%) (98.3%) (99.5%) (96.7%) (99.0%) (97.9%) (99.2%) (98.3%) (98.7%) (98.0%) (94.3%) (97.7%) (98.7%) (97.6%) (97.5%) (99.5%) (96.2%)

(100.0%) (99.9%) (100.0%) (99.3%) (100.0%) (98.5%) (98.9%) (98.8%) (98.8%)

0 2662 5050 5398 7496 7563 7669 9329 9782 9802 9803 11232 11274 11292 11704 12570 12608 12680 12924

J

irrep

(3) (3) (3) (1) (1) (2) (3) (3) (2)

0 1 2 2 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 0 6 6 0 6 1 6 6 6 6 6 6 2 2

(99.7%) (99.4%) (99.5%) (97.3%) (99.3%) (94.8%) (98.4%) (96.6%) (98.3%) (98.3%) (97.4%) (97.1%) (91.4%) (96.6%) (97.9%) (96.4%) (96.6%) (99.3%) (19.2%) (80.7%) (94.0%) (96.8%) (19.2%) (80.1%) (99.9%) (99.0%) (99.9%) (99.6%) (100.0%) (99.7%) (98.1%) (97.1%) (99.8%)

a

Experimental data are taken from ref 55. The electronic states are identified with respect to their symmetry, using the irreducible representations of the octahedral double group. Experimentally, states are also labeled with respect to the 2S+1LJ term they stem from. For the computed spectrum, the quantum number J, alone, of the state having the largest conbribution (J mixing in parentheses) is preferred. The associate wave function, expressed in terms of the eigenstates of JZ (i.e., |J,MJ⟩ or simply |MJ⟩), are shown in section S3.2.4.

closest excited state of the same symmetry (about 4000 cm−1 higher in energy). For the CmIII derivative, all levels from the ground-state manifold are fully degenerate and, thus, equally populated whatever the temperature range. Therefore, the measured and computed Curie constants are very similar to that of the free ion. This is also supported by the first excited state (with the same symmetry), being more than 20000 cm−1 above the ground state, thus minimizing J mixing. Finally, for the PuIII derivative, the CASSCF and NEVPT2 values obtained in the 150−300 K temperature range (i.e., 0.112 and 0.121 cm3 K mol−1, respectively) are close to those obtained for the free ion (0.142 and 0.170 cm3 K mol−1, respectively). This again can be explained by the absence of J mixing, with the closest excited states with either Γ7 or Γ8 symmetry about 3000 cm−1 higher than the ground state (Table 7). The slight difference is probably due to perturbation of the SOC by the ligand-field interaction (as suggested by the relativistic nephelauxetic reduction).

An−Cl Bonding and Covalency Trends in the [AnIIICl6]3− Complexes (AnIII = ThIII−NoIII). Understanding bonding and covalency trends in actinide complexes is of utmost importance. This is particularly true in the field of spent nuclear fuel reprocessing,4 where important steps toward simpler and safer storage of spent nuclear fuel will be made only once one is able to efficiently separate the minor actinides from the postPUREX104 waste.3 The minor actinides indeed represent the smallest but also the most toxic part of the waste. To make such extraction possible, ligands that specifically bind much more (or less) covalently to the minor actinides (than to the other components of the post-PUREX waste) must be used. However, designing such ligands requires, first and foremost, a comprehensive understanding of the nature of the actinide− ligand (An−L) bond. By definition, a covalent bond is a bond that relies on the sharing of electron pairs between two atoms. As such, it can be divided into two contributions: (i) a symmetry-restricted covalency and (ii) a central-field covalency. If ones focuses 8808


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Inorganic Chemistry Table 6. Effective Magnetic Moments μeff (μB), Curie Constants C (cm3 K mol−1), and Temperature-Independent Magnetic Term χTIP (cm3 mol−1) for [AnIIICl6]3− (AnIII = UIII−PuIII and CmIII)a

parameters and trends upon comparison of the covalencies derived from different models and/or spectroscopic techniques and/or types of calculations. Indeed, not all models are equivalent. For instance, considering covalency in terms of orbital overlap and near degeneracy, the model by Neidig et al.,39 which is based on extended Hueckel ideas only focuses on the symmetry-restricted aspect of the covalency. Herein, we adopt a different strategy: We analyze the An−L covalency by assessing the changes observed in the ligand-field parameters upon complexation of the trivalent ions. In that context, bond formation and covalency are thus considered starting from the ionic limit [i.e., An3+−(Cl6)6−].115 More precisely, we investigate to which extent the interelectronic (Fk) and effective one-electron spin−orbit (ζ) parameters are reduced by orbital mixing and radial wave-function expansion through bond formation. Considering the covalency as a one-electron property, this approach allows us to account for both symmetry-restricted and central-field covalencies. Therefore, both the nephelauxetic reduction and the relativistic nephelauxetic reduction in the [AnIIICl6]3− complexes (Figure 2a) are good indicators of the level of covalency in the An−Cl bond. However, because we only consider the ligand-field parameters associated with the 5f shell, we are only able to assess the contribution of the 5f orbitals to bonding. In the [AnIIICl6]3− series, both the nephelauxetic and relativistic nephelauxetic reductions are larger in the first half than in the second half of the [AnIIICl6]3− series (Figure 2a). This indicates that chlorine binds more covalently to the early actinides than to the late actinides. However, because the computed reductions remain small (i.e., lower than 6%), all An−Cl bonds remain predominantly ionic in character. This covalency trend might appear counterintuitive knowing that the An−Cl bond lengths (Table 4) decrease across the [AnIIICl6]3− series, especially when this decrease is greater (on average) than the decrease observed in the 5f radial expectation values of the corresponding free-ion series (see section S3.1.2). Under such conditions, one would expect the shorter An−Cl bonds to be more covalent. However, as mentioned earlier in this section, the covalency does not depend explicitely on the length of the An−Cl bond but rather on the radial distribution of the An(5f) orbitals and their propensity to mix with ligand orbitals. In other words, no matter how close two atoms approach one another, they will not bind covalently if their valence orbitals do not mix. Thus, the trend in both the nephelauxetic and relativistic nephelauxetic reductions indicates that orbital mixing is stronger for the early actinides and that

a

These quantities are determined using the same Curie law and the same procedure as those in Table 5 in the temperature range given by T. Experimental data are always given on the first line of each row (in black) and were taken from ref 60. The second (blue) and third (red) lines correspond to the CASSCF and NEVPT2 results, respectively. The values in parentheses correspond to the free-ion case.

only on the participation of the 5f shell to bonding, the first contribution is induced by the mixing of An(5f) orbitals with the valence orbitals of the ligands, while the second one is characterized by a change (expansion) in the radial wave function of the 5f orbitals upon interaction with the ligands. Because neither orbitals nor wave functions are observables, covalency is not either. It is thus very difficult to quantify to which extent a chemical bond holds the covalency, both experimentally and computationally. However, in recent years, spectroscopic techniques such as electron paramagnetic resonance (EPR), Mössbauer, photoelectron spectroscopy, and K-edge X-ray absorption spectroscopy (XAS) have been successfully used to provide a quite detailed picture of An−L bonding and to assess the covalency.37−40,105−108 Computationally, An−L bonding has mostly been investigated by means of population analysis and topological approaches,4,69,105,109−113 even though some more orbital-related approaches are also available, such as those presented in the recent work by Nakajo et al.114 In all of the aforementioned spectroscopic measurements and calculations, bonding and covalency are usually assessed by means of parameters or quantities that are derived from a given electronic structure model. Because different models with different physical and chemical interpretations of bonding are available, one should not be surprised to find different

Table 7. Electronic Structure of the Ground-State Manifold of [AnIIICl6]3− (AnIII = UIII−PuIII and CmIII)a [UIIICl6]3− 9

J

/2 (99.5%)

CASSCF

0 180 515

(4) (2) (4)

Γ8 Γ6 Γ8

NEVPT2

0 196 576

(4) (2) (4)

Γ8 Γ6 Γ8

0 492 497 498 0 553 565 605

[NpIIICl6]3−

[PuIIICl6]3−

[CmIIICl6]3−

4 (100.0%)

5

7

/2 (99.9%)

Γ5 Γ4 Γ3 Γ1 Γ5 Γ3 Γ4 Γ1

(3) (3) (2) (1) (3) (2) (3) (1)

/2 (100.0%)

0 47

(2) (4)

Γ7 Γ8

0 0 0

(2) (4) (2)

Γ7 Γ8 Γ6

0 91

(2) (4)

Γ7 Γ8

0 4 6

(2) (4) (2)

Γ7 Γ8 Γ6

a The energy (with degeneracy in parentheses) is given in cm−1 together with the associate quantum number J and irrep of the octahedral double group. The percentage in parentheses after the J value gives the contribution of J mixing with respect to this very J (J mixing is only considered within the limit of the three first manifolds).

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Figure 3. (a) Orbital mixing coefficient (%) and (b) natural charges (e) in all [AnIIICl6]3− complexes. In part a, blue squares correspond to the nonbonding a2u molecular orbitals, red circles to the π antibonding t2u molecular orbitals, and green triangles to the σ and π antibonding t1u molecular orbitals. The orbital mixing coefficient α is defined as ψ = αϕM − (1 − α 2) ϕL with ψ the delocalized molecular orbital, ϕM the metal contribution, and ϕL the ligand contribution. In part b, positive values indicate a charge excess, while negative values indicate a charge defect.

of the An(5f) orbitals with the Cl(3p) orbitals is mostly of antibonding character, the AOM parameters cannot be directly related to the level of covalency in the An−Cl bonds. Yet, because AOM is applied to active orbitals, which are not pure 5f orbitals but also have some (very little) ligand character, in the weakly covalent limit, the AOM parameters can, to a certain extent, account for the covalency of the An−Cl bond, hence suggesting that the An−Cl f covalency is mostly driven by σ interactions. Last, to gain deeper insight into the contribution of the 6d orbitals to bonding, the nephelauxetic and relativistic nephelauxetic reductions associated with the 6d shell are computed for the very early derivatives for the trivalent hexachloride series. Indeed, the Th3+−U3+ free ions and corresponding [AnIII Cl6]3− complexes are the only derivatives for which it was possible to converge CASSCF/NEVPT2 + SOC calculations with only the 6d orbitals in the active space, i.e., CAS(n,5). The computed Slater−Condon parameters for interelectronic repulsion, F2 and F4,116 as well as the effective one-electron spin−orbit parameter, ζ, for the 6d shell are shown in section S3.2.5. Both the nephelauxetic and relativistic nephelauxetic reductions associated with the 6d shell (about 20%) are more than two times larger than those associated with the 5f shell. This suggests a much stronger covalency from the 6d orbitals than from the 5f orbitals, which is in line with NPA (Figure 3b) but also with the recent work by Cross et al.117 In addition and contrast to the trend in the 5f shell, the nephelauxetic and relativistic nephelauxetic reductions do not seem to decrease along the series but remain fairly constant. This suggests the constant covalent contribution of the 6d orbitals (at least in the beginning of the series). Comparison of the Tetravalent Actinides An4+ with An = Pa(f1)−Bk(f7). Herein we only consider the first half of the actinide series because the tetravalent oxidation state is not stable for the second half of the series. Similar to the previous section, we perform a comparative study of the ligand-field parameters and covalency trends between the first half of the trivalent and tetravalent actinides. Free Ions. First, we investigate the ligand-field parameters of the tetravalent free ions, i.e., the Fk Slater−Condon parameters for interelectronic repulsion in the 5f shell and the effective

the 5f radial wave function is more affected by the chloride ions in the beginning of the series. Our ligand-field-based analysis of the bonding and covalency is reflected in both the orbital mixing coefficients (Figure 3a) and population analysis (Figure 3b and section S3.7). Indeed, An−L orbital mixing is larger in the first half of the series than in the second half. In our opinion, this result proves that our “simple” but chemically intuituive ligand-field considerations allow for valuable insight into the An−Cl covalency. Also, in more detail, while the a2u orbitals are not affected (because of their nonbonding character), orbital mixing appears to be larger for the t1u orbitals than the t2u orbitals. This observation is fully in line with the work by Löble et al.36 and suggests that at least the ligand t1u orbitals should be added to the active space to improve the description of bonding in this hexachloride series. Yet, this is going beyond the scope of the present discussion, which only focuses on An−Cl bonding from the perspective of the 5f orbitals, and will be part of a separate investigation. On a different note, the observed trend in the An−Cl f covalency is also supported by the natural population analysis (NPA; Figure 3b) and the Löwdin reduced orbital charges (see section S3.7). The excess [defect] of charge in the An(5f) [Cl(3s,3p)] orbitals decreases across the [AnIIICl6]3− series. The mirror symmetry in the natural charge associated with the An(5f) and Cl(3s,3p) orbitals indicates a charge transfer from the Cl(3s,3p) orbitals to the An(5f) orbitals following the same trend as the 5f covalency inferred from our AILFT analysis. NPA also informs us that there is a nonnegligible charge transfer from the Cl(3s,3p) orbitals to the An(6d) and An(7s) orbitals. The magnitudes of these charge transfers stay roughly the same along the whole series. Interestingly, the intensity of the Cl(3s,3p) → An(6d) charge transfer, which is about two times larger than that of the Cl(3s,3p) → An(5f) charge transfer, suggests that the 6d orbitals also contribute to bonding, probably making the An−Cl bond even more covalent (see below for further details). Also, the changes in the f covalency along the [AnIIICl6]3− series are in line with the decrease of the eσ and eπ AOM parameters (Figure 2b) with increasing atomic number. Indeed, the strength of the σ and π An−Cl antibonding interactions correlates directly with the radial expansion of the 5f orbitals. Nevertheless, because interaction 8810


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Figure 4. (a) Nephelauxetic and relativistic nephelauxetic reductions (%) and (b) AOM parameters (cm−1) associated with all [AnIVCl6]2− complexes. In part a, the blue, red, and green colors are for the reductions of F2, F4, and F6, respectively, and the black color is for the reduction of ζ. All reductions are defined as (1 − αcomplex/αfree ion) × 100, with α one of the aforementioned parameters. In part b, the blue open circles correspond to eσ, and the red filled circles to eπ. Dashed lines correspond to the same quantities for the associate trivalent derivatives.

one-electron spin−orbit parameter ζ (see section S3.3.1). These parameters follow the same trends as those seen for the trivalent free ions, i.e., increasing across the series. Also, similar to the trivalent ions, (i) the CASSCF Fk parameters are larger than the experimental ones by anywhere from 15−30% (Bk4+) to 55−65% (U4+). (ii) When accounting for the dynamical correlation, the agreement between the experimental and computed parameters becomes almost quantitative, with the Fk parameters overestimated by 2−10% (Bk4+) to 25−35% (U4+). Finally, (iii) the spin−orbit parameter is never overestimated by more than 10%. The overall experimental/computational agreement is better for the tetravalent ions than for the trivalent ions. This is due to correlation effects having a different origin in the tetravalent ions (see section S3.3.2). On the other hand, similar to the trivalent free-ion case, this overestimation of the AILFT parameters is responsible for an overestimation of the energy splittings in the associate excitation spectra (see sections S3.3.3 and S3.3.4). With respect to the trivalent ions, the tetravalent Slater−Condon and spin−orbit parameters are found to be larger regardless of whether we compare ions with the same number of electrons in the 5f shell or ions with the same atomic number. In the former case, this is due to the nuclear charge being larger for tetravalent ions than for trivalent ions (for a given number of 5f electrons), while in the latter case, it is due to the number of 5f electrons being greater for trivalent ions than for tetravalent ions (for a given nuclear charge). In either case, the 5f shell is more contracted for the tetravalent ion, and thus interelectronic repulsion is stronger. This is in line with the changes in both the radial expectation values and Shannon’s ionic radii for both series (see sections S3.3.2 and S3.4.1). For the same reasons, the SOC constants are larger for the tetravalent ions than for the trivalent ions, thus also leading to a significant departure from Hund’s rules (see sections S3.3.3 and S3.3.4). Bonding Trend in the [AnIVCl6]2− Complexes. As for the trivalent complexes, bonding is investigated by assessing changes in the AILFT parameters upon complexation. (The An−Cl distances are optimized for the tetravalent hexachloride complexes using the same approach as that for the trivalent derivatives (see section S1.3). They are reported and further discussed in section S3.4.1.) From Figure 4a, it appears that the

nephelauxetic and relativistic nephelauxetic reductions are larger and decrease more slowly for the [AnIVCl6]2− complexes than for the [AnIIICl6]3− complexes. This indicates not only that the An−Cl bonds in the tetravalent derivatives are more covalent than the trivalent derivatives but also that the covalency of the An−Cl bonds is more stable in the tetravalent series. This result shows that the An−Cl covalency strongly depends on the oxidation state of the actinide ion, which is in line with the work of Beekmeyer and Kerridge.69 This is of particular importance when it comes to the selective extraction of actinide ions from nuclear wastes. Indeed, if the An−L bond is more covalent for a given metal ion, it means that it will be easier to selectively extract this particular ion from other actinide ions. In the present case, for instance, the further we go along the series, the easier it should be to separate (i) tetravalent ions from trivalent ions (i.e., CmIII from CmIV) and (ii) early trivalent ions from each other than tetravalent ions from each other. However, one should not forget that we presently only consider the 5f covalency and that the 6d covalency may also play an important (or even more important) role here. The observed differences in the covalency of the An−Cl bond in the tetravalent and trivalent hexachloride complexes are also supported by the natural charges, the mixing coefficients of the active orbitals (see section S3.4.2), and the Löwdin reduced orbital charges (see section S3.7). In more detail, charge transfers are of the same kind as those observed in the trivalent series, i.e., Cl(3s,3p) → An(5f) and An(6d). They also have the same features, with the Cl(3s 3p) → An(5f) charge transfer decreasing toward the center of the series and the Cl(3s,3p) → An(6d) charge transfer remaining fairly unchanged. The major difference between the two series comes from all charge transfers being about two times bigger in the tetravalent series. The same holds true for the trend in orbital mixing, which is much larger (about 5%) and fairly constant for the tetravalent series. Finally, concerning the ligand-field splitting of the f orbitals and the associate eσ and eπ AOM parameters (Figure 4b and section S3.4.3), it appears that both sets of parameters follow the same trend, i.e., decreasing across the series. Interestingly, the AOM parameters of the tetravalent complexes are about twice that of the corresponding trivalent complexes, while the 8811


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Figure 5. (a) Nephelauxetic and relativistic nephelauxetic reductions (%) and (b) AOM parameters (cm−1) associated with all [LnIIICl6]3− complexes. In part a, the blue, red, and green colors are for the reduction of F2, F4, and F6, respectively, and the black color is for the reduction of ζ. All reductions are defined as (1 − αcomplex/αfree ion) × 100, with α one of the aforementioned parameters. In part b, blue open circles correspond to eσ and red filled circles to eπ. Dashed lines correspond to the same quantities for the associate actinide derivatives.

the other hand, differ significantly. The Slater−Condon parameters of the lanthanide ions are about 3/2 that of the corresponding actinide ions. This is due to the smaller radial distribution of the 4f orbitals with respect to the 5f orbitals, bringing the f electrons closer to one another and increasing interelectronic repulsion in the valence f shell of the lanthanide ions. Values for the spin−orbit parameter of the lanthanide ions are about 1/2 that of the actinide ions. This is expected because the effective nuclear charge is larger in the actinide derivatives. As a consequence, the computed excitation spectra are in much better agreement with the experiment,88 the Russell−Saunders coupling scheme can be used to describe the electronic levels of the lanthanide free ions and the ground state complies with Hund’s rules (see section S3.5.2). Bonding Trend in the [LnIIICl6]3− Complexes. For the [LnIIICl6]3− series. (The Ln−Cl distances are taken from ref 88 without further optimization because they nicely reproduce the lanthanide contraction (see section S3.6.1).) Concerning the evolution of the interelectronic and SOC parameters, the nephelauxetic and relativistic nephelauxetic reductions are much smaller in the [LnIIICl6]3− complexes than in the [AnIIICl6]3− complexes (Figure 5). In particular, while Fk are reduced by up to 6% at the beginning of the actinide series, they remain almost unchanged in the lanthanide equivalents, with all computed reductions lower than 2%. The overall trend is, however, the same for both lanthanides and actinides because there is a decrease in all computed reductions with increasing atomic numbers. These features suggest (i) that the AnIII−Cl bonds are more covalent than their Ln−Cl analogues, (ii) that the M−Cl bonds are more covalent in the first half of both series of complexes, and (iii) that the M−Cl bond remains predominantly ionic in both series. This is in agreement with the 5f orbitals being more diffuse and thus more inclined to interact with the surrounding chloride ions relative to the 4f orbitals. This is also in line with the covalency trend evidenced above. These bonding tendencies are supported by NPA (see section S3.6.2). While the population of the 5d, 6s, and 6p orbitals in lanthanides is more or less identical with that of the 6d, 7s, and 7p orbitals in the actinides, there are noticeable changes in the population of the M(4f/5f) orbitals and the Cl(3s/3p) orbitals. In particular, we do not see any ligand-to-

average 2/1 ratio between eσ and eπ remains unchanged. This trend is in agreement with the An−L covalency (Figure 4a). Comparison with the Trivalent Lanthanides Ln3+ with Ln = Ce−Yb. Even though lanthanides and actinides belong to the f block, their electronic structures are rather different. They are thus expected to have different spectroscopic, magnetic, and bonding properties. Typically, in the case of lanthanides, interelectronic repulsion is larger than SOC and again larger than the ligand field. This particular hierarchy makes it possible to use Hund’s rules, the LS coupling scheme, and the Landé interval rule.95 Finally, because the 4f orbitals are more contracted than the 5f orbitals, the ligand field is expected to be smaller in lanthanides than in actinides. Thus, we do not expect any J mixing. Hereafter, we report on a comparative AILFT analysis of the trivalent lanthanide and actinide free ions and their associate hexachloride complexes. This is done based on the work of Aravena et al., who, using the same AILFT and AOM approaches, managed to rationalize regularities in the structural, thermodynamical, and spectroscopic properties of the [LnCl6]3− complexes.88 Free Ions. As previously performed for the trivalent and tetravalent actinide free ions, we first consider the Slater− Condon and SOC parameters of the lanthanide trivalent free ions (see section S3.5.1). The agreement between the experimental and computed parameters is much better for the lanthanide free ions than for their actinide analogues. The Fk parameters are only overestimated by 5−10% at the NEVPT2 level, and the spin−orbit parameter is never overestimated by more than 10%. This level of accuracy is the consequence of a more balanced description of the electron correlation effects in lanthanides when using a CAS(n,7) (see section S3.5.3). Probably the excited-state configurations outside the f shell and with proper g−u symmetry to interact with the fn configurations are further away from the fn configurations in lanthanides than in actinides. This is actually supported by the energy difference between the ground and first excited states being about twice as large in the actinide trivalent free ions (Table 2 and section S3.1.7) than in their lanthanide analogues (see section S3.5.2). As for the actinide trivalent free ions, the AILFT parameters increase with increasing atomic number. Their amplitudes, on 8812


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functions can still be accurately described using the CASSCF/ NEVPT2 + SOC approach, with an error of a few hundred wavenumbers for the lower excited states (i.e., Δ (with Δ the overall splitting of the ground-state manifold), which is the case here. (99) Finley, J.; Malmqvist, P.-A.; Roos, B. O.; Serrano-Andrés, L. The multi-state CASPT2 method. Chem. Phys. Lett. 1998, 288, 299−306. (100) Sauri, V.; Serrano-Andrés, L.; Shahi, A. R. M.; Gagliardi, L.; Vancoillie, S.; Pierloot, K. Multiconfigurational Second-Order Perturbation Theory Restricted Active Space (RASPT2) Method for Electronic Excited States: A Benchmark Study. J. Chem. Theory Comput. 2011, 7, 153−168. (101) Seth, M.; Dolg, M.; Fulde, P.; Schwerdtfeger, P. Lanthanide and Actinide Contractions: Relativistic and Shell Structure Effects. J. Am. Chem. Soc. 1995, 117, 6597−6598. (102) In the context of ligand-field theory, the bonding or covalency is considered to be a one-electron property. The NEVPT2 approach, on the other hand, makes use of two-electron operators to account for the dynamical correlation. Analyzing the covalency using the NEVPT2 AILFT parameters would thus, most likely, be biased by unphysical two-electron effects. (103) Barbanel, Y. A.; Kotlin, V. P.; Kolin, V. V. Absorption spectra of curium (III) chloride complexes. Sov. Radiochem. 1977, 19, 497− 501. (104) Herein, PUREX stands for Plutonium Uranium Redox EXtraction. (105) Formanuik, A.; Ariciu, A.-M.; Ortu, F.; Beekmeyer, R.; Kerridge, A.; Tuna, F.; McInnes, E. J. L.; Mills, D. P. Actinide covalency measured by pulsed electron paramagnetic resonance spectroscopy. Nat. Chem. 2016, 9, 578−583. (106) Gianopoulos, C. G.; Zhurov, V. V.; Minasian, S. G.; Batista, E. R.; Jelsch, C.; Pinkerton, A. A. Bonding in Uranium(V) Hexafluoride Based on the Experimental Electron Density Distribution Measured at 20 K. Inorg. Chem. 2017, 56, 1775−1778. (107) Kozimor, S. A.; Yang, P.; Batista, E. R.; Boland, K. S.; Burns, C. J.; Clark, D. L.; Conradson, S. D.; Martin, R. L.; Wilkerson, M. P.; Wolfsberg, L. E. Trends in Covalency for d- and f-Element Metallocene Dichlorides Identified Using Chlorine K-Edge X-ray Absorption Spectroscopy and Time-Dependent Density Functional Theory. J. Am. Chem. Soc. 2009, 131, 12125−12136. (108) Clark, J. P.; Green, J. C. An investigation of the electronic structure of bis(η-cyclo-octatetraene)-actinoids by helium-(I) and -(II) 8816


Ab Initio Ligand-Field Theory Analysis and Covalency Trends in

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