Kinetic energy dependent reactions of Th+ with O2 and CO are studied using a guided ion beam tandem mass spectrometer. The formation of ThO+ in the reaction of Th+ with O2 is observed to be exothermic and barrierless with a reaction efficiency at low energies of k/kLGS = 1.21 ± 0.24 similar to the efficiency observed in ion cyclotron resonance experiments. Formation of ThO+ and ThC+ in the reaction of Th+ with CO is endothermic in both cases. The kinetic energy dependent cross sections for formation of these product ions were evaluated to determine 0 K bond dissociation energies (BDEs) of D0(Th+–O) = 8.57 ± 0.14 eV and D0(Th+–C) = 4.82 ± 0.29 eV. The present value of D0 (Th+–O) is within experimental uncertainty of previously reported experimental values, whereas this is the first report of D0 (Th+–C). Both BDEs are observed to be larger than those of their transition metal congeners, TiL+, ZrL+, and HfL+ (L = O and C), believed to be a result of lanthanide contraction. Additionally, the reactions were explored by quantum chemical calculations, including a full Feller-Peterson-Dixon composite approach with correlation contributions up to coupled-cluster singles and doubles with iterative triples and quadruples (CCSDTQ) for ThC, ThC+, ThO, and ThO+, as well as more approximate CCSD with perturbative (triples) [CCSD(T)] calculations where a semi-empirical model was used to estimate spin-orbit energy contributions. Finally, the ThO+ BDE is compared to other actinide (An) oxide cation BDEs and a simple model utilizing An+ promotion energies to the reactive state is used to estimate AnO+ and AnC+ BDEs. For AnO+, this model yields predictions that are typically within experimental uncertainty and performs better than density functional theory calculations presented previously.

Actinides (An) are of interest because of their use in nuclear power and because of national security concerns, however, research is hampered because of the radioactive nature of most members of the actinide series (except Th and U), which make them difficult and potentially dangerous to investigate. Therefore, it is highly desirable to employ theoretical methods to study these systems. In order to evaluate potential basis sets and theoretical methods, key fundamental experimental benchmarks are necessary. Gas-phase studies that are absent solvent effects can provide these benchmarks, and an increasing number of gas-phase studies of actinide systems have been reported.1–17 These have been accompanied by an increasing number of theoretical reports.14–26 Several examples of discrepancies between experimental and theoretical results exist,14,23,24 such that this field remains in its infancy.

Of these studies, oxidation reactions are probably the best studied. Previously, Marçalo and Gibson13 have determined that there is a correlation between bond dissociation energies (BDEs) of AnOp+ (p = 0–2) and the promotion energy (Ep) of Anp+ to the lowest level having a 6d2 configuration (Usual ground state configurations for An are 5fn−36d7s2 and 5fn−27s2, and for An+ are 5fn−27s2 and 5fn−17s.) Th and Th+ are unique among the actinides because they do not populate the 5f-orbitals in their ground states having 6d27s2 and 6d27s ground level configurations, respectively. One interesting aspect of these configurations is that they compare directly to transition metal systems that are better understood.

Because of its 6d27s ground state configuration, the Ep of Th+ is zero and thus Th+ is the most reactive of the actinide series and has been described as oxophilic. Because of the reactivity of the thorium cation, it is difficult to make a direct measurement of D0 (Th+–O) with thermal methods such as ion cyclotron resonance (ICR) mass spectrometry. Currently, only the lower limit D0(Th+–O) ≥ D0(H2C–O) = 7.85 eV can be established by direct measurement on the basis of ICR results.5 Indirectly, D0 (Th+–O) can be determined using the thermochemical cycle in

D0(Th+– O)=D0(ThO)+IE(Th)IE(ThO),
(1)

where the ionization energies IE(Th) = 6.306 92 eV9,27,28 and IE(ThO) = 6.602 63 ± 0.0002 eV9 are well established. Evaluations of previous thermochemical data by Pedley and Marshall29 from high temperature methods such as Knudsen effusion experiments provide D0 (Th–O),30–34 but such data are dependent on the free energy functions (and molecular parameters) used to adjust energies to 0 K values. Choice of parameters can have a large impact on the reported BDE,29,32 a thorough discussion of which is presented below.

A unique aspect of guided ion beam tandem mass spectrometry (GIBMS) is the ability to control reactant energies over a large range of kinetic energies, which allows the study of the energy dependences of endothermic reactions to establish direct measurements of key thermodynamic information. Furthermore, no knowledge of product molecular parameters is needed. Accurate experimental determination of such BDEs is critical for establishing reliable benchmarks to which theoretical methods can be evaluated. Previously, MO+ BDEs have successfully been measured using GIBMS for first,35–38 second,39,40 and third41–44 row transition metals. Here we present the absolute kinetic energy dependent cross sections of the reactions of Th+ with O2 and CO measured using GIBMS and analyze these to determine D0 (Th+–O) and D0 (Th+–C). We also compare theoretically derived BDEs to these experimental benchmarks and discuss what implications knowledge of Th+ thermochemical values has for the An+ series.

The thermochemistry of ThOn+ (n = 0, 1) is seemingly well established. Pedley and Marshall29 reevaluated data primarily from Ackermann and Rau,30,31 Hildenbrand and Murad,32 and Murad and Hildenbrand33 and established D0(Th–O) = 9.06 ± 0.125 eV. (Pedley and Marshall cite values of 9.08 ± 0.11 eV, 9.04 ± 0.03 eV and 9.09 ± 0.15 eV,30 9.09 ± 0.17 eV and 9.06 ± 0.17 eV,31 8.97 ± 0.20 eV and 9.08 ± 0.20 eV,32 and 9.06 ± 0.11 eV.33) Originally, Ackermann and Rau reported D0(Th–O) ≤ 8.3 eV30 in weight loss experiments and D0(Th–O) = 9.0 ± 0.1 eV in Knudsen cell effusion experiments,31 and Hildenbrand and Murad reported D0(Th–O) = 8.79 ± 0.13 eV.32 Because all data were extrapolated from high temperature regimes, both Pedley and Marshall and Hildenbrand and Murad caution that significant errors can occur in the use of free energy functions because of poorly established molecular parameters.29,32 In this case, errors are plausible because the parameters used by Pedley and Marshall list a 3Δ first excited state with all spin-orbit (SO) levels at ∼5000 cm−1 above the ground state (presumably on the basis of work by Edvinsson45). In earlier work, Huber and Herzberg46 identify the same state as 1Φ. A more recent compilation47 of experimental data identifies the state as 3Δ but with levels found at 5317, 6128, and 8600 cm−1 above the ground level, which agree well with theoretical work by Paulovic et al.47 and Küchle et al.48 

The Gas-phase Ion and Neutral Thermochemistry (GIANT) compilation49 references Pedley and Marshall but extrapolates to 0 K differently so that D0(Th–O) = 9.04 ± 0.11 eV. Marçalo and Gibson later adopt the value of Pedley and Marshall, but list the uncertainty as two standard deviations of the mean, D0(Th–O) = 9.06 ± 0.25 eV.13 Most recently, Konings et al.50 evaluated the previously reported values of the ThO BDE and concluded D0(Th–O) = 9.00 ± 0.10 eV (we report the uncertainty as 2 standard deviations of the mean), where the Ackermann and Rau30 values are excluded for reasons unstated and an additional value of D0(Th–O) = 8.89 ± 0.17 eV reported by Neubert and Zmbov34 is included. A weighted average of the values originally reported by their respective authors (9.0 ± 0.1 eV,31 8.79 ± 0.13 eV,32 and 8.89 ± 0.17 eV34) excluding the upper limit (≤8.3 eV30) yields D0(Th–O) = 8.92 ± 0.14 eV where the uncertainty is two standard deviations of the mean.

In general, reports of ThO+ BDEs have been derived using Eq. (1). Data found in the GIANT compilation lead to a BDE of D0(Th+–O) = 9.03 ± 0.14 eV,49 but GIANT utilizes older values of IE(Th) = 6.08 eV51 and IE(ThO) = 6.1 ± 0.1 eV52 (electron ionization values). More recent spectroscopic determinations of these values are IE(Th) = 6.306 92 eV9,27,28 and IE(ThO) = 6.602 63 ± 0.0002 eV.9 Using the updated IEs, a value of 8.74 ± 0.14 eV can be established from the D0 (Th–O) given in the GIANT tables and Eq. (1). Marçalo and Gibson13 report D0(Th+–O) = 8.74 ± 0.25 eV on the basis of the Pedley and Marshall neutral BDE and the spectroscopic IEs. A value of D0(Th+–O) = 8.70 ± 0.10 eV can be derived using Eq. (1) with the neutral BDE value reported by Konings et al.50 Finally, a value of D0(Th+–O) = 8.62 ± 0.14 eV can be derived from the weighted average of the original reports of D0 (Th–O). All values are consistent with the lower limit established in ICR studies, D0(Th+–O) ≥ D0(H2C–O) = 7.85 eV.5 

Unlike ThOp+, the thermochemistry of ThCp+ (p = 0, 1) is much less well established. The only report of the ThC BDE is D298(Th–C) = 4.70 ± 0.18 eV determined in Knudsen cell effusion experiments.53 An electron ionization energy of IE(ThC) = 7.9 ± 1.0 eV was reported in the same study and is similar to a prior value of 8.0 ± 0.1 eV.54 Neglecting the difference between the 298 and 0 K BDEs, D0(Th+–C) = 3.1 ± 1.0 eV can be determined using Eq. (1) and the lower IE value, a value that is probably best expressed as a lower limit for reasons discussed below.

The GIBMS used in these experiments has been described in detail previously.55 Briefly, ions are created in a direct current discharge/flow tube source (DC/FT) described in more detail below.56 After exiting the source, ions are focused through a magnetic momentum analyzer where the reactant 232Th+ ion beam is mass selected. These ions are decelerated to a well-defined kinetic energy and passed into a radio frequency (rf) octopole ion guide57,58 that constrains the ions radially. The octopole passes through a static pressure reaction cell that contains the neutral reaction partner (O2 or CO). To ensure that the probability of multiple collisions between Th+ and the neutral gas is sufficiently small, the pressure in the reaction cell is maintained at typical pressures of 0.10–0.40 mTorr. Independent measurements at several pressures are performed to ensure that measured cross sections are independent of neutral reactant pressures. Reaction cross sections are calculated from product ion intensities relative to reactant ion intensities after correcting for background ion intensities measured when the neutral gas is no longer directed into the gas cell.59 Uncertainties in the calculated absolute cross section are estimated to be ±20%, with relative uncertainties of ±5%.

Laboratory (lab) ion energies are converted to the center-of-mass (CM) frame using the relationship ECM = Elab × m/(m + M) where m and M are the masses of the neutral reactant and ion, respectively. At very low energies, the conversion includes a correction for the truncation of the ion kinetic energy distribution as described previously.59 Cross sections are known to be broadened by the kinetic energy distribution of the reactant ions and the thermal (300 K) motion of the neutral reactant.60 The absolute zero of energy and the full width at half-maximum (fwhm) of the ion beam are determined by using the octopole guide as a retarding potential analyzer.59 Typical fwhms of the energy distribution for these experiments were 0.4–0.6 eV (lab). Uncertainties in the absolute energy scale are 0.1 eV (lab). All energies reported below are in the CM frame.

The DC/FT source has been described in detail previously.56 Briefly, a cathode containing the thorium powder sample (232Th, 100% abundance) is held at ∼2.5 kV. The resultant electric field ionizes Ar gas that flows over the cathode in a 9:1 He/Ar mixture. The ionized Ar collides with the cathode and Th+ ions are sputtered off and swept into the flow tube by the He/Ar flow at typical pressures of 0.3–0.4 Torr. In the flow tube, ions thermalize by ∼105 collisions with carrier gas. In this work and previous work with Th+,16,17 there is no evidence of excited state species. Previous experiments61–65 utilizing the DC/FT source with transition metal ions have indicated that the internal of temperature of ions generated is 300–1100 K. A population analysis at 300 K indicates that 99.89% of Th+ is in its ground level (4F3/2, 6d27s), whereas at 1100 K, 76% is in the ground level.28,66 Conservatively, we estimate the internal temperature to be 700 ± 400 K, where Th+ has an average electronic energy of Eel = 0.02 ± 0.03 eV. The average excitation energy and its uncertainty are incorporated into all threshold and bond dissociation energies reported here.

The kinetic energy dependence of endothermic reactions is modeled using58,67,68

σ(E)=σ0Σgi(E+Eel+EiE0)n/E,
(2)

where σ0 is an energy-independent scaling factor, E is the relative kinetic energy of the reactants, Ei is the internal energy of the neutral reactants having populations gigi = 1), n is an adjustable parameter, and E0 is the 0 K reaction threshold. Before comparison to the data, Eq. (2) is convoluted over the kinetic energy distributions of the reactants, and the σ0, n, and E0 parameters are optimized using a nonlinear least-squares method to best reproduce the experimental cross section.59,69 Uncertainties in E0 are calculated from the threshold values from several independent data sets over a range of acceptable n values and combined with the absolute uncertainties in the kinetic energy scale and internal energies of reactant ions (0.02 ± 0.03 eV). At high energies, cross sections decline because of product dissociation, so Eq. (2) is modified to include a statistical model of the dissociation probability, as discussed in detail elsewhere.70 Briefly the dissociation probability is controlled by two adjustable parameters: p, which is similar to n, can hold only integer values and Ed, the energy at which product cross sections begin to decline. In most cases, inclusion of the high-energy model does not significantly alter the analysis of E0, however, when a significant deviation is observed, the model that most accurately reproduces the experimental cross section in the threshold region is used. Consequently, for Th+ + CO → ThO+ + C, the high energy model is not included in the present threshold analysis.

E0 obtained from Eq. (2) is used to determine the bond dissociation energy (BDE), D0 (Th+–L), using Eq. (3) where L = O or C

D0(Th+L)=D0(CO)E0,
(3)

Eq. (3) assumes that there are no barriers in excess of the endothermicity of the reaction. No experimental evidence was found to suggest that such a barrier is present in either system studied here, and potential energy surfaces (PESs) presented below confirm that no barriers are present.

The majority of the quantum chemical calculations were performed using the Gaussian 09 suite of programs.71 For Th+, a polarized correlation consistent core-valence quadruple-ζ (20s17p12d11f7g4h1i)/[9s9p8d8f7g4h1i] basis set72 was used with the Stuttgart-Cologne multiconfigurational Dirac Hartree-Fock (MDF) small core (60 electron) relativistic effective core potential73 (ECP), cc-pwCVQZ-PP. The cc-pwCVTZ-PP72 and atomic natural orbital ANO-VQZ-MDF73 basis sets were also used in combination with the MDF ECP. Additionally, Stuttgart-Dresden (SDD-VDZ-MWB), segmented quadruple-ζ (Seg. SDD-VQZ-MWB), and atomic natural orbital (ANO-VQZ-MWB) basis sets48,74 were used in combination with the Stuttgart-Dresden small core quasi-relativistic ECP (MWB). The aug-cc-pwCVQZ,75 cc-pwCVXZ (X = T, Q),76 and Pople77 6-311+ G(3df) basis set were used for C and O. Extrapolation to the complete basis set (CBS) limit for the cc-pwCVXZ (X = T, Q) basis sets was performed using the Karton-Martin method,78 Eq. (4), for Hartree Fock (HF) energies (where Y = 3 for T and Y = 4 for Q)

EX=ECBS+A(Y+1)e6.57Y.
(4)

For coupled-cluster singles and doubles with perturbative (triples) [CCSD(T)]/cc-pwCVXZ calculations, Eq. (5)79 is used to extrapolate the correlation energy

EX=ECBS+B(Y+12)4.
(5)

The use of these basis sets has previously yielded reasonable results for other Th+ and Th systems.16,17,72,80–82

Structures were optimized using density functional theory functionals, B3LYP,83,84 B3PW91,85 BH and HLYP (BHLYP),86 M06,87 and PBE088 with unrestricted wavefunctions. B3LYP and B3PW91 have proven reliable in actinide theoretical calculations by us and others.16,17,20,24 PBE0 and M06 have also yielded reasonable results and M06 was indicated as a promising functional in studies of the ThO2+ BDE.17,26 BHLYP has previously performed well in actinide systems when the molecule is singly bound16,17 but performs poorly in systems with higher bond orders.16,89 Nevertheless, it is included here because it appears to perform well in energy spacing between electronic states in previous studies of Th+.16,17 Additionally, single point energies using a spin unrestricted coupled cluster method that mixes in single and double excitations and perturbative triple excitations, CCSD(T),90–93 are performed using the B3LYP optimized structures. For electron correlation calculations using CCSD(T), the Th+ 5s and 5p and the C/O 1s electrons are frozen. All energies discussed below are corrected by the zero point energy using the frequencies generated at their respective optimized structure after scaling by 0.989.94 Potential energy surfaces are generated by performing relaxed potential scans along the ∠LThO+ coordinate (L = C or O).

For the above theoretically calculated BDEs, a semi-empirical approach that corrects for spin-orbit (SO) splitting is employed. This model is described in detail elsewhere.16,43,44,95,96 Briefly, the uncorrected theoretical BDE is a value averaged over all spin-orbit states of the molecule and the dissociation asymptote. To correct for the SO splitting of the asymptote, the contributions of L are considered negligible, and contributions of Th+ are corrected by the difference in energy of the ground level and the energy of the ground state averaged over all SO levels. For Th+, the J = 3/2 ground level is a mixture of the 4F3/2 and 2D3/2 levels. For the purpose of comparing experimental energies to theoretical energies, we have previously assigned the ground level as 4F3/2.16 Experimentally, the 4F3/2 ground level lies 0.40 eV below the SO averaged 2D ground state, which lies 0.06 eV below the SO averaged 4F state. The SO energy of ThL+ can be corrected using a model described elsewhere,16,43,44,95,96 however, for the present systems, ThL+ ground states are Σ states (as discussed below) that do not exhibit first-order SO splitting, such that no additional correction is necessary, although this ignores any potential second-order effects from interacting states. For the present systems, potential interacting states are separated sufficiently in energy that second-order effects are not believed to be significant. Furthermore, the empirical spin-orbit correction used here is comparable to the spin-orbit contributions calculated using a composite thermochemistry approach described below.

Accurate composite thermochemistry, as outlined in the Feller-Peterson-Dixon (FPD) method,97,98 was used to describe the numerous contributions to the atomization energies at 0 and 298 K for ThC+, ThC, ThO+, and ThO. The majority of these calculations were carried out at the CCSD(T) level of theory with the third order Douglas-Kroll-Hess (DKH3) Hamiltonian99,100 utilizing aug-cc-pVXZ-DK basis sets76,101,102 on the O and C atoms and the all-electron cc-pVXZ-DK3 basis sets on Th (X = D, T, Q)72 (denoted cc-pVXZ-DK3 below). Core-valence correlation (1s on C and O with 5s5p5d on Th) was also considered, and in these cases, the aug-cc-pwCVXZ-DK (O, C)103 and cc-pwCVXZ-DK3 (Th)72 basis sets were used. Geometries were optimized at the CCSD(T)/cc-pVQZ-DK3 level of theory and were consistently used as the reference geometries for all single point calculations except in the case of the zero point energy (ZPE) described below. Open-shell calculations employed restricted open-shell HF (ROHF) orbitals but the spin restriction was relaxed in the CCSD(T) calculations, i.e., the R/UCCSD(T) method104–106 was used. Because of the highly multi-reference character of the 4F3/2 ground state of Th+ when SO is included, all dissociation energy calculations below were carried out relative to the ground electronic state of neutral Th atom (3F2). The calculated dissociation energies were then corrected to the ground state of the Th+ cation using the accurate experimental ionization energy (IE) of Th atom (50 868.71(8) cm−1 or 6.306 92(1) eV).9 All of these calculations, excluding the SO contributions, were carried out using the MOLPRO quantum chemistry package.107 

The final FPD calculated dissociation enthalpies at 0 K consisted of the following contributions:

D0=EV QZDK+ΔECBS+ΔECV+ΔESO+ΔEQED+ΔET+ΔEQ+ΔEIE+ΔEZPE,
(6)

where EV QZDK is the equilibrium dissociation energy at the frozen-core CCSD(T)/cc-pVQZ-DK3 level of theory. The HF energies were then extrapolated to the CBS limit using Eq. (4) with cc-pVTZ-DK3 and cc-pVQZ-DK3 basis sets, while the correlation energies were extrapolated to their CBS limits using Eq. (5). The results of these two extrapolations for the molecules and atoms were combined to yield the total CBS limit dissociation energies, with the difference between the latter values and EV QZDK yielding ΔECBS. ΔECV is the core correlation contribution, ECV − Evalence, both in the same cc-pwCVXZ-DK3 basis sets (X = T and Q), extrapolated to the CBS limit using Eq. (4). The value of ECV was obtained by correlating the 5s, 5p, and 5d electrons of Th and the 1s electrons of C and O in addition to the valence electrons.

SO contributions, ΔESO, were calculated using the exact two-component (X2C) Hamiltonian108,109 with uncontracted cc-pVDZ-DK3 basis sets and 2-component open-shell coupled cluster, CCSD(T),109 for Th, ThC+, ThO, ThO+ and Fock space CCSD (FS-CCSD)110 for ThC. The open-shell calculations for the Th atom utilized average-of-configuration HF orbitals (1 electron in the 10 spinors arising from the 6d orbitals). All SO calculations involving coupled cluster correlated only the valence electrons and were carried out using a virtual orbital cutoff of 12.0 a.u. The FS-CCSD calculations on ThC involved calculating the (0,2) sector from the closed-shell ThC2+ reference state. Analogous calculations were also carried out for ThC+ [from sector (0,1)] in order to obtain a consistent SO correction for the ionization energy of ThC. The SO calculations were carried out using the DIRAC program.111 The SO contributions for the O and C atoms (0.009 67 and 0.003 67 eV, respectively) were obtained from their experimental energy levels.112 

ΔEQED is a contribution for quantum electrodynamic (QED) effects, namely, the Lamb shift. When considering molecules that contain heavy atoms such as actinides, this contribution can begin to become significant.113 In this work, the local potential approach of Pyykkö has been used for both the vacuum polarization and self-energy contributions.72,114 The latter were carried out with the MOLPRO program at the frozen-core CCSD(T) level of theory with the cc-pwCVDZ-DK3 basis sets at the frozen-core cc-pVQZ-DK3 geometries.

The next two terms, ΔET and ΔEQ, account for valence electron correlation effects beyond the CCSD(T) level of theory. The ΔET term is defined as the difference between CCSD with iterative triples (CCSDT)115–117 and CCSD(T) in the cc-pVTZ-DK3 basis set. The effects of quadruple excitations, ΔEQ, were defined as the difference between coupled-cluster singles and doubles with iterative triples and quadruples (CCSDTQ)118–121 and CCSDT using cc-pVDZ-DK3 basis sets. The CCSDTQ/cc-pVDZ-DK3 calculations on ThO involved just under 4.3 × 109 configurations. The MRCC program122 as interfaced to MOLPRO was used for all the higher-order electron correlation calculations. After correcting the dissociation energies of ThC+ and ThO+ to the Th+ dissociation asymptote using the experimental IE of Th,9 ΔEIE, harmonic frequencies at the frozen-core CCSD(T)/cc-pVDZ-DK3 level of theory were used to define the zero point vibrational energy of each molecule, yielding ΔEZPE. Thermal corrections, ΔH(0−298), were calculated using standard ideal gas partition functions to adjust the 0 K dissociation energies to 298 K, i.e., D298 = D0 − ΔH(0−298).

The cross sections as a function of kinetic energy for the reaction of thorium cation with molecular oxygen at a pressure of 0.3 mTorr are presented in Figure 1. The following reactions were observed:

Th++O2ThO++O,
(7)
ThO++O2ThO2++O.
(8)

The energy dependence of the cross section for reaction (7) declines with increasing energy, consistent with an exothermic, barrierless reaction. At low energies, the reaction efficiency is k/kLGS = 1.21 ± 0.24, where the collision limit, kLGS, is the Langevin-Gioumousis-Stevenson (LGS) formula.123 This result is consistent with the results of two separate FT-ICR studies where the reaction efficiency was observed as k/kLGS = 1.12 ± 0.223 and k/kLGS = 0.86 ± 0.43.5 The cross section declines with an energy dependence of E−0.40±0.1, consistent with the energy dependence (E−1/2) of the LGS cross section (σLGS) until approximately 0.6 eV where the cross section levels until 2 eV. At 2 eV, the cross section begins to decline more rapidly until dropping off even faster beginning near 6 eV. The rapid decline starting near 6 eV can be attributed to there being sufficient energy present to dissociate the ThO+ product, a process that can begin at D0(O–O) = 5.117 eV.124 

FIG. 1.

The absolute cross section for the reaction of Th+ + O2 as a function of kinetic energy in the laboratory (upper x-axis) and center-of-mass (lower x-axis) frames. The solid line represents the Langevin-Giomousis-Stevenson collision limit. The arrow shows D0(O–O) = 5.117 eV.

FIG. 1.

The absolute cross section for the reaction of Th+ + O2 as a function of kinetic energy in the laboratory (upper x-axis) and center-of-mass (lower x-axis) frames. The solid line represents the Langevin-Giomousis-Stevenson collision limit. The arrow shows D0(O–O) = 5.117 eV.

Close modal

The energy dependence of reaction (7) from 0.6 to 2 eV is unusual because the cross section deviates from σLGS and has a shallower energy dependence. This energy dependence cannot be a transition to the hard sphere collision limit, which we estimate as 16 Å2 calculated using the atomic radii reported by Waber and Cromer125 (Th = 1.186 Å and O = 0.450 Å) and r(O–O) = 1.208 Å reported by Huber and Herzberg.126 (Note that the atomic radius is used as an estimate of the Th+ ionic radius, but the expected error will be minimal.) A similar energy dependence has been observed previously in the reactions of Zr+ and Nb+ with O2.40 A possible explanation, explored and previously presented in detail,40 is that these reactions couple with the M2+ + O2 asymptote thus creating a Coulombic interaction (V1 ∝ r−1) that is more attractive than the ion–induced dipole interaction (V4 ∝ r−4). In the Zr+ and Nb+ cases, the M2+ + O2 asymptotes are too high in energy to influence the reaction dynamics.40 By contrast, the deviation from σLGS occurs at higher energies for Th+ and IE(Th+) = 11.65 ± 0.35 eV13 is significantly smaller than IE(Zr+) = 13.1 eV and IE(Nb+) = 14.0 eV.28 Thus, the Coulombic interaction may be significant in the Th+ case. Calculations of the V1 and V4 surfaces following the procedure in Ref. 40 are explained in detail in the supplementary material.129 These results indicate that near 0.5 eV, the distance r at the V1 = V4 crossing point exceeds the r value at the maximum along the V4 surface that defines the LGS cross section. Consequently, the reaction may crossover and proceed along the more attractive V1 potential at larger intermolecular distances, yielding a larger cross section.

The ThO2+ cross section in Figure 1 is dependent on the O2 neutral reactant gas pressure, indicating that ThO2+ forms in a sequential process, reaction (8). The cross section for reaction (8) has an energy dependence of E−1.1±0.2, consistent with that expected for product formation in sequential reactions occurring at the LGS rate. The observation of reaction (8) is interesting because direct measures of the reaction ThO+ + O2 yield no products in FT-ICR experiments at thermal energies.3,12 Consistent with these ICR observations, GIBMS studies of the ThO+ + O2 reaction in our lab yield an energy dependence inconsistent with a simple exothermic reaction. Because of these unusual reaction dynamics, a more complete analysis of this reaction is beyond the scope of the present text and will be published elsewhere.127 Ultimately, the explanation for this dichotomy is that reaction (8) is observed because the ThO+ products from reaction (7) are not thermalized.

The cross sections of the reaction of Th+ with CO as a function of kinetic energy are presented in Figure 2. Both reactions 9 and 10 are observed,

Th++COThO++C,
(9)
ThC++O.
(10)

Reaction (9) has an apparent threshold of 2.5 eV with a cross section that increases with increasing energy until it peaks near 8 eV. Reaction (10) has an apparent threshold near 7.5 eV that corresponds with the initial decline of the ThO+ cross section. The ThC+ cross section peaks near D0(C–O) = 11.109 ± 0.005 eV.124 Although not apparent on the logarithmic scale on Figure 2, the total Th+ + CO reaction cross section peaks near D0 (C–O) where sufficient energy is available to allow both ThO+ and ThC+ product to dissociate, equivalent to atomizing CO according to reaction 11,

Th++COTh++O+C.
(11)
FIG. 2.

The absolute cross section for the reaction of Th+ + CO as a function of kinetic energy in the laboratory (upper x-axis) and center-of-mass (lower x-axis) frames with model cross sections, Eq. (2), convoluted over the reactant internal and kinetic energy distributions (solid lines) and unconvoluted (dashed lines). The arrow shows D0(C–O) = 11.109 eV.

FIG. 2.

The absolute cross section for the reaction of Th+ + CO as a function of kinetic energy in the laboratory (upper x-axis) and center-of-mass (lower x-axis) frames with model cross sections, Eq. (2), convoluted over the reactant internal and kinetic energy distributions (solid lines) and unconvoluted (dashed lines). The arrow shows D0(C–O) = 11.109 eV.

Close modal

The barrierless cross section for reaction (7) indicates that D0(Th+–O) ≥ D0(O–O) = 5.117 ± 0.001 eV, consistent with previous ICR results3,5 and reported literature values.13,50 Modelling the ThO+ cross section of reaction (9) reproduces the experimental cross section over the entire energy range, Figure 2. The measured threshold is E0 = 2.54 ± 0.14 eV with other modelling parameters used in Eq. (2) listed in Table I. This yields D0(Th+–O) = 8.57 ± 0.14 eV from Eq. (3). The present value is lower than (but within experimental uncertainty of) the BDE adopted by Marçalo and Gibson, 8.74 ± 0.25 eV,13 and is within combined experimental uncertainties of the value derived from Konings et al., 8.70 ± 0.10 eV.50 Interestingly, when combined with IE(Th) and IE(ThO) in Eq. (1), the present value leads to D0(Th–O) = 8.81 ± 0.16 eV, which agrees very well with D0(Th–O) = 8.79 ± 0.13 eV, originally reported by Hildenbrand and Murad,32 and with D0(Th–O) = 8.89 ± 0.17 eV originally reported by Neubert and Zmbov.34 Thus, the present work suggests that the lower values of D0 (Th–O) in the literature are probably more accurate.

TABLE I.

Fitting parameters from Eq. (1) for the indicated reaction cross section.

Reactionnσ0E0 (eV)D0(Th+–L)
Th+ + CO → ThO+ + C 1.4 ± 0.2 3.5 ± 1.0 2.54 ± 0.14 8.57 ± 0.14 
Th+ + CO → ThC+ + O 1.6 ± 0.2 0.8 ± 0.3 6.29 ± 0.29 4.82 ± 0.29 
Reactionnσ0E0 (eV)D0(Th+–L)
Th+ + CO → ThO+ + C 1.4 ± 0.2 3.5 ± 1.0 2.54 ± 0.14 8.57 ± 0.14 
Th+ + CO → ThC+ + O 1.6 ± 0.2 0.8 ± 0.3 6.29 ± 0.29 4.82 ± 0.29 

The ground and excited states of ThO+ calculated with the cc-pwCVQZ-PP basis sets are listed in Table II. Values obtained using additional basis sets are listed in Table S1 in the supplementary material.129 The calculated ground state of ThO+ is 2Σ+ with a (1σ)2(2σ)2(1π)4(3σ)1 molecular orbital occupation. The 1σ-orbital is the O 2s-orbital, the 2σ-bonding orbital is formed as two O 2pz-electrons are donated into an empty Th+ 6dz2-orbital, and the 1π-bonding orbitals are formed as the remaining O 2p-electrons pair with the two Th+ 6-electrons. Note that this configuration corresponds to a triple bond, consistent with the very strong bond. The radical electron is found in the 3σ-orbital, which is largely composed of the Th+ 7s-orbital, indicating that the 2Σ+ state forms from the atomic asymptote, Th+ (4F, 6d27s) + O (3P), the ground level configuration. A natural bond orbital (NBO) analysis using the B3LYP/cc-pwCVQZ-PP-MDF/aug-cc-pwCVQZ approach is listed in Table S2 and verifies this bonding model. Previous theoretical reports also find a 2Σ+ ground state with the singly occupied σ-orbital having 93% 7s-character.22,26 A low-lying 2Δ state is also found 0.35–0.58 eV higher in energy than the ground state where the lone electron moves to a 6-orbital (1δ molecular orbital). A relaxed potential scan of the diabatic potential energy surface for dissociation of ThO+ (2Δ), Figure S2, suggests that the 2Δ state correlates with the Th+ (4F, 6d3) + O (3P) asymptote. A third doublet state (2Π) where an electron from the 1πb bonding orbital is moved to the Th+ 7s-orbital, (1σ)2(2σ)2(1π)3(3σ)2, is also found 0.73–1.22 eV higher in energy and is most likely formed from Th+ (2D, 6d7s2) + O (3P). Because the J = 3/2 level has 4F3/2 and 2D3/2 mixed character, the 2Π state presumably can form directly from the Th+ ground level. Additional states were also found but were at least 4 eV higher in energy, Table II.

TABLE II.

Molecular parameters and relative energies (in eV) of low lying states of ThO+ and ThC+.a

ThL+Stater(Th+–L)bνbExpt.cCCSD(T)dB3LYPB3PW91BHLYPM06PBE0
ThO+ 2Σ+ (1σ2243σ) 1.808(1.807) 950(955) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
 2Δ (1σ2241δ) 1.825 902(918e0.58f 0.58 0.51 0.41 0.48 0.36 0.41 
 2Π (1σ22321.834 884(904) 0.92g 1.21 0.96 0.84 1.01 0.73 0.86 
 4Δ (1σ22σ1π41δ3σ) 2.003 672  4.48 4.25 4.24 3.95 4.56 4.21 
 4Π (1σ22σ1π43σ2π) 2.007 666  4.91 4.39 4.42 4.20 4.57 4.41 
 4Σ+ (1σ22σ1π43σ4σ) 2.006 669  5.07 4.49 4.52 4.33 4.70 4.52 
 4Σ (1σ22σ1π422.017 653  5.35 5.20 5.13 4.87 5.36 5.09 
ThC+ 2Σ+ (1σ242σ) 1.902 903  0.00 0.00 0.00 0.00 0.00 0.00 
 4Π (1σ232σ3σ) 2.049 766  0.80 0.61 0.69 0.52 0.94 0.70 
 2Π (1σ2322.023 668  0.82 0.64 0.77 0.55 0.85 0.77 
 4Φ (1σ232σ1δ) 2.068 737  1.32 1.07 1.09 0.95 1.22 1.09 
 4Σ (1σ2221δ) 2.124 696  1.55 1.31 1.46 1.27 1.50 1.50 
 2Δ (1σ241δ) 1.962 828  1.79 1.68 1.76 1.68 1.67 1.79 
 2Π (1σ242π) 1.970 814  2.41 2.08 2.12 2.17 2.02 2.18 
ThL+Stater(Th+–L)bνbExpt.cCCSD(T)dB3LYPB3PW91BHLYPM06PBE0
ThO+ 2Σ+ (1σ2243σ) 1.808(1.807) 950(955) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
 2Δ (1σ2241δ) 1.825 902(918e0.58f 0.58 0.51 0.41 0.48 0.36 0.41 
 2Π (1σ22321.834 884(904) 0.92g 1.21 0.96 0.84 1.01 0.73 0.86 
 4Δ (1σ22σ1π41δ3σ) 2.003 672  4.48 4.25 4.24 3.95 4.56 4.21 
 4Π (1σ22σ1π43σ2π) 2.007 666  4.91 4.39 4.42 4.20 4.57 4.41 
 4Σ+ (1σ22σ1π43σ4σ) 2.006 669  5.07 4.49 4.52 4.33 4.70 4.52 
 4Σ (1σ22σ1π422.017 653  5.35 5.20 5.13 4.87 5.36 5.09 
ThC+ 2Σ+ (1σ242σ) 1.902 903  0.00 0.00 0.00 0.00 0.00 0.00 
 4Π (1σ232σ3σ) 2.049 766  0.80 0.61 0.69 0.52 0.94 0.70 
 2Π (1σ2322.023 668  0.82 0.64 0.77 0.55 0.85 0.77 
 4Φ (1σ232σ1δ) 2.068 737  1.32 1.07 1.09 0.95 1.22 1.09 
 4Σ (1σ2221δ) 2.124 696  1.55 1.31 1.46 1.27 1.50 1.50 
 2Δ (1σ241δ) 1.962 828  1.79 1.68 1.76 1.68 1.67 1.79 
 2Π (1σ242π) 1.970 814  2.41 2.08 2.12 2.17 2.02 2.18 
a

Structure optimized at respective level of theory (except CCSD(T)) using cc-pVQZ-PP/6-311+ G(3df) basis sets.

b

Bond lengths (in Å) or frequencies (in cm−1) from B3LYP/cc-pVQZ-PP/6-311+ G(3df) optimized structures. Frequencies scaled by 0.989. Values in parentheses are experimental values from Ref. 9.

c

Reference 9.

d

Single point energies calculated using B3LYP/cc-pVQZ-PP/6-311+ G(3df) optimized structures.

e

Frequency of the 2Δ3/2 level. Frequency of the 2Δ5/2 level is 915 cm−1.

f

Average energy of the 2Δ state weighted over all spin-orbit degeneracies.

g

Energy of the 2Π1/2 level.

Experiments performed using pulsed field ionization-zero kinetic energy (PFI-ZEKE) photoelectron spectroscopy have determined a 2Σ+ ground state for ThO+.9 Low-lying levels of 2Δ3/2, 2Δ5/2, and 2Π1/2 at 0.36, 0.72, and 0.92 eV, respectively, above the ground state were also found. For reference to the theoretical values, which are an average energy of all SO levels, the experimental average of the 2Δ levels weighted over the SO degeneracies is 0.58 eV. Table II indicates that the present calculations are in very good agreement with the experimental electronic state energies.

Bond lengths and vibrational frequencies calculated using B3LYP/cc-pwCVQZ-PP/aug-cc-pwCVQZ are listed in Table II. Molecular parameters calculated using additional levels of theory are listed in Tables S3 and S4 in the supplementary material.129 In general, the B3LYP/cc-pwCVQZ-PP/aug-cc-pwCVQZ values agree very well with the experimental values. This is particularly true for the 2Σ+ ground state where theory is in very good agreement with the experimental bond length, 1.808 Å,9 and the calculated frequency differs from experiment by only 5 cm−1.

The breakdown of the contributions to the FPD dissociation energies of ThC+, ThC, ThO+, and ThO is shown in Table III. Among the CCSD(T) contributions, the CBS extrapolations increased the cc-pVQZ-DK3 dissociation energies by about 5.8–6.7 kJ/mol, whereas core-valence correlation provided further increases of 10–13 kJ/mol for the oxides. The effects of electron correlation beyond CCSD(T) were a bit surprising. The difference between full iterative and perturbative triples was about −5 kJ/mol in all four cases (decreasing the CCSD(T) De values), but the contributions of connected quadruples were relatively small, only about ±1.2 kJ/mol for the two cations and about +2.0 and +0.3 kJ/mol for ThC and ThO, respectively. As expected, the spin-orbit contributions mostly result from the atomic splitting in Th atom and range from −38 to −44 kJ/mol. After inclusion of the Th/Th+ correction for ThC+ and ThO+, as well as the ZPE corrections of 5.0–5.4 kJ/mol, the final 0 K BDEs are calculated to be 5.00 (ThC+), 4.97 (ThC), 8.68 (ThO+), and 8.98 (ThO) eV, respectively. Given the magnitude of the higher order correlation contributions, as well as the other terms, these dissociation energies are expected to be accurate to within about 0.05 eV.

TABLE III.

Calculated contributions to the dissociation energies within the FPD scheme in kJ/mol (eV). See Eq. (6) for details.

MoleculeEV QZ−DKΔECBSΔECVΔEQEDΔETΔEQΔESOΔEIEΔEZPED0D298
ThC+(2Σ+)a −88.89 (−0.92) +6.56 (+0.07) +1.37 (+0.01) +3.52 (+0.04) −5.04 (−0.05) +1.22 (+0.01) −40.02 (−0.41) 608.53 (6.31) −4.94 (−0.05) 482.30 (5.00) 485.89 (5.04) 
ThC (3Σ+)b 517.86 (5.37) +7.22 (+0.07) +5.31 (+0.06) +1.95 (+0.02) −5.79 (−0.06) +1.80 (+0.02) −44.05 (−0.46) N/A −4.80 (−0.05) 479.51 (4.97) 483.35 (5.01) 
ThO+(2Σ+)c 264.13 (2.74) +5.79 (+0.06) +10.59 (+0.11) +1.41 (+0.01) −5.05 (−0.05) −1.09 (−0.01) −41.00 (−0.42) 608.53 (6.31) −5.49 (−0.06) 837.81(8.68) 841.92 (8.73) 
ThO(1Σ+)d 894.82 (9.27) +6.56(+0.07) +12.84(+0.13) +0.11(+0.00) −5.13 (−0.05) +0.30(+0.00) −37.89 (−0.39) N/A −5.14 (−0.05) 866.47e(8.98) 870.55(9.02) 
MoleculeEV QZ−DKΔECBSΔECVΔEQEDΔETΔEQΔESOΔEIEΔEZPED0D298
ThC+(2Σ+)a −88.89 (−0.92) +6.56 (+0.07) +1.37 (+0.01) +3.52 (+0.04) −5.04 (−0.05) +1.22 (+0.01) −40.02 (−0.41) 608.53 (6.31) −4.94 (−0.05) 482.30 (5.00) 485.89 (5.04) 
ThC (3Σ+)b 517.86 (5.37) +7.22 (+0.07) +5.31 (+0.06) +1.95 (+0.02) −5.79 (−0.06) +1.80 (+0.02) −44.05 (−0.46) N/A −4.80 (−0.05) 479.51 (4.97) 483.35 (5.01) 
ThO+(2Σ+)c 264.13 (2.74) +5.79 (+0.06) +10.59 (+0.11) +1.41 (+0.01) −5.05 (−0.05) −1.09 (−0.01) −41.00 (−0.42) 608.53 (6.31) −5.49 (−0.06) 837.81(8.68) 841.92 (8.73) 
ThO(1Σ+)d 894.82 (9.27) +6.56(+0.07) +12.84(+0.13) +0.11(+0.00) −5.13 (−0.05) +0.30(+0.00) −37.89 (−0.39) N/A −5.14 (−0.05) 866.47e(8.98) 870.55(9.02) 
a

At a FC-CCSD(T)/cc-pVQZ-DK3 equilibrium bond length of 1.9269 Å. Combining the current results for ThC+ and ThC yields a final FPD ionization energy at 0 K of 610.22 kJ/mol (6.24 eV).

b

At a FC-CCSD(T)/cc-pVQZ-DK3 equilibrium bond length of 1.9617 Å.

c

At a FC-CCSD(T)/cc-pVQZ-DK3 equilibrium bond length of 1.8127 Å. Combining the current results for ThO+ and ThO yields a final FPD ionization energy at 0 K of 637.18 kJ/mol (6.60 eV) compared to the experimental value9 of 637.0570(25) kJ/mol.

d

At a FC-CCSD(T)/cc-pVQZ-DK3 equilibrium bond length of 1.8488 Å.

e

Using the experimental IEs of both ThO and Th atom, this yields a D0 for ThO+ of 837.93 kJ/mol (8.68 eV).

All of the theoretical BDEs for ThO+ calculated in this work are presented in Table IV. After inclusion of spin-orbit effects, theoretical BDEs are 8.61–8.76, 8.55–8.88, 8.70–8.80, and 8.89–8.99 eV for the SDD-VDZ-MWB, Seg. SDD-VQZ-MWB, and cc-pwCVQZ-PP/aug-cc-pwCVQZ basis sets, and CBS limit, respectively. Likewise the FPD composite thermochemistry approach yields a BDE of 8.68 eV. BDEs calculated with additional basis sets and methods are listed in Table S5 in the supplementary material.129 In general, B3LYP and CCSD(T) values (except CCSD(T)/CBS) are in good agreement with the experimental BDE reported here. Previous work by Pereira et al.14 has calculated theoretical BDEs for ThO+ using the Seg. SDD-VQZ-MWB basis set for Th+ and a 10s6p basis contracted to 5s3p by Dunning for O in conjunction with B3LYP and MPW1PW91 DFT functionals. They find D0 (Th+–O) of 9.02 and 9.29 eV for B3LYP and MPW91PW91 approaches, respectively, with no consideration for spin-orbit energy. Applying the spin-orbit correction used here yields values of 8.62 and 8.89 eV, respectively, where the B3LYP value is in very good agreement with the present experimental work.

TABLE IV.

Comparison of theoretical D0 (Th+–L) to experimental values (in eV).a

ExperimentalTheoretical
ThL+LiteratureThis workBasis setCCSD(T)bB3LYPB3PW91PBE0
ThO+ 8.74 ± 0.25c 8.57 ± 0.14 SDD 8.76 8.61 8.73  
 8.70 ± 0.10d  Seg. SDD 8.55 8.75 8.88 8.82 
 ≥7.85e  cc-pwCVQZ-PPf 8.70 8.70 8.80 8.75 
   CBS-PPg 8.94 8.89 8.99 8.94 
   FPDh 8.68    
ThC+ ≥3.12 ± 0.21i 4.82 ± 0.29 SDD 4.90 4.84 5.21  
   Seg. SDD 4.72 4.92 5.30 5.33 
   cc-pwCVQZ-PP 4.92 4.81 5.16 5.20 
   CBS-PP 5.18 4.99 5.34 5.38 
   FPDh 5.00    
MADj   SDD 0.14 0.03 0.28  
   Seg. SDD 0.06 0.14 0.40 0.38 
   cc-pwCVQZ-PP 0.11 0.07 0.29 0.28 
   CBS-PP 0.36 0.25 0.47 0.46 
   FPD 0.15    
ExperimentalTheoretical
ThL+LiteratureThis workBasis setCCSD(T)bB3LYPB3PW91PBE0
ThO+ 8.74 ± 0.25c 8.57 ± 0.14 SDD 8.76 8.61 8.73  
 8.70 ± 0.10d  Seg. SDD 8.55 8.75 8.88 8.82 
 ≥7.85e  cc-pwCVQZ-PPf 8.70 8.70 8.80 8.75 
   CBS-PPg 8.94 8.89 8.99 8.94 
   FPDh 8.68    
ThC+ ≥3.12 ± 0.21i 4.82 ± 0.29 SDD 4.90 4.84 5.21  
   Seg. SDD 4.72 4.92 5.30 5.33 
   cc-pwCVQZ-PP 4.92 4.81 5.16 5.20 
   CBS-PP 5.18 4.99 5.34 5.38 
   FPDh 5.00    
MADj   SDD 0.14 0.03 0.28  
   Seg. SDD 0.06 0.14 0.40 0.38 
   cc-pwCVQZ-PP 0.11 0.07 0.29 0.28 
   CBS-PP 0.36 0.25 0.47 0.46 
   FPD 0.15    
a

From structures optimized at the respective level of theory (except CCSD(T)) with the indicated basis set. Energies include estimated spin-orbit correction of −0.40 eV. See text and Ref. 16.

b

Energies from single point calculations using B3LYP optimized structures with the indicated basis set for Th+.

c

Reference 13.

d

Calculated from D0 (Th–O) from Ref. 50 utilizing IE(ThO) from Ref. 9 and IE(Th) from Ref. 28. See text.

e

Reference 5.

f

cc-pwCVQZ-PP/aug-cc-pwcVQZ basis sets.

g

Complete basis set limit extrapolated from pwCVXZ-PP/cc-pwCVXZ (X = T, Q) basis sets using Eqs. (4) and (5), see text.

h

Feller-Peterson-Dixon model for composite thermochemistry. See text and Eq. (6). See Table III.

i

Reference 53.

j

Mean absolute deviation from experimental value.

Modeling of the cross section from reaction (10) indicates that E0 = 6.29 ± 0.29 eV, leading to D0(Th+–C) = 4.82 ± 0.29 eV. (Note that the large uncertainty compared to the ThO+ BDE results from the smaller overall cross section, which leads to a lower signal-to-noise ratio.) To the best of our knowledge, no determination of the ThC+ BDE has been reported in the literature, although the value, D0(Th+–C) = 3.1 ± 1.0 eV, can be calculated from Eq. (1) using data from Knudsen cell experiments.53 For reasons discussed below, the latter value is almost certainly too low because IE(ThC) = 7.9 ± 1.0 eV is inaccurate. A better estimate of IE(ThC) can be established as 6.19 ± 0.34 eV using our value of D0(Th+–C) along with D0(Th–C) = 4.70 ± 0.18 eV53 in Eq. (1) (neglecting the difference between the 0 K and 298 K BDE).

Theoretical calculations establish 2Σ+ (1σ242σ) as the ground state of ThC+ where the 1σ is primarily the C 2s-orbital, the 1π orbitals are an interaction of the C 2 and Th+ 6-orbitals, and the 2σ is a bonding interaction between C 2pσ and a sd-hybridized Th+ orbitals, consistent with the NBO analysis presented in Table S2. Thus, ThC+ has a bond order of 2.5, explaining its weaker bond compared to ThO+. A 4Π excited state is found 0.52–0.94 eV higher in energy where one π-electron is moved to the 3σ-orbital (Th+ 7s). A second excited state (2Π) is found 0.55–0.86 eV higher than the ground state where one π-electron is moved to the 2σ-orbital. Other states found were at least 0.96 eV higher in energy than the ground state and are listed in Table II. Energies calculated using additional basis sets are listed in Table S1 in the supplementary material.129 

B3LYP bond lengths for ThC+ ground and excited states are listed in Table II and additional levels of theory are listed in Table S2. The calculated bond length is r(Th+–C) = 1.903 Å, which is shorter than that calculated for ThCH2+, r(Th+–C) = 2.05 Å (B3LYP/SDD-VDZ-MWB), but similar to that for ThCH+, 1.92 Å (B3LYP/SDD-VDZ-MWB).16 These calculations also show that ThCH+ has a triple bond, whereas ThCH2+ has a double bond, compared to the bond order for ThC+ found here of 2.5. In our previous work, we also determined bond strengths of D0(Th+–CH2) ≥ 4.54 ± 0.09 eV and D0(Th+–CH) = 6.19 ± 0.16 eV. On the basis of the bond orders, one expects that D0 (Th+–C) should lie between these values, which is consistent with our 4.82 ± 0.29 eV BDE but inconsistent with the much lower 3.1 ± 1.0 eV.

Theoretical BDEs for ThC+ are listed in Table IV. When including spin-orbit energy corrections, the BDEs are 4.84–5.21 eV (SDD-VDZ-MWB), 4.72–5.33 eV (Seg. SDD-VDZ-MWB), 4.81–5.20 eV (cc-pwCVQZ-PP), 4.99–5.38 eV (CBS-PP), and 5.00 eV (FPD). BDEs calculated with additional basis sets and methods are listed in Table S5 in the supplementary material.129 In general, all levels of theory are in reasonable agreement with the experimental BDE. In particular, CCSD(T) (except non-composite CCSD(T)/CBS) values are within experimental uncertainty and B3LYP values are in excellent agreement with the experimental value. These values are similar to those calculated for ThCH2+, 4.44–5.04 eV (Seg. SDD-VQZ-MWB), and smaller than those reported for ThCH+, 5.57–6.21 eV (Seg. SDD-VQZ-MWB).16 This is again consistent with a bond order in between that of ThCH2+(2) and ThCH+(3).

In order to calculate IE(ThC), additional calculations were performed for ThC using the B3LYP/cc-pwCVQZ-PP/aug-cc-pwCVQZ approach (as well as FPD). Additional single point energies were calculated using CCSD(T)/cc-pwCVQZ-PP/aug-cc-pwCVQZ with the B3LYP optimized structures. Results are listed in Table S6 in the supplementary material.129 We find that the ThC ground state is 3Σ+ (1σ242σ3σ) where orbital compositions are similar to those of ThC+ discussed above. Thus, the electron removed upon ionization of ThC is from the non-bonding 3σ (largely Th 7s) orbital so that it is likely that IE (ThC) ≈ IE(Th) and that D0(Th+–C) ≈ D0 (Th–C). Calculated ionization energies of ThC are 6.40 (B3LYP) and 6.33 (CCSD(T)) eV within experimental uncertainty of the 6.19 ± 0.34 eV value determined here and well below the electron impact values previously reported. The value of IE (ThC) obtained from the composite FPD approach is calculated to be 6.32 eV, whereas that of ThO is determined to be 6.60 eV. The former is well within the experimental uncertainty of the value determined experimentally here, while the FPD IE for ThO is in excellent agreement with the accurately known experimental value of 6.602 63 eV.9 

Relaxed potential energy scans of the electronic surfaces for reaction (7), Th+ + O2, were calculated at the B3LYP/SDD-VDZ-MWB/6-311+G(3df) level and are presented in Figure 3. Notably, these surfaces (as well as those for Th+ + CO) do not include consideration of spin-orbit effects. Initially, the reaction originates on a 2A′ surface where the minimum at very small angles can be understood as a linear Th+–O–O intermediate that lies ∼3 eV below the reactants. 2A″ and 4A′ surfaces lie slightly higher in energy. This 2A′ (2Σ+) intermediate has r(Th+–O) = 1.92 Å, 0.11 Å longer than the 1.81 Å bond length in ThO+ (2Σ+). The r(O–O) bond length of 1.32 Å is elongated from 1.20 Å in unbound O2. The stability of these intermediates appears to come from a covalent interaction of the Th+ 6d-electrons with the O2 π antibonding electrons. At slightly larger angles, there is a slight barrier relative to the 2A′ intermediate along both doublet surfaces as Th+ begins to insert into the O–O bond. At ∼35°, there is an apparent crossing with the 2A1, 2A2, and 2B1 surfaces that lead to potential wells near an angle of 45°. These are all still adducts of Th+ with O2, which pass over barriers as the ∠OThO angle increases, until dropping slightly as a linear thorium dioxide cation is approached. This leads to several excited states of linear ThO2+. The 2A1 surface crosses that for the 2B2 surface (avoided in CS symmetry) which leads to a linear global minimum that is at least 3 eV lower in energy than all other surfaces and 8.5 eV below ground state reactants. This 2B2 surface leads to the linear 2Σu+ ground state of ThO2+, which has been previously characterized.22 This intermediate should readily dissociate to the ThO+ (2Σ+) + O (3P) product asymptote, 4.8 eV higher in energy than ThO2+ (2Σu+). Clearly, these surfaces show that reaction (7) can occur with no barrier above the reactants on doublet surfaces, consistent with experiment. Further, the attractive nature of the surfaces is consistent with the efficiency of the reaction observed.

FIG. 3.

B3LYP/SDD-VDZ-MWB/6-311+ G(3df) relaxed potential energy surface calculations of the Th+ + O2 reaction as a function of ∠OTh+O in degrees. Energies are relative to Th+ (2D, 6d7s2) + O2. In C2v symmetry, doublet surfaces are represented by solid lines and quartet surfaces by dashed lines. Surfaces with Cs symmetry are represented by dotted lines. No spin-orbit interactions are included.

FIG. 3.

B3LYP/SDD-VDZ-MWB/6-311+ G(3df) relaxed potential energy surface calculations of the Th+ + O2 reaction as a function of ∠OTh+O in degrees. Energies are relative to Th+ (2D, 6d7s2) + O2. In C2v symmetry, doublet surfaces are represented by solid lines and quartet surfaces by dashed lines. Surfaces with Cs symmetry are represented by dotted lines. No spin-orbit interactions are included.

Close modal

A number of quartet surfaces were also explored and are shown in Figure 3. These could also lead to the ThO+ (2Σ+) + O (3P) products with no barriers but are clearly higher energy pathways.

If spin-orbit effects are considered, the main change to the surface is the ground level of the reactants (and the principally occupied level) becomes Th+ (4F3/2) + O2 (3Σg), which lies 0.40 eV lower than the 2D state shown in Figure 3. These reactants can combine to form doublet, quartet, and sextet surfaces (where the latter should be largely repulsive as they include no covalent interactions). Thus, evolution from reactants to products along the doublet (or quartet) surfaces remains barrierless and efficient.

Surfaces from relaxed potential surface scans calculated using B3LYP/SDD-VDZ-MWB/6-311+ G(3df) for reactions (9) and (10), Th+ + CO, are presented in Figure 4. At small angles, the lowest energy reaction pathway evolves along the 4A″ surface where the initial intermediate is linear Th+–C–O lying 1.4 eV below ground state reactants. A 2A″ state is also observed 1.2 eV below the reactants. Notably, at small angles near the reactant asymptote, s(s + 1) values for this 2A″ state are typically ∼1.76, suggesting that there is considerable spin-contamination for this doublet surface. At slightly larger angles, barriers to Th+ insertion into the C–O bond are observed. In this vicinity, a second 4A″ surface becomes the lowest energy surface and leads to an intermediate at 115°. This 4A″ intermediate has r(Th+–O) = 1.82 Å, similar to the bond length of ThO+, and r(Th+–C) = 2.34 Å, which is significantly longer than the bond length of ThC+. Thus, this intermediate can be viewed as an adduct between ThO+ (2Σ+) and C (3P), where the quartet spin indicates no covalent coupling between the two, such that this intermediate can readily dissociate to the ThO+ (2Σ+) + C (3P) asymptote 2.1 eV higher in energy than the reactant asymptote with no barrier above the product asymptote. Additionally, no barrier beyond the endothermicity of reaction (9) exists such that the global minimum can readily dissociate to the ThC+ (2Σ+) + O (3P) product asymptote that is ∼6 eV higher in energy than the reactants. Overall, the potential energy surface (PES) in Figure 4 is very similar to the analogous Hf+ + CO reaction PES.42 Like Th+ + CO, Hf+ has low-lying 4A″ and 2A″ surfaces that approach linear at small angles, and a 4A″ intermediate is found near 115°.

FIG. 4.

B3LYP/SDD-VDZ-MWB/6-311+G(3df) relaxed potential energy surface calculations of the Th+ + CO reaction in Cs symmetry as a function of ∠CTh+O in degrees. Energies are relative to Th+ (2D, 6d7s2) + CO. Doublet surfaces are represented by solid lines and quartet surfaces by dashed lines. No spin-orbit interactions are included.

FIG. 4.

B3LYP/SDD-VDZ-MWB/6-311+G(3df) relaxed potential energy surface calculations of the Th+ + CO reaction in Cs symmetry as a function of ∠CTh+O in degrees. Energies are relative to Th+ (2D, 6d7s2) + CO. Doublet surfaces are represented by solid lines and quartet surfaces by dashed lines. No spin-orbit interactions are included.

Close modal

If spin-orbit effects are considered, the Th+ (4F3/2) reacts with CO (1Σ+) along a quartet spin surface, however, because the J = 3/2 ground level is actually a mixture of 4F3/2 and 2D 3/2, both doublet and quartet surfaces should be accessible in the reaction. Furthermore, this presumably permits switching between surfaces of different spin with some facility.

In previous calculations of actinide species,72 the FPD composite approach without any higher-order correlation contributions reproduced the atomization energies of ThO and UFn species within 0.05 eV of the mean experimental values. The present values calculated for ThO+ and ThC+ BDEs, which now include correlation contributions beyond just CCSD(T), are 8.68 and 5.00 eV, respectively, which are both within the experimental uncertainty of the experimental values presented here. The composite D0 (Th+–O) value also agrees very well with the values reported by Marçalo and Gibson13 and Konings et al.50 Interestingly, to confidently obtain an accuracy of a few kJ/mol as acquired here, CCSDTQ calculations are necessary (although the latter contributions were small relative to CCSDT, this was not known a priori). This is very similar to the situation for transition metal species (see, for instance, Ref. 128) and probably indicates that high level theoretical calculations are necessary to accurately reproduce experimental BDEs for actinide systems.

Assuming that the accuracy of the composite values is similar or higher to that previously reported, then the true values of D0 (Th+–O) and D0 (Th+–C) lie at the upper range of the uncertainty of the experimental values reported here. However, it must be noted that the estimates of the accuracy of the FPD composite approach for actinide systems are derived primarily from experimental values with stated uncertainties of 0.1–0.2 eV. Additional work, by both experiment and theory, is likely necessary to further evaluate the accuracy of the FPD composite approach for actinide thermochemical values but the present results are certainly promising.

In past work,16,17 we have compared values calculated using a number of theoretical methods and various basis sets to the experimentally determined thermochemistry of several ThL+ species. Here we evaluate the performance of these methods and additional approaches in order to facilitate future computational work for Th and other actinide species.

Previously ThH+ and ThCH3+ BDEs calculated utilizing CCSD(T)/cc-pVQZ-PP/cc-pVTZ yielded results that overestimated the experimental values by 0.2–0.6 eV, whereas the calculated BDE for ThCH+ reproduced the experimental value reasonably well.16 Here, CCSD(T)/cc-pwCVQZ-PP/aug-cc-pwCVQZ calculations overestimate the experimental BDE of ThO+ and ThC+ by 0.48 and 0.26 eV, respectively. Meanwhile, CCSD(T)/Seg. SDD-VQZ-MWB calculations reproduce the experimental value for both ThO+ and ThC+ within experimental uncertainty, and CCSD(T)/SDD-VDZ-MWB calculations reproduce the experimental ThC+ BDE within uncertainty and overestimate the ThO+ BDE by 0.25 eV. Mean absolute deviations (MADs) between the two experimental values and theory at the CCSD(T) level using cc-pwCVQZ-PP, Seg. SDD-VQZ-MWB, and SDD-VDZ-MWB basis sets are 0.11, 0.06, and 0.14, respectively, indicating that, at least for calculating BDEs, there is no advantage in using the larger cc-pwCVQZ-PP basis set compared to the smaller Seg. SDD-VQZ-MWB basis set for these systems. Indeed, use of a complete basis set approach (CBS-PP) yields even worse results with a MAD of 0.36 eV. Similarly, MADs for these three basis sets using CCSD(T) are 0.28, 0.27, and 0.37, respectively, when comparing ThH+, ThCH+, ThCH3+, and ThCH4+ theoretical BDEs to experimental values. (Only a lower limit can be established for ThCH2+.)16 As pointed out by a reviewer, this does not necessarily indicate that the smaller basis sets perform better, but rather may indicate that the valence electron only approach to the CCSD(T) calculation used here may be inadequate and that additional corrections may be needed.

For DFT methods, there is little difference between calculations using the cc-pwCVQZ-PP and Seg. SDD-VQZ-MWB basis sets for all methods, although the larger basis set yields values that average 0.09 ± 0.04 eV lower and closer to experiment. In general, these calculations generally overestimate the experimental BDE with the exception of BHLYP calculations. BHLYP has previously been shown to perform poorly for multiply bound species.89 Calculations utilizing the SDD basis set also tend to overestimate the experimental BDEs, but with the exception of BHLYP calculations, perform better compared to the other two basis sets. MADs listed in Table IV indicate that there is little advantage to using the larger cc-pVQZ-PP and Seg. SDD basis sets in DFT calculations for these systems, with the latter basis set yielding the worst results.

In general, B3PW91 and PBE0 methods yield similar BDEs to each other regardless of the basis set used and tend to overestimate the experimental BDE with MADs of 0.29 and 0.28 eV, respectively (cc-pwCVQZ-PP). However, B3LYP calculations perform reasonably well with a MAD of 0.07 eV (0.14 and 0.03 eV for the Seg. SDD-VQZ-MWB and SDD-VDZ-MWB basis sets, respectively). In all calculations, the inclusion of spin-orbit effects improves accuracy compared to the experimental values. MADs comparing the lower order methods to the FPD values are similar to the MADs determined by comparing to experimental values.

In the present work, BHLYP/cc-pwCVQZ-PP calculations successfully predict the order and magnitude of the ThO+ excited states, Table II. B3LYP/cc-pwCVQZ-PP and CCSD(T)/cc-pwCVQZ-PP perform similarly whereas the other methods predict the correct ordering but underestimate the energy gaps between excited states, Table II. In previous work,16,17 BHLYP calculations reproduced the experimental order of Th+ ground and excited states with a high degree of accuracy in the energy spacing whereas all other methods, including B3LYP and CCSD(T), either ordered the states incorrectly or spaced the states too closely. Although BHLYP performs poorly in predicting the absolute BDEs for higher bond order species, it appears to perform reasonably well at predicting relative energies of excited states.

One interesting aspect of Th+ is that its ground level does not populate the 5f-orbitals in the ground state, unlike the other members of the actinide series. Its J = 3/2 ground level configuration, a mixture of 4F (6d27s) and 2D (6d7s2), can be directly compared with those for the group 4 transition metal cations, Ti+ (4F, 3d24s), Zr+ (4F, 4d25s), and Hf+ (2D, 5d6s2), which also have three valence electrons. Such a comparison has profitably been included in our analysis of the Th+ + CH416 and Th+ + H217 systems, and others have noted the similarities in the electronic structures of Th+ and Hf+ species.9,15 Here, we also include a comparison to the lanthanide Ce+ (4H, 4f5d2), which also has three valence electrons and is the lanthanide congener to actinide Th+. With the exception of CeO+, all thermochemical values in this discussion are measured from guided ion beam experiments.38,40,42 A brief description of the thermochemical values used to evaluate D0 (Ce+–O) can be found in the supplementary material.129 

MO+ BDEs for the group 4 transition metals, Ce, and Th are listed in Table V in order of increasing atomic number. With the exception of HfO+, the group 4 and Th+ BDEs increase moving down the periodic table consistent with the trend expected for the lanthanide contraction where the (n + 1) s orbital is preferentially stabilized compared to nd orbitals such that sd-hybridization is more efficient and leads to stronger σ-bonds.89,130–132 The lower than expected BDE for Hf+ is likely an electronic effect because the Hf+ ground state (2D, 5d6s2) has a doubly occupied 6s orbital that cannot readily form an M+–O triple bond, which requires the metal cation to have two unpaired electrons that can adopt π-symmetry.133 Therefore, Hf+ must be promoted to a higher state, thereby reducing the BDE. The first level with the correct symmetry (4F3/2, 5d26s) is 0.45 eV higher in energy than the ground level. Similarly, it might have been expected that D0(Th+–O) > D0(Ce+–O), however, this discrepancy can potentially be explained by the considerable 2D (6d7s2) character mixed into the J = 3/2 ground level.66 Furthermore, the lone electron in ThO+, located in the 3σ nominally non-bonding orbital, may be pushed up in energy through configuration interaction with the 2σ-orbital, thereby weakening the overall bond energy. Such an interaction in CeO+ is less likely to occur because the lone electron resides in a 4f-orbital that is decidedly non-bonding.

TABLE V.

Comparison of D0 (M+–L) bond dissociation energies (eV) for M = group 4, Ce, and Th and L = O and C.

LD0(Ti+–L)aD0(Zr+–L)bD0(Ce+–L)D0(Hf+–L)cD0(Th+–L)
6.88 ± 0.07 7.76 ± 0.11 8.82 ± 0.21d 6.91 ± 0.11 8.57 ± 0.14 
4.05 ± 0.24 4.72 ± 0.11 4.11 ± 0.82e 3.19 ± 0.03 4.82 ± 0.29 
LD0(Ti+–L)aD0(Zr+–L)bD0(Ce+–L)D0(Hf+–L)cD0(Th+–L)
6.88 ± 0.07 7.76 ± 0.11 8.82 ± 0.21d 6.91 ± 0.11 8.57 ± 0.14 
4.05 ± 0.24 4.72 ± 0.11 4.11 ± 0.82e 3.19 ± 0.03 4.82 ± 0.29 
a

Reference 38.

b

Reference 40.

c

Reference 42.

d

This work, see supplementary material.129 

e

References 28 and 134.

The periodic trends in the BDEs for MC+ for the group 4 transition metal cations, CeC+, and ThC+ parallel the trends for MO+. All MC+ BDEs are listed in Table V and are taken from guided ion beam experiments except D0 (Ce+–C), which can be calculated using Eq. (1) from D0(Ce–C) = 4.57 ± 0.12 eV,134 IE(CeC) = 6.0 ± 0.8 eV,134 and IE(Ce) = 5.5387 eV28 as D0(Ce+–C) = 4.11 ± 0.81 eV. However, the large uncertainty makes comparison to the other metal cations inexact. Comparison with the oxide values suggests that the true CeC+ BDE is close to the upper limit in this range.

With the exception of HfC+, all BDEs increase moving down the periodic table consistent with the trend expected for the lanthanide contraction.89,130–132 Like HfO+, the lower than expected BDE for HfC+ can be explained as a result of the Hf+ ground state configuration (2D, 5d6s2) that is not conducive to bonding.42 This is largely supported by quantum chemical calculations that indicate that HfC+ has a 2Σ+ (1σ242σ) ground state that cannot be formed directly from the ground state Hf+ (2D, 6d7s2) + C (3P, 2s22p2) asymptote.

Another interesting aspect of the present results is the potential insight into the thermochemistry of other actinides that are more dangerous to work with experimentally. For comparison to the present work, we adopt the AnO+ BDEs reported by Marçalo and Gibson13 but note several potential discrepancies in the supplementary material.129 BDEs are D0(Th+–O) = 8.57 ± 0.14 eV > D0(Pa+–O) = 8.29 ± 0.52 eV > D0(U+–O) = 8.02 ± 0.13 eV > D0(Np+–O) = 7.88 ± 0.10 eV > D0(Cm+–O) = 6.94 ± 0.39 eV > D0(Pu+–O) = 6.75 ± 0.20 eV > D0(Am+–O) = 5.80 ± 0.29 eV. Gibson has previously shown6 that this trend can be explained by a promotion energy argument where the AnO+ BDE is lowered by the cost of promotion from the ground level to the first level with a 6d2 configuration. Thus an AnO+ intrinsic BDE can be defined using6 

D0(An+O)=D0(An+O)+Ep(6d2).
(12)

Notably, because Ep(Th+) = 0 eV, the ThO+ BDE should act as a good estimate of the intrinsic actinide oxide cation BDE.

If the present value for D0(Th+–O) = 8.57 ± 0.14 eV is used in Eq. (12) to predict D0 (An+–O), the predicted values have a MAD of 0.3 ± 0.3 eV compared to the experimental values. This offers no improvement over predictions obtained using Marçalo and Gibson’s13 proposed value of D0(Th+–O) = 8.74 ± 0.26 eV in Eq. (12), where the MAD = 0.2 ± 0.2 eV. However, the comparison above of the ThO+ BDE to group 4 transition metals and Ce+ indicates that D0 (Th+–O) may be depressed by electronic effects, such that Ep(Th+)≠0 eV, in which case D0(Th+–O) = D0(An+–O) is not an exact approximation. Nevertheless, this model more accurately predicts D0(An+–O) than reported DFT calculations where MADs of 0.7 ± 0.4 eV (B3LYP) and 0.6 ± 0.5 eV (MPW1PW91) are obtained.

Presumably a similar model to Eq. (12) can be used to explain AnC+ BDE trends. However, unlike AnO+ BDEs, the AnC+ BDEs are unknown with the exception of ThC+ reported here and D0(U+–C) = 4.8 ± 0.5 eV derived from D0(U–C) = 4.72 ± 0.16 eV,135 IE(UC) = 6.1 ± 0.5 eV,136 and IE(U) = 6.1914 eV28 utilizing Eq. (1). Assuming that D0(Th+–C) = D0(An+–C) because Ep(Th+, 6d2) = 0 eV, one can estimate that D0(U+–C) = D0(Th+ − C) – Ep(U+, 6d2) = 4.25 ± 0.3 eV given Ep(U+, 6d2) = 0.57 eV, which underestimates the experimental BDE of UC+ but lies within the combined uncertainties.

Analysis of the kinetic energy dependence of the cross section from reaction (9), Figure 2, indicates that D0(Th+–O) = 8.57 ± 0.14 eV. Although within experimental uncertainty of previously accepted literature values, the present value is ∼0.2 eV lower, closely matching D0(Th+–O) = 8.49 ± 0.13 and 8.59 ± 0.17 eV derived from Eq. (1) using the values originally reported by Hildenbrand and Murad32 and Neubert and Zmbov,34 respectively, coupled with updated IEs.9,28 The discrepancy likely arises from the choice of parameters used to extrapolate data from high temperatures to 0 K for D0(Th–O), which can be a significant source of error as noted by Murad and Hildenbrand.29,32 D0 (Th+–O) is larger than its transition metal cation congeners, Ti+, Zr+, and Hf+, consistent with the result expected because of lanthanide contraction, however, D0(Th+–O) is weaker than its lanthanide counterpart, D0(Ce+–O), which can be explained in part by the mixed character of the Th+ J = 3/2 ground level and by the apparent, slightly anti-bonding character of the 3σ (largely 7s) orbital (a result of configuration interaction) compared to the 4f non-bonding orbital in CeO+.

Analysis of the cross section in reaction (10), Figure 2, provides the first experimental report of D0(Th+–C) = 4.82 ± 0.29 eV. IE(ThC) = 6.19 ± 0.34 eV can also be calculated using Eq. (1) and the neutral ThC BDE reported by Gupta and Gingerich.53 The ThC+ BDE value agrees well with theoretical calculations and is a significant improvement over previous values reported on the basis of appearance energies in Knudsen effusion cell studies.53 Like ThO+, D0 (Th+–C) is larger than its transition metal cation congeners consistent with that expected because of lanthanide contraction.

In general, the approximate quantum chemical calculations overestimate the ThL+ BDEs, but significant improvement is observed when including spin-orbit corrections using a semi-empirical approach. Using larger basis sets in coupled cluster calculations does not lead to any improvement in performance. This may be a result of the necessity of including higher order terms (i.e., CCSDT, CCSDT(Q), etc.) as demonstrated in composite chemistry calculations performed here. Indeed, FPD composite calculations lead to BDEs in good agreement with experimental values, suggesting that FPD represents an accurate approach to calculating actinide thermochemical values.

This work is supported by the Heavy Element Chemistry Program, Office of Basic Energy Sciences, U.S. Department of Energy, through Grant Nos. DE-SC0012249 (P.B.A.) and DE-FG02-12ER16329 (K.A.P.). R.M.C. and P.B.A. also thank the Center for High Performance Computing at the University of Utah for the generous allocation of computer time. Dr. Bert de Jong is thanked for his helpful advice on spin-orbit calculations.

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