Spectroscopic constants (Te, re, B0, ωe, and ωexe) have been calculated for the low-lying electronic states of UF, UF+, UCl, and UCl+ using complete active space 2nd-order perturbation theory (CASPT2), with a series of correlation consistent basis sets. The latter included those based on both pseudopotential (PP) and all-electron Douglas-Kroll-Hess Hamiltonians for the U atom. Spin orbit (SO) effects were included a posteriori using the state interacting method using both PP and Breit Pauli (BP) operators, as well as from exact two-component methods for U+ and UF+. Complete basis set (CBS) limits were obtained by extrapolation where possible and the PP and BP calculations were compared at their respective CBS limits. The PP-based method was shown to be reliable in calculating spectroscopic constants, in particular when using the state interacting method with CASPT2 energies (SO-CASPT2). The two component calculations were limited by computational resources and could not include electron correlation from the nominally closed shell 6s and 6p orbitals of U. UF and UCl were both calculated to have Ω = 9/2 ground states. The first excited state of UCl was calculated to be an Ω = 7/2 state at 78 cm−1 as opposed to the same state at 435 cm−1 in UF, and the other low-lying states of UCl showed a similar compression relative to UF. Likewise, UF+ and UCl+ both have Ω = 4 ground states and the manifold of low-lying excited Ω = 3, 2, 1, 0 states was energetically closer together in UCl+ than in UF+, ranging up to 776 cm−1 in UF+ and only 438 cm−1 in UCl+. As in previous studies, the final PP-based SO-CASPT2 results for UF+ and UF agree well with experiment and are expected to be predictive for UCl and UCl+, which are reported here for the first time.

Actinide-containing molecules are of significant interest due to their inherent importance in the nuclear fuel cycle and relatively unique chemistry.1,2 These molecules pose significant challenges both experimentally and computationally due to the low-lying 5f and 6d shells of actinide elements that can produce a plethora of accessible molecular electronic states. For actinide-containing molecules with ionic bonding like UO or UF, however, the low-lying states have been reliably predicted3 using ligand-field theory (LFT).4 In this manner, the atomic spectrum of the underlying uranium cation, U+ or U2+, is perturbed by the ligand (F and O2−, respectively), and the atomic configurations and their lowest J levels determine the low-lying spin-free, ΛS, and spin-orbit, Ω, states of the resulting molecule. The accurate description of their spectroscopic properties by ab initio methods is much more difficult, however, since this requires a comprehensive treatment of both relativistic and electron correlation effects.

The first spectroscopic study of the UF/UF+ system was reported by Antonov and Heaven5 using laser induced fluorescence (LIF) for UF and zero kinetic energy photoelectron (ZEKE) spectroscopy for UF+. Analysis of the rotationally resolved spectrum for UF identified the electronic ground state as |Ω| = 9/2 and yielded an accurate value for the ground state rotational constant. The first two excited electronic states, lying at 435 and 650 cm−1, were assigned to |Ω| = 7/2 and 5/2, respectively. The ground state of UF+ was determined to be |Ω| = 4, just as in the isoelectronic UO molecule.6,7 In this case, a total of 15 electronic states were characterized with |Ω| values ranging from 0 to 6, and both rotational and vibrational constants were reported. Recently, the [18.6]3.5–X(1)4.5 transition of UF has been studied8 under high resolution by measuring the Stark and Zeeman effects. These experiments resulted in an improved rotational constant for the ground state and confirmation that the latter state arises primarily from the 4I9/2 state of U+. Last, using matrix isolation infrared spectroscopy, laser ablated U atoms were reacted with gaseous HF diluted in argon, and two bands centered around 565 cm−1 were attributed to UF.9 This was consistent with both density functional theory (DFT) and coupled cluster calculations of the harmonic frequency of the lowest quartet state of UF in this same work.

Previous ab initio calculations on the UF molecule include the early effective core potential (ECP) self-consistent field (SCF) calculations of Krauss and Stevens,10 where the ionic character of its bonding was first described. Somewhat later, Federov et al.11 applied spin-orbit multiconfiguration quasidegenerate perturbation theory (SO-MCQDPT) along with the 3rd-order Douglas-Kroll scalar relativistic Hamiltonian (DK3) and the Breit-Pauli (BP) SO Hamiltonian in a study of the lowest 18 Ω states of the UF molecule. Their work correctly predicted a |Ω| = 9/2 ground state (predominately consisting of 4I9/2) for UF. Following a pair of studies of the dissociation energy of UF by Peralta et al.12 and Pantazis and Neese13 using DFT, a set of complete active space SCF (CASSCF) and 2nd-order complete active space perturbation theory (CASPT2) calculations for both UF and UF+ near their equilibrium bond lengths were reported by Antonov and Heaven5 in support of their spectroscopy study. Their calculations were based on the new relativistic ECP (or pseudopotential, PP) of Dolg and Cao14 with its accompanying basis set for the U atom. Spin-orbit effects were included via the state interacting approach,15 i.e., SO coupling was treated as a perturbation using unperturbed (spin-free) zeroth-order wavefunctions—CASSCF for the off-diagonal matrix elements and CASPT2 for the diagonals. The previous work of Federov et al.11 used an analogous approach at the MCQDPT level of theory. The resulting vertical excitation energies of Antonov and Heaven were in good agreement with their experimental values for both UF and UF+. Previous work on the UCl molecule has been limited to the calculation of its dissociation energy using DFT methods.12,13

In the present work, near-equilibrium potential curves for UX and UX+ (X = F, Cl) have been calculated primarily using the CASPT2 method with basis set extrapolations to the complete basis set (CBS) limit. These are the first calculations reported for the excited electronic states of either UCl or UCl+. Similarly to the previous study of Antonov and Heaven,5 SO effects were recovered using the state interacting method with CASSCF SO matrix elements, but in the present work, this has been carried out for bond distances other than just at equilibrium. A limited number of 2-component (2-C) Kramers restricted configuration interaction (KRCI) calculations were also carried out in an attempt to benchmark the state interacting approach. Spectroscopic constants Te, re, ωe, and ωexe were evaluated for each state where possible. Full details of the calculations are described in Sec. II, with results and discussion in Sec. III. Finally, some conclusions are given in Sec. IV.

The basis sets used for uranium in PP calculations corresponded to cc-pVnZ-PP (n = D,T,Q), while all electron (AE) calculations used cc-pVnZ-DK3.16 The energy consistent 60 electron PP of Dolg and Cao,14 which was adjusted to multiconfiguration Dirac-Hartree-Fock reference data with a perturbative estimate of the Breit interaction, was used in the present work with cc-pVnZ-PP. The basis sets for the lighter elements corresponded to aug-cc-pVnZ for F17 and aug-cc-pV(n + d)Z for Cl,18,19 as well as their DK-contracted versions.20 These combinations will be denoted as VnZ-PP and VnZ-DK (n = D, T, Q), respectively, below. For the calculations in which outer-core electrons were correlated, cc-pwCVnZ-PP16 was used on U with aug-cc-pwCVnZ utilized on F and Cl.21 The outer core is defined here as 5s5p5d on U, 1s on F, and 2s2p on Cl. The CBS limits have been determined by separately treating the HF and correlation contributions to the energy for each bond length and species. The Hartree-Fock CBS limits were obtained by using the Karton and Martin22 formula with the TZ and QZ energies,

EnHF=ECBSHF+An+1e6.57n,
(1)

where the cardinal number of the basis set, n, was used instead of ℓmax, e.g., 4 for QZ, as was previously done for the U(VI) closed shell species.23 The correlation energies were extrapolated to their CBS limits using TZ and QZ via24,25

Encorr=ECBScorr+Bn+124.
(2)

The molproab initio suite of programs26 was used throughout this work for the relativistic one-component calculations. Initially, state-averaged CASSCF calculations27,28 were carried out to represent the lowest spin-free, ΛS, states. These states were calculated in the highest abelian group available, D2h for the atoms and C2v for the molecules. For the molecular cases, expectation values of Lz2 were calculated in all cases to ensure that both components of each Λ state were correctly included. The CASSCF active space for UF and UCl included 5 e in 13 orbitals (5 × a1, 3 × b1, 3 × b2, 2 × a2) that had predominantly U 5f, 6d, and 7s character, while all lower energy orbitals (including U 6s and 6p, F 2s and 2p, and Cl 3s and 3p) were constrained to be doubly occupied. Additional higher-lying orbitals, including the 7p of U and beyond, did not appreciably contribute to the CASSCF wavefunctions of the low-lying states. This is consistent with previous investigations of UH and UF.5,29

Post-CASSCF calculations were carried out at either the internally contracted multireference configuration interaction (MRCI)30–32 or CASPT233,34 levels of theory using the same active spaces as the preceding CASSCF calculations. The resulting energies including correlation of each component of the 2L + 1 atomic states or the 2 components of the molecular states for Λ≠0 were averaged to ensure exact degeneracies. The frozen-core definition in post-CASSCF correlation treatments included all orbitals of U through the 5d (i.e., 6s-7s valence), the 1s of F, and the 1s through 2p of Cl. To avoid problems with intruder states in the CASPT2 calculations, level shifts were utilized using the smallest possible IPEA shift35 for all included states, as well as each bond length for the diatomic molecules. For UCl and UF, this shift corresponded to 0.34, while for UCl+ and UF+, it was 0.28. For U+, an IPEA level shift of 0.25 was sufficient to converge all of the states.

The state interacting method for the treatment of spin-orbit coupling was used as implemented in molpro15 to calculate the molecular Ω states. In this method, the spin-orbit eigenstates are obtained by diagonalizing Hel + HSO in a basis of eigenstates of Hel. The matrix elements of HSO are constructed using either the BP SO operator or the spin-orbit operator from the U PP. In the present work, the SO matrix elements have been calculated throughout at the CASSCF level of theory, whereas the diagonal terms of Hel + HSO have been replaced by either CASPT2 or MRCI energies. The CASSCF SO matrix elements were calculated using the same basis set as used for the diagonal terms or VQZ when the extrapolated CBS limits of the diagonals were used.

Of course in the state interacting approach, the number and identity of the relevant spin-free states (denoted ΛS) that might contribute to the final relativistic |Ω| states of interest must be determined. A potential starting point for predicting the relevant low-lying states is to select the lowest atomic asymptotes and use the molecular states that arise from coupling these. In the present cases, UX molecules and their cations, an ionic model is adopted whereby the X anion is treated as closed shell in these couplings, so that the U+ (for UX) or U2+ (for UX+) ions determine the resulting molecular states. As a result, in the absence of spin orbit coupling, the unsigned projection of the atomic orbital angular momentum L on the diatomic axis, Λ, is equal to |ML| of a given state of U+, where ML ranges from L to 0. Thus, for example, the 4I ground state term of U+ will result in 7 molecular quartet states of UF ranging from Λ = 6 to 0 (I, H, Γ, Φ, Δ, Π, Σ). In the presence of spin orbit coupling, the unsigned projection of the atomic total angular momentum J on the diatomic axis, |Ω| , is equal to |MJ| of the atom, where |MJ| ranges from J to 0 or 12 depending on if J is integer or half integer. Hence, just the 4I9/2 ground state of U+ will lead to molecular |Ω| = 9/2, 7/2, 5/2, 3/2, and 1/2 states.

In the present work, a total of 16 ΛS electronic states were calculated for UX which arise from X (1S) and U+, 7 quartet states arising from the 4I (5f37s2), and 9 sextet states from the 6L (5f36d7s). A total of 23 states were calculated for UX+ arising from U2+, 7 quintets from the 5I (5f37s), 7 triplets from 3I (also 5f37s), and 9 quintets from the 5L (5f36d). This same strategy was also used previously by Antonov and Heaven5 to select the identical states for their CASSCF and CASPT2 calculations on UF and UF+.

After diagonalization of Hel + HSO, the values of |Ω| for the molecules were assigned by converting from the Cartesian eigenfunction basis to a spherical basis, and then adding the projection of the spin angular momentum S on the diatomic axis, Σ, to Λ to obtain |Ω| . For the atoms, the quantum number J was assigned by inspection of the number of degenerate eigenstates.

2-c based methods offer the advantage of including spin from the outset but are computationally much more expensive.36–38 In the present work, the DIRAC program39 was used to run all 2-component calculations. The AE calculations used the exact two-component (X2C) Hamiltonian, which includes atomic-mean-field 2-electron spin-same-orbit contributions.37,38,40 The available 2-c methods for the present open-shell systems were limited to average of configuration self consistent field (aoc-SCF), Kramers pair restricted multiconfiguration SCF (KR-MCSCF), and KRCI. The KR-MCSCF module in DIRAC is limited to only the lowest root of a single Ω value, and thus was too limiting for the current work. Hence to treat all of the excited states with a common set of orbitals, the KRCI calculations utilized aoc-SCF orbitals. The specific details of the orbitals for each species are given below in Sec. III. In all cases to allow sufficient flexibility, the VTZ-DK basis sets were completely uncontracted. At the aoc-SCF level of theory, the individual SCF energies of each J or Ω level were resolved by carrying out a full CI just within the open shell manifold. Full valence KRCI calculations were attempted on UF and UF+ as benchmarks (correlating 6s through 7s of U and 2s2p of F), but the need for large amounts of memory limited these calculations to using very few virtual spinors, specifically those with spinor energies under 4.0 a.u. However even with this limitation, a full valence KRCI calculation for UF+ consisted of 500 × 106 to 1 × 109 determinants and required more than 50 GB of RAM per core. As a result, KRCI calculations were not further pursued with this full valence active space, and instead only a few valence orbitals were correlated, specifically those consisting of predominately U 7s, 6d, and 5f character.

Diatomic potential energy functions of UF, UF+, UCl, and UCl+ were obtained by calculating 7 energies distributed around the equilibrium value of their ground states (rre = − 0.3, − 0.2, − 0.1, 0.0, + 0.1, + 0.3, + 0.5 a.u.). Spectroscopic constants were obtained from the usual Dunham analysis41 after first fitting to polynomials in simple displacement coordinates or by solving the nuclear Schrödinger equation for each ΛS or Ω state using the program package LEVEL 8.2.42 The equilibrium bond length re and equilibrium state separation Te were obtained from the Dunham analysis, while all other spectroscopic constants were obtained from LEVEL. The harmonic frequencies and anharmonicity constants were obtained from fitting the lowest 5 vibrational levels to the simple harmonic oscillator expression

EV=ωev+1/2ωexe(v+1/2)2.
(3)

Since the diatomic potentials sample the lower vibrational levels better than the higher ones, the residuals of each level were scaled by their energy in the fits to Eq. (3). Anharmonicity constants were not reported for an individual state when the standard deviation of the fit was larger than 5 cm−1, the anharmonicity was negative, or when the Dunham analysis and the LEVEL program produced B0 constants that differed by more than a few thousandths of a wavenumber. This commonly occurred when two Ω states were strongly mixed and switched ΛS composition as a function of geometry.

With a goal of obtaining a better understanding of the interplay of spin-orbital coupling and correlation treatment for the molecules of this study, the lowest two electronic configurations of the U+ cation were investigated using both 1- and 2-component methods. In Table I the excitation energy of the lowest SO level of the 6L (5f36d7s) term (J = 11/2) relative to the ground 4I9/2 (5f37s2) level is shown for several levels of theory. In the 1-c calculations within the state-interacting approach using the PP Hamiltonian, the CASSCF method was used for the spin-orbit matrix elements with different levels of theory used for the diagonal elements of Hel + HSO. Only the components of the 4I and 6L states were included in these calculations. The CASSCF method yields a 6L11/2 ground state, which is qualitatively incorrect. All of the 1-c results using correlated energies for diagonal elements correctly place the 4I (5f37s2) state lower than the 6L (5f36d7s), albeit by very different amounts: 228 cm−1, 1376 cm−1, and 2954 cm−1 by MRCI, MRCI + Q, and CASPT2, respectively, with the VTZ basis set. Upon inclusion of spin-orbit coupling, however, only CASPT2 yielded a 4I9/2 ground state resulting from the 5f37s2 configuration, although the excitation energy to the 6L11/2 is larger than experiment by about 850 cm−1. Inclusion of outer-core correlation (5s5p5d) also stabilizes the ground 4I (5f37s2) state, and at the CASPT2 level, this yields an energy separation of 2210 cm−1 at the CBS limit compared to the 289 cm−1 experimental value.43–46 Such a large effect due to correlating the 5d electrons of U was not unexpected since similar large effects on relative energies has been previously calculated at the Fock-space coupled cluster level of theory by, e.g., Réal et al.47 in the context of UO22+, Infante et al. for UO2,48 and Denis et al.. for ThF+.49 As seen in Table I, the effect of the Davidson correction (+Q) on the frozen-core MRCI excitation energies is significant, with the MRCI + Q energies being much closer to the experimental separation, however still with an inverted order. Since core-valence correlation at the MRCI + Q level was prohibitively expensive with current computational resources, the CASPT2 CV correction was added to the MRCI + Q relative energies (before diagonalization of the SO matrix). In this manner, the resulting two states are not only in the correct order but also the excitation energy of 404 cm−1 is in very good agreement with experiment.

TABLE I.

Calculated excitation energy of 6L11/2 (5f36d7s) relative to the 4I9/2 (5f37s2) ground state (in cm−1) for the U+ cation.a Values in square brackets correspond to the term separation before inclusion of SO.

MethodbBasis setValencecValence + outer-cored
CASSCF VTZ-PP −4493 [−2651]  
VQZ-PP −4328  
CBS −4284  
CASPT2 VTZ-PP 1119 [2954] 2354 
VQZ-PP 1164 2278 
CBS 1138 2210 
MRCI VTZ-PP −1608 [228]  
VQZ-PP −1680  
CBS −1769 −700e 
MRCI + Q VTZ-PP −472 [1376]  
VQZ-PP −572  
CBS −682 404e 
2c-SCFf VTZ-DK  287  
2c-KRCIg VTZ-DK −439  
Experimenth  289 
MethodbBasis setValencecValence + outer-cored
CASSCF VTZ-PP −4493 [−2651]  
VQZ-PP −4328  
CBS −4284  
CASPT2 VTZ-PP 1119 [2954] 2354 
VQZ-PP 1164 2278 
CBS 1138 2210 
MRCI VTZ-PP −1608 [228]  
VQZ-PP −1680  
CBS −1769 −700e 
MRCI + Q VTZ-PP −472 [1376]  
VQZ-PP −572  
CBS −682 404e 
2c-SCFf VTZ-DK  287  
2c-KRCIg VTZ-DK −439  
Experimenth  289 
a

Only the degenerate components of the 6L and 4I states were included in the SO matrix. The CASSCF method was used for the spin-orbit matrix elements in all 1-component cases.

b

This refers to the diagonal elements of Hel + HSO in the state interacting approach.

c

Frozen core calculation, i.e., only 6s, 6p, 5f, 7s, 6d correlated.

d

All electrons outside the PP were correlated using the wCVnZ-PP basis sets.

e

The CASPT2 CBS core correlation energies (defined as ECV-Eval) were added to the respective MRCI or MRCI + Q energies.

f

Using orbitals obtained from aoc-SCF calculations of the 5f3 7s2 configuration. See the text.

g

KR-CI(SD) calculation with the 7s6d5f spinors correlated with virtuals up to 4 a.u.

h

References 43–46.

The results of 2-c calculations on the 4I and 6L states of U+ with the X2C Hamiltonian are also shown in Table I. The aoc-SCF calculation for the 5f37s2 configuration was straightforward to construct, with all spin-orbitals constrained to be doubly occupied except for 3 electrons in 14 ungerade spinors (5f). For the 5f36d7s case however, both the 4I and 6L atomic states had to be included since 2 electrons were placed in 12 gerade spinors (7s, 6d) with 3 electrons in 14 ungerade spinors (5f). The aoc-SCF calculation, however, assigns equal weights to all possible determinants/configurations; hence, the 7s2 configuration will be biased against compared to the 7s6d and 6d2 configurations since the latter comprise 65 of the 66 possible configurations. The impact of this is shown explicitly in Table S-I of the supplementary material,50 where the resolved individual states of the aoc-SCF and KR-CI(SD) calculations both incorrectly yield a J = 11/2 (5f36d7s) ground state when these 5f36d7s aoc-SCF orbitals are used. Hence, the orbitals obtained from the 5f37s2 aoc-SCF calculations were used for both the 2c-SCF and KRCI results shown in Table II. The resulting splitting at the 2c-SCF level of theory is in excellent agreement with experiment, but the two states are still slightly inverted at the KRCI level. As shown in Table S-I,50 however, single excitations from the 6p orbitals change this separation by more than 100 cm−1, suggesting that not including 6s and 6p correlation in the KRCI calculations is partially responsible for the relative poor agreement with experiment. This is in addition to the lack of a Davidson correction and the neglect of core-valence correlation, both of which stabilize the 5f37s2 configuration.

TABLE II.

NBO analysis for UX and UX+ (X = F, Cl) molecular systems.

Natural electron configurationb
U+ or U2+ atomic asymptotesStatesaUFNatural charge on U
UF 6L (5f36d7s) 6Λ, 6K, 6I, 6H, 6Γ 6Φ, 6Δ, 6Π, 6Σ+ 7s0.925f2.966d1.27 2s1.972p5.82 +0.82 
4I (5f37s24I, 4H, 4Γ, 4Φ, 4Δ, 4Π, 4Σ 7s1.695f2.966d0.38 2s1.972p5.84 +0.85 
UF+ 5I (5f37s) (1) 5I, (1) 5H, (1) 5Γ, (1) 5Φ, (1) 5Δ, (1) 5Π, 5Σ 7s0.875f2.986d0.35 2s1.972p5.81 +1.81 
5L (5f36d) 5Λ, 5K, (2) 5I, (2) 5H, (2) 5Γ, (2) 5Φ, (2) 5Δ, (2) 5Π, 5Σ+ 7s0.025f2.956d1.25 2s1.982p5.78 +1.78 
3I (5f37s) 3I, 3H, 3Γ, 3Φ, 3Δ, 3Π, 3Σ 7s0.875f2.976d0.35 2s1.972p5.81 +1.81 
UCl 6L (5f36d7s) 6Λ, 6K, 6I, 6H, 6Γ, 6Φ, 6Δ, 6Π, 6Σ+ 7s0.925f2.996d1.31 3s1.963p5.74 +0.72 
4I (5f37s24I, 4H, 4Γ, 4Φ, 4Δ, 4Π, 4Σ 7s1.665f2.996d0.467p0.12 3s1.963p5.77 +0.76 
UCl+ 5I (5f37s) (1) 5I, (1) 5H, (1) 5Γ, (1) 5Φ, (1) 5Δ, (1) 5Π, 5Σ 7s0.825f3.006d0.52 3s1.963p5.65 +1.65 
5L (5f36d) 5Λ, 5K, (2) 5I, (2) 5H, (2) 5Γ, (2) 5Φ, (2) 5Δ, (2) 5Π, 5Σ+ 7s0.045f2.996d1.33 3s1.973p5.64 +1.64 
3I (5f37s) 3I, 3H, 3Γ, 3Φ, 3Δ, 3Π, 3Σ 7s0.845f2.996d0.50 3s1.963p5.65 +1.65 
Natural electron configurationb
U+ or U2+ atomic asymptotesStatesaUFNatural charge on U
UF 6L (5f36d7s) 6Λ, 6K, 6I, 6H, 6Γ 6Φ, 6Δ, 6Π, 6Σ+ 7s0.925f2.966d1.27 2s1.972p5.82 +0.82 
4I (5f37s24I, 4H, 4Γ, 4Φ, 4Δ, 4Π, 4Σ 7s1.695f2.966d0.38 2s1.972p5.84 +0.85 
UF+ 5I (5f37s) (1) 5I, (1) 5H, (1) 5Γ, (1) 5Φ, (1) 5Δ, (1) 5Π, 5Σ 7s0.875f2.986d0.35 2s1.972p5.81 +1.81 
5L (5f36d) 5Λ, 5K, (2) 5I, (2) 5H, (2) 5Γ, (2) 5Φ, (2) 5Δ, (2) 5Π, 5Σ+ 7s0.025f2.956d1.25 2s1.982p5.78 +1.78 
3I (5f37s) 3I, 3H, 3Γ, 3Φ, 3Δ, 3Π, 3Σ 7s0.875f2.976d0.35 2s1.972p5.81 +1.81 
UCl 6L (5f36d7s) 6Λ, 6K, 6I, 6H, 6Γ, 6Φ, 6Δ, 6Π, 6Σ+ 7s0.925f2.996d1.31 3s1.963p5.74 +0.72 
4I (5f37s24I, 4H, 4Γ, 4Φ, 4Δ, 4Π, 4Σ 7s1.665f2.996d0.467p0.12 3s1.963p5.77 +0.76 
UCl+ 5I (5f37s) (1) 5I, (1) 5H, (1) 5Γ, (1) 5Φ, (1) 5Δ, (1) 5Π, 5Σ 7s0.825f3.006d0.52 3s1.963p5.65 +1.65 
5L (5f36d) 5Λ, 5K, (2) 5I, (2) 5H, (2) 5Γ, (2) 5Φ, (2) 5Δ, (2) 5Π, 5Σ+ 7s0.045f2.996d1.33 3s1.973p5.64 +1.64 
3I (5f37s) 3I, 3H, 3Γ, 3Φ, 3Δ, 3Π, 3Σ 7s0.845f2.996d0.50 3s1.963p5.65 +1.65 
a

The density matrices and orbitals of these states were averaged and analyzed.

b

Any contributions smaller than 0.05 are not shown.

Table II shows the results from a natural bond orbital (NBO v.6)51 analysis of CASSCF/VDZ calculations on UF, UF+, UCl, and UCl+. Analysis of the individual degenerate components of each state showed natural electron configurations that differed by less than 0.04 electrons from those obtained from the average density matrices, so only results corresponding to the latter are reported in Table I. As expected, the bonding in all four molecules can be characterized as ionic, with natural charges on U calculated to be about +0.8 in UX and +1.7–1.8 in UX+. NBO however does predict a single bonding molecular orbital (MO) for UF, albeit very ionic and only in the sextet states. This bonding MO has roughly 3.3% 6d character, 1.4% 5f character, and the rest consisting of a sp hybrid on F. While there is some donation observed from the halogen atom into empty 6d orbitals on U, the low-lying ΛS states of UX can be easily assigned to one of the two lowest atomic asymptotes of U+, either the 4Iu (5f37s2) or 6Lu (5f36d7s) with of course the ground state of X, 1Sg. The lowest two U+ atomic states are quite close in energy, e.g., 4I9/2 and 6L11/2 differ by only 289 cm−1.43–46 The center of gravity of these two terms (using an estimated J = 21/2 level for the 6Lu from the observed progression of J = 11/2 to J = 19/2) is within 1700 cm−1 of each other, so it is unsurprising that both would yield low-lying molecular states when coupled with X. The lowest calculated doublet states of UF do result from spin pairing in the 5f37s2 configuration of U+ but were more than 10 000 cm−1 above the quartet ground state, thus confirming the present choice of not including these states, or the low-spin doublets or quartets from the 5f36d7s configuration, in the current calculations.

When UX is ionized to UX+, the additional positive charge is nearly completely localized on U. All of the states of UX+ have a single bonding MO with a character very similar to that observed in the sextet states of UX. The dominant configurations of the uranium center in UX+ are 5f37s and 5f36d, neither of which corresponds to the ground state of U2+, 5I4 (5f4). Experimentally, the 5L6 (5f36d) state is only 210 cm−1 above 5I4 (5f4) in U2+, but the 5I4 (5f37s) state has an excitation energy of 3744 cm−1.45 However, as discussed previously by Antonov and Heaven,5 because the electron in the 7s orbital can more easily polarize away from the X ligand, this state is much more stable in the molecule than those arising from either the 5f36d or particularly the 5f4 configurations. Overall, UCl and UCl+ show very similar trends to UF and UF+, respectively, with about 0.1 less electrons transferred from U to Cl. The same bonding MOs are observed in UCl as in UF but the 6d character is increased to 5%–6% while the 5f character is essentially equivalent.

The basis set convergence of the spectroscopic constants of the first few ΛS states of UF+ is shown in Table III for the case of a PP on U. The VDZ bond lengths are longer than the estimated CBS limits by nearly 25 mÅ; however, the VTZ results are fairly well converged, within 7 mÅ. Despite the poor bond lengths with the VDZ basis set, the harmonic frequencies for these spin-free states at this level are fortuitously closer to the CBS limit than even the VTZ frequencies, presumably due to basis set superposition errors (BSSE). The Te values for low-lying states rapidly converged to the CBS limit, with VTZ results being within 100 cm−1 of the CBS limit for states under 1000 cm−1. CBS limit values of both Te and re for all the spin-free, ΛS states of Table II are explicitly shown in Table S-II50 for UF+ using both PP and DK3 approaches. Both the PP and DK calculations result in an 5I ground state. The Te values obtained by the PP method were all slightly larger than those from the all-electron DK3 approach, averaging 45 cm−1 with a maximum difference of just 135 cm−1 for the highest energy state included, 5Σ+. The bond lengths were also consistently longer from the PP approach, but only by an average of 3 mÅ with a maximum of 5 mÅ, the latter also for the high-lying 5Σ+ state. It is noteworthy that higher excited states, especially those arising from an excited U+ atomic configuration, were more slowly convergent with respect to basis set for both Te and other molecular properties.

TABLE III.

Basis set convergence of the CASPT2 molecular properties of UF+ for selected ΛS states.

ΛS stateBasis setTe (cm−1)re (Å)B0 (cm−1)ωe (cm−1)
X5VDZ-PP 2.000 0.2392 649.6 
VTZ-PP 1.983 0.2436 645.5 
VQZ-PP 1.979 0.2442 649.5 
CBS 1.977 0.2447 651.3 
(1) 5VDZ-PP 304 2.000 0.2390 650.4 
VTZ-PP 357 1.984 0.2428 645.9 
VQZ-PP 368 1.980 0.2438 650.0 
CBS 374 1.978 0.2443 651.8 
(1) 3VDZ-PP 2037 1.999 0.2392 649.5 
VTZ-PP 2410 1.982 0.2435 647.4 
VQZ-PP 2344 1.977 0.2445 651.4 
CBS 2308 1.975 0.2450 653.3 
ΛS stateBasis setTe (cm−1)re (Å)B0 (cm−1)ωe (cm−1)
X5VDZ-PP 2.000 0.2392 649.6 
VTZ-PP 1.983 0.2436 645.5 
VQZ-PP 1.979 0.2442 649.5 
CBS 1.977 0.2447 651.3 
(1) 5VDZ-PP 304 2.000 0.2390 650.4 
VTZ-PP 357 1.984 0.2428 645.9 
VQZ-PP 368 1.980 0.2438 650.0 
CBS 374 1.978 0.2443 651.8 
(1) 3VDZ-PP 2037 1.999 0.2392 649.5 
VTZ-PP 2410 1.982 0.2435 647.4 
VQZ-PP 2344 1.977 0.2445 651.4 
CBS 2308 1.975 0.2450 653.3 

Turning to the inclusion of SO effects, CASPT2 equilibrium excitation energies within the state interacting approach and the frozen-core approximation are compared to 2-component and experimental results for UF+ in Table IV. The state interacting approach was investigated using both all electron DK3 and PP calculations for the energies of the spin-free states as well as the BP or PP operator for the off-diagonal SO matrix elements. The BP and PP operators gave similar results for Te when the same spin-free energies were used, with the maximum and mean absolute differences (MAD) being 300 cm−1 and 81 cm−1, respectively. Table IV also shows a comparison of the PP-based results of the present work with the previous CASPT2 values of Antonov and Heaven5 who used the same PP and approach as the present work but with a different basis set and at a fixed bond length of 2.00 Å. Not surprisingly, the excitation energies for the lower energy states agree very well between the two sets of calculations, and it is only at higher energies when the equilibrium bond lengths of the states have substantially changed that this agreement becomes worse.

TABLE IV.

Equilibrium excitation energies, Te, in cm−1 for selected Ω states of UF+.

CASPT2 state interactingPrevious calculationa
2-componentDK3 DiagdPP diagePP diagf
Ω stateSCFbKRCIcBP SOgPP SOhPP SOhPP SOiExpt.a
(X) 4 0  
(1) 3 337 490 479 463 486 494 510.78 
(1) 2 628 696 782 764 777 725 737.53 
(1) 0 708 821 819 793 815 802 771.81 
(1) 1 706 791 845 822 839 813 776.12 
(1) 5 1189 950 1106 1054 1061 1100 1005.62 
(2) 4 1737 1716 1582 1522 1548 1575 1508.7 
(2) 0 2113 2065 1836 1798 1826 1820 1784.3 
(2) 1 2122 2073 1853 1805 1831 1831 1804.91 
(2) 3 2039 1969 1905 1839 1861 1855 1833.36 
(2) 2 2121 2065 1957 1896 1919 1891 1777.83 
(3) 4 2486 2028 3689 3746 3787  2593.30 
(1) 6 653 78 2026 2184 2217 2929 2604.31 
(3) 3 3485 2889 4430 4164 4388  3059.52 
(3) 2 4270 3683 4856 4557 4538  3301.03 
(2) 5 1373 2065 2720 2833 2877 3511 3313.51 
CASPT2 state interactingPrevious calculationa
2-componentDK3 DiagdPP diagePP diagf
Ω stateSCFbKRCIcBP SOgPP SOhPP SOhPP SOiExpt.a
(X) 4 0  
(1) 3 337 490 479 463 486 494 510.78 
(1) 2 628 696 782 764 777 725 737.53 
(1) 0 708 821 819 793 815 802 771.81 
(1) 1 706 791 845 822 839 813 776.12 
(1) 5 1189 950 1106 1054 1061 1100 1005.62 
(2) 4 1737 1716 1582 1522 1548 1575 1508.7 
(2) 0 2113 2065 1836 1798 1826 1820 1784.3 
(2) 1 2122 2073 1853 1805 1831 1831 1804.91 
(2) 3 2039 1969 1905 1839 1861 1855 1833.36 
(2) 2 2121 2065 1957 1896 1919 1891 1777.83 
(3) 4 2486 2028 3689 3746 3787  2593.30 
(1) 6 653 78 2026 2184 2217 2929 2604.31 
(3) 3 3485 2889 4430 4164 4388  3059.52 
(3) 2 4270 3683 4856 4557 4538  3301.03 
(2) 5 1373 2065 2720 2833 2877 3511 3313.51 
a

Experimental results and calculated values from Ref. 5. The calculated values correspond to vertical excitation energies at r = 2.000 Å.

b

The resolved energies from an aoc-SCF calculation.

c

The energies taken from a KR-CI(SD) calculation with the 5f6d7s spinors correlated.

d

Using the CASPT2/CBS energies generated with the DK3 approach.

e

Using the CASPT2/CBS energies generated with the PP approach.

f

Using the CASPT2 energies generated with the same PP on U as the present work using aVTZ on F and a (14s13p10d8f6g)/[6s6p5d4f3g] basis set on U.

g

SO matrix elements calculated using the BP spin-orbit operator as implemented in MOLPRO at the CASSCF/VQZ-DK level of theory.

h

SO matrix elements calculated using the PP spin-orbit operator of Ref. 14 at the CASSCF/VQZ-PP level of theory.

i

SO matrix elements calculated using the PP spin-orbit operator of Ref. 14 at the CASSCF level of theory. See footnote (f) for the basis set.

In regards to the 2-component calculations shown in Table IV, the molecular spinors were selected by a Mulliken population analysis and inspection of the Lz expectation values. Assignment of approximate projection quantum numbers λ was carried out as previously done by Fleig et al.52 for UO2. The analysis showed that while most spinors of UF+ could be assigned to an integer value of λ, there was a large degree of mixing between near-degenerate spinors of mostly atomic U2+ character, especially for p and f spinors. As an example, the Lz expectation values of the spinors corresponding to 5f on U were −2.9, 1.8, −0.6, −0.4, 1.2, −2.1, 3.0 (compared to “pure” integers of −3 to +3), while those corresponding to 6d were −2.0, −0.9, −0.2, 1.0, and 2.0. Overall, the 2-c calculations yielded results that were very similar to the 1-c values, except for the first Ω = 6 state that was only 78 cm−1 above the ground state at the KRCI level of theory (653 cm−1 by 2-c, aoc-SCF), while both CASPT2 and experiment place this state above 2000 cm−1. The lack of correlation of the 6p electrons of U as well as the 2p of F, together with the lack of a +Q correction, is presumably responsible for this.

Relative to experiment, the current PP results are within 75 cm−1 of the experimental Te values for states below 1500 cm−1 with a MAD of 110 cm−1 when excluding the third Ω = 4, 3, and 2 states. In Ref. 5, the latter three states were observed between 2500 and 3300 cm−1 and were attributed to the 7s25f2 configuration of U2+ as observed in UO.53 Unfortunately, states associated with this asymptote were not included in the present CASPT2 calculations since they were expected to lie very high in energy. The third Ω = 4, 3, 2 states from the current CASPT2 calculations shown in Table IV lie about 1000 cm−1 higher in energy and arise from high-lying quintet states of U2+. These transitions obtained at the KRCI level are a much better match energetically, but analysis of the wavefunctions indicates they arise, not from the 7s25f2, but a low-spin pairing of the 5f36d1 configuration, which is expected to lie very high in energy at the CASPT2 level of theory based on exploratory CASSCF calculations. Ignoring these three transitions, the excitation energies of Antonov and Heaven exhibit a very similar MAD of 81 cm−1 to the present work, while the all-electron BP CASPT2 excitation energies had a MAD of 157 cm−1. Excluding the second Ω = 5 state and the Ω = 6 state from the two-component results, the MAD between experiment and the 2c-SCF results was 271 cm−1 while the KRCI method yielded a MAD of 190 cm−1.

Table V shows the calculated SO-CASPT2/CBS(PP) spectroscopic constants of UF+ for states with Te < 4000 cm−1 together with experimental values.5 A tabulation of spectroscopic constants for all the states included in this work is given in Table S-III.50 The harmonic frequencies are in particularly good agreement with experiment, showing the correct qualitative trends and were within 1-5 cm−1 from the values derived from the experimental spectra in Ref. 5. The anharmonicity constants (ωexe) exhibit larger differences. In particular, the calculated ωexe values were very sensitive to the number of vibrational states used in the level fits, as well as the choice of energy expression used. The energy expression used here, Eq. (3), was slightly different from what was used for the experimental results in Ref. 5 and this could be responsible for some of the discrepancies. Even when using experimental frequencies to calculate the ωexe constants, the number of states included in the fit strongly changed the resulting values.

TABLE V.

SO-CASPT2/CBS molecular properties of UF+ for Ω states below 4000 cm−1. Experimental values from Ref. 5 in square brackets.

Ω stateTe (cm−1)Re (Å)B0 (cm−1)ωe (cm−1)ωexe (cm−1)% ΛS compositiona
(X) 4 1.978 0.2443 [0.250(3)] 650 [649.92] 1.837 [1.831] 78% (1) 5I + 18% (1) 5
(1) 3 486 [510.78] 1.982 0.2434 [0.243] 648 [648.26] 2.003 [2.01] 56% (1) 5H + 30% (1) 5Γ + 10% (1) 5Φ 
(1) 2 777 [737.53] 1.984 0.2429 [0.224] 646 [646.32] 2.036 [1.859] 36% (1) 5Φ + 35% (1) 5Γ + 20% (1) 5Δ + 7% (1) 5Π 
(1) 0 815 [771.81] 1.981 0.2437 647 [646.11] 1.911 [1.77] 49% (1) 5Π + 33% 5Σ + 18% (1) 5Δ 
(1) 1 839 [776.12] 1.982 0.2434 [0.241] 646 [646.39] 1.958 [1.928] 33% (1) 5Π + 31% (1) 5Δ + 19% (1) 5Φ + 17% 5Σ 
(1) 5 1061 [1005.62] 1.978 0.2444 [0.256] 651 [649.53] 1.790 [1.71] 40% (1) 5I + 35% 3I + 16% (1) 5H + 5% 3
(2) 4 1548 [1508.7] 1.980 0.2438 [0.247] 650 [648.54] 1.950 [1.75] 27% 3H + 24% (1) 5Γ + 18% (1) 5H + 10% (1) 5Φ + 10% (1) 5I + 8% 3Γ 
(2) 0 1826 [1784.3] 1.981 0.2435 645 2.012 35% (1) 5Δ + 23% (1) 5Π + 21% 3Π + 19% 3Σ 
(2) 1 1831 [1804.91] 1.982 0.2434 [0.249] 645 [651.77] 2.027 [3.41] 22% (1) 5Φ + 18% 5Σ + 18% (1) 5Π + 16% 3Π + 12% 3Δ + 7% 3Σ5% (1) 5Δ 
(2) 3 1861 [1833.36] 1.983 0.2432 [0.273] 647 [645.7] 2.085 [1.45] 22% (1) 5Φ + 20% 3Γ + 18% (1) 5Δ + 17% (1) 5H + 10% 3Φ + 5% (1) 5Π 
(2) 2 1919 [1777.83] 1.983 0.2433 [0.254] 646 [647.2] 2.070 [1.85] 22% (1) 5Π + 20% (1) 5Γ + 16% 3Φ + 12% (1) 5Δ + 12% 3Δ + 11% 5Σ 
(1) 6 2217 [2604.31] 1.997 0.2398 [0.24] 624 [620.05] 2.159 [1.17] 87% 5Λ + 12% 5
(2) 5 2877 [3313.51] 1.997 0.2398 [0.241] 620 [618] 2.190 66% 5K + 18% (2) 5I + 6% (1) 5I + 5% 3
(3) 4 3787 1.999 0.2394 608 1.729 45% (2) 5I + 22% (2) 5H + 6% (2) 5Γ + 6% (1) 5H + 6% 3
(2) 6 3784 1.978 0.2445 651 1.792 53% (1) 5I + 22% (1) 5H + 20% 3
Ω stateTe (cm−1)Re (Å)B0 (cm−1)ωe (cm−1)ωexe (cm−1)% ΛS compositiona
(X) 4 1.978 0.2443 [0.250(3)] 650 [649.92] 1.837 [1.831] 78% (1) 5I + 18% (1) 5
(1) 3 486 [510.78] 1.982 0.2434 [0.243] 648 [648.26] 2.003 [2.01] 56% (1) 5H + 30% (1) 5Γ + 10% (1) 5Φ 
(1) 2 777 [737.53] 1.984 0.2429 [0.224] 646 [646.32] 2.036 [1.859] 36% (1) 5Φ + 35% (1) 5Γ + 20% (1) 5Δ + 7% (1) 5Π 
(1) 0 815 [771.81] 1.981 0.2437 647 [646.11] 1.911 [1.77] 49% (1) 5Π + 33% 5Σ + 18% (1) 5Δ 
(1) 1 839 [776.12] 1.982 0.2434 [0.241] 646 [646.39] 1.958 [1.928] 33% (1) 5Π + 31% (1) 5Δ + 19% (1) 5Φ + 17% 5Σ 
(1) 5 1061 [1005.62] 1.978 0.2444 [0.256] 651 [649.53] 1.790 [1.71] 40% (1) 5I + 35% 3I + 16% (1) 5H + 5% 3
(2) 4 1548 [1508.7] 1.980 0.2438 [0.247] 650 [648.54] 1.950 [1.75] 27% 3H + 24% (1) 5Γ + 18% (1) 5H + 10% (1) 5Φ + 10% (1) 5I + 8% 3Γ 
(2) 0 1826 [1784.3] 1.981 0.2435 645 2.012 35% (1) 5Δ + 23% (1) 5Π + 21% 3Π + 19% 3Σ 
(2) 1 1831 [1804.91] 1.982 0.2434 [0.249] 645 [651.77] 2.027 [3.41] 22% (1) 5Φ + 18% 5Σ + 18% (1) 5Π + 16% 3Π + 12% 3Δ + 7% 3Σ5% (1) 5Δ 
(2) 3 1861 [1833.36] 1.983 0.2432 [0.273] 647 [645.7] 2.085 [1.45] 22% (1) 5Φ + 20% 3Γ + 18% (1) 5Δ + 17% (1) 5H + 10% 3Φ + 5% (1) 5Π 
(2) 2 1919 [1777.83] 1.983 0.2433 [0.254] 646 [647.2] 2.070 [1.85] 22% (1) 5Π + 20% (1) 5Γ + 16% 3Φ + 12% (1) 5Δ + 12% 3Δ + 11% 5Σ 
(1) 6 2217 [2604.31] 1.997 0.2398 [0.24] 624 [620.05] 2.159 [1.17] 87% 5Λ + 12% 5
(2) 5 2877 [3313.51] 1.997 0.2398 [0.241] 620 [618] 2.190 66% 5K + 18% (2) 5I + 6% (1) 5I + 5% 3
(3) 4 3787 1.999 0.2394 608 1.729 45% (2) 5I + 22% (2) 5H + 6% (2) 5Γ + 6% (1) 5H + 6% 3
(2) 6 3784 1.978 0.2445 651 1.792 53% (1) 5I + 22% (1) 5H + 20% 3
a

Coefficients smaller than 5% are not shown, ΛS compositions are calculated at r = 2.000 Å.

Also shown in Table V are the ΛS compositions of each Ω state that were analyzed at a bond length near equilibrium of the ground state (r = 2.000 Å). The Ω = 4 ground state agrees with previously analyses,5 namely, that it is fairly well described as a 5I4 state. The majority of the excited states, however, are very multireference in character, involving several ΛS states with mainly quintet character, but a few with important contributions from triplet states as well, e.g., the (2)4 and (2)0 states.

The results of analogous SO-CASPT2/CBS(PP) calculations on the UF molecule are shown in Table VI together with ΛS compositions. The basis set convergence of the ΛS states of UF is available in Table S-IV.50 Experimentally, only the ground state, as well as excited states at 18 624.3 and 19 932.1 cm−1, was rotationally resolved.5 Additionally, two states were observed via LIF at 435 and 650 cm−1. The calculated ground state B0 value from this work agrees with the experimental value to within 0.002 cm−1 and the calculated r0 agrees to within 8 mÅ of experiment. The calculated re for the ground state is only 2.5 mÅ less than the calculated r0. The molecular properties for states below 4000 cm−1 and a few selected states above 18 000 cm−1 are shown in Table VI, while results for all Ω states included in this work are in Table S-V.50 The two low-lying excitation energies calculated for Ω = 7/2 and 5/2 agree well with experiment, deviating by just 15 and 17 cm−1, respectively. The high-lying excited state with Ω = 7/2 calculated at 18 624 cm−1 agrees well with the transition measured at 18 442 cm−1, and its associated calculated re agrees to within 2 mÅ of the experimental r0. This excited state arises from the sextet ΛS states. The excited state observed experimentally at 19 932 cm−1 was above the highest energy state observed in the SO-CASPT2 calculations. Similar to UF+, while the ground state of UF is well described by a single ΛS state, i.e., 4I9/2, essentially all the excited Ω states of UF are admixtures of 2 or more ΛS states, nearly exclusively involving quartets up through 4000 cm−1 except for the (2)11/2 state at 3730 cm−1, which can be assigned as 6Λ11/2. Given the very similar methods used, it is not surprising that the current results are in very good agreement with the previously calculated vertical transitions of Antonov and Heaven.5 However, as also in this latter work, the low-lying excitation energies of UF+ obtained in this work are in better agreement with experiment than the previous SO-MCQDPT calculations of Fedorov et al.11 The adiabatic ionization potential of UF calculated in this work, i.e., Ω = 9/2 (v = 0) of UF to Ω = 4 (v = 0) of UF+, was calculated to be 6.337 eV, which can be compared to the experimental value of 6.341 59(6) eV.5 

TABLE VI.

SO-CASPT2/CBS molecular properties of UF for selected Ω states. Experimental values from Ref. 5 in square brackets.

Ω stateTe (cm−1)Re (Å)B0 (cm−1)ωe (cm−1)ωexe (cm−1)% ΛS compositiona
(X) 9/2  0 2.026 [2.020] 0.2328 [0.2348] 584 2.499 81% 4I + 16% 4
(1) 7/2  450 [435] 2.030 0.2320 582 2.546 61% 4H + 30% 4Γ + 8% 4Φ 
(1) 5/2  667 [650] 2.032 0.2314 580 2.548 45% 4Γ + 37% 4Φ + 15% 4Δ 
(1) 1/2  769 2.029 0.2320 580 2.540 49% 4Π + 33% 4Σ + 18% 4Δ 
(1) 3/2  799 2.031 0.2316 580 2.544 38% 4Δ + 30% 4Φ + 24% 4Π + 7% 4Σ 
(1) 11/2  3061 2.026 0.2329 588 3.830 76% 4I + 21% 4
(2) 9/2  3400 2.029 0.2322 583 2.538 40% 4H + 36% 4Γ + 16% 4I + 8% 4Φ 
(2) 7/2  3627 2.030 0.2319 582 2.543 39% 4Φ + 31% 4H + 16% 4Γ + 15% 4Δ 
(2) 5/2  3741 2.030 0.2319 581 2.539 38% 4Γ + 35% 4Δ + 24% 4Π 
(2) 3/2  3759 2.029 0.2321 581 2.535 43% 4Φ + 33% 4Σ + 24% 4Π 
(2) 11/2  3730 2.048 0.2279 551 0.217 84% 6Λ + 13% 6
(2) 1/2  3851 2.029 0.2321 581 2.536 43% 4Δ + 43% 4Π + 14% 4Σ 
(9) 11/2 18 125 2.048 0.2279 555 2.540 46% 6Γ + 32% 6Φ + 19% 6
(10) 9/2 18 314 2.046 0.2282 556 2.549 45% 6Φ + 27% 6Γ + 22% 6Δ + 6% 6
(10) 7/2 18 442 [18 624.3] 2.046 [2.044] 0.2283 [0.2293] 556 2.549 40% 6Δ + 34% 6Φ + 14% 6Π + 10% 6Γ 
(10) 5/2 18 504 2.045 0.2284 556 2.545 39% 6Δ + 34% 6Π + 17% 6Φ + 8% 6Σ 
(10) 1/2 18 508 2.045 0.2286 555 2.540 49% 6Π + 39% 6Σ + 12% 6Δ 
(10) 3/2 18 516 2.045 0.2285 555 2.542 45% 6Π + 24% 6Δ + 24% 6Σ + 5% 6Φ 
Ω stateTe (cm−1)Re (Å)B0 (cm−1)ωe (cm−1)ωexe (cm−1)% ΛS compositiona
(X) 9/2  0 2.026 [2.020] 0.2328 [0.2348] 584 2.499 81% 4I + 16% 4
(1) 7/2  450 [435] 2.030 0.2320 582 2.546 61% 4H + 30% 4Γ + 8% 4Φ 
(1) 5/2  667 [650] 2.032 0.2314 580 2.548 45% 4Γ + 37% 4Φ + 15% 4Δ 
(1) 1/2  769 2.029 0.2320 580 2.540 49% 4Π + 33% 4Σ + 18% 4Δ 
(1) 3/2  799 2.031 0.2316 580 2.544 38% 4Δ + 30% 4Φ + 24% 4Π + 7% 4Σ 
(1) 11/2  3061 2.026 0.2329 588 3.830 76% 4I + 21% 4
(2) 9/2  3400 2.029 0.2322 583 2.538 40% 4H + 36% 4Γ + 16% 4I + 8% 4Φ 
(2) 7/2  3627 2.030 0.2319 582 2.543 39% 4Φ + 31% 4H + 16% 4Γ + 15% 4Δ 
(2) 5/2  3741 2.030 0.2319 581 2.539 38% 4Γ + 35% 4Δ + 24% 4Π 
(2) 3/2  3759 2.029 0.2321 581 2.535 43% 4Φ + 33% 4Σ + 24% 4Π 
(2) 11/2  3730 2.048 0.2279 551 0.217 84% 6Λ + 13% 6
(2) 1/2  3851 2.029 0.2321 581 2.536 43% 4Δ + 43% 4Π + 14% 4Σ 
(9) 11/2 18 125 2.048 0.2279 555 2.540 46% 6Γ + 32% 6Φ + 19% 6
(10) 9/2 18 314 2.046 0.2282 556 2.549 45% 6Φ + 27% 6Γ + 22% 6Δ + 6% 6
(10) 7/2 18 442 [18 624.3] 2.046 [2.044] 0.2283 [0.2293] 556 2.549 40% 6Δ + 34% 6Φ + 14% 6Π + 10% 6Γ 
(10) 5/2 18 504 2.045 0.2284 556 2.545 39% 6Δ + 34% 6Π + 17% 6Φ + 8% 6Σ 
(10) 1/2 18 508 2.045 0.2286 555 2.540 49% 6Π + 39% 6Σ + 12% 6Δ 
(10) 3/2 18 516 2.045 0.2285 555 2.542 45% 6Π + 24% 6Δ + 24% 6Σ + 5% 6Φ 
a

Coefficients smaller than 5% are not shown, ΛS compositions are calculated at r = 2.015 Å.

Spin-orbit CASPT2 results of UCl for Ω states below 5000 cm−1 are shown in Table VII, while results for all Ω states included in this work are in Table S-VII.50 The spin-free basis set convergence for both UCl and UCl+ are similar to what was observed for UF+ (cf. Table III) and are shown in Tables S-VII and IX,50 respectively. The separation between the quartet and sextet states in UCl is less than in UF, as shown in Figure 1. In addition, within each of these manifolds, the states are closer together than they were in UF. As shown in Figure 2, this results in the lowest set of SO-CASPT2 Ω states to be closer in energy (below 400 cm−1) compared to UF. As shown in Table VII, the ΛS compositions of these states are very similar to the analogous states in UF (see Table VI). In general, the low-lying states of UCl can be characterized as either quartets or sextets, i.e., there is little mixing between states of different spin multiplicity; however, spin-orbit coupling does generally mix several states of a particular multiplicity to yield a given Ω state. The ground state is also Ω = 9/2, as shown in Figure 2, despite the compression of states. As shown in Fig. 2(b), the lowest two Ω = 11/2 states in UCl also exhibit a weak avoided crossing, in which the sextet states that comprise the lowest Ω = 11/2 state switch with quartet states that comprise the second Ω = 11/2 state as a function of geometry. This leads to poor agreement between the LEVEL and polynomial fit values of B0 and a negative anharmonicity constant. This also occurs in the second and third Ω = 9/2 states. Despite the compression of states that occurs in UCl, Ω states up to 17 000 cm−1 were still calculated. The Te values for higher energy excited states reported in Table S-VII50 may be useful both for LIF and two photon processes such as ZEKE.

TABLE VII.

SO-CASPT2/CBS molecular properties of UCl for selected Ω states.

Ω stateTe (cm−1)Re (Å)B0 (cm−1)ωe (cm−1)ωexe (cm−1)% ΛS compositiona
(X) 9/2 2.489 0.0891 336 0.812 77% 4I + 19% 4
(1) 7/2 76 2.487 0.0893 336 0.760 60% 4H + 30% 4Γ + 8% 4Φ 
(1) 5/2 259 2.490 0.0890 335 0.842 44% 4Γ + 37% 4Φ + 15% 4Δ 
(1) 1/2 326 2.486 0.0893 334 0.751 49% 4Π + 33% 4Σ + 18% 4Δ 
(1) 3/2 367 2.489 0.0891 334 0.817 38% 4Δ + 29% 4Φ + 25% 4Π + 8% 4Σ 
(1) 11/2 2688 2.581 0.0868 309  79% 6Λ + 17% 6
(2) 11/2 3015 2.446 0.0891 365  72% 4I + 24% 4
(2) 9/2 3043 2.518 0.0877 290  63% 6K + 23% 6I + 5% 6
(3) 9/2 3142 2.490 0.0884 360  34% 4H + 33% 4Γ + 18% 4I + 8% 4Φ + 6% 6
(2) 7/2 3239 2.488 0.0892 337 1.158 38% 4Φ + 32% 4H + 15% 4Δ + 15% 4Γ 
(2) 5/2 3334 2.488 0.0892 335 0.782 39% 4Γ + 35% 4Δ + 24% 4Π 
(2) 3/2 3338 2.487 0.0893 335 0.758 43% 4Φ + 33% 4Σ + 24% 4Π 
(2) 1/2 3423 2.486 0.0893 334 0.745 43% 4Π + 43% 4Δ + 14% 4Σ 
(3) 7/2 3738 2.522 0.0867 315  58% 6I + 31% 6H + 9% 6Γ 
(3) 5/2 4586 2.522 0.0868 320 1.064 49% 6H + 34% 6Γ + 13% 6Φ 
(1) 13/2 4925 2.520 0.0869 322 1.067 69% 6Λ + 25% 6K + 5% 6
Ω stateTe (cm−1)Re (Å)B0 (cm−1)ωe (cm−1)ωexe (cm−1)% ΛS compositiona
(X) 9/2 2.489 0.0891 336 0.812 77% 4I + 19% 4
(1) 7/2 76 2.487 0.0893 336 0.760 60% 4H + 30% 4Γ + 8% 4Φ 
(1) 5/2 259 2.490 0.0890 335 0.842 44% 4Γ + 37% 4Φ + 15% 4Δ 
(1) 1/2 326 2.486 0.0893 334 0.751 49% 4Π + 33% 4Σ + 18% 4Δ 
(1) 3/2 367 2.489 0.0891 334 0.817 38% 4Δ + 29% 4Φ + 25% 4Π + 8% 4Σ 
(1) 11/2 2688 2.581 0.0868 309  79% 6Λ + 17% 6
(2) 11/2 3015 2.446 0.0891 365  72% 4I + 24% 4
(2) 9/2 3043 2.518 0.0877 290  63% 6K + 23% 6I + 5% 6
(3) 9/2 3142 2.490 0.0884 360  34% 4H + 33% 4Γ + 18% 4I + 8% 4Φ + 6% 6
(2) 7/2 3239 2.488 0.0892 337 1.158 38% 4Φ + 32% 4H + 15% 4Δ + 15% 4Γ 
(2) 5/2 3334 2.488 0.0892 335 0.782 39% 4Γ + 35% 4Δ + 24% 4Π 
(2) 3/2 3338 2.487 0.0893 335 0.758 43% 4Φ + 33% 4Σ + 24% 4Π 
(2) 1/2 3423 2.486 0.0893 334 0.745 43% 4Π + 43% 4Δ + 14% 4Σ 
(3) 7/2 3738 2.522 0.0867 315  58% 6I + 31% 6H + 9% 6Γ 
(3) 5/2 4586 2.522 0.0868 320 1.064 49% 6H + 34% 6Γ + 13% 6Φ 
(1) 13/2 4925 2.520 0.0869 322 1.067 69% 6Λ + 25% 6K + 5% 6
a

Coefficients smaller than 5% are not shown, ΛS compositions are calculated at r = 2.503 Å.

FIG. 1.

CASPT2/CBS potential energy curves of ΛS states for (a) UF and (b) UCl.

FIG. 1.

CASPT2/CBS potential energy curves of ΛS states for (a) UF and (b) UCl.

Close modal
FIG. 2.

SO-CASPT2/CBS potential energy curves of the lowest 2 sets of Ω states for (a) UF and (b) UCl.

FIG. 2.

SO-CASPT2/CBS potential energy curves of the lowest 2 sets of Ω states for (a) UF and (b) UCl.

Close modal

The results shown in Table VI for UF and those of Table VII for UCl can also be qualitatively compared to the electronic spectrum of the UH molecule, which has been computed previously at the MRCI + Q + SO (state-interaction approach) level of theory by Dolg and co-workers.14,29 Perhaps not surprisingly, the order of the low-lying Ω states, as well as their ΛS compositions, is very similar since UH can be considered as a U+ cation perturbed by H just as UX (X = F, Cl) is U+ perturbed by X. In the UH case, its calculated excitation energies lie intermediate between those of UF and UCl, but the slightly different levels of theory (CASPT2 in this work vs. MRCI + Q for UH) make it difficult to draw any particular conclusions about this latter trend. Unfortunately, the electronic spectrum of UH has yet to be reported by experiment.

The calculated Ω states of UCl+ below 5000 cm−1 are shown in Table VIII with all Ω states given in Table S-IX.50 As above in UCl, the ΛS states arising from the quintet 5I4 (7s15f3) configuration of the U2+ atom are closer in energy in UCl+ compared to UF+ (cf. Figure 3). This results in spin-orbit states from the ground Ω = 4 state to the Ω = 1 spanning a range of just 438 cm−1, which is roughly half the range that was calculated in UF+. The ΛS compositions of UCl+ generally involve more states compared to UF+, i.e., the spin-orbit states exhibit more multireference character. Contrary to the neutral species, there are several Ω states that involve significant mixing of ΛS states with different spin multiplicities, i.e., quintet and triplet states in this case. As shown in Figure 4, qualitatively UF+ and UCl+ are very similar, but the states of UCl+ are bunched into a lower energy range. States for UCl+ are calculated up to 17 650 cm−1. The CASPT2 adiabatic ionization potential of UCl, i.e., Ω = 9/2 (v = 0) of UCl to Ω = 4 (v = 0) of UCl+, was 6.335 eV. This is very similar to the ionization potential of UF, and both are not far above the first ionization potential of U atom, 6.194 eV.5,44,46,53

TABLE VIII.

SO-CASPT2/CBS molecular properties of UCl+ for selected Ω states.

Ω stateTe (cm−1)Re (Å)B0 (cm−1)ωe (cm−1)ωexe (cm−1)% ΛS compositiona
(X) 4 2.415 0.0947 395 0.919 66% (1) 5I + 26% (1) 5H + 7% (1) 5Γ 
(1) 3 132 2.415 0.0947 394 0.913 53% (1) 5H + 32% (1) 5Γ + 12% (1) 5Φ 
(1) 2 370 2.418 0.0945 393 0.935 36% (1) 5Φ + 35% (1) 5Γ + 20% (1) 5Δ + 7% (1) 5Π 
(1) 0 418 2.414 0.0948 392 0.898 49% (1) 5Π + 33% 5Σ18% + (1) 5Δ 
(1) 1 438 2.415 0.0946 392 0.913 34% (1) 5Π + 31% (1) 5Δ + 18% (1) 5Φ + 17% 5Σ 
(1) 5 1082 2.414 0.0948 396 0.884 38% (1) 5I + 30% 3I + 20% (1) 5H + 6% 3H + 5% (1) 5Γ 
(2) 4 1384 2.415 0.0947 392 1.033 23% 3H + 21% (1) 5Γ + 20% (1) 5I + 11% (1) 5H + 11% (1) 5Φ + 8% 3Γ 
(2) 1 1508 2.417 0.0945 389 0.980 23% (1) 5Φ + 18% 5Σ + 18% (1) 5Π + 15% 3Π + 12% 3Δ + 6% 3Σ5% (1) 5Δ 
(2) 0 1520 2.417 0.0945 389 0.960 36% (1) 5Δ + 23% (1) 5Π + 20% 3Π + 17% 3Σ 
(2) 3 1562 2.418 0.0944 390 1.040 21% (1) 5Φ + 20% (1) 5H + 18% (1) 5Δ + 17% 3Γ + 9% 3Φ + 5% (1) 5Π 
(2) 2 1578 2.418 0.0944 389 1.003 22% (1) 5Π + 21% (1) 5Γ + 14% 3Φ + 12% (1) 5Δ + 11% 3Δ + 11% 5Σ 
(1) 6 2054 2.449 0.0920 378 0.983 85% 5Λ + 13% 5
(2) 5 2421 2.446 0.0923 374 0.982 67% 5K + 20% (2) 5
(3) 4 3200 2.489 0.0922 374 0.838 49% (2) 5I + 25% (2) 5H + 8% (2) 5Γ + 5% (1) 5
(2) 6 3789 2.413 0.0948 396 0.901 49% (1) 5I + 26% (1) 5H + 17% 3
(3) 3 3802 2.441 0.0927 369 0.739 29% (2) 5H + 21% (2) 5Γ + 11% (1) 5H + 10% (2) 5Φ + 7% (1) 5Φ + 6% 3Φ + 6% 3Γ 
(3) 5 3992 2.414 0.0947 395 0.953 34% 3I + 29% (1) 5Γ + 24% (1) 5H + 8% (1) 5Φ 
(4) 5 4131 2.416 0.0946 394 0.956 42% (1) 5I + 20% 3I + 18% 3H + 9% 5
(4) 4 4181 2.415 0.0947 392 1.029 28% (1) 5Φ + 28% (1) 5H + 18% (1) 5Δ + 7% 3Γ + 6% 3Φ + 5% 3
(3) 2 4185 2.421 0.0938 356  20% 3Φ + 19% 5Σ + 15% (2) 5Γ + 14% (2) 5Φ + 13% (1) 5Π + 6% 3Δ + 5% (1) 5Φ + 5% (2) 5Δ 
(4) 2 4252 2.411 0.0941 403  29% (1) 5Γ + 14% 5Σ14% + (1) 5Δ + 13% 3Π + 11% (1) 5Φ + 10% 3Δ 
(3) 1 4240 2.393 0.0946 378 0.919 35% (1) 5Π + 34% (1) 5Φ + 18% 3Σ + 6% 3Π 
(4) 3 4261 2.417 0.0945 393 0.936 26% (1) 5Π + 23% (1) 5Γ + 19% (1) 5Δ + 9% (1) 5H + 7% 3Δ + 5% 3Γ + 5% (2) 5
(3) 0 4292 2.394 0.0946 382 0.988 56% (1) 5Δ + 21% 3Π + 19% 5Σ 
(4) 1 4329 2.418 0.0940 390 1.261 21% 3Δ + 19% (1) 5Π + 15% 3Π + 10% 5Σ + 8% (2) 5Δ + 7% (2) 5Φ + 7% (1) 5Δ + 5% (2) 5Π 
(4) 0 4334 2.408 0.0943 389 1.210 25% 3Π + 23% (1) 5Δ + 22% 3Σ + 16% (1) 5Π + 7% (2) 5Π + 5% (2) 5Δ 
(5) 4 4387 2.417 0.0945 399 0.845 47% 3H + 14% 3Γ + 11% (2) 5I + 9% (1) 5Γ + 5% (1) 5Φ 
(5) 3 4872 2.426 0.0938 402 1.182 36% 3Γ + 18% 3Φ + 18% (2) 5H + 9% (2) 5Γ + 5% (1) 5Δ 
Ω stateTe (cm−1)Re (Å)B0 (cm−1)ωe (cm−1)ωexe (cm−1)% ΛS compositiona
(X) 4 2.415 0.0947 395 0.919 66% (1) 5I + 26% (1) 5H + 7% (1) 5Γ 
(1) 3 132 2.415 0.0947 394 0.913 53% (1) 5H + 32% (1) 5Γ + 12% (1) 5Φ 
(1) 2 370 2.418 0.0945 393 0.935 36% (1) 5Φ + 35% (1) 5Γ + 20% (1) 5Δ + 7% (1) 5Π 
(1) 0 418 2.414 0.0948 392 0.898 49% (1) 5Π + 33% 5Σ18% + (1) 5Δ 
(1) 1 438 2.415 0.0946 392 0.913 34% (1) 5Π + 31% (1) 5Δ + 18% (1) 5Φ + 17% 5Σ 
(1) 5 1082 2.414 0.0948 396 0.884 38% (1) 5I + 30% 3I + 20% (1) 5H + 6% 3H + 5% (1) 5Γ 
(2) 4 1384 2.415 0.0947 392 1.033 23% 3H + 21% (1) 5Γ + 20% (1) 5I + 11% (1) 5H + 11% (1) 5Φ + 8% 3Γ 
(2) 1 1508 2.417 0.0945 389 0.980 23% (1) 5Φ + 18% 5Σ + 18% (1) 5Π + 15% 3Π + 12% 3Δ + 6% 3Σ5% (1) 5Δ 
(2) 0 1520 2.417 0.0945 389 0.960 36% (1) 5Δ + 23% (1) 5Π + 20% 3Π + 17% 3Σ 
(2) 3 1562 2.418 0.0944 390 1.040 21% (1) 5Φ + 20% (1) 5H + 18% (1) 5Δ + 17% 3Γ + 9% 3Φ + 5% (1) 5Π 
(2) 2 1578 2.418 0.0944 389 1.003 22% (1) 5Π + 21% (1) 5Γ + 14% 3Φ + 12% (1) 5Δ + 11% 3Δ + 11% 5Σ 
(1) 6 2054 2.449 0.0920 378 0.983 85% 5Λ + 13% 5
(2) 5 2421 2.446 0.0923 374 0.982 67% 5K + 20% (2) 5
(3) 4 3200 2.489 0.0922 374 0.838 49% (2) 5I + 25% (2) 5H + 8% (2) 5Γ + 5% (1) 5
(2) 6 3789 2.413 0.0948 396 0.901 49% (1) 5I + 26% (1) 5H + 17% 3
(3) 3 3802 2.441 0.0927 369 0.739 29% (2) 5H + 21% (2) 5Γ + 11% (1) 5H + 10% (2) 5Φ + 7% (1) 5Φ + 6% 3Φ + 6% 3Γ 
(3) 5 3992 2.414 0.0947 395 0.953 34% 3I + 29% (1) 5Γ + 24% (1) 5H + 8% (1) 5Φ 
(4) 5 4131 2.416 0.0946 394 0.956 42% (1) 5I + 20% 3I + 18% 3H + 9% 5
(4) 4 4181 2.415 0.0947 392 1.029 28% (1) 5Φ + 28% (1) 5H + 18% (1) 5Δ + 7% 3Γ + 6% 3Φ + 5% 3
(3) 2 4185 2.421 0.0938 356  20% 3Φ + 19% 5Σ + 15% (2) 5Γ + 14% (2) 5Φ + 13% (1) 5Π + 6% 3Δ + 5% (1) 5Φ + 5% (2) 5Δ 
(4) 2 4252 2.411 0.0941 403  29% (1) 5Γ + 14% 5Σ14% + (1) 5Δ + 13% 3Π + 11% (1) 5Φ + 10% 3Δ 
(3) 1 4240 2.393 0.0946 378 0.919 35% (1) 5Π + 34% (1) 5Φ + 18% 3Σ + 6% 3Π 
(4) 3 4261 2.417 0.0945 393 0.936 26% (1) 5Π + 23% (1) 5Γ + 19% (1) 5Δ + 9% (1) 5H + 7% 3Δ + 5% 3Γ + 5% (2) 5
(3) 0 4292 2.394 0.0946 382 0.988 56% (1) 5Δ + 21% 3Π + 19% 5Σ 
(4) 1 4329 2.418 0.0940 390 1.261 21% 3Δ + 19% (1) 5Π + 15% 3Π + 10% 5Σ + 8% (2) 5Δ + 7% (2) 5Φ + 7% (1) 5Δ + 5% (2) 5Π 
(4) 0 4334 2.408 0.0943 389 1.210 25% 3Π + 23% (1) 5Δ + 22% 3Σ + 16% (1) 5Π + 7% (2) 5Π + 5% (2) 5Δ 
(5) 4 4387 2.417 0.0945 399 0.845 47% 3H + 14% 3Γ + 11% (2) 5I + 9% (1) 5Γ + 5% (1) 5Φ 
(5) 3 4872 2.426 0.0938 402 1.182 36% 3Γ + 18% 3Φ + 18% (2) 5H + 9% (2) 5Γ + 5% (1) 5Δ 
a

Coefficients smaller than 5% are not shown, ΛS compositions are calculated at r = 2.426 Å.

FIG. 3.

CASPT2/CBS potential energy curves of ΛS states for (a) UF+ and (b) UCl+.

FIG. 3.

CASPT2/CBS potential energy curves of ΛS states for (a) UF+ and (b) UCl+.

Close modal
FIG. 4.

SO-CASPT2/CBS potential energy curves of the lowest Ω states for (a) UF+ and (b) UCl+. Only states with equilibrium excitation energies less than 4000 cm−1 are shown.

FIG. 4.

SO-CASPT2/CBS potential energy curves of the lowest Ω states for (a) UF+ and (b) UCl+. Only states with equilibrium excitation energies less than 4000 cm−1 are shown.

Close modal

The SO-CASPT2 method was used to calculate the electronic states of UX and UX+ (X = F, Cl) up to 18 000 cm−1. The results for UF+ showed excellent agreement with experimental harmonic frequencies and anharmonicities, as well as experimental and previous ab initio excitation energies. The BP and PP 1-c approaches and X2C 2-c approaches all yielded similar results for UF+, with a few notable exceptions, for the experimentally observed transitions. The separations between states were all slightly higher in BP compared to PP results, leading to a slightly larger deviation from experiment for the BP values, with MADs of 110 and 157 cm−1 respectively, compared to those from the PP of 75 cm−1. The 2-c SCF results had a slightly larger MAD from experiment, 271 cm−1, and while the KRCI calculations reduced this to 190 cm−1. The limited active spaces and a lack of a Davidson correction in the latter 2-c calculations are presumed to have limited the improvement of results compared to CASPT2. The agreement shown for UF and UF+ gives confidence that the analogous values for UCl and UCl+ should be valuable as predictions for future experimental studies. Despite the chemical differences between Cl and F, the UF and UCl molecules and their cations are both very similar. The principle difference between the Cl and F cases was an overall decrease in the separation of the electronic states for UCl0,+ compared to UF0,+.

The authors gratefully acknowledge the support of the U.S. Department of Energy, Office of Basic Energy Sciences, Heavy Element Chemistry Program through Grant No. DE-FG02-12ER16329. The authors would like to thank Timo Fleig, Lucas Visscher, and Stefan Knecht for responses to questions on the usage of the Dirac program. The authors would also like to thank Michael Heaven for discussions on UX/UX+ systems.

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Supplementary Material