The first 6 ionization potentials (IPs) of the uranium atom have been calculated using multireference configuration interaction (MRCI+Q) with extrapolations to the complete basis set limit using new all-electron correlation consistent basis sets. The latter was carried out with the third-order Douglas-Kroll-Hess Hamiltonian. Correlation down through the 5s5p5d electrons has been taken into account, as well as contributions to the IPs due to the Lamb shift. Spin-orbit coupling contributions calculated at the 4-component Kramers restricted configuration interaction level, as well as the Gaunt term computed at the Dirac-Hartree-Fock level, were added to the best scalar relativistic results. The final ionization potentials are expected to be accurate to at least 5 kcal/mol (0.2 eV) and thus more reliable than the current experimental values of IP3 through IP6.

The electronic structure and resulting chemistry of molecules containing uranium are of particular interest not only for their role in the nuclear fuel cycle but also for their complexity due in part to the numerous low-lying electronic states arising from partially filled s, d, and f shells. This complexity extends even to the isolated atom, where the neutral atom ground state (5L) with electron configuration [Rn]5f36d7s2 has many excited states separated by less than 1 eV even in the absence of spin-orbit coupling.1 The complexity is no less challenging for singly and multiply charged uranium atoms.2 Even considering the challenging electronic spectra of the atom, it is still surprising, however, that there have been very few studies of the ionization potentials (IPs) of the U atom. Experimentally, only four of the first six IPs have been reported. The first IP has been very accurately determined via spectroscopic methods3,4 with stated error bars of ±0.5 cm−1 (±0.001 kcal/mol), but the 2nd through 4th IPs, which were determined by the electron impact method,5 carry relatively large uncertainties of about 10-20 kcal/mol. The 5th IP has not been reported from experiment while the 6th (resulting in U6+ with a Rn-like configuration) is accompanied only by a semi-empirical estimate2 with a stated uncertainty of about 40 kcal/mol. Accurate knowledge of the character and energetics of the atomic ions is essential in the application of ligand field theory for actinide molecules,6,7 as well as in the study of the covalency in f-element chemistry by, e.g., ligand K-edge spectroscopy.8,9

The most extensive theoretical calculations on the IPs of U have been previously carried out by Dolg and co-workers.10,11 They utilized the multireference averaged coupled pair functional method (MR-ACPF) with complete basis set (CBS) extrapolations and relativistic pseudopotentials (PPs) to determine the first 4 IPs of U. These same IPs have also been investigated using density functional theory, both in scalar-only12 and 4-component Dirac-Kohn-Sham (DKS) calculations.13 Recently, the 5th and 6th IPs have also been reported using the relativistic intermediate Hamiltonian Fock space coupled cluster method (IH-FSCCSD).14,15 Last, the first 3 ionization potentials were reported at the multireference configuration interaction (MRCI) level without the inclusion of spin-orbit coupling effects by one of the present authors16 to assess the basis set convergence of the new cc-pVnZ-DK3 and cc-pwCVnZ-DK3 correlation consistent basis sets.

The present work is a significant extension of the IP calculations of Ref. 16 to the first 6 IPs of U atom, but in particular now with additions of accurate spin-orbit coupling contributions from multireference calculations based on the 4-component Dirac-Coulomb (DC) Hamiltonian, as well as the first estimates of the contributions due to core-valence (CV) (5s5p5d) correlation and the Lamb shift on all the lower IPs of U. Based on the resulting agreement with experiment for the first IP, as well as the consistency with the previous IH-FSCCSD results for the 5th and 6th IPs, the theoretical error bars of the current IPs are conservatively placed at ±5 kcal/mol (about 0.2 eV), making them much more accurate than experiment for particularly the 3rd through 6th IPs. The computational methodology of the present work can be found in Sec. II with results and discussion in Sec. III. The results are summarized in Sec. IV.

In the present work a composite approach in the spirit of the FPD17–20 composite thermochemistry methodology was used to determine the first six ionization potentials of uranium atom. Specifically each IP was calculated as

IP=IPMRCI+Q(FC)/CBS+ΔCV5s5p5d+ΔQED+ΔSO,
(1)

where the first three contributions (defined below) on the right-hand-side of Eq. (1) were performed on the lowest scalar relativistic, spin-free, LS state associated with the ground state configuration of each oxidation state of the U atom. The 3rd-order Douglas-Kroll-Hess Hamiltonian21–23 (DKH3) was used for these calculations. These results were then adjusted to the ground state J level by adding a correction for spin-orbit coupling (ΔSO).

The molproab initio suite of programs24 was used throughout this work for all calculations not involving SO coupling. State-averaged complete active space self-consistent field (CASSCF)25,26 calculations were first carried out to represent the lowest spin-free states that are specified in Table I. In each case all degenerate spatial components for a given state were averaged within D2h symmetry, e.g., all 17 quintet states (4 × 5B2u + 4 × 5B3u + 4 × 5B1u + 5 × 5Au) for the 5L term of neutral U. The CASSCF natural orbitals from these calculations were then used in subsequent internally contracted MRCI27,28 calculations, where the multireference Davidson correction29 was also applied (MRCI+Q). Three different active spaces were used in this work. One was used in all cases, namely, a CAS involving the 5f, 6d, and 7s orbitals where the 1s through 6p orbitals were fully optimized but constrained to be doubly occupied in all configurations. A second active space consisted of adding the 7p orbitals in order to account for near-degeneracy effects that occur in the cases where the 7s electrons are involved (IP1 and IP2). The effect of including the 7p orbitals into the active space for processes defined by just a change in 5f occupation (IP3, etc.) was found to be completely negligible. The last active space released the constraint of 6p double occupation, which had non-negligible effects for the highest IPs studied here. In order to decrease the size of the subsequent MRCI calculations in these latter cases, the 6p orbital occupation in the MRCI reference function was restricted to reside between 4 and 6 electrons.

TABLE I.

States of uranium atom used in the present work.

Uranium stateConfigurationGround state
[Rn]5f36d7s2 5L6 
U+ [Rn]5f37s2 4I9/2 
U2+ [Rn]5f4 5I4 
U3+ [Rn]5f3 4I9/2 
U4+ [Rn]5f2 3H4 
U5+ [Rn]5f 2F5/2 
U6+ [Hg]6p6 1S0 
Uranium stateConfigurationGround state
[Rn]5f36d7s2 5L6 
U+ [Rn]5f37s2 4I9/2 
U2+ [Rn]5f4 5I4 
U3+ [Rn]5f3 4I9/2 
U4+ [Rn]5f2 3H4 
U5+ [Rn]5f 2F5/2 
U6+ [Hg]6p6 1S0 

Unless otherwise noted, the MRCI calculations employed the frozen-core (FC) approximation whereby the 1s–5d electrons were not correlated (6s6p5f6d7s in valence). The effects of core-valence correlation were calculated by including the 5s5p5d electrons into the correlation treatment using the standard 5f6d7s active space. In order to facilitate these calculations, the new internally contracted MRCI method30 based on the Celani-Werner (CW) contraction scheme (denoted below as MRCIC)31 was utilized since it handles closed-shell orbitals much more efficiently. This method was also used for the frozen-core calculations described above with the 7p active space.

The basis sets used in this work corresponded to the cc-pVnZ-DK3 (n = D, T, Q) sets,16 which exhibit systematic convergence to both the CASSCF and MRCI CBS limits. The CV correlation effects were calculated using the cc-pwCVnZ-DK3 basis sets, both in frozen-core and 5s5p5d correlated calculations. The CBS limits were then determined (for FC and CV) for each species by separately treating the CASSCF and MRCI+Q correlation contributions to the energy. The CASSCF CBS limits were extrapolated using the Karton and Martin32 formula (originally formulated for the HF energy, but used here for CASSCF) with the VTZ and VQZ energies,

EnCASSCF=ECBSCASSCF+An+1e6.57n,
(2)

where n is the cardinal number of the basis set, which was used instead of lmax, i.e., n = 3 for TZ and n = 4 for QZ, as done previously for various Th and U species.16,33 The variable A in Eq. (2) is just a fitting parameter, while ECBSCASSCF is the resulting CASSCF CBS limit. The correlation energies at the MRCI+Q level were extrapolated to their CBS limits using n = 3 and 4 (TZ and QZ basis sets) via34,35

Encorr=ECBScorr+Bn+124,
(3)

where B is a fitting parameter and ECBScorr is the CBS limit of the correlation energy. An identical approach was used previously16 to calculate the first three ionization potentials of U at the frozen-core MRCI+Q level of theory.

Contributions to the IPs from the leading quantum electrodynamic (QED) effects, i.e., the Lamb shift (vacuum polarization and self-energy contributions), were calculated using the model potential approach of Pyykkö and Zhao.36 The local PP was constructed using the parameters of Table II from Ref. 36 for the self-energy term and a 5-term Gaussian fit to the vacuum polarization potential given by Eq. (15) of this same reference. For Z = 92,

VSE+VV P=Beβr21ri=15bieαir2,
(4)

with B = 3.4764 × 102, β = 1.2044 × 105, b1 = 3.6067 × 10−1, b2 = 2.2597 × 10−1, b3 = 3.1423 × 10−1, b4 = 1.2081 × 10−1, b5 = 5.8315 × 10−2, α1 = 4.2022 × 1010, α2 = 8.0395 × 106, α3 = 4.1972 × 108, α4 = 4.9610 × 105, α5 = 6.3602 × 104 (all in a.u.). This local PP was added to the 1-electron DKH3 Hamiltonian with the cc-pwCVDZ-DK3 basis set, and the resulting frozen-core MRCI+Q energy (standard 5f6d7s active space) was compared to the DKH3 result without the local PP to obtain a QED correction for each oxidation state of U.

TABLE II.

Summary of DKH3 scalar relativistic and Lamb shift contributions to the first six ionization potentials of U atom (kcal/mol).

IP1IP2IP3IP4IP5IP6
5f6d7s activea 137.3 273.6 438.9 750.0 1079.2 1429.1b 
+7pc +0.7 +2.1 +0.01    
+6pd   −0.3 −0.4 +0.7 +1.0 
+5s5p5de −3.6 −3.5 −1.4 +2.4 +5.7 +9.3b 
+Lambf +0.3 −1.5 +0.4 +0.6 +0.6 +0.7 
Total 134.8 270.6 437.6 752.6 1086.2 1440.1 
IP1IP2IP3IP4IP5IP6
5f6d7s activea 137.3 273.6 438.9 750.0 1079.2 1429.1b 
+7pc +0.7 +2.1 +0.01    
+6pd   −0.3 −0.4 +0.7 +1.0 
+5s5p5de −3.6 −3.5 −1.4 +2.4 +5.7 +9.3b 
+Lambf +0.3 −1.5 +0.4 +0.6 +0.6 +0.7 
Total 134.8 270.6 437.6 752.6 1086.2 1440.1 
a

Valence electrons (6s6p5f6d7s) correlated at the DKH3-MRCI+Q/CBS level with a 5f6d7s active space.

b

The DKH3-CCSD(T)/CBS result is 1430.3 kcal/mol with a 5s5p5d core-valence correlation contribution of +10.6 kcal/mol.

c

Contribution from including the 7p orbitals into the 5f6d7s active space at the MRCIC+Q/CBS level.

d

Contribution from expanding the reference space by allowing excitations from the 6p orbitals into the 5f6d7s space.

e

Contribution from 5s5p5d correlation at the MRCIC+Q/CBS level using a 5f6d7s active space. IP6 includes the contributions of 6p excitations into the reference space (the latter had no effect for IP5).

f

Contribution from the Lamb shift at the MRCI+Q/cc-pVDZ-DK3 level of theory using a model potential approach. See the text.

The calculations of spin-orbit coupling effects on the IPs were carried out at the 4-component level using the DIRAC program.37 These calculations used the DC Hamiltonian, and the contributions of spin-orbit coupling to a given atomic energy were then assessed by comparing the DC results to analogous calculations using Dyall’s spin-free Hamiltonian.38 In addition, the Gaunt interaction was added to the Hamiltonian and its effect was assessed at the Dirac-Hartree-Fock (DHF) level of theory.

The 4-component correlation consistent basis sets of Dyall were used for all calculations in DIRAC, both VDZ and VTZ,39 and these were used in their uncontracted forms. The small component basis was generated by restricted kinetic balance. The open shell systems were first calculated using the average of configuration self-consistent field (aoc-SCF) method. The aoc-SCF calculations were carried out using the spinors associated with the electron configuration corresponding to the ground state of each oxidation state (cf. Table I). At the aoc-SCF level of theory, the individual SCF energies of the lowest J or LS levels were resolved by carrying out a full CI just within the open shell manifold. Using the aoc-SCF spinors, correlated calculations were then carried out with the Kramers-restricted configuration interaction (KRCI) method40–42 for the maximum Mj projection, e.g., Mj = 9/2 for J = 9/2. The KRCI calculations utilized generalized active spaces (GASs)41,43 to restrict the reference configurations. Each GAS space consisted of a given set of orbitals to which electrons were sequentially added to represent the ground state configuration for each oxidation state of U. These orbital groups corresponded to 6s+6p, 5f, 7s, and 6d. The 7s and 6d were partitioned primarily to allow unambiguous specification of U+ as 5f37s2. The definitions of the GAS shells are explicitly given in the supplementary material.44 Single and double excitations into the virtual orbitals were then allowed out of each GAS space. A virtual orbital cutoff of 10 a.u. was employed throughout.

As can be observed from the electron configurations of Table I, the first IP of U involves the ionization of a 6d electron, while IP2 results from both the loss of a 7s electron and promotion of a 7s electron to the 5f shell to yield the 5f4 configuration of U2+. The next 4 IPs all involve successive removal of an electron from the 5f shell until a Rn-like configuration is reached at U6+. It should be noted that the ground states calculated in this work and shown in Table I are identical to those reported by experiment2 as well as in the previous work on IP1 through IP4 by Dolg and co-workers.10,11

A summary of the scalar relativistic results of the present work is shown in Table II. A comprehensive tabulation of the individual basis set results, including MRCI and MR-ACPF, can be found in the supplementary material.44 As can be seen in Table II, frozen-core MRCI+Q calculations with the “standard” 5f6d7s active space account for the majority of the final calculated IPs. As has been noted previously,11 including the 7p orbital into the active space is important to account for near-degeneracy effects for IPs involving occupation of the 7s orbital. This is clearly seen for IP1 and IP2 where adding the 7p increased the ionization potentials by 0.7 and 2.1 kcal/mol, respectively. In the case of IP3, however, which involves only ionization of an electron from the 5f, this extension of the active space has a negligible effect. For the higher IPs, particularly IP6, it was found to be important to relax the constraint of double occupancy of the 6p orbital in the reference function; this increased IP5 and IP6 by 0.7 and 1.0 kcal/mol, respectively.

Correlation of outer-core electrons is known in general to be critical for high accuracy relative energies, and as shown in Table II, the effects of correlation of the 5s5p5d electrons of U are certainly not negligible for its IPs, ranging from decreasing IP1 by 3.6 kcal/mol to increasing IP6 by 9.3 kcal/mol. It should be noted at this point that only the previous MR-ACPF calculations (IP1–IP4) of Cao and Dolg10 and the IH-FSCCSD calculations of Kaldor and co-workers (IP5 and IP6)15,45 accounted for correlation of electrons below the 6s, and neither specifically addressed the role of this outer-core correlation on the IPs. In addition to the MRCI+Q values of ΔCV shown in Table II, a number of other levels of theory were investigated to assess the reliability of these contributions since MRCI+Q is not rigorously size extensive. DKH3 CBS limit results using complete active space 2nd-order perturbation theory,31,46 CASPT2 (IP1–IP6), ACPF47,48 (IP2–IP6), and coupled cluster with singles, doubles, and perturbative triples,49 CCSD(T) (IP6, using ROHF-CCSD(T)50–52 for U5+), are compared to MRCI+Q in Table SIII of the supplementary material.44 The CASPT2 results for ΔCV differ from MRCI+Q by +0.6 to +7.8 kcal/mol, somewhat smoothly increasing from IP1 to IP6. The analogous ACPF results are much more erratic, sometimes falling in between MRCI+Q and CASPT2 (IP3 and IP5), but the CV effect for IP2 is even larger than the CASPT2 result, IP4 is opposite in sign to either MRCI+Q or CASPT2, and ΔCV for IP6 is nearly identical to CASPT2, which is much larger than MRCI+Q. The one CCSD(T) result, ΔCV for IP6, differs from the present MRCI+Q result by just 1.3 kcal/mol, indicating that the CASPT2 and ACPF methods strongly overestimate ΔCV for IP6. Given the overall trend in the CASPT2 results, these seem to be uniformly too large. The ACPF result of −3.2 kcal/mol for IP4 is difficult to reconcile since this IP simply involves the loss of a 5f electron just as in IP3, IP5, and IP6 which all have positive values of ΔCV. All together, while the MRCI+Q results for ΔCV are expected to be the most reliable, the largest uncertainty in the presently calculated IPs (mainly IP2–IP5) arises from the ΔCV contribution, which could lead to our final calculated values being consistently too small in magnitude by a few kcal/mol.

The last contribution shown in Table II, due to the Lamb shift, has not to the authors’ knowledge been reported before for the IPs of U. For all but IP2, this contribution is less than 1 kcal/mol. For the former, this process involves a change in 7s occupation, which is known to potentially lead to non-negligible Lamb shift effects.53 In the present case, the effect of the Lamb shift for IP2 (−1.5 kcal/mol) is 2-3 times larger than the values calculated for any of the other IPs.

In order to test the choice of the DKH3 Hamiltonian in this work, calculations were also carried out at the CCSD(T) level of theory for IP6 using the DKHn Hamiltonian at both n = 3 and n = 8 with an uncontracted cc-pVDZ-DK3 basis set and including 5s5p5d correlation. While the total energies were significantly different between DKH3 and DKH8 (∼5.077 Eh), the resulting IPs differed by just 0.1 kcal/mol. In contrast, analogous calculations using the DKH2 Hamiltonian yielded a value for IP6 that was 0.7 kcal/mol larger than DKH3.

Obviously for quantitative predictions of ionization potentials, spin-orbit coupling must be accurately taken into account, particularly for actinide atoms. Table III shows results at the HF level of theory, both all-electron and PP-based, compared to values presented previously by Weigand et al.11 Focusing first on the all-electron 4-component DHF/DC+Gaunt IPs of this work, excellent agreement (within at most about 0.2 kcal/mol) is observed with the 4-component multiconfiguration DHF (MCDHF) reference IPs of Ref. 11, which also included a perturbative estimate of the Breit interaction. The IPs of Ref. 11 resulting from applying their MCDHF/DC+B PPs in variational 2-component multiconfiguration HF calculations are also shown in Table III and demonstrate the expected high accuracy of these PPs, with the largest difference (from the all-electron MCDHF/DC+B reference data) being observed for IP1 at about 0.7 kcal/mol. Difficulties arise, however, when the SO effects are separated from the total IPs. Table III shows that the all-electron results for ΔSO are quite different from the PP-based values for IP2 through IP4. This is not due to the present HF treatment since an analogous 2c-PP aoc-HF calculation nearly exactly reproduces the 2c-PP-MCHF results of Ref. 11 (also shown in Table III). Instead, this seems to point to the ambiguity in separating the scalar and spin-orbit contributions from the PP, noting that the PP adjustment is originally made on fully relativistic energy levels. It should be stressed again that the total (scalar+SO) IPs from the PP are very accurate. It seems clear, however, that one should hesitate to match a SO contribution from a PP calculation with an all-electron scalar relativistic result (or vice versa).

TABLE III.

Comparisons of current IPs and SO contributions at the HF level of theory with those of Ref. 11 (in kcal/mol).

IPnIP 4c-DHF+GauntaIP 4c-MCDHF+BreitbIP 2c-PP-MCHFcΔSO 4c-DHF+GauntdΔSO 2c-PP-aoc-HFeΔSO 2c-PP-MCHFc
IP1 127.64 127.75 127.08 +3.61 +3.36 +3.378 
IP2 273.39 273.16 273.15 +1.82 +4.35 +4.365 
IP3 390.74 390.71 390.58 −1.13 −2.86 −2.866 
IP4 718.79 718.81 719.15 +2.64 +0.42 +0.413 
IPnIP 4c-DHF+GauntaIP 4c-MCDHF+BreitbIP 2c-PP-MCHFcΔSO 4c-DHF+GauntdΔSO 2c-PP-aoc-HFeΔSO 2c-PP-MCHFc
IP1 127.64 127.75 127.08 +3.61 +3.36 +3.378 
IP2 273.39 273.16 273.15 +1.82 +4.35 +4.365 
IP3 390.74 390.71 390.58 −1.13 −2.86 −2.866 
IP4 718.79 718.81 719.15 +2.64 +0.42 +0.413 
a

This work. All-electron 4c-aoc-DHF level of theory (resolved to ground states) with an uncontracted (Dyall) VTZ basis set +Gaunt.

b

All-electron reference data of Weigand et al.;11 multiconfigurational Dirac-Hartree-Fock with a perturbative estimate of the Breit interaction.

c

Reference 11, their values were obtained by applying the PP of Ref. 54 in variational 2-component multiconfiguration HF calculations.

d

This work. SO contribution at the all-electron 4c-aoc-DHF level of theory with an uncontracted (Dyall) VTZ basis set with inclusion of the Gaunt contribution.

e

This work. SO contribution at the 2c-PP-aoc-HF level of theory with an uncontracted cc-pVTZ-PP basis set16 and the same PP as in Ref. 11.

A breakdown of the all-electron SO contributions of this work to the first 6 IPs for U is shown in Table IV. The overall values of ΔSO range from nearly zero (IP3, −0.13 kcal/mol) to just over 12 kcal/mol (IP6, 12.11 kcal/mol), and these values are all smaller in magnitude than the current experimental uncertainties in IP3 through IP6. As seen in Table IV, the basis set dependence of ΔSO is nearly negligible in all cases, with the largest effect observed upon extending the basis set from uncontracted VDZ to uncontracted VTZ at the DHF level for IP6, −0.22 kcal/mol. Compared to the HF contributions (see also Table III), electron correlation yielded only modest changes to ΔSO, from a total of about 1.4 kcal/mol for IP1 to just −0.3 for IP6. It should be noted that the correlation effects also implicitly include possible important orbital relaxation effects due to using aoc-SCF rather than MCDHF orbitals in the KRCI. To some extent, some hints of this can be obtained by comparing the present aoc-DHF+Gaunt results with the MCDHF/DC+Breit SO contributions of Weigand et al.11 for IP1 through IP4. As discussed above and shown in Table III, the differences are nearly negligible. For the most part, the addition of a Davidson correction (+Q) to the KRCI results was non-negligible, particularly for IP2, IP3, and IP5 where its inclusion nearly doubled the total correlation contribution compared to KRCI alone. Also shown in Table IV is the contribution to the SO due to the Gaunt term, which includes the spin-other-orbit interaction of the Dirac-Coulomb-Breit (DCB) Hamiltonian. Its magnitude ranges from essentially zero for IP1 to 2-3 kcal/mol for IP2 through IP6 (negative only in the case of IP2). These values can be compared to contributions from the full Breit interaction, which were calculated previously in the works of Weigand et al. by perturbation theory11 and Infante et al.,15 the latter of which used both the DC and full DCB Hamiltonian in IH-FSCCSD calculations. For IP2 through IP4, the present Gaunt contributions are larger in magnitude than the perturbative estimates of the Breit interaction by Weigand et al. by about 0.6 kcal/mol. As in this work, the latter Breit calculations also obtained a contribution of nearly zero for IP1. For IP5 and IP6, the present Gaunt contributions shown in Table IV can be compared to the difference in DC-IH-FSCCSD and DCB-IH-FSCCSD calculations,15 +2.4 kcal/mol for IP5 and +2.5 kcal/mol for IP6, both of which are also about 0.6-0.7 kcal/mol smaller in magnitude than the Gaunt contributions calculated in this work.

TABLE IV.

Breakdown of final SO contributions to each IP of uranium atom (in kcal/mol).

4c-DHFa4c-KRCI correlation contributionΔQ (KRCI+Q–KRCI)ΔGaunt
VDZVTZVDZVTZVDZVTZVDZVTZFinal ΔSOb
IP1 3.61 3.62 1.18 … 0.20 … −0.01 −0.01 4.98 
IP2 4.75 4.71 −0.91 … −0.44 … −2.88 −2.89 0.53 
IP3 −3.73 −3.58 0.63 0.56 0.46 0.44 2.45 2.45 −0.13 
IP4 −0.25 −0.21 0.69 0.65 0.01 0.01 2.85 2.85 3.30 
IP5 4.97 4.85 0.10 0.11 0.26 0.28 3.04 3.03 8.27 
IP6 9.40 9.18 −0.18 −0.22 −0.03 −0.04 3.20 3.20 12.11 
4c-DHFa4c-KRCI correlation contributionΔQ (KRCI+Q–KRCI)ΔGaunt
VDZVTZVDZVTZVDZVTZVDZVTZFinal ΔSOb
IP1 3.61 3.62 1.18 … 0.20 … −0.01 −0.01 4.98 
IP2 4.75 4.71 −0.91 … −0.44 … −2.88 −2.89 0.53 
IP3 −3.73 −3.58 0.63 0.56 0.46 0.44 2.45 2.45 −0.13 
IP4 −0.25 −0.21 0.69 0.65 0.01 0.01 2.85 2.85 3.30 
IP5 4.97 4.85 0.10 0.11 0.26 0.28 3.04 3.03 8.27 
IP6 9.40 9.18 −0.18 −0.22 −0.03 −0.04 3.20 3.20 12.11 
a

aoc-DHF/DC (resolved to ground states).

b

4c-DHF/VTZ+4c-KRCI/VTZ (VDZ for IP1 and IP2) + ΔQ/V TZ (VDZ for IP1 and IP2) + ΔGaunt/V TZ.

The final values of ΔSO of the present work for IP1 through IP4, including correlation at the 4c-KRCI+Q level of theory and the Gaunt term, differ from the MCDHF/DC+B finite difference PP results of Weigand et al. by +1.6, −3.8, +2.7, +2.9 kcal/mol, respectively (see the discussion above however in regards to Table III). In regards to IP5 and IP6, DC-FSCCSD/VDZ calculations have also been carried out in this work (using an active space of the seven 5f spinors), and the resulting SO contribution at that level of theory for IP5 and IP6 differs from the present KRCI+Q results by just −0.6 and +0.2 kcal/mol, respectively.

Table V compares the final composite ionization potentials from this work with other theoretical values as well as the available experimental results. Focusing first on IP1, the present value, 139.8 kcal/mol, is smaller than the accurately known experimental result by 3 kcal/mol. The calculated value of IP2 is well within the 9 kcal/mol reported error bars. Both IP3 and IP4 are well outside the experimental error bars, and as previously noted by Dolg and co-workers,10,11 the results derived from the electron impact ionization experiments5 would seem to be too large in these cases. The calculated value of IP6 in this work is also in good agreement with the semi-empirical value from the literature,2 although the quoted uncertainty in the latter is very large. Overall the agreement with previous high level calculations is very good—the present values for IP1 through IP4 differ from those of Weigand et al.11 by no more than 4.6 kcal/mol, but it should be noted that the latter calculations did not include 5s5p5d correlation, which as shown in Table II decreases IP1 by nearly 4 kcal/mol. Applying the present CV corrections to the IPs of Weigand et al. generally improves the level of agreement between the two sets of calculations to an averaged unsigned deviation of just 1.7 kcal/mol. As shown in the supplementary material,44 MRCI+Q and ACPF generally agree to within 1 kcal/mol at their frozen-core CBS limits (5f6d7s active space), except for IP2 and IP4 where they differ by +5 and −4 kcal/mol, respectively. The origin of these differences is not clear, but the MRCI+Q results were deemed more reasonable based on comparison to MRCI alone. Compared to the DCB-FS-CCSD results of Infante et al.15 and Eliav and Kaldor,45 the present results for IP5 and IP6 differ by just 1-5 kcal/mol. Also shown in Table IV are the results of two sets of density functional theory studies, the small, scalar-only SARC B3LYP calculations of Pantazis and Neese12 and the all-electron DKS calculations of Liu et al.13 with the Becke-Perdew exchange-correlation functional. In particular, the DKS calculations yielded good agreement with the present values for IP2 through IP4, differing by just a few kcal/mol. The difference for IP1, however, was 2-3 times larger than the other IPs.

TABLE V.

Comparison of calculated IPs of uranium with other calculated values and experiment (kcal/mol with values in eV in parentheses).

This workSARC B3LYPaDKS BPbPP-MR-ACPFcPP-MR-ACPFdDC-FSCCeDCB-FSCCeDCB-FSCCfDCB-CASPT2eExpt.
IP1 139.8 125.7 129.4 138.6 144.4     142.838 ± 0.001g 
 (6.06) (5.45) (5.61) (6.01) (6.26)     (6.194 05) 
IP2 271.2 275.3 274.2 274.9 274.2     267 ± 9h 
 (11.76) (11.94) (11.89) (11.92) (11.89)     (11.6 ± 0.4) 
IP3 437.4 445.1 432.8 433.5 434.9     457 ± 7h 
 (18.97) (19.30) (18.77) (18.80) (18.86)     (19.80 ± 0.30) 
IP4 755.9 761.7 753.6 755.5 752.0     846 ± 23h 
 (32.78) (33.03) (32.68) (32.76) (32.61)     (36.7 ± 1.0) 
IP5 1094.5     1087.1 1089.5  1151.2 1060 ± 44i 
 (47.46)     (47.14) (47.25)  (49.92) (46.0 ± 1.9) 
IP6 1452.2     1450.5 1453.0 1455.9  1430 ± 37j 
 (62.97)     (62.90) (63.01) (63.13)  (62.0 ± 1.6) 
This workSARC B3LYPaDKS BPbPP-MR-ACPFcPP-MR-ACPFdDC-FSCCeDCB-FSCCeDCB-FSCCfDCB-CASPT2eExpt.
IP1 139.8 125.7 129.4 138.6 144.4     142.838 ± 0.001g 
 (6.06) (5.45) (5.61) (6.01) (6.26)     (6.194 05) 
IP2 271.2 275.3 274.2 274.9 274.2     267 ± 9h 
 (11.76) (11.94) (11.89) (11.92) (11.89)     (11.6 ± 0.4) 
IP3 437.4 445.1 432.8 433.5 434.9     457 ± 7h 
 (18.97) (19.30) (18.77) (18.80) (18.86)     (19.80 ± 0.30) 
IP4 755.9 761.7 753.6 755.5 752.0     846 ± 23h 
 (32.78) (33.03) (32.68) (32.76) (32.61)     (36.7 ± 1.0) 
IP5 1094.5     1087.1 1089.5  1151.2 1060 ± 44i 
 (47.46)     (47.14) (47.25)  (49.92) (46.0 ± 1.9) 
IP6 1452.2     1450.5 1453.0 1455.9  1430 ± 37j 
 (62.97)     (62.90) (63.01) (63.13)  (62.0 ± 1.6) 
a

Reference 12, scalar relativistic only results.

b

Reference 13, all-electron Dirac-Kohn-Sham (DC Hamiltonian) with Becke-Perdew exchange-correlation.

c

Reference 10, extrapolated frozen-core CBS limits using quasirelativistic Wood-Boring PPs with a PP correction based on all-electron calculations.

d

Reference 11, extrapolated frozen-core CBS limits using MCDHF/DC+B PPs with SO calculated at the CASSCF level.

e

Reference 15, large basis set, extrapolated intermediate Hamiltonian FSCCSD results with the DC or DCB 4-component Hamiltonians.

f

Reference 45, large basis set, extrapolated intermediate Hamiltonian FSCCSD results with the Dirac-Coulomb-Breit 4-component Hamiltonian.

g

References 3 and 4.

h

Electron impact ionization experiments of Ref. 5.

i

Theoretical estimate from Ref. 55. See also Ref. 56.

j

Semi-empirical value from J. F. Wyart, unpublished, as cited in Ref. 2.

Using a consistent and accurate treatment for both electron correlation and relativistic effects, the first 6 ionization potentials of the U atom have been calculated. In scalar relativistic DKH3 calculations, extrapolation to the CBS limits was facilitated by the use of all-electron correlation consistent basis sets. The effect of correlation of the outer-core 5s5p5d electrons of U was found to contribute 1.4–9.3 kcal/mol in magnitude, while inclusion of the Lamb shift decreased IP2 by 1.5 kcal/mol. Spin-orbit effects on the IPs based on 4-c KRCI calculations ranged from nearly zero (IP3) to about 12 kcal/mol (IP6). The Gaunt interaction included in this work was found to agree well with previous calculations that included the full Breit term of the DCB Hamiltonian, with contributions to the IPs averaging about 2.5 kcal/mol. This is nearly twice as large as the effects on SO due to electron correlation as calculated in this work. Based on the agreement with the accurately known experimental value for the first IP of U, as well as with previous FSCCSD calculations for IP5 and IP6, the IPs from this work are expected to be accurate to within about 5 kcal/mol (0.2 eV). Most of this uncertainty arises from the calculated values of the core correlation contribution ΔCV, where MRCI+Q might lead to a systematic underestimation by a few kcal/mol. This is expected to be much larger than any errors due to additivity assumptions in the present composite method. As in previous calculations, the experimental electron impact values for IP3 and particularly IP4 are judged to be too high and warrant re-investigation. Overall the calculated values of this work for IP2 through IP6 are expected to be more reliable than the currently recommended experimental values.

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 the generous help provided by the user group of the DIRAC program as well as Dr. Stefan Knecht for his development of the Davidson correction module in DIRAC.

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Supplementary Material