Relativistic Small-Core Pseudopotentials for Actinium, Thorium, and

Article pubs.acs.org/JPCA

Relativistic Small-Core Pseudopotentials for Actinium, Thorium, and Protactinium Anna Weigand,* Xiaoyan Cao, Tim Hangele, and Michael Dolg* Institute for Theoretical Chemistry, University of Cologne, Greinstrasse 4, 50939 Cologne, Germany S Supporting Information *

ABSTRACT: Small-core pseudopotentials for actinium, thorium, and protactinium have been energy-adjusted to multiconfiguration Dirac− Hartree−Fock reference data based on the Dirac−Coulomb−Breit Hamiltonian and the Fermi nucleus model. Corresponding optimized valence basis sets of polarized valence quadruple-ζ quality are presented. Atomic test calculations for the first four ionization potentials show satisfactory results at both the Hartree−Fock and the multireference averaged coupled-pair functional level. Highly correlated Fock-space coupled cluster calculations demonstrate that the new pseudopotentials yield ionization potentials, which are in excellent agreement with corresponding all-electron results and experimental data. The pseudopotentials and basis sets supplement a similar set previously published for uranium.

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INTRODUCTION Quantum chemical investigations of actinide systems are nowadays still a great challenge because of the large number of electrons, the significant relativistic effects, and the distinct electron correlation. Furthermore, the nearly degenerate 5f, 6d, 7s, and 7p orbitals give rise to a multitude of possible configuration interactions and a high density of low-lying electronic states, which complicates computations.1,2 There are two possible ways to perform ab initio calculations on molecules including actinides: either the relativistic effective core potential (ECP) approach, i.e., model potential (MP) and pseudopotential (PP) methods,3,4 or the more rigorous, albeit in practice still approximate, relativistic all-electron (AE) schemes.5 In contrast to AE methods MPs and even more PPs lead to significant computational savings at the scalarrelativistic level, especially if d or f shells can be attributed to the PP core. Furthermore, the implicit treatment of relativistic effects in the case of PPs also allows us to include contributions such as the Breit interaction,4 which is neglected by standard AE schemes based solely on the full four-component Dirac− Coulomb (DC) Hamiltonian or its two-component forms, e.g., the exact two-component relativistic (X2C) Hamiltonian6,7 or the Douglas−Kroll−Hess (DKH)8−11 Hamiltonian. With regard to electron correlation PPs allow for large active spaces and accurate correlation methods, because the computational effort is concentrated on the important valence part of the system using smaller valence basis sets compared to AE basis sets.12 Moreover, the combination of PPs and effective corepolarization potentials accounting for static and dynamic corepolarization admits to include core−valence correlation effects. PPs mainly come in two varieties, i.e., shape-consistent ones based on one-particle properties such as orbitals and orbital © 2014 American Chemical Society

energies, as well as energy-consistent ones based on total valence energies as AE reference data. Small-core twocomponent shape-consistent PPs for several light actinides have been generated by Mosyagin and co-workers.13 A comparison to the present energy-consistent PPs in terms of accuracy was published in a previous paper.4 PPs for selected actinides using larger, albeit less accurate cores14 were published by Hay and Martin15 and by Ermler and coworkers.16 For actinides there exist two kinds of energy-consistent PPs with different core definitions, i.e., 5f-in-valence small-core PPs (SPPs) and 5f-in-core large-core PPs (LPPs), both adjusted to scalar-relativistic Wood−Boring (WB) reference data. The SPPs17 include the 1s−4f shells in the PP core, whereas all orbitals with main quantum numbers larger than 4 (29−43 electrons) are treated explicitly. Furthermore, additional effective valence spin−orbit (SO) operators18 adjusted to AE state-averaged multiconfiguration Dirac−Hartree−Fock (MCDHF) data based on the Dirac−Coulomb Hamiltonian with a perturbative treatment of the Breit interaction (DC+B) were provided. A variational treatment of the Breit interaction together with the Dirac−Coulomb Hamiltonian (DCB) leads for valence energy differences to very similar results as the perturbative treatment. The LPPs include the 1s−5f shells in the PP core, whereas all orbitals with main quantum numbers larger than 5 (10−14 electrons) are treated explicitly. Because here the open 5f shell is treated implicitly, one PP for each Received: January 8, 2014 Revised: March 14, 2014 Published: March 14, 2014 2519

dx.doi.org/10.1021/jp500215z | J. Phys. Chem. A 2014, 118, 2519−2530

The Journal of Physical Chemistry A

Article

possible 5f occupation was adjusted; i.e., di-,19 tri-,20 tetra-,19 penta-,21 and hexavalent21 PPs are available. The WB SPPs have been quite successfully applied in many calculations but showed less satisfactory results in the case of the atomic third and fourth ionization potentials (IPs); i.e., the finite-difference PP multiconfiguration Hartree−Fock (MCHF) results deviate on average by 0.28 (1.4%) and 0.53 eV (1.5%) from AE MCDHF values based on the DC+B Hamiltonian for IP3 and IP4, respectively.22 In the case of IP1 and IP2 these deviations amount only to 0.03 (0.6%) and 0.06 eV (0.5%), respectively, mainly because in contrast to IP3 and IP4 these ionizations in most cases take place without changes in the 5f occupation. Test calculations indicated that the larger deviations for the higher IPs do not arise from the PP approximation itself, but rather from the deviations between the underlying AE WB from the corresponding AE MCDHF/DC +B average energies.22 The WB SPPs were supplemented by valence SO operators to be used in first-order perturbation theory18 and are not recommended for variational twocomponent treatments. Therefore, an adjustment of new PPs directly to four-component AE MCDHF/DC+B reference data within the intermediate coupling scheme appeared to be desirable.3 In such a way, besides the dominating Dirac oneelectron relativistic contributions, the SPPs can also implicitly include the Breit interaction and thus can model relativistic AE approaches, which go beyond the DC Hamiltonian or approximations thereof.4 Relativistic SPPs including the Breit interaction are already available for heavy main group elements,23−27 transition metals,28−30 and the superheavy elements with nuclear charges 111−120.31−33 Recently, a MCDHF PP based on the DC Hamiltonian perturbatively including the low-frequency limit Breit interaction has been adjusted for uranium and tested for the cations U4+ and U5+ 3 as well as for uranium monohydride UH.4 The deviations of molecular results obtained with this improved MCDHF/DC+B PP from AE reference data are slightly smaller than those for the WB PP except for the vibrational constant. We present improved SPPs analogous to this uranium PP for actinium, thorium, as well as protactinium together with polarized valence quadruple-ζ basis sets and test them for the first to fourth ionization potentials. Molecular tests were also completed and will be published elsewhere.

nuclear charge decreased by the number of core electrons), and P̂lj is the projection operator onto the Hilbert subspace with angular momentum quantum numbers l and j = l ± 1/2 j

Plĵ =

(3)

The ansatz for the radial potentials is a linear combination of k Gaussians with coefficients Bklj and exponents βklj. The pseudopotential ΔVPP can be separated in a spin-averaged (SA) scalar-relativistic part to be used in one-component calculations L−1 PP ΔVSA ( ri ⃗) =

∑ l=0

PP lVlPP , |l − 1/2|(ri) + (l + 1)Vl , l + 1/2(ri)

2l + 1

Pl(̂ i) (4)

and a spin−orbit part to be used subsequently in a variational or perturbative two-component treatment L−1 PP ΔVSO ( ri ⃗) =

∑ l=1

PP 2(VlPP , l + 1/2(ri) − Vl , |l − 1/2|(ri))

2l + 1

Pl(̂ i)liŝ îPl(̂ i) (5)

The 5f-in-valence SPPs for actinium, thorium, and protactinium treat 29, 30, and 31 valence electrons in shells with main quantum numbers n ≥ 5 explicitly, whereas the 1s− 4f shells (60 electrons) are included in the PP core. The PP parameters up to f symmetry were adjusted in two-component MCHF calculations to four-component AE MCDHF/DC+B reference data based on the Fermi nucleus model (masses in atomic mass units: Ac 227, Th 232, Pa 231) using the atomic structure code GRASP,35 which works at the finite-difference level avoiding basis set effects. The same intermediate coupling scheme was used at the PP and AE level. The reference data comprised 38, 57, and 81 nonrelativistic configurations yielding a total of 333, 3673, and 16186 J levels for Ac, Th, and Pa, respectively. These configurations were obtained for Ac−Ac4+, Th−Th5+, and Pa−Pa6+, respectively, and included not only a wide spectrum of occupations in the 5f, 6d, 7s, and 7p valence shells but also additional configurations with holes in the core/ semicore orbitals 5s, 5p, 5d, 6s, and 6p as well as configurations with electrons in the 7f−9f (only for Pa), 7d−9d, 8p−9p, and 8s−9s shells (cf. the Supporting Information for lists of reference configurations). Due to convergence problems, mainly occurring for energetically higher and relatively diffuse f orbitals, some configurations could not be included into the reference set. Because the energetic position of the bare inner core relative to valence states is not expected to be notably relevant for chemical processes, the fit was restricted to the chemically more significant energy differences between valence states; i.e., a global shift was applied to all reference energies and treated as an additional parameter to be optimized29

METHOD In the following the pseudopotential adjustment and the basis set optimization are described. Pseudopotential Adjustment. The method of relativistic energy-consistent ab initio pseudopotentials and in particular their adjustment to MCDHF/DC+B reference data is described in detail elsewhere4,24,29,34 and will be outlined only briefly. The atomic valence-only model Hamiltonian is given as 1 1 Ĥ v = − ∑ Δi + ∑ + ∑ VPP( ri ⃗) 2 i r i < j ij i (1)

∑ (ωI [EIPP − EIAE + ΔEshift]2 ):= min I

EPP I

(6)

EAE I

Here, and are the PP total valence energies and the corresponding AE valence energies, respectively. The weights ωI were chosen to be equal for all J levels arising from a nonrelativistic configuration, and all nonrelativistic configurations were assigned to have equal weights. The global shift ΔEshift allows for the usage of configurations including core/ semicore holes and can improve the accuracy of the fit by one or 2 orders of magnitude. For all PPs the global shift amounts to less than 1% of the ground state total valence energy.

VPP (ri⃗ ) denotes a semilocal ECP and its second term implicitly includes all relativistic effects Q Q + ΔVPP( ri ⃗) = − + ri ri

|ljmj⟩⟨ljmj|

mj =−j

■

VPP( ri ⃗) = −

∑

∑ Bljk exp(−βljkri 2)Plĵ ljk

(2)

whereas all other terms in eq 1 and eq 2 are formally nonrelativistic. Q corresponds to the core charge (i.e., the 2520



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Table 1. Accuracy of the PP Adjustmenta no. An Ac Th Pa U

conf 38 57 81 100

mae J 333 3673 16186 30190

conf 0.0026 0.0067 0.0083 0.0015

max dev J

conf +

0.004 0.013 0.021 0.024

Ac Th Pa U

2

5f 5f26d2 5f5 5f47s17p1

n

eV

%

13 442 188 1218

0.027 0.095 0.208 0.269

1.8 1.6 3.2 2.2

a Given are the number (no.) of nonrelativistic configurations (conf) and corresponding J levels (J) used for the adjustment of the s, p, d, and f projectors, the mae (eV) in the total valence energies for J levels and nonrelativistic configurations (a [Rn] core is assumed). Furthermore, the maximum deviations (max dev) in the total valence energies of the J levels are listed, with the corresponding nonrelativistic configuration and the number of the J level (n) in the configuration. For comparison the results of the uranium PP are also shown.4

figure shows that the 5f and 6d shells are spatially split in 5f5/2/ 5f7/2 and 6d3/2/6d5/2 subshells, respectively, indicating nonnegligible SO effects. Basis Set Optimization. The Gaussian type orbital (GTO) valence basis sets were constructed analogous to the (14s13p10d8f6g)/[6s6p5d4f3g] basis set for the uranium MCDHF/DC+B PP.4 First, (14s11p8d8f) basis sets were HF energy-optimized36 for the [Rn] 5fn+17s2 (n = 0−2 for Ac−Pa) states of the neutral actinide atoms. In the case of Ac the strong indirect relativistic destabilization and expansion leads to a very diffuse 5f shell in the Ac [Rn] 5f17s2 state of the neutral atom. The f exponents HF energy-optimized36 for this state are much smaller than those for the heavier actinides and are unable to describe an occupied 5f shell in a charged Ac system, e.g., the largest f exponents are 5.01, 32.81, and 38.89 for Ac, Th, and Pa, respectively. Therefore, we decided to derive the f exponents for the Ac standard basis set for the [Rn] 5f1 state of the doubly charged cation, which results in a largest f exponent of 32.95. However, if one is interested to study a case, where the energetically high-lying Ac [Rn] 5f17s2 state may be important, one should either increase the actinium standard basis set by diffuse f exponents or use the set optimized for the neutral atom (cf. Supporting Information). Second, two diffuse d and p exponents for the description of the 6d and 7p shells were HF energy-optimized for the [Rn] 5fn6d17s2 and [Rn] 5fn7s27p1 (n = 0−2 for Ac−Pa) states, respectively. Third, the generalized contracted basis sets (14s13p10d8f)/ [6s6p5d4f] were obtained by using coefficients from atomic natural orbitals (ANOs). To ensure a reasonable description of states with different 5f occupations, the contraction coefficients were taken from averaged density matrices for the lowest LS states of the [Rn] 5fn6d17s2 and [Rn] 5fn+17s2 (n = 0−2 for Ac− Pa) configurations received from state-averaged complete active space self-consistent field (CASSCF) calculations followed by a multireference configuration interaction (MRCI) treatment in MOLPRO.37 In the case of Ac the averaged density matrix for the lowest LS states of the Ac2+ [Rn] 5f1 configuration was included as well by mixing 50% Ac [Rn] 6d17s2, 25% Ac [Rn] 5f17s2, and 25% Ac2+ [Rn] 5f1. It is hoped that thus the 5f shell of Ac bearing a charge intermediate to 0 and +2 in a molecular environment is reasonably well described. Symmetry-breaking at the CASSCF level was avoided by averaging over all components of each LS state. Whereas no orbitals were frozen in the CASSCF calculations, the 5s, 5p, and 5d shells were kept frozen in the MRCI treatment, i.e., the basis sets are able to correlate the 5f and n > 5 shells. If the correlation of 5s, 5p, and 5d orbitals is also needed, additional functions should be added. Finally, (6g) exponents identical to the 6 largest f exponents were added and contracted as described above.

Due to the SO splitting of the shells with angular momentum l > 0 the pseudopotential VPP is divided into two components Vl,j=l−1/2 and Vl,j=l+1/2. For each lj term up to f symmetry four Gaussians were optimized yielding 56 parameters. The g-parts of the PPs consist of three Gaussians for each j term and were adjusted to the following eight single-valence-electron configurations Ac28+, Th29+, and Pa30+ [Kr] 4d104f14ng1 (n = 5−12), respectively. Whereas the fit of the g-part can be considered as exact, Table 1 demonstrates the accuracy for the adjustment of the s-, p-, d-, and f-parts (including the results of the previously published uranium PP4). Detailed results for each J level are provided in the Supporting Information. The mean absolute errors (mae) for J levels of Ac, Th, and Pa amount at most to 0.021 eV (Pa) and those for nonrelativistic configurations stay below 0.0085 eV, which is clearly below the error of 0.025 eV introduced by the basis set incompleteness (vide infra). The maximum deviation between the AE reference and PP energies amounts to 0.208 eV corresponding to 3.2% and occurs for the 188th, i.e., a very high-lying, J level of the chemically not too important configuration [Rn] 5f5 of Pa. Overall, the errors increase with increasing complexity of the electronic structure of the atoms, i.e., from Ac to Pa, and fit quite well to those observed for U. Because the PPs were adjusted to energetic reference data, it is interesting to check how well the AE valence orbitals are reproduced by the PP pseudovalence orbitals in the valence region. Figure 1 gives such a comparison for the Pa ground state [Rn] 5f26d17s2. One can see that the radial distribution of the 5f, 6d, and 7s PP pseudovalence orbitals are identical to the AE orbitals in the spatial valence region. Furthermore, the

Figure 1. Radial densities of the 5f, 6d, and 7s AE and pseudovalence orbitals of Pa in the [Rn] 5f26d17s2 ground state configuration obtained from numerical MCDHF calculations.35 2521



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Table 2. Experimentally Observed Lowest Electronic Configurations and LSJ Labels for the Actinides Ann+ (An = Ac−U and n = 0−4)39,40 An+

An Ac Th Paa U a

1

2

6d 7s 6d27s2 5f26d17s2 5f36d17s2

2

D3/2 F2 4 K11/2 5 L6 3

2

7s 6d27s1 5f27s2 5f37s2

An2+ 1

1

S0 F3/2 3 H4 4 I9/2

7s 5f16d1 5f26d1 5f4

4

An3+ 2

6

S1/2 H4 4 I11/2 5 I4

6p 5f1 5f2 5f3

3

An4+ 1

S0 F5/2 3 H4 4 I9/2 2

5

2

6p 6p6 5f1 5f2

P3/2 S0 2 F5/2 3 H4

1

The present calculations find a Pa2+ 5f26d1 4K11/2 ground state.

Table 3. Ionization Potentials IP1−IP4 of Ac, Th, Pa, and U (eV) from Finite-Difference MCSCF Calculations Using PPs in Comparison to AE MCDHF/DC+B Data35a IP1 PP

IP2 AE

PP

AE

An

MCDHF/DC+B

WB+SOper

MCDHF/DC+B

MCDHF/DC+B

WB+SOper

MCDHF/DC+B

Ac Th Pa U mae mre

4.3641 4.8540 5.4218 5.5108 0.0135 0.26

4.4057 4.8754 5.4552 5.5625 0.0235 0.48 IP3

4.3729 4.8581 5.4339 5.5399

10.5506 11.8791 10.2402 11.8451 0.0083 0.08

10.5654 11.8321 10.2595 11.6142 0.0779 0.67 IP4

10.5535 11.8651 10.2241 11.8455

PP

AE

PP

AE

An

MCDHF/DC+B

WB+SOper

MCDHF/DC+B

MCDHF/DC+B

WB+SOper

Ac Th Pa U mae mre

16.6232 17.5484 18.0637 16.9372 0.0127 0.07

16.6393 17.6402 18.1435 17.1710 0.0967 0.56

16.6150 17.5662 18.0829 16.9430

43.8635 26.8126 29.2500 31.1853 0.0084 0.03

27.0372 29.5437 31.5465 0.3037 1.04

MCDHF/DC+B 43.8652 26.8058 29.2400 31.1704

a

The MCDHF/DC+B PPs were applied in two-component MCHF calculations, whereas the WB PPs of Küchle et al.17 were used in onecomponent MCHF calculations, followed by a perturbative treatment of the MCDHF/DC SO valence operators of Cao and Dolg.18 Mean absolute errors (mae, eV) and mean relative errors (mre, %) are given for each IP with respect to the AE reference values.

as corresponding results obtained using finite basis sets will be discussed first. Besides the IPs also SO contributions to the IPs are given. Comparison is made to corresponding results obtained with the WB PPs and their perturbative or valence variational SO operators,17,18 AE finite-difference MCDHF/DC +B results,35 and AE finite basis set results using the secondorder Douglas−Kroll−Hess Hamiltonian and the Breit−Pauli SO operator in first-order perturbation theory (DKH2+BP).8−11,37 Second, results of correlated calculations will be discussed. Scalar-relativistic state-averaged CASSCF and subsequent multireference averaged coupled-pair functional (MRACPF) calculations as well as for some cases coupled cluster calculations with single and double excitations and perturbative triple excitations calculations [CCSD(T)] were performed. Corresponding SO-corrected results are compared to experimental and semiempirical data,39,41−47 if available, and corresponding basis set extrapolated AE DKH2+BP CASSCF/ MRACPF results.37 In addition, results of intermediate Hamiltonian Fock-space coupled cluster (IHFSCC) PP calculations48,49 for selected cases are compared to corresponding fully relativistic AE data reported in the literature.50,51 The computational details of these calculations will be given in each section separately. Finite-Difference MCHF Calculations. To test the performance of the PPs without bias introduced by finite basis sets, we first performed one- and two-component PP MCHF as well as four-component AE MCDHF/DC+B

The resulting (14s13p10d8f6g)/[6s6p5d4f3g] basis sets correspond to polarized valence quadruple-ζ quality. The loss in total valence energies of the ground states for the primitive basis sets with respect to the HF limits amounts at most to 0.025 eV and that of the contracted basis sets with respect to the primitive ones is below 0.0015 and 0.08 eV at the CASSCF and MRCI level, respectively. The parameters of PPs and basis sets are compiled in the Supporting Information of this publication and they are also available from the authors.38

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RESULTS AND DISCUSSION Atomic Test Calculations. To test the new MCDHF/DC +B PPs and corresponding valence basis sets for atoms, the first four ionization potentials of Ac, Th, Pa, and U have been calculated. The experimentally found energetically lowest electronic configurations and LSJ states of the neutral atoms and their cations are listed in Table 2.39,40 We note that all present calculations yield at the highest level, i.e., correlated results including SO corrections, a Pa2+ 5f26d1 4K11/2 ground state, in contrast to the experimental assignment to 4I11/2. The previous assignment to 2H11/2 was based on the fact that, when the SO coupling is omitted, the 2H is lower in energy than the 4 K state.18,22 However, when the SO interaction is accounted for in perturbation theory, 4K11/2 becomes lower than 2H11/2 and thus dominates when both LS states are considered in intermediate coupling. In the following the results of uncorrelated calculations at the finite-difference level as well 2522



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Table 4. SO Contributions (eV) for the First to Fourth Ionization Potentials of Ac, Th, Pa, and U from Finite-Difference PP Calculations35a IP1 MCDHF/DC+B An Ac Th Pa U

0.1101 −0.000026 0.0803 0.1465

An 0.0321 −0.0064 0.1099 −0.1243

MCDHF/DC+B

+SOvar

+SOper

0.1137 −0.0117 0.0948 0.1407 IP3

0.1094 −0.0193 0.0764 0.1303

MCDHF/DC+B Ac Th Pa U

IP2 WB

WB

0.0280 0.1157 −0.0555 0.1893

WB +SOvar

+SOper

0.0000 0.0817 −0.1045 0.0525 IP4

0.0000 0.0708 −0.0921 0.0672

MCDHF/DC+B

+SOvar

+SOper

0.0000 −0.0230 0.0906 −0.0650

0.0000 −0.0423 0.0768 −0.0818

−2.2074 0.2516 0.1536 0.0179

WB +SOvar

+SOper

0.2723 0.2133 0.0802

0.2752 0.2090 0.0692

a

MCDHF/DC+B PPs were applied in variational two-component MCHF calculations, whereas scalar-relativistic HF calculations were performed with the WB PPs of Küchle et al.17 followed by a variational (var) and a perturbative (per) treatment of the MCDHF/DC SO operators of Cao and Dolg.18

calculations using the finite-difference atomic structure program GRASP.35 Table 3 lists results for the first to fourth ionization potentials of Ac, Th, Pa, and U. The SO contributions listed in Table 4 were derived as the differences between multiconfiguration self-consistent field (MCSCF) results for the IPs obtained with and without ΔVPP SO. All LS states and J levels arising from the configurations listed in Table 2 were simultaneously optimized at the finite-difference MCSCF level. The MCDHF/DC+B PPs were used in fully variational one- and two-component MCSCF calculations, whereas the two sets of MCDHF/DC SO operators18 associated to the WB PPs17 were treated in first-order perturbation theory based on scalar-relativistic MCSCF solutions obtained with the WB PPs (+SOper) and variational two-component MCSCF calculations keeping the 5s, 5p, 5d, 6s, and 6p shells frozen in their scalarrelativistic form (+SOvar), respectively. Because the SO corrections for the variational treatment of SO coupling are very similar to those obtained by a perturbative one (cf. Table 4), the PP WB+SOvar results resemble those of the simpler PP WB+SOper approach closely and are thus not included in Table 3 but are given in the Supporting Information. The present MCDHF/DC+B PPs account implicitly for the Breit interaction, which is neglected in the WB PPs and their MCDHF/DC SO operators. At the AE MCDHF level, the Breit interaction contributes noticeably to the IPs; i.e., the mean absolute (eV), maximum (eV), and relative (%) deviations between MCDHF/DC and MCDHF/DC+B results for the four IPs are 0.009, 0.020, 0.07 for Ac, 0.046, 0.089, 0.30 for Th, 0.027, 0.091, 0.12 for Pa, and 0.070, 0.100, 0.42 for U, respectively (cf. the Supporting Information). We note that especially for the 5f transitions, occurring for IP2 and IP4 of Th, IP4 of Pa, and IP2 to IP4 of U, the Breit interaction should not be neglected. For example, differences from AE MCDHF calculations with and without Breit interaction show that this effect amounts at most to 0.020 eV [IP4(Ac)] for s, p, and d transitions but ranges from 0.081 [IP3(U)] up to 0.100 eV [IP2(U)] for f transitions (cf. the Supporting Information). It should be noted that the two sets of SO operators associated to the WB PPs are only designed for a treatment of SO effects in the 5f, 6d, and 7p valence shells, whereas the present SO operators can also be applied for the inner 5p, 5d,

and 6p shells. Therefore, the SO operators adjusted for the WB PPs can only account for SO effects in states with L > 0 and S > 0, whereas the usage of the new MCDHF/DC+B PPs also leads to scalar-relativistic SO contributions, which can be easily detected by differences of the energies of states with L = 0 and/ or S = 0 in calculations with and without ΔVPP SO. For example, for Ac+ 7s2 1S0 the energy of a two-component MCDHF/DC +B PP calculation is by 0.5127 hartree lower than for a corresponding one-component calculation, whereas in the case of Ac2+ 7s1 2S1/2 the energy lowering amounts to 0.5117 hartree. Thus, a net increase of IP2 of Ac of 0.0280 eV results for the MCDHF/DC+B PP, whereas no SO contribution is obtained for the calculation using the WB PP and one of the two valence SO operators. Finally, it can be expected that the Breit interaction also has an effect on the SO contributions. Due to these reasons, a perfect agreement between the SO contributions listed in Table 4 cannot be expected; however, it is observed that the corrections lead in the same direction and often are also of similar magnitude. The PP MCDHF/DC+B results agree, due to the adjustment of the PPs, excellently with the AE reference data; i.e., mean absolute and relative errors for each IP are below 0.014 eV and 0.3%, respectively. The PP WB+SOper approach models the DC Hamiltonian only but still the results show a reasonable agreement with the AE reference data. The mean absolute and relative errors for each IP are below 0.1 eV and 0.7% for IP1 to IP3. As also already mentioned above, the WB scalar-relativistic AE scheme does not describe energy differences between states with different f occupations very accurately. Therefore, the WB +SOper results for ionization potentials involving changes of the f occupation tend to deviate more strongly from the AE MCDHF/DC+B values. Another, more obvious, reason is that the WB PPs were only adjusted to neutral atoms and their monocations,17 whereas the MCDHF/DC+B PPs comprised at least up to 4-fold charged cations. Taking all these deficiencies into account it is not surprising that for the WB PPs in IP4 the mean absolute and relative errors of 0.3 eV and 1.0%, respectively, are more than an order of magnitude larger than for the MCDHF/DC+B PPs. With exception of the ionization from the Ac 6p shell the SO corrections amount to less than 3% of the total ionization 2523



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Table 5. Ionization Potentials IP1−IP4 of Ac, Th, Pa, and U (eV) Calculated with MCDHF/DC+B PPs by Quasi-Degenerate Perturbation Theory at the State-Averaged MCSCF Level with GRASP35 Given as (1) and Corresponding Results for the Lowest LS State with MOLPRO37 Given as (2), Respectivelya IP1

IP2

PP

AE

PP

AE

MCDHF/DC+B

DKH2+BP

MCDHF/DC+B

DKH2+BP

An

(1)

(2)

Ac Th Pa U mae mre

4.4601 4.8253 5.5067 5.5844 0.0593 1.20

4.4380 4.8459 5.5041 5.6306 0.0595 1.17 IP3

4.5089 4.8169 5.5180 5.6667 0.0970 1.95

(1)

(2)

10.5226 11.8699 10.0827 11.7528 0.0674 0.62

10.5235 11.8777 10.1475 11.6887 0.0690 0.62 IP4

10.5244 11.8963 10.1032 11.7130 0.0784 0.71

PP

AE

PP

AE

MCDHF/DC+B

DKH2+BP

MCDHF/DC+B

DKH2+BPb

An

(1)

(2)

Ac Th Pa U mae mre

16.5911 17.6143 18.1460 16.9426 0.0339 0.19

16.5905 17.5046 18.0515 16.9784 0.0382 0.22

16.5785 17.5840 18.1240 16.9916 0.0360 0.21

(1)

(2)

43.7677 26.8926 29.3281 31.2304 0.0831 0.26

43.7663 26.8967 29.2427 31.2151 0.0593 0.18

41.9084 26.9102 29.2345 31.2042 0.5251 1.24

a

All-electron DKH2+BP results obtained analogous to (2) are also listed. Mean absolute errors (mae, eV) and mean relative errors (mre, %) are given for each IP with respect to the AE MCDHF/DC+B reference values of Table 3. bmae and mre for IP4 AE DKH2+BP excluding Ac 0.0479 eV and 0.17%, respectively.

Table 6. Comparison between SO Contributions (eV) for the First to Fourth Ionization Potentials of Ac, Th, Pa, and U Calculated with MCDHF/DC+B PPs by Quasi-Degenerate Perturbation Theory at the State-Averaged MCSCF Level with GRASP35 Given as (1) and Corresponding Results for the Lowest LS State with MOLPRO37 Given as (2), Respectivelya IP1

IP2

PP

AE

PP

AE

MCDHF/DC+B

DKH2+BP

MCDHF/DC+B

DKH2+BP

An

(1)

(2)

Ac Th Pa U

0.2062 −0.0287 0.1652 0.2201

0.2056 −0.0381 0.1641 0.1897 IP3 PP

0.2270 −0.0320 0.1893 0.2170

(1)

(2)

0.0000 0.1065 −0.2130 0.0970

0.0000 0.1714 −0.0891 0.0900 IP4

AE

MCDHF/DC+B

PP

DKH2+BP

An

(1)

(2)

Ac Th Pa U

0.0000 0.0595 0.1922 −0.1188

0.0000 −0.0714 0.0716 −0.1063

0.0000 −0.0413 0.2469 −0.0961

0.0000 0.1622 −0.2619 0.0815 AE

MCDHF/DC+B

DKH2+BP

(1)

(2)

−2.3033 0.3317 0.2317 0.0630

−2.3051 0.3352 0.1408 0.0272

−4.1734 0.3420 0.1565 0.0444

a

All-electron DKH2+BP results obtained analogous to (2) are also listed. The MOLPRO results for IP2 and IP3 refer to the Th2+ 5f16d1 1G LS state ( H4 J level), the Pa 5f26d17s2 2H LS state (4K11/2 J level), and the Pa2+ 5f26d1 2H LS state (4K11/2 J level). 3

where the AE MCDHF/DC(+B) SO contributions of 0.293 (0.278) eV are only slightly larger than the PP results of 0.252, 0.272, and 0.275 eV. Also here a perfect agreement cannot be expected, because averaging over a j dependent potential and averaging over the energies of J levels may lead to different SO contributions. A drastic example is IP4 of Ac (ionization from the 6p shell of [Rn] semicore), where a comparison of results for two- and one-component calculations with PP MCDHF/ DC+B yields a contribution of −2.207 eV in contrast to the

potentials (cf. Table 3). In a few cases averaging over the J levels of LS states is possible and a comparison to AE MCDHF/DC(+B) results can be made. For IP1 of Ac (ionization from the d shell of [Rn] 7s26d1) the AE MCDHF/DC(+B) SO contributions of 0.107 (0.112) eV agree quite well with the corresponding PP results of 0.110, 0.114, and 0.109 eV for PP MCDHF/DC+B, PP WB+SOvar, and PP WB+SOper, respectively. A reasonable agreement is also observed for IP4 of Th (ionization from the f shell of [Rn] 5f1), 2524



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Figure 2. Deviations (eV) of AE DKH2 ionization potentials (a) IP1, (b) IP2, (c) IP3, and (d) IP4 of Ac, Th, Pa, and U from state-averaged CASSCF with subsequent MRACPF calculations37 for near-complete basis sets including basis functions up to a given angular momentum from corresponding results of a basis set extrapolation.

result of −2.481 eV obtained by averaging AE MCDHF/DC+B energies. However, averaging over the energies of J levels obtained in a two-component PP MCDHF/DC+B calculation yields a value of −2.481 eV in excellent agreement (deviation 2 × 10−4 eV) with the AE result. Algebraic MCSCF Calculations. Correlated atomic and molecular calculations are usually carried out with finite basis sets. Thus, in addition to the performance of the MCDHF/DC +B PPs tested in the finite-difference atomic calculations reported above, the quality of the optimized valence basis sets was tested at the uncorrelated and correlated level. Due to computational limitations set by the size and/or the complexity of the electronic structure of a specific system correlated calculations in an intermediate coupling scheme based on an interaction of all states arising from one or several configurations are often not feasible. We therefore investigated the performance of the derived PPs, SO operators, and (14s13p10d8f)/[6s6p5d4f] valence basis sets in scalarrelativistic PP calculations for the lowest LS states at the state-averaged MCSCF level followed by quasi-degenerate perturbation theory for SO effects using MOLPRO.37 Here, the Hamiltonian matrix including SO contributions is built on the basis of the LS states and then diagonalized. State-averaging always comprised all components of a given LS state in the D2h point group and was necessary to avoid symmetry breaking; i.e., the MCSCF code in MOLPRO was used here to obtain all space- and spin-restricted components of the LS states, rather than including, e.g., static correlation effects. We compare these results obtained with the MCDHF/DC+B PPs to AE DKH2+BP results obtained in the same way using the uncontracted (27s24p18d14f) and (26s23p17d13f) basis sets of Roos et al.52 for Ac, Th, Pa, and U, respectively. To separate the effects of orbital relaxation from those of restricting the space of interacting states as well as basis set effects, we also report results obtained at the finite-difference level with the atomic code GRASP,35 where the scalar-relativistic part of PP MCDHF/DC+B was used in a state-averaged MCSCF over all

LS states of a given configuration, followed by a SO configuration interaction in the basis of these states using the full PP MCDHF/DC+B. The corresponding results for the IPs are listed in Table 5, and those for the SO contributions are given in Table 6. The mean absolute and mean relative errors listed in Table 5 for each IP are taken with respect to the AE MCDHF/DC+B reference values of Table 3. A comparison of these tables reveals that a perturbative treatment of SO contributions using the MCDHF/DC+B PP at the finite-difference level is inferior to the corresponding variational treatment; however, the errors amount at most to 2% and are still tolerable. Very similar results are obtained when finite basis sets are used and only the LS state leading to the lowest J level is included for the treatment of SO coupling. Here, it has to be noted that not always the lowest LS state gives rise to the lowest J level; e.g., for Th2+ [Rn] 5f16d1, the 1G state is found below the 3H state. However, because the perturbative SO contribution for the former state equals zero, the LSJ ground level is 3H4. Similarly, for Pa [Rn] 5f26d17s2 the LS ground state is 2H, whereas the LSJ ground level is 4K11/2. The same LS and LSJ ground levels are obtained for Pa2+ [Rn] 5f26d1. Finally, AE DKH2+BP results are slightly worse than the PP MCDHF/DC+B values. A significant overestimation of the fine structure splitting of Ac4+ 6p5 is observed at the AE DKH2+BP level (12.5201 eV), which is 68% larger than the AE MCDHF/DC+B reference value (7.4416 eV) and thus the main reason for the disagreement of the AE and PP values for IP4 of Ac. The PP MCDHF/DC+B values obtained perturbatively at the finitedifference (6.9098 eV) and finite basis set (6.9153 eV) levels are much more reasonable, although not as accurate as the variationally obtained PP MCDHF/DC+B result (7.4421 eV). We note that even the valence 7p SO operator of the WB PP yields with 5.5387 eV a less wrong result for the very large 6p splitting than the AE DKH2+BP value. Table 6 compares the SO contributions from GRASP and MOLPRO for all ionization potentials considered here. As one 2525



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Table 7. Ionization Potentials IP1−IP4 of Ac, Th, Pa, and U (eV) from Spin−Orbit Corrected State-Averaged CASSCF with Subsequent MRACPF Calculations37 Using WB17,18 and MCDHF/DC+B PPs in Comparison to AE DKH2+BP Results as Well as Experimental and Semiempirical Data39,41−47,a IP1 PP An

WB +SOper

IP2 AE

PP

MCDHF/DC+B

DKH2+BP

5.38 5.25 6.30 6.26 6.13 5.97 6.26 6.08 0.08 0.09 1.3 1.6

5.44

5.38

6.27

6.31

Ac 5.19 Th 6.27 Pa 5.86 U 5.99 mae 0.11 mre 2.0

exp

WB+SOper

DKH2+BP

exp

11.59 11.57 12.41 12.52 11.64 11.71 11.89 12.10 0.24 0.35 2.0 3.0

11.59

11.75 ±0.03

11.58 12.50 6.10 6.27

5.89 ±0.12 6.19

AE

MCDHF/DC+B

11.88 11.99

0.10 0.29 1.7 2.4 IP3

PP An

WB+SOper

Ac 17.29 Th 17.96 Pa 18.41 U 18.69 mae 0.74 mre 3.8

12.46 11.64 11.94

11.59 ±0.37

0.26 2.2 IP4

AE

MCDHF/DC+B

DKH2+BP

17.29 17.28 18.10 17.97 18.64 18.48 18.86 18.53 0.58 0.82 3.0 4.2

17.28 18.18

PP exp

18.33 ±0.05

WB+SOper

28.15

18.70 30.49 18.88

19.80 ±0.25

32.57

0.54 2.32 2.7 6.5

AE

MCDHF/DC+B

DKH2+BP

44.52 44.40 28.27 28.01 30.58 30.29 32.61 32.29 2.23 2.53 6.2 7.1

42.64 28.28

exp

28.65 ±0.02

30.57 32.61

36.70 ±0.99

2.23 6.2

a Results from basis set extrapolations and for standard basis sets are given in the first and second lines, respectively. Mean absolute errors (mae, eV) and mean relative errors (mre, %) are given for each IP with respect to experimental and semiempirically derived reference values. The error bars of the experimental values are smaller than 0.01 eV unless otherwise noted. Semiempirical estimates are Ac IP2,42 Th IP3,44 and Pa IP1.46

also the question of orbital relaxation under the SO operator. This is obvious, e.g., by comparing the PP MCDHF/DC+B SO contributions from Tables 4 and 6, which differ only in their variational respectively perturbative treatment of the SO operators. CASSCF/MRACPF Calculations. The ionization potentials were also determined with state-averaged CASSCF and subsequent MRACPF calculations with MOLPRO. 37 (14s13p10d8f6g)/[6s6p5d4f3g] standard valence basis sets were applied for the WB17,18 and the MCDHF/DC+B PPs (this work). To perform for the new PPs also basis set extrapolations according to E(l) = E(∞) + c/l3, with l denoting the highest angular quantum number included in the basis set, sets of (8g), (8g8h), and (8g8h8i) were added to the uncontracted (14s13p10d8f) sets, adopting the exponents from the (8f) set. The parameter c as well as E(∞) were determined by a least-squares fit. For each angular quantum number one additional diffuse function was added, yielding (15s14p11d9f9g9h9i) basis sets. Corresponding AE DKH2 calculations used the uncontracted (27s24p18d14f) and (26s23p17d13f) basis sets of Roos et al.52 for Ac, Th, Pa, and U, respectively, to which (ng), (ngnh), and (ngnhni) sets were added, adopting the exponents from the (nf) set (n = 14 for Ac, Th, Pa; n = 13 for U). One additional diffuse function

can see, the deviations between the two PP methods amount usually to less than 0.04 eV, except for IP2 of Th and Pa as well as IP3 and IP4 of Pa, where deviations of up to 0.12 eV occur. Whereas in GRASP all LS states of a given configuration are optimized at the MCSCF level and allowed to interact in the subsequent diagonalization of the Hamiltonian matrix including SO effects, only the lowest LS state in the highest spin multiplicity was considered in MOLPRO. If in the latter case all LS states with term energies up to 1 eV are allowed to interact for Th, the SO contributions for PP MCDHF/DC+B and AE DKH2+BP, respectively, become 0.0973 and 0.0930 eV for IP2 as well as 0.0677 and 0.0995 eV for IP3. These results are in reasonable agreement with the GRASP results for PP MCDHF/DC+B of 0.1065 and 0.0595 eV. The corresponding values for Pa are −0.2192 and −0.2551 eV for IP2 and 0.1984 and 0.2375 eV for IP3, to be compared to −0.2130 and +0.1922 eV. Finally, at the PP MCDHF/DC+B and AE DKH2+BP level the SO contributions to IP4 of Pa become 0.2458 and 0.2711 eV, respectively, to be compared to 0.2317 eV. Although these values clearly agree better with the PP MCDHF/DC+B finitedifference results, a slightly worse overall agreement of the IPs with AE MCDHF/DC+B results than reported in Table 5 is observed. We note here that not only the origin of the orbitals and the number of interacting LS states seem to play a role but 2526



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AE DKH2+BP data, one observes absolute (relative) deviations of 0.03 eV (0.2%), 0.04 eV (0.2%), and 0.01 eV (0.04%) for IP2, IP3, and IP4, respectively. We note that the semiempirical estimate of IP1 of Pa (5.89 ± 0.12 eV) published by Sugar46 in 1973 may be somewhat too low analogous to the corresponding estimates for Ac (5.17 ± 0.12 eV), Th (6.08 ± 0.12 eV), and U (6.05 ± 0.07 eV). These were later replaced by about 0.14−0.23 eV higher measured values, i.e., 5.38, 6.31, and 6.19 eV for Ac, Th, and U, respectively.41,43 A correction of similar magnitude for Pa would lead to an IP1 of about 6.03−6.12 eV and mae (mre) of 0.04−0.06 eV (0.8−1.1%) for the basis set extrapolated AE DKH2+BP results, and of 0.02−0.05 eV (0.4−0.7%) for the corresponding MCDHF/DC+B PP results. In the case of U the very large deviations of all calculated values from the experimental ones of about 1 and 4 eV for IP3 and IP4, respectively, most likely cannot be completely attributed to deficiencies of the CASSCF/MRACPF correlation treatment but rather suggest that the experimental values determined by electron impact techniques53 might be somewhat too high. This view is also supported by fully relativistic AE Dirac−Kohn− Sham (DKS) calculations yielding values of 18.77−18.94 and 32.68−32.87 eV for IP3 and IP4, respectively,54 which agree very well with the corresponding basis set extrapolated AE and PP ACPF results of 18.86−18.88 and 32.61 eV obtained here. CCSD(T) and IHFSCC Calculations. A previously published study of the higher ionization potentials of the lanthanides revealed that CCSD(T) yielded better agreement with experimental data than corresponding ACPF calculations.55 We therefore performed such calculations using the WB and MCDHF/DC+B PPs with MOLPRO37 for those cases where the involved electronic states can be reasonably well described by a single-reference approach, i.e., for all IPs of Ac and IP4 of Th. In the case of IP1 and IP4 of Ac as well as of IP4 of Th these CCSD(T) results are based on symmetry broken HF solutions, whereas IP2 and IP3 of Ac can be calculated without symmetry breaking (cf. Table 2). Two kinds of basis sets were applied, i.e., the standard (14s13p10d8f6g)/ [6s6p5d4f3g] and the extended uncontracted (16s15p12d10f8g7h7i) basis sets obtained from the former primitive sets by adding diffuse and higher angular momentum functions. Because it is not possible to evaluate SO corrections for the symmetry broken open-shell cases in a well-defined way, we added the corrections listed in Table 4. For all IPs of Ac and IP4 of Th, which can be calculated at the CCSD(T) level of theory, calculations using the new MCDHF/ DC+B PPs in a modified version of the four-component DHF and relativistic FSCC program of Kaldor, Ishikawa, Eliav, and co-workers48,50 were feasible as well. Both the standard (14s13p10d8f6g)/[6s6p5d4f3g] and the extended (16s15p12d10f8g7h7i) basis sets were applied. Two sets of calculations were carried out to calculate IP1−IP3 of Ac analogous to the AE DCB FSCC calculations of Eliav et al.50 The first set of calculations started from the closed-shell configuration Ac+ [Rn] 7s2, where electrons were added in the 6d and 7p orbitals (primary model space) to give the states of the neutral atom, according to the scheme

was added for each angular quantum number, yielding (28s25p19d15f15g15h15i) and (27s24p18d14f14g14h14i) sets for Ac, Th, Pa, and U, respectively. Parts a−d of Figure 2 show for the AE DKH2 ACPF level for each basis set size the deviations of the ionization potentials IP1, IP2, IP3, and IP4 from the extrapolated values. The accuracy of the extrapolation for the IPs is better than ±0.02 eV, as estimated from the difference of extrapolations with l = 4, 5, 6 and l = 5, 6. In most cases the difficulty to obtain converged results increases with the complexity of the electronic states involved (e.g., the presence of an open d shell besides the open f shell),14 i.e., usually from the lighter to the heavier actinides and from the higher to the lower ionization potentials. It can be expected that, for the PP standard basis sets, basis set incompleteness errors of similar magnitude remain as for the AE reference calculations including up to g functions; i.e., errors of about 0.15−0.3 eV have to be expected for U. The active space in the CASSCF wave functions comprised all open-shell orbitals for the case under consideration (possibly 6p, 5f, 6d, 7s). In addition, to account for near degeneracy effects, the 6d, 7s, and 7p shells were always included in the active space for a 7s2 subshell. SO contributions were evaluated at the state-averaged CASSCF level and added to the best correlated results obtained in LS coupling. We note that the magnitude of some SO contributions depends quite strongly on the active space of the CASSCF wave function it is evaluated with; e.g., at the AE DKH2+BP level, SO contributions of 0.20, 0.17, and 0.23 eV have been calculated for IP1 of Ac for when excitations from 7s2 are allowed to 6d and 7p, only to 6d, and to no orbitals, respectively. In the MRACPF calculations, excitations from all occupied valence orbitals (5f, 6d, 7s, 7p) as well as the 6s and 6p semicore orbitals were allowed; i.e., the 5s, 5p, and 5d shells were kept frozen. Allowing excitations from 7s2 to 6d and in addition to 7p in the CASSCF reference wave function increases the first ionization potentials at the ACPF level by about 0.15−0.20 and 0.03−0.06 eV, respectively (cf. the Supporting Information). Table 7 lists the IPs calculated at the MRACPF level in comparison to the available experimental data and semiempirical estimates.39,41−47 In contrast to the MCSCF results at this level of theory, the WB PPs seem to perform slightly better than the MCDHF/DC+B PPs when the standard valence basis sets are used. The good performance of the WB PPs is certainly partly due to an error cancellation effect, because the electron correlation is not described accurately enough using the MRACPF method, leading to an underestimation of the IPs, which is counteracted by the tendency of the WB PPs to overestimate most of the IPs (cf. Table 3). It has to be noted, however, that except for IP1, where in fact the MCDHF/DC+B PPs perform somewhat better (mae (mre) 0.09 eV (1.6%) vs 0.11 eV (2.0%)), there is only little and in some cases probably also not very accurate experimental reference data available for calibration. In addition, both sets of PP results obtained with standard basis sets are much closer to the basis set extrapolated AE DKH2+BP values than to the experimental and semiempirical values for the higher ionization potentials; e.g., the mean absolute (relative) deviations stay below 0.1 eV (0.7%) for IP2, 0.2 eV (1.1%) for IP3, and 0.7 eV (1.7%) for IP4. In fact, excluding Ac where the application of the BP Hamiltonian leads to much too large SO contributions at the AE DKH2 level as discussed above, yields mean absolute (relative) deviations of at most 0.3 eV (1.0%) for IP4. When comparing the basis set extrapolated MCDHF/DC+B PP results to the corresponding

Ac+(0,0) → Ac(0,1)

(7)

yielding IP1. The intermediate model space covered the 5−8f, 7−9d, 8−10s, 8−10p, and 5−6g orbitals. 2527



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Table 8. Ionization Potentials of Ac and Th (eV) Calculated at Various Levels of Theory, i.e., Using Different PPs, Methods, and Basis Sets, in Comparison to AE DCB FSCC Results50,51 and Experimental Data41,42,45 Ac methoda

IP1

IP2

IP3

WB PP MRACPF + SOperb std WB PP CCSD(T) + SOperb std WB PP CCSD(T) + SOperb ext WB PP IHFSCC + SOperb ext DC+B PP MRACPF + SOb std DC+B PP CCSD(T) + SOb std DC+B PP CCSD(T) + SOb ext DC+B PP IHFSCC + SOb ext DC+B PP IHFSCC std DC+B PP IHFSCC ext AE DCB IHFSCCc DC+B PP FSCC ext AE DCB FSCCd expe

5.19 5.19 5.27 5.16 5.25 5.18 5.29 5.18 5.15 5.25 5.27 5.30 5.32 5.38

11.58 11.65 11.67 11.87 11.57 11.66 11.69 11.89 11.84 11.89 11.91 11.87 11.90 11.75 ± 0.03

17.29 17.33 17.35 17.47 17.28 17.35 17.37 17.49 17.46 17.49 17.49 17.51 17.51

IP4

44.40 44.51 44.60 44.91 44.83 44.98 44.99

Th IP4 28.15 28.25 28.47 28.85 27.92 28.02 28.23 28.62 28.36 28.66 28.65

28.65 ± 0.03

a

Standard (std) PP basis sets (14s13p10d8f6g)/[6s6p5d4f3g], extended (ext) PP basis sets (16s15p12d10f8g7h7i), AE DCB FSCC basis sets (35s26p21d16f10g6h5i) for Ac IP1−IP3 and (35s30p25d20f11g9h9i7k7l) for IP4 of Ac and Th. bThe spin-averaged scalar-relativistic PPs corresponding to eq 4 were applied and the SO contributions from Table 4 were added to the results. cReference 51 for IP4 of Ac and Th, IP1−IP3 of Ac this work. dReference 50 for Ac IP1−IP3. eReferences 41 and 42 for Ac IP1 and IP2; ref 45 for Th IP4.

The second set of calculations started from the Ac3+ [Rn] configuration, where two electrons were added to the reference states, one at a time, to get the sequence Ac 3 +(0,0) → Ac 2 +(0,1) → Ac+(0,2)

further by about 0.1 and 0.2 eV for IP1 of Ac and IP4 of Th, respectively, as expected from the trends displayed in Figure 2. For IP1 of Ac both PPs yield results that differ from the experimental value by about 0.1 eV, whereas the deviations for IP4 of Th still amount to 0.18 and 0.42 eV for the WB and MCDHF/DC+B PP, respectively. Replacing the scalar-relativistic CCSD(T) approach by a FSCC treatment, maintaining the additive SO corrections, yields values for IP1 of Ac that are about 0.1 eV smaller and in worse agreement with experimental data, whereas for IP4 of Th an increase of about 0.4 eV is observed, bringing the calculated values in much better agreement with experimental data. Taking the AE FSCC results of Kaldor and co-workers50,51 as a reference, the PP results show mean absolute (relative) deviations for IP1−IP3 of Ac and IP4 of Th of 0.11 eV (1.1%) and 0.05 eV (0.8%) for the WB and MCDHF/DC+B PPs, respectively. Note that IP4 of Ac had to be excluded, because the WB PP SO operator is not valid for the Ac 6p semicore shell. It is also noteworthy that the results for both PPs agree within 0.02 eV for IP1−IP3 of Ac, whereas a large deviation of 0.23 eV occurs for IP4 of Th. The agreement with AE FSCC reference data becomes even better when the twocomponent MCDHF/DC+B PP is applied as intended by the fitting procedure already at the HF level. The mean absolute (relative) deviation is reduced to 0.03 eV (0.4%). AE DCB FSCC results from literature50,51 and PP IHFSCC results of this work agree within 0.02 eV for all IPs, except for IP1 of Ac where a larger difference of 0.07 eV is observed. However, using the MCDHF/DC+B PP with the extended basis sets within the FSCC approach without intermediate Hamiltonian formalism leads to an IP1 of Ac of 5.30 eV in excellent agreement with the corresponding AE DCB FSCC value of 5.32 eV.50 Similarly, redoing the AE DCB calculation with the newer IHFSCC approach yields a value of 5.27 eV that also compares favorably to the MCDHF/DC+B PP result of 5.25 eV. Thus, PP and AE DCB results agree within ≈0.02 eV if the same FSCC formalism is used. The remaining disagreement of about 0.06−0.13 eV of both AE and PP results with respect to the experimental value of 5.38 eV remains unexplained at present.

(8)

yielding IP2 and IP3. The primary model space consisted of the 6d and 7s orbitals, whereas the intermediate model space comprised the 5−8f, 7−9d, 8−10s, 7−10p, and 5−6g orbitals. To obtain IP4 of Ac and Th, calculations starting from the closed-shell rare-gas [Rn] configuration of Ac3+ and Th4+ were performed following the work of Eliav et al.51 Ac 3 +(0,0) → Ac 4 +(1,0)

and

Th4 +(0,0) → Th3 +(0,1) (9)

The primary model space included the 6p, 5f, 6d, 7s, and 7p orbitals, whereas the intermediate model space comprised the 6s, 6−8f, 7−9d, 8−10s, 8−10p, and 5−6g orbitals. For all these calculations the explicitly treated electrons were correlated and excitations in all virtual orbitals were allowed. At least all orbitals with negative energy for the standard contracted basis sets in the Ac3+ and Th4+ reference systems were included in the model space. Due to convergence problems the intermediate model space in each angular symmetry had to be reduced by one orbital when the extended basis sets are used. The CCSD(T), FSCC, and IHFSCC results are summarized in Table 8. Using CCSD(T) instead of CASSCF/MRACPF with standard basis sets increases the IPs slightly by at most 0.1 eV for Th IP4. Unfortunately, only IP1 of Ac and IP4 of Th are measured values,41,45 which can be used for calibration purposes, whereas IP2 of Ac is a semiempirical estimate.42 Although the CCSD(T) approach improves the results, the calculated values still differ too much from the experimental ones to draw any conclusions regarding the accuracy of the WB and MCDHF/DC+B PPs. Here one has to note that the correlation contributions of IP4 of Th converge very slowly with the highest angular quantum number present in the basis set.14 Moreover, the results for IP1 of Ac are very similar for the two PPs. The application of the extended basis sets increases the IPs 2528



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The results presented in Table 8 show that the errors arising from the use of the PPs are typically smaller than those resulting from basis set incompleteness. Although the WB PP performs reasonably well, the experimental value of IP4 of Th bears such a small error bar that one can conclude that the Th MCDHF/DC+B PP is superior to the WB PP. In view of the data collected in Table 3 we assume that the MCDHF/DC+B PPs will be in general preferable over the WB PPs, especially for situations where the 5f occupancy changes. Furthermore, the comparison between FSCC calculations applying the relativistic MCDHF/DC+B PP directly in an one-step procedure and the two-step procedure of using the spin-averaged part of the PP with subsequent addition of the separately calculated SO effect shows that the two-step procedure is legitimate, because the deviation amounts at most to 0.07 eV. This is especially important for routine calculations on larger systems containing actinides, where one may be forced to keep the computational effort in reasonable limits.

CONCLUSION Relativistic small-core pseudopotentials for Ac, Th, and Pa energy-adjusted to multiconfiguration Dirac−Hartree−Fock/ Dirac−Coulomb−Breit Fermi nucleus reference data (MCDHF/DC+B PPs) have been presented together with valence basis sets of polarized valence quadruple-ζ quality. State-averaged finite-difference multiconfiguration Hartree− Fock (HF) test calculations using the proposed MCDHF/ DC+B PPs for the first to fourth ionization potentials show clearly better agreement with all-electron (AE) reference data than the application of the previously published Wood−Boring (WB) PPs, augmented by valence spin−orbit operators. A seemingly better performance of the WB PPs at the complete active space self-consistent field and subsequent multireference averaged coupled-pair functional (CASSCF/MRACPF) level most likely results from an error cancellation between deficiencies of the correlation treatment and the PPs, as could be shown in large scale Fock-space coupled cluster calculations for selected cases. Yet unpublished HF test calculations for small closed-shell molecules as monohydrides and monofluorides yield satisfactory agreement with AE DHF/ DC calculations; i.e., bond distances and force constants deviate at most by 0.012 Å (0.6%) and 8.4 N/m (3.0%), respectively. Thus, the new MCDHF/DC+B PPs are a valuable addition to the already existing PP tool box for actinides. ASSOCIATED CONTENT

S Supporting Information *

All reference configurations used for the PP adjustment, the obtained PP and basis set parameters, and additional finitedifference and CASSCF/ACPF basis set extrapolation results. Output files of fits. This material is available free of charge via the Internet at http://pubs.acs.org/.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*A. Weigand: e-mail, [email protected]; phone, +49 (0)221 4706887; fax, +49 (0)221 4706896. *M. Dolg: e-mail, [email protected]; phone, +49 (0)221 4706893; fax, +49 (0)221 4706896. Notes

The authors declare no competing financial interest. 2529



Article

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Relativistic Small-Core Pseudopotentials for Actinium, Thorium, and

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