Correlation consistent basis sets for lanthanides. The atoms La Lu

Transcription

1 Correlation consistent basis sets for lanthanides. The atoms La Lu Qing Lu and Kirk A. Peterson a) Department of Chemistry, Washington State University, Pullman, WA , USA Abstract Using the 3rd-order Douglas-Kroll-Hess (DKH3) Hamiltonian, all-electron correlation consistent basis sets of double-, triple-, and quadruple-zeta quality have been developed for the lanthanide elements La through Lu. Basis sets designed for the recovery of valence correlation (defined here as 4f5s5p5d6s), cc-pvnz-dk3, and outer-core correlation (valence+4s4p4d), cc-pwcvnz-dk3, are reported (n = D, T, Q). Systematic convergence of both Hartree-Fock and correlation energies towards their respective complete basis set (CBS) limits are observed. Benchmark calculations of the first three ionization potentials (IPs) of La through Lu are reported at the DKH3 coupled cluster singles and doubles with perturbative triples, CCSD(T), level of theory, including effects of correlation down through the 4s electrons. Spin-orbit coupling is treated at the 2-component HF level. After extrapolation to the CBS limit, the average errors with respect to experiment were just 0.52, 1.14, and 4.24 kcal/mol for the 1st, 2nd, and 3rd IPs, respectively, compared to the average experimental uncertainties of 0.03, 1.78, and 2.65 kcal/mol, respectively. The new basis sets are also used in CCSD(T) benchmark calculations of the equilibrium geometries, atomization energies, and heats of formation for Gd 2, GdF, and GdF 3. Except for the equilibrium geometry and harmonic frequency of GdF, which are accurately known from experiment, all other calculated quantities represent significant improvements compared to the existing experimental quantities. With estimated uncertainties of about ±3 kcal/mol, the 0 K atomization energies (298 K heats of formation) are calculated to be (all in kcal/mol): 33.2 (160.1) for Gd 2, ( 36.6) for GdF, and ( 295.2) for GdF 3. a) Electronic mail: kipeters@wsu.edu 1

2 I. INTRODUCTION It is now commonly recognized that one of the main sources of error in electronic structure calculations is the incompleteness of the 1-particle basis set. One method to address this is to just use very large basis sets, but a more efficient and reliable approach is to use a sequence of basis sets that systematically approach the complete basis set (CBS) limit. In particular, with an accurate estimate of the CBS limit, the intrinsic error of the underlying correlation method can be unambiguously assessed. By far the most common choice of basis sets for this purpose is the correlation consistent (cc) family of basis sets first introduced by Dunning. 1 These sets are designed to yield regular convergence to both the Hartree-Fock (HF) and correlated CBS limits as they increase in size, 2,3 leading to numerous efforts to exploit this behavior via extrapolation in the basis set index or the maximum angular momentum present in the basis set (see, e.g., Ref. 4). Combined with the use of highly correlated wavefunction methods, these sets are the cornerstone of nearly all composite ab initio thermochemistry and spectroscopy methodologies. 5 While cc basis sets are available throughout most of the periodic table, they have only recently been developed for f-block elements, namely the early actinides. 6,7 The current contribution extends the availability of cc basis sets to the entire lanthanide row. The extension to the remainder of the actinide elements will be the subject of a separate report. Similar to the actinide elements, the lanthanides start (La) and end (Lu) like d-block transition metals with the ground states occupying the 5d orbital with the 4f either empty (La) or completely filled (Lu). Partial occupation of the 4f begins with Ce with a 4f5d configuration and the 5d is then unoccupied until the middle of the row where single occupation of the 5d in Gd is favored over adding an additional f electron beyond a half-filled 4f shell. Analogous to the actinides, the 6s orbital is doubly occupied in the ground state configurations of all the lanthanides. As discussed in Ref. 8, the radial extent (root-mean-square and 95% density) of the lanthanide atomic spinors are in shell order, 4d < 4f << 5s < 5p < 5d << 6s. In addition and contrastingly to the actinides, the radii containing 95% of the 4f spinor density are always less than those of the 5s, implying that the 4f is not very accessible for bonding since it lies inside the semi-core 5s and 5p orbitals. The 4f orbital does, however, lie energetically above the 5s and 5p orbitals, and since for nearly all of the lanthanides the 4f is partially occupied, it should be treated as a valence orbital from the point of view of the basis set (as well as the 5s, 5p, 5d, and 6s orbitals). Last, even though the 6p orbital is not occupied in the ground states of any of the 2

3 lanthanides, it is often important in molecular systems and should be considered as part of a balanced valence basis set. This then leaves the 4s, 4p, and 4d orbitals as the outer-core, which should be treated in the development of core-valence basis sets. As noted previously by Gomes et al., 9 correlation of the 4d may be particularly important in the lanthanides due to the compactness of the 4f. The currently available (before this work) basis sets and their construction for the lanthanide elements was the subject of a recent review. 8 If the discussion is limited to all-electron basis sets designed for correlated calculations, the choices are fairly limited. Several consist of just one size the compact SARC basis sets of Pantazis and Neese (using both the 2nd-order Douglas-Kroll-Hess (DKH2) and zero-order regular approximation 13 (ZORA) Hamiltonians) designed for density functional theory calculations, 14 the comparably-sized DKH2 sets of Dolg, 15 and the 3rd-order Douglas-Kroll (DK3) sets of Hirao and co-workers 16 that are between doubleand triple-zeta quality. Three other families of sets could in principle be used to obtain estimates of the CBS limit since they systematically range in size from double- through at least quadruplezeta. The basis sets of Gomes et al. 9 were developed in the style of correlation consistent basis sets but explicitly for full 4-component Dirac-Coulomb relativistic calculations. They range in size from valence double-zeta (DZ) to quadruple-zeta (QZ) with additional tight functions available for correlation of the outer-core 4s4p4d electrons as well as functions for dipole polarization. These sets are most often used in their completely uncontracted forms. The Sapporo DK-nZP sets (denoted SAP-nZP in this work) of Sekiya et al. 17 were optimized in conjunction with the 3rd-order DKH (DKH3) scalar relativistic Hamiltonian and range from DZ to QZ. These sets were developed for correlation down through the 4s electrons of the lanthanide elements. Extensions of these sets obtained by adding an additional diffuse function in each angular momentum are also available and are denoted in this work as SAP-nZP+diffuse. The last basis sets to mention here are the atomic natural orbital (ANO) basis sets of Roos et al., 18 denoted ANO-RCC (relativistic semi-core correlation), which range in size from DZ to beyond QZ ("large") by adding additional ANO contractions from a common primitive set. These sets were optimized with the DKH2 Hamiltonian and are designed for correlation down through the 4f and 5s5p semi-core electrons of the lanthanide elements. In the present work new correlation consistent basis sets ranging in size from double- to quadruple-zeta quality have been developed using the DKH3 scalar relativistic 3

4 Hamiltonian 10,11,19,20 for both valence and outer-core electron correlation of the lanthanides La through Lu. The details of the basis set construction are given in Sec. II. To demonstrate the efficacy of the resulting sets, they are then used in both atomic and molecular benchmark calculations using the coupled cluster single and doubles with perturbative triples, CCSD(T), level of theory. These results are shown and discussed in Sec. III. Conclusions are presented in Sec. IV. II. BASIS SET DEVELOPMENT The development of correlation consistent basis sets for lanthanide elements has been carried out in a very similar manner as recently reported for the first few actinides. 6 All calculations pertaining to the basis set development utilized the MOLPRO suite of ab initio programs. 21,22 The orbitals from the HF calculations below were fully symmetry equivalenced, which generally required a state-averaged multiconfigurational self-consistent field (MCSCF) treatment. The DKH3 scalar relativistic Hamiltonian has been used throughout. This order of the DKH Hamiltonian was chosen in order to be consistent with earlier work on the 6p elements, 23 as well as considering the recent results 6 of CCSD(T) atomization energy calculations of the heavier ThO 2 and UF 6 molecules where an extension from DKH3 to DKH4 resulted in changes of only 0.1 kcal/mol. A. Hartree-Fock primitive sets Just as in the actinide correlation consistent basis sets, 6 the HF spdf primitive exponents were bootstrapped from the primitive sets of Dyall and co-workers, 9 which were optimized in 4- component Dirac-Hartree-Fock calculations using the Dirac-Coulomb Hamiltonian. Since they employed a finite nucleus model, these sets avoid the problem of heavy element basis sets optimized in a point charge nucleus framework, whereby particularly the s-type primitive exponents become large enough to cause numerical difficulties in the integral evaluation. In the present work the innermost spd functions from the valence double-zeta (VDZ), triple-zeta (VTZ), and quadruple-zeta (VQZ) sets of Gomes et al. 9 were retained unchanged, and the exponents representing approximately the outer two radial maxima of the valence orbitals, as well as all the f exponents, were re-optimized at the HF level using the DKH3 scalar relativistic Hamiltonian. Specifically, the total sizes of the HF primitives sets are (24s19p13d8f), (30s24p16d11f), and 4

5 (35s30p19d13f) for DZ, TZ, and QZ, respectively, from which the most diffuse (5s5p7d), (7s7p8d), (9s9p10d) exponents (and all the f functions) were reoptimized in this work. For Ce through Yb the s-type exponents were optimized for the lowest electronic state with a 4f n 6s 2 configuration (in this context n=1-14 for La through Yb). In the case of Lu, the lowest electronic state with configuration 4f 14 5d 6s 2 was used. For the p-type exponents, the average energy from the lowest states with configurations 4f n 6s 6p and 4f n-1 6s 2 6p were used for Ce through Yb (4f 14 5d6s6p and 4f 14 6s 2 6p for Lu). Likewise the d-type exponents for Ce through Yb were optimized for the energy average of states associated with 4f n-1 5d 6s 2 and 4f n-1 5d 2 6s. Those for Lu utilized states with configurations 4f 14 5d 6s 2 and 4f 14 5d 2 6s. Finally, for the f-type exponents, states with configurations 4f n 6s 2 were used for Ce through Yb and 4f 14 5d 6s 2 for Lu. As noted previously by Weigand et al., 24 optimization of f-type primitives for the Ac atom using the 5f 7s 2 configuration leads to exponents that are too diffuse, hence they utilized optimizations on Ac 2+. In the present work, the f-type exponents for La were optimized for the 3 F state of La +, which corresponds to a 4f 6s configuration. This yielded f exponents consistent with the trend of exponents from nearby elements of the row. In addition, the s-type primitives for La were optimized for the 5d 6s 2 configuration of neutral La, while the d-type exponent optimizations involved both the 5d 6s 2 and 5d 2 6s configurations (La). Finally, the p-type exponents of La were optimized for the state-average of both the 6s 2 6p of La and the 6s 6p of La +. In all cases in order to get a proper distribution of f-type primitives, i.e., a primitive set that well described the 4f orbital in a HF sense, as well as providing functions appropriate for correlation (see below), the optimizations of the f exponents were carried out in two steps. First the most diffuse f exponent from the 4f-dipole polarization set of Gomes et al. 9 was removed and the remaining exponents optimized at the HF level as described above. Then the diffuse f exponent of Gomes et al. was combined with the most diffuse f resulting from the first HF optimization and reoptimized with all tighter exponents now frozen. The resulting HF spdf primitives were generally contracted to [6s5p3d1f] using atomic orbital (AO) coefficients from state-averaged MCSCF calculations. Of these contracted functions, all but the last p and d contractions were obtained for Ce Yb by state averaging the 4f n 6s 2 and 4f n-1 5d 6s 2 states of the neutral atoms. An additional d contraction was added from the 5d AO of 5

6 the 4f n-1 5d 6s 2 state and a p-type contracted function was added from the 6p AO of a stateaverage of the 4f n 6s 6p and 4f n-1 6s 2 6p states. For Lu all of the contraction coefficients except the 6p orbital were taken from the 4f 14 5d 6s 2 configuration. The Lu 6p contraction was taken from a state average of the lowest states corresponding to the 4f 14 6s 2 6p and 4f 14 5d 6s 6p configurations. Finally, for La all contraction coefficients except those for the 6p and 5d AOs were taken from a state-average of states arising from the 5d 6s 2 of La and 4f 6s of La +. The 5d contraction coefficients were taken from the 5d 6s 2 (La) while the 6p utilized a state-average of the 6s 2 6p of La and the 6s 6p of La +. B. Correlating functions valence correlation (5s5p4f5d6s) Correlating functions for valence correlation, defined here to be the 4f, 5d, and 6s electrons together with the 5s and 5p semi-core electrons, were also optimized in the same manner as the actinide elements, assuming identical correlation consistent groupings. 6 The latter choice is borne out by inspection of the incremental correlation recovery as a function of angular momentum functions at the frozen-core multireference configuration interaction (MRCI) level of theory 25,26 for the Gd atom as shown in Figure 1. To construct this figure, the minimallycontracted QZ basis set described above was used with the 5 most diffuse s and p functions uncontracted. Angular momentum functions of d, f, g, h, and i symmetries were then optimized for 5s5p4f correlation on the lowest energy state corresponding to the 4f 8 6s 2 configuration ( 7 F g ) one at a time following an even-tempered prescription; first 1-4 d-type functions, then 1-4 f-type functions were added and optimized in the presence of the optimal 4d set, then 1-4 g-type functions were optimized in the 4d4f set, etc. In all cases the resulting optimized values of the exponents followed a very regular progression. As seen in Fig. 1, the incremental correlation energy lowerings within each angular momentum show nearly an exponential trend. It is observed, however, that the g-type functions contribute more correlation energy than the h or i angular momenta, making the correlation consistent groupings for DZ and TZ somewhat ambiguous. However, to remain consistent with previous work, only 1 g-type function is included for 5s5p4f correlation at the DZ level and 2g1h for TZ. The QZ level set should consist then of 3g2h1i functions for 5s5p4f correlation. 6

7 The higher angular momentum correlating functions were optimized at the MRCI level separately for 5s5p4f and 5d6s correlation due to the very different radial extents of these AOs. As described above for Gd (see also Fig. 1), functions added for 5s5p4f correlation (utilizing states associated mainly with the 4f n 6s 2 configurations, except for the 4f 6s configuration for La + and 4f 14 5d 1 6s 2 for Lu) corresponded to 1g for DZ, 2g1h for TZ, and 3g2h1i for QZ. Additional sets of 1g and 2g1h functions were added to the TZ and QZ sets, respectively, for 5d6s correlation (using states associated with the 5d 6s 2 configurations). In the TZ and QZ cases, the two sets of exponents for these different correlation spaces were optimized together and iterated until consistency was reached. In all cases the ratios between consecutive exponents were constrained to be greater than a factor of 1.6 in order to minimize linear dependency issues. Correlating functions for the occupied angular momenta were represented as ANO contractions obtained using averaged density matrices from MRCI calculations on the lowest states associated with the 4f n 6s 2 and 4f n-1 5d 6s 2 configurations. Those for La involved the 4f 6s configuration of La + and the 5d 6s 2 configuration of La, while Lu utilized only the 4f 14 5d 6s 2 configuration. The contractions of [1s1p1d1f] for DZ, [3s3p3d3f] for TZ, and [4s4p4d4f] for QZ were supplemented by uncontracting the most diffuse exponent of each angular momentum to provide additional flexibility. Note that the ANO sets for TZ and QZ are one function larger (in each angular momentum) than the previously reported sets for Th U. 6,7 The larger sets were found to be necessary to reduce basis set superposition error (BSSE) in initial benchmark calculations on Gd 2. The total contracted sizes of the resulting cc-pvnz-dk3 basis sets were [8s7p5d3f1g], [10s9p7d5f3g1h], and [11s10p8d6f5g3h1i] for n=d, T, and Q, respectively. C. Correlating functions - core correlation (4s4p4d) As noted previously by Gomes et al., 9 due to the compactness of the 4f orbital, correlation of the 4d could be very important for the lanthanide elements. In this work additional functions were optimized for 4s4p4d correlation using the weighted core-valence scheme, 27 which emphasizes intershell core-valence correlation over intrashell core-core correlation. These additional groups of correlating functions corresponded to 1s1p1d1f for DZ, 2s2p2d2f1g for TZ, and 2s2p2d3f2g1h for QZ (the 3 f-type functions were constrained to an even-tempered sequence). The MRCI optimizations utilized states associated with the 4f n 6s 2 configurations (with the same modifications for La and Lu as described above). The resulting basis sets are 7

8 denoted cc-pwcvnz-dk3 with n=d, T, Q. To avoid linear dependency issues in the DKH integral evaluation, many of the optimized correlating functions corresponding to occupied angular momenta were replaced by functions obtained by simply uncontracting underlying HF primitives based on their proximity to the optimal functions. As in the actinide elements, due to the possibility of linear dependency between the high angular momentum core correlating functions with the exponents of inner valence functions (from cc-pvnz-dk3), the innermost valence correlating functions were reoptimized (for 5s5p4f correlation) along with the 4s4p4d correlating g- and h-type functions at the TZ and QZ basis set levels. Hence the final ccpwcvnz-dk3 sets (TZ and QZ) are not exactly equivalent to simply adding tight functions to the cc-pvnz-dk3 valence basis sets. III. RESULTS AND DISCUSSION A. Atomic ionization potentials As an initial assessment of the new basis sets, the first three ionization potentials (IPs) of the lanthanide atoms were calculated at the CCSD(T) level of theory. Previously there have been several benchmark quality ab initio studies of these IPs, which have been fairly well characterized by experiment. In particular Cao and Dolg 28,29 have carried out large basis set average coupled pair functional (ACPF) pseudopotential (PP) calculations on the first 4 IPs of the lanthanide atoms, as well as CCSD(T) calculations on a subset of these (all including correlation from the 6s through the 4s electrons). Extrapolation to the CBS limit, spin-orbit corrections, and PP corrections were applied, resulting in average errors of ~3 to ~8 kcal/mol. Where CCSD(T) was applied, this typically resulted in lower errors by up to a factor of 6. In a latter study, Roos et al. 18 calculated the 1st IPs of the lanthanide atoms using the complete active space 2nd-order perturbation theory (CASPT2) method with the DKH2 scalar relativistic Hamiltonian and their new ANO-RCC basis sets. Spin-orbit effects were not included but comparisons to J-averaged experimental data exhibited agreement to generally within 0.1 ev (2 kcal/mol) or better. Results for Ce and Gd, however, differed from experiment by about 4 kcal/mol. Recently Wilson and co-workers 30 reported an investigation of the 3rd IPs of La through Eu using the DKH3 scalar relativistic Hamiltonian with CCSD(T) and the Sapporo sequence of basis sets. Somewhat similar to the earlier work of Cao and Dolg, they included 8

9 spin-orbit (SO) corrections based on 4-component DHF and complete open-shell configuration interaction (COSCI) calculations. With an uncontracted basis set of quadruple-zeta quality, they reported an average deviation from experiment of just above 5 kcal/mol when the 4s through 6s electrons were correlated. 1. Computational details In the present work the CCSD(T) calculations employed restricted open-shell HF (ROHF) orbitals (symmetry broken) and an open-shell variant of CCSD that allows for small amounts of spin contamination, i.e., ROHF-CCSD(T) or R/UCCSD(T), using the MOLPRO program. 21,22 The electronic states corresponded to those shown in Table I, except for the case of neutral Ce atom where the low-spin open-shell 1 G state is not amenable to single determinant CCSD(T) calculations. In that case the low-lying 3 F state (which like the 1 G ground state level corresponds to the 4f5d6s 2 configuration) was employed, and the final CCSD(T) ionization potential was corrected using the experimental excitation energy. 35 The final CCSD(T) IPs were calculated in a Feller-Peterson-Dixon (FPD) composite scheme 5,36,37 via IP = IP CBS + CV/CBS + SO (1) where IP CBS is the ionization potential calculated at the frozen-core (FC) CCSD(T) level of theory at the extrapolated CBS limit, CV/CBS is the effect of correlating the outer-core electrons (4s4p4d) at the CCSD(T) level also extrapolated to the CBS limit, and SO is the contribution to the IP from spin-orbit coupling. In the FC calculations the CBS limits were obtained by separate extrapolation of the HF total energies and CCSD(T) correlation energies using cc-pvtz-dk3 and cc-pvqz-dk3 basis sets. The HF extrapolations used the Karton- Martin formula originally developed for molecules containing light elements, 38 E n E CBS An 1 e 6.57 n (2) with n=3 and 4, while the CCSD(T) correlation energy utilized 39 E n E CBS B 4 (3) n 1 2 These choices have been previously used to good effect in similar calculations for actinide atoms and actinide-containing molecules. 6,40-42 In particular Eq. (3) was chosen for correlation 9

10 energies based on its robustness for yielding accurate CBS limits in calculations involving main group elements. 4 The CCSD(T) calculations for CV used the cc-pwcvnz-dk3 basis sets, both for frozencore and outer-core (4s4p4d) correlated calculations. The difference in these two calculations for each atom yielded the CV correction to the IP, from which the TZ and QZ results were extrapolated to the CBS limit via Eq. (3). Of course by extrapolating CV to the CBS limit a decomposition of FC and CV effects is not actually needed, but is very useful in order to assess both the importance of outer-core correlation and the impact of additional tight basis functions on the FC results (compared to the normal cc-pvnz-dk3 basis sets). Last, contributions from SO coupling were calculated at the 2-component HF level of theory with the Exact 2-component (X2C) Hamiltonian 43 using uncontracted cc-pvdz-dk3 basis sets. Two-electron spin-same-orbit and spin-other orbit (Gaunt) corrections were included using the atomic-mean-field integral (AMFI) approximation. 44,45 Specifically 2-component average-of-configuration HF calculations were first carried out by distributing just the open-shell electrons among the 4f, 6s, or 5d spinors, the choice of which corresponded to the minimal set depending on the electronic term required. The energy of the ground state configuration (or 3 F in the case of Ce atom) was then recovered from a COSCI calculation. The SO contribution was then derived from calculations both including and excluding SO terms in the Hamiltonian. All SO calculations were carried out with the DIRAC program package Results and discussion The results for the first, second, and third IPs of the lanthanide atoms are given in Tables II, III, and IV, respectively. In each case in addition to the FC and CV CBS limits and SO contributions, the basis set convergence is represented by the difference of a FC IP or CV contribution calculated from a finite basis set from that at the extrapolated CBS limit, e.g., TZ is the difference between a TZ quantity from its CBS limit. In nearly all cases the FC CCSD(T) IPs converge to their CBS limits from below, i.e., the IPs increase with increasing basis set cardinal number. The only two exceptions are the first IPs of both La and Ce, in which the ionization involves removal of a 6s electron accompanied by an excitation from the 6s to 5d orbital. All other first IPs, except for Lu where the 1st IP also involves ionization of a 5d electron, involve ionization of a 6s electron and the basis set convergence is very rapid. The different 10

11 ionization processes between Yb and Lu (6s in Yb and 5d in Lu) is reflected by the much slower basis set convergence of IP 1 in the case of Lu where the FC cc-pvqz-dk3 value is still 1.2 kcal/mol below the CBS limit. At this same basis set level, all the first IPs involving 6s ionization had QZ values of just 0.2 to 0.3 kcal/mol. In regards to the effect of correlating the outer-core 4s4p4d electrons on the 1st IPs, this is calculated to be nearly negligible in the cases involving 6s ionization, i.e., generally +0.3 kcal/mol or less. Contrastingly, the CV values for La and Ce lie between +2 and +3 kcal/mol while that of Lu is the only IP 1 where outer-core correlation decreases the IP, albeit by just -0.7 kcal/mol. In all cases the basis set convergence is rapid. Where ionization involves just the 6s electron, not surprisingly the SO contribution to IP 1 is nearly negligible, on the order of 0.1 kcal/mol. The largest effect is for IP 1 of Lu (5d ionization), which is calculated to be 2.4 kcal/mol at the present 2c-HF level of theory. The other two IP 1 values that involve the 5d electrons, i.e., those of La and Ce, also have non-negligible SO values, -0.5 and -1.2 kcal/mol, respectively. All together the calculated first IPs are in excellent agreement with experiment 47 with a mean unsigned deviation (MUD) of just 0.52 kcal/mol. Not surprisingly the largest differences are observed for La, Ce, and Lu which range from 1 to about 3 kcal/mol. Even so, the present results represent an improvement over the previous extrapolated-cbs averaged coupled pair functional (ACPF) investigations of Cao and Dolg 28, as well as the complete active space 2nd-order perturbation theory (CASPT2) values of Roos et al. 18, where average deviations ranged from 2-3 kcal/mol. Cao and Dolg 28 did note however that the use of CCSD(T) did improve upon their ACPF average errors by more than a factor of 2 in several cases. Inspection of the calculated 2nd IPs of Table III reveals only a slightly stronger basis set dependence of the FC results, but the CV effects are still small and rapidly convergent. The only exception for the latter trend is the Ce atom, where IP 2 involves the removal of two 5d electrons (see Table I), one by ionization and one by promotion to the 4f. This leads to a relatively large outer-core correlation contribution of just over 10 kcal/mol with a very slow basis set convergence. The 2nd IP of La also involves ionization of a 5d electron, but this results in only a slightly stronger basis set dependence of the FC result compared to those involving 6s ionization and a small negative CV contribution. All atoms involving 6s ionization have small (less than 1 kcal/mol) positive CV contributions. The final average errors with respect to 11

12 experiment are still very good, just over 1.1 kcal/mol, with the largest errors (1-3 kcal/mol) observed for the early lanthanides La, Pr, Nd, and Pm. The average experimental uncertainty is 1.8 kcal/mol, however, and the present calculations yield 2nd IPs within 2 kcal/mol of the experimental values in all cases except La. In the case of the 3rd IPs, except for La, Gd, and Lu, which involve ionization of a 5d (La and Gd) or a 6s (Lu) electron, the ionization process corresponds to removal of a 5f electron. As shown in Table IV, this is marked by a much stronger basis set dependence of the FC IPs compared to IP 1 and IP 2 and relatively large CV and SO contributions. As shown in Figure 2, the relatively slow, but regular, basis set convergence of the FC CCSD(T) IPs is dominated by the slow convergence of the correlation contribution. Even at the DZ level the HF contribution is within 2 kcal/mol of the HF limit and within about 0.2 kcal/mol with just cc-pvtz-dk3. The convergence of the FC correlation contribution is much slower, with DZ values ranging from about -5 to -7 kcal/mol for La and Lu, respectively, which involve 5d ionization, to much larger values when a 5f electron is removed, i.e., up to -30 kcal/mol for Er. Obviously accurate calculation of IP 3 strongly benefits from systematically convergent basis sets yielding the ability to extrapolate to the CBS limit. For example for Er the large cc-pvqz-dk3 basis set still yielded a FC CCSD(T) IP 3 value that was 4 kcal/mol below the extrapolated CBS limit. For the cases involving ionization of a 4f electron, outer-core correlation effects on IP 3 were calculated to be relatively large (4 to 9 kcal/mol) for the early lanthanides, dropping to at most 1 to 2 kcal/mol from Eu onwards. Not surprisingly SO effects were much larger when a 4f electron was ionized, ranging up to nearly 13 kcal/mol for Yb. Upon comparing the final composite values to experiment, the average error increases to 4.2 kcal/mol for these 3rd IPs, which is still less than 1% and suprisingly about half the average error as the ACPF treatment of Cao and Dolg. 28 Some individual errors, however, reached values as large as nearly 11 kcal/mol (Tb) and this can be attributed to using CCSD(T) in situations where non-dynamical correlation effects are becoming important. Simply inspecting the sizes of the T 1 diagnostic or the largest doubles amplitudes was not sufficient, however, to determine when the CCSD(T) results would lead to larger errors. For the most part the present results are similar to the recent CCSD(T) results of Wilson and co-workers 30 (La-Eu) who used the Sapporo basis sets, specifically their FC1 (analogous to our FC results) and FC3 (analogous to our FC+ CV) values. The average errors, 12

13 however, resulting from the present composite treatment were about 1 kcal/mol smaller than those reported in their work, presumably due to the use of CBS extrapolations in the present work that were enabled by the new cc-pvnz-dk3 and cc-pwcvnz-dk3 basis sets. B. Molecular calculations For representative molecular benchmark calculations, CCSD(T) calculations have been carried out on three gadolinium-containing molecules, Gd 2, GdF, and GdF 3. A composite scheme analogous to that described above for the atomic IPs has been utilized. Namely the DKH3 Hamiltonian is used in conjunction with ROHF-CCSD(T) and sequences of cc-pvnz- DK3 and cc-pwcvnz-dk3 basis sets (n=d-q) for Gd. The standard diffuse-augmented aug-ccpvnz-dk and aug-cc-pwcvnz-dk sets (n=d-q) are used for F. 1,27,48,49 Below the combinations cc-pvnz-dk3/aug-cc-pvnz-dk and cc-pwcvnz-dk3/aug-cc-pwcvnz-dk are denoted simply as VnZ-DK and wcvnz-dk. In each case frozen-core calculations (correlating 4f and 5s through 6s on Gd with 2s and 2p on F) were first carried out with the VnZ-DK sets while core correlation effects were determined using the wcvnz-dk basis sets. In each case the final composite energies were determined as E Final = E CBS[TQ] + CV CBS[wTQ] + SO + QED (4) where E CBS[TQ] represents the frozen-core CCSD(T) energy at the CBS limit obtained by extrapolating the HF and CCSD(T) correlation contributions separately via Eq. (2) and (3), respectively, with VTZ-DK and VQZ-DK basis sets. The 2nd term, CV CBS[wTQ], is the contribution due to outer-core correlation (4s4p4d on Gd and 1s on F), obtained as the difference in two CCSD(T) calculations, one with FC and one with outer-core correlated, both in the same wcvnz-dk basis sets. The contributions due to SO coupling were determined by a 2c-X2C-HF calculation together with COSCI in the case of Gd 2 or from 2c-X2C-CCSD(T) calculations 50 for GdF and GdF 3. Each of these used uncontracted VDZ-DK basis sets. The atomic spin orbit correction for F atom was obtained by J-averaging the experimental SO energy levels. 51 Last, the final term in Eq. (3), QED (quantum electrodynamics), is an estimate of the Lamb shift on the Gd atom and was calculated at the CCSD(T) level of theory with wcvdz-dk basis sets with valence electrons correlated. These calculations used the model potential approach for both the 13

14 vacuum polarization and self-energy contributions as first described by Pyykkö and Zhao. 52 The current implementation has been previously described in detail. 6 Spectroscopic properties of the two diatomic molecules were determined by fitting 7 calculated energies distributed about the equilibrium bond lengths ( 0.3 a o r - r e +0.5 a o ) to 6th-order polynomials in internal displacement coordinates. The derivatives of these fits were then used in the usual 2nd-order perturbation theory expressions for e, e x e, etc. (see, e.g., Ref. 53). 1. Gd 2 The gadolinium dimer is an interesting molecule in its own right since it has the honor of having the highest experimentally confirmed 54 spin multiplicity ground state of any diatomic molecule, 19 (18 unpaired electrons). The 19 g ground state corresponds to the superconfiguration (4 f 7 )(4 f 7 ) 2 g 1 u 1 g 2 u. As previously shown in the work of Cao and Dolg, 55 the doubly occupied g is a bonding orbital arising from the 6s atomic orbitals (AOs), while the singly occupied g and doubly occupied u can be attributed to the 5d AOs. The singly occupied u has predominately 6s character with strong contributions from the 6p AOs. The spectroscopic properties of the Gd dimer have been previously determined from Raman spectra in argon matrices by Lombardi and co-workers. 56 In addition to deriving values for the harmonic frequency (138.7±0.4 cm -1 ) and anharmonicity constant (0.3±0.1 cm -1 ), a dissociation energy of 48±16 kcal/mol was derived based on the dimer force constant. Their dissociation energy can be compared to the value reported by Kant and Lin 57 based on mass spectrometry of 41.1 ± 8.1 kcal/mol. The previous PP-based CCSD(T) calculations of Cao and Dolg yielded a dissociation energy of 31.8 kcal/mol. CCSD(T) results using both VnZ-DK and wcvnz-dk basis sets from the present work using the composite treatment of Eq. (4) are shown in Table V. Comparisons of identical calculations using the ANO-RCC basis sets of Roos et al. 18 as well as the Sapporo basis sets (SAP-nZP) of Sekiya et al., 17 are also given. The ANO-RCC sets range from double-zeta to quadruple-zeta ANO contractions with an additional ANO-RCC-Large set that is intermediate between QZ and 5Z. These sets are only designed for valence electron (4f and 5s through 6s) 14

15 correlation and hence have not been employed in this work for outer-core correlation. The Sapporo sets also range from DZ to QZ but were optimized for inclusion of outer-core correlation. Also shown in Table V are the SAP-nZP sets augmented with an additional diffuse function in each angular momentum, SAP-nZP+diffuse. The ANO-RCC sets are much more compact than either the VnZ-DK or SAP-nZP basis sets, while the SAP-nZP sets are only slightly larger than the VnZ-DK ones. The wcvnz-dk sets of the present work are similar in size to the SAP-nZP+diffuse basis sets. In order to assist the interpretation of the resulting spectroscopic properties, particularly r e and D e, the basis set superposition error (BSSE) has also been calculated in each case using the standard function counterpoise method. 58 Focusing first on the frozen-core CCSD(T) results, the VnZ-DK basis sets result in spectroscopic properties that exhibit a regular convergence towards their CBS limits. As also shown in Figure 3, particularly the ANO-RCC basis sets yield an irregular convergence pattern. In the latter case, the dissociation energy initially converges from above and this can be attributed to the very large BSSE with the ANO-RCC-DZP set, which is nearly an order of magnitude larger than either VDZ-DK or SAP-DZP. While both the ANO-RCC-TZP and -QZP basis sets yield D e values close to the estimated CBS limit, this is due to fortuitous BSSE contributions that are still 4.5 kcal/mol at the QZP level. In fact the ANO-RCC-Large result, where the BSSE is calculated to be below 1 kcal/mol, falls below the estimated CBS limit by about 2 kcal/mol. Without additional diffuse functions, the SAP-nZP basis sets are associated with small BSSE values similar to the VnZ-DK sets, but the convergence with increasing basis set size is very slow. This is remedied by adding additional diffuse functions, SAP-nZP+diffuse. Presumably this occurs due to a bias in the standard SAP-nZP sets towards 4s4p4d correlation. The FC spectroscopic constants obtained using the very large SAP-QZP+diffuse basis set are intermediate between the VQZ-DK and CBS[TQ] results. The effects of correlating the 4s4p4d electrons are shown in Table V for the wcvnz-dk and SAP-nZP+diffuse basis set sequences. The convergence of r e, e, and D e are fairly regular for the correlation consistent basis sets but are somewhat irregular for the SAP sets. At the estimated CBS limit, the effect of outer-core correlation is only Å, 0.13 cm -1, and kcal/mol for r e, e, and D e, respectively. The effects of SO coupling is calculated to be nearly negligible for r e and e, but predominately due to atomic SO it lowers the dissociation energy by 15

16 4.2 kcal/mol. The latter correction is about 0.9 kcal/mol smaller than what would be expected due to just the atomic fine structure (estimated from experiment, 5.1 kcal/mol). To the authors' knowledge this is the first ab initio calculation of the effects of SO coupling on the Gd dimer. (The work of Cao and Dolg 55 utilized the experimental atomic SO splitting.) The impact of the Lamb shift is nearly negligible, increasing the dissociation energy by only ~0.2 kcal/mol. The final composite results are in good agreement with both experiment and the previous ab initio results of Cao and Dolg. In particular the composite harmonic frequency differs from (matrix) experiment result by less than 4 cm -1. The final D 0 of 33.2 kcal/mol is consistent with the previous result of Cao and Dolg (31.8 kcal/mol) and at the lower end of the large experimental uncertainties. The current composite value, however, is estimated to be accurate to within about 3 kcal/mol, representing a significant improvement over the current experimental values. This error estimate is motivated by the ionization potential results discussed above in the cases where CCSD(T) is particularly valid, as well as previous CCSD(T) calculations on 3d transiton metal molecules 59 and those involving Th and U. 6,60,61 Applying 298 K thermal corrections from standard harmonic oscillator-rigid rotor statistical mechanical partition functions yields a D 298 value of 34.4 kcal/mol. Combining this with the experimental 298 K formation enthalpy of Gd atom 62 (97.25 ±0.48 kcal/mol) yields H f (298) for Gd 2 of kcal/mol. 2. GdF Table VI displays the results of the current CCSD(T) calculations for the 8 7/2 ground state of the GdF molecule, which is well described as [Gd + ][F ] where Gd + is in a 4f 7 6s 2 configuration. Both the FC and CV results show good convergence towards their respective CBS limits, however some slight irregularities in e are observed at the DZ level presumably from BSSE effects. In both cases the QZ quality basis set results are fairly close to the extrapolated CBS limits. The effects of outer-core correlation are again calculated to be very small for r e and e, but nearly +0.9 kcal/mol for D e. The contributions due to SO coupling are also small, except for D e where it is lowered by 2.6 kcal/mol, which is nearly entirely due to atomic SO on Gd. The effects of the Lamb shift are completely negligible in this case. Compared to the accurate values 16

17 from spectroscopic measurements, 63 the present composite results for r e and e are in excellent agreement, being within about Å and 2 cm -1, respectively. The values of D 0 and D 298 from the present work, and kcal/mol, respectively, are larger than the experimental value (298 K) of kcal/mol derived from mass spectrometric measurements, 64 which has a stated uncertainty of ±4 kcal/mol. The present results, however, are consistent with the previous ab initio calculations of Dolg et al. 65 The uncertainty of the present result for D 0 is also estimated to be about 3 kcal/mol. Combining the present D 298 value with the experimental formation enthalpies at 298 K of both Gd(g) (97.25 ± 0.48 kcal/mol) 62 and F(g) (18.97 ± 0.07 kcal/mol) 66 leads to a H f (298) value for GdF of 36.6 kcal/mol. The analogous experimental value from Ref. 64 is -40 ± 6 kcal/mol. 3. GdF 3 The lanthanide trihalides have been the subject of many experimental and ab initio studies. In these molecules the central Ln atom is in its +3 oxidation state and for GdF 3 this leads to a 4f 7 electron configuration on Gd +3. While there is still considerable uncertainty on whether their true equilibrium geometries are planar or pyramidal, it is clear that the out-of-plane bending mode is very soft with a concomitant small barrier to inversion. 67 The current DKH3 CCSD(T) calculations are the most extensive to date, however the present structural results are qualitatively similar to the recent PP-based (predominately MP2) calculations of Lanza and Minichino. 68 The CCSD(T)/VTZ-DK level of theory yields a pyramidal C 3v equilibrium structure with r e = Å and a F-Gd-F bond angle of 119.3º. The latter corresponds to the F atoms being just 4.9º below the D 3h plane. An analogous optimization with the VDZ-DK basis set yielded a planar D 3h structure with r e = Å. Experimentally the structure has been characterized by electron diffraction, 69 giving a pyramidal structure with a thermally averaged bond length of ±0.003 Å. An equilibrium distance has also been estimated in the same experimental work as ±0.006 Å (see Ref. 70). Based on the present calculations, the experimentally-derived correction to the thermal average yielding the equilibrium bond length seems to be strongly overestimated. 17

18 Results pertinent to the CCSD(T) atomization energy of GdF 3 are given in Table VII. Both in the frozen-core calculations with the VnZ-DK basis sets and those with the outer-core electrons correlated with the wcvnz-dk sets, the calculated atomization energy smoothly converges from below their respective CBS limits. The CBS extrapolation is not negligible in this case, with the CBS limit being more than 3 kcal/mol larger than the QZ result. Correlation of the outer-core electrons (including 1s correlation on F) increases the atomization energy over the frozen-core value by 0.88 kcal/mol, which is identical to the GdF case (see Table VI). After inclusion of SO and QED (the latter again negligible), the equilibrium atomization energy is calculated to be kcal/mol. Employing the CCSD(T)/VDZ-DK harmonic frequencies, the 0 K and 298 K values are and kcal/mol, respectively (with estimated uncertainies of ±3 kcal/mol). The latter can be compared to the experimentally derived value of Myers, ± 5 kcal/mol or the experimental value of Zmbov and Margrave, ± 10 kcal/mol. The enthalpy of formation at 298 K is calculated in this work to be kcal/mol. IV. CONCLUSIONS New correlation consistent basis sets have been developed in conjunction with the DKH3 scalar relativistic Hamiltonian for the lanthanide elements La through Lu. The basis sets range from double- to quadruple-zeta, encompassing those for valence correlation (4f5s5p5d6s, denoted cc-pvnz-dk3) and outer-core correlation (valence + 4s4p4d, denoted cc-pwcvnz- DK3). Both series of basis sets systematically converge the HF and correlation energies towards their respective CBS limits. Coupled cluster benchmark calculations were carried out for the 1st three ionization potentials of the lanthanide atoms, as well the Gd 2, GdF, and GdF 3 molecules. The ability to converge both the valence and outer-core correlation energies with respect to basis set was found to be essential for the accurate determination of the 3rd IPs, which involved the ionization of a 4f electron. The new correlation consistent basis sets of this work were compared to the ANO-RCC and SAP-nZP basis sets in CCSD(T) calculations on the Gd dimer. Both the new correlation consistent and SAP sets were found to yield a more systematic convergence towards the CBS limit for the dissociation energy in comparison to the ANO-RCC sets, the latter of which suffer from relatively large BSSE. The most comparable results to the present ccpvnz-dk3 or cc-pwcvnz-dk3 values were obtained by using the larger SAP-nZP+diffuse 18

19 basis sets. The final CBS limit CCSD(T) results for atomization energies, which are estimated to be accurate to within 3 kcal/mol, are proposed to be an improvement over the current experimental values in each case. This is particularly true for Gd 2 where the current experimental dissociation energies have uncertainties of 8 16 kcal/mol. The effect of correlating the outercore 4s4p4d electrons was found to be negligible for the bond lengths and frequencies of Gd 2 and GdF, but contributed up to 0.9 kcal/mol to the atomization energies. The final 0 K atomization energies (298 K heats of formation) are calculated to be (all in kcal/mol): 33.2 (160.1) for Gd 2, ( 36.6) for GdF, and (-295.2) for GdF 3. The new basis sets are provided in the Supplemental Material 72 and will also be made available on the authors' website, 73 the Basis Set Exchange, 74 as well as the MOLPRO basis set library. 21 ACKNOWLEDGMENTS The support of the U.S. Department of Energy Office of Science, Office of Basic Energy Sciences, Heavy Element Chemistry Program through Grant No. DE-FG02-12ER16329 is gratefully acknowledged. 19

20 REFERENCES T. H. Dunning Jr, J. Chem. Phys. 90, 1007 (1989). J. Dunning, T H, J. Phys. Chem. A 104, 9062 (2000). J. Dunning, T H, K. A. Peterson, D. E. Woon, P. Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, P. Kollman, H. F. Schaefer III, and P. R. Schreiner, Correlation consistent basis sets for molecular calculations. (1998). D. Feller, K. A. Peterson, and J. G. Hill, J. Chem. Phys. 135, (2011). K. A. Peterson, D. Feller, and D. A. Dixon, Theor. Chem. Acc. 131, 1079 (2012). K. A. Peterson, J. Chem. Phys. 142, (2015). M. Vasiliu, K. A. Peterson, J. K. Gibson, and D. A. Dixon, J. Phys. Chem. A 119, (2015). K. A. Peterson and K. G. Dyall, in Computational Methods in Lanthanide and Actinide Chemistry, edited by M. Dolg (John Wiley & Sons, 2015), pp A. P. Gomes, K. G. Dyall, and L. Visscher, Theor. Chem. Acc. 127, 369 (2010). M. Douglas and N. M. Kroll, Ann. Phys. 82, 89 (1974). B. A. Hess, Phys. Rev. A 33, 3742 (1986). G. Jansen and B. A. Hess, Phys. Rev. A 39, 6016 (1989). E. van Lenthe, J. G. Snijders, and E. J. Baerends, J. Chem. Phys. 105, 6505 (1996). D. A. Pantazis and F. Neese, J. Chem. Theory Comp. 5, 2229 (2009). M. Dolg, J. Chem. Theory Comp. 7, 3131 (2011). T. Tsuchiya, M. Abe, T. Nakajima, and K. Hirao, J. Chem. Phys. 115, 4463 (2001). M. Sekiya, T. Noro, T. Koga, and T. Shimazaki, Theor. Chem. Acc. 131, 1247 (2012). B. O. Roos, R. Lindh, P.-A. Malmqvist, V. Veryazov, P.-O. Widmark, and A. C. Borin, J. Phys. Chem. A 112, (2008). A. Wolf, M. Reiher, and B. A. Hess, J. Chem. Phys. 117, 9215 (2002). M. Reiher and A. Wolf, J. Chem. Phys. 121, (2004). MOLPRO, version , a package of ab initio programs, H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz, P. Celani, T. Korona, R. Lindh, A. Mitrushenkov, G. Rauhut, K. R. Shamasundar, T. B. Adler, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, E. Goll, C. Hampel, A. Hesselmann, G. Hetzer, T. Hrenar, G. Jansen, C. Köppl, Y. Liu, A. W. Lloyd, R. A. Mata, A. J. May, S. J. 20

21 McNicholas, W. Meyer, M. E. Mura, A. Nicklass, D. P. O'Neill, P. Palmieri, D. Peng, K. Pflüger, R. Pitzer, M. Reiher, T. Shiozaki, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinsson, and M. Wang,, see H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, and M. Schütz, WIREs Comput. Mol. Sci. 2, 242 (2012). D. H. Bross and K. A. Peterson, Theor. Chem. Acc. 133, 1434 (2014). A. Weigand, X. Cao, T. Hangele, and M. Dolg, J. Phys. Chem. A 118, 2519 (2014). P. J. Knowles and H.-J. Werner, Chem. Phys. Lett. 145, 514 (1988). H. J. Werner and P. J. Knowles, J. Chem. Phys. 89, 5803 (1988). K. A. Peterson and T. H. Dunning, Jr., J. Chem. Phys. 117, (2002). X. Cao and M. Dolg, Mol. Phys. 101, 961 (2003). X. Y. Cao and M. Dolg, Chem. Phys. Lett. 349, 489 (2001). C. Peterson, D. A. Penchoff, and A. K. Wilson, J. Chem. Phys. 143, (2015). K. Raghavachari, G. W. Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys. Lett. 157, 479 (1989). G. E. Scuseria, Chem. Phys. Lett. 176, 27 (1991). J. D. Watts, J. Gauss, and R. J. Bartlett, J. Chem. Phys. 98, 8718 (1993). P. J. Knowles, C. Hampel, and H.-J. Werner, J. Chem. Phys. 99, 5219 (1993). W. C. Martin, R. Zalubas, and L. Hagan, Atomic Energy Levels - The Rare Earth Elements. (Natl. Stand. Ref. Data Ser., Natl. Bur. Stand., U.S., 1978). D. Feller, K. A. Peterson, and D. A. Dixon, J. Chem. Phys. 129, (2008). D. A. Dixon, D. Feller, and K. A. Peterson, in Annual Reports in Computational Chemistry, edited by R. A. Wheeler and G. Tschumper (Elsevier, Amsterdam, 2012), Vol. 8, pp. 1. A. Karton and J. M. L. Martin, Theor. Chem. Acc. 115, 330 (2006). J. M. L. Martin, Chem. Phys. Lett. 259, 669 (1996). D. H. Bross and K. A. Peterson, J. Chem. Phys. 141, (2014). D. H. Bross and K. A. Peterson, J. Chem. Phys. 143, (2015). R. M. Cox, M. Citir, P. B. Armentrout, S. R. Battey, and K. A. Peterson, J. Chem. Phys. 144, (2016). M. Iliaš and T. Saue, J. Chem. Phys. 126, (2007). B. A. Hess, C. M. Marian, U. Wahlgren, and O. Gropen, Chem. Phys. Lett. 251, 365 (1996). 21