1 Introduction

In recent work [1] on TiO\(_x\)(OH)\(_y\), we found that the geometries and harmonic frequencies computed using an B3LYP [2] hybrid [3] functional and coupled cluster single and doubles [4, 5], including a perturbational estimate of the triples [6], (CCSD(T)) approaches were in very good agreement, and even the computed reaction energies were in reasonable agreement for the two treatments. Thus optimizing the geometries and computing the frequencies at the B3LYP level and improving the energetics at the CCSD(T) level are very cost-effective. For the second transition row the inclusion of scalar relativistic effects during the geometry optimization becomes more important than for the first transition row. For all electron basis sets, including scalar relativistic effects adds the complication that analytic derivatives are not available in most codes. Switching to effective core potentials (ECPs) allows the inclusion of scalar relativistic effects and allows the use of analytic first and second derivatives; in addition, using small core ECPs with correlation consistent basis sets in the CCSD(T) calculations yields accurate energetics. For example, we found that CCSD(T) reaction energies for YO\(_x\)(OH)\(_y\) and ZrO\(_x\)(OH)\(_y\) [7] using an ECP treatment agreed with all-electron treatment (both using triple zeta basis sets) to within 1 kJ/mol for systems with only single bonds and 2–3 kJ/mol for systems with double bonds.

Table 1 Summary of the B3LYP calculations on MF, for M= Eu, Gd, Tb, and Yb
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Given our success of using the B3LYP approach with an ECP to optimize the geometries and compute the harmonic frequencies for several species containing second and third transition row atoms, this is the approach we used in a recent study [8] of GdO\(_x\)(OH)\(_y\) and YbO\(_x\)(OH)\(_y\) species. The results for the YbO\(_x\)(OH)\(_y\) systems were consistent with our previous work, namely the B3LYP and Brueckner doubles [9] with triples (BD(T)) reaction energies were in reasonable agreement (a maximum difference of 23 kJ/mol). However, for GdO\(_x\)(OH)\(_y\) the difference between the B3LYP and BD(T) levels increased with number of bonds, so that while GdOH shows reasonable agreement between the B3LYP ECP and BD(T) AE atomization energies, Gd(OH)\(_3\) shows a 248 kJ/mol difference between the B3LYP/TZ ECP and the BD(T)/QZ AE calculations. If an all electron treatment is used in the B3LYP calculations, the difference between the B3LYP/TZ AE and BD(T)/QZ AE calculations is reduced to 2.6 kJ/mol. These results suggest that the Gd ECP is not performing as well as hoped. In this work we report on some additional tests of lanthanide atoms and molecules to assess if the problem encountered for Gd is unique or if other atoms are similarly affected. We study EuF, EuF\(_3\), GdF, GdF\(_3\) TbF, TbF\(_3\), YbF, and YbF\(_3\). We also look at the energy levels of Pr, Nd, Eu, Gd, Tb, Dy, Ho, Er, Tm, and Yb atoms.

Table 2 Summary of the B3LYP calculations on MF\(_3\), for M= Eu, Gd, Tb, and Yb
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2 Methods

The MF and MF\(_3\), for M=Eu, Gd, Tb, and Yb, geometries are optimized and harmonic frequencies computed using the B3LYP hybrid functional. The metal atoms are described with both ECP and all-electron treatments. The metal ECPs include the inner most 28 electrons in the ECP [10] and explicitly include the 4f electrons in the valence space. The (14s13p10d8f6g)/[10s8p5d4f3g] segmented and (14s13p10d8f6g)/[6s6p5d4f3g] atomic natural orbital (ANO) basis sets [11, 12] are used; the calculations are denoted ECP(seg) and ECP(ANO), respectively. In the all-electron calculations, metal atoms use the Douglas-Kroll correlation consistent polarized weighted core valence triple zeta basis sets of Lu and Peterson [13], i.e., the cc-pwCVTZ-DK3 set and the F uses the aug-cc-pVTZ set of Dunning and coworkers [14, 15]. In the AE calculations, the scalar relativistic effects are included using second-order Douglas-Kroll-Hess (DKH) method [16], which results in the AE frequencies being computed using only energy differences. In order to eliminate imaginary frequencies and bring the two components of the e vibrations into agreement, we use a superfine grid, four displacements instead of the default two, and a larger step size, i.e., freq=(step=500,fourpoint) in Gaussian16 [17].

Table 3 Summary of the computed atomization energies for GdF\(_3\)
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CCSD(T) calculations are performed using two approaches: in the first, all of the orbitals are taken from the HF calculations and, in the second, the core orbitals are taken from the HF and the valence and virtual orbitals are taken from the B3LYP calculations [18], denoted B3-CCSD(T). The same basis sets are used as were used in the DFT calculations. In the AE calculations the third-order DKH approach is used to include scalar relativistic effects. The metal 4f, 5s, 5p, 5d, and 6s electrons and the F 2s and 2p electrons are correlated in the CCSD(T) calculations. The CCSD(T) calculations are performed using Molpro [19].

The atomic energy levels are studied using the spin restricted Hartree-Fock, with and without symmetry and equivalence restrictions, the B3LYP, and the CCSD(T) levels of theory. The cc-pwCVQZ-DK3 basis sets [13] are used in the atomic calculations. In the AE calculations, the third-order DKH approach is used to include scalar relativistic effects. The metal 4f, 5s, 5p, 5d, and 6s electrons are correlated in the CCSD(T) calculations. All of the atomic calculations were performed with Molpro [19].

The spin-orbit splitting is quite large for these systems; in some cases, the order of the states reverse between the lowest spin-orbit level and the \(m_j\) averaged energies. When all the spin-orbit levels are available from experiment, we use experiment [20]. When there are missing spin-orbit levels, we use theory to compute the missing levels. This is done using the state interaction approach with the Breit-Pauli Hamiltonian, where the orbitals are determined using the state-averaged complete active space self-consistent field (SA-CASSCF) approach. The specific choice of active space and states included in the SA-CASSCF and the configuration interaction calculations in the state-interaction approach are discussed below and/or given in the Supplemental Information.

Table 4 Summary spin-orbit splitting, in cm\(^{-1}\)
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Table 5 Summary Ho spin-orbit splitting, in cm\(^{-1}\)
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Table 6 Summary atomic energy levels, in cm\(^{-1}\)
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3 Results

3.1 MF and MF\(_3\)

The results of the MF calculations are summarized in Table 1. The computed \(D_e\) values are converted to \(D_0\) values using the computed \(\omega _e\) values and experimental spin-orbit splittings (the difference between the lowest \(m_j\) level and the weighted average of the \(m_j\) levels). The AE and ECP approaches yield the same ground state. The agreement between the ECP and AE treatments for \(D_0\), \(r_e\), and \(\omega _e\) values is good for EuF and YbF. For GdF and TbF, the ECP(seg) harmonic frequencies are smaller than the AE results. Using the ECP(ANO) basis set brings the GdF \(\omega _e\) value into good agreement with the AE result, while for TbF it improves the agreement, but the ECP(ANO) value is still smaller. For TbF, the ECP(ANO) increases the \(D_0\) disagreement with the AE value. There is limited experimental data [21], and there is reasonable agreement between theory and experiment. The AE and ECP(Seg) Mulliken population are given in the Supplemental Information Tables S1–S4. The largest difference is for TbF, where the AE f\(\sigma \) population is 0.2 electrons larger than the ECP(seg) population. While it is tempting to attribute that difference between the AE and ECP(seg) results to this change in the population, the ECP(Seg) for GdF also yields a longer \(r_e\) and smaller \(\omega _e\) than the AE, but the AE and ECP(seg) populations are more similar than those for TbF.

The results for XF\(_3\) species are given in Tables 2 and 3 along with the limited experimental data [22, 23]. The populations are in Tables S5–S8. We have included both components of the e vibrations, which differ slightly for the AE treatment due to the use of numerical differences for the calculation of the harmonic frequencies. The small differences for the AE treatment support the use of the larger displacements and 4 displacements, as the defaults show larger difference. We also note that using this same numerical approach for the ECP(seg) approach shows a maximum difference with the analytic approach of 1.2 cm\(^{-1}\). There are also small differences for some of the ECP(ANO) calculations even though analytic derivatives are used.

We first note that there is good agreement with the limited experimental results for all three treatments. Comparing the three levels of theory among themselves shows that the ECP(seg), ECP(ANO), and AE results for EuF\(_3\) and YbF\(_3\) are very similar. For GdF\(_3\), the ECP(seg) atomization energy differs with the AE by 2.3 eV and the lowest frequency by 50.9 cm\(^{-1}\). The ECP(ANO) reduces the difference with the AE treatment for the atomization energy and the lowest frequencies. While the lowest frequency at the ECP(sep) level is actually in better agreement with experiment than the AE approach, we suspect that this is fortuitous. All three treatments are similar for TbF\(_3\) except for the atomization energy, where the ECP(ANO) value is larger than the other two results. This is similar to TbF, where the ECP(ANO) yielded a somewhat larger binding energy than the other two treatments.

The results for the XF\(_3\) suggests that the ECP(seg) treatment is not as reliable as either AE or ECP(ANO) treatments at the B3LYP level. We next compute the GdF\(_3\) atomization energy using the CCSD(T) approach at the B3LYP optimized geometry to see if the problem with the ECP(seg) is unique to B3LYP or if it also occurs for other highly correlated methods. The results are summarized in Table 3, and the B3LYP results from Table 2 are included for comparison. The CCSD(T) atomization energy using the AE treatment is 18.91 eV compared with 18.68 eV for the ECP(ANO) treatment, which is good agreement. This is in contrast with the ECP(seg) case where the CCSD fails. The results for the B3-CCSD(T) approach show the same agreement for the AE and ECP(ANO) treatments and again the ECP(seg) CCSD fails. The B3LYP and CCSD(T) results suggest that the problems with the ECP(seg) treatment can be largely avoided by using the ANO basis set instead of the segmented basis set. However, as we show below for the atomic energy levels there are cases where the use of either the segmented or the ANO basis sets leads to failure.

3.2 Atomic calculations

The spin-orbit splitting in the lanthanides is sufficiently large that it must be accounted for before comparing theory with experiment. For the ground states of all atoms that we consider in this work, and for excited states of Pr, Nb, Eu, Gd, Tb, and Dy, the spin-orbit levels can clearly be attributed to the splitting of a specific LS state. For the excited states of Ho, Er, Tm, and Yb, this is not the case. We therefore break our discussion in two cases. The results for Pd, Nb, and Eu are given in Table 4. The results for Gd, Tb, and Dy, which are similar, are given in supplemental data Tables S9 and S10. For the ground states, the SA-CASSCF calculation is performed including all components of the LS state; this corresponds to a symmetry and equivalence restricted SCF calculation. The \(m_j\) levels are computed in a state interaction calculation including all of the LS components. For the excited states, SA-CASSCF are performed for the states listed in the tables, in most cases this is one state, while for a few, two nearby states are included. The theoretical values for the excited states are shifted so that the lowest \(m_j\) component agrees with experiment. Overall the agreement of theory and experiment is very reasonable.

Ho is representative of the second class of atoms, and the spin orbit levels are shown in Table 5. The results for the ground \(^4I^o\) state are consistent with those for Pr to Dy, namely the levels appear to be derived from a specific LS state. The lowest excited LS state is the \(^6I\), and the computed \(m_j\) levels show no correspondence to experiment. The calculations are repeated adding one LS state at a time. When all of the sextet states arising from the coupling of the shell 5d with the 4f\(^{10}\)(\(^4I_4\)) shell are included in the calculation, the agreement between theory and theory is acceptable. Adding an equal number of quartet states as sextet states makes a small improvement. The results of the calculations suggest that the lowest spin-orbit component is derived from the \(^6I\) state.

We next compute the excitation energies between the ground and lowest excited state at the HF, B3LYP, CCSD(T), and B3-CCSD(T) levels of theory using the all electron and ECP approaches; the results are summarized in Table 6. To compare with experiment we need to account for the experimental spin orbit splittings. For the states that clearly arise from the splitting of one LS state, we use a weighted average of the experimental \(m_j\) energies. In those cases where there are missing experimental lines, the theoretical values are used, for example theory is used for Pr \(^4I\) \(m_j\)=15/2 level. As noted above, the excited states of Pr, Nb, Eu, Gd, Tb, and Dy are not derived from a single LS state. For these cases, we shift the lowest observed line by the difference in energy between our lowest \(m_j\) component and our lowest LS state included in the spin orbit calculation. We compare our computed LS separations with these “corrected” experimental values.

For the AE treatment, the CCSD(T) and B3-CCSD(T) results are in good agreement with each other and in good agreement with experiment. The ECP(seg) results vary significantly. For Pr, Nd, Ho, Er, Tm, and Yb the AE and ECP(seg) results show good agreement. For Eu, the HF-based CCSD(T) shows agreement between the AE and ECP(seg), but the B3LYP and B3-CCSD(T) results are in poor agreement. For Gd, Tb, and Dy, the ECP(seg) B3LYP results show very poor agreement with the AE results and the B3-CCSD(T) calculations fail. We assume that the very poor results for the ECP(seg) treatment of Gd are related to the problems observed in GdF\(_3\). Tb also shows poor results and the only indication of a problem in the molecular results was the slightly poorer agreement of the \(r_e\) value for TbF and an 4f\(_\sigma \) population that shows some differences between the AE and ECP(seg) treatment. We also note that while we did not see any indication of a problem with EuF or EuF\(_3\), the Eu atom results show some significant differences between the AE and ECP(seg) treatments, but not as bad as for Gd or Tb.

Given that the worst agreement between the AE and ECP(seg) results occurs for Gd, Tb, and Dy, we consider the ECP(ANO) for these three cases. For Gd, the HF-sym and HF results are very similar for the ECP(ANO) and ECP(seg) results. The ECP(ANO) and ECP(seg) CCSD(T) results are in good mutual agreement and in reasonable agreement with the AE results and with experiment. At the B3LYP level, the ECP(ANO) is in much better agreement with the AE than the ECP(seg). The B3-CCSD(T), which failed for the ECP(seg), yields a ECP(ANO) separation in good agreement with the AE results. Thus for Gd, switching to the ECP(ANO) basis set results in good agreement with the AE result; this is similar to the XF\(_3\) results described above. For Tb and Dy, we again find the HF and HF-sym results to be very similar for the ECP(seg) and ECP(ANO) approaches, and to be in good agreement with the AE results. Unlike Gd, the CCSD(T) fails for Tb and Dy for both the ECP(ANO) and ECP(seg) approaches. For the B3LYP, the ECP(ANO) is a significant improvement over the ECP(seg) separation, but unlike Gd, the difference with the AE results is still quite large. The B3-CCSD(T) fails for both the ECP(ANO) and ECP(seg) approaches. The Tb and Dy results show that even using the ECP(ANO) approach does not solve all of the problems encountered using the ECP(seg) approach.

4 Conclusions

We have studied EuF, EuF\(_3\), GdF, GdF\(_3\) TbF, TbF\(_3\), YbF, and YbF\(_3\) and the excitation energies of Pr, Nd, Eu, Gd, Tb, Dy, Ho, Er, Tm, and Yb atoms. In many cases we find good agreement between the all-electron and ECP treatments, while in others the agreement is less than satisfactory. We find that in many of the cases using the ANO basis set instead of the segmented basis set avoids the ECP problem. However, there are cases where even using ANO does not avoid problems associated with the ECP treatment. We suggest that additional studies and new effective core potentials may be required for the lanthanide atoms.