Electronic structure of lanthanide dimers

Transcription

1 MOLECULAR PHYSICS, 10 July 2003, VOL. 101, NO. 13, Electronic structure of lanthanide dimers XIAOYAN CAO 1,2 and MICHAEL DOLG 1, * 1 Institut fu r Theoretische Chemie, Universita tzuko ln, D-50939, Germany 2 Biochemistry Department, Zhongshan University, Guangzhou, , PR China (Received 2 October 2002; revised version accepted 25 October 2002) Relativistic energy-consistent small-core lanthanide pseudopotentials of the Stuttgart Bonn variety and extended valence basis sets have been used for the investigation of some selected lanthanide dimers with open 4f shell, that is Ce 2,Pr 2 and Gd 2. Comparison is made with results of corresponding previous studies of La 2 and Lu 2 as well as to available experimental data. The trends in the molecular constants of the dimers of the lanthanide series are discussed. The ground state candidates of Ce 2 ð4f 1 4f 1 s 2 g p4 u 1 þ g, 1 u, 3 g, 3 þ u,1 6 3 g, 6 u Þ and Pr 2 ð4f 2 4f 2 s 2 g p4 u 5 þ g, 5 u,5 10 g Þ are degenerate within 20 cm 1 and have the same valence subconfiguration s 2 g p4 u, which was previously found to give rise to the La 2 ð4f 0 4f 0 s 2 g p4 u 1 þ g Þ ground state. In the case of Gd 2 the 4f 7 4f 7 s 2 g s1 u s1 g p2 u 19 g ground state found previously is confirmed. The derived molecular constants are the best theoretical estimates available at present and show a satisfactory agreement with experimental data. Discrepancies in the vibrational constants of La 2,Ce 2 and Pr 2 are shown to be probably related to quite large Ar-matrix shifts. 1. Introduction The investigation of the dimers of the f transition metals, for example the lanthanide dimers considered in this paper, is still a challenge for both experimentalists and theoreticians [1 4]. From the theoretical point of view, the large number of low-lying molecular electronic states arising from the multiple possible combinations of ground and low-lying excited states of the separated atoms makes the ground state assignment very difficult. Since atomic states with different 5d and/or 4f occupation may mix in the molecules, the most difficult part of ab initio studies is to account precisely for electron correlation. Relativistic effects are also notable for elements as heavy as the lanthanides, but their inclusion in practical calculations seems to be somewhat easier than the correlation treatment. From the experimental point of view, the limited possibilities to produce and isolate lanthanide dimers are additional obstacles. Therefore, up to now both theoretical and experimental knowledge of the lanthanide dimers is very limited, and further studies are desirable, also as a first step to investigations of larger lanthanide clusters [5]. Despite the problems mentioned above, the method of ab initio pseudopotentials (PPs) has become increasingly successful during the past decade in its application to systems containing one or even two lanthanide atoms [6 14]. Recently we developed new valence basis sets (14s13p10d8f6g)/[6s6p5d4f3g] for relativistic *Author for correspondence. dolg@thch.uni-bonn.de energy-consistent small-core (28 core electrons) lanthanide pseudopotentials [13], which include the 4f shell explicitly in the valence. A generalized contraction scheme based on atomic natural orbital coefficients was adapted and several g functions are included to polarize and correlate the 4f shell. Results from test calculations using the new standard basis sets agree very well with experimental data for numerous diatomic lanthanide compounds (LnX, Ln ¼ La, Lu, Gd, Yb, Eu; X ¼ H, O, F, S) [13, 14]. Excellent agreement with experimental data was also obtained for the dimers at the beginning and the end of the lanthanide series, that is La 2 and Lu 2 [15], when the standard basis sets were augmented by diffuse functions. In addition to these covalently bonded systems, results for the mainly van der Waals dimers Eu 2 and Yb 2 are also available [15]. Besides these systems, it is also interesting to study Ce 2, Pr 2 and Gd 2, not only because they have seldom been theoretically investigated before [6, 7] but also because they are typical representatives of the rather complex homonuclear lanthanide dimers containing partially occupied 4f shells on both atoms, that is 4f is empty (La 2 ), partially occupied (Ce 2,Pr 2 ), half-filled (Eu 2,Gd 2 ) and completely filled (Yb 2,Lu 2 ). At the same time we may study the change in bonding owing to the lanthanide contraction, which is known to be mainly a shell structure effect owing to the incomplete shielding of the valence electrons from the increasing nuclear charge by the 4f shell and to be enhanced further by the increase in relativistic effects along the series. Molecular Physics ISSN print/issn online # 2003 Taylor & Francis Ltd DOI: /

2 1968 X. Cao and M. Dolg The most recent experimental data on La 2,Ce 2,Pr 2, Gd 2 and Lu 2 were obtained in 2000 by Lombardi and co-workers [16 19], using Raman and absorption spectroscopy in Ar matrices. The measured ground state vibrational constants! e are cm 1, cm 1, cm 1, cm 1 and cm 1 for La 2,Ce 2,Pr 2,Gd 2 and Lu 2, respectively. From these data and the anharmonicities x e! e the spectroscopic dissociation energies of ev, ev and ev for La 2,Gd 2 and Lu 2, respectively, have been estimated. However, these values differ substantially, even in their trend along the row, from the thermochemically determined dissociation energies of ev for La 2, ev for Gd 2 and ev for Lu 2 [20], respectively. To our knowledge the bond distance has not been determined experimentally for these dimers. For La 2 it was estimated by Verhaegen et al. [21] to be 2.80 A. The first theoretical paper for lanthanide dimers was published in 1992 by one of the present authors [6], and reported quantum chemical configuration interaction (CI) and correlation energy density functional theory (DFT) calculations using scalar-relativistic energyadjusted ab initio pseudopotentials modelling lanthanide elements with fixed integral 4f occupation numbers. The ground state superconfigurations predicted for La 2,Ce 2, Pr 2 and Gd 2 were ð4f n Þð4f n Þ s 2 g s1 u s1 g p2 u ðn ¼ 0,1,2,7Þ and for Lu 2 ð4f 14 Þð4f 14 Þ s 2 g s2 u s2 g. The spectroscopic constants of the proposed ground states at the CI(SD) level for La 2 and Lu 2 and at the DFT level for Ce 2,Pr 2 and Gd 2 (La 2 : R e ¼ 3.25 A,! e ¼ 130 cm 1, D e ¼ 1.17 ev; Ce 2 : R e ¼ A, D e ¼ 1.54 ev; Pr 2 : R e ¼ A, D e ¼ 0.35 ev; Gd 2 : R e ¼ A, D e ¼ 0.98 ev; Lu 2 : R e ¼ 3.79 A,! e ¼ 74 cm 1, D e ¼ 0.55 ev) are in clear disagreement with the current set of experimental values. Later, the same author published more accurate theoretical results for Gd 2 (R e ¼ A,! e ¼ 117 cm 1, D e ¼ 1.41 ev) [7], which are closer to the experimental values, but not perfect. In 1999 and 2002 two theoretical studies were published for Lu 2 by Bastug et al. [22] and by Yang et al. [23], respectively. Bastug et al. obtained the spectroscopic constants R e ¼ 2.51 A,! e ¼ 174 cm 1 and D e ¼ 2.39 ev with relativistic gradient-corrected density functional theory (RDFT), but a corresponding ground state assignment has not been made [22]. However, Yang et al. recently reported R e ¼ 3.24 A,! e ¼ 113 cm 1 and D e ¼ 1.79 ev for a 3 g ground state, with ab initio quadratic CI calculations using an energy-consistent large-core Lu pseudopotential (60 core electrons) and a (7s6p5d) valence basis set [23]. Analysis of an output provided by the authors [24] revealed that the complete term symbol is a 3 g and a contracted (7s6p5d)/ [5s4p3d] þ (5s)/[3s] basis set was used. The additional three s functions result from the use of cartesian instead of spherical Gaussians for the d basis. Pseudopotential and basis sets were taken from [25] and [26], not from the reference cited in the paper. The most recent theoretical results for lanthanide dimers were obtained by the present authors using relativistic energy-consistent small-core lanthanide pseudopotentials of the Stuttgart Bonn variety and extended valence basis sets at the coupled cluster level with single, double and perturbative triple excitation operators (CCSD(T)), as well as at the complete active space self-consistent field (CASSCF) and the multireference configuration interaction (MRCI) level [15]. It was found that the ground states for La 2 and Lu 2 are most likely 1 þ g ðs2 g p4 u Þ and 3 g ð4f14 4f 14 s 2 g s2 u p2 uþ, respectively. The theoretical values for La 2 (R e ¼ A, D e ¼ ev,! e ¼ cm 1 ) show good agreement with the experimental binding energy (D e ¼ ev [20]), but the experimental vibrational constant in the Ar matrix (! e ¼ cm 1 [16]) is significantly higher. For Lu 2 the theoretical values (R e ¼ A, D e ¼ ev,! e ¼ 123 1cm 1 ) are in overall excellent agreement with experimental data (D e ¼ ev [16],! e ¼ 122 1cm 1 [19]). It was argued that owing to the different ground state configurations Ar-matrix effects might increase the value of the! e of La 2 by 22 cm 1, but leave that of Lu 2 almost unchanged. In this paper, we report the calculated spectroscopic constants of Ce 2,Pr 2 and Gd 2 using basis sets of the same quality as used previously for La 2 and Lu 2 in connection with large-scale correlation treatments. For Ce 2 and Pr 2, which were not investigated theoretically before, we assign the same ground state valence subconfiguration s 2 g p4 u as for La 2. Differences in the spectroscopic constants of these three dimers arise solely from the different nuclear charges and 4f occupation numbers of the constituent atoms. For the convenience of comparison with homonuclear dimers of different lanthanide elements the results for La 2 and Lu 2 taken from previous work [15] are also included. 2. Method The relativistic energy-consistent small-core pseudopotentials used in this work for La, Ce, Pr, Gd and Lu have been published elsewhere [13, 27]. The 1s 3d shells were included in the pseudopotential core, while all shells with main quantum number larger than 3 were treated explicitly. Recently developed Gaussian (14s13p10d8f6g)/[6s6p5d4f3g] atomicnatural orbital valence basis sets were applied [13]. In the case of the ground states the standard basis sets (14s13p10d8f6g)/[6s6p5d4f3g] were augmented by a diffuse (3s3p3d3f3g) set.

3 Electronic structure of lanthanide dimers 1969 For Ce 2 and Pr 2 the CASSCF method [28 30] was used to generate the orbitals for the subsequent MRCI calculations. In the CASSCF calculations the orbitals 4s, 4p, 4d, 5s, 5p were kept doubly occupied. CASSCF and subsequent MRCI calculations which use an active space allowing for a proper dissociation of the dimers into the neutral atoms (Ce 4f 1 5d 1 6s 21 G, Pr 4f 3 6s 24 I), treated in a state average in order to avoid symmetry breaking, are not feasible at present: for Ce 2 and Pr 2 active spaces of 26 (Ce 4f, 5d, 6s) and at least 16 (Pr 4f, 6s) orbitals would be needed to accommodate eight (Ce 2 ) and 10 (Pr 2 ) valence electrons, respectively. The small active space for Pr 2 would allow a study of van der Waals-type states with 4f 3 4f 3 s 2 g s2 u superconfiguration, whereas for covalent bonding the large active space used for Ce 2 is needed. In order to establish the ground state and to generate restricted active orbital spaces suitable for the molecules near R e, we took the following steps. First, only the lowest 4f orbitals and the lowest s, p and d valence orbitals of gerade and ungerade parity, arising mainly from the 5d, 6s and 6p atomicorbitals, were chosen as active space for the 4f and valence electrons (eight electrons for Ce 2 and 10 electrons for Pr 2 in 15 orbitals). According to the superconfiguration model advocated by Field [31] all states belonging to a superconfiguration (i.e., a valence substate combined with a 4f subconfiguration) have nearly identical spectroscopic constants. From our CASSCF calculations assuming a high-spin coupling in the 4f subsystems we were able to establish a s 2 g p4 u ð1 þ g Þ valence subconfiguration (substate) for the electronic ground states of both Ce 2 and Pr 2. Therefore, we secondly kept the ðs 2 g p4 uþ valence subconfiguration doubly occupied, and chose all 4f orbitals as active space for the 4f electrons (two electrons for Ce 2 and four electrons for Pr 2 in 14 orbitals). From these CASSCF calculations we derived the 4f substate and identified the leading contributions of the configurations. Thirdly, we then selected all 4f natural orbitals with significant occupations (total contribution >99% for Ce 2 and Pr 2 ) as a new reduced active space and generated the corresponding reference wavefunctions for the subsequent internally contracted MRCI calculations. For the molecular ground states two d g, two d u, two j g,twoj u (eight orbitals) orbitals with predominant 4f character accommodating two (Ce 2 ) and four (Pr 2 ) electrons defined the new active spaces. Clearly this active space is not what one would really like to choose on the basis of physical arguments, but at least near the equilibrium distance it is a fairly good approximation. The MRCI configuration space was obtained by single and double excitations with respect to the CASSCF reference wavefunction. The 4s, 4p orbitals were frozen in the MRCI calculations using the standard basis set for Ce 2 and Pr 2, whereas for the extended basis sets the 4s, 4p, 4d orbitals were frozen. For the restricted active spaces used in the CASSCF/ MRCI calculations of Ce 2 and Pr 2 it is not possible to derive binding energies avoiding largely size-consistency errors by calculating the separated atoms at large distance using an unchanged active space. Therefore we use two sets of atomicreference energies leading to an upper and a lower limit. In the first case the energy of the separated atoms in a high-spin configuration at 50 au distance obtained at the CISD level was subtracted from the molecular CASSCF/MRCI energy. In the case of Ce the experimental 3 F 1 G energy difference (0.24 ev [32]) was subtracted twice to correct from the calculated 3 F state to the 1 G ground state. In the second case two times the CASSCF/MRCI energy of the atoms in the full active space was subtracted. Symmetry breaking was avoided by state-averaging over all components of the atomicground state in D 2h symmetry. Sizeextensivity errors were corrected by adding the energy difference between CISD calculations for the isolated atom (times two) and the separated atoms at large distance. Many of the low-lying electronic states of Ce 2 and Pr 2 have a significant multireference character. In a few cases, however, for example for the 3 þ u ground state candidate of Ce 2, high-level single-reference treatments appear reasonable. We therefore performed additional coupled cluster singles and doubles calculations with a perturbative treatment of triples (CCSD(T)) as an alternative to the CASSCF/MRCI approach. Excitations were allowed from the s 2 g p4 u valence orbitals and the 4f, 4d, 5s and 5p inner shells of both atoms for Ce 3 2 þ u. A similar active orbital space was chosen for the study of some selected excited states. For Gd 19 2 g only CCSD(T) calculations were performed, since a MRCI with an active space larger than required by the 18 unpaired electrons was not feasible owing to the limit of our current hardware capabilities. Excitations were allowed from the s 2 g s1 u s1 g p2 u valence orbitals and the 4f, 4d, 5s and 5p inner shells of both atoms. In addition we provide CCSD(T) results for largecore pseudopotentials keeping the 4f shell in the core [25, 26]. A core polarization potential (CPP) was applied to account for non-frozen core effects [8]. (7s6p5d)/ [5s4p3d] valence basis sets augmented by a (2s2p2d5f3g) polarization and diffuse set were applied and all electrons outside the pseudopotential cores were correlated. This approach was also adopted to investigate possible Ar-matrix effects on the vibrational frequencies. The program system MOLPRO [33 37] was used to perform the calculations reported here. D 2h symmetry

4 1970 X. Cao and M. Dolg was applied for the dimers and the atoms. The spectroscopic constants were derived by fitting a fifth-degree polynomial in the interatomicdistance R times a factor 1/R for six points on the potential curve near the equilibrium distance. A spacing of 0.1 au between the points was used. Tests with different fitting functions, including least-squares fits, indicate that the accuracy of the derived molecular constants is better than A for bond lengths, ev for binding energies and 1cm 1 for vibrational frequencies. 3. Results and discussion The elements La, Ce, Gd and Lu are the only lanthanides that possess a 4f n 5d 1 6s 2 (n ¼ 0, 1, 7, 14) ground state configuration [32]. All these elements also have a low-lying 4f n 5d 2 6s 1 excited state, which is much more favourable for bonding. However, the experimental 4f n 5d 1 6s 2 4f n 5d 2 6s 1 promotion energy is much lower for La (0.33 ev) and Ce (0.29 ev) than for Gd (0.79 ev) and especially Lu (2.34 ev) [32] (see figure 1). Therefore La 2 and Ce 2 are expected to exhibit stronger bonding than Gd 2 and Lu 2. Noting that both Ce and La have very similar 5d 1 6s 2! 5d 2 6s 1 promotion energies, one may expect the same ground state valence subconfiguration for the dimers, that is the electron configurations differ merely by the La 4f 0 and Ce 4f 1 subconfigurations. However, a different electronic structure of Lu 2 compared to La 2, Ce 2 and possibly Gd 2 may be expected, mainly owing to increased relativistic effects. These lead to a contraction and stabilization of the 6s shell and an expansion and destabilization of the 5d shell. From a comparison of non-relativistic Hartree Fock (HF) with scalar-relativistic Wood Boring (WB) all-electron calculations, the increasing impact of relativisticeffects along the lanthanide series is obvious: one finds the relativistic hri expectation values for 6s of La, Ce, Gd, Lu to be by 0.21, 0.22, 0.28, 0.36 au smaller and for 5d by 0.13, 0.13, 0.16, 0.25 au larger than the non-relativisticresults (see table 1). For the valence orbital energies even a reversal of the ordering of the 5d and 6s orbital is observed, that is " 6s is au higher than " 5d for La, au higher for Gd, and au lower for Lu (table 1). This explains the higher 5d 1 6s 2! 5d 2 6s 1 promotion energies of Gd (0.79 ev) and Lu (2.34 ev) compared to La (0.33 ev). For Lu the 6p orbital is also expected to contribute significantly to chemical bonding, since the promotion energy from the ground state 4f 14 5d 1 6s 2 2 D to 4f 14 6s 2 6p 1 2 P is only 0.72 ev [32], which is by 1.62 ev lower than the promotion energy to 4f 14 5d 2 6s 14 F (figure 1). For Pr a low-lying excited state 4f 2 5d 1 6s 2 4 I (about 0.55 ev higher than the ground state 4f 3 6s 24 I [32]) was found (see figure 2). Moreover, the orbital parameters of 5d and 6s for the excited state 4f 2 5d 1 6s 24 I of Pr are very similar to those of the ground state 4f 1 5d 1 6s 21 G of Ce, that is the absolute differences of the hri expectation values, negative orbital energies " for 5d and 6s are lower than 0.07 au and au, respectively (cf. table 1). Figure 1. Term energies of the lowest experimentally known levels of the indicated configurations with 4f n -subconfiguration with respect to the lowest level of the 4f n 5d 1 6s 2 configuration (n ¼ 0 14 for La Lu) [32].

5 Electronic structure of lanthanide dimers 1971 Table 1. hri expectation values (au) and negative orbital energies " (au) for the 5d and 6s valence orbitals of La, Ce, Pr, Gd and Lu (n ¼ 0, 1, 2, 7, 14) in their 4f n 5d 1 6s 2 configurations from non-relativistic (HF) and scalar-relativistic (WB) all-electron calculations. hri d hri s " d " s Metal HF WB HF WB HF WB HF WB La Ce Pr Gd Lu Figure 2. As for figure 1, but for levels resulting from configurations with the 4f n þ 1 -subconfiguration. Therefore the same ground state valence subconfiguration may be expected for La 2, Ce 2 and Pr 2. This is consistent with the experimental results of Lombardi and co-workers, who found the force constants to fall into two groups, k e ¼ mdyne A 1 for La 2,Ce 2 and Pr 2 as well as for Nd 2,Gd 2,Tb 2 and Lu 2 [2]. Pr (4f 3 6s 24 I) has the same ground state valence 6s 2 subconfiguration as Eu (4f 7 6s 28 S) and Yb (4f 14 6s 21 S), which form predominantly van der Waals-type dimers with 4f nþ1 4f nþ1 s 2 g s2 u (n ¼ 6 for Eu, n ¼ 13 for Yb) ground state (super)configurations [8, 15]. It is interesting to note that of all elements with 4f n þ 1 6s 2 ground state configuration, Pr appears to be the only one for which a strongly bound dimer can be formed by promoting on both atoms one electron from the 4f shell to the 5d valence shell. We attribute this to the fact that the 4f 2 5d 2 6s 1 6 L is also very low-lying, that is 0.45 ev above 4f 2 5d 1 6s 24 I La 2 and Lu 2 In our previous work we assigned the ground states of La 2 ðs 2 g p4 u 1 þ g Þ and Lu 2 ð4f 14 4f 14 s 2 g s2 u p2 u 3 g Þ [15]. Except for the vibrational frequency of La 2, the calculated spectroscopic constants for La 2 (R e ¼ A, D e ¼ ev,! e ¼ cm 1 ) and Lu 2 (R e ¼ A, D e ¼ ev,! e ¼ 123 1cm 1 ) are in overall excellent agreement with experimental data (La 2 : R e 2.80 A estimate, D e ¼ ev,! e ¼ cm 1 ; Lu 2 : D e ¼ ev,! e ¼ 122 1cm 1 ). We will not go into the details of these calculations here since they have been extensively discussed in the previous publication [15].

6 1972 X. Cao and M. Dolg However, we want to mention that the disagreement in the! e value of La 2 might be at least partly due to large Ar matrix shifts, which were estimated to be þ22 cm 1 and þ3cm 1 from model calculations on Ar La 2 and Ar Lu 2, respectively Ce 2 and Pr 2 The low-lying electronic states of Ce 2 and Pr 2 were first investigated at the CASSCF level and possible ground state candidates were selected. These were then investigated at a higher level of correlation treatment, that is CASSCF/MRCI and if appropriate CCSD(T). The results for the most likely ground states are listed in table 2. A ð4f n 4f n s 2 g p4 u (n ¼ 1,2) ground state superconfiguration (similar to the La 2 ground state) was obtained both for Ce 2 and Pr 2. For Ce 2 the 4f 1 4f 1 s 2 g p4 u 1 þ g, 1 u, 3 g,3 þ u,1 6 g and 3 6 u states were calculated to be virtually degenerate (within 4 and 20 cm 1 at the CASSCF and MRCI þ Q level, respectively) and the same spectroscopic constants were obtained (within A and 0.04 cm 1 for R e and! e ). The practical degeneracy of singlet and triplet states is attributed to the core-like properties of the 4f orbitals (hri 4f 0.55 A ), which owing to the long bond distance (R e 2.6 A ), lead to a negligible spin correlation for the two 4f electrons localized on the two Ce atoms. For further studies of Ce 2 with extended basis sets, we considered 1 þ g and 1 u as representatives. From our results obtained at the MRCI þ Q and CCSD(T) level (table 2) we obtain the following intermediate estimates and error bars of the molecular constants: R e ¼ A,! e ¼ 206 8cm 1, D e ¼ ev. It is not possible for us at present to estimate the magnitude of the counterpoise correction (CPC) at the MRCI þ Q level, owing mainly to the necessary restrictions of the active space and the need to calculate D e with respect to the separated atoms at large distance in order to roughly achieve size-extensivity. Therefore, the CPC was extracted at only the CCSD(T) level (R e ¼ A, D e ¼ 0.22 ev,! e ¼ 10 cm 1 ). Since it is known that the CPC tends to overestimate the basis set superposition error (BSSE), we believe the true results to be between the estimates with and without CPC. The average of these values, as well as error bars derived from their maximum difference, that is also taking into account the difference between MRCI þ Q and CCSD(T) results, led us to our final scalar-relativistic results for the Table 2. Bond lengths R e (A ), vibrational constants! e (cm 1 ), and binding energies D e (ev) for the possible ground states of Ce 2, Pr 2 and Gd 2 from small-core pseudopotential calculations. Metal State a R e (Å) D e (ev)! e (cm 1 ) Method b Ce a 1 þ g 2.582/ / /244 MRCI/MRCI þ Q c 2.604/ / /213 MRCI/MRCI þ Q d b 3 þ u 2.582/ / /216 CCSD/CCSD(T) c 2.592/ / /210 CPC, CCSD/CCSD(T) c 2.600/ / /198 CCSD/CCSD(T) d 2.615/ / /188 CPC, CCSD/CCSD(T) d 2.582/ / /244 MRCI/MRCI þ Q c 2.604/ / /213 MRCI/MRCI þ Q d [20] [17] expt. Pr c 5 þ g 2.524/ / /215 CCSD/CCSD(T) c d 5 þ g 2.541/ / /243 MRCI/MRCI þ Q c 2.563/ / /213 MRCI/MRCI þ Q d [20] [17] expt. Gd e 19 g 2.868/ / /145 CCSD/CCSD(T) c 2.886/ / /145 CPC, CCSD/CCSD(T) c 2.842/ / /151 CCSD/CCSD(T) d 2.878/ / /147 CPC, CCSD/CCSD(T) d [20] [18] expt. a Leading (reference) configuration a, 4f 1 4f 1 s 2 g p4 u ; b, 4f1 ðj 1 g Þ4f1 ðj 1 u Þs2 g p4 u ; c, 4f1 ðd 1 g Þ4f1 ðd 1 u Þ4f1 ðd 1 g Þ4f1 ðd 1 u Þs2 g p4 u ; d, 4f2 4f 2 s 2 g p4 u (no leading configuration for 4f occupation); e, 4f 7 4f 7 s 2 g s1 u s1 g p2 u (all 4f orbitals singly occupied). b CPC, counterpoise correction of the basis set superposition error. MRCI þ Q: MRCI with cluster correction of Siegbahn. c Standard basis set (14s13p10d8f6g)/[6s6p5d4f3g]; 4s, 4p frozen in MRCI and CCSD(T). d Standard basis set (14s13p10d8f6g)/[6s6p5d4f3g] augmented by a diffuse (3s3p3d3f3g) set, 4s, 4p, 4d frozen in MRCI; 4s, 4p frozen in CCSD(T).

7 Electronic structure of lanthanide dimers P þ g =3 P þ u ground state of Ce 2 (R e ¼ A, D e ¼ ev,! e ¼ cm 1 ). For Pr 2 4f 2 4f 2 s 2 g p4 u 5 þ g (at 0 cm 1 ), 5 u (at 0cm 1 ) and 5 10 g (at 20 cm 1 ) with virtually identical spectroscopic constants were obtained as ground state candidates. State average CASSCF calculations yield the lowest triplet and singlet states at term energies of 300 cm 1 and 400 cm 1, respectively. This different behaviour for Pr 2 may be attributed to the increasing number of 4f electrons (four 4f electrons for Pr 2 and two 4f electrons for Ce 2 ), and the concomitant increased multireference character of the wavefunction. Whereas the two unpaired electrons in Ce 2 are essentially localized in 4f j orbitals and direct coupling is not mediated by other occupied shells of j symmetry, in Pr 2 the four unpaired electrons are in 4f d, 4fj orbitals and the direct coupling is possible via the occupied valence shells having partly the same symmetry. The calculated binding energy of Pr 2 obtained at the MRCI level with the cluster correction of Siegbahn (MRCI þ Q) and extended basis sets (D e ¼ ev) is in excellent agreement with the available experimental result (D e ¼ ev [20]); however, similar to La 2 and Ce 2 the vibrational frequency (213 cm 1 )isby31cm 1 lower than the recent experimental value ( cm 1 [17]) obtained by Raman spectroscopy in an Ar matrix. Since CCSD(T) calculations are not appropriate for Pr 2, we cannot derive estimates for the CPC in a similar way as for Ce Gd 2 For Gd 2 a ground state configuration with 18 unpaired electrons ð4f 7 4f 7 s 2 g s1 u s1 g p2 uþ was theoretically predicted [6] and later confirmed by experiment [38]. The spectroscopic constants (R e ¼ A, D e ¼ ev,! e ¼ 149 2cm 1 ) calculated with an extended basis set at the CCSD(T) level are in good agreement with the available experimental data (D e ¼ ev,! e ¼ cm 1 [18, 20]). Again the error bars result from calculations with and without CPC of the BSSE Spin-orbit effects Spin-orbit effects cannot be neglected for atoms as heavy as Ce, Pr and Gd. A simple estimate based on the experimental fine structure splittings of the ground states of Pr and Gd [32] would yield a maximum bond destabilization of 0.60 ev for Pr 2 and 0.22 ev for Gd 2. First-order spin-orbit effects vanish for the molecular ground states of symmetry as well as for the Ce 1 G ground state and one might be tempted to neglect spinorbit corrections for these systems. However, owing to the high density of states, considerable spin-orbitinduced configuration interaction can occur. In addition, the 4f shells in molecules are to a good approximation spherical, that is atomic-like, and spinorbit effects may not be significantly quenched. In order to estimate spin-orbit effects on the dissociation energies, we therefore performed atomic/molecular spin-orbit configuration interaction calculations in the basis of the energetically lowest LS/ CASSCF states and using the pseudopotential spin-orbit terms. In the case of Ce 2 in the 4f 1 4f 1 s 2 g p4 u ground configuration, we included all low-lying singlet and triplet states that correspond to two unpaired electrons being in different atom-centred 4f shells. At the stateaveraged CASSCF level a total of 56 states with term energies up to 0.17 ev splits into 111 levels with term energies up to 0.62 ev. The 0 þ g ground state has contributions of 40% 1 þ g, 40% 3 g and 20% 3 g. The 0 u state is degenerate to 0þ g within 1 cm 1 and is composed of 40% 1 u, 40% 3 þ u and 20% 3 u. A 5 u state at a term energy of 2cm 1 is constituted of 80% 3 6 u, 10% 3 5 u and 10% 1 5 u. All other states have term energies above 300 cm 1. The molecular energy lowering by spin-orbit interaction was calculated to be 0.27 ev. For Ce in the 4f 1 5d 1 6s 2 ground state configuration we considered the 1 G, 3 F and 3 H states. The 1 G 4 ground level is found to be 0.07 ev below the 1 G state leading to a total spin-orbit stabilization of Ce 2 of 0.13 ev. The corrected binding energy is therefore ev (table 3). For Pr 2 in the 4f 2 4f 2 s 2 g p4 u ground state configuration, too many low-lying states arise and a physical choice as for Ce 2 was not possible. Therefore only the 125 lowest quintet, triplet and singlet states with term energies below 1.1 ev were selected. The molecular energy lowering by spin-orbit effects was calculated to be 0.42 ev for the 5 10 g state. The 4 I 9/2 level of Pr is calculated to be 0.26 ev below the 4 I state. The total spin-orbit destabilization of Pr 2 derived from these values is 0.10 ev and the corrected binding energy is ev (table 3). Owing to the large number of unpaired electrons we were not able to derive spin-orbit corrections for Gd 2. We therefore base our estimate of the spin-orbit corrected binding energy of ev (table 3) on the above-mentioned experimental atomicfine structure splitting. Owing to the very large number of low-lying electronic states the prediction of the correct ground state for lanthanide dimers with open 4f shells becomes somewhat uncertain. Nevertheless we assume that the states found by us are, if not the true ground states, quite low in energy and that owing to the validity of the superconfiguration model [31] their molecular constants are close to those of the true ground states.

8 1974 X. Cao and M. Dolg Table 3. Bond lengths R e (A ), vibrational constants! e (cm 1 ), and binding energies D e (ev) for selected lanthanide dimers. Metal State R e (Å) a D e (ev) b! e (cm 1 ) c Ref. La s 2 g p4 u, 1 þ g [15] expt. Ce 4f 1 4f 1 s 2 g p4 u, 3 þ u this work expt. Pr 4f 2 4f 2 s 2 g p4 u, 5 10 g this work Gd 4f 7 4f 7 s 2 g s1 u s1 g p2 u, 19 g this work expt. Lu 4f 14 4f 14 s 2 g s2 u p2 u, 3 g [15] expt. a The estimated experimental bond lengths for La 2 are from [21]. b Experimental values are from [20]. Spin-orbit corrections of 0.16 ev (La 2 ), þ 0.13 ev (Ce 2 ), 0.10 ev (Pr 2 ), 0.22 ev (Gd 2 ) and 0.30 ev (Lu 2 ) were added to the scalar-relativistic results. c The experimental values for La 2,Ce 2,Pr 2,Gd 2 and Lu 2 are from [16], [17], [17], [18] and [19], respectively. Table 4. Mulliken population analysis of the CASSCF wavefunction of La 2,Ce 2,Pr 2,Gd 2 and Lu 2 in their ground states. The virtually integral 4f occupation (La 2 0, Ce 2 2, Pr 2 4, Gd 2 14, Lu 2 28) is not included. Contributions of atomic natural orbital basis functions corresponding to the atomic valence orbitals (La, Ce, Pr: 6s, 6p, 5d, 4f; Gd: 6s, 7s, 6p, 5d, 4f; Lu: 6s, 6p, 5d, 5f) are given in parentheses (in %). s (6s,7s) p (6p) d (5d) f (4f, 5f) La 2 s g 95 (93) 5 (5) p u 1 (1) 96 (95) 3 (2) Ce 2 s g 95 (93) 5 (5) p u 2 (1) 95 (94) 3 (2) Pr 2 s g 95 (91) p u 2 (2) 94 (93) 4 (3) Gd 2 s g 100 (99) s g 5 (1,3) 4 (4) 91 (90) s u 68 (64) 23 (15) 9 (8) p u 14 (11) 86 (82) Lu 2 s g 96 (96) 1 (1) 3 (3) s u 81 (80) 17 (10) 2 (2) p u 31 (22) 69 (67) According to our results the ground states of homonuclear dimers composed of lanthanide elements with ground state valence subconfiguration 5d 1 6s 2 (6 s 2 for Pr), are most likely La 2 1 þ g s2 g p4 u, Ce 2 1 þ g, 1 u,3 g, 3 þ u, 1 6 g, 3 6 u 4f 1 4f 1 s 2 g p4 u, Pr 2 5 þ g, 5 u, 5 10 g 4f 2 4f 2 s 2 g p4 u, Gd 2 19 g 4f7 4f 7 s 2 g s1 u s1 g p2 u and Lu 2 3 g 4f14 4f 14 s 2 g s2 u p2 u. Our best estimates of their spectroscopic constants and available experimental data are listed in table 3. For La 2,Ce 2 and Pr 2 the analysis of the valence orbitals for the ground state exhibits a weak contribution of the 4f shell to chemical bonding (table 4). Therefore, the 4f electrons should be treated as valence electrons in highly accurate calculations. For Lu 2, as expected in the introduction, relatively large contributions of the 6p shell to bonding are found (table 4). Noting their binding energies, it is concluded that La 2 and Ce 2 exhibit a stronger bonding than Pr 2,Gd 2 and Lu Large-core pseudopotential calculations Our large-core pseudopotential results summarized in table 5 are in quite good agreement with the best values from small-core pseudopotential calculations listed in table 3. Whereas for La 2 small differences are observed in R e and! e, the agreement is almost perfect for Lu 2 owing to the inert 4f 14 subshell. The large-core value for D e of Ce 2 is better than the small-core result, owing

9 Table 5. Electronic structure of lanthanide dimers 1975 Bond lengths R e (A ), vibrational constants! e (cm 1 ), and binding energies D e (ev) for selected lanthanide dimers in their ground state superconfiguration from large-core pseudopotential CCSD(T) calculations. R e (A ) D e (ev) D e (ev) a! e (cm 1 ) La / / /206 Ce / / /205 Pr / / /198 Gd / / /136 Lu / / /120 a SO-corrections of 0.16, 0.16, 0.20 and 0.30 ev were added to the scalar-relativistic CCSD(T) results for La 2,Ce 2, Gd 2 and Lu 2. mainly to the absence of technical problems caused by the open 4f shells in the latter case. For Pr 2 D e can only be estimated since Pr and Pr 2 have different 4f occupations in their ground states. A reasonable result is obtained when the energy difference with respect to the atoms in the (4f 2 )5d 1 6s 2 2 D valence substate is calculated and the result is reduced by the experimental energy difference between the lowest levels of the 4f 3 6s 2 ground and 4f 2 5d 1 6s 2 excited configurations. As observed in previous work, relatively large differences occur for Gd 2 [7] Matrix effects The theoretical results for La 2,Ce 2,Pr 2,Gd 2 and Lu 2 summarized in table 3 are in reasonable overall agreement with the available experimental data. A notable exception is the! e values of La 2,Ce 2 and Pr 2. In order to get some idea of possible matrix effects in the experimental work (Raman spectroscopy in the Ar matrix [16, 19]), we investigated a simple model system. A linear complex between a single Ar atom and the lanthanide dimers Ln 2 was geometry-optimized at the CCSD(T) level using large-core pseudopotentials for Ln [5, 25] as well as Ar [39]. A (6s6p3d)/[4s4p3d] valence basis set was chosen for Ar [26]. Compared to free La 2 (206 cm 1 ), Ce 2 (205 cm 1 ), Pr 2 (198 cm 1 ), Gd 2 (136 cm 1 ), and Lu 2 (120 cm 1 ) we found substantially higher frequencies (228 cm 1 for La 2, 223 cm 1 for Ce 2 cm 1, and 216 cm 1 for Pr 2 ) in the Ar Ln 2 complexes (Ln ¼ La, Ce, Pr). In contrast to this, the Ar Gd 2 and Ar Lu 2 complexes exhibit vibrational frequencies (136 cm 1 for Gd 2 and 123 cm 1 for Lu 2 ) which are almost the same as for the free Gd 2 and Lu 2 systems. Matrix shifts of vibrational frequencies with respect to gas phase values are typically of the order of 0 3%, but a few exceptions exist (XeF gas phase 204 cm 1, Ne matrix 227 cm 1 ) [40, 41]. As has been noted by Lombardi and co-workers [2, 16] the force constants of La 2,Ce 2 and Pr 2 derived from Raman spectroscopy in the Ar matrix are anomalously large. Although our simple model does not fully explain the 50 cm 1 disagreement in! e for La 2 and Ce 2, it indicates that La 2, Ce 2 and Pr 2 could be significantly affected by Ar-matrix effects, whereas Gd 2 and Lu 2 behave quite normally. We attribute this different behaviour mainly to the absence/ presence of an occupied s u orbital (strongly polarized away from the bond) in (La 2, Ce 2, Pr 2 )/(Gd 2, Lu 2 ) (table 4). Whereas the Pauli repulsion between Ar 3p 6 and Lu 2 s 2 u (Gd 2 s 1 u ) allows only for a very weak interaction in Ar Lu 2 (distance 4.9 A, force constant au) and Ar Gd 2 (distance 5.09 A, force constant au), the electron deficiency owing to the unoccupied Ln 2 (Ln ¼ La, Ce, Pr) s u leads to a more stable Ar Ln 2 complex. The Ar Lu 2 distance and force constant are 3.3 A,610 3 au for Ar La 2 and 3.3 A, au for Ar Ln 2 (Ln ¼ Ce, Pr), respectively. A search for a possible change of the La 2 ground state under the influence of the Ar-matrix model was not successful. Further experimental work, that is measurements in a less polarizable Ne matrix, could help us to understand the high! e of La 2,Ce 2, and Pr Conclusion Extended Gaussian atomicnatural orbital valence basis sets and relativistic energy-consistent small-core lanthanide pseudopotentials have been used to establish the ground states for the La 2 ð 1 þ g s2 g p4 u Þ, Ce 2ð 1 þ g, 1 u, 3 g, 3 þ u, 1 6 g, 3 6 u 4f 1 4f 1 s 2 g p4 u Þ, Pr 2ð 5 þ g, 5 u 4f2 4f 2 s 2 g p4 u Þ, Gd 2ð 19 g 4f7 4f 7 s 2 g s1 u s1 g p2 u Þ and Lu 2 ð 3 g 4f14 4f 14 s 2 g s2 u p2 uþ dimers and to derive their spectroscopic constants. The molecular parameters (La 2 : R e ¼ A, D e ¼ ev,! e ¼ cm 1 ; Ce 2 : R e ¼ A, D e ¼ ev,! e ¼ cm 1 ; Pr 2 : R e ¼ A, D e ¼ ev,! e ¼ 213 cm 1,Gd 2 : R e ¼ A, D e ¼ ev,! e ¼ 149 2cm 1 ;Lu 2 :R e ¼ A, D e ¼ ev,! e ¼ 123 1cm 1 ) derived from multireference configuration interaction and coupled cluster calculations, including corrections for the atomic fine structure and basis set superposition errors, are in reasonable agreement with experimental data

10 1976 Electronic structure of lanthanide dimers (La 2 : D e ¼ ev,! e ¼ cm 1 ; Ce 2 : D e ¼ ev,! e ¼ cm 1 ; Pr 2 : D e ¼ ev,! e ¼ cm 1 ; Gd 2 : D e ¼ ev,! e ¼ cm 1 ;Lu 2 : D e ¼ ev,! e ¼ cm 1 ), except for the vibrational frequencies of La 2,Ce 2, and Pr 2. Model calculations point to possible large positive matrix shifts of! e for La 2,Ce 2 and Pr 2, but normal behaviour for Gd 2 and Lu 2.Itis concluded that La 2 and Ce 2 exhibit stronger bonding than Pr 2,Gd 2 and Lu 2. Weak contributions to chemical bonding of 4f orbitals (about 3%) are found for La 2, Ce 2 and Pr 2, whereas no contributions are observed for Gd 2 and Lu 2. For the latter systems a relatively large contribution from the 6p orbital (23% for Gd 2, 32% for Lu 2 ) is detected. The authors are grateful to J.R. Lombardi for valuable comments concerning the experimental results. The financial support of Fonds der Chemischen Industrie is acknowledged. References [1] MORSE, M.D., 1986, Chem. Rev., 86, [2] LOMBARDI, J.R., and DAVIS, B., 2002, Chem. Rev., 102, [3] DOLG, M., 1998, Encyclopedia of Computational Chemistry (Chichester: Wiley), p [4] DOLG, M., and STOLL, H., 1996, Handbookon the Physics and Chemistry of Rare Earths, Vol. 22 (Amsterdam: Elsevier), p [5] WANG, Y., SCHAUTZ, F., FLAD, H.-J., and DOLG, M., 1999, J. chem. Phys., 103, [6] DOLG, M., STOLL, H., and PREUSS, H., 1992, THEOCHEM, 277, 239. [7] DOLG, M., LIU, W., and KALVODA, S., 2000, Int. J. quantum Chem., 76, 359. [8] WANG, Y., and DOLG, M., 1998, Theor. Chem. Acc., 100, 124. [9] CUNDARI, T.R., and STEVENS, W.J., 1991, Int. J. quantum Chem., 40, 829. [10] CUNDARI, T.R., SOMMERER, S.O., STROHECKER, L.A., and TIPPETT, L., 1995, J. chem. Phys., 103, [11] SEIJO, L., BARANDIARAN, Z., and HARGUINDEY, E., 2001, J. chem. Phys., 114, 118. [12] SAKAI, Y., MIYOSHI, E., and TATEWAKI, H., 1998, THEOCHEM, 451, 143. [13] CAO, X., and DOLG, M., 2001, J. chem. Phys., 115, [14] CAO, X., LIU, W., and DOLG, M., 2002, Science in China, B45, 91. [15] CAO, X., and DOLG, M., 2002, Theor. Chem. Acc., 108, 143. [16] LIU, Y., FANG, L., SHEN, X., and LOMBARDI, J.R., 2000, Chem. Phys., 262, 25. [17] SHEN, X., FANG, L., CHEN, X., and LOMBARDI, J.R., 2000, J. chem. Phys., 113, [18] CHEN, X., FANG, L., SHEN, X., and LOMBARDI, J.R., 2000, J. chem. Phys., 112, [19] FANG, L., CHEN, X., SHEN, X., and LOMBARDI, J.R., 2000, J. chem. Phys., 113, [20] CONNOR, J.A., 1986, Metal Clusters in Catalysis, Studies in Surface Science and Catalysis, Vol. 29 (Amsterdam: Elsevier). [21] VERHAEGEN, G., SMOES, S., DROWART, J., 1964, J. chem. Phys., 40, 239. [22] BASTUG, T., ERKOC, S., HIRATA, M., and TACHIMORI, S., 1999, Phys. Rev., A59, [23] YANG, C.L., ZHANG, Z.H., and REN, T.Q., 2002, J. chem. Phys., 116, [24] YANG, C.L., 2002, personal communication. [25] DOLG, M., STOLL, H., SAVIN, A., and PREUSS, H., 1989, Theor. Chim. Acta, 75, 173. [26] DOLG, M., STOLL, H., and PREUSS, H., 1993, Theor. Chim. Acta, 85, 441. [27] DOLG, M., STOLL, H., and PREUSS, H., 1989, J. chem. Phys., 90, [28] ROOS, B.O., TAYLOR, P.R., and SIEGBAHN, P.E.M., 1980, Chem. Phys., 48, 157. [29] ROOS, B.O., 1980, Int. J. quantum Chem., S14, 175. [30] ROOS, B.O., 1987, Advances in Chemical Physics; Ab Initio Methods in Quantum Chemistry II (Chichester: Wiley), p [31] FIELD, R.W., 1982, Ber. Bunsenges. Phys. Chem., 86, 771. [32] MARTIN, W.C., ZALUBAS, R., and HAGAN, L., 1978, Atomic Energy Levels The Rare Earth Elements, Natl. Bur. Stand. Ref. Data Ser. 60 (Washington, DC: US Department of Commerce). [33] WERNER, H.-J., KNOWLES, P.J., AMOS, R.D., BERNING, D.L., COOPER, D.L., DEEGAN, M.J.O., DOBBYN, A.J., ECKERT, F., HAMPEL, C., HETZER, G., LEININGER, T., LINDH, R., LIOYD, A.W., MEYER, W., MURA, M.E., NICKLASS, A., PALMIERI, P., PETERSON, K., PITZER, R.M., PULAY, P., RAUHUT, G., SCHU TZ, M., STOLL, H., STONE, A.J., and THORSTEINSSON, T., 2000, Molpro 2000, a package of ab initio electronic structure programs. [34] KNOWLES, P.J., and WERNER, H.-J., 1985, Chem. Phys. Lett., 115, [35] WERNER, H.-J., and KNOWLES, P.J., 1988, J. chem. Phys., 89, [36] KNOWLES, P.J., and WERNER, H.-J., 1988, Chem. Phys. Lett., 145, 514. [37] KNOWLES, P.J., and WERNER, H.J., 1992, Theor. Chim. Acta, 84, 95. [38] VAN ZEE, R.J., LI, S., and WELTNER, W., 1994, J. chem. Phys., 100, 994. [39] NICKLASS, A., DOLG, M., STOLL, H., and PREUSS, A., 1995, J. chem. Phys., 102, [40] JACOX, M.E., 1985, J. molec. Spectrosc., 113, 286. [41] JACOX, M.E., 1994, Chem. Phys., 189, 149.