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Homonuclear diatomic lanthanoid compounds: a Pseudopotential configuration interaction and correlation energy density functional study
Journalof Molecular Structure (Theo&m), 271 (1992) 23%249 Elsevier Science Publishers B.V., Amsterdam
239
Homonuclear diatomic lanthanoid compounds: a pseudopotential configuration interaction and correlation energy density functional study Michael Dolg, Hermann Stoll and Heinzwerner Preuss Znstitut fiir Theoretische Stuttgart 80 (Germany)
Chemie, Universitiit
Stuttgart,
Pfaffenwaldririg
55, W 7000
(Received 27 March 1992)
Abstract Quasi-relativistic energy-adjusted ab initio pseudopotentials modelling lanthanoid elements with fixed integral 4f occupation numbers have been used for the investigation of the properties of homonuclear rare earth dimers in their electronic ground state configuration. Valence correlation effects have been included by means of correlation energy density functionals and by configuration interaction calculations. The trend in the derived dissociation energies agrees well with available experimental data, whereas the calculated bond lengths and vibrational constants are predictions.
INTRODUCTION
The investigation of the d transition metal dimers is a formidable challenge for ab initio quantum chemistry owing to the large number of low lying electronic states arising from the multiple possible combinations of ground states and low lying excited states of the separated atoms [l-3]. Owing to the open d shell it is necessary to include for even higher angular momentum functions in the basis sets to account properly for electron correlation in multi configuration self-consistent field (MCSCF) and subsequent multi reference configuration interaction (MRCI) calculations. The inclusion of relativistic effects seems to be somewhat easier than the correlation treatment: in all-electron calculations on the dimers of the first row transition elements the mass-velocity and Darwin correction terms may be applied in first-order perturbation theory; in the case of second and third row transition elements the use of quasi-relativistic ab initio pseudoCorrespondence to: M. Dolg, Institut fur Theoretische Pfaffenwaldring 55, W 7000 Stuttgart 60, Germany.
Chemie, Universitat Stuttgart,
0166-1260/92/$05.00 0 1992 Elsevier Science Publishers
B.V. All rights reserved.
240
M. LWg et al.lJ. Mol. Struct. ~Th~o~~~)
277 (1992) 239-249
potentials [4,5] is usually superior, both with respect to accuracy and economy. Looking at the dimers of the f transition metals, e.g. the lanthanoid dimers considered in this paper, the situation with regard to ab initio electronic structure treatments seems to be even more complicated than ford transition elements: Eu, for example is probably a van der Waals molecule with 14 unpaired electrons, seven in each of the 4f shells localized on the two atoms [6]. Highly accurate quantum chemical ab initio calculations on dimers or even clusters of rare earth elements with open 4f shells at this time seem to be out of the question for routine investigations, and approximate but reliable methods are necessary. A simplified approach to the electronic molecular structure of the rare earth dimers (or even larger clusters) is possible with recently developed quasi-relativistic energy-adjusted pseudopotentials for the rare earth elements [7]. The pseudopotentials are based on the chemically well-known fact that the rare earth elements usually occur trivalent, sometimes also divalent (Eu, Yb) or tetravalent (Ce, Tb), in their compounds. In a one particle picture each valency corresponds to a specific fixed integral 4f occupation number at the rare earth centre in the molecules: trivalent rare earth atoms are characterized by a 4f” subconfiguration (n = O-14 for La to Lu), whereas the 4fnt1 and 4f”-’ subconfigurations correspond to the divalent and tetravalent situation respectively. All molecular electronic states with equal 4f subconfigurations on the lanthanoid centres and a specific valence subconfiguration are expected to have very similar rotational and vibrational constants and are considered to belong to a so-called superconfiguration [S]. We therefore adjusted quasi-relativistic pseudopotentials for each of the most common valencies of the rare earth atoms, keeping the 4f orbitals in the pseudopotential core, Our pseudopotentials and valence basis sets have already been successfully applied in quantum chemical investigations of a large number of lanthanoid compounds and were found to yield reliable average values for molecular constants of all states belonging to a superconfiguration, see, for example, refs. 9-15. In this paper we present ground state configuration assignments together with estimates of molecular constants for the homonuclear lanthanoid diatom&. With the exception of the dissociation energies, for which experimental values are known, our results are predictions. METHOD The pseudopotentials and valence basis sets used in this work have been published elsewhere [7]. For the lanthanoid elements only the 5s, 5p, 5d and 6s orbitals are explicitly treated in the valence space, whereas the chemically inactive 4f orbitals are included in the pseudopotential core. The quasi-relativistic energy-adjusted pseudopotentials are designed to model
lanthanoid atoms with a fixed integral 4f occupation number and a corresponding valency in a molecular environment. Within this approach, divalent and trivalent rare earth atoms are treated as 10. and ll-valenceelectron systems with (4f”-‘)6s2 and (4f”)5d’6s2 ground state configurations respectively. Energy-optimized (~s6p~~~~5s4p3d] Gaussian-type orbital (GTO) valence basis sets, for the quotation interaction calculations augmented by one f function, were applied. Valence correlation was accounted for by means of the correlation energy density functional of Vosko, Wilk and Nusair (VWN) [16], the corresponding self-interactioncorrected form by Stoll, Pavlidou and Preuss (SPP) [17], the gradientcorrected functionals by Perdew (P) [18] and by Hu and Langreth (HL) [19]. The self-consistent field (SCF) density was used to calculate the correlation energy. For the molecules with empty, half-filled or fully occupied 4f shell, i.e. La,, Eu,, Gdz, Yb, and Lu,, we also performed configuration interaction calculations including all single and double substitutions (CISD) frum the SCF reference configuration. All results were approximately corrected for size-consistency errors ( f SCC) using the Langhoff-Davidson formula [ZO]. In order to check the results for size-consistency errors we also performed calculations within the coupled electron-pair approximation (CEPA-1) for some cases. Dissociation energies have been calculated with respect to the separated atoms in the ground state at a distance of 1000 a.u. Because most of the molecules considered here are not bound with respect to the separated atoms at the SCF level we only list results from calculations in which electron correlation was included. Energetic separations between two su~rco~~rations with differing 4f su~on~~ations ~~~-~~ have been estimated by calculating the dissociation energy of each superconfiguration with respect to the atomic reference states with the same 4f subconfiguration (DJ4f”) and DQ(4fnt1)) and correcting twice the experimental atomic 4fn+‘6s2 + 4f”5d*6s2 excitation energy (TztOm)by these values, i.e. !l?lmer = D,(4f”+‘) e
+ 2 x T;tOm- Q(4f”)
The necessary atomic corrections are given in ref. 7. All calculations have been performed with the MOLPRO [21] and MELD [22] program systems,
Experimental information on the homonuclear lanthanoid dimers exists only for the dissociation energies which are known with error bars ranging from 10% (Ce,) to 100% (Yb,) [23]. Rotational and vibrational constants may be roughly estimated by a suitable scaling of values known for model molecules like Ca, (e.g. for Eu, and Yb,) or SC, (e.g. for La, and Gd,, but not for Lu, as discussed below) for the divalent or trivalent case respectively. In the following we pcill discuss the protokype systems La, and Lu, as well as Eu, and Yb, in more detail.
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M. BoZg et d/J. Mol. Struct. (Theochem) 277 (1992) 239-249
In principle, the situation for La, (empty 4f shell) and Lu, (closed 4f shell) appears to be similar to the one encountered for the light rare earth dimers SC, and Y,, although relativistic effects are expected to be more important. The atoms SC, Y, La and Lu have d’s’ 2D ground states and rather low lying d’s1 4F states from which a large number of molecular electronic states result. The co~esponding experimental excitation energies [24] averaged over spin-orbit component of SC (1.43 eV) and Y (1.36 eV) are considerably larger than for La (0.36 eV) but significantly lower than for Lu (2.33 eV). Although La and Lu are isovalent the term energies of their d2s1‘F states differ by 1.97 eV and a different electronic structure may be expected for La, and Lu,. This is mainly due to shell-structure and relativistic effects, whereas the correlation contributions are similar for both atoms. The non-relativistic Hartree-Fock (HF) all-electron excitation energy increases from La to Lu by 1.47eV. Besides this shell-structure effect, differential relativistic effects of 0.63 eV and 1.15 eV for La and Lu may be obtained from a comparison of the HF results with quasi-relativistic woodboring (WB) all-electron values [7]. The corresponding correlation contributions, derived as the difference between experimental data and quasirelativistic WB all-electron results, are 0.66 eV and 0.69eV for La and Lu respectively. The contribution of the s and d valence orbitals to the chemical bond in the homonuclear dimers depends on their relative availability for molecule formation. This may be characterized by the corresponding orbital energies and the (r)-expectation values which are listed in Table 1. In studies of transition metal dimers, Walch and Bauschlicher [I] discussed the ratio of the (r)-expectation values of the d and s valence orbitals as a measure for the relative importance of both orbitals in bonding in transition metal dimers. At the non-relativistic HF level we find Y, La and Lu to be rather similar (SC, 0.42; Y, 0.57; La, 0.56; Lu, 0.58), whereas in quasi-relativistic WB calculations, only Y and La are similar, but somewhat different from Lu (SC, 0.43; Y, 0.60; La, 0.61; Lu, 0.70). These values reflect the increased availability of the d orbitals for chemical bonding owing to their relativistic expansion and the corresponding contraction of the s orbitals. The importance of relativistic effects can also be seen from the orbital energies: at the non-relativistic level the s orbital energy is always higher than the d orbital energy; at the quasi-relativistic level this is only true for SC, Y and La, whereas for Lu the energetic ordering is reversed. The relativistic destabilization of the d orbitals opposes their increasing availability for the formation of a chemical bond. The predicition of the correct molecular ground state configurations for the homonuclear rare earth dinners on the basis of atomic data therefore seems to be rather difficult. Several theoretical calculations on SC, partly led to controversial ground state assi~en~ in the past, see, for example, refs++25and 26. Extensive
M. Dolg et al@. Mol. Stmct. tT~eoche~~ 277 (2992) 239-249
243
TABLE 1 The (r)-expectation values (a.u.) and negative orbit& energies -E (a.u.) for the d and s valence orbital8 of the rare earth metals SC, Y, La and Lu in their d’s’ eD ground state from non-relativistic (HF) and quasi-relativistic (WB) all-electron calculations, and experimental term energies T, (eV) of the excited d%’ state with respect to the d’s’ ground state Metal
(r),
-&d
-E*
T,
Exp.”
HF
WB
HF
WB
HF
WB
HF
WB
SC
1.675
1.691
3.966
3.938
0.344
0.336
0.210
0.211
1.43
Y
2.345
2.513
4.300
4.211
0.250
0.231
0.196
0.201
1.36
La
2.768
2.890
4.931
4.722
0.269
0.236
0.170
0.180
0.36
Lu
2.486
2.738
4.259
3.993
0.243
0.188
0.199
0.222
x2.33
“Ref. 24.
complete active space self-consistent field and CISD calculations [26] yield a +~G~Tc%5; ground state in agreement with experimental evidence, e.g. ESR data [27]. In similar theoretical investigations the same ground state assignment was made for Y, [l]. However, since no ESR spectrum could be obtained, Y, might also have a singlet ground state. No theoretical or TABLE 2 Bond lengths R, (A), vibrational constants OJ,(cm-‘) and term energies T, (eV) for low lying electronic states of La, from CISD + SCC calculations State
R, (A)
w, (cm-‘)
T, (eV)
o$:a~?r~ %; tin’ 1X.F6A, &c$;
3.247 3.033 2.830
130 145 167
0.00 0.21 0.11
o2aZn2 B g (1TXrPa’ n2S’ b ” gU”g 4,2,,2,%, dLaZti gug ‘Z:+ k? 02P?? @UU 5x+ g (r’(r’n~6~ B;BUG 7x+ g o2,a~is;~~3A, u$;o;s; 62; c?a’dlcP 3Au gugg 02026= ggg $?-g
3.073 3.371 3.580 3.890 3.095 3.157 3.995 3.862 3.632 3.363
167 125 115 96 118 112 88 107 108 123
0.53 0.64 0.73 1.00 1.03 1.14 1.46 1.51 1.54 2.52
M. Dolg et al./J. Mol. Struct. (Theochem) 277 (1992) 23%249
244 TABLE 3
Bond lengths, R, (A), vibrational constants, o, (cm-l), and dissociation energies, D, (eV), from CISD + SCC calculations (experimental values for dissociation energies are given in parentheses) Metal
State”
R. (A)
0, (cm-‘)
;
3.241 2.830
130 167
1.17 1.06
:
3.032 3.896
145 96
0.53 0.17
e
5.390
16
0.04
(0.30 * 0.17)
a
3.040 3.683
133 90
0.52 0.06
(1.78 f 0.34)
C
e
5.308
13
0.05
(0.17 f 0.18)
; :
2.938 2.644 2.979 3.786
110 172 134 74
0.36 - 1.50 - 0.55 0.16
(1.43 + 0.34)
La
Eu Gd Yb Lu
aa, (4f”)(4f”)a:o:a~n~ ‘2;; e, (4P”) 4P+‘)+~ ‘Xl. bRef. 23.
b, (4f”)(4f”)tix:
D, (eV)
Ex~.~
(2.52 + 0.22)
‘Z: ; c, (4P)(4f”)a~a~a~ ‘El; d, (4f”)(4f”)a~u:n~6~ 5AU;
experimental ground state assignments exist for La, to Lu, to our knowledge. In CISD + SCC calculations on a large number of low lying electronic states of La, we also obtained a o~a~o~n~ 5E; ground state (La 5d0°.515dn0,88 6sa’.416pa0.086p710.‘2) in agreement with the experimental and theoretical evidence for SC, and Y,. Our results are summarized in Table 2. As for Y,, the ground state [29]. Since at the CISD + SCC level a crzrct‘Ci state (La 6s~0.g76pa0.02 5daO.O’ 6px”.‘05dx”.%) is located only 0.11 eV above the G”,at 0: xx 5X- state it is also a possible candidate for a ground state. In similar calculations on Lu, all states with significant contribution of 5d atomic orbitals to occupied molecular orbitals were found to be rather high in energy owing to the relativistic destabilization of the 5d orbitals with respect to the 6s orbital, (see Table 3). The Lu, ground state is found to be the <~J:G”, ‘Z:,’ state (Lu 5d0°.456su1.826pcr0.73) which has a term energy of l.OOeV for La,. We suspect that the rather large difference in the experimental dissociation energies of La, (2.50 eV [24]) and Lu, (1.43 eV [24]) is due to the differing molecular ground states, i.e. a double or a triple bond for La, and a single bond for Lu, in terms of simple molecular orbital theory. The bond lengths and vibrational frequencies listed in Tables 2 and 3 are predictions. Our CISD + SCC results for Lu, (R, = 3.786 A; w, = 74 cm-‘,
245
M. Dolg et aE.fJ. Mol. Struct. (Theochem) 277 (1992) 239-249 TABLE 4 Bond lengths, R, (A), from SCF and correlation experimental values are not known
energy density functional calculations;
Metal
State”
VWNb
SPPb
Pb
HLb
La
a C
3.293 3.892
3.325 3.933
3.245 3.866
3.196 3.832
Ce
a
3.240
3.282
3.203
3.176
Pr
a e
3.198 5.403
3.239 5.935
3.164 5.242
3.132 4.847
Nd
a e
3.159 5.408
3.204 5.949
3.131 5.220
3.106 4.835
Pm
e
5.398
5.944
5.152
4.819
Sm
e
5.409
5.951
5.073
4.795
Eu
e
5.394
5.928
5.058
4.783
Gd
a c
3.060 3.717
3.099 3.769
3.037 3.656
3.015 3.550
Tb
a c
3.044 3.671
3.084 3.751
3.020 3.631
2.993 3.526
BY
e
5.411
6.006
4.923
4.747
Ho
e
5.411
6.002
4.913
4.727
Er
e
5.407
5.991
4.904
4.729
Tm
e
5.434
5.994
4.912
4.632
Yb
e
5.412
5.994
4.879
4.536
LU
a c
2.990 3.779
3.050 3.816
2.963 3.755
2.917 3.636
“See footnote a to Table 3. bSee text.
D, = 0.55 eV) are in good agreement with corresponding CEPA-1 values (R, = 3.797A; W, = 71 cm-*; D, = 0.69eV), indicating a relatively small influence of size-consistency errors in the CISD + SCC values. Unfortunately, in the case of the La, r$~~~~x~ ?Z- ground state, the CEPA-1 calculations did not converge. For the very low lying 0:~: ‘Z+ state the CEPA-1 results (R, = 2.869A; w, = 143cm-‘; D, = l.OZeV) are again close to our CISD + SCC values (R, = 2.830 A; o, = 167 cm-‘; 0, = 1.06 eV). As an alternative to the CISD treatment of correlation effects, we applied various correlation energy density functionals using the SCF density. This approach proved to be competitive to CISD calculations in previous cal-
M. Ddg et cd/J. Mol. Struct. (Theo&em)
246
277 (1992) 239-249
TABLE 5 Binding energies, D, (eV), from SCF and correlation energy density functional calculations in comparison to experimental vahes Metal
State”
VWNb
SPPb
La
a c
0.57 - 0.06
1.12 0.06
1.63 0.49
2.52* 0.22
-0.17
Ce
a
0.49
0.26
1.04
1.54
2.47f 0.22
Pr
a e
- 0.71 0.04
-0.93 0.02
-0.16 0.11
0.35 0.32
1.31+ 0.30
Nd
a e
- 1.39 0.33
- 1.61 0.01
-0.83 0.11
-0.33 0.31
0.83+ 0.30
Pm
e
0.03
0.01
0.10
0.31
Sm
e
0.03
0.01
0.10
0.31
0.52+-0.21
EU
e
0.03
0.01
0.10
0.30
0.30+ 0.17
a
- 0.09 - 0.29
-0.28 -0.30
0.49 -0.13
0.98 0.28
1.78F 0.34
- 0.22 - 0.34
-0.40 -0.33
0.36 -0.16
0.85 0.24
1.32+ 0.26
C
DY
e
0.02
0.01
0.09
0.28
0.69& 0.30
Ho
e
0.02
0.01
0.08
0.27
0.82& 0.18
Er
e
0.02
0.01
0.08
0.27
0.74t 0.30
Tm
0.02
0.00
0.08
0.27
0.52rt0.18
Yb
0.02
0.00
0.08
0.27
0.17+ 0.18
Lu
- 0.35 0.26
-0.45 0.37
0.26 0.47
0.69 0.78
1.43F 0.34
Gd
C
Tb
a
0.35
Pb
HLb
Exp.”
BSee footnote a to Table 3. bSee text. “Ref. 23.
culations for atomic excitation and ionization energies [‘7]. The calculated bond lengths and dissociation energies for the rare earth dimers are summarized in Tables 4 and 5 respectively. The density functional results usually bracket the CISD + SCC values. The numbers obtained with the gradient-corrected form given by Perdew [18] are in the closest agreement with the CISD results. We do not list vibrational constants (which are close to the corresponding CISD values of Table 3) because errors due to small inaccuracies in the numerical evaluation of the correlation energies are certainly larger than the slight variations in these values from La, to Lu,.
M. Do& et tsl.jJ.Mol. Struct.(Theochem)277 (1992) 239-243
247
Whereas the calculations on the dimers of the trivalent lanthanoid atoms are complicated by the large number of low lying electronic states, the corresponding calculations on the dimers of the divalent lanthanoid atoms have to account for the rather weak van der Waals bond in the (4f~~l)(4~+l)~~~ ground state configuration. The derived CISD + SCC dissociation energies of 0.04 eV and 0.05 eV for Eu, (Eu 6s01.956p(r0.046daa.or.) and Yb, (Yb 6s01.976poo.026dao.o’) are in reasonable agreement with the experimental values of 0.30eV ( + 0.17 eV) [S] and 0.17 eV ( + 0.18eV) [30] respectively. Because the potential curves have extremely shallow minima the derived bond lengths (Eu, 5.390 A; Yb, 5.308 A) and vibrational frequencies (Eu, 16 cm-‘; Yb, 14 cm-‘) might be affected by errors. Nevertheless the derived vibrational constants agree reasonably with estimates used in experimental work (Eu, 35 cm-“; Yb, 22 cm-‘) [31]. In corresponding CEPA1 calculations we obtained results very similar to the CISD + SCC values: L), = 0.06 eV, R, = 5.267 A, w, = 16 cm-’ for Eu,; De = 0.05 eV, R, = 5.203 A, w, = 14cm-’ for Yb,. The correlation energy density functional results for the van der Waals bonded superconfigurations of Prz to Eu, and Dy, to Yb, again bracket the CISD + SCC values; however, owing to the flat potential curves, the range of bond lengths is considerably larger than for the dimers with formally trivalent lanthanoid metals. The gradient-corrected functional forms yield the shortest bond lengths and highest dissociation energies. The latter are again in reasonable agreement with experimental data. It should be noted, however, that local density functional methods are known to overestimate bonding due to van der Waals type interactions. From our CISD and correlation energy density functional calculations we predict for La,, Ce,, (probably) Pr, , Gd, and Tb, a (4~)(4~)~~~~~~~~ and for Lu, a (4f~)~4~)~~~~~~ ground state superconfiguration (n = 0, 1, 2, 7, 8, 14 for La, Ce, Pr, Gd, Tb and Lu respectively). For Eu, and Yb, the van der Waals type (4fn+l)(4fnt1) .i,i superconfiguration should give rise to the molecular ground state. The assignment of the ground state superconfiguration for the other diatomics is less easy. The experimental dissociation energies of Nd,, Smtf Dy,, Ho,, Er, and Tmz range from 0.52 to 0.83 eV and appear to be somewhat too large for van der Waals type bonding, which was found to yield the highest dissociation energies for these molecules in our calculations. Because supercon~gurations with 4f” subcon~gurations on both atoms are considerably higher in energy, a mixed-valence situation arising from a superposition of 4p and 4f”+’ subconfigurations cannot be ruled out. Our study omitted spin-orbit effects and did not account for possible mixed-valence situations. In order to treat both effects in an ab initio level one would have to use pseudopotentials that include the 4f shell explicitly in the calculations, e.g. those of ref. 32. A smaller pseudopotential core in
248
M. Dolg et al@. Mol. Struct. (Theochem) 277 (1992) 239-249
order to avoid frozen core errors, larger basis sets including g functions to polarize and correlate the 4f shell, inclusion of spin-orbit operators and long and computationally demanding CI expansions would be necessary to obtain reliable results. According to our experience from calculations on diatomics with a single lanthanoid centre with both types of pseudopotential[11,12,15], we suppose that the effort to treat a single lanthanoid dimer with an open 4f shell would be considerably higher than it was for the whole study presented here. CONCLUSION
Quasi-relativistic energy-adjusted pseudopotentials for lanthanoid elements with a fixed valency have been used in quantum chemical calculations to derive for the first-time theoretical estimates for molecular constants of the homonuclear dimers of the 4f elements. Possible electronic ground state superconfigurations have been discussed. The derived dissociation energies are in satisfactory agreement with experimental values whereas other parameters and the assignments of ground state superconfigurations are predictions, ACKNOWLEDGEMENTS
The authors thank H.-J. Werner for providing the program system and A. Savin for the correlation energy density functional code.
MOLPRO. Thanks are also due to B. ~iehlich
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2
3 4 5 6 7 8
9 10
11
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239
Homonuclear diatomic lanthanoid compounds: a pseudopotential configuration interaction and correlation energy density functional study Michael Dolg, Hermann Stoll and Heinzwerner Preuss Znstitut fiir Theoretische Stuttgart 80 (Germany)
Chemie, Universitiit
Stuttgart,
Pfaffenwaldririg
55, W 7000
(Received 27 March 1992)
Abstract Quasi-relativistic energy-adjusted ab initio pseudopotentials modelling lanthanoid elements with fixed integral 4f occupation numbers have been used for the investigation of the properties of homonuclear rare earth dimers in their electronic ground state configuration. Valence correlation effects have been included by means of correlation energy density functionals and by configuration interaction calculations. The trend in the derived dissociation energies agrees well with available experimental data, whereas the calculated bond lengths and vibrational constants are predictions.
INTRODUCTION
The investigation of the d transition metal dimers is a formidable challenge for ab initio quantum chemistry owing to the large number of low lying electronic states arising from the multiple possible combinations of ground states and low lying excited states of the separated atoms [l-3]. Owing to the open d shell it is necessary to include for even higher angular momentum functions in the basis sets to account properly for electron correlation in multi configuration self-consistent field (MCSCF) and subsequent multi reference configuration interaction (MRCI) calculations. The inclusion of relativistic effects seems to be somewhat easier than the correlation treatment: in all-electron calculations on the dimers of the first row transition elements the mass-velocity and Darwin correction terms may be applied in first-order perturbation theory; in the case of second and third row transition elements the use of quasi-relativistic ab initio pseudoCorrespondence to: M. Dolg, Institut fur Theoretische Pfaffenwaldring 55, W 7000 Stuttgart 60, Germany.
Chemie, Universitat Stuttgart,
0166-1260/92/$05.00 0 1992 Elsevier Science Publishers
B.V. All rights reserved.
240
M. LWg et al.lJ. Mol. Struct. ~Th~o~~~)
277 (1992) 239-249
potentials [4,5] is usually superior, both with respect to accuracy and economy. Looking at the dimers of the f transition metals, e.g. the lanthanoid dimers considered in this paper, the situation with regard to ab initio electronic structure treatments seems to be even more complicated than ford transition elements: Eu, for example is probably a van der Waals molecule with 14 unpaired electrons, seven in each of the 4f shells localized on the two atoms [6]. Highly accurate quantum chemical ab initio calculations on dimers or even clusters of rare earth elements with open 4f shells at this time seem to be out of the question for routine investigations, and approximate but reliable methods are necessary. A simplified approach to the electronic molecular structure of the rare earth dimers (or even larger clusters) is possible with recently developed quasi-relativistic energy-adjusted pseudopotentials for the rare earth elements [7]. The pseudopotentials are based on the chemically well-known fact that the rare earth elements usually occur trivalent, sometimes also divalent (Eu, Yb) or tetravalent (Ce, Tb), in their compounds. In a one particle picture each valency corresponds to a specific fixed integral 4f occupation number at the rare earth centre in the molecules: trivalent rare earth atoms are characterized by a 4f” subconfiguration (n = O-14 for La to Lu), whereas the 4fnt1 and 4f”-’ subconfigurations correspond to the divalent and tetravalent situation respectively. All molecular electronic states with equal 4f subconfigurations on the lanthanoid centres and a specific valence subconfiguration are expected to have very similar rotational and vibrational constants and are considered to belong to a so-called superconfiguration [S]. We therefore adjusted quasi-relativistic pseudopotentials for each of the most common valencies of the rare earth atoms, keeping the 4f orbitals in the pseudopotential core, Our pseudopotentials and valence basis sets have already been successfully applied in quantum chemical investigations of a large number of lanthanoid compounds and were found to yield reliable average values for molecular constants of all states belonging to a superconfiguration, see, for example, refs. 9-15. In this paper we present ground state configuration assignments together with estimates of molecular constants for the homonuclear lanthanoid diatom&. With the exception of the dissociation energies, for which experimental values are known, our results are predictions. METHOD The pseudopotentials and valence basis sets used in this work have been published elsewhere [7]. For the lanthanoid elements only the 5s, 5p, 5d and 6s orbitals are explicitly treated in the valence space, whereas the chemically inactive 4f orbitals are included in the pseudopotential core. The quasi-relativistic energy-adjusted pseudopotentials are designed to model
lanthanoid atoms with a fixed integral 4f occupation number and a corresponding valency in a molecular environment. Within this approach, divalent and trivalent rare earth atoms are treated as 10. and ll-valenceelectron systems with (4f”-‘)6s2 and (4f”)5d’6s2 ground state configurations respectively. Energy-optimized (~s6p~~~~5s4p3d] Gaussian-type orbital (GTO) valence basis sets, for the quotation interaction calculations augmented by one f function, were applied. Valence correlation was accounted for by means of the correlation energy density functional of Vosko, Wilk and Nusair (VWN) [16], the corresponding self-interactioncorrected form by Stoll, Pavlidou and Preuss (SPP) [17], the gradientcorrected functionals by Perdew (P) [18] and by Hu and Langreth (HL) [19]. The self-consistent field (SCF) density was used to calculate the correlation energy. For the molecules with empty, half-filled or fully occupied 4f shell, i.e. La,, Eu,, Gdz, Yb, and Lu,, we also performed configuration interaction calculations including all single and double substitutions (CISD) frum the SCF reference configuration. All results were approximately corrected for size-consistency errors ( f SCC) using the Langhoff-Davidson formula [ZO]. In order to check the results for size-consistency errors we also performed calculations within the coupled electron-pair approximation (CEPA-1) for some cases. Dissociation energies have been calculated with respect to the separated atoms in the ground state at a distance of 1000 a.u. Because most of the molecules considered here are not bound with respect to the separated atoms at the SCF level we only list results from calculations in which electron correlation was included. Energetic separations between two su~rco~~rations with differing 4f su~on~~ations ~~~-~~ have been estimated by calculating the dissociation energy of each superconfiguration with respect to the atomic reference states with the same 4f subconfiguration (DJ4f”) and DQ(4fnt1)) and correcting twice the experimental atomic 4fn+‘6s2 + 4f”5d*6s2 excitation energy (TztOm)by these values, i.e. !l?lmer = D,(4f”+‘) e
+ 2 x T;tOm- Q(4f”)
The necessary atomic corrections are given in ref. 7. All calculations have been performed with the MOLPRO [21] and MELD [22] program systems,
Experimental information on the homonuclear lanthanoid dimers exists only for the dissociation energies which are known with error bars ranging from 10% (Ce,) to 100% (Yb,) [23]. Rotational and vibrational constants may be roughly estimated by a suitable scaling of values known for model molecules like Ca, (e.g. for Eu, and Yb,) or SC, (e.g. for La, and Gd,, but not for Lu, as discussed below) for the divalent or trivalent case respectively. In the following we pcill discuss the protokype systems La, and Lu, as well as Eu, and Yb, in more detail.
242
M. BoZg et d/J. Mol. Struct. (Theochem) 277 (1992) 239-249
In principle, the situation for La, (empty 4f shell) and Lu, (closed 4f shell) appears to be similar to the one encountered for the light rare earth dimers SC, and Y,, although relativistic effects are expected to be more important. The atoms SC, Y, La and Lu have d’s’ 2D ground states and rather low lying d’s1 4F states from which a large number of molecular electronic states result. The co~esponding experimental excitation energies [24] averaged over spin-orbit component of SC (1.43 eV) and Y (1.36 eV) are considerably larger than for La (0.36 eV) but significantly lower than for Lu (2.33 eV). Although La and Lu are isovalent the term energies of their d2s1‘F states differ by 1.97 eV and a different electronic structure may be expected for La, and Lu,. This is mainly due to shell-structure and relativistic effects, whereas the correlation contributions are similar for both atoms. The non-relativistic Hartree-Fock (HF) all-electron excitation energy increases from La to Lu by 1.47eV. Besides this shell-structure effect, differential relativistic effects of 0.63 eV and 1.15 eV for La and Lu may be obtained from a comparison of the HF results with quasi-relativistic woodboring (WB) all-electron values [7]. The corresponding correlation contributions, derived as the difference between experimental data and quasirelativistic WB all-electron results, are 0.66 eV and 0.69eV for La and Lu respectively. The contribution of the s and d valence orbitals to the chemical bond in the homonuclear dimers depends on their relative availability for molecule formation. This may be characterized by the corresponding orbital energies and the (r)-expectation values which are listed in Table 1. In studies of transition metal dimers, Walch and Bauschlicher [I] discussed the ratio of the (r)-expectation values of the d and s valence orbitals as a measure for the relative importance of both orbitals in bonding in transition metal dimers. At the non-relativistic HF level we find Y, La and Lu to be rather similar (SC, 0.42; Y, 0.57; La, 0.56; Lu, 0.58), whereas in quasi-relativistic WB calculations, only Y and La are similar, but somewhat different from Lu (SC, 0.43; Y, 0.60; La, 0.61; Lu, 0.70). These values reflect the increased availability of the d orbitals for chemical bonding owing to their relativistic expansion and the corresponding contraction of the s orbitals. The importance of relativistic effects can also be seen from the orbital energies: at the non-relativistic level the s orbital energy is always higher than the d orbital energy; at the quasi-relativistic level this is only true for SC, Y and La, whereas for Lu the energetic ordering is reversed. The relativistic destabilization of the d orbitals opposes their increasing availability for the formation of a chemical bond. The predicition of the correct molecular ground state configurations for the homonuclear rare earth dinners on the basis of atomic data therefore seems to be rather difficult. Several theoretical calculations on SC, partly led to controversial ground state assi~en~ in the past, see, for example, refs++25and 26. Extensive
M. Dolg et al@. Mol. Stmct. tT~eoche~~ 277 (2992) 239-249
243
TABLE 1 The (r)-expectation values (a.u.) and negative orbit& energies -E (a.u.) for the d and s valence orbital8 of the rare earth metals SC, Y, La and Lu in their d’s’ eD ground state from non-relativistic (HF) and quasi-relativistic (WB) all-electron calculations, and experimental term energies T, (eV) of the excited d%’ state with respect to the d’s’ ground state Metal
(r),
-&d
-E*
T,
Exp.”
HF
WB
HF
WB
HF
WB
HF
WB
SC
1.675
1.691
3.966
3.938
0.344
0.336
0.210
0.211
1.43
Y
2.345
2.513
4.300
4.211
0.250
0.231
0.196
0.201
1.36
La
2.768
2.890
4.931
4.722
0.269
0.236
0.170
0.180
0.36
Lu
2.486
2.738
4.259
3.993
0.243
0.188
0.199
0.222
x2.33
“Ref. 24.
complete active space self-consistent field and CISD calculations [26] yield a +~G~Tc%5; ground state in agreement with experimental evidence, e.g. ESR data [27]. In similar theoretical investigations the same ground state assignment was made for Y, [l]. However, since no ESR spectrum could be obtained, Y, might also have a singlet ground state. No theoretical or TABLE 2 Bond lengths R, (A), vibrational constants OJ,(cm-‘) and term energies T, (eV) for low lying electronic states of La, from CISD + SCC calculations State
R, (A)
w, (cm-‘)
T, (eV)
o$:a~?r~ %; tin’ 1X.F6A, &c$;
3.247 3.033 2.830
130 145 167
0.00 0.21 0.11
o2aZn2 B g (1TXrPa’ n2S’ b ” gU”g 4,2,,2,%, dLaZti gug ‘Z:+ k? 02P?? @UU 5x+ g (r’(r’n~6~ B;BUG 7x+ g o2,a~is;~~3A, u$;o;s; 62; c?a’dlcP 3Au gugg 02026= ggg $?-g
3.073 3.371 3.580 3.890 3.095 3.157 3.995 3.862 3.632 3.363
167 125 115 96 118 112 88 107 108 123
0.53 0.64 0.73 1.00 1.03 1.14 1.46 1.51 1.54 2.52
M. Dolg et al./J. Mol. Struct. (Theochem) 277 (1992) 23%249
244 TABLE 3
Bond lengths, R, (A), vibrational constants, o, (cm-l), and dissociation energies, D, (eV), from CISD + SCC calculations (experimental values for dissociation energies are given in parentheses) Metal
State”
R. (A)
0, (cm-‘)
;
3.241 2.830
130 167
1.17 1.06
:
3.032 3.896
145 96
0.53 0.17
e
5.390
16
0.04
(0.30 * 0.17)
a
3.040 3.683
133 90
0.52 0.06
(1.78 f 0.34)
C
e
5.308
13
0.05
(0.17 f 0.18)
; :
2.938 2.644 2.979 3.786
110 172 134 74
0.36 - 1.50 - 0.55 0.16
(1.43 + 0.34)
La
Eu Gd Yb Lu
aa, (4f”)(4f”)a:o:a~n~ ‘2;; e, (4P”) 4P+‘)+~ ‘Xl. bRef. 23.
b, (4f”)(4f”)tix:
D, (eV)
Ex~.~
(2.52 + 0.22)
‘Z: ; c, (4P)(4f”)a~a~a~ ‘El; d, (4f”)(4f”)a~u:n~6~ 5AU;
experimental ground state assignments exist for La, to Lu, to our knowledge. In CISD + SCC calculations on a large number of low lying electronic states of La, we also obtained a o~a~o~n~ 5E; ground state (La 5d0°.515dn0,88 6sa’.416pa0.086p710.‘2) in agreement with the experimental and theoretical evidence for SC, and Y,. Our results are summarized in Table 2. As for Y,, the ground state [29]. Since at the CISD + SCC level a crzrct‘Ci state (La 6s~0.g76pa0.02 5daO.O’ 6px”.‘05dx”.%) is located only 0.11 eV above the G”,at 0: xx 5X- state it is also a possible candidate for a ground state. In similar calculations on Lu, all states with significant contribution of 5d atomic orbitals to occupied molecular orbitals were found to be rather high in energy owing to the relativistic destabilization of the 5d orbitals with respect to the 6s orbital, (see Table 3). The Lu, ground state is found to be the <~J:G”, ‘Z:,’ state (Lu 5d0°.456su1.826pcr0.73) which has a term energy of l.OOeV for La,. We suspect that the rather large difference in the experimental dissociation energies of La, (2.50 eV [24]) and Lu, (1.43 eV [24]) is due to the differing molecular ground states, i.e. a double or a triple bond for La, and a single bond for Lu, in terms of simple molecular orbital theory. The bond lengths and vibrational frequencies listed in Tables 2 and 3 are predictions. Our CISD + SCC results for Lu, (R, = 3.786 A; w, = 74 cm-‘,
245
M. Dolg et aE.fJ. Mol. Struct. (Theochem) 277 (1992) 239-249 TABLE 4 Bond lengths, R, (A), from SCF and correlation experimental values are not known
energy density functional calculations;
Metal
State”
VWNb
SPPb
Pb
HLb
La
a C
3.293 3.892
3.325 3.933
3.245 3.866
3.196 3.832
Ce
a
3.240
3.282
3.203
3.176
Pr
a e
3.198 5.403
3.239 5.935
3.164 5.242
3.132 4.847
Nd
a e
3.159 5.408
3.204 5.949
3.131 5.220
3.106 4.835
Pm
e
5.398
5.944
5.152
4.819
Sm
e
5.409
5.951
5.073
4.795
Eu
e
5.394
5.928
5.058
4.783
Gd
a c
3.060 3.717
3.099 3.769
3.037 3.656
3.015 3.550
Tb
a c
3.044 3.671
3.084 3.751
3.020 3.631
2.993 3.526
BY
e
5.411
6.006
4.923
4.747
Ho
e
5.411
6.002
4.913
4.727
Er
e
5.407
5.991
4.904
4.729
Tm
e
5.434
5.994
4.912
4.632
Yb
e
5.412
5.994
4.879
4.536
LU
a c
2.990 3.779
3.050 3.816
2.963 3.755
2.917 3.636
“See footnote a to Table 3. bSee text.
D, = 0.55 eV) are in good agreement with corresponding CEPA-1 values (R, = 3.797A; W, = 71 cm-*; D, = 0.69eV), indicating a relatively small influence of size-consistency errors in the CISD + SCC values. Unfortunately, in the case of the La, r$~~~~x~ ?Z- ground state, the CEPA-1 calculations did not converge. For the very low lying 0:~: ‘Z+ state the CEPA-1 results (R, = 2.869A; w, = 143cm-‘; D, = l.OZeV) are again close to our CISD + SCC values (R, = 2.830 A; o, = 167 cm-‘; 0, = 1.06 eV). As an alternative to the CISD treatment of correlation effects, we applied various correlation energy density functionals using the SCF density. This approach proved to be competitive to CISD calculations in previous cal-
M. Ddg et cd/J. Mol. Struct. (Theo&em)
246
277 (1992) 239-249
TABLE 5 Binding energies, D, (eV), from SCF and correlation energy density functional calculations in comparison to experimental vahes Metal
State”
VWNb
SPPb
La
a c
0.57 - 0.06
1.12 0.06
1.63 0.49
2.52* 0.22
-0.17
Ce
a
0.49
0.26
1.04
1.54
2.47f 0.22
Pr
a e
- 0.71 0.04
-0.93 0.02
-0.16 0.11
0.35 0.32
1.31+ 0.30
Nd
a e
- 1.39 0.33
- 1.61 0.01
-0.83 0.11
-0.33 0.31
0.83+ 0.30
Pm
e
0.03
0.01
0.10
0.31
Sm
e
0.03
0.01
0.10
0.31
0.52+-0.21
EU
e
0.03
0.01
0.10
0.30
0.30+ 0.17
a
- 0.09 - 0.29
-0.28 -0.30
0.49 -0.13
0.98 0.28
1.78F 0.34
- 0.22 - 0.34
-0.40 -0.33
0.36 -0.16
0.85 0.24
1.32+ 0.26
C
DY
e
0.02
0.01
0.09
0.28
0.69& 0.30
Ho
e
0.02
0.01
0.08
0.27
0.82& 0.18
Er
e
0.02
0.01
0.08
0.27
0.74t 0.30
Tm
0.02
0.00
0.08
0.27
0.52rt0.18
Yb
0.02
0.00
0.08
0.27
0.17+ 0.18
Lu
- 0.35 0.26
-0.45 0.37
0.26 0.47
0.69 0.78
1.43F 0.34
Gd
C
Tb
a
0.35
Pb
HLb
Exp.”
BSee footnote a to Table 3. bSee text. “Ref. 23.
culations for atomic excitation and ionization energies [‘7]. The calculated bond lengths and dissociation energies for the rare earth dimers are summarized in Tables 4 and 5 respectively. The density functional results usually bracket the CISD + SCC values. The numbers obtained with the gradient-corrected form given by Perdew [18] are in the closest agreement with the CISD results. We do not list vibrational constants (which are close to the corresponding CISD values of Table 3) because errors due to small inaccuracies in the numerical evaluation of the correlation energies are certainly larger than the slight variations in these values from La, to Lu,.
M. Do& et tsl.jJ.Mol. Struct.(Theochem)277 (1992) 239-243
247
Whereas the calculations on the dimers of the trivalent lanthanoid atoms are complicated by the large number of low lying electronic states, the corresponding calculations on the dimers of the divalent lanthanoid atoms have to account for the rather weak van der Waals bond in the (4f~~l)(4~+l)~~~ ground state configuration. The derived CISD + SCC dissociation energies of 0.04 eV and 0.05 eV for Eu, (Eu 6s01.956p(r0.046daa.or.) and Yb, (Yb 6s01.976poo.026dao.o’) are in reasonable agreement with the experimental values of 0.30eV ( + 0.17 eV) [S] and 0.17 eV ( + 0.18eV) [30] respectively. Because the potential curves have extremely shallow minima the derived bond lengths (Eu, 5.390 A; Yb, 5.308 A) and vibrational frequencies (Eu, 16 cm-‘; Yb, 14 cm-‘) might be affected by errors. Nevertheless the derived vibrational constants agree reasonably with estimates used in experimental work (Eu, 35 cm-“; Yb, 22 cm-‘) [31]. In corresponding CEPA1 calculations we obtained results very similar to the CISD + SCC values: L), = 0.06 eV, R, = 5.267 A, w, = 16 cm-’ for Eu,; De = 0.05 eV, R, = 5.203 A, w, = 14cm-’ for Yb,. The correlation energy density functional results for the van der Waals bonded superconfigurations of Prz to Eu, and Dy, to Yb, again bracket the CISD + SCC values; however, owing to the flat potential curves, the range of bond lengths is considerably larger than for the dimers with formally trivalent lanthanoid metals. The gradient-corrected functional forms yield the shortest bond lengths and highest dissociation energies. The latter are again in reasonable agreement with experimental data. It should be noted, however, that local density functional methods are known to overestimate bonding due to van der Waals type interactions. From our CISD and correlation energy density functional calculations we predict for La,, Ce,, (probably) Pr, , Gd, and Tb, a (4~)(4~)~~~~~~~~ and for Lu, a (4f~)~4~)~~~~~~ ground state superconfiguration (n = 0, 1, 2, 7, 8, 14 for La, Ce, Pr, Gd, Tb and Lu respectively). For Eu, and Yb, the van der Waals type (4fn+l)(4fnt1) .i,i superconfiguration should give rise to the molecular ground state. The assignment of the ground state superconfiguration for the other diatomics is less easy. The experimental dissociation energies of Nd,, Smtf Dy,, Ho,, Er, and Tmz range from 0.52 to 0.83 eV and appear to be somewhat too large for van der Waals type bonding, which was found to yield the highest dissociation energies for these molecules in our calculations. Because supercon~gurations with 4f” subcon~gurations on both atoms are considerably higher in energy, a mixed-valence situation arising from a superposition of 4p and 4f”+’ subconfigurations cannot be ruled out. Our study omitted spin-orbit effects and did not account for possible mixed-valence situations. In order to treat both effects in an ab initio level one would have to use pseudopotentials that include the 4f shell explicitly in the calculations, e.g. those of ref. 32. A smaller pseudopotential core in
248
M. Dolg et al@. Mol. Struct. (Theochem) 277 (1992) 239-249
order to avoid frozen core errors, larger basis sets including g functions to polarize and correlate the 4f shell, inclusion of spin-orbit operators and long and computationally demanding CI expansions would be necessary to obtain reliable results. According to our experience from calculations on diatomics with a single lanthanoid centre with both types of pseudopotential[11,12,15], we suppose that the effort to treat a single lanthanoid dimer with an open 4f shell would be considerably higher than it was for the whole study presented here. CONCLUSION
Quasi-relativistic energy-adjusted pseudopotentials for lanthanoid elements with a fixed valency have been used in quantum chemical calculations to derive for the first-time theoretical estimates for molecular constants of the homonuclear dimers of the 4f elements. Possible electronic ground state superconfigurations have been discussed. The derived dissociation energies are in satisfactory agreement with experimental values whereas other parameters and the assignments of ground state superconfigurations are predictions, ACKNOWLEDGEMENTS
The authors thank H.-J. Werner for providing the program system and A. Savin for the correlation energy density functional code.
MOLPRO. Thanks are also due to B. ~iehlich
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2
3 4 5 6 7 8
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11
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