ARTICLE pubs.acs.org/JPCA
Theoretical Study of the Structure and Bonding in ThC2 and UC2 Peter Pogany,† Attila Kovacs,*,†,‡ Zoltan Varga,‡ F. Matthias Bickelhaupt,§ and Rudy J. M. Konings† †
European Commission, Joint Research Centre, Institute for Transuranium Elements, P.O. Box 2340, 76125 Karlsruhe, Germany Materials Structure and Modeling Research Group of the Hungarian Academy of Sciences, Budapest University of Technology and Economics, H-1111 Budapest, Szt. Gellert ter 4, Hungary § Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands ‡
bS Supporting Information ABSTRACT: The electronic structure and various molecular properties of the actinide (An) dicarbides ThC2 and UC2 were investigated by relativistic quantum chemical calculations. We probe five possible geometrical arrangements: two triangular structures including an acetylide (C2) moiety, as well as the linear AnCC, CAnC, and bent CAnC geometries. Our calculations at various levels of theory indicate that the triangular species are energetically more favorable, while the latter three arrangements proved to be higher-energy structures. Our SO-CASPT2 calculations give the ground-state molecular geometry for both ThC2 and UC2 as the symmetric (C2v) triangular structure. The similar and, also very close in energy, asymmetric (Cs) triangular geometry belongs to a different electronic state. DFT and singledeterminant ab initio methods failed to distinguish between these two similar electronic states demonstrating the power of multiconfiguration ab initio methods to deal with such subtle and delicate problems. We report detailed data on the electronic structure and bonding properties of the most relevant structures.
1. INTRODUCTION Research of actinide carbides dates back to the 1950s, when they were first considered as potential nuclear fuels for fast reactors. However, later the interest in carbide fuel declined because of the selection of mixed uraniumplutonium oxides in fast reactor designs. An exception is the Indian Fast Breeder Test Reactor, which is operated with plutonium rich UPu mixed carbide fuel. In the past decade, the need for novel fuel types in Generation IV nuclear plant systems has renewed the interest in actinide carbides. Their suitability for high burnup and high linear power in future fast reactors13 recently initiated active research on these systems. These studies have been performed almost exclusively on the solid phases.47 The extreme conditions of the planned future fast reactors, however, result in a very high temperature in the middle zone of the fuel (up to 2500 K). At this temperature, the vapor pressure of carbides is sufficient for them to escape into the cracks in the fuel. Hence, for an accurate thermodynamic description of the system, the solidgas equilibrium has to be taken into account. The molecular parameters are essential input data for the evaluation of thermal contributions (rotational, vibrational, and electronic) in the Gibbs free energies. Information on gaseous actinide carbides is very scarce. Uranium812 and thorium carbides11,1316 AnCn (n = 16) have been detected in the vapors above the solid carbides or metal alloys in graphite systems. Partial pressures of the carbide molecules were measured by mass spectrometry, and thermodynamic data were calculated. Recently, the IR absorption bands of UC and UC2 prepared by laser evaporation of carbon-rich r 2011 American Chemical Society
uranium/carbon alloys followed by atom reactions in solid argon have been identified by matrix-isolation IR spectroscopy.17,18 The first theoretical study of actinide carbides was that of UC2 by the simple discrete variational Xα method.19 Among linear symmetric, linear asymmetric, and triangular structures, the first one was claimed to be the ground-state. After a long break, calculations using the B3LYP exchange-correlation functional in conjunction with a relativistic effective core potential (RECP) were performed on UC2,20 PuC, and PuC2.21 In these studies, the dicarbides were found to have a triangular ground-state geometry. Vibrational frequencies and dissociation energies were also reported. The molecular structure of ThC2 was investigated recently by MP2 calculations using RECP for thorium.22 In these calculations, the singlet state emerged as more stable than the triplet state, and for the electronic ground-state, the optimizations converged to an asymmetric (L-type, Cs symmetry) triangular geometry. At the applied MP2 level, the singlet C2v triangular structure was found to be a first-order saddle-point lying 7.7 kJ/mol above the ground-state. Advanced multiconfigurational calculations have been performed only for UC and for two structures of UC2 (symmetric linear and symmetric triangular) molecules to aid the interpretation of the matrix-IR spectra.17,18 The goal of the present study is to elucidate the electronic, structural, and bonding properties of the ThC2 and UC2 Received: October 24, 2011 Revised: December 2, 2011 Published: December 22, 2011 747
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molecules by advanced quantum chemical calculations. As outlined above, the actinide dicarbides have complex and interesting structural and electronic features. The various electronic states can lead to different structures, which can be fully uncovered only by sophisticated multiconfigurational methods. Therefore, we performed complete active space second-order perturbation theory (CASPT2) calculations probing all the possible structures in the relevant spin multiplicity states. We determined the electronic ground and low-lying excited states as well as the bonding interactions in the most relevant structures.
2. COMPUTATIONAL DETAILS The all-electron relativistic multireference calculations were performed using the code MOLCAS 7.4, patch 097.2325 The complete active space (CAS) SCF method26 was used to generate molecular orbitals and reference functions for subsequent multiconfigurational second-order perturbation theory calculations of the dynamic correlation energy (CASPT2).27,28 Ideally, the active space should include all the valence orbitals of the actinides (7s, 6d, and 5f) and carbon (2s and 2p) giving altogether 21 orbitals for AnC2. However, presently the computationally feasible number of activespace orbitals is 16; therefore, some compromises were necessary. In test calculations, we checked the contribution of the low-energy bonding orbitals of the C2 moiety to the static correlation energy. We found one CC bonding orbital of ThC2 and two CC and one UC bonding orbital of UC2 giving negligible contributions to the static correlation; therefore, we moved them into the inactive space. In addition, some antibonding orbitals (due to their negligible populations in the state averaged calculations) were also not considered in the active space. These optimizations left 10 electrons in 14 orbitals for the active space of ThC2 and 8 electrons in 16 orbitals for UC2. All-electron basis sets of atomic natural orbital type developed for relativistic calculations (ANO-RCC) with the DouglasKroll Hess Hamiltonian29,30 were used for all the atoms. For Th and U, the primitive sets of 27s24p18d14f6g3h and 26s23p17d13f5g3h were contracted to 9s8p6d4f2g1h and to 9s8p6d5f2g1h, respectively,31 whereas for carbon a primitive set of 14s9p4d3f2g was contracted to 4s3p2d1f32 achieving VTZP quality. The Douglas KrollHess Hamiltonian was used in the CASSCF calculations to account for scalar relativistic effects. In the second-order perturbation treatment (CASPT2), the orbitals up to 5d of the actinides and 1s of carbon were kept frozen, while the remaining valence and semivalence orbitals (including 6s and 6p of the actinides and 2s of carbon) were correlated. Multireference and other post-HartreeFock calculations on ThC2 using the small-core relativistic effective core potential of the CologneStuttgart group (ECP60MWB)33,34 in conjunction with a 14s13p10d8f6g valence basis set contracted to 6s6p5d4f3g (ECP60MWB_ANO basis)34 were performed using the MOLPRO 2010.1 code.35 For carbon, the correlationconsistent cc-pVTZ basis set36 was applied. The multiconfigurational methods included MRCI+Q37,38 and CASPT239 treating 20 electrons in the valence orbitals (including 6s and 6p of thorium) in the dynamic correlation. These calculations were successful only with the small (8/7 and 8/10) active spaces. The other postHartreeFock methods were MP2 and CCSD(T),40 both correlating 20 valence electrons. In addition, calculations with the PBE0 exchange-correlation functional41 have been performed. Additional density functional theory (DFT) calculations with the B3LYP exchange-correlation functional42,43 were carried out
Figure 1. Structures of AnC2 molecules.
with the Gaussian03 code44 using the correlation-consistent cc-pVTZ basis set36 for carbon and the same ECP60MWB core potential as above33 but in conjunction with a 14s13p10d8f6g valence basis set contracted to 10s9p5d4f3g (ECP60MWB_SEG basis),34 which is more suitable for this code. In the case of the open-shell systems, the DFT calculations often failed to find the electronic ground-states obtained by the SO-CASPT2 calculations. In such cases, we altered the orbital populations in order to be consistent with the multiconfigurational results. The energy decomposition analysis was performed with the Amsterdam Density Functional (ADF) program45,46 using allelectron relativistic DFT calculations. Scalar relativistic effects were accounted for using the zeroth-order regular approximation (ZORA).47 The PBE0 exchange-correlation functional41 was used, in combination with an uncontracted set of Slater-type orbitals (STOs) of triple-ζ quality augmented with polarization and diffuse functions for all elements (TZ2P). An auxiliary set of s, p, d, f, and g STOs was used to fit the molecular density and to represent the Coulomb and exchange potentials accurately in each SCF cycle.45 Closed-shell and open-shell systems were treated using the spinrestricted and spin-unrestricted formalism, respectively. The bonding interactions between the actinide and C2 fragments have been analyzed by means of the energy decomposition analysis (EDA) developed by Ziegler and Rauk48 for DFT methods and incorporated in ADF 2.3.46,49 In EDA, we analyze the instantaneous interaction energy between the two fragments in the molecule, ΔEint, consisting of the following main components: ΔEint ¼ ΔV elstat þ ΔEPauli þ ΔEorb
ð1Þ
where ΔVelstat is the electrostatic interaction energy, and ΔEPauli refers to the Pauli repulsive interactions, while ΔEorb represents the orbital interactions between the fragments.
3. RESULTS AND DISCUSSION 3.1. Structure and Ground-State Properties. We investigated five structural isomers with different arrangements of the two carbon atoms around the actinide (Figure 1): the symmetric triangle (I) obtained by DFT calculations for rare earth dicarbides,5052 the asymmetric triangle (II) computed previously for 748
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experiments by Wang et al.17,18 and/or the less preferred formation of C2 under the given conditions. The electronic ground state of structure I of ThC2 has singlet spin multiplicity, but the triplet state is only higher in energy by 33 kJ/mol. The most stable linear structures of ThC2 have triplet spin multiplicities. Wang et al. reported the electronic ground states and some geometrical parameters of I and III of UC2 using a similar theoretical level as in our study.17,18 Our results for these two structures (using a somewhat larger active space) are in good agreement with their work. We note that the symmetric linear (III) structure of ThC2 is much higher in energy with respect to structure I than found for UC2. Apparently, the four valence electrons of thorium cannot establish as strong AnC bonds in III (reported to be triple bonds in UC217) compared to the six valence electrons of uranium. Selected data on various electronic states from our CASPT2 calculations are given in the Supporting Information. We note that the above results on UC2 are in disagreement with studies using less sophisticated methods found in the literature. Wang et al.20 reported the electronic ground state of UC2 to be 5B1 with UC and CC bond distances of 2.128 and 1.525 Å, respectively. This CC distance would correspond to a CC single bond, while the UC bond in their results is shorter by ca. 0.1 Å than found in our results. The calculations of Wang et al. might have been converged to an excited state or may be an artifact of the used large-core RECP.57 It has been shown that large-core RECPs in conjunction with DFT can provide unrealistic structures.58 3.2. Close-Lying Structures I and II of ThC2. The first quantum chemical calculations on ThC2 were performed recently at the MP2 level using a RECP and valence triple-ζ basis set.22 The study resulted in a double-well potential with structure II at the minimum energy and structure I as a first order saddlepoint. Our first multireference calculations at the CASPT2 (10/14) level indicated both structures to correspond with energy minima with structure I being lower in energy (cf. Table 1). This contradiction and the small energy difference motivated a more detailed investigation using additional calculations at various theoretical levels. We probed DFT with B3LYP and PBE0 hybrid exchange-correlation functionals, the singlereference HF, MP2 and CCSD(T) methods, as well as the multiconfigurational MRCI and CASPT2 theories applying different active spaces. In most cases, the small-core relativistic ECP of the StuttgartCologne group33,34 was used, but with PBE0, we applied also an uncontracted all-electron basis set of Slater-type orbitals. The computed relative energies, geometries, and vibrational frequencies are compiled in the Supporting Information. All the applied DFT and single-determinant ab initio levels predicted a double-well potential energy surface, where the asymmetric triangular structure (II) is the minimum and the symmetric triangular structure (I) is a first-order saddle-point (the latter is confirmed by the single imaginary frequency obtained in the frequency analyses). However, the energy barrier between the two II minima was small, the same order of magnitude as the bending ground-state vibrational energy (ca. 2 kJ/mol). The relationship of I vs II does not seem to be affected by the character of the basis set (RECP vs all electron) nor by the level of dynamic electron correlation (second-order perturbation vs coupled-cluster). Analysis of the molecular orbitals did not reveal any significant difference between the electronic structures of the two geometries. However, a different picture emerged from the multiconfigurational CASPT2 (10/14) calculations. We computed the
Table 1. Characteristics of Electronic Ground States of the Five Possible Structures of ThC2 and UC2a I
II
III
IV
V
ΔE
0.0
8.8
ΔE (SO) state
0.0 1 A1
496.6
151.3
644.7
8.9 1 0 A
495.7 3 + Σg
151.9 3 0 A
a
2.281
2.143
2.058
2.158
2.141
1.336
3.547
ThC2
a0
645.3 A2
1
2.484
b
1.272
1.275
α
32.4
30.9
180.0
111.8
UC2 ΔE
0.0
28.1
285.1
94.5
ΔE (SO)
0.0
64.9
280.4
119.1
1053.1
state a
5
5 00
3 + Σu
5 00
A 2.149
3
1.276
2.289
A2 2.257
a0
A 2.151
1.825
962.5 B2 2.358
2.487
b
1.271
1.272
α
32.7
30.8
180.0
58.1
a
The relative energies (kJ/mol) and geometrical parameters (Å, deg) were obtained by all-electron spin-free CASPT2 calculations. For the symbols of geometrical parameters, see Figure 1. The spinorbit corrections of the energy were determined on the spin-free optimized geometries.
ThC222 and GeC2,53,54 the symmetric linear CAnC arrangement (III) characteristic for actinide dioxides and other triatomic actinide molecules (CUO, NUN, NUO),55 and found recently for UC2 in Ar matrix,17,18 the asymmetric linear arrangement (An-CC, IV) suggested in early thermodynamic calculations,8,9,1116,56 and the bent CAnC arrangement (V) with a large bond angle like the structure of ThO2 and reported recently from DFT calculations on PuC2.21 The electronic properties of these five structures were assessed by all-electron CASPT2 calculations probing all relevant spin multiplicities and symmetry species. For the lowest-energy states of each structure, the obtained relative energies and optimized geometries are compiled in Table 1. In our CASPT2 calculations, the electronic ground states of ThC2 and UC2 adopted the symmetric triangular molecular structure I (similar to those found for rare earth dicarbides,5052 while the asymmetric triangular structures II lie very close in energy. The bent and the two linear molecular arrangements are considerably (generally by a few hundreds of kJ/mol) higher in energy. However, the high energy structures remain of interest because they can be readily formed under certain conditions. As mentioned in the introduction, the linear structure (III) was observed recently for UC2 in Ar and Ne matrices, while the thermodynamically more stable triangular structures were not detected.17,18 The preferential formation of III in this case results from the reaction conditions used for the preparation of the carbide molecules in the gaseous phase: the solid sample was evaporated (and atomized) by a laser beam; hence, the molecules form from the atoms. UC2 can arise by the consecutive reactions U + C f UC and UC + C f CUC, where the second carbon approaches UC from the metal side, establishing a linear CUC structure. The reaction of U and gaseous C2 would likely produce a triangular UC2 molecule. Alternative reasons why the triangular structure was not detected are that its characteristic IR absorption is near the detection limit in the IR 749
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There are some slight differences in the σ-type ThC2 bonding orbitals (see Figures 3 and 4): MO-1 of I includes minor contributions from the 7s and 6dδ orbitals of Th, while contributions of Th atomic orbitals in II are negligible. In MO-4 of I minor contributions from Th orbitals include 6dπ, 6pπ, and 5fπ, while in MO-4 of II appears 6dσ, 7s, and 6dπ instead. While the probed DFT and single-determinant ab initio methods failed to distinguish between the two close-lying electronic states, their performance on the geometrical parameters and vibrational frequencies of ThC2 was quite good: the bond distances were within 0.02 Å of the CASPT2 (10/14) values, while there was a proper agreement among the computed vibrational frequencies. Properties analogous to the uncovered electronic characteristics of ThC2 can be expected for structures I and II of UC2, where the CASPT2 calculations indicated a larger energy difference than in the case of ThC2 (cf. Table 1). Therefore, we skipped a detailed study of UC2. We performed only B3LYP calculations, which failed to find structure II; the few probed asymmetric initial geometries converged to the symmetric one I. 3.3. Electronic Structure. In Table 2, we compiled the characteristics of the most significant spin-free states of ThC2 and UC2. We considered here the low-energy triangular structures I and II (the latter for ThC2 only) as well as the linear structure III. The states of ThC2 have one major electron configuration with contributions of around 80%. The most stable states for both structures I and II of ThC2 are singlets. The thorium atom is divalent in the low-energy states of I and II as two electrons form a lone pair mainly of the 7s character in the singlet states, while they occupy the 7s and 6dδ orbitals, respectively, in the triplet states. The triplets are somewhat higher in energy than the corresponding singlet states. The ground state of structure III of ThC2 is a triplet, which is ca. 500 kJ/mol higher in energy than the most favored triangular structures. There are two unpaired electrons in both the triplet and singlet states of this linear
potential energy curves around structures I and II with the ThCC angle as abscissa using the SUPSYM keyword in MOLCAS. This latter option supports to keep the electronic state during the change of the geometry. In contrast to the other above-mentioned results, the CASPT2 calculations did not confirm the flat double-well potential around structure II. The independence of the resulting curves in Figure 2 indicate that structures I and II represent different electronic states from which the symmetric triangular I is the global minimum. Varying the computational parameters (size of active space from 8 to 14 orbitals, probing the 6p orbitals of Th in the active space, introducing spinorbit interactions) did not change the lower energy character of structure I. Analysis of the valence orbital populations in the two electronic states revealed only small differences: according to the CASPT2 (10/14) results, the nonbonding electron pair of Th consists of 7s (major) and 6dσ (minor) components in both electronic states.
Figure 2. Potential energy curves for structures I (solid) and II (dashed) of ThC2.
Figure 3. Characteristic molecular orbitals of the actinide-C2 interactions in symmetric triangle (I) AnC2 molecules. Presented are the ThC2 orbitals obtained from CASPT2 calculations. 750
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Figure 4. Characteristic molecular orbitals of the actinide-C2 interactions in the asymmetric triangular structure (II) of ThC2.
Table 2. Selected Spin-Free Electronic States of ThC2 and UC2 structure ThC2
UC2
term I I IIc II III III III Ic I I I I I I I III III III III III III
1
A1 A2 1 0 A 3 00 A 3 + Σg 1 + Σg 3 Πu 5 A2 5 A1 5 B2 5 B1 3 A1 3 A2 3 B2 3 B1 3 + Σu 3 Πu 3 Πg 1 Πg 1 + Σg 1 Πu 3
T (kJ/mol)
main configurationa
Geometry (a, a0 , b in Å; α in deg)b
0.0 19.6 8.8 39.1 496.6 534.6 545.2 0.0 6.9 11.0 13.9 31.1 31.2 45.2 37.6 285.1 370.7 370.8 374.4 375.2 375.9
79% (7s,6dσ)2 87% 7s1, 6dδ1 80% (7s,6dσ)2 93% 7s1, 6dδ1 78% πu1, πu1 74% πu1, πuβ 89% πu1, σg1 51% 7s1, 5fπ1, 5fδ1, 5fϕ1 63% 7s1, 5fπ1, 5fδ1, 5fϕ1 70% 7s1, 5fσ1, 5fδ1, 5fϕ1 49% 7s1, 5fσ1, 5fδ1, 5fϕ1 30% 7s β, 5fπ1, 5fδ1, 5fϕ1 30% 7s β, 5fπ1, 5fδ1, 5fϕ1 26% 5fδ1, 5fϕ1 29% 7s β, 5fσ1, 5fδ1, 5fϕ1 89% σg1, σu1 59% σg1, πu1 80% σu1, πu1 76% σu1, πuβ 86% σg2, σu0, πu2, πu2, πg2, πg2 46% πg1, σuβ
2.281, 1.272, 32.4 2.228, 1.265, 33.0 2.143, 2.484, 1.275, 30.9 2.153, 2.482, 1.277, 31.0 2.058, 180.0 2.041, 180.0 2.046, 180.0 2.257, 1.271, 32.7 2.257, 1.271, 32.7 2.279, 1.270, 32.3 2.258, 1.269, 32.7 2.262, 1.270, 32.6 2.262, 1.270, 32.6 2.280, 1.269, 32.3 2.276, 1.269, 32.4 1.825, 180.0 1.847, 180.0 1.848, 180.0 1.850, 180.0 1.838, 180.0 2.010, 180.0
a The symbol β means beta spin polarization of the given electron. For the σ and π bonding orbitals, see Figure 5. b For definition of the geometrical parameters, see Figure 1. c The harmonic vibrational frequencies obtained with B3LYP/RECP+cc-pVTZ calculations are 1789, 616, and 165 and 1829, 510, and 243 cm1 for ThC2 and UC2, respectively.
structure; in the singlet states, one of the electrons has beta spin polarization. These singly populated orbitals are mixed from atomic orbitals of carbon and thorium, forming one-electron bonding orbitals (vide infra). In agreement with the recent studies of Wang et al. on triangular UC2,18 we found the quintet 5A2 as the electronic ground-state for the molecule. The three next lowest quintet states of structure I are within 15 kJ/mol. The low-energy triplet states are higher in energy by 3045 kJ/mol. The spin-free states of UC2 have a stronger multiconfigurational character than those
of ThC2, the major electron configurations appearing as 4970% in the quintet states, while only as 2630% in the triplet states. The linear UC2 structure III is higher in energy by 285 kJ/mol than the triangular ground-state. In ThC2, we observed a much larger energy difference between the triangular and linear structures (vide supra). In UC2, the six valence electrons of uranium (compared to the four of thorium) can facilitate more actinide carbon covalent interactions. The considerably stronger character of the UC bonds in III is manifested also in the UC bond distances, which are shorter by 0.2 Å than the ThC bonds 751
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The 2-electron bonding molecular orbitals in structure I of ThC2 (identical with those of UC2) are shown in Figure 3. The related molecular orbitals of structure II are similar (see Figure 4); they look somewhat distorted because of the asymmetric geometry. The lowest energy MO-5 corresponds to a σ(CC) bond formed by carbon 2s orbitals. This MO has no actinideC2 bonding. A σ*(CC) antibonding orbital appears in MO-4, which, however, establishes σ-type bonding interactions with the 6dπ orbitals of the actinide. MO-3 and MO-2 represent π(CC) bonding overlapping with the 6dπ and 6dσ orbitals of the actinide atom, respectively. MO-1 consists of a σ(CC) bond formed mainly from carbon 2pσ orbitals interacting with a 7s/6d hybrid of the actinide. Hence, we identified altogether four molecular orbitals with a CC bonding character and four molecular orbitals with an actinideC2 bonding character. We note that MO-2, MO-3, and MO-4 were shown also in ref 18. These mixed CC/AnC interactions complicate the determination of the bond orders. Taking into account the four bonding and one antibonding CC interactions, the CC bond should be between a double and triple bond (the same conclusion was drawn by Wang et al. on the basis of the CC bond distance18), while the four weak AnC interactions suggest a bond order not larger than two. The latter suggestion is supported by the AnC distances compared with the covalent radii of carbon and the actinides.59 Utilizing the single bond covalent radius of sp carbon (0.67 Å), we obtain radii values of 1.59 Å and 1.57 Å for Th and U in ThC2 and UC2, respectively. These values are intermediate between the single and double bond covalent radii of these actinides.59,60 As can be seen in Figures 3 and 4, the C2 orbitals have generally larger contributions in the bonding molecular orbitals than those of the actinide atoms. This is in agreement with the charge distribution in the molecules assessed using the Voronoi Deformation Density (VDD) atomic charges.61 The values of +0.4 and +0.5 for Th and U, respectively, indicate a small charge separation. A quantitative characterization of bonding in terms of the electrostatic and covalent interactions in ThC2 was carried out by energy decomposition analysis using the ZieglerRauk model48 in the DFT framework. Such an analysis is particularly straightforward for the triangular structures, which can be separated into an actinide and a C2 fragment. Thus, these results can be closely related to the above discussion of the molecular orbitals. A prerequisite for the proper orbital interactions in the analysis is that the fragment orbitals should be occupied according to their effective valence state in the AnC2 molecule. This requirement could be managed for the Th fragment in ThC2 but not for the U fragment of UC2. Therefore, we report here the energy decomposition analysis of structure I of ThC2 only. Because of the aforementioned small charge separation (vide supra), we chose the neutral actinide atom and triplet C2 radical as fragments for our energy decomposition analysis. This model is related to the formation (Th + C2 f ThC2) and dissociation reactions (ThC2 f Th + C2) in the gaseous phase. The dissociation energy differs from the total interaction energy (consisting of the energy contributions from the various interactions between the fragments) by the preparation energy, i.e., the energy associated with the geometry and electronic relaxation of the fragments after dissociation. Our energy decomposition results are compiled in Table 4. They show that the bonding in ThC2 is mainly of covalent character: the orbital interaction energy is ca. 65% of the total attraction energy. The main contributions
Table 3. Low-Lying Spin-Orbit States of Selected Structures of ThC2 and UC2 structure ThC2
I
II
III
UC2a
I
III
T (cm1) 0
96% A1 + 4% 3A2
1455 1455
100% 3A2 100% 3A2
1508
96% 3A2 + 4% 1A1
0
99% 1A0 + 1% 3A00
2620
100% 3A00
2620
100% 3A00
2657
99% 3A00 + 1% 1A0
0
100% 3Σ+g
3199 4067
100% 1Σ+g 100% 3Πu
0
38% 5A2(1) + 37% 5A1(1)
9
38% 5A2(1) + 38% 5A1(1)
215
27% 5B1(1) + 26% 5B2(1)
278
17% 5A1(3) + 14% 5A2(3) + 11% 5B1(3)
321
28% 5B2(1) + 24% 5B1(1) + 11% 5A1(3)
451
20% 5A1(3) + 13% 5A1(2) + 11% 5A2(3)
562 944
21% 5B2(2) + 20% 5B1(3) + 16% 5A1(2) 21% 5B2(2) + 14% 5B1(3) + 13% 5A1(2)
0
95% 3Σ+u + 5% 3Πu
0
95% 3Σ+u + 5% 3Πu
97
97% 3Σ+u + 3% 1Σ+u
5490
53% 3Πg + 47% 1Σ+g
7699
100% 3Πu
7706
100% 3Πg
7708 8038
100% 3Πg 100% 1Πg
8078
95% 3Πu + 5% 3Σ+u
8078
95% 3Πu + 5% 3Σ+u
9650
97% 1Σ+u + 3% 3Σ+u
10 184 a
composition 1
53% 1Σ+g + 47% 3Πg
For additional spin-orbit states of I, see the Supporting Information.
(cf. Table 2). The bonding situation in these structures will be discussed in the next section. The most characteristic spin-orbit states of ThC2 and UC2 are presented in Table 3. In the low-energy states of ThC2, the spinorbit coupling is not important, a minor effect can be observed only between the 1A1 and 3A2 spin-free states of structure I. However, strong spin-orbit coupling was found in all the presented structures of UC2. A more detailed list of the spinorbit states of structure I of UC2 is given in the Supporting Information. 3.4. Bonding. Some characteristics of the bonding orbitals in structures I and III of UC2 have been reported by Wang et al.17,18 We performed an extensive analysis on the basis of the molecular orbitals, atomic charges, and energy decomposition. From the five studied molecular structures of ThC2 and UC2, we focus here on the most relevant structures, I, II, and III, starting with I and II. The highest energy occupied orbitals are nonbonding in both dicarbides. In ThC2, the HOMO is a lone pair on thorium consisting mainly of the 7s atomic orbital. In UC2, the highest-energy occupied orbitals are four 1-electron orbitals with uranium 7s and 5f characters. 752
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doubly occupied MOs (two σ and four π) as well as two singly occupied orbitals (both σ) characterizing the bonding scenario. The molecular orbitals obtained in our study for UC2 are consistent with those of Wang et al.;17 however, a detailed analysis of the orbital compositions led to some differences compared to the interpretation of Wang et al. First, one of the doubly occupied σ orbitals is a nonbonding one, as it includes an antibonding combination of C2s and U6p orbitals. Because of the quite low energy of this orbital and to the minor contribution of 6p, we suggest a nonbonding (instead of antibonding) character, representing essentially lone electrons on the two carbons. Second, the assigned antibonding character of the one-electron σ* orbital is not correct, as it involves a bonding combination of C2p and U5fσ orbitals. As the two errors cancel each other, we arrive at the same bond order of 3 reported by Wang et al.17 The bonding scenario is somewhat different in structure III of ThC2 due to fewer valence electrons in thorium with respect to uranium. The bonding molecular orbitals of the ground state (3Σ+g ) are presented in Figure 5. We find here the same orbitals as reported previously for UC2,17 but with somewhat different occupations. The three σ orbitals are doubly occupied in ThC2 (the second σg orbital is singly occupied in the ground state of UC217), as are the two πg orbitals. In contrast to UC2, the πu orbitals are singly occupied in ThC2. There are no occupied antibonding orbitals in ThC2, but the σu orbital has a nonbonding composition of C2s with minor contribution of thorium 6p orbitals and represents lone electrons on the carbons (similarly to the reinterpreted UC2 orbital, vide supra). The shown occupation pattern of ThC2 corresponds to a bond order of 2.5 explaining the somewhat lower stability and longer bond distances compared to structure III of UC2.
in structure I come from the molecular orbitals of A1 symmetry (MO-1 and MO-2 in Figure 3). Contributions from the other two orbitals are one order of magnitude less of which the out-ofplane π interaction (MO-4, B2) is the weaker. The bonding in structure III of UC2 has been reported in ref 17 on the basis of the molecular orbitals. Wang et al. presented six Table 4. Results of the Energy Decomposition Analysis and Charge Transfer Characteristics in Form I of ThC2a energies (kJ/mol) total interaction
I (C2v) 728.9
Pauli repulsion
1285.9
electrostatic attraction
700.6 (35%)
orbital interaction
1314.1 (65%)
A1/A0
1084.2
A2 B1/A00
0.0 97.5
B2
132.4
dissociation energy (kJ/mol)a ThC2 f Th + C2
794.6
charge transferb Th charge
0.37
A1/A0 (Th f C2)
0.75
A2 B1/A00 (C2 f Th)
0.00 0.21
B2 (C2 f Th)
0.17
a
The dissociation energies of forms I of ThC2 and UC2 were computed to be 739 and 691 kJ/mol, respectively, at the SO-CASPT2 level of theory. b Net charge (electron) transferred between the C2 and Th fragments in terms of VDD atomic charges.61
Figure 5. Selected occupied molecular orbitals of structure III of ThC2. 753
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’ CONCLUSIONS Actinide carbides have reemerged as important subjects of nuclear research as they are potential fuel materials in the next generation of nuclear reactors. The properties of the vapor species are important for the safety analysis of fast reactors with fuel operating at high temperature at which vaporization of material can occur in the core of the fuel pin. In the present study, we have performed an extensive analysis of the molecular and electronic structures of ThC2 and UC2 molecules. We have considered five possible structures: two triangular structures including an acetylide (C2) moiety, as well as the linear An-CC and CAnC and the bent CAnC (with bond angle of ca. 150°) geometries. Our calculations at various levels of theory showed the preference of the triangular species, while the latter three geometries proved to be high-energy structures. The distinction of the similar and energetically very close-lying electronic states represented by the symmetric (C2v) and asymmetric (Cs) triangular structures highlights the superiority of multiconfigurational ab initio theory over DFT and post-HF singledeterminant methods regarding the treatment of the subtle and delicate problems in this work. The latter computational levels failed to find the C2v state of ThC2 and the Cs state of UC2. According to our SO-CASPT2 calculations, the symmetric (C2v) triangular structure is the ground-state molecular geometry for both ThC2 and UC2. It can be characterized by a C2 moiety bounded by various σ and π interactions to the actinide atom. The energy decomposition analysis and the computed atomic charges showed that the bonding is mainly covalent in character with the electrostatic attraction playing a secondary role. Though the symmetric linear CAnC structure lies 300500 kJ/mol higher in energy, it can be formed under certain experimental conditions where the kinetics of formation would be important. Thus, its experimental relevance (CUC has been observed in matrix-isolation IR experiments) calls for theoretical data on this structure as well. CUC is characterized by five doubly and two singly occupied bonding orbitals, which leads to a UC bond order of 3. The bonding in CThC is somewhat weaker; in its ground state, we observed only four doubly and two singly occupied bonding orbitals leading to a ThC bond order of 2.5. In addition, we have determined the electronic structure and geometries of several spin-free states and characterized the electronic structures of the low-energy spin-orbit states.
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Hungarian Scientific Research Foundation (OTKA No. 75972), and The Netherlands Organization for Scientific research (NWO/CW and NWO/NCF) are acknowledged for financial support.
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’ ASSOCIATED CONTENT
bS
Supporting Information. Characterization of several high-energy electronic states of ThC2 and UC2 not discussed in the article, data on the symmetric and asymmetric triangular structures of ThC2 obtained at various levels of theory, extended list of spin-free and spin-orbit states of structure I of UC2 from CASPT2 calculations. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT The seventh Framework Programme of the European Commission (Collaboration Project No. 211690, F-BRIDGE), the 754
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