Relative stabilitiesof the [AnX]+ & [LnX]+ systems
The energy gaps between HOMO/SOMO and LUMO may be considered as a marker for discussing the stability of a molecular system (in Table 2). Higher the difference in the energies of the HOMO/SOMO and LUMO, higher will be the stability.[42, 43] The HOMO/SOMOs are of more negative energies for all the oxides as compared to the other chalcogenides (in Table S1 in SI), indicating a stability order as oxides > sulphides > selenides. The gaps between the frontier molecular orbitals (i.e. HOMO/SOMO & LUMO) are mostly higher for the [LnX]+ systems than the corresponding [AnX]+ systems, indicating higher stability for the former, with only a few exceptions.
Table 2
HOMO/SOMO-LUMO energy gaps for all the systems
|
HOMO/SOMO-LUMO Gap (eV)
|
System
|
[AnX]+
|
[LnX]+
|
0-O
|
4.23
|
5.74
|
5-O
|
3.17
|
3.99
|
6-O
|
3.43
|
3.61
|
7-O
|
3.60
|
4.72
|
0-S
|
3.11
|
4.02
|
5-S
|
2.94
|
2.02
|
6-S
|
1.84
|
1.90
|
7-S
|
2.91
|
3.30
|
0-Se
|
2.73
|
3.61
|
5-Se
|
2.55
|
1.73
|
6-Se
|
1.62
|
1.70
|
7-Se
|
2.54
|
2.86
|
Bonding energies between the An3+/Ln3+ and X2− fragments could be used to analyse the thermodynamic stability of the systems instead of the conventionally used free energies of formation, as in this case only gas phase two centred systems are considered. The interaction (bonding) energies and their components were calculated in Energy Decomposition Analysis.
Energy decomposition analysis (EDA)
The bonding energies were found to be the largest for all the oxides, followed by sulphides and selenides for a given metal ion. And for a given chalcogenide, most of the [LnX]+s had higher negative interaction energies than the [AnX]+s, indicating higher stability of the former over the latter. This is in line with the observation from HOMO/SOMO-LUMO gap analysis. Delving into the components contributing to the bonding energies, it was noted that the electrostatic contribution was nearly three times more than the orbital contribution for all the systems (in Table S2 in SI) suggesting strong ionic bonding, with small covalent contributions, which we want to zoom into. The bond lengths of the systems vis-à-vis the ionic radii of the interacting ions were compared to understand the extent of bonding between them. Significant bonding interaction is expected to be reflected in the difference of the sum of the radii of the interaction ions and the An-X or Ln-X bond lengths.
Variation in bond lengths of the [AnX]+ & [LnX]+ systems
The calculated bond lengths of the [AnO]+ and [AnS]+ systems were benchmarked with the data presented by Pereira et al[24].Calculated bond lengths were in good agreement with the experimental data (deviation of maximum 2%). The difference in the bond lengths of the [AnX]+ andthe corresponding [LnX]+ system with same number of f-electrons is presented in Fig. 1 and the complete bond length data are presented in Table S3 in SI.
The bond lengths of [AnX]+ systems are expected to be longer than the corresponding [LnX]+ systems due to the larger ionic size of the An3+ ions. This trend is nicely followed by all the metal oxides and the f0& f7 (half filled) metal-chalcogenide systems. But for the sulphides and selenides, there is a relative shrinkage of some of the [AnX]+ bond lengths w.r.t their [LnX]+ counterparts. In case of f5 (Pu3+/Sm3+) and f6 (Am3+/Eu3+) systems the shrinkage in the An-donor bond is observed for the sulphides and selenides. This hinted that the presence of unpaired ‘f’ electrons those are partially filling the f-orbitals (f5 and f6 in the present case), barring f7 (half-filled f-orbital) has some role in the better overlap of actinide ions with softer donors viz., S2− and Se2−. Charge analysis on each of the fragments was assumed to be useful for understanding bonding features of these ‘f’ block metal ions with the chalcogenides. Hence it was carried out on the fragment atoms for all the studied systems.
Partial charge analysis
The computed natural (in Table 3) and AIM charges (in Table S4 in SI) are in good agreement with each other with regression coefficients 0.97 for [AnX]+ and 0.94 for [LnX]+ systems (Figure S1 in SI). The charge analyses reflect that the extent of charge neutralization on the metal centres is higher for most of the [LnX]+ systems than their corresponding [AnX]+ systems. The higher charge transfers from the chalcogenides to the Ln3+ ions can be attributed to the higher polarizing power of the Ln3+ ions with respect to the corresponding An3+ ions by virtue of their difference in ionic radii. Also the increased polarizability of the softer donors from sulphide to selenide ion, renders charge donation to the charged metal ion subsequently easier.
Table 3
The natural charges on An/Ln and X centres and extent of ligand to metal charge transfer (LMCT)
System
|
An
|
X
|
LMCT (X->An)
|
Ln
|
X
|
LMCT (X->Ln)
|
0-O
|
2.175
|
-1.175
|
0.825
|
1.984
|
-0.984
|
1.016
|
5-O
|
1.891
|
-0.891
|
1.109
|
1.896
|
-0.896
|
1.104
|
6-O
|
1.903
|
-0.903
|
1.097
|
1.863
|
-0.863
|
1.137
|
7-O
|
1.971
|
-0.971
|
1.029
|
1.947
|
-0.947
|
1.053
|
0-S
|
1.850
|
-0.850
|
1.150
|
1.660
|
-0.660
|
1.340
|
5-S
|
1.616
|
-0.616
|
1.384
|
1.590
|
-0.590
|
1.410
|
6-S
|
1.632
|
-0.632
|
1.368
|
1.583
|
-0.583
|
1.417
|
7-S
|
1.678
|
-0.678
|
1.322
|
1.670
|
-0.670
|
1.331
|
0-Se
|
1.765
|
-0.765
|
1.235
|
1.596
|
-0.596
|
1.404
|
5-Se
|
1.580
|
-0.580
|
1.420
|
1.551
|
-0.551
|
1.449
|
6-Se
|
1.573
|
-0.573
|
1.427
|
1.545
|
-0.545
|
1.455
|
7-Se
|
1.599
|
-0.599
|
1.401
|
1.605
|
-0.605
|
1.395
|
After obtaining an insight on the charge transfer, the extent of electron density shared between the metal and donor ions, thereby an indication of the nature of the bond between them was obtained from the QTAIM analysis.
QTAIM analysis
(a) Analyses of the metrices at bond critical point In QTAIM analysis the covalent interactions are quantified by analysing the electron density, its laplacian values and total energy densities at the bond critical point. Negative values of the total energy density H(r) at BCP is a descriptor of stabilization through chemical bond formation. ρBCP, a marker for the extent of deposition of electron density at the bond centre, may be interpreted as a measure of covalent interaction manifested by orbital overlap. The thumb rule states, ρBCP > 0.2 a.u. &∇2ρBCP < 0 indicate pure covalent bonding. Closed shell interactions are indicated by ρBCP < 0.1 a.u. &∇2ρBCP > 0. ρBCP with intermediate values indicate partial covalent character.[36–38] For all the An-X and Ln-X bonds, the ∇2ρBCP > 0 and H(r) < 0 implying a partial covalent character. More negative H(r) values for all the Ln-X bonds in case of f0 systems indicate the higher stabilization than the corresponding An-X. For rest of the systems, H(r) values are more negative for the An-X bonds in comparison to the Ln-X bonds, indicating stabilization of the former bonds with respect to the latter. Absolute values of the electron densities (ρBCP) (in Table S5 in SI) for oxides of Pu, Am and Cm are in well agreement with the data quoted by Huang et al.[27] The ρBCPvalues are the highest for the An/Ln oxides, followed by sulphides and selenides. The An-O and Ln-O bonds have ρBCP more than 0.2 e/bohr3, indicating significant covalent contribution to bonding. The sulphides and some of the selenides of both [AnX]+ and [LnX]+ have ρBCPvalues higher than 0.1 e/bohr3 indicating partial covalent interactions. Only for the f0 systems, the values are higher for the Ln-X bonds than the An-X bonds, indicating higher covalent character of the former. Rest of the systems follow the opposite order. In case of the partially filled unpaired f-electron systems (i.e. f5 and f6), higher ρBCPwas observed for the ‘An-X’ bonds with X = S2− and Se2−. Similar trend was also noted from the H(r) values.
(b) Extent of electron sharing between the atomsThe delocalisation index [δ(A,B)] in QTAIM is a measure of the number of electrons shared between two atoms. δ(A,B) is a generic indicator for covalency as it encompasses the contributions from both orbital overlap and energetic degeneracy. Large values of ρBCP are accompanied by large values of δ(A,B), the converse is not always true.[44] The delocalisation index, in the present case is defined as δ(M,X), where M is the trivalent An/Ln ion and X is the chalcogenide ion. The values of the δ(M,X) are tabulated in (Table S6 in SI). The Wiberg bond indices (WBI) obtained from natural population analysis show good correlation with the δ(M,X) for all the systems (Table S6 in SI). The WBI and δ(M,X) values are very close for most of the [AnX]+/ [LnX]+ pairs. In case of the f0 systems, both these indices are higher for the Ln-X bonds than the An-X bonds. Rest of the systems, however, follow opposite trend. Higher values of both these indices are observed for the [PuS]+, [AmS]+, [PuSe]+and [AmSe]+ than their corresponding [LnX]+s.
Consolidated plots of the electron densities at BCP and delocalisation indices are shown in Fig. 2.
Upon summarizing the results of the QTAIM analysis, we see that, the electron density values at BCP indicate significant covalent bonding for only the oxides. For most of the cases the ρBCP and δ(M,X) are higher for the [AnX]+ systems than the [LnX]+ counterparts, with the f0 systems being exceptions. The similar values of electron densities at BCP and delocalisation indices for An-O and Ln-O bonds indicate similar degree of covalency in these systems. Interestingly, the difference in the delocalisation indices is significant for the f5, f6 sulphides and selenides. Electron densities at BCP for all the [AnX]+ systems are slightly more (max 25%) than the corresponding [LnX]+. f5, f6 sulphides and selenides have the highest differences in these metrices between the [AnX]+/ [LnX]+ systems. These findings corroborate with similar observations from bond length and partial charge analyses.
Higher delocalisation index, inspite of smaller ρBCP values is indicative of near degeneracy driven covalency for the bonding of actinides with progressively softer donors, due to the lowering of energy gaps of the donor and acceptor orbitals. It is reported that such covalent contributions are not expected to render energetic stabilization of the systems, as was also observed from the EDA.[22, 45]
For understanding the involvement of the valence (n-1)d and nf orbitals in the bonding between the trivalent An/Ln ions with the chalcogenide ions, the HOMO/SOMOs of the systems were studied.
Frontier Molecular Orbital Analysis
Most of the HOMO/SOMOs of the [AnX]+ or [LnX]+ systems reflect interaction between the (n-1)-d-orbitals of the actinide or lanthanide ions with the np-orbitals of the chalcogenide donors indicating significant involvement of the metal ‘d’-orbitals in bonding. Representative HOMO/SOMOs for the systems are presented in Fig. 3. Detailed presentation is available on Table S1 in SI.
This indicates a predominant metal d orbital directed bonding.[21–23] But, the effect of the metalf orbitals is yet to be understood. For obtaining a deeper insight into the relative involvement of the valence d/f orbitals of the An3+ or Ln3+ ions with that of the chalcogenide ions, Natural Bond Orbitals (NBO) analysis was taken up.
NBO Analysis of the [AnX]+/ [LnX]+ systems
Natural Bond Orbitals[46] of the bonding type with the highest orbital overlap were analysed to understand the extent of the metal orbital participation during their bonding with the chalcogenides. A quantity ‘relative d/f orbital contribution’ is defined as the ratio of the respective d/f orbital contributions of the An3+ to that of the Ln3+ ions in order to provide a comparative account of bonding in the [AnX]+ and [LnX]+ systems [Figure 4].
The relative d orbital contribution for all the pairs is close to unity, indicating insignificant difference in participation of the d-orbitals for An-X and Ln-X bonding. For all the f0 systems the relative d and f orbital contributions are nearly equal having values less than 1. This is indicative of slightly higher d and f orbital participation in [LaX]+ systems than [AcX]+ systems. In the sulphides and selenides of f5, f6 actinides, 4–8 times larger f-orbital contribution was observed than their corresponding [LnX]+ counterparts. Even the d orbital contributions for these actinide systems are also slightly higher than their corresponding [LnX]+s.
Delocalisation energies from the second order perturbation theory are indicative of the prominent routes of donor to acceptor electron transfer. The prominent delocalisation energies in Table S7 in SI indicate the contribution from the metal orbital donation to M-L bond. The delocalisation energies corresponding to chalcogenide to metal donation is of the similar range for most of the systems, but the contribution from metal orbital donation to M-L bond is significantly larger for the [AnS]+ and [AnSe]+ systems than the corresponding [LnX]+. The magnitude of these energies is higher for the oxides and of the similar order (decreasing trend) for the rest of the chalcogenide systems. All the f0 systems have the highest of both the delocalisation energies for varying chalcogenide ions. Prominent metal f orbital participation in f5, f6 [AnS]+& [AnSe]+ systems with respect to the corresponding [LnX]+s is noted. To magnify upon the metal f-orbital participation in the bonding of the [AnX]+s, the natural electronic configurations (NEC) were analysed. The NECs for the actinide chalcogenides obtained by summing up the natural atomic orbital occupancies are presented in Fig. 5. These are expected to provide an idea on the metal f-orbital participation in bonding. The increase in ‘5f’ population on complexation with chalcogenide ions (5f excess) is significantly less for the f0 and f7 systems than the rest of the systems, which indicates the resistance of these systems to acquire more electrons. Similar observation was also reported in literature where the 5f excess population of the trivalent Am (f6) was the highest in comparison to that of f3, f4, f5, f7 centres upon complexation. This was addressed by the authors as the allure of the stable half-filled f7 configuration, which renders the 5f excess populations of f6 formal configuration the maximum and that of f7, the minimum.[26, 45]