Bonding of isovalent homologous actinide and lanthanide pairs with chalcogenide donors: effect of metal f-orbital participation and donor softness

Chemistry of f-element gains importance in several fields due to the extensive scope of their applications. The aim of this study is to understand the subtle differences in bonding in the structurally similar actinides (An) and their homologous (isovalent) lanthanides (Ln) with several donors, which may lead to their covalency mediated separation. Several experimental and theoretical studies have been reported to address this aspect. However, to the best of our knowledge, a systematic study on the variation in the bonding patterns of the isovalent ‘Ln’ and ‘An’ pairs encompassing the effect of valence f-orbital participation was not attempted. In this study, the minute differences in covalent interactions of these An/Ln pairs having same number of f-electrons with chalcogenide ions of varying softness was probed using relativistic density functional theory (DFT). The f-electronic configurations of the metal ions were observed to play an important role in the f-orbital participation. [AnX]+/[LnX]+ pairs with f0, f7 (half-filled) configurations show resistance to f-orbital directed bonding, unlike the systems having f5, f6 electronic configurations where the f-orbital directed covalency is more pronounced in the [AnX]+ systems. The nature of covalency was identified to be near degeneracy-driven. This enhanced covalency was found to increase with the subsequent increase in the softness of the donor centres.


Introduction
The rich and fascinating chemistry of the f-elements and wide range of their applications excite theoreticians, experimentalists as well as technologists. The understanding of the bonding of f-elements, i.e., the actinides (An) and the lanthanides (Ln) with various donor centres may be helpful in designing suitable extractants for their successful separation to reduce the load of long-lived highly active waste in the nuclear industry [1]. Moreover, some of the lanthanides and actinides present in the nuclear wastes find their utility in numerous forms such as misch-metal [2], phosphors, laser materials [3], catalysts [4], high temperature superconductors [5], ionisation and thermoelectric sources, and permanent magnets [6]. Lanthanides mostly exhibit trivalent oxidation state due to the unavailability of their deep seated 4f-orbitals for bonding. Unlike the lanthanides, early actinides exhibit variable oxidation states due to participation of the 5f-orbitals in bonding to some extent, whereas heavier actinides (beyond Pu) predominantly exhibit + 3 oxidation state due to significant stabilisation of the 5f-orbitals. Separation of these trivalent 'An' and 'Ln' ions employing conventional 'O' donor based extractants is not feasible due to their very similar chemistry [7,8]. To achieve their separation, the minute differences in the thermodynamic stability of homologous 'Ln' and 'An' complexes as a result of small differences in their covalent interactions with soft donors like N, P, S, and Cl are exploited [9,10]. Affinity of the relatively softer actinide ions (with respect to lanthanides) with soft donors is in line with the well-established hard soft acid base (HSAB) principle [11]. Nature of bonding in the actinide and lanthanide complexes can be probed both experimentally and computationally [12,13]. Several experimental techniques like photoelectron [14], Mössbauer [15], electron paramagnetic resonance [16] spectroscopies are extensively used for this purpose. X-ray absorption near edge spectroscopy (XANES) is an unparalleled technique used to unfold the covalency of in the complexes of the f-block elements [17]. Computational studies including orbital and density-based analyses contribute significantly towards understanding the nature of covalency where experiments are difficult to perform.
Traditionally, a covalent bond is defined as the sharing of electrons between two atoms followed by accumulation of electron density in between the nuclei. But from a quantum chemical perspective, covalency may be achieved by two means viz., (i) orbital overlap driven or (ii) by minimising the energy gap between the interacting orbitals, i.e., (near) degeneracy-driven. It is reported that the thermodynamic stability can only be acquired through the orbital overlap driven covalency [18,19]. In the standard molecular orbital theory, atomic orbitals combine (through spatial overlap) to produce molecular orbitals. The associated atomic orbitals should have same symmetry and similar energy (near degeneracy). It is because the linear combination of nondegenerate eigen functions of an operator is not an eigen function of that operator and hence the generated molecular orbitals will lose their connection with the Hamiltonian. In the bonding molecular orbitals, electron density gets accumulated in between the nuclei and gets depleted elsewhere providing essentially the covalent bonding.
An interesting review article elaborated on most of the contemporary computational studies which dealt with An/ Ln-donor bonding. It indicated predominant ionic nature of the f-element-donor bond is indicated with relatively less covalent contribution. The covalent bonding was successfully probed with topological study of electron density in the bond by quantum theory of atoms-in-molecules (QTAIM) approach, albeit with the support of orbital based analyses [19]. Reports pertaining to the bonding of the An/Ln ions with different donor centres indicated that the covalency in these bonds is mainly metal d-orbital directed, with subtle f-orbital contribution. The nature of covalent bonding between the actinide ions and softer donors is mostly energy degeneracy-driven, as the difference in energy between the filled donor and the empty metal-centred orbitals are significantly less in these cases [20][21][22][23]. Moreover, such bonding is also sensitive to the coordination environment of these metal ions and the ligand structure where the information of the actual interactions between the metal ion and the isolated donor is lost. Hence, in these systems, it was considered relevant to understand the bonding between the lanthanide or actinide ion with donor atom, through diatomic gas phase model systems, so that their exclusive interactions can be zoomed in. The nature of bonding of trivalent lanthanide or actinide ion with varying (n-2)f configurations interacting with divalent donor ion forming an [MX] + type unit is investigated in this work. Dinegative donors of the chalcogenide series were chosen to cover a wide spectrum of softness, which is expected to tune the covalent interactions significantly, thereby their separation behaviour. Systems like [AnS] + and [AnO] + have been thoroughly studied experimentally using mass spectrometry linked thermochemical analysis [24] and theoretically, using high end computation [25][26][27][28]. These studies have set up a yardstick for our theoretical calculations. Also, such units maybe conceived to form the primary interacting module for real ligand and metal ion bonding [29,30]. Rigorous computational studies have been carried out on [AnO] + like systems elucidating the extent and nature of covalency by analysing the QTAIM metrices at bond critical points which served as a stencil for our problem formulation [31]. Similar studies on corresponding lanthanide based systems are rare in the literature. To the best of our knowledge, no systematic study highlighting the differences in the bonding of Ln and An pairs having same number of 'f' electron is reported in literature.
In this work, a comparative study on the bonding patterns in [AnX] + and their corresponding [LnX] + systems having same number of 'f' electrons is carried out by applying the established methods like the orbital and QTAIM analyses. The energetics components of the formation of such systems was studied using energy decomposition analysis (EDA). This study is expected to be helpful in understanding the minute differences in the f-orbital directed bonding in the complexes of these homologous trivalent An and Ln pairs, and thereby serving as a guide towards designing ligands for their covalency-driven separation.

Computational details
In this study, we have made an attempt to understand the bonding interactions of trivalent actinides, viz., Pu 3+ , Am 3+ , Cm 3+ and their homologous lanthanide analogues, viz. Sm 3+ , Eu 3+ , Gd 3+ with dinegative chalcogenide ions, viz., O 2− , S 2− , Se 2− using unrestricted relativistic density functional theory calculations to shed light on the difference in affinity of the An/Ln ions towards different donors of varying 'softness'.
[AnX] + or [LnX] + systems of trivalent Pu/Sm, Am/Eu and Cm/Gd were considered to have ground state high spin configurations of f 5 , f 6 and f 7 respectively. Interaction of f 0 systems, viz., trivalent Ac/La with the chalcogenide ions were also studied to understand the impact of unpaired f-electrons in bonding. Systems were named as (n) [AnX] + or (m) [LnX] + where n or m are the number of unpaired f-electrons and An or Ln represent the actinide or lanthanide corresponding to those many unpaired f-electrons, e.g. monopositive plutonium (III) sulphide system with 5 unpaired f-electrons will be represented as (5) [PuS] + system. A consolidated table consisting of the mapping of number of unpaired electrons with the An 3+ /Ln 3+ ions is presented below in Table 1.
Geometry optimisation and single point calculations for all the studied systems were performed at the all electron B3LYP [32,33]/TZ2P [34] level. Relativistic corrections were incorporated with scalar zeroth-order regular approximation (ZORA) [35,36]. Absence of imaginary frequencies confirmed that the optimised structures correspond to the minima on their respective potential energy surfaces.
Energetic contributions associated with the binding of the trivalent An or Ln ions with the chalcogenide ions were analysed using energy decomposition analysis (EDA) as prescribed by Ziegler and Rauk [37]. As it was previously reported that the energetics trend varies minimally with basis sets [23], the analysis was performed on the optimised structures obtained at the B3LYP/TZ2P/scalar-ZORA level. Within the framework of EDA, the bonding energy (ΔE bond ) of the complex is decomposed into the following interaction terms: stabilising electrostatic interaction (ΔE ion ), orbital interaction (ΔE orb ) and repulsive Pauli exchange interaction arising from the fragments (i.e. Ln 3+ or An 3+ with X 2− ) of the same spin (ΔE Pauli ).
Natural population analysis (NPA) [38,39], which is reported to work better in describing the bonds with minimal basis-set dependence, along with computation of atoms-inmolecule charges were carried out for partial charge analysis.
QTAIM descriptors like electron density (ρ), laplacian of the electron density (∇ 2 ρ) and total energy density (H(r)) were analysed to understand the ionic or covalent characters in metal-donor bonding. Nature of the bond is analysed by partitioning the molecular electron density (ρ(r)) region at the zero flux surface, thereby explaining bond formation by a (3, − 1) bond critical point (BCP). The properties at the BCP provide a quantitative understanding on the extent of electron sharing between the metal and the donor centres. Additionally, Wiberg bond indices (WBIs) and the delocalization indices (DI) were computed as a measure of the shared electron density between the bonded atom [40][41][42][43].
Frontier molecular orbitals (MO) and natural bond orbitals (NBO) [44] were analysed to understand the extent of participation of metal and ligand orbitals during bonding. All the abovementioned calculations, unless stated otherwise, were carried out in gas phase using ADF 2017 software [45]. NBO analysis was carried out using the GENNBO module of the ADF software, which links the same to NBO 6.0 program [46]. Negligible spin contamination was noted for all the systems.

Relative stabilities of the [AnX] + and [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 [47,48]. 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 and LUMO) are mostly higher for the [LnX] + systems than the corresponding [AnX] + systems, indicating higher stability for the former.
The interaction (bonding) energies between the An 3+ / Ln 3+ adn X 2− fragments and their components were calculated in energy decomposition analysis.

Variation in bond lengths of the [AnX] + and [LnX] + systems
The calculated bond lengths of the [AnO] + and [AnS] + systems were benchmarked with the data in the referred articles [24][25][26][27][28]. Calculated bond lengths were in good agreement with the literature data (deviation of maximum 2%). The bond lengths of the [AnX] + and the corresponding [LnX] + system with same number of f-electrons is presented in Fig. 1.
The bond lengths of [AnX] + systems are expected to be longer than the corresponding [LnX] + systems due to the larger ionic size of the An 3+ ions. This trend is nicely followed by all the metal oxides and the f 0 and f 7 (half-filled) metal-chalcogenide systems. Interestingly, in case of f 5 (Pu 3+ /Sm 3+ ) and f 6 (Am 3+ /Eu 3+ ) systems, significant elongation of the Ln-donor bond is observed for the sulphides and selenides. This hinted that the presence of unpaired 'f' electrons in case of partially filled f-orbitals (f 5 and f 6 in the present case), except f 7 (halffilled f-orbital) has some role in the differential bonding of the trivalent An/Ln ions with softer donors viz., S 2− and Se 2− .
Charge analysis on each of the fragments was anticipated to be useful to understand bonding features of these 'f' block metal ions with the chalcogenides, and hence 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 S3 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 neutralisation 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 Ln 3+ ions can be attributed to the higher polarising power of the Ln 3+ ions with respect to the corresponding An 3+ ions by virtue of their difference in ionic radii. Also, the increased polarisability of the softer donors from sulphide to selenide ion renders charge donation to the metal ion subsequently easier.
After obtaining an insight on the charge transfer, further understanding of the nature of bonding could be obtained from the calculation of electron density shared between the metal and donor ions and their Laplacians, by the QTAIM analysis.

Metrices at bond critical point (BCP)
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. ρ BCP , 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, ρ BCP > 0.2 a.u. and ∇ 2 ρ BCP < 0 indicate pure covalent bonding. Closed shell interactions are indicated by ρ BCP < 0.1 a.u. and ∇ 2 ρ BCP > 0. ρ BCP with intermediate values indicate partial covalent character [40][41][42]. For all the  Table S4 in SI) for oxides of Pu, Am and Cm are in well agreement in the referred text [31]. The ρ values are the highest for the An/Ln oxides, followed by sulphides and selenides. The An-O and Ln-O bonds have ρ more than 0.2 e/bohr 3 , indicating significant covalent contribution to bonding. The sulphides and some of the selenides of both [AnX] + and [LnX] + have ρ values higher than 0.1 e/bohr 3 indicating partial covalent interactions. Only for the f 0 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. Whereas, in case of the partially filled unpaired f-electron systems (i.e. f 5 and f 6 ), higher ρ was observed for the 'An-X' bonds with X = S 2− and Se 2− . Similar trend was also noted from the H(r) values.

Extent of electron delocalisation
The delocalisation index [δ(A,B)] obtained 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 ρ BCP are accompanied by large values of δ (A,B), the converse is not always true [49]. 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 S5 in  In case of the f 0 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 (5) [PuS] + , (6) [AmS] + , (5) [PuSe] + and (6) [AmSe] + than their corresponding [LnX] + s.
Consolidated plots of the electron densities at BCP and delocalisation indices are shown in Fig. 2.
Upon summarising the results of the QTAIM analysis, we see that the electron density values at BCP indicate significant covalent bonding only in case of the oxides. In most of the cases, the ρ BCP and δ(M,X) values are higher for the [AnX] + systems than the [LnX] + counterparts, except the f 0 systems. 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 f 5 , f 6 sulphides and selenides. Electron densities at BCP for all the [AnX] + systems are slightly more (max 25%) than the corresponding [LnX] + . Sulphides and selenides of metal ions with f 5 , f 6 electrons 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.

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 with the MO of the other systems is provided on Table S1 in SI.
This indicates a predominant metal d-orbital directed bonding [21][22][23]. But, the effect of the metal f-orbitals is yet to be understood. In order to have a deeper insight into the relative involvement of the valence d/f-orbitals of the An 3+ or Ln 3+ ions with the chalcogenide ions, natural bond orbitals (NBO) analysis was performed.

NBO analysis of the [AnX] + / [LnX] + systems
Natural bond orbitals [51] 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 An 3+ to that of the Ln 3+ ions in order to provide   (6) [AmS] + / (6) [EuS] + ( 7) [CmO] + / (7) [GdO] + π type d-p overlap σ type d-p overlap a comparative account of bonding in the [AnX] + and [LnX] + systems [ Figure S2 in SI].
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 f 0 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 (0) [LaX] + systems than (0) [AcX] + systems. In the sulphides and selenides of Pu and Am, 4-8 times larger f-orbital contribution was observed than their corresponding [LnX] + counterparts, as is expected from the known low energy (and the consequent relatively inert character) of the Ln 4f-orbitals. 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 Fig. 4 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. Both the delocalisation energies are the highest for the f 0 systems irrespective of the chalcogenide ions. Prominent metal f-orbital participation in (5)/ (6) [AnS] + and (5)/ (6) [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] + / [LnX] + s, the natural electronic configurations (NEC) were analysed. The NECs for these metal chalcogenides were obtained by summing up the natural atomic orbital occupancies. The increase in metal 'f' population upon bonding with chalcogenide ions (f-excess) is defined as the difference between f-population from NEC and f-population in the trivalent ion in Fig. 5.
The f-excess is observed to be significantly less for the f 0 and f 7 systems than the rest, which indicates the resistance of these systems to accept more electrons in the f-orbitals. Significantly high f-excess populations on the Sm and Eu centres in the sulphide and selenide series along with poor delocalisation indices. This indicate that in order to approach this favourable 4f 7 case, electron density from the other valence orbitals of Sm and Eu moves to the 4f-orbital, rather than engaging in electron sharing with the donor atoms. The localisation indices on the An 3+ /Ln 3+ centres (Table S6 in SI) also indicated corroborative trend. This electronic structure of the aforementioned systems might have led to the significant elongation of the respective bonds, as was also observed in Fig. 2. Similar observation has also been reported in the literature for actinide systems where the 5f excess population of the trivalent Am (f 6 ) was the highest in comparison to that of U (f 3 ), Np (f 4 ), Pu (f 5 ) and Cm (f 7 ) centres upon complexation. This was addressed by the authors as the allure of the stable halffilled f 7 configuration, which renders the 5f excess populations of f 6 formal configuration the maximum and that of f 7 , the minimum [30,50].

Conclusions
A number of features of the bonding between trivalent An/Ln ions and the chalcogenides of varying softness were highlighted in this study using the results from Fig. 4 Delocalisation energies from the second order perturbation theory the DFT-based calculations. The analyses reflected that the interaction between the ions is mostly ionic with a small covalent contribution rendering the [LnX] + systems more stable than the [AnX] + systems. This covalency for both set of systems is predominantly metal d-orbital directed with minor participation of the metal f-orbitals. While zooming into the f-orbital directed covalency, it was noted that the f-orbital participation is significantly higher for the [AnX] + than the [LnX] + systems. Interestingly, the initial f-orbital electronic configurations of the metal ions also seem to play an important role in their f-orbital directed bonding.
[AnX] + /[LnX] + pairs having f 0 , f 7 (half-filled) configurations behave in a similar fashion, whereas the systems having f 5 , f 6 configurations behave in a completely different manner. The former set expresses resistance to f-orbital directed bonding. QTAIM study, on the other hand, indicated that the f-orbital directed covalent bonding is pronounced for the f 5 and f 6 [AnX] + than the corresponding [LnX] + systems, and its nature is near degeneracy-driven. The extent of near degeneracydriven covalency increases with the subsequent increase in the softness of the donor centres, i.e. from oxides through sulphides to selenides. However, this enhanced near degeneracy-driven covalency, did not contribute to energetic stabilisation of the systems. These results are anticipated to provide deeper understanding on exclusive differences in bonding of the homologous An 3+ /Ln 3+ ions with the chalcogenide donors in gas phase, where the solvent or ligand environment is not expected to affect the metal-donor bonding.