Hund’s three rules in actinide-containing superatoms with spin-orbit coupling calculations

The intriguing and challenge issue in magnetic superatoms is searching for the suitable candidates to validate the Hund’s rules. Here, early actinide elements (An: Ac, Th, Pa, U, Np, Pu, Am) whose 5f electrons may crossover the localization and delocalization characteristics have been chosen to alloy with Al atoms in designing magnetic An@Al12 superatoms. By doing the global minimum structure search and the spin-orbital coupling density functional theory calculations, we provide an original idea to give theoretical argument that Hund’s three rules are still applicable in superatoms, which can be related to the fillings of highly localized An-5f orbitals into large exchange-splitting 2 F superatom orbitals. Specifically, selective 5f sub-orbitals of several An dopants can exhibit a dual nature in superatomic bonding, i.e. partial 5f electrons of Pa, U and Pu are reactive whereas all 5f electrons of Np and Am are highly localized. The molecular orbital analyses, combined with the qualitative interpretation of the phenomenological superatom sub-shell model, address the intricate interplays between the structure symmetry, electronic structure, spin and orbital magnetic moments. These findings have important implications for understanding the bonding and magnetic behaviors of An-containing superatoms and pave the way for designing novel magnetic superatoms.

For open-shell atoms in the ground state, the spin-orbit coupling (SOC) determines the total angular momentum according to the formula J = L-S/L+S when the atom is below/above the half-filling (Hund's third rule), after maximizing total spin angular moment S = ∑ s i (first rule) and total orbital angular moment L = ∑ l i (second rule).For magnetic superatom, different from the structureless atom, the situation is obviously much more complicated because neither of three quantum numbers (S, L, J) is well defined due to the breaking of atom-like spherical symmetry.Nevertheless, some of the basic mechanisms of Hund's rules are still relevant.For example, high spin moments have been reported in magnetic superatoms of Mg 8 Fe [14], Na 8 V [35], Al 12 Cu [36], BLn [37], etc. [38][39][40] in accordance with the Hund's first rule of maximum spin moment by filling the sub-shell in one spin channel.The stabilities of these open-shell superatoms are generally attributed to large energy gap between splitting D/F sub-shells induced by the crystal field, and their high spin magnetic moments are found to be depend on the unpaired d/f electrons that are locally occupied in D/F superatomic shell.Given the evidences that only Hund's first rule applies approximately and the feature of sizable orbital moment is undescribed in magnetic superatoms [41,42], one thus wonders whether Hund's second and third rules are still valid in magnetic superatoms in their ground states.
The An atoms in yielding the chemical bonding interaction, partially due to the inherent characteristics of 5f electrons, can exhibit either localization similar to 4f electrons in lanthanide (Ln), or partial delocalization similar to d electrons in TM, or the dual-character of localization and itinerant [43].When An atom as a dopant is introduced into the sp-block element clusters, it will be a fascinating structural model to provide insight into the fundamental understanding of unusual characters of 5f bonding in many intermetallic actinide compounds [44][45][46].Even if these 5f electrons do not participate in chemical bonding and only contribute to magnetic moment in the unpaired manner, 6d7s electrons of An atoms can interact inter-atomically with sp valance electrons of host atoms.In a rational term, it can guarantee An-containing clusters undertaking high structural stability and large magnetic moment, being as potential magnetic superatoms.Since heavy An atoms within strong SOC have large orbital magnetic moments in nanostructures, inclusion of SOC effect in magnetic clusters allows us to construct a platform for inspecting the Hund's second and third rules.
To extend Hund's first rule to second and third rules in magnetic superatoms, we focused on spin and orbital magnetic moments of An-containing clusters by constructing the spherical jellium model structures.Various magnetic superatoms are correctly predicted by considering the structure symmetry, in addition to the number of valence electrons and the number of constitutive atoms.We demonstrated that large exchange-splittings of atomic 5f states of An dopant lead to the hybridization with the F superorbitals to construct the 2 F superorbitals, which further undergo a spin-splitting by the ligand-field actions and form the sub-superorbitals.The fillings of 5f electrons in partial 2 F sub-superorbitals can maximize the spin moment and give the symmetry-allowed high orbital moment, together with the antiparallel coupling between spin and orbital moments, which are in accordance with the Hund's rules.

Computational methods
Our calculations were performed using the density functional theory (DFT) method with generalized gradient approximation (GGA) for exchange-correlation energy functional in the form introduced by Perdew, Burke and Ernzerhof (PBE) [47,48], which was implemented in the Vienna ab initio simulation package (VASP) [49].The plane-wave cutoff energy for the projected augmented wave (PAW) pseudopotentials was set to 450 eV, and reciprocal space integrations were carried out at the gamma point only.For the improvement of the self-consistent field (SCF) procedure, we initially imposed the largest spin magnetic moments on each An atom that are equal to the numbers of unpaired 5f electrons in terms of the first Hund's rule, and we adopted the Gaussian smearing method with broadening parameter 0.1 eV.The projected density of states (PDOS) is constructed with very accurate 13 × 13 × 13 k-point grids.The valence electrons of An atoms were considered as the ground electronic configurations (6s 2 6p 6 5f n 6d 0−2 7s 2 , n = 0-7).The An@Al 12 cluster was placed in a 15 × 15 × 15 Å cubic supercell and thus the interaction between two neighboring cluster images was negligible.
The CALYPSO (crystal structure analysis by particle swarm optimization) code combined with the DFT calculations were utilized to search for the global minima of An@Al 12 [50,51].For the structural predictions, (i) we have created 50 loop generations within 30 structures per generation, where 70% structures were generated by the PSO algorithm and the others were generated randomly; (ii) symmetry constraints were incorporated during the structure generation to narrow down the search space and enhance structural diversity; (iii) 1500 generated original structures were optimized with the convergence criterion 10 −4 eV and the maximum residual force on each atom less than 0.02 eV A −1 .Subsequently, the top thirty low-lying structures were re-optimized with 10 −5 eV and the maximum residual force less than 0.01 eV A −1 .Among them, the first nine low-energy structures were shown in figure S1 in the supplementary material.The global minima searches indicate that An@Al 12 clusters have an icosahedral structure and An atom always prefer to exterior sit at the surface (C 5v , figure 1).
Due to strong correlation of An-5f orbitals, we taken into account the on-site Hubbard correction by employing the Dudarev's GGA+U scheme [52], and adopted previously reported values of U eff = U-J = 4.0-0 = 4.0 eV for An atoms [53][54][55][56].To obtain the spin and orbital magnetic moments, the spin-orbital coupling (SOC) effect was included in the SCF calculations within GGA+U+SOC method.This approach can correct the errors associated with standard DFT-GGA calculations, particularly in systems with the highly localized 5f electron states of An atoms.Although high-level ab initio methods such as the multiconfigurational quantum chemical methods (CASSCF/CASPT2) are more reliable than DFT to predict accurate properties [57], the DFT based method strikes an balance between accuracy and computational cost.Typically, our calculations need to consider the SOC effect for obtaining the orbital moments, which further increase the difficulty and computational cost of using ab initio methods.So far, experimental and DFT characterizations have been performed on the Al-based TM@Al 12 superatoms [23,58], endohedrally doped cage clusters [4] as well as other superatomic clusters [2].These studies have confirmed the reliability of cluster models and GGA+U+SOC methodologies in predicating the electronic structures and magnetic moments.
To verify the reliability of our adopted parameters, we taken Pa@Al 12 as a test example and increased the cutoff energy from 450 eV to 500 eV and to 600 eV.The results indicate that the energy change are less than 0.1% (increase less than 0.001 eV).To investigate the influence of the Hubbard correction U parameters on the electronic structure and magnetic moments (Hund's rule filling), we chosen U@Al 12 as the testing system.It was found that spin magnetic moments remain unchanged while orbital magnetic moments increase gradually with the increase of U value (table S1).Nevertheless, the maximum change in orbital magnetic moment is less than 9%.With the increase of U correction values, the arrangement of superorbitals as well as the filling rules of 1 F and 2 F superorbitals do not changed, except for the shifting of 2 F superorbitals towards the low-energy side, as shown in figure S2 in the supplementary material.Consequently, the cutoff energy and Hubbard U values do not affect the general conclusion we obtained.The analyses of An-Al bonding interactions were done with the help of VESTA [59], Bader [60] and Vaspkit codes [61], from which we can derive the electronic localization function (ELF), charge transfer and crystal orbital Hamiltonian population (COHP).To reinforce our obtained lowest-energy structures and further examine their thermal dynamical stabilities, we also performed the vibrational frequencies and ab inito molecular dynamic (AIMD) calculations in NVT ensemble to assess their thermodynamical stability, as shown in figure S3 in the supplementary material.For confirming the magnetic ground-states of different clusters, spin constraint calculations were done by considering the possible spin multiplicities within the optimal structure.

Geometric stability
The structural parameters, stability, electronic and magnetic properties of An@Al 12 clusters are summarized in table 1.The icosahedral structure within short chemical bonds were found, i.e, the peripheral Al-Al bond-lengths range from 2.66 Å to 2.67 Å and the An-Al bond-lengths rang from 3.04 Å to 3.27 Å.The cluster stability can be estimated by calculating the binding energy E b and formation energy E fo , via , where E Al and E An are energies of Al and An atom, respectively, E cluster and E Al12 are total energies of An@Al 12 cluster and Al 12 fragment, respectively.With the progression across the period table of the An series, in figure 2, both energy curves display the increasing trends from Ac to U dopant, followed by gradually decreasing trends to Am dopant.The E b of all systems are generally greater than Al 12 TM (TM = W, Zr, Sc, Y), which are also larger than the measured values of W@Au 12 (2.48 eV), Zr@Au 14 (2.62 eV), Sc@Au 15 (2.50 eV) and Y@Au 15 (2.53 eV) [62].The largest values of E b = 3.14 eV and E fo = 8.83 eV at U dopant suggest its highest stability.By inspecting the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), we found the transition trend that increasing first and then decreasing occurs at Th dopant, giving a largest HOMO-LUMO gap about 1 eV.The 12 Al atoms have 36 valence electrons and Th atom in 5f 0 6d 2 7s 2 configuration has 4 valence electrons, both of which contribute to 40 electrons count for a closure shell filling with the magic assignment.Nevertheless, other An@Al 12 clusters lack this specific configuration but still demonstrate significant stability.For example, although U atom in 5f 3 6d 1 7s 2 configuration has only 3 valence electrons if 5f electrons are not included, U@Al 12 is the most stable one even it has a modest HOMO-LUMO gap of 0.56 eV.On the whole, the gaps of An@Al 12 are comparable to the values of magnetic Mg 8 TM superatoms (0.2-0.65 eV) [63] and magnetic Al 12 X superatoms (0.24-0.49eV) [64].Understanding how these valence electrons fill up in the molecular orbitals of An@Al 12 clusters is worthy of the exploration for underling their stabilities.

Electronic structure
We analyzed the one electron levels and their associated orbital wavefunctions for the majority (α) and minority (β) spin states.Based on the point group theory of C 5v symmetry, the perfective degenerate superatom electronic states labeled by S, P, D, F . . .will split into one non-degenerate state and a set of degenerate doublet states, i.e.P→P z ⊕ P x/y , D→D . By inspecting the energy levels and the Kohn-Sham wave function shapes of An@Al 12 clusters from our DFT+U calculations, we can identify each molecular-orbitals and display their 3D visualizations in figure 3 and figure S4 in the supplementary material.Typically, molecular-orbitals such as 1 F and 2 F can be distinguished in terms of their nodes, shapes and energetic sequence analogous to that of atomic orbitals.On the whole, the first and second lowest energy states are spread out over the entire cluster but show spherical and dumbbell characters, respectively, which can be regarded as the 1S 2 and 1P 6 superorbitals.These states are orderly followed by four 1D 4 sub-superorbitals, 2S 1 sub-superorbital and 1D z sub-superorbital in both spin channels.Unexpectedly, the next filling orbitals are a set of 2 F states instead of the 1 F states due to the presence of An atom, which is different from the findings of homogeneous simple-metal superatoms in term of the jellium model for a nearly free electron gas [65,66].Previously, a set of 2D states instead of 1 F states were revealed as the sequential superorbitals in Mg 8 TM superatoms (TM = V, Cr, Mn, Fe, Co, Ni) [14].The following occupied states are a group of splitting 1 F superorbitals that are interspersed with 2P 3 superorbitals.
For U@Al 12 cluster from the figure 3, its majority and minority superorbitals have the energy , where Core = 1S 2 1P 6 1D 10 2S 2 2P 6 (26e) and 2 F 3 α 1 F 7 α 1 F 6 β (16e) can be regarded as the inner-shell superorbitals and outer-shell superorbitals, respectively.The arrangements of superatomic shells of other An@Al 12 are fairly robust and in analogy to U@Al 12 counterpart, shown in the figure S4 in the supplementary material, where we can note the same arrangements in inner-shell superorbitals but the different arrangements in outer-shells such as 1 F 7 for Am, respectively.The Th@Al 12 superatom is the first cluster in the series with a 40-electron closed-shell configuration 1S 2 1P 6 1D 10 2S 2 2P 6 1 F 14 .The open shells with highly degenerated superorbitals are split into a group of sub-shells with lowly degenerated or un-degenerated superorbitals when atom-like high symmetry is broken, namely, the numbers of electrons of the closure shells for non-spherical superatoms deviate from 2, 8, 18, 20, 34, 40, 58 • • • for the spherical superatoms [67][68][69].These superatoms with seemingly open-shells such as 2 F 7 α 1 F 6 α 1 F 6 β still be stable due to the sub-shell closures.This is why these clusters can maintain the C 5v structure and they do not undergo further Jahn-Teller distortions to stabilize the structures [70].
Table 1.Average bond-lengths between An and Al atoms (R An-Al ) and between Al atoms (R Al-Al ), binding energy (E b ), formation energy (E fo ), HOMO-LUMO gap (gap), charge transfer (Q), spin/orbital magnetic moment of An atom (µ An S /µ An L ) and An@Al12 (µS/µL).The bond-lengths, energy, charge and magnetic moment are in the unit of Å, eV, |e| and µB, respectively.

System (Configuration
Ac@Al12(5f 0 6d 1 7s 2 ) 3.27 2.67 2.93 6.02 0.42 −1.27 5f 0.03 6d 0.64 1 (0) 0 (0) Th@Al12(5f 0 6d 2 7s 2 ) 3.10 2.67 3.02 7.62 0.96 −0.69 5f 0.09 6d 1.21 0 (0) 0 (0) Pa@Al12(5f   Consequently, the 2 F superorbitals exclusively contributed from atomic 5f orbitals undergo significantly large spin exchange splitting, resulting in high stabilization of superatoms with unpaired electrons. In figures 4(a)-(g), we depict the PDOS of An@Al 12 superatoms to resolve the atomic-orbital contributions for each superorbital.Generally speaking, 1S 2 1P 6 1D 10 2S 2 2P 6 inner-shells are dominatingly composed of Al-3s3p states with a minor contribution from An-6d states; 1 F valence-shells are mainly originated from Al-3s and An-6d states with tiny An-7s state; 2 F valence-shells are almost entirely contributed by An-5f states.To reveal the bonding interactions between An atom and Al 12 cage and underline why 1 F valence-shells are composed of Al-3s3p and An-6d7s states, in the figure 4(h) and figure S5 within the supplementary material, we illustrated the atomic-orbital levels of isolated An atom, molecular-orbital levels of Al 12 fragment and superatomic-levels of An@Al 12 superatoms, respectively.Our calculations indicate that the 5f majority and minority spin channels are separated by 4.33-6.53eV in isolated An atom.The energy levels of 7s5f6d atomic orbitals and 1 F 14 2P 2 molecular orbitals match with each other.The introductions of 7s6d electrons from An dopants will firstly fill into the 2P molecular orbitals.If the 2P molecular orbitals is not fully filled, additional An-5f electrons will be filled into; the residual 5f electrons are highly localization and populate into 2 F molecular orbitals.Although energy levels of 1S1P1D2S2P inner-shells are downshifted by doping An atom because of more attractive natures of the potential of An dopants, there is an energy upshift of 1 F molecular orbitals due to a Coulomb repulsive barrier between 1 F and 2 F molecular orbitals that are occupied by remaining 5f electrons.The degrees of energy shift depend on the electron distributions of individual molecular orbitals.
From the table 1, we found that U(5f 2.96 6d 0.85 ), Np(5f 4.01 6d 0.86 ) and Am(5f 7.03 6d 0.48 ) dopants nearly maintain the numbers of 5f electrons as that in isolated An atom, indicative of highly localized characteristic of 5f electrons, while Pa(5f 1.10 6d 1.28 ) and Pu(5f 5.06 6d 0.87 ) dopants loss one 5f electron with respect to the values of isolated atoms, indicative of a certain delocalized characteristic of 5f electrons.The occupations of 5f orbitals were listed in table S2 within the supplemental material , where the values close to 0 or 1 indicate the high localization of 5f electrons and vice versa.Clearly, 5f electrons in Pa and U are relatively delocalization while 5f electrons in Np, Pu and Am are localization.Regarding the fact one 5f electron of Pu undergoes transfer to 6d orbital, we proposed that partial 5f electrons in Pu also have the characteristic of delocalization.
Since An-5f electrons are exclusively occupied the spin-up channels and exhibit spin magnetic moments µ 5f s , the magnetic differences between An spin moments µ An s and An-5f spin moments µ 5f s are from the contributions of An-6d electrons, i.e. small values of µ 6d s ≃0.20 µ B for U and Np dopants and negligible values for other dopants.It implies that An-6d electrons are substantially involved in the chemical bonding to form the covalent and ionic bonds, they can be roughly regarded as the valence electrons.To examine the charge transfers between atoms, we performed the Bader charge analyses by doing individual calculations with the DS-PAW [71] and VASP softwares, listed the calculated values in table S3 within the supplemental material .It is clear that their values are consistent with each other for every An dopant.Furthermore, a remarkable charge transfers from An to Al atoms can be detected except for Th and Pa dopants, implying the existence of the dominate An-Al ionic character.For other superatoms, charge transfer is modest and the certain covalent bond is assumed to contribute to the structural stability.Note that the amount of transferred values cannot accurately reflect the valence state of An dopants because of the Bader analyses method.Finally, we should emphasize that there is almost no occupied 7s states from the PDOS in figure 4, suggesting the entirely losing of two 7s electrons for An dopants.Thus, it can be concluded that Ac, U, Np, Pu dopants are in the III-valent states, Th and Pa dopants are in the IV-valent states, while Am is in the II-valent state; one 5f electron of Pa and Pu dopants shows delocalization via the excitation to 6d orbital in participating the bonding interactions as discussed in the following.
In figure 5, we plotted the projection map of the ELF on a specific plane within An-Al and Al-Al bonds.The figure S6 in the supplemental material illustrates the ELF obtained via DS-PAW software [71].The clearly divided regions between An and Al atoms indicate the yielding of An-Al ionic bonds.There is red crescent surrounded the Al atoms, suggesting that Al-sp electrons are already being as the cluster itineration electrons.Furthermore, Ac, Th and Am dopants exhibit the green near-spherical distribution with a light yellow-ring in it, while U, Np and Pu dopants display the green hexapetal shapes.The former evidences correspond to the losing of 7s inner-shell, while latter evidence indicates the changes of population electrons on 5f orbitals.To inspect the covalent bonds between An and Al atoms, we performed the computations on the COHP, and displayed the orbital-resolved An-Al bonding states (positive values) as well as the anti-bonding states (negative values) in figure 6.For the dopants from Ac to Np, their bonding states come dominantly from 6d-3p hybridizations (around −1 eV) and partially from 6d-3s hybridizations (around −3.5 eV); from Pu to Am, their bonding states are exclusively related to 7s-3s hybridizations.Intriguingly, Pa/U-5f states also participate in the bonding interactions with Al-3p states (around −2 eV), again suggesting the delocalization of their 5f electrons.Regarding the anti-bonding states (around −3 eV), due to the electrostatic repulsions between An-7s and Al-3s states, their peak magnitudes and energy positions decrease gradually from Ac to Pa and subsequently increase from Pa to Am, which enhance and suppress the An-Al interactions, respectively.The cancellation between the bonding and antibonding states lead to a variational energy trend (first increasing and then decreasing), as shown in figure 2. Therefore, An dopants also exhibit the covalent bonding characteristics and give positive contribution to structural stability.

Magnetic moment and Hund's rules
The calculated spin (µ An S ) and orbital magnetic moments (µ An L ) are listed in table 1 and displayed in figures 7 and 8.The fillings of the outer-shell superorbital are presented to give the interpretation of spin moments in terms of the distribution of unpaired electrons in frontier orbitals.From figure 7, there is a configuration transition from an open-shell to a close-shell (1 F 13 →1 F 14 , 1µ B →0µ B ) as the 1 F shell is fully filled at Th dopant.Subsequently, the gradually introduced electrons stemming from the localized An-5f electrons have a priority to occupy the 2 F inner-shell superorbitals and exhibit the spin magnetic moments.Nevertheless, if both of two 5f electrons of Pa atom occupy the 2 F superorbitals, the frontier 1 F superorbitals still need one electron to be fully filled.Under the premise of ensuring electronic structural stability within the 1 F close-shell, 5f electrons exhibit a 'dual' characteristics of itinerant and localization.Consequently, one 5f electron of Pa dopant is assumed to excite to 6d states in hybridizing with the sp state of Al atoms for fully filling of the 1 F shell, while one residual 5f electron is still localized in 5f orbital and contribute to the inner 2 F superorbitals, showing 2 F 1 1 F 14 electronic configuration and exhibiting 1µ B spin moment.Starting from U dopant, there is a relatively large energy gap between 5f and 6d orbitals (figure S5 in the supplementary material), making it difficult to be excited between them.As a result, three/four 5f electrons of U/Np atoms entirely occupy in the inner 2 F superatomic orbital, which give an open-shell superorbitals in 2 F configuration and exhibit spin moment of n 5f +1 = 4 and 5 µ B , respectively.For Pu dopant, its two 7s electrons are inter-atomically transferred to Al atoms in filling the 1 F frontier superorbitals (N = 12 × 3+2 = 38).One more electron is required to get the 1 F 7 α 1 F 6 β half-filling close-shell (or two more electrons for 1 F 7 α 1 F 7 β close-shell, because the 1 F frontier sub-superorbital is a splitting singlet state in both spin channels (half filling is possible but does not seem to be important).Then, two 5f electrons of Pu atom are intra-atomically transferred to empty 7s orbitals regarding tiny energy level gap between 5f and 7s atomic orbitals.Subsequently, one is transferred to Al atoms again in occupying the 1 F α superorbital and the other one occupies the 2D α superorbital.This gives 2 F β configuration and spin moment of n 5f−2 +1+1 = 6 µ B .The unusual 'dual nature' of the 5f orbitals has been reported in An@Au 14 superatoms [72], UPt 3 compounds [73].In an earlier study, we have found that Pa and U adatoms on the cavity site of porous graphene-like gh-C 3 N 4 also lose their 5f electrons to present the itinerant characteristics [74], and these delocalized 5f electron occupy the 5f -suborbitals that are matching and conducive to hybridize with the ligand atom orbitals.Here, Pa and Pu dopants in An@Al 12 superatoms exhibit the similar behaviors.These results demonstrate that whether 5f electrons of An atoms present the delocalization in materials are element-dependent selective to give the closure sub-shells and also orbital-dependent selective to sustain the covalent hybridizations.
For Am dopant, the characteristic of half-filled 5f orbital renders its seven 5f electrons high localization and thus they occupy the 2 F 7 inner-shell.Compared to Pa dopant that two 6d electrons originate from the inherent owner one, Am dopant with the absence of 6d electrons presents the 1 F occupy of 1 F 6 α 1 F 6 β .Correspondingly, 2 F n 5f α 1 F 6 α 1 F 6 β configuration of Am@Al 12 gives the spin moment of n 5f = 7 µ B .Here, we should emphasize that 6d electrons mentioned above do not actually fill the An-6d orbitals, and they are partially transferred to Al-3p orbitals through the ionic bonds with Al atoms, making the actual number of 6d electrons being less than 2 in the superatomic electronic configuration (table 1).On the whole, the Hund's rule filling of 2 F shells is still suitable in open-shell An@Al 12 superatoms, which is in accordance with the Hund's first rule of maximizing spin multiplicity.
Since the superatomic orbital moment primarily originates from the An-5f states, in figure 8, we showed the orbital moments µ An L of An dopants.For Ac and Th dopants, nearly no 5f electrons as well as basically negligible unpaired 6d electrons lead to zero µ An L .For Pa and U dopants within n 5f = 1.1 and 2.96, respectively, their µ An L = −1.43 and −4.40 µ B are correlated to the population occupancies of one and three 5f electrons in 5f −2 and 5f −3,−1,0 sub-orbitals, respectively, as illustrated in the figure S7 of the supplementary materials .Note that the numerical calculations using the conventional GGA+U+SOC method often underestimate the orbital moments and the derived µ An L correspond to ⟨L⟩ instead of L.
Consequently, we argued that the resultant orbital angular momenta L = ∑ l i = −2, −4 of Pa and U dopants can be approximated by summing the orbital angular momenta of the individual electrons, respectively.Under the even occupation of four electrons in 5f ±3,±1 sub-orbitals, Np dopant gives a zero µ An L (L = 0).Followed by analogy, the fillings of 5f electrons in 5f −3,±2,±1 and 5f ±3,±2,±1,0 lead to Pu and Am having µ An L = -2.81(L = -3) and 0.14 µ B (L = 0), respectively.Finally, let us discuss the validation of the Hund's second and third rules in An@Al 12 superatoms.The Hund's rules hold for isolated atoms without any perturbation of spherical symmetry and under the weak SOC coupling.In the presence of ligand field actions on An atoms and thus chemical bondings in An@Al 12 superatoms, the energy orders of the splitting 5f sub-orbitals are in the sequence different from that in isolated An atoms.For example, U dopant has ,−1 occupations according to the Hund's second rule for isolated U atom.Based on the symmetry-adapted orbital model for clusters [75], the order of these sub-superorbital levels should obey a certain law with respect to the structural symmetry.Here, different charge distributions on An dopants give rise to different level shifts as well as splitting widths of 5f sub-orbitals due to different effective ion potentials acting on the 2 F superorbitals.To stabilize the structures, suitable numbers of valence electrons are required to coincide with the filling conditions of 5f sub-orbitals.Since the energy differences among these sub-orbitals are delicate, their energy orders would be adjusted appropriately to meet this requirement.This is reason why 5f sub-orbitals are in different energy sequence in different An@Al 12 superatoms even they are in icosahedral structure with same C 5v symmetry.Note that our magnetization is constrained along the C 5v symmetry axis, ensuring L z component comparable to total orbital angular momentum L.Although 5f electrons in An@Al 12 superatoms do not sequentially fill in seven sub-orbitals of 5f −3 , 5f −2 , 5f −1 • • • 5f +3 in the majority spin channel to maximize the total orbital angular moment due to the breaking of spherical symmetry, orbital moments ⟨L z ⟩ = ∑ ⟨l i z ⟩ obtained from our GGA+U+SOC calculations are very close to the allowable maximum values L = ∑ l i derived from the sub-orbital filling within non-SOC calculations.Generally speaking, the Hund's second rule L = ∑ l i is approximately applicable if we discard the inevitable influence of orbital splitting determined by non-spherical symmetry of superatoms.Consequently, the validation of the Hund's first rule in superatoms is quantitative whereas the Hund's second rule is qualitative in nature.
The Russell-Saunders (LS) coupling and jj coupling are two limiting cases for the coupling of angular momenta in multielectronic atoms, and LS coupling ( 2S+1 L J ) refers to the Hund's third rule.Historically, LS/jj approach is generally appreciated for light/heavier atoms with weak/strong SOC interaction, because strong SOC interaction will lead to the mixtures of the Hund's rule ground states with other LS coupled states having the same J value.The jj coupling scheme with large splitting sub-shells of 5f 6  5/2 and 5f 8 7/2 can give the reasonable interpretation of the metallic or insulator characters of An-based compound like AnO 2 bulk [54,76].To derive the orbital moment, practically, we have to perform the self-consistent DFT calculations by including the SOC interaction, which in turn results in S and L no longer being good quantum numbers and unacceptable Hund's rules.Nevertheless, from the PDOS of An@Al 12 superatoms under the SOC calculations figure 9, we found that the SOC splitting is negligibly faint with respect to the ligand-field splitting.Moreover, if one applies jj scheme to Pu dopant, the 5f 5 occupations of 5f 5/2 or 5f 7/2 sub-shells will definitely result in spin/orbital magnetic moment being remarkably less than ⟨S z ⟩ = 5  2 /⟨L z ⟩ = 3 due to the paired filling manner in the lowest 5f 5/2 spinor orbitals.This is contrast to our findings under the SOC calculations, i.e. the maximum unpaired electron spins and large angular momentum.It means that the Russell-Saunders SOC coupling scheme can provide a good approximation.In agreement with the predictions of the Hund's third rule, we found that the orbital moment and spin moment of An dopants are always coupled antiparallel in An@Al 12 superatoms.In addition, 5f orbitals play a pivotal role in facilitating 2 F superatomic orbital, because An-5f orbitals can induce the superorbitals just like 2 F shapes, which are thinly dispersed on the whole clusters as we adopt a finer contour surface for presentation (figure S7 of the supplementary materials ).Considering the feasible approximation of Russell-Saunders SOC as well as the antiferromagnetically coupling of µ An S and µ An L , we proposed that the Hund's third rule J = L-S is formally obeyed in superatoms.Nevertheless, we should mentioned that DFT magnetic moments µ An total = µ An S +µ An L are not strictly equal to the values derived from total angular momentum J given by Hund's third rule, and thus the validation of Hund's third rule holds only in form.
Although there is currently no experimental study on the preparation of An@Al 12 clusters, a large number of TM embedded magnetic superatoms have been produced by using different methods such as pulsed laser vaporization source or magnetron gas aggregation source.Their magnetic moments can be measured via the Stern-Gerlach experiments [77][78][79] or x-ray absorption and XMCD spectra [80].Similar to the preparations and measurements of pure and doped Al clusters TM@Al 12 [4], our designed An@Al 12 superatoms can be synthesized by these laser vaporization and condensation methods.Typically, remarkable stability as well as size selectivity of An@Al 12 superatoms render them high abundance in the mass spectrum.Once they are formed, they can be easily screened out via time-of-flight mass spectrometer for further property measurements; their electronic structures can be measured by using the electronic spectra; their magnetic moment can be measured by Stern-Gerlach molecular beam deflection.These An@Al 12 superatoms, being as ideal building blocks, can replace atoms to achieve different functions directly or by forming devices for bottom-up building materials with specific properties.For instance, they can be self-assembled to form one-dimensional nanowires, two-dimensional sheet, and three-dimensional crystals such as the magnetic molecular crystal [8].In addition, superatoms can be regarded as a research platform to inspect the localization-vs-delocalization of 5f orbitals, which is a fundamental question in actinide chemistry.We hope that the present findings will stimulate further efforts to assemble such An-containing magnetic superatoms.

Conclusion
In summary, inspired by the discover of bimetallic quasi-spherical superatoms, we have proposed a series of An@Al 12 clusters as magnetic superatoms by doping the early series of An atoms, and confirmed the icosahedron as the ground state structure via the global minimum search and the following DFT calculations.It was demonstrated that these clusters are thermodynamically and chemically stable with large HOMO-LUMO gaps (0.4-0.96 eV) and high formation energies (3.17-8.8eV), which originates from the primary ionic An-Al bonds and a degree of covalent bonding interactions, together with the closures of the splitting sub-shells in electronic configurations.The 5f electrons of An atoms such as Pa, U and Pu can exhibit the 'dual nature' of localization and itinerant, which are not only element-dependent due to the electron accounting rule for giving the closure sub-shells but also orbital-dependent due to the orbital matching rule for forming the covalent hybridizations.These superatoms share the similar electronic patterns: the filled 1S 2 1P 6 1D 10 2S 2 2P 6 core-shells as well as the closure outer-shells of 1 F sub-superorbitals and the inner-shells of 2 F sub-superorbitals.The exchange split 5f -states in An atoms can induce large exchange splitting in 2 F superatomic shells, breeding the magnetic moments in An@Al 12 superatoms.If we discard the inevitable changes of orbital moment due to variable populations of the electrons on the 5f sub-superorbitals, which are the results of orbital splitting determined by non-spherical symmetry, the filling of superatomic 2 F orbitals do generally follow Hund's three rules, giving the maximum spin moment and the summed orbital moments of the individual electrons as well as the antiparallel between them.

Figure 1 .
Figure 1.Optimal structure of An@Al12 with C5v symmetry.The pink/blue balls represent Al/An atoms.

Figure 3 . 3 ⊕
Figure 3.One electron energy levels and orbital wave function of U@Al12 cluster with isosurface values (1-4) × 10 −8 |e| Å −3 .Left and right columns indicate the majority (α) and minority (β) spin states, respectively.Continuous lines correspond to filled levels while dotted lines correspond to unfilled states.For each energy level, the angular momentum of supershells was marked by S, P, D, F, respectively.Note that non-degenerate superorbitals belong to the same shell and with close energies are represented in merging manner such as 1D 4 α,β (D xz/yz ⊕ D xy/x 2 −y 2 ) and 1 F 3 α (F z 3 ⊕ F xz 2 /yz 2 ).

Figure 4 .
Figure 4. (a)-(g) The projected density of states (PDOS) of An@Al12, where the uppercase and lowercase letters represent the superatomic and atomic orbitals, respectively.Some peaks are overlapped with each other in showing a broader 'envelope' due to their tiny splitting gaps.(h) The orbital energy levels of isolated U atom, Al12 fragment and U@Al12 superatoms with the dots lines representing the unoccupied orbitals.

Figure 6 .
Figure 6.The COHP analysis for the orbital-resolved bonding and antibonding states between An and Al atoms.

Figure 7 .
Figure 7. Spin magnetic moment of An@Al12.The occupy of the outer-shell superorbital is given, where α and β denote spin up and spin down channels, respectively.

Figure 8 .
Figure 8. Local spin and magnetic moment of An dopants.

Figure 9 .
Figure 9. PDOS of An-5f in An@Al12 cluster, where blue and red curves indicates the PDOS without and with SOC.