Научная статья на тему 'Electronic structure and stabilization of C60 fullerenes encapsulating actinide atom'

Electronic structure and stabilization of C60 fullerenes encapsulating actinide atom Текст научной статьи по специальности «Физика»

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FULLERENES / ACTINIDES / AB INITIO METHODS / RELATIVISTIC CALCULATIONS / MOLECULAR STRUCTURE / STABILITY

Аннотация научной статьи по физике, автор научной работы — Ryzhkov M.V., Ivanovskii A.L., Delley B.

The geometry optimization of the neutral molecules An@C 60 (An = Th Md) was carried out using the DFT based Dmol 3 method. In order to perform calculations for these complexes' electronic structures, the fully relativistic discrete variational method (RDV) was used. Two types of stable position of metal atom inside the C 60 cage were obtained. The most stable non-central positions are favored over the position of actinide in the fullerene center for all An@C 60 complexes. Systems containing light actinides have considerable energetic stability, which is noticeably greater than that of corresponding exohedral and "networked" complexes. The 5f-orbitals' contribution to chemical bonding was found to be noticeably less than that of the 6d-states, even for the complexes at the beginning of An@C 60 row. The e ective charges on the actinide atoms were calculated using integral scheme incorporated in RDV and Hirshfeld procedure of DMol 3 code.

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Текст научной работы на тему «Electronic structure and stabilization of C60 fullerenes encapsulating actinide atom»

ELECTRONIC STRUCTURE AND STABILIZATION OF C60 FULLERENES ENCAPSULATING ACTINIDE ATOM

M. V. Ryzhkov1'*, A. L. Ivanovskii1 , B. Delley:

institute of Solid State Chemistry, Ural Branch of the Russian Academy of Sciences,

620990, Ekaterinburg, Russia

2Paul Scherrer Institut WHGA 123, CH-5232, Villigen PSI, Switzerland

*ryz@ihim.uran.ru

PACS 31.15A, 31.15ae, 31.15aj, 31.15E

The geometry optimization of the neutral molecules An@C6o (An = Th - Md) was carried out using the DFT based Dmol3 method. In order to perform calculations for these complexes' electronic structures, the fully relativistic discrete variational method (RDV) was used. Two types of stable position of metal atom inside the C60 cage were obtained. The most stable non-central positions are favored over the position of actinide in the fullerene center for all An@C60 complexes. Systems containing light actinides have considerable energetic stability, which is noticeably greater than that of corresponding exohedral and "networked" complexes. The 5f-orbitals' contribution to chemical bonding was found to be noticeably less than that of the 6d-states, even for the complexes at the beginning of An@C60 row. The effective charges on the actinide atoms were calculated using integral scheme incorporated in RDV and Hirshfeld procedure of DMol3 code.

Keywords: fullerenes, actinides, ab initio methods, relativistic calculations, molecular structure, stability. 1. Introduction

The earlier investigations of charged and neutral endohedral fullerenes An@C28 (An = Th - Md) [1,2] as well as An@C40 (An = Th - Md) [3] showed that some of these clusters can be very stable and therefore, may be useful for nuclear applications such as medicine or nuclear waste disposal. Since the discovery of C60 [4], these fullerenes have been widely studied both experimentally and theoretically. To date, we know only one actinide endohedral system U@C60, which has been obtained experimentally, Diener et al. [5] reported that U@C60 was produced by subliming fullerenes from arc-produced soot onto a mass spectrometry target. Chang et al. [6] theoretically predicted its properties, according to these Restricted Hartree-Fock calculations the U@C60 and U+@C60 complexes should be less stable than separated C60 and U. However, in Ref. [6] only one central position of various atoms inside the icosahedral carbon cage was considered. On the other hand, the radius of this fullerene (nearly 3.6 A) is evidently too large for one actinide atom or ion, so the An@C60 structure with metal site just in the center of the cage could be less favorable than those where actinide atom is shifted to the cage wall and interacts with only few carbon neighbors.

Interest in C60 fullerene-encapsulated actinide atoms is also due to the possibility of 5f - states participation in bonding. It is evident that theoretical study of the electronic structure and chemical bonding of any systems containing actinides requires the inclusion of all relativistic effects in the computational method. The fully relativistic calculations of An@C28 clusters (An = Th - Md) [2] showed that 5f orbitals participate in chemical

bonding of the first half of this series from Th@C28 to Cm@C28• Conversely, in the fully relativistic calculations of An@C40 complexes (An = Th - Md) [3], we determined that An5f contributions to bonding are nearly three times less than that of the main An6d-C2p interaction even in the clusters at the beginning of this row (Th@C40 and Pa@C40). These results are in agreement with the known sensitivity of 5f bonding features to the variation of bond-lengths, because an average radius of fullerene cage increases from 2.5 A (C28) to 3 A (C 40)• Since the radius of C60 molecule is close to 3.6 A, one can expect that in a case of central position of metal atom the role of 5f states in bonding will be noticeably less than in An@C40. On the other hand, some shift of actinide atom from the center to the cage wall can reduce the An-C bond lengths to the values, which are more typical for the An-C interactions in the molecules and solids.

The aim of the present paper was the search for most stable positions of actinide atom inside a C60 cage, the evaluation of geometrical parameters of the neutral An@C60 complexes for almost all actinides from Th to Md, the investigations of the chemical bonding and the role played by 5f states in the interaction between metal atom and the carbon shell. Another aim of the work presented here was the comparison of binding energies of the three competitive types of structures: (1) endohedral An@C60; (2) exohedral AnC60, where the actinide atom is bound to C60 from the outside of the cage; (3) "networked" C59An, where the actinide atom is incorporated into the C60 cage, at least in the cases of most stable complexes. The results of present calculations also allow us to evaluate the electron density redistribution for various positions of metal atom and across this series.

2. Objects and methods of calculations

Geometry optimization of "empty" C60 molecule with Ih symmetry and C60 interacting with actinide atoms was performed using the DMol3 method [7] in the scalar relativistic approach [8] and with the largest double numerical atomic basis set ("dnp"). The Coulombic potential was computed with the use of model density obtained as decomposition of charge density into multipolar components including those with l = 3. The generalized gradient approximation (GGA) in "BLYP" [10,11] and "PBE" [9] forms was used in all calculations. In a previous paper [2], we used "PBE" as well as "BLYP" [10,11] functionals, as was shown, the energetic and geometrical parameters obtained using both approaches were similar. Optimization of the molecular structures was done until the change in the maximum energy gradient values was less than 0.001 atomic unit, the global orbital cutoff was 8.0 A. To confirm that the stationary points correspond to minima, the vibrational frequencies were computed for all clusters. To test the reliability of parameters used in the calculations, we performed geometry optimization for Th@C60, Am@C60 and Md@C60 systems using the multipolar components with l = 4 and 10 AA for orbital cutoff. Though the computer time increased considerably, the changes in energy were less than 0.05 eV and the shifts in atomic coordinates were less than 10-3 A.

For the investigation of the nature of chemical bonding and the role played by 5f electrons in An@C60 complexes, we also used the fully relativistic discrete variational method (RDV) [12,13]. For the most stable geometrical structures obtained by DMol3, RDV calculations were performed. The RDV method is based on the solution of the Dirac-Slater equation for four-component wave functions, transforming according to irreducible representations of the double point group (CSS and C2V in the present calculations). For calculation of the symmetry coefficients, we used the original code which realizes the projection-operators technique [12] and includes the matrices of irreducible representations of double point groups obtained in Ref. [14] and the transformation matrices presented in Ref. [15]. The extended

bases of four-component numerical atomic orbitals (AO) obtained as the solution of the Dirac-Slater equation for isolated neutral atoms also included An7pi/2 and 7p3/2 functions in addition to occupied AOs. Numerical Diophantine integration in matrix elements calculations was done for 126000 sample points, which provided the convergence of valence MO energies within 0.1 eV. The effective charges on atoms (Qe//) were computed as integrals of electron density inside the domains bounded by the points of its minimum [16].

3. Results and Discussion

3.1. Results of geometry optimization

It is well known that optimized structure of C60(Ih) consists of 12 regular pentagons and 20 hexagons (Fig. 1). In our DMol calculations, we obtained that the radius for the empty fullerene is close to 3.55 A. This value for the cage radius corresponds to the C-C bond lengths in pentagons and hexagons in the range 1.46 - 1.40 A. In the calculations of Chang et al. [6] the interval of C-C bond lengths in C60(Ih) was slightly wider: 1.45 - 1.37 A. For the evaluation of the relative stability of various An@C60 complexes, it is reasonable to consider the binding energy Eb (which is sometimes called the "total bond energy" or the "atomization energy"). The binding energy of a cluster is usually defined as Eb = Etot -Esum, where Etot is the total energy of a cluster and Esum is the sum of total energies of all isolated atoms in the cluster. The binding energies for all investigated fullerenes are summarized in Table 1.

Fig. 1. The geometrical structure of fullerene Обо

We started by modeling the most symmetrical central position of actinide atom inside the C60 cage, the values of Eb(Ih) obtained in these calculations are shown in the second column of Table 1. Comparison of the Eb values for C60 and An@C60 allows one to evaluate the stability of An@C60 clusters with respect to the dissociation limit An + C60 for all actinides. According to our results, the symmetrical complexes of Pa, U and Np are stable systems and the most stable one is Np@C60 with Np + C60 dissociation energy near 3.1 eV. In the cases of smaller endohedral fullerenes An@C28 [2] and An@C40 [3], the Pa@Cn were the most stable complexes with corresponding dissociation energies near 8.1 and 6.4 eV respectively. Although the stability of An@C60 (Ih) obtained in the present calculations is considerably less than that of the corresponding An@C28 and An@C40, the uranium encapsulation reaction U + Сб0 ^ U@C60 (Table 1) is still exothermic (2.6 eV), whereas according to the results of Chang et al. [6] the U@C60 complex should be noticeably less stable than separated C60 and U (by ~ 3.5 eV). The analysis of geometrical parameters obtained for An@C60 (Ih) shows

Table 1. Binding and dissociation energies (eV), distances (Â) between the actinide atom and the nearest C atoms and effective charges (e) on actinide atoms in the investigated complexes

Complex Eb (Ih) Eb ^ C2v) Ed C2v) RAn-C (Cs, C2v) QAn C2v)

Hirshfeld Integral

C60 -452.7 - - - -

Th@Ceo -454.3 -458.5(CS) 5.8 2.50 / 2.50 / 2.50 0.68 2.17

Pa@C60 -455.0 -459.3(Cs) 6.6 2.44 / 2.44 / 2.44 0.71 1.93

U@C60 -455.3 -458.9(Cs) 6.2 2.41 / 2.43 / 2.44 0.63 1.83

Np@C60 -455.8 -458.4(Cs) 5.7 2.42 / 2.43 / 2.44 0.57 1.74

Pu@C60 -453.5 -455.2(C2v ) 2.5 2.36 / 2.51 / 2.51 0.66 1.41

Am@C60 -453.3 -454.3(C2v ) 1.6 2.36 / 2.52 / 2.52 0.62 1.34

Cm@C60 -453.4 -454.5(C2v ) 1.8 2.46 / 2.61 / 2.61 0.60 1.35

Bk@C60 -453.4 -454.4(Cs) 1.7 2.60 / 2.61 / 2.63 0.75 1.22

Cf@C60 -453.4 -454.4(Cs) 1.7 2.64 / 2.65 / 2.66 0.73 1.23

Es@C60 -453.4 -454.2(Cs) 1.5 2.65 / 2.67 / 2.69 0.72 1.15

Fm@C60 -453.4 -453.6(Cs) 0.9 2.65 / 2.67 / 2.70 0.62 1.14

Md@C60 -453.4 -453.5(Cs) 0.8 2.78 / 2.79 / 2.79 0.64 1.06

PaC60 - -455.7 3.0 2.34 0.58 1.73

C59Pa - -448.8 -3.9 2.17 / 2.25 / 2.25 0.78 2.44

that the deformation of the cage due to addition of any actinide atom is small: the radial expansion of the cage is less than 0.01 A in all clusters.

The search for less symmetrical An@C60 structures was undertaken in a few ways: the various shifts of metal atom from the center in different directions were considered. These initial configurations were subjected to geometry optimization, leading to new stable, but less symmetrical An@C60 structures. The values of initial shift of actinide atom varied from 0.1 to 0.6 A. However, small shifts (0.1 - 0.3 A) led to a relaxation of the system into structures with An atom located near the center of C60 shell. Conversely, a shift of metal atom by 0.4 A or greater led to rearrangement of the complex into a structure with an actinide atom located near the cage wall. This result means that the potential barrier for transformation of the geometry with central actinide position is quite small. The optimized structures with various symmetries were obtained (C2v, Cs, Ci). According to results of DMol calculations, we can predict that there are two types of most stable geometry for "distorted" endohedral fullerenes, corresponding to Cs and C2v symmetries. The structures of these complexes are illustrated in Figure 2. Though the difference in binding energy for these two isomers for each An@C60 cluster is within 0.1 eV, we can conclude that for complexes of Th, Pa, U and Np, the former type (Cs) is slightly more stable and should be considered as the ground structure. For the three clusters in the middle of the row (Pu, Am, Cm) the C2v - isomer appeared to be slightly more stable. For the "end part" of the row from Bk to Md the Eb for Cs structures is slightly lower than that for C2v isomers. The binding and dissociation energy values obtained for these structures are shown in the third and the fourth columns of Table 1 respectively. As can be seen from Fig. 2, in both "distorted" isomers, the actinide

atom has six nearest carbon neighbors, which belong to one hexagon in Cs - structure and to the two adjacent hexagons ("X" - shape of carbon vicinity) in the C2v structure.

Fig. 2. The geometrical structures of endohedral complexes An@C60 corresponding to Cs (left) and C2v (right) symmetry

According to our results, complexes of Th, Pa, U and Np are more stable systems with An + C60 dissociation energy near 6 eV. The most stable cluster is Pa@C60 with a dissociation energy close to 6.6 eV. As mentioned above, in the cases of smaller endohedral fullerenes the most stable complexes were also formed by encapsulation of protactinium atom [2,3]. It is interesting that stable systems were also obtained in the second half of the An@C60 row: the complexes of Cm, Bk Cf and Es are noticeably more stable than separated C60 and An. Note that stable clusters for heavier actinides were also predicted for An@C40 systems [3], moreover, for An@C60 clusters the similar non-monotonic trend is obtained, particularly, in the case of Am@C60 the absolute value of Eb is less than that for Cm@C60. As can be seen, the values of Eb for all isomers of Fm@C60 and Md@C60 (near -453.5 eV) are lower than that of empty C60 (-452.7 eV).

The values of An-C bond lengths for the nearest carbon neighbors in the Cs and C2v structures are shown in the fifth column of Table 1. Though the shapes of hexagons and pentagons in Cs and C2v structures are very close to regular geometry of an empty C60, the six nearest neighbors of actinide atom belong to the three and two nonequivalent types in Cs and C2v isomers respectively. These atoms are labeled later as C1, C2 and C3 (Cs), while in C2v clusters the two pairs of carbon atoms of C2 and C3 types are equivalent. In each complex of the Cs - type containing Th, Pa, U, Np and from Bk to Md, the distances between metal atom and C1, C2 and C3 are close to each other (Table 1). These results mean that the actinide atom is located just under the center of one hexagon. For the molecules of C2v type (Pu@C60, Am@C60, Cm@C60) we obtained noticeable variation of these bond lengths (by 0.15 - 0.16 A). As can be seen, there is no complete correlation between the variation of average values of An-C bond lengths in An@C60 row and the main trend of Eb variation, however, the increase of stability from Th@C60 to Pa@C60 and its decrease from Pa@C60 to Am@C60 and from Cm@C60 to Md@C60 are accompanied by the decrease and increase of corresponding An-C distances, as could be expected from general consideration. There is one noticeable exception to this rule: the An-C bond lengths in the less stable Am@C60 are less than those in the more stable Cm@C60, Bk@C60 and Cf@C60 complexes.

In Table 1 we also show the results obtained for the most stable example of exohedral fullerene (PaC60) and "networked" complex (C59Pa). The exohedral structures were generated from empty C60 particle by the addition of Pa atom from several spatial directions. These initial configurations were subjected to geometry optimization, leading to a few stable

structures. The lowest-energy C2v isomer is a distorted C60 fullerene, with the metal atom bonded to a pair of carbon atoms, another isomer, in which the actinide atom interacts with carbon hexagon appeared to be slightly less stable. The structure of this complex is illustrated in Figure 3. We obtained that in these complexes Pa-C distances are near 2.34 A. The absolute value of binding energy of this most stable exohedral complex was found to be 455.7 eV, this value is noticeably less than that for the non-central endohedral position of Pa inside C60 cage (459.3 eV). However, the value of |Eb| for exohedral cluster was greater than that for endohedral position in the center of fullerene (455.0 eV). Thus, the dissociation energy of PaC60 complex with respect to the PaC60 ^Pa + C60 reaction is more than two times less than that obtained for the non-central endohedral position of protactinium atom. The distortion of fullerene cage in the PaC60 (C2v) molecule due to interaction with metal atom led to increase of the number of non-equivalent carbon sites. However, the average value of r(C) in PaC60 system is close to the corresponding parameter of endohedral complex.

The "networked" C5gPa structure was generated from an empty C60 particle by the substitution of an actinide atom for a carbon site. Then, this initial structure was subjected to geometry optimization, leading to the new equilibrium positions for the metal and carbon atoms (Fig. 3). The parameters of this cluster were shown in Table 1. In contrast to endohedral and exohedral systems, the "networked" complex is less stable than an empty C60 cage (Table 1). This result is quite expected because the C-C bonds are certainly stronger than C-An bonds in such systems. We also obtained that during geometry optimization of C5gAn structure the initial distances between metal atom and the cage center increased from 3.55 A to 4.96 A. This considerable increase in the distances between metal atom and the center of the cage is accompanied by a corresponding increase in the bond lengths between actinide atom and its three nearest neighbors: these distances were found to be 2.17 A, 2.25 A and 2.25 A. Nevertheless, the interatomic Pa-C distances in C5gPa as well as in PaC60 are noticeably less than those in more stable endohedral complex (2.44 A).

9

Fig. 3. The geometrical structures of exohedral PaC60 (left) and "networked" C59Pa (right) complexes

As mentioned above, the present DMol results were obtained in the scalar relativistic approach, however, the authors [1] evaluated the role of spin-orbit coupling in the Pu4+@C28 cluster and showed that the addition of these effects caused a maximum expansion of the Pu-C distance of 0.027 A with respect to the scalar relativistic results. They also reported

that the scalar relativistic bonding energy of the Pu4+@C28 increased by less than 8% when the spin-orbit coupling effects were included in the calculations.

3.2. RDV calculations

In Figure 4 we show the partial densities of states (DOS) obtained in the fully relativistic RDV calculations for the ground state isomers of Th@Ceo and Pa@Ceo clusters. In the cases of C2s and C2p DOS we show only the contributions from C1, C2 and C3 atoms, which are the nearest neighbors of metal ion. As can be seen from Fig. 4, the spin-orbital interaction is small for the "outer" bands, particularly the 0.9 eV splitting was obtained for An5f5/2 and An5f7/2 main peaks. The relativistic effects become considerable for deeper orbitals, that is, the main peak of An6pi/2 band is shifted to the lower energies by ~ 9 eV (Th) and ~ 10 eV (Pa) from the highest intensity peak of An6p3/2 DOS. In both complexes the most intensive part of occupied valence C2p band in the energy region from 0 to -10 eV (Fermi level is used as a zero of energy scale) contains some contributions from An6d AOs (near -6 - -4 eV) and An5f AOs (near -5-0 eV). The vacant MOs are formed by C2p, An5f, 6d, 7s and 7p orbitals in the energy region from 0 to 19 eV.

C2p ^jyAiWAt^Aj.

C2s

Th7p Ji.

Th7s ,!,.....,..........

Th5f

............... Tii6d ........ 11' ......... 111

Th6p ,,, 11111111111 it,™.»/ 1 M 1 | 1 1 1 . | 1 1 1 1 | 1 M 1 |

±iUwLi.

C2s . L-AAii ih i .....,

Pa7p .u.

Pa7s 1 A

Pa5f

J PaSd ,1

Pa6p l 11 - - i 11 | i -■. lAAf-rvW. V. ...........

-45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 MO energy, eV

-45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 MO energy, eV

Fig. 4. Partial densities of states for the ground isomers of Th@C60 (left) and Pa@C60 (right) clusters. The broken lines correspond to p1/2, d3/2 , and f5/2 DOS, the solid lines correspond to p3/2, d5/2 and f7/2 DOS (vertical line is the border between occupied and vacant states)

The energy gap between occupied and vacant molecular states, defined as the difference of energy of the highest occupied (HOMO) and the lowest unoccupied (LUMO) molecular orbitals, obtained in our relativistic calculations for Th@C60, is near 0.4 eV. In similar calculations for Th@C28 [2] and Th@C40 [3], we obtained considerably greater HOMO - LUMO gaps (2.3 and 1.4 eV respectively). The latter gap values were noticeably greater than those for the empty C40 and C28 cages (near 0.05 and 0.6 eV respectively), on the other hand, the energy gap in empty C60 cage (1.6 eV) is considerably greater than that in any An@C60 cluster studied in the present work. In Th@C60 and Pa@C60 complexes both the HOMO and LUMO are of predominantly C2p character with small contributions from

An5f states. According to our results, the Fermi level in Pa@C60 is located at the small C2p subband consisting of two MOs corresponding to r3 and r4 irreducible representations of Cs double point group, in this cluster one orbital is occupied and one is vacant. The energy splitting between this C2p band (the admixtures of Pa5f AOs are near 10%) and the next unoccupied levels, corresponding to the states with main contribution from 5f5/2 AOs (— 80%), is nearly 0.8 eV (Fig. 4). Although the Th5f, 6d and Pa5f, 6d molecular levels are vacant, the covalent mixing between 5f5/2, 5f7/2, 6d3/2, 6d5/2 and C2p orbitals in the occupied valence band leads to noticeable Mulliken population of 5f and 6d AOs, which were obtained as 5fg/423, 5f°/423, 6d3/524, 6d5/624 (Th@Ceo) and 5f5/224, 5f°/922, 6d3/522, 6d°/529 (Pa@Ceo).

In the next molecule of this row, U@C60, the HOMO is still of C2p character, but the admixtures of U5f AOs increase up to 20%, while the LUMO in this cluster is of 5f5/2 character (82%). The energy gap between occupied and vacant levels is near 0.4 eV. Though the molecular orbitals containing main contributions from 5f AOs are still vacant in U@C60 cluster, the hybridization of 5f5/2, 5f7/2 and C2p AOs in the occupied molecular orbitals are responsible for the essential Mulliken populations of U5f AOs (5fi/928 and 5fi/225), which are greater than those in Pa@C60. In contrast, the populations of U6d orbitals (6d3/5216d0/527) are nearly the same as in protactinium complex. In the neptunium cluster, both HOMO and LUMO contain main contributions from Np5f5/2 AOs (80%), i.e. the Fermi level is located in the 5f5/2 band. In addition, the admixtures of Np5f states in the occupied MOs of C2p character considerably increase (up to 35-40%). The filling of 5f5/2 states and increasing hybridization of 5f5/2, 5f7/2 and C2p AOs in the valence band are responsible for the essential Mulliken population of Np5f AOs: 5f5/7265fi/525, on the other hand, the populations of Np6d orbitals decrease (6d3/4286d0/524) as compared to the uranium cluster.

The ground structures of the next three molecules of this row Pu@C60, Am@C60 and Cm@C60 correspond to C2v -type. In Figure 5, we show the partial DOS obtained in RDV calculations of Pu@C60 and Cm@C60 clusters. In the cases of C2s and C2p DOS we show only the contributions from two C1 and four C2 atoms, which are the nearest neighbors of metal ion. As can be seen from Fig. 5, the peaks corresponding to An6p1/2 bands are shifted to lower energies by — 11 eV (Pu) and — 12 eV (Cm) from the highest intensity peak of An6p3/2 DOS. In both complexes, the An5f5/2 bands are completely occupied, however, between 5f5/2 and 5f7/2 states the small band of C2p character is located. This C2p band contains six MOs, two of which are occupied in Pu@C60 and three and four orbitals are occupied in Am@C60 and Cm@C60 clusters respectively. As a result, the An5f7/2 MOs are vacant in all these systems, however, the admixtures of An5f7/2 AOs in the highest occupied molecular states increase from 17% in Pu@C60 to 35% in Am@C60 and to 39% in Cm@C60. Thus, the HOMO and LUMO in these three complexes are the mixture of C2p and An5f5/2, 5f7/2 AOs. The strong hybridization of both 5f5/2 and 5f7/2 AOs with C2p orbitals in occupied MOs is responsible for the non-integer Mulliken population of 5f AOs, which increases from 5f3/215f7/124 in Pu@C60 to 5f4/3255f2/25 in Am@C60 and to 5f4/9255f3/022 in Cm@C60. As mentioned above, the populations of An6d orbitals decreases for heavier actinides, the same trend was obtained for Pu@C60 (6d3/3256d5/323), Am@C60 (6d°/3226d5/29) and Cm@C60 (6d3/256d°/21) complexes.

The occupied molecular orbitals of 5f7/2 character (51%, 57% and 73%) are firstly achieved in Bk@C60 (Fig. 6), as a result, the Mulliken populations of Bk5f states are 5f5/4235f7/622, i.e. the total number of 5f electrons in Bk@C60 (-9.05) is greater than that in Cm@C60 (-7.97) by 1.08. The LUMO in Bk@C60 is also a mixture of Bk5f7/2 (75%)

C2p , L||L

cis.....ImaWa.J^ , ...A „VIIMKA,

Pu7p 1 ........"T^1 1 1 „ i

~ Pu7s I .

- Pu5f J

- Pu6d . jl Y

Pu6p

...... J^kkUlfij'Mlkl

C2s k^JVUm^..... .....k r J.

Cm7p

Cm 7s 1

| - Cm5f j ■ l A.

i i i i 11 i i i | i 11 11 i i 11 | i i i i 11 i i i | 11 i i | i , i, J Cm6d ......I 1 1 1 1 I M 1 1 ......

CmSp , iJi.

-40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 MO energy, eV

-40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 MO energy, eV

Fig. 5. Partial densities of states for the ground isomers of Pu@C60 (left) and Cm@C60 (right) clusters

and C2p AOs. On the other hand, the hybridization of Bk6d and C2p orbitals in the occupied C2p band is less than that in Cm@C60, and a noticeable decrease of the Mulliken populations of Bk6d AOs was obtained (6d3/1286d5/122). Further filling of 5f shell and considerable depopulation of 6d shell were obtained for Cf@Ceo (5f5/215f|/4206d3/1216d0/o24) and Es@Ceo (5f5/8265f5/2286d0/12o6d0/o22). The complete occupation of both 5f5/2 and 5f7/2 molecular orbital types was achieved in the Fm@C60 complex. The partial DOS obtained in RDV calculations of Fm@C60 are shown in Fig. 6. The LUMO in Fm@C60 are of almost purely C2p character (95%), the contributions from Fm5f AOs are nearly 3%. However, as a result of covalent mixing between the Fm atomic orbitals and the C60 cage orbitals the Mulliken populations of 5f AOs are 5f5/9235f7/220, i.e. the total number of 5f electrons in the Fm@C6o cluster (12.1) is close to that of the isolated Fm atom. The energy gap between the occupied and vacant levels in Fm@C60 is near 0.9 eV, which is the greatest value obtained in our calculations for endohedral fullerenes An@C60. In the "last" investigated Md@C60 complex, one more molecular orbital of C2p subband is occupied, thus both HOMO and LUMO are of almost purely C2p character and lie above the completely occupied Md5f7/2 band by 0.4 eV. As expected, the admixtures of C2p AOs in the molecular orbitals of Md5f7/2 type decrease and the Mulliken populations of Md5f shell are 5f5/9285f7/126. As in the case of the fermium cluster, the total number of 5f electrons in the Md@C60 molecule (13.1) is close to that of isolated Md atom.

To compare the effects of spin-orbit coupling for the valence orbitals of the systems at the beginning and at the end of An@C60 series, one can use the splitting of the main peaks corresponding to 5f5/2 and 5f7/2 states. According to RDV calculations, the 5f5/2 -5f7/2 peaks separation for Th@C60, Pa@C60, U@C60 and Np@C60 is close to 0.9 - 1.1 eV. For the Pu and Am complexes we obtained ~1.2 eV, for Cm and Bk these values are close to 1.3 - 1.5 eV. In Cf@C60 and Es@C60 this spin-orbit splitting can be evaluated approximately as 1.6 - 1.7 eV. The noticeable increase of this value was obtained for the end of the row, i.e.

-40 -35 -3D -25 -20 -15 -10 -5 0 5 10 15 20 25 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25

MO energy, eV MO energy, eV

Fig. 6. Partial densities of states for the ground isomers of Bk@C60 (left) and

Fm@C60 (right) clusters

for Fm@C60 and Md@C60 the splitting between 5f5/2 and 5f7/2 peaks is near 2.2 and 2.3 eV respectively (Fig. 6).

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The molecular orbital structure of the most stable exohedral complex PaC60 noticeably differs from that of the corresponding endohedral system. In Figure 7, we show the partial DOS obtained in RDV calculation of PaC60 cluster. In the cases of C2s and C2p DOS we show only the contributions from two C1 atoms, which are the nearest neighbors of metal ion. As can be seen from Fig. 7, the hybridized C2s - Pa6p3/2 subband in the energy region from -23 to -20 eV is absent in PaC60, resulting in a more narrow distribution of 6p3/2 DOS. The positions of Pa6d, 5f and 7s levels become closer to the Fermi level (Fig. 7). These shifts of the 6d, 5f and 7s states lead to noticeable increases in their contributions to molecular orbitals in this energy region, particularly the admixtures of Pa5f and 6d AOs in HOMO and LUMO achieve 19% and 10% respectively. Although the Pa5f and 6d orbitals are also vacant in the PaC60 cluster, the hybridization of these states with occupied C2p orbitals leads to some Mulliken population of these AOs. According to our calculations, the occupations of these orbitals can be defined as 5f1/3275f0/7246d3/6256d0/528. As seen, the populations of 5f AOs in PaC60 are slightly less than those in Pa@C60, while the populations of 6d AOs in PaC60 are greater than those in Pa@C60 by 0.13. Surprisingly, in the case of a "networked" complex C59Pa, we did not obtain the principal transformation of electronic structure (Fig. 7). The C2s - Pa6p3/2 subband in the energy region from -23 to -20 eV is also absent in the C59Pa cluster. The Pa6d, 5f and 7s DOS also shift to lower energies, however, their contributions in the valence band in the energy region from 0 to -4 eV become more important than those in endohedral and exohedral clusters. We can also note, that "networked" position of metal atom in the fullerene cage leads to some broadening of Pa6d, 7s and 7p bands in the energy region from 1 to 11 eV (Fig. 7).

The considerable hybridization between An5f and C2p AOs was obtained in all investigated fullerenes. However, the coefficients corresponding to An5f contribution in the molecular orbitals are not directly connected with the degree of 5f states delocalization. The

.....^.^Miji tt*. .wLta^iW^

c¿s 1 1 ÍLÍIju,.

- I I I I I I I I I I I I I I I I I I I [I I I I I 11 I I I I tt I I I r I r Pa7p i

Pa7s i, „

' M 1 | 1 I I I | I 1 1 I | I I I I | I I I 1 | I I 1 I | I I 1 I | I M Pa5f

- PaSd i

PaSp : -....................K.............

-40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 MO energy, eV

' ' [^MkJ A

Pa7p À .bk..

■ Pa7s I /L . .A

Pa5f

Pa6d /i.J Ak À.

Pa6p ; A 1.......

-40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 MO energy, eV

Fig. 7. Partial densities of states for the PaCeo (left) and C5gPa (right) clusters

latter effect could be evaluated qualitatively by an analysis of the shapes of corresponding molecular orbitals or more quantitatively by the values of overlap populations of various pairs of the metal and carbon AOs (n^). The values of n^ can also give the bond orders of these states [17]. The values of overlap populations for C2p and 5f, 6d, 7s AOs of actinides obtained in our RDV calculations of endohedral clusters are listed in Table 2. As expected, the role of An7p-C2p interaction (not shown in Table 2) is anti-bonding in all investigated complexes.

Table 2. Overlap populations of An5f, 6d, 7s and C2p orbitals (10-3e, per one pair of interacting atoms) in An@C60 complexes

Orbitals Th Pa U Np Pu Am Cm Bk Cf Es Fm Md

An5f - Ci2p 36 41 37 32 22 18 12 8 3 2 1 0

C22p 35 42 39 34 18 15 11 8 4 2 1 0

Cs2p 35 39 36 31 - - - 7 3 1 1 0

An6d - Ci2p 98 84 83 84 64 62 60 50 40 38 35 29

C22p 97 83 85 84 50 49 49 46 41 37 35 27

Cs2p 96 83 84 84 - - - 47 40 37 34 28

An7s - Ci2p 15 16 16 16 9 9 10 12 9 9 10 3

C22p 15 16 16 16 18 19 18 13 7 9 8 2

Cs2p 14 16 16 16 - - - 13 8 7 9 2

Examination of Table 2 reveals that An6d orbitals play the main role in the chemical bonding of a metal atom and Ceo cage in all of An@C60 clusters. This result is quite expected, because in our earlier relativistic calculations of actinide oxides, fluorides [18,19] and An@C40 clusters [3], the main contributions to bonding was also provided by An6d AOs and the next An5f contributions to bonding were considerably less. However, the earlier calculations of small endohedral fullerenes An@C28 [2] showed that An6d and 5f orbitals play comparable roles in the chemical bonding of a metal atom and C28 cage in the first

half of the An@C28 row. This difference in the bonding behavior of An5f states in An@C40, An@C60 and An@C28 can be explained by the well known fact that An5f AOs participation in bonding is sensitive to the distances between actinide atom and its neighbors [19]. As mentioned above, the shortest An-C bond lengths in An@C40 and non-symmetrical An@C60 clusters are greater than the longest An-C bond lengths in An@C28 systems. As shown in Table 2, the overlap populations of An5f-C2p states increase when going from Th@C60 to Pa@C60 clusters and then monotonously decrease to zero at the end of this row. However, noteworthy is the considerable decrease of An5f contributions to bonding on going from Np@C60 to Pu@C60. On the other hand, the slightly non-monotonous bonding behavior at the beginning of An@C60 row is detected for An6d orbitals. As can be seen from Table 2, a small increase of overlap populations for An6d-C2p states takes place for the U@C60 and Np@C60 molecules in comparison to the Pa@C60 cluster.

Comparison of the stability of endohedral complexes (Table 1) and corresponding bond orders (Table 2) shows that there is good correlation between DMol and RDV results. The increase of the dissociation energy (An@C60 ^ An + C60) when going from Th@C60 to Pa@C60 is in agreement with the increase of contribution to bonding from Pa5f states, note that this contribution is the greatest in An@C60 row (Table 2). Further decrease of stability from Pa to Am correlates with a decrease of the overlap populations for An5f-C2p as well as for An6d-C2p AOs. The small increase of |Eb| for Cm@C60 and Bk@C60 clusters (Table 1) can be explained by the role of spin-polarization, which is included in the scalar-relativistic DMol calculations, but is not treated in the standard Dirac theory. Note, that similar results were also obtained for An@C40 complexes [3].

The values of overlap populations for C2p and 5f, 6d, 7s AOs of protactinium obtained in our RDV calculations for the exohedral and "networked" complexes are listed in Table 3. The corresponding parameters for endohedral cluster are also shown in Table 3 for comparison. Note, that n^ values presented in Tables 2 and 3 correspond to one pair of interacting atoms, hence, the sum of these values for the six nearest neighbors in endohedral cluster will be considerably greater than total overlap populations in exohedral system. However, as can be seen from Table 3, the interactions between actinide and a specific carbon atom in the C59Pa and PaC60 are stronger than that in Pa@C60 complex. Examination of Table 3 reveals that Pa6d orbitals also play the main role in chemical bonding of metal atom and carbon cage in these clusters. The variation of n^ for the Pa5f-C2p and Pa6d-C2p bonds also correlates with Pa-C bond lengths (Table 1). As shown in Table 3, the interaction between protactinium atom and each of three nearest neighbors in "networked" complex is considerably stronger than that in more stable endohedral cluster. However, as mentioned above, the actinide - carbon substitution in C60 cage is energetically disfavored process and the formation of C5gPa complex does not provide a better way for the extraction of the actinides from some species.

The effective charges on actinide atoms (QAn) obtained in our scalar relativistic (DMol) and fully relativistic calculations (RDV) are listed in Table 1. As shown, the effective charges obtained in DMol calculations using Hirshfeld scheme correlate with neither the variation of binding energy nor with the variation of An-C bond lengths. It is quite evident that the increase in charge transfer for the Pu@C60 and Am@C60 in comparison with Np@C60 as well as for the second half of the series (from Bk to Md) in comparison with Cm@C60 and the weaker interaction between cage and actinide atom are contradictory. On the other hand, the integral charges obtained in RDV calculations (Table 1) agree with the variation in the binding energy. Although the QAn obtained in relativistic calculations are considerably greater than the DMol values, the effective charges on actinide atoms are less

Table 3. Overlap populations of An5f, 6d, 7s and C2p orbitals (10-3e, per one pair of interacting atoms) in exohedral PaC60, "networked" C5gPa and endohedral Pa@C60 complexes

Orbitals PaC60 (C2v) C59Pa (Cs) Pa@C6o (Cs)

Pa5f - C12p 46 67 41

C22p - 51 42

C32p - - 39

Pa6d - C12p 90 148 84

C22p - 112 83

C32p - - 83

Pa7s - C12p 24 36 16

C22p - 34 16

C32p - - 16

than their formal valencies in solid compounds. The comparison of Tables 1 and 2 shows that the variation of the integral charges on going from Th@C60 to Md@C60 is mainly due to the decrease of An6d-C2p covalent mixing and overlap of these orbitals.

Although the chemical bonding between actinide atom and the carbon cage is weak for the clusters at the end of An@C60 row, both the DMol and RDV calculations exhibit essential charge transfers of more than 0.6 electron from the actinide orbitals to the cage. Similar results were obtained in our calculations of An@C40 row [3]. From our point of view, there is no contradiction between binding energy and charge transfer because a change in binding energy on going from superposition of isolated C60 (or C40) cage and actinide atom to endohedral complex is due to contributions from: (1) interaction between actinide and the carbon atoms and (2) deformation of the initial fullerene structure. The latter contribution is certainly unfavorable and even in the case of Md@C60 complex the interaction between metal and carbon atoms has the bonding character. On the other hand, the strong C-C bonds in the C60 (or C40 and C28) cage are not accompanied by noticeable charge transfer, whereas in the C59Pa complex the considerable charge transfer (more than 2 electrons) was obtained for the weaker Pa-C bonds.

According to our calculations of endohedral complexes, the additional charge on the cage orbitals is distributed over all carbon atoms, i.e. the deformation of electron density takes place even for the carbon atoms, which are the most distant from actinide site. However, as expected, variation of the charge density on the nearest actinide neighbors is slightly greater than that for other atoms of the cage. Nevertheless, the effective charges on C1, C2 and C3 atoms are within -0.1 in endohedral fullerenes of the first half of the row and within -0.05 for the complexes of heavy actinides. In contrast, the main part of charge transfer from the metal atom in the exohedral and especially in the "networked" clusters is localized on the nearest carbon neighbors. For instance, in the C5gPa complex, the integral charges on one C1 and two C2 atoms are -0.76 and -0.67 respectively, i.e. the total electron density, which is accepted by these carbon atoms is close to 2.1.

4. Conclusions

Our investigations of the new class of organometallic nanoparticles An@C60 confirm earlier results [1-3] that endohedral fullerenes can be very stable. Moreover, the stable character of such species for the light actinides obtained for the An@C28 and for the major part of An@C40 clusters (from Th to Fm) is also predicted for the neutral endohedral fullerenes An@C60 from Th@C 60 to Md@C60. The dissociation energy for the most stable complex Pa@C60 (Pa@C60 ^ Pa + C60) was found to be 6.6 eV, which is noticeably greater than that for the exohedral position of Pa atom on the outer surface of C60 cage. Comparison of the electronic structure of endohedral complexes based on Cs and Ih isomers showed that the former type is more favorable for the encapsulation of an actinide atom.

Analysis of the molecular orbital structure showed that strong hybridization takes place between C2p and An5f states for all An@C60 complexes. However, the calculated values of overlap populations of C2p-An5f AOs showed that the An5f contribution to bonding was more than two times less than that of the main An6d-C2p interaction, even in the neutral clusters at the beginning of An@C60 row. The analysis of chemical bonding of endohedral, exohedral and "networked" fullerenes showed that the strongest interaction between actinide atom and specific carbon atom in the C60 cage takes place for "networked" complex. However, the stability of the latter cluster is less than that of the corresponding endohedral systems, because: (1) in C5gPa molecule the three Pa-C bonds substitute for the three stronger C-C bonds; (2) in An@C60 complexes the An-C interactions are additional to the C-C bonds in the cage. Two schemes for atomic effective charge calculations give quite different results for An@C60, however, more realistic values were obtained using the spatial numerical integration procedure incorporated into the RDV method [16].

In this paper, we discussed the stability of endohedral fullerenes An@C60 with respect to the An + C60 fragmentation only. However, for the understanding of stability of these systems, it is important to consider the other competitive processes which can happen when An@C60 systems will ionize. It is interesting to calculate the ionization energy and the dissociation energy of An@Cn+ systems, this work is in progress now and will be published in the separate paper.

Acknowledgments

This work was supported by the Russian Foundation for Basic Research, grant 1003-00152.

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