J. Phys. Chem. C 2010, 114, 10775–10781
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Theoretical Predictions of a New Family of Stable Bismuth and Other Group 15 Fullerenes Aristides D. Zdetsis Department of Physics, UniVersity of Patras, GR-26500 Patras, Greece ReceiVed: April 8, 2010; ReVised Manuscript ReceiVed: May 3, 2010
Motivated by the structure and properties of isoelectronic carbon “fulleranes” and the recent synthesis of bismuth nanotubes and nanowires, we have studied, by density functional theory using the hybrid B3LYP functional and large doubly polarized valence triple-ζ (TZVPP) basis sets, Bin cages, n ) 4-80, similar to CnHn. It is shown that such cages have very high binding energies, high symmetry, and large highest occupiedlower unoccupied molecular orbital (HOMO-LUMO) energy gaps. The larger cages are considerably stabilized through bond-puckering, similarly to CnHn and SinHn. In addition endohedral doping of these “fullerenes” with similar or different group 15 atoms leads to structures with well localized spin at the central atom, which is highly suggestive of possible technological applications in electronic and optoelectronic nanodevices and quantum computing. Energetically, the optimum cage size for efficient doping with similar or different group 15 atom corresponds to the n ) 50. On the basis of binding energy per atom, the puckered icosahedral Bi80 is more stable than the n ) 4, 20, 50, and 60 puckered and nonpuckered cages, in full analogy to CnHn and SinHn. Analogous fullerene-like cages for other group 15 elements, such as antimony, arsenic, and phosphorus, are also examined and discussed, in order to facilitate possible future synthesis and functionalization of these cages. 1. Introduction The discovery of buckminsterfullerene1 (C60) and other smaller and larger fullerenes stimulated the search for analogous fullerene-like cages made of other than carbon atoms. Obviously the most favorable and “natural” choice was silicon2-5 due to its “similarity” with carbon. On the contrary, bismuth would have been considered as one of the least favorable choices, although, as will be shown bellow, the conceptual route to bismuth fullerenes examined here and to bismuth nanotubes, which have been already synthesized6 goes through the same search for silicon fullerenes. However, the direct approach of “devising” Si60 and other Sin cages in full analogy to C60 and other Cn fullerenes was of very little or marginal success, with conflicting results and distorted or unstable structures due to the dangling bonds.2 On the other hand, the indirect approach, based on either endohydral doping or exohedral coverage with other atoms or group of atoms has been more successful and promising.2 In particular exohedral (together with, sometimes, partially endohedral) hydrogenation is very successful in stabilizing highly symmetric SinHn cages with large highest occupied-lowest unoccupied molecular orbital (HOMO-LUMO) energy gaps, similar to CnHn fulleranes.2,7-10 Hydrogenated silicon cages of varying sizes and symmetries have been considered in rather different contexts by Zdetsis,7 by Kumar and Kawazoe,8 and by Kartunenn.9 Kumar and Kawazoe8 have considered small hydrogenated cages for n ) 8-28, examining their doping with various metal atoms. Karttunen et al.9 have considered the stability of large icosahedral SinHn “fullerenes” with n ) 20, 60, up to 540, in relation to corresponding polysilanes. The stabilization of large hydrogenated silicon fullerenes (n ) 60, 70, 76, 80, and 180) through oligo(poly)merization and puckering has been examined by Zdetsis10 very recently. Also recently the present author10b has suggested the connection of Si-doped silicon fulleranes (in particular the Si28H28) with the experimentally discovered 1 nm
luminus Si nanoparticle, thus expanding the idea and scope of hydrogenated silicon cages. These cages, as would be expected, are very similar both structurally and electronically with the isoelectronic (isovalent) CnHn analogues.11,12 It is very interesting both scientifically and technologically to examine (at least theoretically) the possibility of developing additional similar “fullerenes” besides SinHn and CnHn. An obvious but trivial answer would be GenHn (and perhaps SnnHn or even PbnHn) fullerenes. One more interesting and intriguing possibility arises by a closer examination of a recent work by the present author on Si-Bi and Ge-Bi clusters,13 which shows that Bi2Sin-2 and Sin-2(CH)2 clusters are very similar structurally and electronically and in fact they are “isolobal”. This assertion is based on an isoelectronic and isolobal analogy under the scoptical name “the boron connection”.14,15 More information on this analogy can be found in refs 14 and 15 and references therein. What is most important here is the “equivalence” of the CH and Bi subunits, which are isovalent, with 5 valence electrons each. The replacement CH f Bi (or equivalently BH1- f CH f Bi) is not purely academic but it has been tested in practice, through the synthesis of the well-known bisboranes.16 Also available theoretical13 and experimental17 results on Bi-Si clusters (in particular Bi2Si5), as well as theoretical analysis on very stable Sn10Bi2 clusters18 fully support this idea. Through this replacement it becomes rather obvious that Bin cages analogous to CnHn (and SinHn) fullerenes should be expected to be very stable and could possibly be synthesized. In favor of such expectations is the recent synthesis of bismuth nanotubes6 which could have also been conceived through the same type of reasoning. The aim and scope of the present investigation is to explore this possibility. Apparently the same reasoning applies also, besides bismuth, to other group 15 clusters and fullerenes, with possible small variations, as will be seen below. Anionic and neutral clusters of bismuth, which is the heaviest group 15 element, up to 24 atoms have attracted considerable
10.1021/jp103179z 2010 American Chemical Society Published on Web 05/27/2010
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interest recently,19 although photofragmentation20 and other experimental and theoretical data21,22 have been known for more than 10 years (see ref 19 for more details). Similar work (and further information) on small Asn (n e 28) clusters can be found in the work of Zhao et al.,23 whereas antimony clusters have been studied among others by Rayane et al.24 and by Sattler et al.25 Finally, Karttunen et al.26 have examined (and classified) small and large phosphorus clusters and fullerenes of various kinds. Here we examine Bin fullerenes with n ranging from n ) 4 to 80, although the extension to larger fullerenes is rather straightforward. These results are presented and discussed in section 3, after a short description of the technical details of the calculations in section 2. The results and discussion for the embedded fullerenes of the form Bi@Bin (and related fullerenes) are presented in section 4 together with a comparison of their differences and similarities. In section 5 some selected results for Sb, As, and/or P fullerenes are briefly discussed and compared. Finally, the conclusions and future plans of this work are summarized in section 6. 2. Some Technical Details of the Calculations The general theoretical and computational method followed for all cages is density functional theory (DFT) with the hybrid, nonlocal exchange and correlation functional of Becke-Lee, Parr, and Yang (B3LYP).27 These calculations were performed with the TURBOMOL program package28 using the triple-ζ singly polarized29 (TZVP) and the TZVPP doubly polarized basis set30 for the smaller cages. For bismuth and antimony, compatible effective core potentials (ECPs) were used which include scalar relativistic effects.31 For the very large structures (n ) 60, 76, and 80), the initial geometries were optimized using the nonhybrid BP86 functional28,32 employing the “resolution of the identity” (RI) approximation28 which greatly improves computational cost. The BP86 functional includes the Becke exchangea and the Vosko-Wilk-Nussair27 (VWN), and Perdew32b correlation functional. Both B3LYP and BP86 functional were used for several medium size clusters to allow the comparison and extrapolation to large structures. Also in several critical cases involving very small energy differences the def2-TZVPP basis set30 was used which includes the small-core ECPs.33 For the lowest energy structures of the small-medium Bi20, Bi28, and Bi60, second order Møller-Plesset (MP2) perturbation theory was also used34 employing the same basis sets and the same program package. 3. Results and Discussion for the Bin “Fullerenes” The general results on Bin fullerenes in full analogy to SinHn and (CnHn) fullerenes7,9 are summarized in Figure 1 and Table 1 which describe their structural (form in Figure 1 and symmetry, SYM, in the table), cohesive (binding energy per atom Eb), and electronic (HOMO-LUMO gap) characteristics. We can see, as in the SinHn and (CnHn) fullerenes,7,9 that n ) 20 is the turning point, which also reflects very high binding energy per atom (the highest from the cages in Figure 1) and very large HOMO-LUMO gaps (the largest among medium and large size fullerenes). Bellow we briefly discuss each case separately. a. Bi4, Bi6, and Bi8. The lowest energy (and highest symmetry) structures of Bi4, Bi6, and Bi8 obtained here, as shown in Figure 1 (n ) 4, 6, and 8) are exactly similar to the ones obtained7 for Si4H4, Si6H6, and Si8H8. These structures in Figure 1 (n ) 4 and 6) are also predicted by Yuan et al.19 to be the lowest energy structures with binding energies per atom 1.88
Figure 1. Lowest energy (high symmetry) bismuth fullerenes structures Bin, n ) 4, 6, 8, 10, 12, 20, 24, 28, 32, 36, 50, 60, 76, and 80.
TABLE 1: Binding Energy Per Atom (Eb/n), HOMO-LUMO Gap (H-L), in eV, and Symmetry (sym) of Bin Cages, Characterized by n and Figure Number (in Parentheses) n
Eb eV/atom
H-L eV
sym
4 (1) 6 (1) 8 (1) 10 (1) 12a (1) 12b (1) 20 (1) 24 (1) 28 (1) 30 (1) 32 (1) 36 (1) 50 (1) 50 (3a) 50 (3c) 60 (1) 60 (3) 76 (1) 80 (1) 80 (3)
1.94 1.89 1.88 1.92 (1.88) 2.00 2.01 1.99 1.97 1.96 1.95 1.93 1.89 1.97 2.05 1.87 2.00 1.98 1.84 2.07
4.11 2.55 2.22 2.09 1.89 2.64 2.31 2.27 2.15 2.20 2.01 1.82 1.65 1.81 2.01 1.53 1.86 1.38 1.38 1.90
Td D3h Oh D5h D6h C2 Ih D6d Td D5h D3d D6h D5h C5V D5h Ih D5d Td Ih Ih
and 1.89 eV/atom, respectively, comparable to the values of 1.94 and 1.88 eV/atom obtained here (see Table 1). The tetramer is the most prominent species of the group 15 elements and has been extensively studied. The tetrahedral structure in Figure 1 (n ) 4) is the lowest energy structure for all three of them.19,23-26 It is also the structural unit of white phosphorus.26 As a matter of fact all three structures serve as primary building blocks for larger clusters, in particular of As.23 The Oh structure in Figure 1 (n ) 8) for Bi8 according to Yuan et al. is the second lowest by only 0.04 eV/atom from the first.
Bismuth and Other Group 15 Fullerenes
Figure 2. Comparison of the Bi20 (a) and Si20H20 (a′) cages and their HOMO (b and b′) and LUMO (c and c′) orbitals.
These differences, which however are not systematic (in ref 19, for instance, Bi4 is not the most stable of the three), are attributed to the different DFT functionals and/or basis sets used in the two works. This is also reflected in the HOMO-LUMO gaps (1.94, 1.89, and 1.88 eV respectively in ref 19), which are a measure of the kinetic stability35 and which are drastically underestimated in that work (ref 19), in comparison to the B3LYP results here (in agreement with known trends36 in hydrogenated Si nanostructures36 and fullerenes7 of nonhybrid functionals to overestimate binding energies and underestimate HOMO-LUMO gaps). b. Bi10 and Bi12. The cage structures of Figure 1 (n ) 10 and 12a), fully analogous to Si10H10 and Si12H12, are not the global minima, and yet they are very low lying local minima. This becomes easily apparent by comparing the best binding energies per atom in the literature19 (1.99 and 2.03 eV/atom respectively) with the values quoted in Table 1, especially after taking into account the expected overestimation of these values7,34 by the nonhybrid functionals. The C2 symmetric Bi12 structure of Figure 1 (n ) 12b) which is of octahedral origin (obtained after geometry optimization and distortion according to imaginary frequency vibrational modes) is of much lower energy. c. Bi20. This is the smallest stable icosahedral bismuth structure (fullerene), which is fully analogous and “isolobal” to the isovalent Si20H20 and C20H20 cages7,12 as we can see in Figure 2. Two structural units or fragments are isolobal if the number, symmetry properties, approximate energy, and shape of the frontier orbitals (and the number of electrons in them) are similar18,35). On the basis of the binding energy per atom Bi20 is the most stable fullerene from the fullerenes in Figure 1 and the third most stable isomer in Table 1. This is analogous to the stability of Si20H20 (see refs 7 and 9) and C20H20 (refs 11 and 12) cages. The lowest energy structure(s) of Bi20 in the literature19 is(are) open nonsymmetric (C1 symmetry) “amorphous”-like structures composed mainly of Bi4 and Bi6 subunits, with binding energy about 2.06 eV/atom and with average coordination number equal to 3. These characteristics (binding energy and coordination) are not far from the ones shown in Table 1 for the highly symmetric (Ih) fullerene (considering also the expected overestimation of the binding in ref 19) which by construction is 3-fold coordinated. This is certainly true for the other medium and large size Bi fullerenes examined here. Yuan et al.19 have concluded that cage-like and compact Bin structures are metastable, whereas open nonsymmetrical struc-
J. Phys. Chem. C, Vol. 114, No. 24, 2010 10777 tures are the lowest energy configurations (at least up to n ) 24), similarly to the corresponding P, As, and Sb counterparts.19 The present results which are, at least aesthetically, more appealing present alternative possibilities and reveal that the fullerene structures are true very low lying minima with real frequencies. In this respect they are not metastable. d. Medium Size Bi24, Bi28, Bi30, Bi32, and Bi36 Fullerenes. Between Bi20 and the smallest of the large (Bi50) fullerenes, we have considered a of series medium size fullerenes: Bi24, Bi28, Bi30, Bi32, and Bi36, in full analogy to Si24H24, Si28H28, Si32H32, and Si36H36 (see ref 7). These clusters represent various (high) symmetries and structural motifs such as D6d, Td, D3d, D5h, etc., with binding energies ranging from 1.89 to 1.99 eV/atom. All of these structures, like Bi20, are true low-lying minima with real frequencies. The calculations for Bi24 and Bi20 in the literature19 seem to favor open nonsymmetrical structures. The same is true for the As20, As24, and As28 structures.23 All of the fullerenes examined here, as Bi20 and As20, have real frequencies and, in this respect, are not metastable. As was mentioned earlier, here we have not performed exhaustive lowest energy structure searches, and therefore, we cannot confirm or exclude the possibility of energetically lower compared to other nonsymmetrical compact, or open “amorphous” isomers. This is not so important for the present study, as we are mainly interested in showing that the fullerene structures are stable (statically and dynamically) and very low-lying energetically (not necessarily the lowest). Furthermore, most of these fullerenes (in Figure 1) can be further stabilized by improving their bonding arrangement and/or by doping, as will be discussed below. e. Large Fullerenes Bi50, Bi60, Bi76, and Bi80. With the exception of n ) 76, which will be discussed separately below, the large size fullerenes in Figure 1 (n ) 50, 60, and 80), have substantially lower binding energies from the medium and small size fullerenes in the same figure, discussed above. This suggests that larger cages get progressively destabilized. However, On the basis of the experience from the corresponding homologous SinHn clusters,9-12 we expect that we can substantially improve the binding by improving bonding through “puckering”, as can be seen in Figure 3. In the icosahedral SinHn and CnHn clusters, n ) 60 and 80, “puckering” is facilitated by partial endohedral hydrogenation.9,11,12 For such clusters (and not only those) the stability can be improved even to values better than n ) 20. This type of endohedral hydrogenation allows each H-Si-Si-H or H-C-C-H fragment, corresponding to a Si-Si (or C-C) shared edge, to adopt a chair like structure with one hydrogen atom pointing inside the cage and one pointing outward9,11,12 leading to a “puckered” cage. This allows for better optimization of the sp3 bond-angles and a resulting energy lowering. For n ) 80 this can be achieved without altering the high Ih symmetry.11 For n ) 60 puckering is achieved9,12 by slightly lowering the symmetry to D5d (the larger subgroup of Ih). In both cases the number, Nedh, of endohedrally bonded hydrogens is crucial. The critical number Nedh for n ) 60 has been shown12 to be at least 10; whereas, for n ) 80, Nedh )20. It has been found also in the course of the present investigation that puckering could be also beneficial for smaller than n ) 60, such as the n ) 50 (Si50H50 and Bi50) fullerene (see structures in Figure 3a,b, and c). Surprisingly enough the lowest energy puckered Bi50 isomer with D5h symmetry is more stable (on the basis of binding energy) than Bi20 and Bi60 and its stability is only second to the puckered icosahedral Bi80 fullerene. In this case the symmetry is lowered to C5V initially (see Figure 3a)
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Figure 4. The spin density (a), HOMO (b), and LUMO (c) orbitals of the lowest energy Bi@Bi60 isomer with D5d symmetry.
Figure 3. Low energy “puckered” structures of n ) 50, 60, 80 Bin fullerenes in two different views: top (a) and side (b). The lowest energy puckered Bi50 isomer is shown in (50c).
and the energy gain is about 5 eV (or binding energy gain of about 0.1 eV/atom). Yet, through further relaxation of the strain followed by geometry optimization we can get the puckered structure of Figure 3c, symmetrized to D5h, with an additional gain of about 0.1 eV per atom, in binding energy. It is clear (in particular for the lower symmetry fullerenes) that several alternative structures can be constructed with the same number Nedh but noticeably different binding energies, depending on the type and geometry of the puckered atoms. As was explained elsewhere,9,11,12 the puckering of hexagons with bond-angles far away from the ideal sp3 bond-angles is more favorable compared to pentagons which are closer to the ideal bond-angles. However, the puckering of the bond angles in hexagons, affects unavoidably also the bond angles of the neighboring pentagons (108°) which were already near the optimal value of 109.5°, thus increasing the corresponding pentagon energy. Therefore, puckered isomers in which more hexagons and fewer pentagons are puckered are energetically more favorable. For the n ) 20 fullerene which consists solely of pentagons puckering is not effective, and so is expected for smaller fullerenes in which the number of hexagons (compared to pentagons) is limited. From the results presented here it seems that the region of efficient puckering starts around n ) 50. The effect of puckering is more general than just partial endohydrogenation which facilitates puckering for SinHn and CnHn fullerenes or even silicon nanotubes37 and nanowires.38 It also works, as is shown here and exemplified in Figure 3 (and Table 1) for the large size (n ) 50, 60, 76, and 80) Bin and other group 15 fullerenes. The role of endohydrogenated and exohydrogenated Si or C atoms is played by Bi atoms pointing inward and outward, respectively, resulting in an “inkpot neck” effect (shape), for the second lowest Bi50 isomer (in the top row of Figure 3a,b) and similar more symmetrical shapes for the D5h puckered Bi50 isomer in Figure 3c (top) and the rest of the structures in Figure 3. The binding energies of these puckered isomers are substantially (around
10%) improved and the best puckered Bi80 isomer, as the homologous Si80H80 fullerene becomes more stable (on the basis of binding energy) than the reference critical Bi20 and other very stable (as the Bi50) fullerenes. This is also true for the for Si80H80 and C80H80 fullerenes. The symmetry of the puckered isomers is always lower than (as in Bi60) or equal to (as in Bi80) the symmetry of the corresponding nonpuckered isomers. The n ) 76 structure in Figure 1 and Table 1, is already the best puckered Bi76 fullerene, as is verified also by its relatively high binding energy. This is why this structure is not included in Figure 3. It becomes clear from this discussion that large size fullerenes can be very stable, and in fact more stable than smaller clusters and fullerenes, due to puckering. The puckered fullerenes are always more stable than the corresponding nonpuckered ones (at least by about 10%). This stability is comparable to or better than the stability of the smallest Bi20 fullerene, the stability of which is not far from the stability quoted in the literature19 for the calculated lowest energy structures, which are open and amorphous-like. Furthermore, in agreement with the finding19 that the average coordination number of the larger Bin clusters is 3, the (average) coordination number of all these fullerenes is strictly 3. 4. Embedded Bi@Bin “Fullerenes” One of the most common applications of fullerenes and hollow cages is endohedral doping with various atomic agents (molecules, ions, etc.). For C20H20 and Si20H20 (and other CnHn and other SinHn “fulleranes”), one of the most popular10,39 “agents” for doping is Si-, which is isovalent with Bi. Doping with Si- of the SinHn fulleranes, and in particular Si28H28 has led to an alternative model for the 1 nm luminous siliconhydrogen nanoparticle10 and, at the same time, to a highly localized spin at the central embedding Si- ion while the Si28H28 cage more or less retains its electronic structure. Such endohedral fullerenes whose dopant atoms (or ions) retain (more or less) their isolated atomic states have received attention recently,40,41 partly because of newly proposed solid-state quantum computers based on such materials. It is expected that the advent of such solution-processed optoelectronic materials would offer the potential for a revolution in optoelectronics, since their solution processability enables low-cost large-area monolithic integration on a variety of electronic read-out platforms. It is natural therefore to expect analogous behavior for bismuth fullerenes doped centrally with Bi atom, which is isovalent to Si-. Figure 4, which shows the spin density and HOMO-LUMO gaps of the D5d symmetric Bi@Bi60 puckered isomer, demonstrates the spin localization (and the localization of the HOMO orbital) at the central atom. This illustrates that Bi@Bi60 is a potential candidate for the kind of optoelectronic applications discussed above. Obviously, it is expected that besides Bi@Bi60, other Bi@Bin fullerenes would be operating in a similar way. Energetically, this possibility is quantified
Bismuth and Other Group 15 Fullerenes through the embedding energy, which is a measure of the total energy difference (the energy gain) before (embedding atom at infinity) and after the insertion of the central Bi atom. There are different ways to define the embedding energy7 (Eemb) depending on the reference state before and after embedding. For relative comparisons the exact definition is not important, provided we use the same definition throughout. Here we simply use for Eemb the binding energy difference of the system of empty cage and the embedding atom at infinity from the optimized doped cage.7 For the Bi@Bi60 embedded fullerene in Figure 4, we find Eemb ) 0.26 eV. This is a very small value indicating the very weak interaction, which could be desirable for certain applications. The value of Eemb very much depends on the size of the atom and of the cage, having the largest value of about 0.40 eV for Bi@Bi50 for the D5h Bi50 isomer in Figure 3c, and 0.33 for the C5V Bi50 isomer in Figure 3a,b. In the later cage, in which there is no center of inversion symmetry, the embedding Bi atom is moved closer to the “neck of the pot” in order to optimize its coordination and its interaction with the cage. It seems that for the n ) 50 cage there is an optimum size matching of the atom and cage. For larger cages the embedding energy decreases because of the larger distance of the embedding atom from the cage. Thus for Bi@Bi76, Eemb ) 0.18 eV, whereas Bi@Bi80 is characterized by practically zero embedding energy, indicating virtually no interaction at all for the central atom with the cage atoms. For smaller than n ) 50 cages the embedding energy is getting smaller due to the increased repulsion of the central atom from the cage, which is maximized for small near n ) 20 cages, leading to negative Eemb. For Bi@Bi20 and Bi@Bi24 we have the largest magnitude negative values (-0.4 and -0.2 eV respectively) of Eemb. For these fullerenes (with negative Eemb) the Bi atom at the center of the cage is in fact under pressure. This could have significant influence on the optical and optoelectronic, among others, properties. Thus, for the n ) 50, 60, and 76 “fullerenes” the doping with Bi stabilizes them better, whereas for n ) 20-36 it tends to destabilize them. For n ) 80 doping has practically no effect in the stability. These properties could be properly taken advantage of for optoelectronic applications, provided that the cages could be synthesized. 5. Other Group 15 “Fullerenes” a. Pn, Asn, and Sbn Cages. The same reasoning and arguments about the stabilization of fullerene structures apply also for other group 15 elements as antimony, arsenic, and phosphorus. It is therefore very interesting to examine also antimony, arsenic, and phosphorus fullerenes. In Table 2 we have compiled some cohesive (binding energy) and electronic (HOMO-LUMO gaps) characteristics of some representative Pn, Asn, and Sbn cages. In this table for n ) 6, we have also performed an interesting comparison between the normal prismatic structure, similar to the “ground state structure” of Si6H6 (hexasilaprismane), and a six-member ring structure (6R), similar to C6H6 (benzene). As we can see, although all P6, As6, and Sb6 and Bi6 structures are similar to Si6H6, the energy differences between the prismatic and the ring structures keeps decreasing as we move from Bi6 to P6 and in fact turns negative for the first row N6 cluster which is similar to C6H6 (see footnote in Table 2). For the larger cages we can see that although P50, like Bi50 is more stable than both P20 and P60, this is not true for the corresponding arsenic and antimony fullerenes, for which n ) 20 is the most stable (on the basis of binding energy) fullerene.
J. Phys. Chem. C, Vol. 114, No. 24, 2010 10779 TABLE 2: Cohesive (Binding Energy Per Atom, Eb in eV/Atom) and Electronic (HOMO-LUMO Gap, H-L, in eV) Characteristics of Selected Pn, Asn, and Sbn Cages Pn
Asn
Sbn
n
Eb
H-L
Eb
H-L
Eb
H-L
4 6a 6Ra 8 20b 28 (1)50 (3)50 60b
2.96 2.76 2.64 2.70 2.71 2.78 2.90c 2.90c 2.80
6.32 3.49 3.11 2.67 2.79 2.70 2.29 2.29 2.08
2.55 2.43 2.26 2.39 2.55 2.47 2.34 2.46 2.48
5.43 3.41 3.10 2.60 2.67 2.65 1.86 2.16 2.07
2.13 2.06 1.83 2.06 2.17 2.09 1.98 2.09 2.12
4.11 2.81 2.57 2.35 2.38 2.35 1.69 1.95 1.97
a The binding energies of N6 pyramid are 2.01 eV/atom and ring 2.88 eV/atom. b The binding energy of N20 is 2.97 eV/atom, and its HOMO-LUMO gap is 5.24 eV. c They both converge to the same structure.
TABLE 3: Structural Characteristics (Minimum, lmin, and Maximum, lmax, Bond Lengths in Angstroms) of Representative Pn, Asn, Sbn, and Bin Cages Pn
Asn
Sbn
Bin
n
lmin
lmax
lmin
lmax
lmin
lmax
lmin
lmax
4 6 6R 8 20 28 (1)50 (3)50 (3)60
2.215 2.269 2.137 2.308 2.290 2.247 2.274a 2.274a 2.255
2.215 2.323 2.137 2.308 2.290 2.292 2.315a 2.315a 2.337
2.455 2.502 2.350 2.540 2.488 2.468 2.447 2.499 2.475
2.455 2.552 2.350 2.540 2.488 2.512 2.679 2.687 2.549
2.842 2.892 2.735 2.922 2.867 2.844 2.842 2.858 2.858
2.842 2.935 2.735 2.922 2.867 2.886 2.969 2.978 2.925
3.007 3.040 2.874 3.083 3.008 2.994 2.995 2.995 3.010
3.007 3.085 2.874 3.083 3.008 3.020 3.055 3.062 3.061
a
They both converge to the same structure.
TABLE 4: Embedding Energy, Eemb, in eV for Several Representative Embedding Atoms (P, As, Sb, and Bi) and Embedded Cages of Group 15 Elements Cage (CG)
P@CG
P20 As20 Bi20 (3a,b) Bi50 (3c) Bi50 (3) Bi60
-1.73 -0.85 -0.07 0.34 0.26 0.13
a
As@CG -1.08 -0.10 0.38 0.33 0.18
Sb@CG
Bi@CG
-1.64 -0.38 0.30 0.31 0.20
-2.30 -1.76 -0.45 0.33 0.39 0.26 (-0.01)a
Value for the nonpuckered Ih isomer.
We can also observe that, unlike Bi50, the C5V P50 isomer is not stable, reverting under optimization to the D5h symmetric isomer. The structural and bonding characteristics of the clusters of Table 2 are summarized and compared in Table 3. b. Doped Pn, Asn, and Sbn Cages. Like Bin, Pn, and Asn, as well as Sbn, cages can be doped by similar or different group 15 atoms. Some representative results for the embedding energies of characteristic doped n ) 20, 50, and 60 cages are listed in Table 4. Due to the small size of the n ) 20 cages and the resulting repulsions with the central atom, all embedded energies for n ) 20 cages, as would be expected are negative. This is also reflected in the HOMO and LUMO orbitals of the Bi@Bi20 and Pi@Bi20 fullerenes shown in Figure 5, which have no visible contributions from the central atom. In agreement with the size restriction interpretation, we can see in Table 4 that the variation of the embedding energy of X@Bi20, X ) P, As, Sb, and Bi, increases (decreases in absolute value) with decreasing size of the embedding atom. Similarly
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Figure 5. HOMO (left) and LUMO (right) orbitals of P@Bi20 (top) and of P@Bi20 (bottom). Spin density for both is shown in the middle right.
Figure 7. Frontier orbitals and spin density of the C5V symmetric second lowest structure of Bi@Bi50 in top (left) and side (right) views.
orbital, has components localized in specific peripheral bismuth atoms, besides the “central” atom. It is also clear that neither the HOMO orbital is localized at the “central” atom. Instead, as we can see in Figure 7, the HOMO-1 orbital is rather well localized at the doping bismuth atom. The energy gain by such doping is comparable but smaller than the corresponding gain of the lowest energy D5h puckered isomer (0.33 eV in comparison to 0.39 eV) which has center of inversion. Figure 6. HOMO and LUMO orbitals together with spin density of P@P50 (left) and Bi@Bi50 cages in top (a, c, e, a′, c′, and e′) and side (b, d, f, b′, d′, and f′) views.
the embedding energy of Bi@X20 (X ) P, As, Sb, and Bi) increases with increasing size of X atom and, consequently, of X20 cage. This is also true for the embedding energy of Sb@X20, As@X20, and P@X20 cages. The increase of Eemb with cage size continues up to n ) 50, from where it starts decreasing again (due to decreasing interactions of the central atom with the cage) up to n ) 80 where it becomes practically zero, implying no interactions at all with the central atom. For the same reason (decreasing doping atom-cage interactions) the X@Bi60 (X ) P, As, Sb, and Bi) embedding energy, oppositely to X@Bi20, decreases with decreasing X atom size. The same trend is true for the variation X atom size of Eemb for the X@Bi50 with a small deviation for the As@Bi50. In contrast to Bi@Bi50, P@P50, which is compared in Figure 6, is characterized by negative Eemb, apparently due to size mismatch. However, as is shown in Figure 6, the orbitals are pretty much similar and the structures could be characterized as isolobal. In both cases, as well as in all similar embedded fullerene cages shown in Figures 4 and 5, where the doping atom resides at the center of symmetry of the cage, the spin is fully localized at the central doping atom. Normally the same is expected to be true for the HOMO orbital (but not for the LUMO). However, this is not always true (or it is partially true), especially for cages without center of symmetry such as the C5V symmetric Bi50 structure in Figure 3c. This is illustrated in Figure 7, which shows the HOMO, HOMO-1, and LUMO orbitals together with the spin density of the C5V symmetric Bi@Bi50 doped fullerene. As was explained earlier, the doping atom in this case moves and resides closer to the “bottle neck”, not in the center. Figure 7 shows that the spin density in this case, similarly to the HOMO
6. Conclusions It has been illustrated and predicted that stable bismuth fullerenes Bin of high binding energies per atom (higher than the ones calculated in the literature for the largest clusters, assumed as global minima) can be formed (and hopefully synthesized) with large HOMO-LUMO gaps. In agreement with earlier predictions that the average coordination of the large bismuth clusters should be 3, all these fullerenes have (by construction) a coordination number of exactly 3. These Bin fullerenes are similar and isolobal with the isoelectronic CnHn fulleranes or SinHn “fullerenes”. The larger Bin fullerenes (n g 50), similarly to CnHn and SinHn “fullerenes” are considerably stabilized by puckering of the bonds which significantly improves the tetrahedral bond angle optimization. The improvement of binding energy by such puckering could be in some cases better than 10%. From all Bin fullerenes examined, the puckered Bi80 isomer in Figure 3 has the highest binding energy (2.07 eV/atom) with second the D5h symmetric puckered Bi50 isomer in Figure 3c. Both of these fullerenes have higher binding energy than both Bi20 and Bi60. The n ) 50 fullerene, in addition, has the higher embedding energy for doping by bismuth or other group 15 atom, indicating that this (n ) 50) is the optimum atom-cage size matching. Such doped fullerenes could have significant applications in optoelectronic devices and possibly quantum computers due to the spin localization on the central atom, which retains to a large degree its atomic-like electronic states in most cases. Similar results are obtained for other group 15 elements besides bismuth, leading to antimony, arsenic and phosphorus fullerenes with homologous structural and electronic properties. Here again the optimum size for homoatomic doping of the
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