Smallest Fullerene-like Structures of Boron with Cr, Mo, and W Encapsulation
SSmallest Fullerene-like Structures of Boron with Cr,Mo, and W Encapsulation
Amol B. Rahane ∗ , Pinaki Saha , N. Sukumar , and Vijay Kumar † Dr. Vijay Kumar Foundation, 1969, Sector 4, Gurgaon - 122 001,Haryana, India, Tel: +91 124 407 9369 Department of Physics, K. R. T. Arts, B. H. Commerce and A. M.Science (KTHM) College, Nashik - 422 002, Maharashtra, India. Department of Chemistry, School of Natural Sciences, Shiv NadarUniversity, NH-91, Tehsil Dadri, Gautam Budhha Nagar 201 314, UttarPradesh, India. Center for Informatics, School of Natural Sciences, Shiv NadarUniversity, NH-91, Tehsil Dadri, Gautam Budhha Nagar 201 314, UttarPradesh, India.
July 31, 2019 ∗ Electronic address: amol [email protected],[email protected] † Electronic address: [email protected], [email protected], [email protected] ; Corre-sponding author a r X i v : . [ phy s i c s . a t m - c l u s ] J u l bstract Using density functional theory calculations, we study doping of a Cr, Mo, andW atom in boron clusters in the size range of 18-24 atoms and report the find-ing of metal atom encapsulated fullerene-like cage structures with 20 to 24 boronatoms in contrast to a fullerene-like structure of pure boron with 40 atoms. Our re-sults show that bicapped drum-shaped structures are favoured for neutral Cr@B ,Mo@B , and W@B clusters whereas a drum-shaped structure is preferred forneutral, cation, and anion of Mo@B and W@B . Further, we find that B is thesmallest cage for Cr encapsulation, while B is the smallest symmetric cage for Moand W encapsulation and it is magic . Symmetric cage structures are also obtainedfor Mo@B and W@B . A detailed analysis of the bonding character and molec-ular orbitals suggests that Cr@B , Cr@B , M@B (M = Cr, Mo, and W) andM@B (M = Mo and W) cages are stabilized with 18 π -bonded valence electronswhereas the drum-shaped M@B (M = Mo and W) clusters are stabilized by 20 π -bonded valence electrons. Calculations with PBE0 functional in Gaussian 09 codeshow that in all cases of neutral clusters there is a large highest occupied molecularorbital-lowest unoccupied molecular orbital (HOMO-LUMO) gap. In some casesthe lowest energy isomer of the charged clusters is different from the one for theneutral. We discuss the calculated infrared and Raman spectra for the neutral andcation clusters as well as the electronic structure of the anion clusters. Also wereport results for isoelectronic anion and neutral clusters doped with V, Nb, and Tawhich are generally similar to those obtained for Mo and W doped clusters. Theseresults would be helpful to confirm the formation of these doped boron clustersexperimentally. The recent finding of fullerene-like empty cage structure[1] of B using a combined ex-perimental and computational study as well as the occurrence of quasi-planar structuresfor many boron clusters has spurred a lot of interest in experimental and theoreticalstudies of boron clusters to understand their growth behavior and stability. Also a layerof boron, called α -sheet as well as quasi-planar structures of clusters are stabilized withhexagonal holes in otherwise triangular structures. These results are exciting because thepossibility of the formation of carbon-like nanotubular or graphene-like planar structuresbased on a hexagonal lattice was ruled out due to the deficiency of electrons in boronwhich is known to favor three center bonding. It has been suggested that the stability of α -sheet as well as quasi-planar structures of clusters is due to a mixture of triangular andhexagonal network that is energetically favorable over only a triangular network whichhas excess of electrons. Several studies on small boron clusters having up to around 36atoms show that they have planar or quasi-planar or tubular structures as their groundstate[2, 3, 4, 5, 6, 7, 8] with the exception of B for which a cage-like structure has beenproposed[9]. A double-ring tubular (DRT) structure has been suggested for neutral B [4]and B [10] while a cage structure has been suggested for B [11] using particle swarmalgorithm combined with density functional theory (DFT) calculation. A recent study2y Chen et al. has also identified a cage structure with axially chiral feature for B [12]and another recent theoretical study[13] on neutral B suggests a filled cage structureto be the lowest in energy while a planar isomer is nearly degenerate. Also, althoughfor neutral B a fullerene-like cage structure is the most stable one, a planar isomer isfavoured for B anion and therefore a quasi-planar isomer competes in energy for B .A similar situation may arise for other sizes as well that charged clusters have anothercompeting structure. We ask the question if smaller cages of boron can be stabilized suchas by metal (M) atom encapsulation.Studies on a large number of bulk boron compounds show that an empty center icosa-hedral cage of B is the major building block of their structures. However, an isolatedB icosahedral cage is not stable due to the availability of 36 valence electrons, while 26electrons are required by Wade’s rules[14] to stabilize this cage. One way to stabilize theicosahedral cage is to attach ligands such as hydrogen atoms exohedrally. As an example,cage-like borane structure B H − is stabilized by 26 valence electrons excluding thosein the B-H bonds.[15] Another way to form cage structures is to dope endohedrally an Matom that may interact with boron atoms strongly and stabilize cage structures. Such astrategy has been successfully used to stabilize non-carbon cage structures such as thoseof silicon and other elements.[16, 17, 18, 19, 20, 21, 22, 23] In particular, exceptionalstability has been suggested for Zr@Si fullerene and Ti@Si Frank-Kasper polyhedronstructures[16] that have been subsequently realized in laboratory and even assemblieshave been formed.[24] The size of a B icosahedron is too small to encapsulate an Matom. As boron atom is smaller in size compared to a silicon atom, a possibility to formboron cages may lie around the size of about 20 atoms. A recent independent study hasindeed proposed stabilization of M doped cages for B by encapsulation of Mo and W,whereas for Cr a less symmetric configuration has been reported for this size.[25] Disk-likeor wheel-shaped structures have also been predicted for boron clusters having up to about11 boron atoms by doping a transition M atom[26, 27, 28, 29, 30, 31, 32, 33] as well as otherelements.[34] The stability of some disk-shaped clusters has been correlated with electronicshell closing at 12 valence electrons.[34] Further, recently bowl-shaped[35, 36, 37, 38, 39]and drum-shaped[40, 39] structures of boron have been predicted to be stabilized by Matom dopants such as M@B drum with M = Cr, Fe, Co, and Ni, and also with 16 boronatoms such as Co@B − .We have studied M atom encapsulated boron clusters in the size range of 18 to 24 atomsin order to find the smallest cage of free boron clusters besides the drum structures. Notethat in this size range, pure boron clusters have quasi-planar or tubular structures. Asthe bonding in boron is stronger compared with silicon, we considered strongly interact-ing transition M atoms with half-filled d states such as Cr, Mo, and W to explore thepossibility of stabilizing boron cages in this size range. Based on our systematic study,we predict fullerene-like cages Cr@B , M@B and M@B with M = Cr, Mo, and W inwhich one M atom is encapsulated in the cage. Also we find bicapped drum structures forCr@B and M@B (M = Mo and W) and a drum structure for M@B , M = Mo andW. The stability of these structures has been studied by analysing the molecular orbitals(MOs) as well as the electronic charge density. Our results suggest that Mo and W are3ell suited to produce endohedrally doped novel cage structures of boron as the dopingof M atom has the effect of increasing the binding energy significantly. The doping ofCr atom also leads to cage formation, but the binding energy of the doped clusters hassimilar values as for the pure boron clusters. We also present results for IR and Ramanspectra of the neutral and cation clusters as well as for the electronic structure of anionclusters that would help to identify the atomic structure of these doped clusters whenexperimental data may become available. We used generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof(PBE)[41] for the exchange-correlation functional and projector augmented wave (PAW)pseudopotential plane wave method[42, 43] in Vienna ab initio simulation package (VASP)[44]to explore several isomers for the doped boron clusters. The calculations were consideredto be converged when the absolute value of the force on each ion was less than 0.005 eV/˚ A with a convergence in the total energy of 10 − eV. Further calculations were performedfor the lowest energy isomers of the neutral clusters using Gaussian09 code[45] and PBE0functional. Also, we have calculated the vibrational modes of cation and neutral clustersusing Gaussian09 code, and in almost all cases we obtained real frequencies suggestingthe dynamical stability of the obtained atomic structures. For these calculations we usedB3PW91 hybrid exchange-correlation functional as well as PBE0 functional and 6-311+Gbasis set[46, 47] for Cr doping and LANL2DZ basis set[48, 49] for Mo and W while 6-311+G basis set has been used for B atoms in all the cases. The atomic structures wereoptimized and used to calculate IR and Raman spectra for the neutral and cation clusters.Calculations using PBE0 exchange-correlation functional on the lowest energy isomers ofthe pure and M doped boron clusters also suggest stabilization of boron cages with Mencapsulation. The bonding characteristics have been studied by performing adaptive nat-ural density partitioning (AdNDP)[50] analysis at the PBE level of the theory and usingthe same basis sets in the Gaussian09 code and also from the Laplacian L = -(1/4)∆ ρ ( r ),of the electronic charge density ρ ( r ) using AIMALL.[51] We also calculated the electronlocalization function (ELF) using the charge density distribution obtained from VASPfor the lowest energy isomers of the neutrals and the electron localization-delocalizationindex using AIMLDM script.[52] Further calculations have been done on cation and anionclusters using PBE0 functional in Gaussian09 code. Some results are also included forthe isoelectronic anions of V, Nb, and Ta doped clusters. The MOs have been analysedusing the Gaussian 09 code. We used VESTA 3,[53] Molekel 5.4,[54], XCrysden 1.5 [55],AIMALL, and Gaussview[56] for visualization. In all cases of charged and neutral clustersthe atomic structures were again optimized when using Gaussian09 code.4 RESULTS and DISCUSSION
We studied doping of an M atom (M = Cr, Mo, and W) in several different structures ofboron clusters in the size range of B to B in the hope of finding the smallest cage ofboron. To begin with we doped an Mo atom inside a 16-atom Frank-Kasper polyhedronof boron because all the faces in this structure are triangular and boron has preferenceto form 3-center bonds as in an icosahedron. But upon optimization it ends up in anopen structure indicating that the formation of a boron cage for this size is unlikely withendohedral doping of an Mo atom. However, recently drum-shaped clusters of 16 boronatoms have been obtained with the doping of a Co atom, while bowl-shaped clusters havebeen shown[39] to be formed for many of the 3d, 4d, and 5d transition M atoms. Also14-atom drum-shaped boron clusters have been shown[39] to be favored by doping ofsome 3d transition M atoms such as Cr, Fe, Co, and Ni. These results suggest that a cagestructure, besides drum-shaped clusters, is unlikely to be favorable for a 16-atom boroncluster with these M atoms.Next we focussed our attention on exploring cage formation with 18 and 20 boronatoms. We added one Mo (and also Cr and W) atom on the planar B isomer and atthe center of the DRT structure of B . Upon optimization, the planar B structuretransforms into a bowl-shaped isomer as shown in Figure S1 (see isomer III and othersimilar bowl-shaped structures IV and VIII) in Supplementary Information. This clearlyshowed that the doping of these M atoms tends to form a cage of boron. We rearrangedthe atoms and tried to construct a cage-like structure of boron but upon re-optimizationit becomes again a bowl-shaped open structure indicating that more boron atoms areneeded to complete a cage. Some such open as well as cage type structures are shownin Fig. S1 in Supplementary Information, but all of these lie higher in energy than thelowest energy isomer. Figure S1(I) in Supplementary Information and Fig. 1(a) show thelowest energy isomer for CrB which is a bicapped 16-atom drum-shaped structure. Onthe other hand for MoB and WB , a tubular drum-shaped structure shown in Fig. 1(b)and (c), respectively, and also in Fig. S1(II) in Supplementary Information, has the lowestenergy, while the bicapped drum-shaped isomer lies 0.36 eV and 0.42 eV higher in energyfor Mo and W, respectively. On the other hand the drum-shaped isomer of Cr@B lies1.25 eV higher in energy compared with the lowest energy isomer. In the case of MB (M= Cr, Mo, and W), we attempted many structures, both open and cage type, and theseare shown in Fig. S2 in Supplementary Information. We added two boron atoms in asymmetric fashion to the bowl-shaped MoB cluster and optimized the atomic structure.The resulting structure was found to be lower in energy than the one obtained from thedoping of an Mo atom in B DRT structure. However, the optimized structure is stillopen, again suggesting that 20 boron atoms are not sufficient to form a cage with Moencapsulation. Among the different structures we explored, it is found that a bicapped18-atom drum-shaped isomer (Fig. 1(e) and (f)) has the lowest energy for Mo and Wdoping. However, as we shall show below, Cr is able to form a cage (Fig. 1(d)) with 205oron atoms, and it is the smallest cage of boron that we have obtained.It is clear from the above that there is room to accommodate a few more B atoms onthe bicapped drum of Mo@B to form a cage. Therefore, we added two more B atoms onthe other side of the drum so that the two capping dimers form a cross leading to Mo@B cluster. Also we added four boron atoms to MoB open structure to form a Mo@B cagestructure in the hope of finding a symmetric isomer, as intuitively a 22-atom cage structureis unlikely to be very symmetric. The optimized structures show that the Mo@B isomerconverges with the same local structure while the optimized atomic structure of Mo@B had a cage form, but it was slightly lower in symmetry. We changed the positions of twoboron atoms in this Mo@B structure in order to create a symmetric structure and re-optimized it. In two such steps we obtained a very symmetric cage structure of Mo@B as shown in Figure 1 (k). This is about 2 eV lower in energy compared with the initiallyoptimized structure (without rearranging boron atoms). The pathway starting from B to the lowest energy isomer of Mo@B is shown in Figure 2. The lowest energy structurefor Mo@B has (D h ) symmetry and a large highest occupied molecular orbital-lowestunoccupied molecular orbital (HOMO-LUMO) gap (GGA) of 2.46 eV indicating its highstability. We tried several other isomers for Mo@B but all of them lie higher in energythan the D h cage isomer, except in one case where we considered an Mo atom doped atthe center of a truncated cube and it converged to the same lowest energy D h isomer. Thisgave us further support that our structure may be the lowest in energy. The pathway isindicated in Figure 3. Interestingly the same isomer has been also independently obtainedfor Mo@B by Lv et al.[25] using particle swarm optimization algorithm implementedin the CALYPSO package, which involved calculations of a few thousand structures. Inthese calculations hybrid PBE0 method has been used for the structure optimizationwith 611G* basis set in Gaussian 09 package. This gave us further confidence that ourstructure may indeed be the lowest energy structure for Mo@B . Further calculations onW doping led to the same isomer to be of the lowest energy. The optimized structuresof some of the isomers we attempted for M@B and M@B (M = Cr, Mo, and W) areshown in Figs. S3 and S4, respectively in Supplementary Information. It can be notedthat for Cr doping many isomers lie in a narrow window of energy.After obtaining neutral Mo@B and W@B doped boron clusters, both of which arealmost identical, we asked ourselves the question if we can stabilize a smaller cage ofboron, and in particular if a 24 boron-atom cage is the smallest cage for Mo and W. It isto be noted that when Cr is doped in the lowest energy isomer of Mo@B , the optimizedstructure is a somewhat distorted cage (Fig. 1 (j)) in which the Cr atom does not interactwith all the boron atoms optimally. This is because the size of a Cr atom is not optimalfor the B cage as it is slightly smaller than Mo and W atoms. The natural question wefurther asked was: what is the smallest cage stabilized by Cr. To address this questionwe further explored different isomers for the size of 22 boron atoms, as in the 24-atomcage we can remove two B atoms and still keep the cage structure, and also some cagestructures for 20 boron atoms. Below we first discuss the formation of a B cage andthen the B cage.In order to study the doping of a Cr, Mo, and W atom in B cluster, we tried6 a) B Cr, C (b) B Mo, C s (c) B W, C s (d) B Cr, D (e) B Mo, C s (f) B W, C s (g) B Cr, D (h) B Mo, D (i) B W, D (j) B Cr, C (k) B Mo, D (l) B W, D Figure 1: Atomic structures of the lowest energy configurations of clusters having 18, 20,22, and 24 boron atoms doped with a Cr, Mo, and W atom. In each case the symmetryis also given. 7 d)(e)(f)(g)(a) (b) (c)
Planar Curved ModifiedLowest Energy B Mo Mo B Mo B Mo B Mo B Mo B Mo Figure 2: Atomic structures showing the pathway from (a) planar B to (g) D h Mo@B cage clusters. (b) is the optimized structure when an Mo atom is placed above the planarisomer (a). (c) is slightly modified structure obtained from (b) by adjusting some atomsin an effort to make a cage. In going from (c) to (d) two boron atoms are added and thenfour boron atoms are added in the configuration (d). The resulting structure (e) is notsymmetric. (f) and (g) are obtained by adjusting B atoms in (e) in order to close the cageand make the cluster symmetric. 8 a) BB Three Ring Tubular (TRT) B B Cr B Cr Lowest EnergyTruncated Cube B (c) B Mo B Mo Lowest EnergyFullerene (C ) BB B (b) B Mo B Mo Lowest Energy
Figure 3: Optimized atomic structures for (a) D Cr@B , (b) D Mo@B , and (c) D h Mo@B clusters from their respective starting atomic configurations, namely 18-atomthree-ring tubular structure which was capped with two B atoms, a 20-atom dodecahedronwhich was capped with two B atoms, and 24-atom truncated cube, respectively. In eachcase Mo atom was placed inside the cage. 9everal different initial structures including C type B fullerene with all pentagonalfaces, quasi-planar, and tubular structures. All of them were optimized and also atomswere rearranged to possibly find symmetric lowest energy structures for all the threeM atom dopants. The optimized structures are shown in Fig. S2 in SupplementaryInformation. First note that for Cr, the CrB structure remains open and forms abicapped-drum structure as discussed above, suggesting that eighteen atoms of boron arenot sufficient to form a cage. The lowest energy isomers for MB , (M = Cr, Mo, andW) show (Figure 1(a)-(c)) that the structure of CrB is different from the one for theMoB and WB clusters. CrB structure has two boron atoms capped on to a B DRTstructure forming a bicapped-drum structure with C v symmetry. This structure has twopentagons arranged in base sharing fashion. For Cr@B , we obtained a D symmetriccage structure as shown in Figure 1(d) to be of the lowest energy. This smallest cagein our calculations has three empty boron hexagons that are inter-linked through three2-atom chains to two capped boron hexagons. However, for MoB and WB , a bicappeddrum structure with two boron atoms capping an 18-atom boron DRT with C s symmetryand M atom at the centre has the lowest energy. This structure has one pentagon andone hexagon arranged in base sharing fashion. For the CrB case, the bicapped drumisomer with C s symmetry is 0.32 eV higher in energy than the cage structure. Also the D fullerene structure for MoB and WB cases is, respectively, 1.79 eV and 2.06 eVhigher in energy than the lowest energy bicapped drum ( C s ) structure. We also obtainedan unsymmetric cage ( C ) structure for Mo@B and W@B (see isomer X in Fig. S2 inSupplementary Information) which is 0.88 eV and 1.02 eV, respectively, higher in energythan the lowest energy structure. Several other isomers that we tried are shown in Fig.S2 in Supplementary Information using PBE in VASP. These results suggest that for Moand W, the cage need to have more than 20 boron atoms.In order to explore a cage of B , we tried several configurations of neutral Mo@B .The lowest energy configuration has D symmetry as shown in Figure 1(h). This isobtained by adding two boron atoms in the C isomer of Mo@B with slight rearrange-ment. This structure has four heptagons that are interlinked through two 2-atom chainsand three boron atoms. Another calculation starting from C fullerene for B and cap-ping two opposite pentagons with two boron atoms converged to the same D structureafter optimization for Mo@B . This is shown in Fig. 3(b). We also tried several isomersfor Cr and W doping and obtained the same D structure for W@B to be of the lowestenergy as shown in Figs. 1(i), but for Cr@B , while in VASP calculation, the same iso-mer is of the lowest energy, a slightly different isomer shown in Fig.1(g) is obtained fromGaussian calculation as the lowest energy one. This structure also shows that less than 22boron atoms are better to have an optimal cage with Cr doping. Interestingly when thisisomer is reoptimized in VASP, it reverts to the same structure as before in VASP. Severalisomers including the lowest energy isomers for Cr, Mo, and W doped B are shown inFig. S3 in Supplementary Information. As can be seen, there are a few isomers for Crdoping that lie close in energy within about 0.30 eV of each other. The isomer in whichthe M atom interacts with a quasi-planar isomer of B (Fig. S3 XVI in SupplementaryInformation) lies more than 2 eV higher in energy than the lowest energy isomer.10 .855.25.45.65.86 18 20 22 24 B i n d i n g E n e r g y ( e V / a t o m ) Cage Size (n)
Cr@B n Mo@B n W@B n B n PBE (VASP)PBE0 (G09)
Figure 4: Calculated binding energies for the lowest energy isomers.For the case of the B cage, we further explored different structures for M@B withM = Cr, Mo, and W. These are shown in Fig. S4 in Supplementary Information. Thelowest energy configuration for Cr@B has C symmetry as shown in Fig. 1(j). Theisomer with D h configuration for Cr lies 0.25 eV higher in energy. Although our result ofthe lowest energy structure of Cr@B is in agreement with that reported by Lv et al. [25],they reported this structure to have quintet spin-multiplicity. But in our calculationsa singlet configuration has the lowest energy. The triplet and quintet configurationsare, respectively, 0.97 eV and 2.02 eV higher in energy. As stated earlier, the lowestenergy isomers for Mo@B and W@B are similar and are shown in Figs. 1(k) and (l),respectively.It is to be noted that neutral pure boron clusters, B n with n = 18-24, prefer quasi-planar or tubular structures in their ground states and these become drum-shaped or formcages after doping of a transition M atom. We calculated the energy of the drums/cagesby removing the M atom and keeping the atomic positions of the boron atoms the sameas in the optimized doped cluster. It is seen that the empty cage structures of B n ( n = 20,22, and 24) lie 1.50-3.50 eV higher in energy than their respective ground state isomerswithin PBE. Therefore, the interaction of the M atom due to endohedral doping lowersthe energy of the cage significantly. The doping energy has been calculated from ∆ E = E ( M atom ) + E ( B n cage ) − E ( M @ B n ) and the values are given in Table 1. In general largevalues ( > E are obtained for W doping and for Mo in B cage. This also leads tosignificant enhancement in the binding energy of the doped boron clusters compared withthat of the undoped clusters. The binding energy is defined as the energy per atom of thecluster with respect to the energy of the free atoms, [ nE ( B )+ E ( M ) − E ( M @ B n )] / ( n +1),and it is shown in Fig. 4 using PBE in VASP and PBE0 in Gaussian 09 program for thelowest energy isomers of all the sizes and with different dopants. Here E ( B ), E ( M ),11nd E ( M @ B n ) are the energies of B atom, M atom, and the doped cluster, respectively.The overall trend of the binding energy in both the methods is similar, but the valuesare slightly lower when PBE0 is used. These results also show that the binding energyincreases in going from Cr doping to Mo and then to W, indicating that W doping isthe strongest among the three dopants. There is a slight increase in the binding energyin going from 18-atom to 20-atom boron clusters for all the dopants. For Cr doping thebinding energy remains nearly the same in going from Cr@B to Cr@B . However, itslightly increases to 5.59 eV for Cr@B cluster within PBE. In the case of Mo and Wdoping the trend shows that the increase in the binding energy in going from Mo@B toMo@B is more than the increase in going from Mo@B to Mo@B . Similarly, in thecase of W doping the binding energy increases in going from W@B to W@B , but itremains nearly the same for W@B and W@B within PBE. Our results suggest thatboth 20 and 22 boron atom clusters doped with these M atoms are likely to be abundant,as the second derivative of energy (∆ E = 2 E (M@B n ) - E (M@B n − ) - E (M@B n +2 ))/4is negative for both these sizes. We have also plotted the values of the binding energyfor the pure boron clusters with 18, 20, 22, and 24 atoms. These results show that thedoping of Mo and W atoms leads to a significant increase in the binding energy of theboron clusters and therefore, Mo and W are favorable to form endohedral cage structuresof boron. However, in the case of Cr doping, the binding energy is either nearly the sameor marginally higher (in the case of CrB ) or lower (Cr@B ) than the value for thecorresponding pure boron cluster within PBE in VASP (see Table 1). But using PBE0in Gaussian 09 program the binding energy of CrB n clusters is slightly lower than thevalues for the pure boron clusters except for the Cr@B case as shown in Fig. 4. Alsowe find that within PBE0, there is a large HOMO-LUMO gap for Cr@B . Therefore,we conclude that the stabilization of boron cages with Cr doping is less than those of Moand W doped clusters, and the magnetic moments on the M atom is quenched. UsingPBE0 the second derivative of energy (∆ E ) for Cr@B is -0.165 eV, while for MoB andWB doping it is -0.018 eV and -0.072 eV, respectively. So M@B is weakly magic forMo and intermediate for W doping. Also for M@B , it is -0.015 eV, -0.043 eV, -0.050eV, respectively for M = Cr, Mo, and W. Accordingly Mo@B and W@B are likely tobe more abundant than Cr@B . Therefore, MoB is magic within PBE0 while W@B is likely to be more abundant than W@B . In the following we discuss Cr as well as Moand W doping to understand the cage stability and bonding characteristics in the dopedclusters.From Table 1 the calculated HOMO-LUMO gaps (E g ) within PBE are 1.24 eV, 0.40 eV,and 0.47 eV for MB , M = Cr, Mo, and W, respectively. Note that GGA underestimatesE g and the values obtained using PBE0 are much larger. The values of E g for these casesare, respectively, 2.72 eV, 1.71 eV and 1.71 eV when PBE0 functional is used. Here notethe large value for the case of Cr doping. For MB , the HOMO-LUMO gaps increaseto 2.14 eV, 1.76 eV, and 1.79 eV, respectively with M = Cr, Mo, and W, and further to2.26 eV, 2.51 eV, and 2.50 eV for MB within PBE. These large values of E g suggest thestability of the cage structures. However, for Cr@B the HOMO-LUMO gap decreases to0.89 eV as the cage becomes less symmetric and Cr is not bonded well with all the B atoms.12able 1: Calculated embedding energy (E n (M)) of the M atom in n-atom boron cage, BE,HOMO-LUMO gap (E g ), vertical electron affinity (VEA) and vertical ionization potential(VIP) for the lowest energy isomers of M@B n (n = 18, 20, 22, and 24; M = Cr, Mo, W)clusters. VEA and VIP are calculated using Gaussian 09 program. For comparison, thebinding energy and HOMO-LUMO gap calculated within PBE0 are also given. Resultshave also been given for the lowest energy isomers of pure B n clusters ( n = 18, 20, 22,and 24). PBE PBE0Cluster E n (M) BE per atom E g E n (M) BE per atom E g VEA VIP(eV) (eV) (eV) (eV) (eV) (eV) (eV) (eV)B - 5.48 0.47 - 4.93 1.39 - -B - 5.56 1.36 - 5.05 2.72 - -B - 5.57 0.29 - 5.06 1.35 - -B - 5.63 1.30 - 5.10 2.18 - -Cr@B and W@B clusters also have large E g (2.46 eV and 2.45 eV, respectively), butthe values are slightly lower than for Mo@B and W@B cages. The calculated E g forthe M doped B and B cage structures using PBE0 are again much larger than thevalues obtained within PBE, and these are given in Table 1. The largest value (4.40 eV)of E g within PBE0 is obtained for Mo@B and a quite similar value for W@B . Thelargest value (5.28 eV/atom) of the binding energy within PBE0 is obtained for W@B and a slightly smaller value (5.26 eV/atom) for W@B . This is in contrast to the valuesof 5.06 eV/atom and 5.10 eV/atom for the binding energies of elemental boron clustersB and B , respectively, within PBE0. Therefore, M doping enhances the stability ofboron clusters and leads to the formation of the cage structures of boron for smaller sizescompared with pure boron clusters. In order to understand the bonding character in these structures, we noted that theshortest B-B bond lengths for Cr@B , Mo@B , and W@B are 1.61 ˚A, 1.58 ˚A, and 1.58˚A respectively, whereas B-Cr, B-Mo, and B-W bond lengths are in the range of 2.12-2.36˚A, 2.32-2.63 ˚A, and 2.33-2.60 ˚A, respectively using PBE. The shortest B-B bonds are thetwo-center (2c) sigma bonds. As the cluster size increases, we find that for Cr@B theB-B and B-Cr bond distances lie in the range of 1.60-1.83 and 2.08-2.30 ˚A, respectivelywhile for Mo@B and W@B the shortest B-B bond length is 1.57 ˚A. The B-Mo andB-W bond lengths are in the range of 2.13-2.57 ˚A and 2.14-2.58 ˚A, respectively. TheseMo-B and W-B bond lengths are shorter than for Mo@B and W@B , respectively. Theshorter bonds are formed by the capping boron atoms with the M atom. For the M dopedB cage, some of the B-B bonds are shorter than those in the M doped B cage. There aretwo shortest B-B bonds for the B cage with bond length 1.56 ˚A for Cr encapsulation,and 1.57 ˚A for Mo and W encapsulation, but the M-boron bond lengths are slightlyelongated and lie in the range of 2.31-2.54 ˚A for Cr encapsulation and 2.38-2.59 ˚A for Moand W encapsulation as compared to those in the respective doped B cages. In the B cage the shortest B-B bond length increases to 1.64 ˚A for Mo and W encapsulation. Alsosome of the M-boron bond lengths are longer and lie in the range of 2.43-2.55 ˚A for Moencapsulation and 2.44-2.55 ˚A for W encapsulation. For Cr encapsulation in B , the B-Bbond lengths are in the range of 1.54-1.84 ˚A while B-Cr bond lengths have wide variationand are in the range of 2.18-2.66 ˚A. This is becuase the Cr atom is not at the center ofthe cage.We further calculated the total charge density isosurfaces, ELF, and molecular orbitalsfor the doped clusters. We also performed AdNDP analysis on some of the clusters. Detailsof these results are presented in Supplementary Information and in Figs. 5 and 6. Earlierstudies on pure boron clusters have suggested the existence of multi-center bonds namely2c, 3c, 4c, 6c, and 7c bonds using AdNDP analysis[50]. However, in our experience theAdNDP analysis is not straightforward as there is not a unique way to identify multi-center bonds. We were greatly assisted by the analysis of the charge density, molecularorbitals, and ELF. In an earlier study[57] on B cluster, the total charge density and14 a) (b) (c) Figure 5: Isosurfaces of ELF for (a) Cr@B , (b) Mo@B , and (c) Mo@B . Two ori-entations are shown in each case. In (a) the lobes shown by purple color have stronglocalization of charge with high value of ELF (0.94) representing 2c-2e bonds. As thevalue of the iso-surface is decreased (0.84), these lobes become bigger and new lobes ap-pear. These other lobes shown with yellowish color are 3c or higher center σ bonds. In(b) there are 2 purple color lobes with ELF = 0.96 (2c) and sixteen yellowish color lobeswith ELF = 0.88 (3c). In (c) there are six 2c (purple color) and eighteen 3c (yellowish)bonds.ELF were used as tools to understand 2c and 3c bonds in boron clusters along with theAdNDP analysis. This was helpful in further calculating the multi-center bonds withAdNDP. We followed a similar strategy for some of the M doped boron clusters studiedhere. Further, we have calculated Laplacian of the electron density and bond- as well asring-critical points to analyse the bonding character.Figure 5 shows ELF for the smallest cage Cr@B and also for Mo@B and Mo@B clusters. For Cr@B the bond length analysis showed that there are six short B-B bondswith 1.60 ˚A length. These bonds are along the three 2-atom chains joining the threeempty heptagons and can be detected in the charge density and ELF plots at a highvalue of the isosurface (see Fig. S5 in Supplementary Information). A high value of ELFshows strong localization of charge as in covalent bonds. These six lobes are indicatedby purple color in Fig. 5(a) at the ELF value of 0.94. We identify these bonds as six2c-2e bonds. Further decreasing the ELF value to 0.84, these six lobes become bigger(shown by yellowish color around the purple color lobes) and twelve more lobes can beseen. These twelve additional lobes (yellowish in color in Fig. 5(a)) represent twelve15 a) (b)
12 x 3c-2e σ-bonds ON=1.92-1.93 |e| (c) (e) (f) (d)
Figure 6: Different multi-center bonds in the lowest energy isomer for Cr@B as obtainedfrom the AdNDP analysis.3c σ bonds. This identification was also supported by the AdNDP analysis as we shalldiscuss below. Figs. 5(b) and (c) show ELF plots for Mo@B and Mo@B , respectively.For Mo@B , the two lobes shown by purple color are obtained at the ELF value of 0.96and they represent two 2c bonds with the bond length of 1.57 ˚A. There are sixteen morelobes shown by yellowish color at the ELF value of 0.88. Similar to the case of the B cage, they represent sixteen 3c bonds. For Mo@B the analysis of the charge density andELF shows that this cluster has six 2c σ -bonds (purple color lobes in ELF) and eighteen3c σ -bonds. Further analysis using AdNDP shows six 5c σ -bonds (see Fig. 5(c)). Theremaining eighteen valence electrons are involved in π -bonding as well as bonding withthe M atom, and can be detected in the AdNDP analysis[25].We consider Cr@B cage cluster as the smallest cage, and its AdNDP analysis shows(Fig. 6 (a)) that there are six 2c-2e bonds, as also concluded from ELF. Further, thereare twelve 3c bonds, six are along the three 2-atom chains, and the remaining six areplaced alternatively on the three edges of each capped hexagon as shown in Figure 6(b).16 B Cr
012 0 1 2 Degeneracy3.48 eV E i g e n v a l u e s ( e V ) LUMO+1LUMOHOMOHOMO-4HOMO-5HOMO-9 HOMO, d xy , d x2-y2 HOMO-3, d z2 HOMO-4, d zx , d yz HOMO-6, p x , p y HOMO-7, p z HOMO-12, s Figure 7: The π -bonded MOs for the bare B cage and Cr@B calculated with Gaussian09 program.These bonds were also detected in the total charge density and ELF analysis as discussedabove. In addition to these eighteen σ -bonds there are four 5c-2e σ -bonds and two 6c-2e σ -bonds as shown in Figure 6(c) and (d), respectively. These bonds connect the atomson the hexagons to the atoms on the three 2-atoms chains. All these 24 σ -bonds involvebonding among boron atoms. There are six 5c-2e π -bonds, each connecting three B atomson one capped hexagon to one B atom on the edge through the Cr atom. There are alsothree 6c-2e π -bonds along the three 2-atom chains, each involving six B atoms on thechain and the Cr atom. It is interesting to see that there are altogether nine π -bondswhich involve bonding of the boron atoms on the cage and the Cr atom. This indicatesthat the stability of the cage due to the Cr atom doping is governed by the completionof an electronic shell with 18 valence electrons, as we shall further show in the followingfrom the analysis of the MOs of Cr@B .In order to understand the electronic origin of the stability of these cages, we calculatedthe MOs for the M doped clusters as well as the corresponding boron cage by removing theM atom and keeping the atomic positions fixed. For the Cr@B case, six electrons of theCr atom and twelve π -bonded electrons from the B cage contribute to the stability withelectronic shell closing at eighteen π bonded valence electrons. Figure 7 shows that thebare B cage has six occupied π -bonded MOs (doubly degeberate HOMO and HOMO-4,HOMO-5, and HOMO-9). The two HOMO levels have d angular momentum character.There are three unoccupied π -bonded MOs (LUMO and LUMO+1) of d angular momen-tum character. For Cr@B these five MOs of d angular momentum character hybridizewith the d orbitals of Cr atom forming bonding and anti-bonding MOs. The six electrons(3d65 4s ) of the Cr atom are accommodated in the bonding MOs and there is a largeHOMO-LUMO gap which results in the stability of the endohedral Cr@B cage cluster.For the Mo@B cage, the π -bonded MOs for the bare B cage and also the MOs forMo@B along with the corresponding eigenvalues are shown in Fig. 8 as obtained fromthe Gaussian calculation with the PBE0 functional. It is seen that the stability of the Moencapsulated B cage is governed by the nine π -bonded MOs with electronic shell closing17igure 8: Some of the π -bonded MOs for the bare B and Mo@B .at 18 valence electrons. For the bare B cage there are six occupied π -bonded MOs andthree empty π -bonded MOs (not shown) of d angular momentum character just above theHOMO. The five π -bonded MOs with d angular momentum character hybridize stronglywith the d orbitals of the Mo atom forming five bonding MOs and five anti-bonding MOs.The bonding MOs are fully occupied and similar to the case of 18 valence electron stabilityof the B cage, the stability of the Mo@B cluster also arises from 18 valence electrons.The ordering of the MOs can be seen in the figure. Note that the bare B cage has verysmall HOMO-LUMO gap but it is very much increased by Mo encapsulation. Further,the stability of the Mo@B cluster also arises from the occupation of nine π bonded MOscorresponding to 18 valence electrons. A similar result will hold for the case of W doping.We have also performed analysis of the MOs of cr@B bicapped drum structure, and inthis case also the stability is associated with 18 π bonded valence electrons. On the otherhand the stability of the disk-shaped M@B , for M = Mo and W arises from 20 π bondedvalence electrons.We have further performed analysis of the electronic charge density ρ ( r ) of Cr@B as representative of the 22-atom boron cage by calculating contours of the Laplacian of ρ ( r ) as well as Laplacian at bond critical points (BCPs) and ring critical points (RCPs).The method of calculation has been discussed earlier.[34] Figure 9 (a) shows the atomicstructure with BCPs (green dots) and RCPs (red dots) while (b) shows a symmetricalview of Cr@B along with the electrostatic potential mapped onto a ρ ( r ) = . / Bohr electron density isosurface. Blue regions indicate negative electrostatic potential asso-ciated with the boron atoms. Of the 22 boron atoms, there are four B atoms (B1,B3, B17, and B19) that have coordination 2 in the middle of the two boron chains,while 18 boron atoms are in two semi-circular bands consisting of BBB triangles and/or18BBB quadrilaterals. The two boron chains appear at the left and right ends of thefigure, while one of the semi-circular bands appears in the foreground running diago-nally from top left to bottom right (the other is partially occluded at the back, run-ning from top right to bottom left). The two bands and the two chains loop aroundand encapsulate the M atom in a tetrahedral coordination. Such bands are a recur-ring motif in many kinds of boron clusters, including drums[34, 39] and quasiplanarstructures.[57] The two bands join each other and the two chains at four tetra-coordinateboron atoms. The critical point analysis requires that the Poincare-Hopf relationship: N umN ACP + N umN N ACP − N umBCP + N umRCP − N umCCP = 1 be satisfied,where
N umN ACP is the number of nuclei (here 23),
N umN N ACP is the number ofnon-nuclear attractors (here 0) critical points,
N umBCP is the number of BCPs (here40),
N umRCP is the number of RCPs (here 20), and
N umCCP is the number of cagecritical points (here 2). In Figs. 9 (c)-(e) red dots indicate the locations of the RCPs ofthe BBBB quadrilaterals. Of the 36 B-B BCPs (green dots), there are 6 that representcovalent bonds in the two boron chains. Figure 9(d) shows a contour plot of the Lapla-cian distribution in a plane passing approximately through the rim of one of these bands(Cr@B with B10-B9-B4-B8-B7-B22). 12 BCPs correspond to bonds at the bases of BBBtriangles in the semi-circular bands, and 10 corresponding to bent bonds spanning thewidth of one or the other of the two bands. Figure 9 (e) (Cr@B with B2-B1-B3-B10-Cr23) shows a contour plot of the Laplacian distribution in a plane passing approximatelythrough one of the boron chains and the M atom. 8 BCPs represent covalent bonds alongrims of the semi-circular bands. The properties of these bonds are shown in Table IIand Fig. 10. Large values of the charge density and large positive Laplacian with lowvalues of bond ellipticity correspond to covalent bonds while high bond ellipticities, lowpositive values of the Laplacian L( r ) and delocalization index (off-diagonal localizationdelocalization matrix (LDM) elements) are characteristic of multi-center bonding in theBBB triangles. On the other hand negative values of Laplacian correspond to the M-Crbonds. Silimar results have been obtained for Cr@B cage and shown in Fig. 10(c).We also performed calculations on cationic and anionic clusters. For these calculationswe have considered the lowest energy and some low-lying isomers of neutral clusters. Thecationic and anionic clusters are calculated using Gaussian09 program. Note that for theneutral and charged clusters the calculations using Gaussian09 program give almost thesame energy order of the low-lying isomers as with the VASP calculations, but we findthat in some cases the lowest energy neutral isomer does not have the lowest energy whencharged. As an example the cation of W@B has a double bicapped drum structure, whilethe anion of the lowest energy neutral isomer continues to have the lowest energy. Also acage structure of anions of M@B (M = Mo and W) has the lowest energy. More detailsare given in Supplementary Information. We also performed calculations on isoelectronicneutral and anionic clusters of B , B , and B with the doping of V, Nb, and Ta atomsusing Gaussian 09 code with PBE0 functional. Broadly the trends are similar as obtainedfor the cases of Cr, Mo, and W doping. For VB anion we obtained a bicapped drumto be most favorable as for isoelectronic Cr doping, but neutral cluster favors a drumstructure over the bicapped drum isomer by 0.27 eV. On the other hand for Nb and Ta19 a)(b) (c)(d) (e) Figure 9: (a) BCPs (green dots) and RCPs of BBBB quadrilaterals (red dots) of Cr@B cluster and (b) electrostatic potential mapped onto a ρ (r) = 0.1 e/Bohr electron densityisosurface. Blue regions indicate negative electrostatic potentials associated with theboron atoms. (c) - (e) Contour plots of the Laplacian in different planes passing throughatoms (c) B7, B8, B15, B16, B21, B22, and Cr, (d) B4, B7, B8, B9, B10, B22, and (e)B1, B2, B3, B10, and Cr. The contour values are 0.0, 0.001, 0.002, 0.004, 0.008, 0.02,0.04, 0.08, 0.2, 0.4, 0.8, 2.0, 4.0, 8.0, 20.0, 40.0, 80.0, 200.0, 400.0, 800.0, -0.001, -0.002,-0.004, -0.008, -0.02, -0.04, -0.08, -0.2, -0.4, -0.8, -2.0, -4.0, -8.0, -20.0, -40.0, -80.0, -200.0,-400.0, -800.0. Pink (grey) balls show B (M) atoms.20igure 10: (a) Laplacian (in units of e/Bohr ) vs electron density (e/Bohr ), (b) bondellipticity as a function of the electron density for Cr@B cluster and (c) Laplacian vselectron density for Cr@B cluster. 21able 2: Different types of bonds, the number of bonds of each type, charge density( ρ ) in units of e/Bohr at the BCP, bond ellipticity, Laplacian (L) in units of e/Bohr at the BCPs, and the delocalization index. Small value of ellipticity and large positivevalue of Laplacian indicates strong 2c covalent bonds while large values of ellipticity andsmall values of Laplacian indicates delocalization of charge (multi-center bonding). Thenumbering of boron atoms is given in Fig. (9). Bond Type No. of Bonds ρ Bond Ellipticity L Delocalization IndexB1-B3, B17-B19
B1-B2, B3-B10, B12-B19, B17-B20
B4-B8, B5-B16, B11-B22, B14-B21
B4-B9, B5-B6, B11-B18, B13-B14
B7-B8, B7-B22, B15-B16, B15-B21
B2-B6, B9-B10, B12-B13, B18-B20
B2-B22, B8-B12, B10-B21, B16-B20
B2-B7, B7-B12, B10-B15, B15-B20
B9-B21, B6-22, B16-B18, B8-B13
B5-B11, B4-B14 doping, the drum isomer is lowest in energy for both neutral and anion similar to the caseof isoelectronic Mo and W, respectively. In the case of VB anion and neutral, a bicappeddrum is 0.375 eV and 1.281 eV, respectively, lower in energy than a cage structure whichis the most favorable for the neutral Cr case. However, for Nb and Ta doped B anionsa bicapped drum is 3.084 eV and 3.065 eV, respectively, lower in energy than a cageisomer similar to the case of Mo and W doping. Also in both cases the neutral NbB andTaB clusters also favor the bicapped isomer over the cage isomer by 2.181 and 1.395eV, respectively. Interestingly for the B cage all the three metal atoms V, Nb, and Tastabilize it without much distortion for both neutral and anions and the electric dipolemoment in all the cases is very close to zero. This is in contrast to Cr in which case theM atom drifts away from the center and does not interact with all the B atoms properly.The density of states (DOS) for the neutral and anionic clusters provide further infor-mation on the nature of the electronic states. The DOS for the anionic clusters (shown inthe Supplementary Information) will be useful for comparison with the results of photo-electron spectroscopy experiments that may become available in the future. The DOSsfor the neutral clusters with M = Cr, Mo, and W are shown in Figure 11. Our resultsshow that all the cages have a large HOMO-LUMO gap as obtained from the PBE0 cal-culations, indicating their very good chemical stability. As seen from the MOs, all theenergy levels near the HOMO (shown by the vertical line) arise from hybridization of theboron cage orbitals with the d orbitals of the M atom. For Cr@B very localized peakscan be seen below the HOMO, indicating the degeneracy arising from the symmetry ofthe structure. In going from Cr@B to Cr@B there is a spread in the distribution of theelectronic energy levels. For Mo@B and W@B , a capped drum-like structure is lowerin energy and the electronic states show more localized peaks compared to the Mo@B r@B Mo@B W@B Cr@B Mo@B W@B W@B Mo@B Energy (eV) D e s n s i t y o f S t a t e s ( a r b . un i t s ) -12 -10 -8 -6 -4 -2 -12 -10 -8 -6 -4 -2 Cr@B Cr@B Mo@B W@B Figure 11: Calculated density of states (DOS) for neutral Cr, Mo, and W encapsulatedB -B clusters. The vertical line shows the HOMO.23nd W@B cases which lack symmetry. For M@B and M@B some sharp peaks canbe noticed. The overall nature of the DOS for Mo and W encapsulation remains similar. We have calculated the vibrational modes for the lowest energy neutral isomers and theircations for all the cases using PBE0 functional in Gaussian09 program. Calculationswere also done using B3PW91 hybrid functional in Gaussian09 code for some cases. Thegeneral features of the spectra using the two functionals are similar, as can be seen fromthe data in Tables S1-S4 in Supplementary Information, though there are some deviationsin intensities and frequencies. Figure 12 shows the calculated IR and Raman spectra forCr, Mo, and W doped neutral clusters obtained using the PBE0 functional. In almostall the cases we find no imaginary frequency, indicating that the cage structures aredynamically stable. The IR and Raman spectra for the cationic clusters are given inFig. S9 in Supplementary Information using the PBE0 functional. Tables S1 and S2 inSupplementary Information give the IR intensities and Raman activities for the neutraland cationic clusters, respectively, using B3PW91 functional. We have also given thebond distances in the neutral case. The results using PBE0 functional are given in TablesS3 and S4. Considering the IR active modes for the Cr@ B cluster, we find a strong peakat 424 cm − corresponding to the breathing mode. In this case the Cr atom vibrates alongthe axis of the drum, which involves breathing of the two B rings. The most dominantmodes occur at 473 cm − and 482 cm − corresponding to the vibration of the Cr atomin two perpendicular directions in the plane of the B disk. In one case the Cr atomvibrates parallel to the B-B bond of the two capped boron atoms.For the drum-shaped structure of Mo@B there are strong bending/breathing modesin the range of 291-320 cm − in the IR spectrum involving M atom and the boron rings.The strong peak at 466 cm − corresponds to the stretching mode of the boron ring.Another stretching mode occurs with high intensity at 1317 cm − . In the case of W@B ,a similar behavior of the modes has been obtained. The strong modes at 250 cm − and269 cm − for W@B involve swinging of the W atom in two perpendicular directions inthe plane of the boron disk, and therefore the frequencies are reduced compared to theMo case. The mode at 456 cm − involves bending of the disk without the movement ofthe W atom, and so it is less affected compared with the Mo case. On the other handthe frequency of the stretching mode at 1332 cm − has increased compared to the Mocase, which suggests stronger bonding between the boron atoms. The Raman spectrumof Cr@B has a very high intensity peak at 707 cm − corresponding to the symmetricbreathing mode of the capped drum boron structure. For Mo@B the high intensityRaman active peak at 676 cm − and for W@B the high intensity peaks at 616 cm − and665 cm − corrpdfond to the breathing modes of the B ring. These vibrations involvepurely boron atoms and not the M atom.For the IR spectrum of Mo@B the strong modes occur at 353 cm − and 357 cm − and correspond to the swing of the Mo atom in the direction perpendicular to the cappingboron dimer and parallel to the dimer, respectively. There is also associated bending of the24oron rings. The mode at 371 cm − is also a swing mode of the Mo atom along the borondimer. For W@B the IR spectrum is similar with two strong modes with frequencies 294cm − and 303 cm − , with the swing of W atom in the direction perpendicular and alongthe capping boron dimer, respectively. There is an imaginary frequency for MoB at -122cm − and for W@B at -100 cm − but in both the cases the intensity is very small. TheRaman spectrum of Mo@B has two strong modes at 633 cm − and 762 cm − while forW@B there is one strong mode at 636 cm − . These correspond to the breathing modeof the outer capped drum structure.For Cr@B , the IR spectrum has several high intensity peaks with the most dominantones at 454 cm − , 455 cm − , 482 cm − , 484 cm − , and 508 cm − corresponding to theswing modes of the Cr atom while the peaks at 577 cm − and 662 cm − correspond tobending modes, and the peak at 807 cm − represents stretching and breathing mode. Ingeneral the IR active peaks involve stretching and bending vibrations of different B-B andB-Cr bonds. However, the Raman spectrum has only one intense peak at 742 cm − whichcorresponds to the breathing mode of the cage. For Cr@B the Raman active peaks comefrom vibrations involving only boron atoms. For Cr@B , the IR spectrum has dominantpeaks at 383 cm − and 391 cm − , whereas the Raman spectrum has dominant peaks at608 cm − , 641 cm − , and 702 cm − . Similar spectra are obtained for Mo and W dopedB cages. For Mo@B , the IR modes at 313 cm − and 362 cm − are swing modes of theMo atom with associated motion of the B atoms. The Raman spectrum of Mo@B hastwo major peaks at 631 cm − and 675 cm − corresponding to breathing mode of the boroncage. For W@B the dominant IR modes occur at frequencies 259, 275, and 309 cm − ,whereas the Raman activity has strong peaks at 630 cm − and 675 cm − correspondingto the breathing modes of the cage. In general the dominant IR peaks shift towards lowerfrequencies compared to Cr encapsulation because of the higher mass of Mo compared toCr atom. The peaks shift further towards lower frequencies for W encapsulation with IRactive modes at 259 cm − and 275 cm − , compared to 313 and 362 for MO@B and 383as well as 391 for Cr@B .A similar behavior has been obtained for Mo and W encapsulated B cages. ForMo@B , the IR active dominant peaks are at 325 cm − and 327 cm − , whereas theRaman spectrum has a strong peak at 632 cm − corresponding to breathing of the cage.For W@B the strong IR modes are at frequencies 258 cm − , 263 cm − , 265 cm − , and1056 cm − , whereas there is one strong Raman peak at 636 cm − similar to the Mo@B cage. In general the highest Raman intensity peak corresponds to the breathing mode,whereas the IR active and other Raman active peaks involve different stretching andbending modes. The IR and Raman spectra have more localized and high intensity peaksfor the symmetric cases compared to the broader spectrum for clusters with low symmetry.We also calculated the vibrational spectra for the cationic clusters (see SupplementaryInformation); Figure S9 in Supplementary Information shows the IR and Raman spectra.For Cr doped cases the IR and Raman spectra differ more significantly for neutral andcationic clusters as compared to those with Mo and W doping. For Cr@B +18 there isone dominant IR peak at 479 cm − corresponding to swinging mode (perpendicular tothe capping boron dimer) of the Cr atom, while the peak at 544 cm − corresponds to25 I R I n t e n s i t y ( a r b . un i t s ) R a m a n A c t i v i t y ( a r b . un i t s ) Frequency (cm -1 ) Cr@B W@BW@BMo@BCr@B
Figure 12: Calculated IR and Raman spectra for neutral clusters.26winging along the boron dimer and the peak at 638 cm − corresponds to the motionof the Cr atom along z direction. There is only one strong Raman mode at 717 cm − corresponding to breathing of the boron cage. In the case of Mo@B +18 the IR spectrumhas dominant peaks at 312 cm − , 313 cm − (swing of the M atom), 481 cm − (swing ofthe cage), and 730 cm − (stretching and bending of the B-B bonds), while the Ramanspectrum has a strong peak at 676 cm − corresponding to the breathing mode of the boroncage. In the case of W@B cation, the strong IR modes are at 268 cm − and 269 cm − corresponding to swinging of the W atom, while the mode at 462 cm − corresponds to theswing of the cage. The Raman spectrum has one strong peak at 667 cm − correspondingto the breathing mode.For Cr@B the dominant IR modes are at 402 cm − corresponding to the swing ofthe M atom, 748 cm − (bending of the boron network), 802 cm − (swing of the cage),960 cm − (stretching of the B-B bonds), and 1208 cm − (stretching of the B-B bonds).The Raman spectrum has strong peaks at 164 cm − (swinging of the M atom) whichis also IR active, and scissor mode at 960 cm − , and stretching/breathing mode at 968cm − . For Mo@B +20 the dominant IR modes are at 338 cm − , 354 cm − , and 359 cm − (allthree swing modes of the Mo atom), while the Raman spectrum has a strong peak at 650cm − corresponding to the breathing of the cage. In the case of W@B +20 , the dominant IRmodes are at 292 cm − and 304 cm − (corresponding to swinging of the W atom in twoperpendicular directions) while the dominant Raman peak is at 651 cm − correspondingto the breathing mode.In the case of Cr@B +22 the main IR peaks occur at 81 cm − , 321 cm − , and 382cm − , all corresponding to the swinging of the Cr atom, while the Raman spectrumhas a strong peak at 693 cm − corresponding to the breathing of the cage. There is aimaginary frequency at 187 cm − but the intensity is very small. For Mo@B +22 , the mainIR peaks are at 326 cm − and 369 cm − corresponding to swinging modes of the Moatom, while the Raman spectrum has two dominant peaks at 615 cm − and 677 cm − corresponding to breathing modes. There is an imaginary frequency at 91 cm − but againthe intensity is low. For W@B the main IR peaks are at 258 cm − , 297 cm − , and 302cm − corresponding to the swinging modes of the W atom, bending mode at 328 cm − ,and stretching mode at 1150 cm − . The Raman spectrum has a dominant peak at 678cm − corresponding to a breathing mode. For Mo@B the dominant IR peaks are at 314cm − , 367 cm − , and 376 cm − corresponding to swinging modes of the Mo atom, whilethe dominant Raman peak is at 636 cm − corresponding to the breathing mode. ForW@B +24 , the IR modes are strong at 253 cm − , 267 cm − , and 297 cm − corresponding toswinging modes, while the peak at 1035 cm − is a stretching mode. The Raman spectrumhas a strong peak at 638 cm − corresponding to the breathing mode. In general forMo@B , Mo@B , W@B and W@B the positions of the dominant peaks occurat similar frequencies as for the corresponding neutral cases.27 CONCLUSIONS
In summary, we have performed a systematic study on M = Cr, Mo, and W dopedboron clusters in the size range of 18 to 24 boron atoms. Our results show that by Mencapsulation, it is possible to have fullerene-like cage structures of boron with about20 atoms in contrast to dominantly planar or quasi-planar or tubular structures of pureboron clusters for less than 40 atoms. Our results suggest that doping of Cr is suitableto produce symmetric small cage clusters of boron with B , whereas B is the smallestsymmetric cage for Mo and W encapsulation, which is magic and is likely to be producedin high abundance in experiments. A symmetric B cage is also formed with Mo andW encapsulation, as also predicted recently, but the variation in the BE of the clusterssuggests that the B cage has the optimal size for Mo and W encapsulation. There is alarge gain in energy when a Mo or W atom is encapsulated in the cage, which also supportsthe strong stability of the doped clusters. We performed an analysis of the bonding naturein these clusters and found that the cage structures are stabilized by strong interactionbetween the M atom d orbitals and the π -bonded MOs of the bare boron cage. Fromthis analysis we find that the stability of the Cr@B , Cr@B , M@B , M = Cr, Mo,and W, and M@B clusters is associated with 18 π bonded valence electrons while forM@B (M = Mo and W) disk-shaped tubular clusters, the stability is associated with20 π bonded valence electrons. In most cases the IR and Raman spectra for the neutraland cationic clusters show that the cages are dynamically stable. These results as well asthose of the electronic levels of the anionic clusters will help to compare our predictionswith experiments. We have also studied isoelectronic anion and neutral B , B , B ,and B clusters doped with V, Nb, and Ta. In general the structural trends are similaras obtained for Cr, Mo, and W but for B we obtained a symmetric cage in all cases.We hope that our results will stimulate experimental work on these M doped disk-shapedand fullerene-like structures of boron. We gratefully acknowledge the use of the high performance computing facility Magus ofthe Shiv Nadar University where a part of the calculations have been performed. ABRand VK thankfully acknowledge financial support from International Technology Center- Pacific. ABR acknowledges international travel support (ITS), from SERB, Govt. ofIndia. We thank Prof. Cherif Matta for providing access to AIMLDM software.
References [1] H.-J Zhai, Y.-F. Zhao, W.-L. Li, Q. Chen, H. Baim, H.-S. Hu, Z. A. Piazza, W.-J.Tian, H.-G. Lu, Y.-B Wu, Mu Y.-W., G.-F. Wei, Z.-P. Liu, J. Li, S.-D. Li, and L.-S.Wang. Observation of an all-boron fullerene.
Nature Chem. , 6:727–731, 2014.282] H. J. Zhai, A. N. Alexandrova, K. A. Birch, A. I. Boldyrev, and L. S. Wang. Hepta-and octacoordinate boron in molecular wheels of eight- and nine-atom boton clusters:observation and confirmation.
Angew. Chem. Int. Ed. , 42:6004–6008, 2003.[3] H. J. Zhai, B. Kiran, J. Li, and L. S. Wang. Hydrocarbon analogues of boron clusters,planarity, aromaticity and antiaromaticity.
Nat. Mater. , 2:827–833, 2003.[4] B. Kiran. Planar-to-tubular structural transition in boron clusters: B as the embryoof single-walled boron nanotubes. Proc. Natl Acad. Sci. USA , 102:961–964, 2005.[5] K. C. Lau, M. Deshpande, and R. Pandey. A theoretical study of vibrational prop-erties of neutral and cationic b clusters. Int. J. Quant. Chem. , 102:656–664, 2005.[6] W. Huang. A concentric planar doubly π -aromatic b − cluster. Nat. Chem. , 2:202–206, 2010.[7] I. A. Popov, Z. A. Piazza, W. L. Li, L. S. Wang, and A. I. Boldyrev. A combinedphotoelectron spectroscopy and ab initio study of the quasi-planar b − cluster. J.Chem. Phys. , 139:144307, 2013.[8] E. Oger. Boron cluster cations: transition from planar to cylindrical structures.
Angew. Chem. Int. Ed. , 46:8503–8506, 2007.[9] L. Cheng. B : An all-boron fullerene. J. Chem. Phys. , 136:104301–104304, 2012.[10] S. Chacko, D. G. Kanhere, and I. Boustani. Ab inition density functional investigationof b clusters: Rings, tubes, planes and cages. Phys. Rev. B , 68:035414, 2003.[11] J. Lv, Y. Wang, L. Zhua, and Y. Ma. B :an all boron fullerene analogue. Nanoscale ,6:11692–11696, 2014.[12] Q. Chen, W.-L. Li, Y.-F. Zhao, S.-Y. Zhang, H.-S. Hu, H. Bai, H.-R. Li, W.-J. Tian,H.-G. Lu, H.-J. Zhai, S.-D. Li, J. Li, and L.-S. Wang. Experimental and theoreticalevidence of an axially chiral borospherene.
ACS Nano , 9:754–760, 2015.[13] J. Zhao, X. Huang, R. Shi, H. Liu, Y. Su, and R. B. King. B : smallest all-boroncage from ab initio global search. Nanoscale , 7:15086–15090, 2015.[14] K. Wade. The structural significance of the number of skeletal bonding electron-pairsin carboranes, the higher boranes and borane anions, and various transition-metalcarbonyl cluster compounds.
J. Chem. Soc. D , 15:792–793, 1971.[15] G. R. Eaton and W. N. Lipscomb.
NMR studies of boron hydrides and related com-pounds . W. A. Benjamin, Inc., 1969.[16] V. Kumar and Y. Kawazoe. Metal-encapsulated fullerenelike and cubic caged clustersof silicon.
Phys. Rev. Lett. , 87:045503, 2001.2917] V. Kumar and Y. Kawazoe. Magic behavior of si m and si m (m=cr, mo, and w)clusters. Phys. Rev. B , 65:073404, 2002.[18] V. Kumar and Y. Kawazoe. Metal-encapsulated icosahedral superatoms of germa-nium and tin with large gaps: Zn@ge and cd@sn . Appl. Phys. Lett. , 80:859,2002.[19] V. Kumar and Y. Kawazoe. Metal-encapsulated caged clusters of germanium withlarge gaps and different growth behavior than silicon.
Phys. Rev. B , 88:235504, 2002.[20] V. Kumar. Novel metal-encapsulated caged clusters of silicon and germanium.
Eur.Phys. J. D , 24:227, 2003.[21] V. Kumar. Novel caged clusters of silicon: Fullerenes, frank-kasper polyhedron andcubic.
Bull. Mat. Sci. , 26:109, 2003.[22] V. Kumar and Y. Kawazoe. Metal-doped magic clusters of si, ge, and sn: The findingof a magnetic superatom.
App. Phys. Lett. , 83:2677, 2003.[23] V. Kumar and Y. Kawazoe. Hydrogenated caged clusters of si, ge, and sn and theirendohedral doping with atoms: Ab initio calculations.
Phys. Rev. B , 75:155425, 2007.[24] K. Koyasu, M. Akutsu, M. Mitsui, and A. Nakajima. Selective formation of ms (m= sc, ti, and v). J. Am. Chem. Soc. , 127:4998–4999, 2005.[25] J. Lv, Y. Wang, L. Zhang, H. Lin, J. Zhao, and Y. Ma. Stabilization offullerene-like boron cages by transition metal encapsulation.
Nanoscale , pageDOI:10.1039/C5NR01659B, 2015.[26] C. Romanescu, T. R. Galeev, W. L. Li, A. I. Boldyrev, and L. S. Wang. Aromaticmetal-centered monocyclic boron rings: Cob − and rub − . Angew. Chem. Int. Ed. ,50:9334–9337, 2011.[27] W. L. Li, C. Romanescu, T. R. Galeev, Z. A. Piazza, A. I. Boldyrev, and L. S. Wang.Transition-metal-centered nine-membered boron rings: M@b and m@b − (m = rh,ir). J. Am. Chem. Soc. , 134:165–168, 2012.[28] C. Romanescu, T. R. Galeev, A. P. Sergeeva, W. L. Li, L. S. Wang, and A. I.Boldyrev. Experimental and computational evidence of octa- and nona- coordinatedplanar iron-doped boron clusters: Fe@b − and fe@b − . J. Organomet. Chem. , 721-722:148–154, 2012.[29] T. R. Galeev, C. Romanescu, W. L. Li, Z. A. Piazza, L. S. Wang, and A. I. Boldyrev.Observation of the highest coordination number in planar species: decacoordinatedta@b − and nb@b − anions. Angew. Chem. Int. Ed. , 51:2101–2105, 2012.3030] C. Romanescu, T. R. Galeev, W. L. Li, A. I. Boldyrev, and L. S. Wang. Geomet-ric and electronic factors in the rational design of transition-metal-centered boronmolecular wheels.
J. Chem. Phys. , 138:134315, 2013.[31] C. Romanescu, T. R. Galeev, W. L. Li, A. I. Boldyrev, and L. S. Wang. Transition-metal-centered monocyclic boron wheel clusters (m@b n ): A new class of aromaticborometallic compounds. Acc. Chem. Res. , 46:350–358, 2013.[32] R.-N. Zhao, Y. Yuan, and J.-G. Han. Transition metal mo-doped boron clusters: acomputational investigation.
J. Theo. Comp. Chem. , 13:1450036, 2014.[33] R.-N. Zhao, Y. Yuan, and J.-G. Han. A computational investigation on boron clusterswith w impurity.
Poly. Arom. Comp. , 0:1, 2015.[34] P. Saha, A. B. Rahane, V. Kumar, and N. Sukumar. Analusis of the electron densityfeatures of small boron clusters and the effect of doping with c, p, al, si and zn: magicb p and b si clusters. Phys. Scr. , 91:053005, 2016.[35] S.-D. Li, C.-Q. Miao, J.-C. Guo, and G.-M. Ren. Transition metal-boron complexesb n m: from bowls ( n =8-14) to tires (n=14). J. Comp. Chem. , 27:1858, 2006.[36] M. B¨oy¨ukata and Z. B. G¨uvenc. Density functional study of alb n clusters for n =114. J. Alloys. Comp. , 509:4214, 2011.[37] J.-B. Gu, X.-D. Yang, H.-Q. Wang, and H.-F. Li. Structural, electronic, and magneticproperties of boron cluster anions doped with aluminum: B n al − (2¡n¡9). Chin. Phys.B , 21:043102, 2012.[38] H. T. Pham and M. T. Nguyen. Effects of bimetallic doping on small cyclic andtubular boron clusters: B m and b m structures with m = fe, co. Phys. Chem.Chem. Phys. , page DOI:10.1039/C5CP01650A, 2015.[39] P. Saha, A. B. Rahane, V. Kumar, and N. Sukumar. Electronic origin of the stabilityof transition-metal-doped b drum-shaped boron clusters and their assembly into ananotube. J. Phys. Chem. C , 121:10728, 2017.[40] I. A. Popov, T. Jian, G. V. Lopez, A. I. Boldyrev, and L.-S. Wang. Cobalt-centredboron molecular drums with the highest coordination number in the cob − clusters. Nature Commun. , 6:8654, 2015.[41] J. P. Perdew, K. Burke, and M. Ernzerhof. Generalized gradient approximation madesimple.
Phys. Rev. Lett. , 77:3865–3868, 1996.[42] P. E. Bl¨ochl. Projector augmented-wave method.
Phys. Rev. B , 50:17953, 1994.[43] G. Kresse and J. Joubert. From ultrasoft pseudopotentials to the projectoraugmented-wave method.
Phys. Rev. B , 59:1758, 1999.3144] G. Kresse and J. Furthm¨uller. Efficient iterative schemes for ab initio total-energycalculations using a plane-wave basis set.
Phys. Rev. B , 54:11169, 1996.[45] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheese-man, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Cari-cato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg,M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima,Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta,F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov,R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyen-gar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross,V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J.Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G.Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels,. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox. Gaussian09Revision D.01. Gaussian Inc. Wallingford CT 2009.[46] A. J. H. Wachters. Gaussian basis set for molecular wavefunctions containing third-row atoms.
J. Chem. Phys. , 52:1033, 1970.[47] P. J. Hay. Gaussian basis sets for molecular calculations - representation of 3d orbitalsin transition-metal atoms.
J. Chem. Phys. , 66:4377–84, 1977.[48] P. J. Hay and W. R. Wadt. Ab initio effective core potentials for molecular calcula-tions - potentials for the transition-metal atoms sc to hg.
J. Chem. Phys. , 82:270–83,1985.[49] P. J. Hay and W. R. Wadt. Ab initio effective core potentials for molecular calcu-lations - potentials for main group elements na to bi.
J. Chem. Phys. , 82:284–98,1985.[50] D. Y. Zubarev and A. I. Boldyrev. Developing paradigms of chemical bonding:adaptive natural density partitioning.
Phys. Chem. Chem. Phys. , 10:5207–5217,2008.[51] T. A. Keith. Aimall. TK Gristmill Software Overland Park KS, USA. (Version 11.03.14), 2012. Available at: http://aim.tkgristmill.com.[52] I. Sumar, R. Cook, P. W. Ayers, and C. F. Matta. Aimldm: A program to gener-ate and analyze electron localization-delocalization matrices (ldms).
Comp. Theor.Chem. , 1070:55–67, 2015.[53] K. Momma and F. Izumi. Vesta 3 for three-dimensional visualization of crystal,volumetric and morphology data.
J. Appl. Crystallogr. , 44:1272–1276, 2011.[54] U. Varetto.
Molekel 5.4, (Swiss National Supercomputing Centre) , 2009.3255] A. Kokalj. Computer graphics and graphical user interfaces as tools in simulationsof matter at the atomic scale.