First Principles Study of Structural and Optical Properties of B 12 Isomers
FFirst Principles Study of Structural and Optical Properties of B Isomers
Pritam Bhattacharyya
Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Ihsan Boustani
Theoretical and Physical Chemistry, Faculty of Mathematics and Natural Sciences, Bergische Universität, Wuppertal,Gauss Strasse 20, D-42097 Wuppertal, Germany
Alok Shukla
Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Abstract
In this work we undertake a comprehensive numerical study of the ground state structures and opticalabsorption spectra of isomers of B cluster. Geometry optimization was performed at the coupled-cluster-singles-doubles (CCSD) level of theory, employing cc-pVDZ extended basis sets. Once thegeometry of a given isomer was optimized, its ground state energy was calculated more accuratelyat the coupled-cluster-singles-doubles along with perturbative treatment of triples (CCSD(T)) levelof theory, employing larger cc-pVTZ basis sets. Thus, our computed values of binding energies ofvarious isomers are expected to be quite accurate. Our geometry optimization reveals eleven distinctisomers, along with their point group, and electronic ground state symmetries. We also performedvibrational frequency analysis on the three lowest energy isomers, and found them to be stable.Therefore, we computed the linear optical absorption spectra of these isomers of B , employinglarge-scale multi-reference singles-doubles configuration-interaction (MRSDCI) approach, and founda strong structure-property relationship. This implies that the spectral fingerprints of the geometriescan be utilized for optical detection, and characterization, of various isomers of B . We also exploredthe stability of the isomer with with the structure of a perfect icosahedron, with I h symmetry. Inbulk boron icosahedron is the basic structural unit, but, our vibrational frequency analysis revealsthat it is unstable in the isolated form. We speculate that this instability could be due to Jahn-Tellerdistortion because five-fold degenerate HOMO orbitals in I h structure are unfilled. Keywords:
MRSDCI; Boron clusters; Optical absorption; CCSD(T)
1. Introduction
Since the eighties, and until now, the structures and energetics of small atomic clusters havebeen of great interest, both experimentally, and theoretically.[1] Initially, the clusters were consid-ered as a bridge, or an accumulation at nanoscale to solids, but over the years, due to sustainedresearch effort, they have become well established as a separate research discipline. Since then, theexploration and synthesis of structures and energetics of pure atomic clusters has acquired both
Email addresses: [email protected] (Pritam Bhattacharyya), [email protected] (Ihsan Boustani ), [email protected] (Alok Shukla)
Preprint submitted to Elsevier May 22, 2019 a r X i v : . [ phy s i c s . a t m - c l u s ] M a y cademic and practical importance. Thereupon, the mass spectra of alkali-metal, non-metal, carbonand boron clusters were investigated as well as the related magic numbers were determined.[1] Forexample, among the allotropes of carbon, all sp -types are closely related and have been extensivelystudied: graphene (flat monoatomic sheet of graphite), spherical fullerenes, and nanotubes.[2] Alsofor boron, carbon’s left neighbor in the periodic table, the landscape of fullerene-like possibilities isjust beginning to emerge: from small quasi-planar clusters, spherical cages and nanotubes [3], thento borophene (single atom-thin monolayer sheet of boron atoms) [4].However, the development of non crystalline boron beyond the icosahedral arrangements beganin the late eighties with theoretical and experimental studies on small boron clusters. Andersongroup [5] carried out the first experimental and theoretical study of bonding and structures in boroncluster ions B n+ for (n ≤
13) in comparison with the well known closo boron hydrides. Furtherimportant study on boron clusters was carried out by Kawai and Weare [6] using Car-Parrinello ab initio molecular dynamics simulation. They found that an open 3D structure is more stablethan the icosahedral boron. However, Kato et al. [7] investigated boron clusters B n for (n=8-11)using ab initio molecular orbital theory, and concluded that the most stable clusters have planaror pseudo-planar cyclic structures. The breakthrough in boron cluster research happened in 1997due to the work of one of us,[8] in which it was discovered that the most stable boron clusters havequasi-planar structures of dovetailed hexagons including the B cluster. In this work, the so-calledAufbau principle was proposed, according to which highly stable novel structures in form of boronsheets (nowadays called borophene), nanotubes, and spheres can be constructed from only two basicunits: pentagonal and hexagonal pyramids B , and B , respectively.[8] These novel structures aredifferent from the conventional allotropes of solid boron.It is well known that the bulk boron exists in several crystalline phases.[9] The most famousboron solids are the rhombohedral α -B and β -B phases with 12 and 106 atoms per unit cells,respectively, The α -rhombohedral phase transforms at 1200 °C into the more stable β -rhombohedralone. Besides the α -tetragonal, and β -tetragonal phases with 190-192 atoms per unit cell, there is anew phase of crystalline boron called orthorhombic γ -B boron [10], with two B clusters, and two B pairs in the unit cell, accompanied by a charge transfer as boron boride (B ) δ + (B ) δ − . Remarkable isthe common basic unit cell of all these structures: the B icosahedron. This regular B icosahedronoperates as a building block for all phases mentioned above. Due to the multi-center covalent inter-atomic bonds between the icosahedra, the electron deficiency is reduced, and the electron distributionsaturates the twelve boron atoms in each icosahedron, leading to high stability. However, as soon asthis B icosahedron is separated from the bulk, losing its neighbors, and becoming a free standingcluster, it loses its stability, and flattens to transform into a quasi-planar structure.[8] Based uponmolecular orbital occupancies, we argue in this work that this flattening of the B - I h icosahedroninto the quasi-planar B - C v structure could be a consequence of Jahn-Teller distortion.Given the importance of the B cluster, in this work, we study all its isomers including theicosahedron, and compute the optical absorption spectra of its most stable isomers. For the purpose,we have utilized standard wave function based first principles quantum chemical methodologies,employing Cartesian Gaussian basis functions. Geometry optimization was carried out using thecoupled-cluster-singles-doubles (CCSD) approach, while the optical absorption spectra of variousisomers were computed using the multi-reference singles-doubles configuration interaction (MRSDCI)approach, which has been extensively utilized in the group of one of us to study optical properties ofconjugated polymers,[11, 12, 13, 14, 15] graphene quantum dots,[16, 17] along with atomic clusterssuch as those of boron,[18] aluminium,[19] sodium,[20] and magnesium.[21]2 . Theoretical approach and Computational details The geometries of various isomers of B cluster were optimized using the coupled-cluster singlesdoubles (CCSD) method, and the cc-pVDZ basis set, as implemented in the Gaussian 09 package.[22]The optimized geometries of all the isomers of B cluster are shown in the Fig. 1. The final totalground state energies of various isomers reported in Table 1, were obtained by performing single-point energy calculations at those optimized geometries, using the coupled-cluster singles doublestriples (CCSD(T)) method, and the cc-pVTZ basis set. Thus, the final ground state total energies ofvarious isomers were computed using a higher level of theory, and a larger basis set. One importantcriteria, i.e., the binding energy per atom which is directly related to the stability of the cluster hasbeen computed using the formula E b n = E − E n n , (1)where E is the energy of a single boron atom, E n is the total energy of the cluster, and n isthe number of atoms present in the cluster. For boron atom E = − . Hartree was used,obtained using CCSD(T) approach as implemented in Gaussian09 program, employing the cc-pVTZbasis set. Computed binding energies per atom of all the eleven isomers are listed in Table 1.We have also computed the optical absorption spectra of the three lowest-energy structures, whichnot only have the highest binding energies, but were also found to be stable from the vibrationalfrequency analysis.
With the aim of understanding structure property relationship, we also computed the opticalabsorption spectra of three of the lowest energy isomers: (a) B - quasi-planar , (b) B - doublering, and (c) B - chain. In order to compute the optical absorption spectra, we utilized the abinitio MRSDCI approach, as implemented in the computer program MELD,[23] and described inrecent works on the optical properties of clusters.[18, 19, 20, 21] For the purpose, we utilized thegeometries of the isomers (quasi-planar, double ring, and chain) presented in Fig. 1, and the cc-pVDZbasis set, which was also used for geometry optimization. Calculations were initiated by performingthe restricted Hartree-Fock calculations, and the resultant orbital set (occupied and virtual), servedas the single-particle basis for the CI calculations. Because the orbital basis set was fairly large,the following approximations were employed to truncate it: (a) frozen-core approximation, and (b)removal of high-energy virtual orbitals. Both these approximations are justified on the physicalgrounds in that the orbitals which are far away from the Fermi level are unlikely to contribute tooptical excitations. Nevertheless, we critically examine the influence of these approximations on ourresults in the following section. After fixing the single-particle basis, the next step involves singles-doubles configuration interaction calculations (SDCI), employing single reference wave functions.Ground and the excited states obtained from these calculations are used to calculate the opticalabsorption spectrum σ ( ω ) , according to the formula σ ( ω ) = 4 πα X i ω i |h i | ˆ e.r | i| γ ( ω i − ω ) + γ . (2)Above, ω denotes frequency of the incident light, ˆ e denotes its polarization direction, r is the positionoperator, α is the fine structure constant, and i denote, respectively, the ground and excited states, ω i is the frequency difference between those states , and γ is the line width, taken as 0.1 eV, inthese calculations. The summation over i in Eq. 2 involves an infinite number of excited states3 a) Quasi-planar( C v ) (b) Double ring( D d ) (c) Chain ( C h )(d) Hexagon ( D h ) (e) Perfect Icosahe-dron ( I h ) (f) Distorted Icosa-hedron ( D h ) (g) Quad ( D h )(h) Pris-3-square-1( D h ) (i) Pris-3 square-2( D h ) (j) Parallel triangular ( D h ) (k) Planar square( D h )Figure 1: Optimized geometries of various isomers of B cluster, along with their point group symmetries. Opti-mizations were performed using CCSD level of theory, and cc-pVDZ basis set, as implemented in the Gaussian 09package.[22] All isomers have singlet multiplicity, and the bond lengths are in Å unit. able 1: All the isomers of B cluster have been listed below along with their point group, ground state symmetry,and the ground state energy (in Hartree). Also their relative energy (in eV) with respect to the lowest isomer, i.e.,quasi-planar structure is presented in the same table. The correlation energy is the difference of the total energies ofthe system at the CCSD(T) and the Hartree-Fock levels. In the last column, the binding energy per atom (in eV) isshown to illustrate the stability of the structure. Isomer Point Ground state Ground state Relative Correlation Bindinggroup symmetry energy † (Ha) energy energy per energy per(eV) atom (eV) atom (eV)Quasi-planar C
3v 1 A -297.208187 0.0 2.92 4.60Double ring D
6d 1 A -297.1399191 1.858 3.00 4.45Chain C
2h 1 A g -297.1029824 2.863 2.92 4.36Hexagons D
2h 1 A g -297.0853492 3.342 2.97 4.33Perfect Icosahedron I h 1 A g -297.0662229 3.863 3.36 4.28Distorted Icosahedron D
2h 1 A g -297.077724 3.550 3.35 4.31Quad D
2h 1 A g -297.0180888 5.173 3.08 4.17Pris-3-square-1 D
4h 1 A -296.8945639 8.534 3.44 3.89Pris-3-square-2 D
4h 1 A -296.8192288 10.583 3.29 3.72Parallel triangular D
3h 1 A -296.7691979 11.945 3.19 3.61Planar square D
4h 1 A -296.7132189 13.468 3.36 3.48 † Ground state energy was computed using CCSD(T) level of theory and cc-pVTZ basis set.5hich are dipole connected to the ground state, while in practice it is restricted to states withexcitation energies up to 10 eV. We carefully examine the excited states contributing to the peaks inthe absorption spectrum, and the configurations contributing to those are included in the referenceset for MRSDCI calculations. A new set of configuration space is generated by considering singlesand doubles excitations from the reference set, and the entire procedure is repeated. This process iscontinued until the calculated absorption spectrum has converged within reasonable limits.[18, 19,20, 21]We make a few brief comments about the line width γ occurring in the expression for the opticalabsorption spectrum (see Eq. 2 above). The line width incorporates all kinds of uncertaintiesassociated with the values of energy levels due to reasons such as: (a) natural line widths, (b)vibrational broadening, because of finite temperature effects, (c) collisional broadening in the gas orliquid phase, (d) impurities and disorder in the samples, and (e) experimental uncertainties. Thevalue γ = 0 . eV chosen by us is fairly reasonable, and has been employed in all our past calculationsof optical absorption spectra. From the structure of Eq. 2 it is obvious that a smaller value of γ leadsto sharper and higher peaks, while a larger value causes the peaks to become broader, and lower inheight. Therefore, as per the requirement, one can use any reasonable value of γ . The computational effort in CI calculations grows ≈ N , where N is the total number of activeorbitals. Thus, the size of the CI expansion, and hence the computational effort involved proliferatesvery rapidly with the increasing N . Therefore, the choice of the active orbitals becomes crucial, anddetermines the feasibility and the quality of the calculations. As discussed earlier, there are two waysto control the value of N : (a) by the choosing how many low-lying orbitals will be frozen during theCI calculations, and (b) how many virtual orbitals will be included in the CI calculations. In Fig. 2,we examine the convergence of the computed optical absorption spectrum of the three isomers of B ,with respect to the choice of the active orbitals. Assuming that N f implies the total number of frozenorbitals in the calculations, we performed three sets of calculations for each isomer: (i) N f = 12 , N = 40 / / , (ii) N f = 12 , N = 46 , and (iii) N f = 18 , N = 40 / / . Here N f = 12 impliesthat 1s core orbital of each boron atom of the cluster was frozen during the CI calculations, whilefor N f = 18 implies that in addition to the atomic core orbitals, six additional orbitals were frozenduring the calculations. Furthermore, for N f = 12 , total number of valence electrons considered inthe CI calculations was N val = 36 , while for N f = 18 , this reduces to N val = 24 . When we examinethe computed spectra, it is obvious that: (a) for the quasi-planar and the chain structures, threesets of calculations are in good qualitative and quantitative agreement with each other, (b) for thecase of the double ring structure, three sets of calculations are in good agreement with each otherfor photon energy E < eV, but at E ≈ eV, results of three sets of calculations differ from eachother. Nevertheless, for the double ring, even in that energy region N f = 12 , N = 42 and N f = 12 , N = 46 calculations agree with each other as far as peak locations are concerned, but disagree onpeak intensities. Therefore, we conclude that the calculations corresponding to N f = 12 , N = 46 ,are well converged, and analyze those results in the next section. As mentioned previously, the MRSDCI procedure adopted in these calculations is iterative innature. Thus, once the number of active/frozen orbitals
N/N f is fixed, MRSDCI procedure isiterated until the absorption spectrum converges within reasonable tolerance levels. In Table 2, wepresent the data regarding the final MRSDCI calculations performed on the three isomers, whichincludes: (a) point group symmetry used in the calculations, and (b) total number of configurations N total in the MRSDCI expansion for various irreducible representations. We note that N total ranges6 I n t e n s it y ( a r b . un it s ) N f = 12, N = 41N f = 12, N = 46N f = 18, N = 41 (a) quasi-planar I n t e n s it y ( a r b . un it s ) N f = 12, N = 42N f = 12, N = 46N f = 18, N = 42 (b) Double ring I n t e n s it y ( a r b . un it s ) N f = 12, N = 42N f = 12, N = 46N f = 18, N = 40 (c) ChainFigure 2: Optical absorption spectra of three most stable isomers of B , calculated using varying numbers of frozenand active orbitals. able 2: The average number of total configurations involved in MRSDCI calculations (N total ) of the optical absorptionspectra of the most stable isomers of B cluster. Isomer Point Symmetry N total groupQuasi-planar † C A † D A A g u u † C s point group symmetry was used during the calculationsroughly from five millions to ten millions, implying that these were large-scale calculations, and,therefore, electron-correlation effects have been included to a high-level both for the ground state,and the excited states.
3. Results and Discussion
Next, we discuss the geometry and the electronic structure of isomers of B considered in thiswork. Furthermore, we also discuss the optical absorption spectra of three of the most stable isomers. This is one of the most studied isomers of the B cluster, perhaps, because of its high-levelof symmetry, i.e., a perfect icosahedral structure, and I h symmetry group.[24, 25, 8] Bambakidisand Wagner used the Slater’s SCF-Xa-SW approach to study the electronic structure and cohesiveproperties of B icosahedral cluster, while Boustani employed first-principles DFT and Hartree-Fock approaches to study it.[25, 8] Furthermore, B icosahedron forms the fundamental units inbulk boron, and other boron-rich solids, therefore, it has always made one curious whether or notit exists in isolated form.[6, 26, 27] Hayami[28] studied the encapsulation properties of the B icosahedral cluster using first-principles DFT based methodology. Kawai and Weare[6] based uponab initio molecular dynamics simulations, and Boustani using first-principles DFT and Hartree-Fockmethods,[25, 8] concluded that the isolated B icosahedral cluster is unstable, and actually stabilizesto a lower-energy open structure.In order to examine the fundamental reasons behind the instability of the singlet I h structureagainst structural distortions, we consider the possibility of Jahn-Teller effect, and to that endwe examine the single-particle energy levels of B perfect icosahedron, obtained from calculationswithout any electrons. These calculations were performed using a STO-3G basis set, and the energylevels obtained, along with their symmetries, are shown in Fig. 3. If we start filling the energy levelsusing auf-bau principle, we note that the five-fold degenerate highest-occupied molecular orbital(HOMO) of H g symmetry is partially filled, thus making the isomer a candidate for Jahn-Tellerdistortion to a structure of lower symmetry.Another manifestation of this instability is that automated geometry optimization procedures,such as the one based on Berny algorithm implemented in Gaussian 09 package,[22] inevitably lead8 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Figure 3: A rough sketch of orbital energies of B - icosahedron ( I h symmetry group), close to the Fermi level. Irre-ducible representations of the corresponding orbitals are indicated next to them. HOMO, with its five-fold degeneracy,is partially occupied, thereby making the system a candidate for Jahn-Teller distortion, to a lower symmetry structure. to a lower symmetry structure, if the starting geomertry is a perfect icosahedron, with I h symmetry.Since we are interested in computing the total energy of the perfect icosahedral structure, we opti-mized corresponding geometry manually, by varying the nearest-neighbor bond length, and lookingfor the minimum of energies computed using the CCSD approach, and cc-PVDZ basis functions. Atthis point, an objection can be raised against the use of single-reference approaches such as CCSDto describe the ground state of the I h structure, given the orbital degeneracy at the Fermi level asdepicted in Fig. 3. However, this is not a problem if one uses the Hartree-Fock (HF) molecularorbitals (MOs) for performing the correlated-electron calculations, because the electron-electron re-pulsion lifts the orbital degeneracies to a great extent. Furthermore, we verified this by performingMRSDCI calculations on the I h structure at its minimum energy geometry using the HF MOs, andfound that the coefficient of the HF reference state is 0.81 in the lowest singlet-state wave function(see Table S16 of Supporting Information, and the related discussion), thereby, justifying the use ofsingle-reference approaches to compute the ground state energy. The minimum energy I h structureobtained from these CCSD/cc-PVDZ calculations is depicted in Fig. 1(e), with the correspondingtotal energy listed in Table 1. Next, we checked the stability of the perfect icosahedral structureby performing vibrational frequency analysis at the CCSD level, using the cc-PVDZ basis set, andfound it to be unstable with five imaginary frequencies.Next, we performed automated geometry optimization[22] on the icosahedral structure, with thestarting atomic coordinates corresponding to the I h structure, and 1.73 Å bond length correspondingto Fig. 1(e). The resultant optimized geometry has a distorted icosahedral structure with D h symmetry, with non-uniform nearest-neighbor distances in the range 1.71 Å—1.80 Å, as shown inFig. 1(f). This distorted structure has an energy 3.55 eV higher than the quasi-planar structure,but 0.31 eV more stable as compared to the I h structure. To explore the stability of this structure,we performed vibrational frequency analysis on it at the CCSD level, using the same cc-pVDZ basis9et, and found that it is a transition state, with one imaginary frequency. This, clearly reveals thatperfect I h structure of B cluster with the singlet spin is totally unstable, and deforms into a distortedicosahedral transition state corresponding to a saddle point on the potential energy surface. Thisindicates that this D h structure is itself unstable, and will undergo further distortions to achievestability. Thus, our calculations have confirmed the instability of the I h structure of the B cluster. This lowest lying quasi-planar convex shaped isomer of B has C v point group symmetry, withthe irreducible representation A of the electronic ground state wave function. Several authorshave theoretically studied this isomer earlier,[29, 30, 31] including one of us.[25, 8] Our geometryoptimization study reveals this to be the most stable one amongst the eleven isomers studied (seeTable 1), a result in good agreement with several earlier studies performed using lower levels oftheory.[6, 25, 8] The optimized geometry of this isomer is shown in Fig. 1(a). This structure iscomposed of one outer ring with nine boron atoms, and an inner triangular ring with three boronatoms. The inner ring is slightly out of plane compared with the outer ring, and has larger bondlength (1.71 Å) as compared to those in the outer ring (1.65 Å and 1.59 Å). Each inner boron atomis surrounded by six other boron atoms, thus this quasi-planar isomer consists of three dovetailedhexagonal pyramids. The reported bond lengths optimized using Hartree-Fock level of theory byone of us earlier are in good agreement with our present work.[8] The bond lengths obtained by usare in good agreement with those reported by Atis et al .[30] Existence of this isomer was verifiedexperimentally by Wang et al ,[29] and recently the unusual stability of this isomer was explainedtheoretically by Kiran and co-workers.[31] To understand the stability of this isomer from the Jahn-Teller perspective, we present the orbital occupancy diagram in Fig. 4 based upon a single-pointHartree-Fock calculation performed at the optimized geometry, using a cc-pVTZ basis set. From thefigure it is obvious that even though the HOMO is two-fold degenerate, it is completely filled, and,therefore, as per Jahn-Teller theorem, it will not undergo any further distortion. To confirm this, weperformed vibrational frequency analysis for this structure at the CCSD level theory using cc-pVDZbasis, and found it to be completely stable, with all the vibrational frequencies being real. The factthat this structure has the lowest-energy of all the isomers considered, also suggests that it is stable.Because this is a stable isomer, we also calculated its ground state linear optical absorptionspectra presented in Fig. 5. During the electron-correlated MRSDCI calculations, C s point groupsymmetry was utilized. HOMO orbitals of this isomer are doubly degenerate, denoted as H andH , both of which are doubly occupied. From Fig. 5, it is obvious that absorption in this isomerstarts with a very small peak (I) at 3.58 eV, followed by three weak peaks at 4.43 eV (II), 4.93 eV(III) and 5.44 eV (IV), which is a shoulder to the most intense peak (V) located at 5.74 eV. Thewave function of the excited state corresponding to this peak is dominated by single excitation (H -3) → L and (H - 3) → L with respect to the closed-shell Hartree-Fock reference configuration.Detailed information about the excited states contributing to the peaks, is presented in Table S1 ofthe supporting information. This structure is composed of two regular hexagons of B atoms, displaced from each other, withoutthe boron atoms eclipsing each other, as shown in Fig. 1(b). As a result, this isomer has the D d point-group symmetry, along with a closed-shell A electronic ground state, with its total energy1.86 eV higher as compared to the lowest energy quasi-planar isomer. The bond length of two closestin-plain and out of plain boron atoms are 1.6458 Å and 1.7349 Å, in good agreement with recentwork of Atis et al. [30] The top view of this isomer, with its two hexagonal rings visible, is presentedin Fig. 6. Based upon the vibrational frequency analysis of this structure, we conclude that this10 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(cid:3) 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(cid:4) 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(cid:4) 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(cid:4) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2) (cid:4) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2) (cid:3) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2) (cid:4) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2) (cid:4) Figure 4: A rough sketch of the energy levels of quasi-planar ( C v ) isomer, along with a few low-lying virtual levels,obtained from the Hartree-Fock calculations performed at the optimized geometry, using a cc-pVTZ basis set. Irre-ducible representations of the orbitals corresponding to these levels, are indicated next to them. It is obvious that thedoubly-degenerate HOMO is fully occupied for this isomer, indicating that it is stable against a Jahn-Teller distortion. I n t e n s it y ( a r b . un it s ) I II III IV V VI VII VIII
Figure 5: Linear optical absorption spectrum of quasi-planar isomer computed using the MRSDCI approach, and thecc-pVDZ basis set. During the calculations forty six active orbitals were used, while all the twelve 1s core orbitalswere assumed frozen. To plot the spectrum, 0.1 eV uniform line-width was used.Figure 6: Double ring structure of B isomer (Top view). isomer is stable, and as a result we have computed its optical absorption spectrm using the MRSDCIapproach, presented in Fig. 7.For the MRSDCI calculations C s (Abelian) point group symmetry was employed, and in theHartree-Fock ground state, doubly degenerate HOMO orbitals denoted as H and H were fullyoccupied. Other degenerate occupied and virtual molecular orbitals were also denoted by suffixes 1and 2 etc. The detailed information about the wave functions of the excited states contributing tovarious peaks, along with the frontier orbitals involved in those excitations, are presented in TableS2, and Fig. 2, respectively, of the supporting information. The computed spectrum starts with avery intense peak (I), which is due to two nearly degenerate excited states around 4.8 eV, with wavefunctions dominated by singly-excited configurations |(H - 1) → (L+1) i , |(H - 1) → (L+1) i ,|(H -1) → (L+1) i , and |(H - 1) → (L+1) i . It is followed by two comparatively less intensepeaks around 5.9 eV (II) and 6.4 eV (III), whose wave functions also consist mainly of the singleexcitations.. The peak IV is the most intense peak near 6.8 eV, which is due to two nearly degenerateexcited states, largely due to singly-excited electrons. We note that the absorption spectrum of thisstructure is sufficiently different, both qualitatively, and quantitatively, as compared to that of thequasi-planar isomer. 12 I n t e n s it y ( a r b . un it s ) I II III IV
Figure 7: Linear optical absorption spectrum of B isomer with the double ring structure, calculated using theMRSDCI approach, and the cc-pVDZ basis set. During the calculations forty six active orbitals were used, while allthe twelve 1s core orbitals were assumed frozen. To plot the spectrum, 0.1 eV uniform line-width is used. Chain-like isomer of B is 2.86 eV higher in energy as compared to the lowest-energy quasi-planarstructure, and has the structure of a ladder with rungs of the ladder not being parallel to each other(see Fig. 1(c)), with bond lengths ranging from 1.55 Å to 1.87 Å. The point group symmetry of theisomer is C h , with the electronic ground state symmetry A g . We also performed the vibrationalfrequency analysis on this isomer, revealing it to be stable. As a result, we have calculated the opticalabsorption spectrum of this structure using the MRSDCI approach, and the results are presented inFigs. 8 and 9. In Fig. 8, we have plotted the component corresponding to the photons polarized inthe plane of the chain, leading to the absorption to the B u type excited states, and this componentcarries the bulk of the absorption intensity. In Fig. 9, we plot the absorption spectrum of thephotons polarized perpendicular to the plane of the chain, corresponding to A u excited states. Wehave plotted two absorption components separately because of the huge difference in the intensitiesinvolved. It is quite understandable that the bulk of the oscillator strength is carried by absorptionpolarized in the plane of the chain, because that is where the atoms, and hence, electrons are, leadingto large transition dipole moments with respect to the ground state.The in-plane absorption begins with a small peak (I) at 1.64 eV dominated by |H → L i and |H-1 → L+1 i . The most intense peak (V) appears near about 4.5 eV dominated by |H → L i and |H-1 → L+1 i single excitations. From Fig. 3 of supporting information it is obvious that the orbitalsinvolved H − / L + 1 have π/π ∗ character, while H/L have σ/σ ∗ character, thus these transitionhave a mixed π − π ∗ + σ − σ ∗ character. The detailed many-particle wave functions of the excitedstates associated with the peaks of the optical absorption spectra are presented in Table S3 of thesupporting information.As far as the out-of-plane component of the absorption spectrum is concerned, it is obvious fromFig. 9 and from Table S4 of the supporting information, that oscillator strengths correspondingto peaks are at least two orders of magnitude smaller than those of “in-plane” component. Thiscomponent of the absorption starts at 2.42 eV, with its most intense peak IV located at 4.85 eV.From Table S4, it is clear that the wave functions of the excited states contributing to the peaksare dominated by singly excited configurations involving orbitals away from the Fermi level, anddoubly excited configurations involving orbitals close to the Fermi level. However, detection of these13 I n t e n s it y ( a r b . un it s ) I II III IV V VI VII VIII IX
Figure 8: Linear optical absorption spectrum of chain-like B isomer using the MRSDCI approach, and the cc-pVDZbasis set, for the photons polarized in the plane of the chain, i.e. , for excited states of B u symmetry. During thecalculations forty six active orbitals were used, while all the twelve 1s core orbitals were assumed frozen. To plot thespectrum, 0.1 eV uniform line-width is considered. resonances will be a difficult task unless it is possible to orient the isomer perpendicular to thepolarization direction of the incident photons.Comparison of the optical absorption spectrum of the chain isomer with those of the quasi-planarand the double ring structures, clearly reveals qualitative and quantitative differences which can beutilized as the fingerprints of these structures, and thus can be used in their optical detection. This planar isomer belongs to D point group with the electronic ground state symmetry A g and has the appearance of a boron dimer surrounded by ten other boron atoms. One of us discoveredthis structure earlier using Hartree-Fock level geometry optimization, and a 3-21G basis set.[8] Inthis work the optimized geometry obtained using the CCSD level of theory, and cc-pVDZ basisset, is shown in Fig. 1(d). For this structure, the average bond length in between two consecutiveperipheral boron atoms is 1.56 Å, while the bond length between any of the central atom, and theclosest peripheral atom is near about 1.8 Å. This isomer has a three-dimensional structure shown in Fig. 1(g), with a D point group, alongwith the electronic ground state of symmetry A g . The structure consists of two six-membered ringsof boron atoms, lying in mutually perpendicular planes. In one plane the B-B bond lengths arealmost equal, i.e., near about 1.65 Å, but in the other plane it has two distinct bond lengths of 1.72Åand 1.83 Å. This isomer is predicted to be 5.17 eV higher in energy as compared to the lowestenergy one. This isomer also has a three-dimensional structure of a cuboidal shape, whose two faces aresquares, while the remaining four are rectangles. The length of the sides of square shaped faces is1.73 Å, and the two squares are 2.53 Å apart from each other. Furthermore, four opposite facesas shown in Fig. 1(h) are capped by a boron atom each in a symmetric manner, with the nearest14 I n t e n s it y ( a r b . un it s ) I II III IV V VI
Figure 9: Linear optical absorption spectrum of chain-like B isomer, corresponding to the excited states of A u symmetry , with the photons polarized perpendicular to plane of the chain. Spectrum was computed using the MRSDCIapproach, and the cc-pVDZ basis set. During the calculations forty six active orbitals were used, while all the twelve1s core orbitals were assumed frozen. To plot the spectrum, 0.1 eV uniform line-width is considered. distance between the capping atom and the cube atom being 1.69 Å. This structure has D pointgroup symmetry, and it is 8.5 eV higher in energy than the lowest energy one, with the electronicground state symmetry A . This isomer has a very similar structure as Pris-3-square isomer discussed in the previous section,except that the length of the sides of the square-shaped faces is slightly larger at 1.79 Å, and thedistance between the opposite squares is slightly shorter at 1.98 Å. As shown in Fig. 1(i), distance ofthe capping atoms from the corresponding faces is also slightly shorter at 1.66 Å. With a structuresimilar to that Pris-3-square-1, this isomer also has D point group symmetry, with the electronicground state symmetry of A . In spite of a structure similar to that of Pris-3-square-1 isomer, it isenergetically about 2 eV higher than that, and 10.58 eV higher than the lowest energy quasi-planarstructure. B - parallel triangular isomer has D point group symmetry, with the electronic ground statesymmetry A . As shown in Fig. 1(j), it consists of four parallel equilateral triangles arrangedsymmetrically about a central plane. The first/last of those triangles has edge length 1.71 Å, whilethe two middle triangles have equal bond lengths of 1.85 Å. The connecting bond length in betweentwo middle triangles have edge length 1.85 Å. The interplanar distances between the first two and thelast two triangles is close to 1.6 Å, while that between the middle triangles is 1.85 Å. This structureis roughly 12 eV higher in energy as compared to the lowest energy structure. This isomer which is about 13.5 eV higher than the lowest energy structure, is strictly a planarone, with the D point group, and the electronic ground state symmetry A . As shown in Fig. 1(k),it is composed of four isosceles trapezoids connecting with each other at a common side, resulting ina square-like symmetry. The equal sides of isosceles trapezoid have the bond length of 1.75 Å, whilethe other two sides have the bond lengths of 1.56 Å, and 1.66 Å.15 . Conclusions In conclusion, we presented a detailed ab initio electron-correlated study of the electronic struc-ture and the ground state geometries of several isomers of B cluster. In agreement with the earlierworks, quasi-planar convex structure was found to be the lowest energy structures, however, severalother structures corresponding to the local minima on the potential energy surface were also identi-fied. In particular, our calculations confirm that B perfect icosahedral structure with I h symmetryis unstable against structural distortions. Furthermore, Jahn-Teller analysis of the icosahedral iso-mer revealed that it may be unstable because at the single-electron level of theory, it has severalunfilled degenerate HOMO orbitals. On the other hand, a similar analysis of the lowest-energy quasi-planar structure reveals that the isomer has completely filled doubly-degenerate HOMO, signalingits stability against Jahn-Teller distortion. These results are also in agreement with the vibrationalfrequency analysis which also predicts the singlet I h structure to be unstable, and the quasi-planarstructure to be stable. However, whether Jahn-Teller distortion causes B perfect icosahedron intothe quasi-planar structure, can be resolved by other calculations, such as the ones based upon linear-transit theory connecting the potential energy surface of the I h structure, to that of the quasi-planarone, which at present is beyond the scope of this work. Vibrational frequency analysis also pre-dicted two other isomers, namely double ring, and the chain structures, to be stable. For these threestable structures, namely, quasi-planar, double ring, and the chain isomer, we also calculated theoptical absorption spectra employing the state-of-the-art MRSDCI methodology, utilizing large-scaleCI expansions. The calculated spectra revealed strong structure-property relationship which can beutilized for detection of different isomers using optical techniques. Acknowledgements
Work of P.B. was supported by a Senior Research Fellowship offered by University Grants Com-mission, India.
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Inorganic Chemistry , 48(21):9965–9967, 2009. PMID: 19785465. 18 upporting Information: First Principles Study of Structural and OpticalProperties of B Isomers
Pritam Bhattacharyya
Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Ihsan Boustani
Theoretical and Physical Chemistry, Faculty of Mathematics and Natural Sciences, Bergische Universität, Wuppertal, Gauss Strasse20, D-42097 Wuppertal, Germany
Alok Shukla
Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Here we present the plots of the frontier orbitals, and the wave functions of important excited states contributingto the calculated optical absorption spectra of three B isomers, namely: (a) quasi-planar, (b) double ring, and(c) chain.
1. Quasi-Planar IsomerH-4 (H-3) (H-3) H-2 (H-1) (H-1) H H L L L+1 L+2 (L+3) (L+3) L+4 (L+6) (L+6) Figure 1: Frontier molecular orbitals of quasi-planar ( C v ) isomer of B (see Fig. 1a, of the main article), contributing to importantpeaks in its optical absorption spectra. The symbols H and L corresponds to HOMO and LUMO orbitals. Subscripts 1, 2 etc. are usedto label the degenerate orbitals. Email addresses: [email protected] (Pritam Bhattacharyya), [email protected] (Ihsan Boustani ), [email protected] (Alok Shukla)
Preprint submitted to Elsevier May 22, 2019 a r X i v : . [ phy s i c s . a t m - c l u s ] M a y able S1: Many-particle wave functions of the excited states associated with the peaks in the optical absorption spectra of quasi-planar( C v ) isomer of B (Fig. 1a of the main article). E denotes the excitation energy (in eV) of the state involved, and f stands for oscillatorstrength corresponding to the electric-dipole transition to that state, from the ground state. In the “Polarization” column, x, y, and zdenote the the direction of polarization of the absorbed light, with respect to the Cartesian coordinate axes defined in the figure below.In the “Wave function” column, | HF i indicates the Hartree-Fock (HF) configuration, while other configurations are defined as virtualexcitations with respect to it. H and L denote HOMO and LUMO orbitals, while the degenerate orbitals are denoted by additionalsubscripts 1, 2 etc. In parentheses next to each configuration, its coefficient in the CI expansion is indicated. Below, GS does not indicatea peak, rather it denotes the ground states wave function of the isomer. Peak E (eV) f Polarization Wave functionGS | HF i (0 . I 3.58 0.6317 y/z | H → L i (0 . | H → L i (0 . II 4.43 0.3851 y/z | ( H − → L + 1 i (0 . | H → L + 2 i (0 . | ( H − → L + 1 i (0 . | H → L + 2 i (0 . III 4.93 0.1464 y/z | ( H − → L + 2 i (0 . | H → L + 2 i (0 . | H − → L i (0 . | H → L + 2 i (0 . IV 5.44 0.7118 y/z | H − → L + 1 i (0 . | H → L + 1; H → L i (0 . V 5.74 8.5755 x | ( H − → L i (0 . | ( H − → L + 2 i (0 . | ( H − → L i (0 . | ( H − → L i (0 . | ( H − → L i (0 . | ( H − → L i (0 . (CONTINUED)2able S1 – CONTINUES FROM THE PREVIOUS PAGEPeak E (eV) f Polarization Wave functionVI 6.07 6.5321 y/z | ( H − → L i (0 . | ( H − → L i (0 . | ( H − → L i (0 . | ( H − → L i (0 . VII 6.46 1.3019 y/z | H → ( L + 3) i (0 . | H − → L i (0 . | H − → L i (0 . | ( H − → L + 1 i (0 . | H → ( L + 3) i (0 . | H → ( L + 3) i (0 . VIII 6.88 0.9803 y/z | H → ( L + 6) i (0 . | ( H − → ( L + 3) i (0 . | H → ( L + 6) i (0 . | H → L ; H → L i (0 . . Double Ring Isomer(H-3) (H-3) H-2 (H-1) (H-1) H H L (L+1) (L+1) (L+2) (L+2) (L+3) (L+3) (L+4) (L+4) (L+5) (L+5) Figure 2: Frontier molecular orbitals of double ring ( D d ) isomer of B (see Fig. 1b, of the main article), contributing to importantpeaks in its optical absorption spectra. The symbols H and L corresponds to HOMO and LUMO orbitals. Subscripts 1,2 etc. are usedto label the degenerate orbitals.Table S2: Many particle wave functions of the excited states associated with the peaks in the optical absorption spectra of the doublering ( D d ) isomer of B (Fig. 1b, main article). E denotes the excitation energy (in eV) of the state involved, and f stands for oscillatorstrength corresponding to the electric-dipole transition to that state, from the ground state. In the “Polarization” column, x, y, and zdenote the the direction of polarization of the absorbed light, with respect to the Cartesian coordinate axes defined in the figure below.In the “Wave function” column, | HF i indicates the Hartree-Fock (HF) configuration, while other configurations are defined as virtualexcitations with respect to it. H and L denote HOMO and LUMO orbitals, while the degenerate orbitals are denoted by additionalsubscripts 1, 2 etc. In parentheses next to each configuration, its coefficient in the CI expansion is indicated. Below, GS does not indicatea peak, rather it denotes the ground states wave function of the isomer. Peak E (eV) f Polarization Wave functionGS | HF i (0 . I 4.78 5.7331 y | ( H − → ( L + 1) i (0 . | ( H − → ( L + 1) i (0 . x | ( H − → ( L + 1) i (0 . | ( H − → ( L + 1) i (0 . z | H → ( L + 4) i (0 . | H − → L i (0 . II 5.90 1.8173 z | ( H − → ( L + 4) i (0 . | ( H − → ( L + 4) i (0 . III 6.41 3.9347 z | ( H − → ( L + 1) i (0 . | ( H − → ( L + 1) i (0 . VI 6.80 6.1390 y | ( H → ( L + 5) i (0 . | ( H − → ( L + 3) i (0 . x | ( H − → ( L + 3) i (0 . | H → ( L + 5) i (0 . . Chain IsomerH-9 H-8 H-7 H-6 H-5 H-4 H-3 H-2H-1 H L L+1 L+2 L+3 L+4 L+5L+6 L+7 Figure 3: Frontier molecular orbitals of chain ( C h ) isomer of B (see Fig. 1c, of the main article), contributing to important peaks inits optical absorption spectra. The symbols H and L corresponds to HOMO and LUMO orbitals. able S3: Many-particle wave functions of the excited states associated with the peaks in the optical absorption spectrum of the chain( C h ) isomer of B [Fig. 1c, main article], corresponding to excited states of B u symmetry. E denotes the excitation energy (ineV) of the state involved, and f stands for oscillator strength corresponding to the electric-dipole transition to that state, from theground state. In the “Polarization” column, x, y, and z denote the the direction of polarization of the absorbed light, with respectto the Cartesian coordinate axes defined in the figure below. In the “Wave function” column, | HF i indicates the Hartree-Fock (HF)configuration, while other configurations are defined as virtual excitations with respect to it. H and L denote HOMO and LUMOorbitals, while the degenerate orbitals are denoted by additional subscripts 1, 2 etc. In parentheses next to each configuration, itscoefficient in the CI expansion is indicated. Below, GS does not indicate a peak, rather it denotes the ground states wave function ofthe isomer. All transitions correspond to photons polarized in plane of the chain. Peak E (eV) f Wave functionGS | HF i (0 . | H → L ; H → L i (0 . I 1.64 2.3438 | H → L i (0 . | H − → L + 1 i (0 . II 3.16 3.1597 | H → L + 5 i (0 . | H − → L + 1 i (0 . | H − → L + 3 i (0 . | H − → L i (0 . III 3.63 9.6596 | H − → L i (0 . | H − → L + 1 i (0 . IV 3.94 1.2794 | H − → L + 2 i (0 . | H → L ; H − → L i (0 . V 4.49 48.6542 | H − → L + 1 i (0 . | H → L i (0 . VI 5.63 10.3584 | H − → L i (0 . | H − → L + 1 i (0 . VII 6.20 1.7806 | H − → L + 4 i (0 . | H → L + 2; H − → L + 1 i (0 . VIII 6.46 1.3157 | H − → L + 3 i (0 . | H − → L + 7 i (0 . IX 6.74 0.8341 | H − → L + 2 i (0 . | H − → L + 7 i (0 . able S4: Many-particle wave functions of the excited states corresponding to the peaks in the optical absorption spectrum of thechain ( C h ) isomer of B [Fig. 1c, main article], corresponding to excited states of A u symmetry. E denotes the excitation energy(in eV) of the state involved, and f stands for oscillator strength corresponding to the electric-dipole transition to that state, fromthe ground state. In the “Polarization” column, x, y, and z denote the the direction of polarization of the absorbed light, with respectto the Cartesian coordinate axes defined in the figure below. In the “Wave function” column, | HF i indicates the Hartree-Fock (HF)configuration, while other configurations are defined as virtual excitations with respect to it. H and L denote HOMO and LUMOorbitals, while the degenerate orbitals are denoted by additional subscripts 1, 2 etc. In parentheses next to each configuration, itscoefficient in the CI expansion is indicated. Below, GS does not indicate a peak, rather it denotes the ground states wave function ofthe isomer. All transitions correspond to photons polarized perpendicular to the plane of the chain. Peak E (eV) f Wave functionGS | HF i (0 . | H → L ; H → L i (0 . I 2.42 0.0170 | H − → L + 2 i (0 . | H → L ; H − → L i (0 . II 3.59 0.0932 | H → L + 4 i (0 . | H − → L + 1 i (0 . III 4.07 0.1111 | H − → L + 1 i (0 . | H → L ; H → L + 1 i (0 . | H − → L + 1 i (0 . | H → L ; H − → L + 1 i (0 . IV 4.80 0.1128 | H → L + 1; H − → L + 1 i (0 . | H → L + 6 i (0 . | H − → L + 3 i (0 . | H → L + 2; H − → L + 3 i (0 . V 5.88 0.0077 | H → L + 1; H − → L + 3 i (0 . | H → L + 5; H → L + 1 i (0 . VI 6.22 0.0153 | H → L + 2; H − → L + 1 i (0 . | H − → L + 4 i (0 . | H → L + 1; H − → L + 2 i (0 . | H → L ; H − → L + 3 i (0 . | H → L + 1; H − → L + 2 i (0 . | H → L + 2; H → L + 4 i (0 . . Atomic Coordinates In this section we present tables containing the atomic coordinates, in Å units, of various isomers studied in thiswork.
Atom x y zB 0.854596 0.493401 0.396007B -0.854596 0.493401 0.396007B 0.000000 -0.986802 0.396007B 0.000000 2.063378 -0.045471B 1.786937 -1.031689 -0.045471B -1.786937 -1.031689 -0.045471B 2.425806 0.485458 -0.175268B 1.633322 1.858081 -0.175268B -1.633322 1.858081 -0.175268B -2.425806 0.485458 -0.175268B -0.792484 -2.343538 -0.175268B 0.792484 -2.343538 -0.175268
Table S5: Optimized coordinates of Quasi-planar (C ) isomer Atom x y zB 0.000000 1.645803 0.755665B -1.425308 0.822902 0.755665B -1.425308 -0.822902 0.755665B 0.000000 -1.645803 0.755665B 1.425308 -0.822902 0.755665B 1.425308 0.822902 0.755665B 0.822902 1.425308 -0.755665B 1.645803 0.000000 -0.755665B 0.822902 -1.425308 -0.755665B -0.822902 -1.425308 -0.755665B -1.645803 0.000000 -0.755665B -0.822902 1.425308 -0.755665
Table S6: Optimized coordinates of Double ring (D ) isomer Atom x y zB -0.430275 4.400542 0.000000B 0.905913 3.618456 0.000000B -0.755353 2.873999 0.000000B 0.914514 2.038590 0.000000B -0.777694 1.210978 0.000000B 0.755353 0.376209 0.000000B -0.755353 -0.376209 0.000000B 0.777694 -1.210978 0.000000B -0.914514 -2.038590 0.000000B 0.755353 -2.873999 0.000000B -0.905913 -3.618456 0.000000B 0.430275 -4.400542 0.000000
Table S7: Optimized coordinates of Chain (C ) isomer .4. Hexagons Atom x y zB 0.000000 0.000000 0.844921B 0.000000 1.517663 0.000000B 0.000000 -1.517663 0.000000B 0.000000 -1.664249 1.586458B 0.000000 -0.768140 2.882657B 0.000000 0.768140 2.882657B 0.000000 1.664249 1.586458B 0.000000 0.000000 -0.844921B 0.000000 -0.768140 -2.882657B 0.000000 0.768140 -2.882657B 0.000000 1.664249 -1.586458B 0.000000 -1.664249 -1.586458
Table S8: Optimized coordinates of Hexagons (D ) isomer Atom x y zB 0.000000 2.015258 0.000000B 0.000000 0.000000 1.268281B 0.000000 0.000000 -1.268281B 0.000000 -2.015258 0.000000B -1.682909 0.000000 -0.912871B -1.490298 -1.389700 0.000000B -1.490298 1.389700 0.000000B -1.682909 0.000000 0.912871B 1.682909 0.000000 0.912871B 1.490298 -1.389700 0.000000B 1.490298 1.389700 0.000000B 1.682909 0.000000 -0.912871
Table S9: Optimized coordinates of Quad (D ) isomer Atom x y zB 0.000000 1.574917 0.000000B 1.574917 0.000000 0.000000B -1.574917 0.000000 0.000000B 0.000000 -1.574917 0.000000B -0.862562 0.862562 1.264540B -0.862562 -0.862562 1.264540B 0.862562 0.862562 1.264540B 0.862562 -0.862562 1.264540B 0.862562 0.862562 -1.264540B 0.862562 -0.862562 -1.264540B -0.862562 0.862562 -1.264540B -0.862562 -0.862562 -1.264540
Table S10: Optimized coordinates of Pris-3-square-1 (D ) isomer .7. Pris-3-square-2 Atom x y zB 0.000000 1.870473 0.000000B 1.870473 0.000000 0.000000B -1.870473 0.000000 0.000000B 0.000000 -1.870473 0.000000B -0.896666 0.896666 0.994197B -0.896666 -0.896666 0.994197B 0.896666 0.896666 0.994197B 0.896666 -0.896666 0.994197B 0.896666 0.896666 -0.994197B 0.896666 -0.896666 -0.994197B -0.896666 0.896666 -0.994197B -0.896666 -0.896666 -0.994197
Table S11: Optimized coordinates of Pris-3-square-2 (D ) isomer Atom x y zB 0.000000 1.069852 0.924715B -0.926519 -0.534926 0.924715B 0.926519 -0.534926 0.924715B 0.000000 1.069852 -0.924715B 0.926519 -0.534926 -0.924715B -0.926519 -0.534926 -0.924715B 0.000000 0.989584 2.529660B -0.857005 -0.494792 2.529660B 0.857005 -0.494792 2.529660B 0.000000 0.989584 -2.529660B 0.857005 -0.494792 -2.529660B -0.857005 -0.494792 -2.529660
Table S12: Optimized coordinates of Parallel triangular (D ) isomer Atom x y zB 0.000000 1.237919 0.000000B 1.237919 0.000000 0.000000B -1.237919 0.000000 0.000000B 0.000000 -1.237919 0.000000B -1.237919 2.340594 0.000000B 1.237919 2.340594 0.000000B 2.340594 1.238828 0.000000B 2.340594 -1.238828 0.000000B -2.340594 -1.238828 0.000000B -2.340594 1.238828 0.000000B 1.238828 -2.340594 0.000000B -1.238828 -2.340594 0.000000
Table S13: Optimized coordinates of Planar square (D ) isomer .10. Perfect Icosahedron Atom x y zB 0.000000 0.865000 1.399599B 0.000000 0.865000 -1.399599B 0.000000 -0.865000 1.399599B 0.000000 -0.865000 -1.399599B 1.399599 0.000000 0.865000B -1.399599 0.000000 0.865000B 1.399599 0.000000 -0.865000B -1.399599 0.000000 -0.865000B 0.865000 1.399599 0.000000B 0.865000 -1.399599 0.000000B -0.865000 1.399599 0.000000B -0.865000 -1.399599 0.000000
Table S14: Optimized coordinates of perfect Icosahedron (I h ) isomer Atom x y zB 0.000000 0.852920 -1.337059B 0.000000 0.852920 1.337059B 0.000000 -0.852920 -1.337059B 0.000000 -0.852920 1.337059B -1.394554 0.000000 -0.843230B 1.394554 0.000000 -0.843230B -1.394554 0.000000 0.843230B 1.394554 0.000000 0.843230B -0.897896 1.466072 0.000000B -0.897896 -1.466072 0.000000B 0.897896 1.466072 0.000000B 0.897896 -1.466072 0.000000
Table S15: Optimized coordinates of distorted Icosahedron (D ) isomer
5. Nature of the Wave Function of the Lowest Singlet State of B Icosahedron
We performed calculations on the lowest singlet state of the B isomer with perfect icosahedral symmetry ( I h ),by employing multi-reference singles-doubles CI (MRSDCI) level of theory, and cc-pVDZ basis set, to understandthe nature of its wave function. For the purpose we used the HF orbitals, and the seven reference configurationslisted in the table below. From the magnitudes of the coefficients of the reference configurations in the CI expansionof the wave function, it is obvious that the closed-shell HF reference state dominates the wave function. Therefore,any single-reference approach such as CCSD, using the HF state as the reference, will give reasonable results forthis state. Table S16: Many-particle wave function of the lowest singlet state of isomer of B with perfect icosahedral symmetry ( I h ) (Fig. 1eof the main article). | HF i indicates the closed-shell Hartree-Fock (HF) configuration, while other configurations are defined as virtualexcitations with respect to it. H and L denote HOMO and LUMO orbitals. In the parentheses next to each configuration, its coefficientin the CI expansion is indicated. Dominant configurations in the MRSDCI wave function | HF i (0 . | H → L + 3; H → L + 3 i (0 . | H → L + 3; H − → L + 6 i (0 . | H → L + 2; H − → L + 3 i (0 . | H → L + 3; H − → L + 3 i (0 . | H → L + 3; H − → L + 1 i (0 . | H − → L + 4; H − → L + 6 i (0 .0901)