The optimized GMs of bilayer C1 B50 (I), C2h B52 (II), C1 B56 (IV), and C2v B58 (V) at the PBE0/6-311 + G(d) [10] level are shown in Fig. 1 in comparison with that of the previously reported C2 B54 (III), with more alternative low-lying isomers depicted in Fig. S1. The configurational energy spectra of (a) B50, (b) B52, (c) B56, and (d) B58 at PBE0/6-311 + G(d) level are collectively shown in Fig. S2. Table S1 tabulates the calculated lowest vibrational frequencies, HOMO-LUMO energy gaps, cohesive energies per atom, NICS values, [11] and the number of interlayer B-B σ bonds formed between the top and bottom layers for the B48–B72 bilayer series. The cohesive energies per atom Ecoh = (E2n–2nEB)/2n for the B2n series between B48–B74 in different structural motifs are compared in Fig. 2. The calculated second-order energy differences Δ2E2n = (E2n + E2(n+1))–2E2(n+1) and HOMO–LUMO energy gaps (ΔEgap/eV) of the bilayer B48–B72 clusters are depicted in Fig. 3. The structural evolution of the bilayer B48–B72 series derived from the experimentally observed B48 is demonstrated in Fig. 4.
Structures and Stabilities
We start from B50, the much concerned even-numbered boron cluster which was predicted in 2017 to have a quasi-planar C2v structure with two adjacent hexagonal holes at the center via unbiased GM searches. [10] However, as shown in Fig. S1a and Fig. S2a, extensive GM searches starting from manually constructed initial structures derived from the experimentally known bilayer D2h B48 performed in this work indicate that the slightly distorted bilayer C1 B50 (I) (1A) is more stable than the planar C2v B50 by 0.52 eV and 0.12 eV at PBE0/6-311 + G(d) and TPSSh/6-311 + G(d) levels, respectively. The second lowest-lying bilayer isomer Cs B50 lies only about 0.01 eV higher in energy than the C1 GM (Fig. S2a). The two close-lying isomers C1 B50 and Cs B50 are practically iso-energetic isomers with extremely similar geometries which are expected to coexist in gas-phase experiments. Both C1 B50 (I) and Cs B50 can be obtained by adding two B atoms on one edge of the observed D2h B48 to form a B5 pentagonal hole at the bottom-left corner (Fig. 1 and Fig. S2a). The third and fourth bilayer isomers with the relative energies of 0.06 eV and 0.16 eV respectively also appear to be obviously more stable than the planar C2v B50. It is interesting to notice that, similar to the previously reported C2 B54, C2h B60, and C1 B62, [18] the four lowest-lying bilayer isomers of B50 all contain a B38 bilayer hexagonal prism at the center (highlighted in a black box in Fig. S1a) surrounded by 12 boron atoms on the waist in different arrangements. More intriguingly, two interlayer B-B σ bonds are formed to function as two “nails” to bind the top and bottom layers together, helping to stabilize the bilayer systems effectively. The commonly concerned tubular C1 B50 and cage-like C1 B50 isomers are found to be much less stable (by 2.68 eV and 3.53 eV respectively) than the bilayer GM at PBE0/6-311 + G(d) (Fig. S1a).
With two more boron atoms added in on the opposite side, the high-symmetry bilayer C2h B52 (II) (1Ag) is generated which is the well-defined GM of system with two B5 pentagonal windows on the waist along the long diagonal. The six lowest-lying isomers of B52 within 1.0 eV all appear to have bilayer structures, while the first planar C1 B52 and first tubular C1 B52 lie 1.21 eV and 2.26 eV higher than the bilayer GM at PBE0/6-311 + G(d), respectively (Fig. S2b and Fig. S1b). Hexagonal holes start to appear for the first time in the second, third, fourth, and fifth bilayer isomers of B52 on both the top and bottom layers.
B56 is another species much concerned in literature. The high-symmetry quasi-planar C2v α-B56 with two adjacent hexagonal holes was predicted in 2015 in Ref.20. However, adding two boron atoms to the previously reported bilayer C2 B54 (III) [18] at one edge generates the slightly distorted bilayer C1 B56 (IV) (1A) which appears to be 0.60 eV and 0.28 eV more stable than the previously predicted quasi-planar C2v B56 (1A1) [20] at PBE0/6-311 + G(d) and TPSSh/6-311 + G(d) levels, respectively (Fig. S2c and Fig. S1c). The second lowest-lying bilayer Cs B56 (1A′) (Fig. S2c) lying 0.01 eV higher than B56 (IV) possesses a small imaginary vibrational frequency at 45.5i cm− 1 (a″ mode) which leads to the bilayer C1 GM during structural relaxation. It has an extremely similar geometry with B56 (IV) and is expected to be the vibrationally averaged structure of the system. The penta-ring tubular C1 B56 and cage-like C1 B56 are found to lie significantly higher than the bilayer GM by 1.64 eV, and 4.33 eV at PBE0/6-311 + G(d) (Fig. S1c), respectively.
With two more boron atoms added in to B56 (IV) on the opposite side, the high-symmetry bilayer C2v B58 (V) with two symmetrically distributed pentagonal windows about the C2 molecular axis is achieved. B58 (V) as the GM of the system is the largest bilayer species considered in this work. It is followed by thirteen closely-lying bilayer isomers within 0.86 eV (Fig. S2d and Fig. S1d). The first quasi-planar and tubular isomers are found to be 0.88 eV and 2.44 eV less stable than GM at PBE0/6-311 + G(d) (Fig. S1d), respectively. The intensive distribution of thirteen lowest-lying bilayer isomers in B58 is unique in medium-sized boron clusters reported to date.
The cohesive energies per atom (Ecoh), second-order energy differences (Δ2E), and HOMO-LUMO energy gaps (ΔEgap) are used as criteria to compare the relative stabilities of B2n clusters in different structural motifs in the size range between B48–B72. As clearly shown in Fig. 2, relative to their core-shell, planar, tubular, and cage-like counterparts, the bilayer isomers possess universally the highest Ecoh values in the whole size range between B48–B74, with the cohesive energies per atom increasing almost monotonously from Ecoh = 5.3433 eV/atom at bilayer D2h B48 to Ecoh = 5.4132 eV/atom at bilayer Ci B72. Even-numbered B2n clusters thus all possess bilayer GMs in thermodynamics in a surprisingly large size range between B48–B72. Figure 3 indicates that the calculated HOMO-LUMO gaps ΔEgap (a) and second-order energy differences Δ2E (b) of the bilayer B2n GMs exhibit similar variation trends with increasing cluster sizes. C2 B54, C2 B64, D2 B68, and Ci B72 as the four local maxima with ΔEgap = 1.98, 1.54, 1.47, and 1.55 eV, respectively, are predicted to be chemically the most stable bilayer species to be targeted in future spectroscopic measurements, while C2h B60 and D2 B66 as two local minima with ΔEgap = 1.14 and 1.13 eV, respectively, are anticipated to be chemically unstable (Table S1).
Extensive BOMD simulations indicate that bilayer B50 (I), B52 (II), B56 (IV), and B58 (V) are also highly dynamically stable, with the small average root-mean-square-deviations of RMSD = 0.15, 0.14, 0.14, and 0.18 Å and the maximum bond length deviations of MAXD = 0.69, 0.60,0.61, and 0.97 Å at 1000 K (Fig. S3), respectively, similar to the situations in the previously reported bilayer C2 B54, C2h B60, C1 B62, C2 B64, D2 B66, D2 B68, C1 B70, and Ci B72. [18, 19] No low-lying isomers were observed during the simulations in 30 ps.
Structural Evolution in Bilayer B48–B72
To better appreciate the bilayer GM geometries of B42–B72 obtained to date, we portray their general structural evolution in Fig. 4, starting from the experimentally observed D2h B48. The theoretically predicted C2 B54, C2h B60, and C1 B62 in the first row all contain a B38 bilayer hexagonal prism at the center, with one, two, and three B6 hexagonal windows on the waist, respectively. More specifically, C2 B54 can be constructed from D2h B48 by adding a B6 hexagonal window on the upper side, C2h B60 is built by adding one more B6 hexagonal window to C2 B54 on the opposite side. With two more B atoms added in, the slightly twisted bilayer C1 B62 is generated with three B6 hexagonal windows at three corners.
Both bilayer C1 B50 and C2h B52 in the first column with a B38 bilayer hexagonal prism at the center can be derived directly from D2h B48. Adding two B atoms to D2h B48 on one edge generates both C1 B50 (I) and the second lowest-lying Cs B50 which contain a B5 pentagonal hole at the bottom-left corner. Addition of two more B atoms to C1 B50 on the opposite side results in the high-symmetry C2h B52 with two symmetrically distributed B5 pentagonal holes along the long diagonal. Similarly, both C1 B56 and C2v B58 in the second column with a B38 bilayer hexagonal prism at the center are derivatives of the previously reported C2 B54. C1 B56 can be constructed by adding two B atoms to C2 B54 on the left side of the C2 molecular axis to form a B5 pentagonal hole at the bottom-left corner, while the high-symmetry C2v B58 is built by adding two more boron atoms on the right side to form two symmetrically distributed B5 pentagonal holes about the C2 molecular axis.
Bilayer C2 B64, D2 B68, C1 B70, and Ci B72 in the third column and D2 B66 in the fourth column with an elongated B46 bilayer hexagonal prism at the center can be generated from the previously reported C2h B60. C2 B64 is built by adding one B4 rhombus to C2h B60 on the upper side, while the high-symmetry D2 B68 contains two symmetrically distributed B4 rhombuses on the upper and lower sides along the C2 main molecular axis. The high-symmetry D2 B66 can be generated by adding two B-B-B chains on the upper and lower sides of C2h B60 over two B6 hexagonal holes simultaneously. The elongated C1 B70 can be obtained from D2 B68 by adding two boron atoms to one edge of the system, while Ci B72, the largest bilayer species reported to date, can be achieved by addition of two more boron atoms on the opposite side of the inversion center to form two symmetrically distributed B6 hexagonal windows on two opposite sides. Overall, the bilayer B48–B72 series all contain a B38 bilayer hexagonal prism or an elongated B46 bilayer hexagonal prism at the center, sealed by suitable numbers of B-B-B chains, B4 rhombuses, B5 pentagonal holes or B6 hexagonal windows or their combinations on the waist.
Bonding Analyses
Detailed AdNDP [28–30] analyses are performed on the high-symmetry C2h B52 (II) and C2v B58 (V) in Fig. 5. As indicated in Fig. 5a, C2h B52 (II) possesses 64 σ-bonds in total, including 2 effective interlayer B-B 2c-2e σ bonds between four inward-buckled B atoms on the top and bottom layers with the occupation number of ON = 1.88 |e|, 26 2c-2e B-B σ bonds on the waist with ON = 1.68–1.79 |e|, 28 3c-2e σ bonds on the top and bottom layers with ON = 1.76–1.93 |e|, 4 4c-2e σ bonds at two corners along one diagonal with ON = 1.86 |e|, and 4 5c-2e along the other long diagonal with ON = 1.88 |e|. The remaining 14 delocalized π bonds could be divided into four groups in the overall symmetry of C2h, including 2 6c-2e π bonds on the left and right corners along one diagonal with ON = 1.73 |e|, 2 11c-2e π bonds at two corners along the other long diagonal with ON = 1.60 |e|, 8 6c-2e π bonds on the top and bottom layers with ON = 1.60–1.70 |e|, and 2 9c-2e π bonds over two close-packed B9 central units on the top and bottom layers with ON = 1.78 |e|.
C 2v B58 (V) exhibits a similar bonding pattern with B52 (II) (Fig. 5b). It contains 3 effective interlayer B-B 2c-2e σ bonds between six inward-buckled B atoms on the top and bottom layers, 22 2c-2e σ bonds on the waist, and 44 3c-2e and 2 5c-2e σ bonds evenly distributed on the top and bottom layers, in an overall symmetry of C2h. The corresponding π system includes 8 6c-2e π bonds on the top and bottom layers, 4 6c-2e π bonds over the central region, 2 11c-2e π bonds over the upper part of the bilayer structure on the left and right, and 2 12c-2e π bonds over the lower part on the left and right.
Obviously, both C2h B52 (II) and C2v B58 (V) follow the universal σ + π double delocalization bonding pattern over the bilayer surface of the systems, similar to the situations observed in bilayer B54, B60, B62, B64, B66, B68, B70, and B72 [18, 19] and cage-like Bnq borospherenes (n = 36–42, q = n-40). [12–17] Similar bonding patterns exist in C1 B50 (I) and C1 B56 (IV). Such bonding patterns render three-dimensional aromaticity to C1 B50 (I), C2h B52 (II), C1 B56 (IV), and C2v B58 (V), as evidenced by the calculated negative nuclear independent chemical shift values of NICS = − 22.9, − 18.9, − 32.9 and − 19.7 ppm at their geometrical centers (Table S1), respectively.
Spectral Simulations
To facilitate future spectral characterizations of these bilayer species, we simulate the infrared (IR), Raman, and UV-vis spectra of C2h B52 (II) and C2v B58 (V) in Fig. S4. B52 (II) exhibits five strong IR peaks at 456 (bu), 655 (bu), 1087 (bu), 1185 (bu), and 1208 (bu) cm− 1, while B58 (V) possesses two intensive IR active modes at 1153 (a1) and 1203 (b2) cm− 1, respectively. B52 (II) has three major symmetrical Raman active vibrational modes at 434 (ag), 695 (ag), and 1171 (ag) cm− 1 and B58 (V) exhibits four strong Raman bands around 266 (a1), 472 (a1), 699 (a1), and 1241 (a1) cm− 1, respectively.
The simulated UV-vis spectra of C2h B52 (II) and C2v B58 (V) lie between 200–650 nm, with the main absorption peaks lying at 268 nm, 282 nm, 313 nm, 432 nm, and 639 nm in C2h B52 (II) and at 291 nm, 311 nm, 352 nm, 378 nm, 577 nm, and 645 nm in C2v B58 (V), respectively (Fig. S4). The most strong UV absorption peak of C2h B52 (II) occurs at 313 nm (Bu) which mainly originates from the S0→S279 electronic excitation with major contributions from HOMO→LUMO + 12 (27%) transition (Table S2). The strongest UV absorption peak of C2v B58 (V) at 291 nm (B1) corresponds to the electronic transition of S0→S395 with major contributions from HOMO–22→LUMO + 1 (26%). (Table S2).