Structure and Electronic Properties of Neutral and Negatively Charged

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Structure and Electronic Properties of Neutral and Negatively Charged RhB Clusters (n = 3-10): A Density Functional Theory Study n

Peifang Li, Tingting Mei, Linxia Lv, Cheng Lu, Weihua Wang, Gang Bao, and Gennady Lavrenty Gutsev J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b06123 • Publication Date (Web): 08 Aug 2017 Downloaded from http://pubs.acs.org on August 11, 2017

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The Journal of Physical Chemistry

Structure and Electronic Properties of Neutral and Negatively Charged RhBn Clusters (n = 3−10): A Density Functional Theory Study Peifang Li1, Tingting Mei1, Linxia Lv2, Cheng Lu*,2,3, Weihua Wang1, Gang Bao1, and Gennady L. Gutsev∗,4 1

College of Physics and Electronic Information, Inner Mongolia University for

Nationalities, Tongliao 028043, China 2

Department of Physics, Nanyang Normal University, Nanyang 473061, China

3

Department of Physics and High Pressure Science and Engineering Center,

University of Nevada, Las Vegas, Nevada 89154, United States 4

Department of Physics, Florida A&M University, Tallahassee, FL 32307, USA

Abstract The geometrical structure and electronic properties of the neutral RhBn and singly negatively charged RhBn– clusters are obtained in the range of 3 ≤ n ≤ 10 using the unbiased CALYPSO structure search method and density functional theory (DFT). A combination of the PBE0 functional and the def2-TZVP basis set is used for determining global minima on potential energy surfaces of the Rh doped Bn clusters. The photoelectron spectra of the anions are simulated using the time-dependent density functional theory (TD-DFT) method. Good agreement between our simulated and experimental obtained photoelectron spectra for RhB9– provides support to the validity of our theoretical method. The relative stability of the ground-state RhBn and RhBn– clusters are estimated using the calculated binding energies, second-order total energy differences, and HOMO−LUMO gaps. It is found that RhB7 and RhB8– are the most stable species in the neutral and anionic series, respectively. The chemical bonding analysis reveals that the RhB8–cluster possesses two sets of delocalized σ and

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π bonds. In both cases, the Hückel 4N + 2 rule is fulfilled and this cluster possesses both σ and π aromaticities.

Introduction Since the discovery of different carbon allotropes such as fullerenes, nanotubes, and graphenes,1-4 the search for similar nanostructures composed of other materials became an increasingly intriguing subject. Special attention was paid to boron which is a lighter neighbor of carbon in the Periodic Table. In attempts of discovering new nanomaterials with novel structures and properties, different boron materials have been intensively studied.5-25 However, unlike carbon, boron is an electron-deficient element which results in instability of large planar structures composed entirely of boron atoms. The structure and chemical properties of a large number of boron clusters Bn0/+/− have been investigated11-25 using different experimental and theoretical methods. The studies revealed a rich structural diversity of the clusters ranging from planar and quasi-planar structures to three-dimensional (3D) cage-type structures. The lowest total energy states of neutral and anionic boron clusters were found to possess planar or quasi-planar geometrical structures up to B2015 and B36–,23 although some low-lying states of B28– and B29− also possess 3D cage-like geometries.20,21 According to the most recent studies, the ground-state B36 and B35– clusters possess planar geometries.22,23 In the positively charged Bn+ series, the transition from planar to 3D geometrical structures of the lowest total energy states occurs at n = 16.25 Doping of pure clusters with transition metal (TM) atoms leads to a new class of compounds with novel geometrical topologies and interesting chemical and magnetic


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properties.26-34 A number of boron clusters doped with a single TM atom, have been observed and studied experimentally. It was found that the TM atoms prefer internal positions in the monocyclic boron rings which leads to the formation of spectacular wheel-type geometrical structures.26,28-30 In joint experimental and theoretical studies, it was shown that both Rh@B90/− and Ir@B90/− clusters possess planar structures.26 Li et al. pointed out that the neutral Rh@B9 and Ir@B9 clusters possess D9h symmetry and their geometrical structures are stabilized by completely delocalized σ and π electrons despite this puts strain between the central TM atom and the outer boron ring.26 The structure and chemical bondings of Ta©B10– and Nb©B10– have been investigated in a joint photoelectron spectroscopy (PES) and theoretical study, as well as by using the Adaptive Natural Density Partitioning (AdNDP) analysis.28 It was found that both of these anions are doubly σ and π aromatic because of satisfying the Hückel 4N + 2 rule.28 Romanescu and co-workers29,30 have optimized the Co@B8– and Ru@B9– anion states with wheel-type geometries and coordination numbers of eight and nine, respectively. The CB62–, CB7– and C3B4 clusters with wheel-type geometries have been studied both experimentally and theoretically; and it was found that carbon atoms avoid the central positions in geometrical structures of these species.35,36 In the case of aluminum-doped boron clusters, it was shown that an aluminum atom occupies peripheral positions in the AlB6-11– systems.37-39 Therefore, one can expect that the ground state of a small-to-medium sized boron cluster doped with a single metal atom possesses a planar wheel-type geometrical structure. It is worth noting that a TM atom generally prefers to occupy the central position in a boron cluster whereas non-TM atoms tend to occupy peripheral positions.



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Rh being a very important TM element has been widely studied in the context of catalysis.40,41 However, relatively little is known about the structural and electronic properties of Rh-doped boron clusters. These interesting findings ignited our interest into the structure of boron clusters doped with an Rh atom, which is isovalent with the Co atom. Of particular interest is to find answers to the following questions: (i) what are the ground-state geometrical structures of RhBn and RhBn– clusters? (ii) is it favorable to replace а central boron atom with a Rh atom? (iii) will a cluster have a planar wheel-type structure if a Rh atom replaces the central boron atom? (iv) can a Rh atom participate in the delocalized bonding and a Rh-doped boron cluster be aromatic? With this aim, we have carried out an extensive search for the ground-state structures of the neutral and singly negatively charged RhBn and RhBn– clusters in the range of n = 3 – 10 using the CALYPSO global minima search on the potential energy surfaces obtained from our density functional theory calculations. Next, we have computed the electronic properties of the RhBn– anions and simulated their photoelectron spectra. We have also explored the stabilizing mechanisms stipulated by the structural properties and peculiarities of chemical bonding of the neutral and anionic RhBn clusters.

Computational Methods Our search for the optimal geometrical structures of the lowest total energy states of the neutral and anionic rhodium-doped boron clusters RhBn and RhBn– (n = 3–10) was performed using the crystal structure analysis by particle swarm optimization (CALYPSO) method.42-47 It provides a global version of the particle swarm optimization (PSO) algorithm intended for the search of global minima on the free-energy surfaces of both periodic and non-periodic systems. The advantage of the CALYPSO is that it can identify the most stable structures based only on the chemical composition. It is found to be successful in predictions of the ground-state geometries


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in variety of different systems.48-51 In the CALYPSO method, trial geometrical structures are ordered in generations. Each generation contains 50 structures, 60% of which is produced by the PSO algorithm, whereas the rest is generated randomly. We performed 30 generations to produce at least 1200 structurally different isomers for each cluster. Among these generated isomers, the isomers whose total energies fell into a 3.00 eV interval from the state with the lowest total energy are reoptimized using the PBE0/def2-TZVP level52,53 of the DFT. The choice of the PBE0/def2-TZVP method is based on the proof of its accuracy in the previous studies of TM-doped boron clusters.31 We used a spin-polarized approach and computed excited states up to septets and octets in the neutral and anionic series, respectively. No symmetry constraints were imposed during geometry optimizations. Each optimization was followed by computations of harmonic vibrational frequencies to make sure that an optimized state corresponds to a minimum on the potential energy surface. All calculations were performed using the Gaussian 09 suite.54 The photoelectron spectra of the RhBn– clusters were simulated using the TD-DFT method. To better understand the chemical bonding, an electron localization analysis was carried out by using the AdNDP method.55

Geometric Structure Using the CALYPSO method, a large number of isomers of the neutral and anionic RhBn clusters are obtained in the range of 3 ≤ n ≤ 10. All geometrical structures reported in the previous experimental and theoretical studies, are taken into account as well. We optimize all prospective candidates for the low-lying isomers, and the lowest total energy isomers found for the neutral and charged species are presented in Figure 1. The neutral and anionic isomers with the number of boron atoms n are labeled as na0/–, nb0/– and nc0/– in the order of decreasing total energy of the corresponding



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isomer states. The assigned electronic states, point group symmetries, and relative energies of all isomers displayed in Figure 1 are presented in Table 1. As can be seen, most of the ground and low-lying structures favor the low-spin states. According to Figure 1, both neutral and anionic RhBn clusters prefer planar structures when n ≤ 5, except for the 3a– case, where the Rh atom is on the top of the boron triangle. A transition from planar to 3D in the ground-state structures occurs at n = 6. According to the results of our optimizations, the ground state of RhB3 is 2B1 and followed by the 1A1 state. The ground state of the RhB3– cluster is found to have a non-planar structure of C3v symmetry whereas the first excited state 3b–,, which is higher in total energy than the ground state by 1.60 eV, has a planar geometrical structure. The ground states of RhB4 and RhB4– belong to the same Cs point group of symmetry and have similar geometries, where the Rh atom is located at the periphery. The 4b, 4c, 4b– and 4c– isomers of RhB4 and RhB4–, respectively, all possess non-planar structures. The ground-state RhB5 cluster possesses a planar geometry, which corresponds to the ground-state geometry of RhB4 with a B atom attached to its left side. The 5b and 5c isomers are higher in total energy than the ground state by 0.71 eV and 1.44 eV, respectively. The ground-state geometry of the RhB5– cluster is presented by a planar frame of Cs symmetry. Its isomers 5b– and 5c– are also of Cs symmetry and are higher in total energy by 0.66 eV and 1.48 eV, respectively. The geometrical structure of ground-state RhB6 is quasi-planar, with one B atom being above the planar ring formed by the rest of atoms. The first excited state 6b, which is higher in total energy


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than the ground state by 0.71 eV, possesses a planar ladder-type geometry. The ground-state geometry of the RhB6– anion is similar to that of its neutral parent, i.e., the attachment of an extra electron does not lead to the change in topology of the neutral parent. This is similar to the preceding pairs with n = 4 and 5, but not with n = 3. The ground state of neutral RhB7 (7a) possesses a bipyramydal C6v geometrical structure which is 1.60 eV more stable than the first excited state 7b with a C2v planar structure and by 2.41 eV than the second excited state 7c. The structures 7a–, 7b– and 7c– of the anion are similar to the corresponding 7a, 7b and 7c structures of neutral RhB7. The C1 symmetry RhB8 cluster can be viewed as a B atom capping the ring of the RhB7 cluster (7a), forming an umbrella-shaped structure. The geometries of the 8b and 8c isomers can be obtained from those of the 7b and 7c isomers in a similar manner. The attachment of an extra electron to RhB8 results in the ground state geometry of the RhB8– anion which is similar to that of the second excited state of neutral RhB8, whereas the 8b– and 8c– geometrical structures are similar to the 8a and 8b structures, respectively. The ground-state geometry of the RhB9 cluster (9a) can be viewed as 3D structure formed by capping the 7b structure with a B2 dimer. The 9b and 9c isomer states are less stable than the ground state by 0.73 eV and 2.91 eV, respectively. The ground state of the RhB9– cluster (9a–) is found to be planar with an Rh atom being inside a monocyclic B9 ring. This is in agreement with the result of Li et al.26 obtained by using the PBE0/Rh/Stuttgart’97/B/aug-cc-pVTZ level of theory. It is worth noting that the 9b– structure of Cs symmetry is similar to the 9a structure of the neutral ground



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state. The global minima on the potential energy surfaces of both RhB10 and RhB10– do correspond to the geometrical structures where the Rh atom is placed above the planar B10 frame. The 10b and 10c geometries of the neutral excited states are similar to the 10b– and 10c– structures of the corresponding anion states. In summary, we can conclude that the preference in planar 3D structures strongly depends on the size and charge of a RhBn cluster.

3.2. Photoelectron Spectra of the RhBn– Clusters Photoelectron spectra of the RhBn– anions are simulated using the excitation energies and oscillator strengths obtained from our TD-DFT computations performed at ground-state geometries of the RhBn– anions. The peaks in the spectra presented in Figure 2 correspond to the transitions from the anion ground state to the neutral ground and excited states. As can be seen in Figure 2, the first peak in each spectrum, which represents the vertical detachment energy (VDE), is located inside the interval of 0-3 eV except for the spectra of RhB8– and RhB10–. The spectrum of RhB3– possesses three major and two weak peaks, with the center of the lowest energy peaks locates at 1.60 eV. The spectrum of RhB4– has three strong and three weak peaks and the lowest energy peak shifts to 2.20 eV, i.e., by 0.60 eV with respect to the first peak of the RhB3– spectrum. The spectrum of RhB5– also contains six peaks with the first major peak located at 2.53 eV which is followed by two peaks located at 3.30 eV and 4.85 eV, respectively. There are three main peaks in the spectrum of RhB6– with the first peak located at 2.09 eV. The second peak is presented by four overlapping features with different amplitudes, and the third peak possesses the largest amplitude. In the spectrum of


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RhB7–, the first two peaks are separated by a large gap of 2.05 eV, and the third peak is presented by a superposition of two features. The VDE on the simulated spectrum of RhB8− emerges very late (at 3.69 eV), followed by a distinct peak at about 5.05 eV. The first peak in our RhB9– spectrum is located at 2.88 eV and is followed by three features at 4.10 eV, 4.40 eV and 5.00 eV. Comparison of our and experimental spectra shows that the essential features of the experimental spectrum26 are successfully reproduced in our TD-DFT simulation. The spectrum of the RhB10– anion is dominated by a strong peak at about 5.25 eV with the first peak of a much smaller intensity appearing at 3.44 eV. Good agreement between simulated and experimental spectra for the RhB9– anion allows one to expect a similar quality for other anions considered in the present work.

3.3. Relative Stabilities The thermodynamic stability of a neutral or anionic RhBn cluster can be related to their average binding energy (Eb) per atom defined as Eb =  E ( Rh Q ) + nE ( B ) − E ( RhBn Q )  / ( n + 1) , Q = 0, ±1.

(1)

where E(Rh0/−), E(B), and E(RhB0/−) are the total energies of the corresponding species. In order to test the reliability of our calculations, we have calculated the bond lengths and the binding energies of the B2 and RhB clusters and compared with the available theoretical and experimental value.56,57 The theoretical results as well as the available experimental data are listed in Table S1 (in the Supporting Information). The calculated results are in excellent agreement with the theoretical and experimental data, which strengthens the reliability of our computational method. The average binding energies per atom computed for the ground states of the



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RhBn and RhBn– clusters for n = 3–10 are presented in Figure 3(a) and listed in Table 2. It can be seen that the average binding energy per atom increases monotonically when the cluster size grows and both neutral and anionic curves show similar size dependence. Besides, it can also be seen that the Eb values of RhBn– are larger than those of the RhBn clusters. This can be related to lower total energies of the anion with respect to the total energies of their neutral parent. Furthermore, it is worth noting that the localized maximum peaks of the Eb curve for RhB7 and RhB8– clusters imply that RhB7 and RhB8– clusters have strong relative stability. Similar results occur in the detachment energy, which have been shown in the Figure S1. An important parameter, which reflects the relative stability of species in the neutral and anionic ground-state series, is the second-order difference of the total energy (Δ2E), which is calculated according to the equation

Δ 2 E = E ( RhBn −1Q ) + E ( RhBn +1Q ) − 2 E ( RhBn Q ) , Q = 0, ±1.

(2)

The Δ2E values for all RhBnQ (n = 3−10; Q = 0, −1) clusters are displayed in Figure 3(b) and summarized in Table 2. It is worth noting that the Δ2E curves of neutral and anionic clusters show a similar oscillating trend in the range of n = 4−7, while the curves show different trends for n = 8 and 9. Two prominent peaks of the Δ2E curves are observed at n = 7 and 8 in the neutral and anionic series, respectively. This indicates that the RhB7 and RhB8– clusters are more stable than their neighbors in the corresponding series. The energy gap between the highest occupied and lowest unoccupied molecular orbitals (Egap) of a species is often considered to be related to its thermodynamic


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stability. In particular, larger Egap values are expected to reflect a higher thermodynamic stability. The values computed for the ground-state RhBn and RhBn– clusters are listed in Table 2 and presented in Figure 3(c) as a function of the cluster size n. As we can seen, there are even-odd oscillations in the range of n = 3−10 for the neutral RhBn series, and the presence of a maximum at n = 7 is in accord with the predicted high stability of RhB7 on the basis of the of Δ2E peak at this n value. The anionic curve shows a different oscillating trend and possesses a peak at n = 8. Note that the anionic Δ2E curve has also the maximum at this value of n.

3.4. Infrared and Raman Spectra As stated above, the ground-state RhB7 and RhB8– clusters are more stable than their neighbors in the corresponding series. In order to gain more insight in the structure of these two clusters, we have simulated their vibrational spectra using the results of our DFT computations of the harmonic vibrational frequencies. The infrared and Raman spectra of the clusters are shown in Figure 4. As can be seen, the IR and Raman spectra of RhB7 have three and two prominent peaks, respectively. The most intense IR peak at the frequency of 766.0 cm−1 corresponds to the swing symmetric stretching of all the B atoms and static modewhich does not involve of the Rh atom. The Raman activity is mainly related with breathing modes and the Raman spectrum in Figure 4(a) displays only two dominant peaks because of high symmetry of the RhB7 cluster. The breathing mode of the boron atoms possesses the highest Raman activity of 11.8 Å4 amu−1 at the frequency of 936.0 cm−1. As shown in Figure 4, there is a single dominant peak and two prominent peaks in the in the IR and Raman spectra of the RhB8– anion, respectively. The scarcity of peaks in the spectra is due to the high C8v symmetry of the anion ground state. The most intense IR peak at 387.2 cm−1 corresponds to the symmetric stretch of the ring composed of eight boron atoms,



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whereas the strongest peak at 758.4 cm−1 in the Raman spectrum corresponds to the cluster breathing mode. In addition, the frequency dependence of the IR and Raman spectra of the most stable RhBnQ (n = 3−10; Q = 0, −1) clusters are displayed in Figure S2. We hope that our theoretical results would provide more available information for further experimental investigation.

3.5. Chemical Bonding Analysis In order to gain insight into the nature of interactions between the Rh and B atoms, we have performed an AdNDP analysis for the ground-state RhB7 and RhB8– and presented the results in Figure S3 and Figure 5, respectively. AdNDP analysis of RhB7 (Figure S3) reveals three set chemical bonds which are consist of three 1c-2e lone pairs, six 2c-2e localized bonds and six 8c-2e delocalized bonds. According to the results of the AdNDP analyses presented in Figure 5, there are three 4d lone pairs (dz2, dxy, dx2-z2) with the occupation numbers (ON) ranging from 1.71 to 1.99. Since the occupation of the dxy and dx2-z2 AOs is 1.71 instead of 2.0 required for true lone pairs, it suggests that these AOs are partially involved in covalent bonding between the Rh atom and the B8 ring. The remaining 28 valence electrons are distributed over eight localized 2c−2e σ-bonds, three completely delocalized 9c−2e σ-bonds, and three completely delocalized 9c−2e π-bonds. There are eight 2c−2e σ-bonds with the ONs of 1.96 between atoms of the B8 ring, which puts strain in the interior between the ring and central Rh atoms. There are two sets of delocalized bonds: three 9c−2e σ-bonds and three 9c−2e π-bonds, each of which obeys the Hückel 4N + 2 rule for aromaticity. In other words, the delocalized bonding in RhB8– give rise to the double aromaticity analogous to that in the valence isovalent CoB8– cluster.29,30 It is worth


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pointing out that both of the two clusters contain the total number-c two-electron bonds (8c−2e for RhB7, 9c−2e for RhB8–), which strengthen the interaction between Rh and B atoms and make them particularly stable. 4. Conclusion Using a combination of the unbiased CALYPSO structure search and density-functional theory optimizations, we have studied the equilibrium structures and electronic properties of the neutral and anionic RhBnQ clusters (n = 3 − 10; Q = 0, −1). The ground-state geometrical structures of the neutral and singly negatively charged RhB4 and RhB5 clusters are found to be planar independently on the charge. According to the results of our study, the Rh atom in both neutral and anionic geometrical structures gradually shifts from peripheral capped sites to the central sites when the cluster size increases. The preference for the 3D geometry of the lowest total energy states depends on the size and charge of a RhBn cluster. Our simulated photoelectron spectrum of RhB8– is in nice agreement with experiment; therefore, it can be expected that the spectra simulated for other RhBn– anions in the range of 3 ≤ n ≤ 10 will provide a useful reference for future theoretical and experimental investigations. A detailed chemical bonding analysis shows that the RhB8– anion possesses double aromaticity, which enhances its stability. Hopefully, the results of our study will further be confirmed by experimental measurements.

Supporting Information Available The detachment energy, calculated infrared and Raman spectra of RhBnQ clusters (n = 3 − 10; Q = 0, −1) clusters, and chemical bonding analysis for the RhB7 cluster. This material is available free of charge via the Internet at http://pubs.acs.org.

Corresponding Authors



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Electronic mail: [email protected] (Cheng Lu). Electronic mail: [email protected] (Gennady L. Gutsev)

Notes The authors declare no competing financial interest.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 11304143, 11304167, 21671114, 11464033 and 11164020), the Program for Science & Technology Innovation Talents in Universities of Henan Provence (Nos. 15HASTIT020) and the Natural Science Foundation of Inner Mongolia (No. 2016BS0107). Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) (No. U1501501). Calculations were performed using the Cherry Creek Supercomputer of the UNLV’s National Supercomputing Institute, as well as the National Supercomputer Center in Guangzhou.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 1. The geometrical structures of the lowest total energy and low-lying excited states of RhBnq (n = 3−10; q = 0, −1) clusters. Rhodium and boron atoms are in pink and yellow, respectively.


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Figure 2. Simulated photoelectron spectra of the lowest total energy states of the RhBn– clusters (n = 3 − 10), together with the available experimental spectrum of RhB9– in the inset.



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Figure 3. The average binding energies per atom Eb(a), second-order energy differences Δ2E(b), and HOMO–LUMO energy gaps Egap(c) for the lowest total energy states of RhBnQ clusters (n = 3 − 10; Q = 0, −1) as a function of n.

Figure 4. Simulated infrared (a) and Raman spectra (b) of the RhB7 and RhB8–clusters.


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Figure 5. The results of the chemical bonding analysis for the RhB8– cluster obtained using the AdNDP method. ON stands for the occupation numbers.



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Table 1. Electronic states, symmetries, and relative energies (ΔE) of the RhBnQ clusters (n = 3−10; Q = 0, −1).

Sym.

ΔE (eV)

Sym.

ΔE (eV)

B1

C2v

0.00

3a−

2

A1

C3v

0.00

A1

C3v

0.30

3b−

6

A

C1

1.60

A’

Cs

0.00

4a−

1

A’

Cs

0.00

A’’

Cs

0.61

4b−

1

A

C1

1.74

B1

C4v

1.49

4c−

1

A

C1

1.80

A1

C2v

0.00

5a−

2

A’

Cs

0.00

Sta.

Sta.

3a

3

3b

1

4a

2

4b

2

4c

2

5a

1

5b

1

A’

Cs

0.71

5b−

2

A’’

Cs

0.66

5c

3A

C1

1.44

5c−

4

A’’

Cs

1.48

6a

2

A’’

Cs

0.00

6a−

1

Cs

0.00

6b

2

A

C1

0.39

6b−

1

A

C1

1.06

6c

2

A’

Cs

1.17

6c−

1

A’

Cs

2.49

7a

1

A1

C6v

0.00

7a−

2

A1

C6v

0.00

7b

1

A1

C2v

1.60

7b−

2

A2

C2v

0.16

7c

1

Cs

2.41

7c−

2

A

C1

0.64

8a

2

A

C1

0.00

8a−

--

C8v

0.00

8b

2

A2

C2v

0.95

8b−

1

A’

Cs

0.71

8c

2

A

C1

1.19

8c−

1

A1

C2v

0.74

9a

1

A’

Cs

0.00

9a−

2

B1

C2v

0.00

9b

1

A

C1

0.73

9b−

2

A

Cs

0.27

9c

1

A

C1

2.91

9c−

2

A

C1

0.43

10a

4

A1

C2v

0.00

10a−

1

A’

Cs

0.00

10b

2

A’

Cs

0.66

10b−

1

A’

Cs

0.73

10c

2

A’

Cs

1.53

10c−

1

C1

1.40

A’

A’

A


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Table 2 Average binding energies (Eb), second-order difference energy (Δ2E) and HOMO−LUMO energy gaps (Egap) of RhBnQ (n = 3−10; Q = 0, −1) clusters. All energies are in eV.

Eb

Δ2E

Egap

Eb

Δ2E

Egap

3.20

3a−

0.133

−0.018

2.70

4a−

0.150

0.010

2.90

0.151

−0.001

3.10

5a−

0.160

0.010

2.67

6a

0.159

−0.051

2.37

6a−

0.166

−0.030

2.09

7a

0.171

0.069

4.26

7a−

0.174

−0.005

1.87

8a

0.173

−0.009

2.12

8a−

0.180

0.046

3.37

9a

0.176

−0.005

3.53

9a−

0.181

−0.033

1.84

10a

0.178

2.05

10a−

0.185

3a

0.129

4a

0.141

5a


2.61

3.18


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TOC Graphic


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Structure and Electronic Properties of Neutral and Negatively Charged

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