摘要： 本文通過(guò)粒子群優(yōu)化算法結(jié)合量子化學(xué)計(jì)算，對(duì)V 摻雜硼團(tuán)簇VB-40 的各種候選結(jié)構(gòu)進(jìn)行了全局預(yù)測(cè). 對(duì)于VB-4 0 團(tuán)簇，我們預(yù)測(cè)2 個(gè)能量最低的結(jié)構(gòu)（結(jié)構(gòu)Ⅰ和Ⅱ）為內(nèi)嵌式金屬硼球烯. 結(jié)構(gòu)Ⅰ具有C2v 對(duì)稱性，繼承了B40 （ D2d 對(duì)稱性）的籠形結(jié)構(gòu)，硼與釩的配位數(shù)為16，而結(jié)構(gòu)Ⅱ之前未見(jiàn)報(bào)道. 值得注意的是，新發(fā)現(xiàn)的結(jié)構(gòu)Ⅱ有Cs 對(duì)稱性，有最高配位數(shù)，為20. 基于這2 種結(jié)構(gòu)，本文利用分子軌道、Wiberg 鍵級(jí)和電子局域函數(shù)分析了其電子性質(zhì). 適應(yīng)性自然密度劃分與化學(xué)鍵分析表明，VB-40 團(tuán)簇（C2v對(duì)稱性）的σ 鍵和π 鍵較好地繼承了B40 （ D2d對(duì)稱性）鍵的特征. 此外，我們還模擬了光電子能譜、紅外光譜、拉曼光譜和紫外可見(jiàn)光光譜，以便于后期的實(shí)驗(yàn)表征.

關(guān)鍵詞： 金屬硼球烯； 光電子能譜； CALYPSO； 密度泛函理論

中圖分類號(hào)： O561. 1 文獻(xiàn)標(biāo)志碼： A DOI： 10. 19907/j. 0490-6756. 2024. 044003

Abstract： Various candidate structures of V-doped boron clusters VB-40 are predicted globally via the particleswarm optimization algorithm combining quantum chemistry calculations. For the VB-40 clusters，the two lowestenergy structures（ C2v structure Ⅰ and Cs structure Ⅱ） are predicted to be endohedral metalloborospherenes.Structure Ⅰ is well inherited in the cage geometry of D2d B40 with the vanadium coordination number ofsixteen，while structure Ⅱ has not been reported before. Notably，the newly discovered structure Ⅱ has 20-fold coordination with the largest coordination state. Based on these two structures，electronic properties areanalyzed by molecular orbitals，Wiberg bond order and electron localization function. The adaptive natural densitypartitioning and chemical bond analysis indicate that both the σ and π bonds of the C2v VB-4 0 cluster are well inherited in the D2d B40 bonding characteristics. In addition，photoelectron spectroscopy，infrared，Raman，and UV-Vis spectra are simulated to facilitate future experimental characterizations.

Keywords： Metalloborospherene； Photoelectron spectroscopy； CALYPSO； Density functional theory

1 Introduction

Owing to the electron deficiency and strongbonding capacity of boron（ 2s22p1），this element aggregatesinto versatile structures to share electronsand produces many structures of bulk boron［1-4］.Metal doping markedly expands the chemistry of boronclusters，and therefore there are unprecedentedstructures and novel chemical bonding for metaldopedboron clusters［5］. In the past decades，the structuresof metal-doped boron clusters have been extensivelyinvestigated via experiments in combinationwith theoretical calculations［5］， underlying the fabricationof novel boron-based nanomaterials. Theseboron-based nanomaterials get extensive applicationin many fields，such as a Wankel motor based onIrB-12，a nanoscale compass based on binary star clustersMg2B8，Earth-Moon rotation simulation systembased on coaxial triple-layered Be6 B-11 cluster，etc［6-8］.On the whole，boron clusters and metal-doped boronclusters have a vast variety of potential applications instructural chemistry and materials science.

Since 2002 quite a few groups have studied sizeselectedBn clusters to elucidate their structures，bondingand chemical properties experimentally and theoreticallyvia photoelectron spectroscopy（ PES），etc.These results showed that the increasing number ofboron atoms leads to a structural evolution from planarstructure to bowl-like and then to drum-like andfinally cage geometry［5，9-13］. A large number of studieshave shown that some transition-metal atoms，alkalimetal atoms doped boron clusters can generate a varietyof interesting structural forms with unusual chemicalbonding patterns，such as mono-ring planar CoB-8 ，RuB-9 ，and TaB-10，half-sandwich CoB-12 and RhB-12，drum-like CrB-16 and CoB-16，and cage-like Sr@B40 andCa@B40［14-19］. Similar to Bn clusters，the growth behaviorsof metal-doped boron clusters evolve from metalcenteredmono-ring to metal-coordinated halfsandwich，then transform to metal-centered drum，and finally endohedral cage as the number of boron atomsincreases.

Vanadium （V） is a refractory transition metalwith a high melting point，and it plays an importantrole in the steel industry，aerospace industry，chemicalindustry，materials industry as a very important alloyingelement. For VB-10 clusters，theoretical studiesat the PBEPBE/B/6-311+G（d）/V/Stuttgart’97level and experimental photoelectron spectroscopyspectra revealed that the global minimum is a threedimensional（3D） boat-like structure［20］. For thegrowth pattern of V2Bn（ n=1～10），Zhang et al. ［21］found that V2Bn=2～4 clusters tend to possess incom ?plete bipyramidal structures while V2B5 cluster preferirregular polyhedral structures，and bipyramidal structuresof V2Bn=6～10 clusters are identified as the globalminimum at the CCSD（T）/6-311+G（d）//B3LYP/6-311+G（d） level. For V-doped B-20 clusters，Li etal.［22］ reported representative Cs，C1，C2v，D2d symmetricstructures，and predicted the lowest energy structureis composed of a V-centered eighteen membereddrum at the PBE0/6-311+G（d） level.

The above studies indicated that vanadium dopingsignificantly affects the growth behaviors of Bnclusters，and the increasing number of B atoms leadsto a significant structural evolution in TM-doped boronclusters. Can V form interesting and meaningfulstructural framework with the larger B40 clusters？Will its global minimum conform to the above evolutionarypattern？ What are the bonding and spectralcharacteristic of such structures？ To achieve theabove goal，inspired by a series of previous studies，we explore the structure and properties of VB-4 0 clustersby particle swarm optimization（ PSO） algorithmcombined with density functional theory（ DFT） calculationsin this paper. In addition，electronic propertiessuch as molecular orbitals，Wiberg bond order，electron localization function and adaptive naturaldensity partitioning（ AdNDP） is also analyzed. Furthermore，photoelectron spectroscopy，infrared（ IR），Raman，and UV-Vis spectra of VB-4 0 clusters are investigatedto facilitate their future spectral characterizations.Our researches can help to better understandboron-boron and metal-boron bonds，enabling us tosynthesize new borides and design new boron-basedmaterials with desired properties.

2 Theoretical methods

We carried out an extensive structural search forVB-40 clusters by crystal structure analysis by particleswarm optimization （CALYPSO） code，which performsstructure evolution via the particle swarm optimization（PSO） algorithm. Currently，CALYPSOhas been successfully applied to the structure predictionof many known systems（ e. g. ，elemental，binaryand ternary compounds） with metallic，ionic or covalentbonding environments［23-26］. PSO technique is differentwith the genetic algorithm and has apparentlyavoided the use of evolution operators（ e. g. ，crossoverand mutation）. As a stochastic global optimizationmethod，PSO［27-29］ is inspired by the choreographyof a bird flock and can be seen as a distributed behavioralgorithm that performs multidimensionalsearch. According to PSO，the behavior of each individualis guided by either the best local or the bestglobal individual，during this process an individualcan learn from its part experiences to adjust its flyingspeed and direction. These behaviors help each individualfly through a hyperspace. Therefore，all the individualsin the swarm can quickly converge to theglobal position. To achieve a high efficiency of structureprediction，bond characterization matrix，thepoint group symmetries，Metropolis criterion，andatom-centered symmetrical function are employed inCALYPSO.

First，each structure in the first generation wasgenerated randomly by CALYPSO，and geometryoptimization was performed at the PBE0/6-31Glevel，including a variety of spin states（ 2S+1=1，3，5）. For comparison，the initial geometric optimizationwas also performed at the HF/6-31G level. For thelatter structural searches，80% of the structures weregenerated by particle swarm optimization methods，and the rest were randomly generated. In addition，the minimal distances of V-V，V-B and B?B duringthe search process were set to 2. 5， 1. 9 and 1. 5 ?，respectively. To ensure as many candidate structuresas possible，this procedure of structure searches continueduntil 30 generations were produced and thepopulation of each generation were 30. So a total of900 low energy structures were predicted for eachcluster size of the VB-4 0 clusters. Second，the most desirablestructures obtained from CALYPSO werefinely re-optimized at the PBE0/def2-TZVP level. Ithas been shown that the PBE0 functional is a reliableand popular choice for their successful performance inthe previous computational studies on the boron clusters［17-19，22］. In order to evaluate the effect of differentfunctionals and basis sets on geometries，the structureswere also re-optimized at TPSSh/6-311+G（d），B3LYP/6-311+G（d），and PBE0/6-311+G（d） levels based on Refs.［21，22］. In order to comparethe ground state structure of neutral VB40 withanionic VB-40，the candidate structures of the neutralVB40 cluster were also obtained at the PBE0/def2-TZVP level. Third，for VB-40 clusters，the vibrationfrequency calculations were used to verify the stabilitiesof the finally optimized structures，and meanwhileobtain the zero-point vibration energies（ZPVE） and infrared（ IR） spectra. At last，the calculationsrelated to electronic properties were at thePBE0/def2-TZVP level. The time-dependent densityfunctional theory（ TD-DFT） was used to simulatephotoelectron spectroscopy （PES） and UV-Visspectra. The simulated spectra were calculated forthe future experimental determination. All quantumchemistry calculations were implemented through theGaussian 16 program［30］. The wave function analysiswas accomplished by Multiwfn［31］，and the results visualizationwas achieved by VMD［32］.

3 Results and discussion

3. 1 Geometric configurations

10 representative low-lying structures for VB-40clusters are shown in Fig. 1 with their point groupsymmetries and relative energies. For the sake of brevity，these structures are marked with Roman numeralsfrom Ⅰ to Ⅹ . For comparison，the moststable cage structure and quasi-plane structure of B-40clusters are also displayed in Fig. 1. The structures ofthe B-40 clusters change markedly when a V atom isdoped. 3 kinds of structures are sorted through structurecomparison： Ⅰ to Ⅲ possess endohedralmetallofullerene structures with the V atom inside aswell as exohedral geometry with a face-capping Vatom；For structure Ⅳ to Ⅸ ，the V atom is in theplane of VB-40 clusters；Ⅹ is an exohedral structurewith the V atom located outside of a less compact boroncluster.

Several theoretical studies of the structural andelectronic properties of metalloborospherene reportedthat the global minima of metal-doped boron clustersfavor overwhelmingly cage-like structures［19，33，34］. Forthe VB-40 clusters studied in this work，the lowest energystructure，namely the cage structure Ⅰ ，is endohedralmetalloborospherene with 2 hexagonal holesat the top and bottom and 4 heptagonal holes on thewaist. The boron cage is made of interwoven doublechainboron ribbons，and the ribbons consist of 6 horizontalB8 ribbons and 4 vertical B9 ribbons. The onlyinterior V atom deviates from the center of the cagebut near the top of the cluster. The structure Ⅰ of theVB-40 clusters has very similar structure to the 2ndstable borofullerene of the B-40 clusters，which is acage structure with 2 hexagonal holes at the top andbottom，4 heptagonal holes at the waist［35］. Differently，the structure Ⅰ slightly deforms at the top. Forthe structure Ⅰ ，the distance between V atom and Batoms is in the range of 2. 182～2. 713 ? with the averagebond length of 2. 395 ?，and the V atom is coordinatedwith 16 B atoms. In order to present theconvergence of structural search more intuitively，theconvergence of the global energy minimum withsteps of population evolution is shown in Fig. 2.

Meanwhile，we also find another cage structure，that is，the structure Ⅱ which has a hexagonal hole atthe top and 2 adjacent pentagonal holes at the bottom.The Ⅴ atom is located inside the cage but awayfrom the center of the cage. The number of closepackedB6 triangles of the cage structure of B-40 clustersis 8［35］，whereas that of the structure Ⅱ of theVB-40 clusters reduces to 4. This is because the additionof V leads to the interaction between the B atomsat the bottom and the V atom，and therefore theshape of the structure Ⅱ changes. This structure hasthe point group symmetry of Cs. In addition，structureⅡ has a higher symmetry of C2v which is unstable，the C2v structure has an imaginary frequency of32. 49i cm?1 （PBE0/def2-TZVP） and 42. 18i cm?1（PBE0/6-311+G（d））. For the structure Ⅱ，the distancebetween V and B is from 2. 051 to 2. 650 ?，and the average bond length is 2. 391 ?. It is worthnoting that the VB-40 cluster in configuration of thestructure Ⅱ has 20-fold coordination，which is thelargest coordination state of the VB-40 clusters.

As for the 2nd structure style，the main feature ofstructures Ⅳ to Ⅸ is that the V atom is located on thesurface of VB-40 clusters. The V atom and B atomsforms 9-membered rings or 10-membered rings on thesurface of the clusters，and the V atom is roughly locatedat the center of the multiple-ring. For this structure type，the structure Ⅳ has the lowest energy with the distancebetween V and B of 2. 248～2. 546 ?，and the V atomis coordinated with 14 B atoms.

The 3rd structure type is composed of the structureⅩ ，and this structure is a closed boron chainstructure，and the ring is found to have an obviousfold at the V atom. The global minimum of B-4 0 clustersis a quasi-planar structure with 2 adjacent hexagons［35］，and our predicted structure Ⅹ may be relatedto the quasi-planar structure of the B-4 0 clusterwith the addition of vanadium atom. The coordinationnumber of V atom in the structure Ⅹ is 6，whichis significantly less than that of the other 2 structuretypes. Besides，the distance between the V atom andB atom of this structure is in the range from 1. 928 to2. 176 ?，and the average bond length is 2. 064 ?.Based on the analysis of the 3 structure types， it isfound that as the V atom moves from the interior ofthe boron clusters to the exterior，the coordinationnumbers of V atoms become fewer and the averageV?B bond length get shorter.

Fig. 1 depicts the energies of the 10 structures atthe level of PBE0/def2-TZVP， PBE0/6-311+G（d），TPSSh/6-311+G（d），and B3LYP/6-311+G（d）. By comparing the energies of 3 structuraltypes at 4 different combination of functionals and basissets，we find the structures and energies of VB-40clusters obtained with the PBE0 functional are consistentwith the results of TPSSh，which predict thestructure Ⅰ and Ⅱ are the lowest energy structures.For comparison， the 2 lowest energy structures arethe structure Ⅰ and Ⅲ with the B3LYP functional.In general，the total energy order predicted by the B3LYP functional is different from the result predictedby the PBE0 functional and the TPSSh functional.The energy evolution of structures Ⅰ and Ⅱwith convergence criteria is shown in Fig. 3. We seethat the energy of structure Ⅰ and structure Ⅱ convergewell at the level of PBE0 and B3LYP to ensurethe reliability of the calculated results.

According to Refs.［26，33-35］，the PBE0 functionalis widely used in the calculation of boron clusters，and this is what we have adopted in the presentstudy of the VB-40 clusters. At the PBE0 level oftheory，the lowest energy structure type is the first type with the vanadium atom residing inside the boroncage mostly or outside the boron cage in a fewcases，and the cage structure Ⅰ with C2v symmetry isthe most stable structure of the VB-40 clusters. Becauseof the addition of the vanadium atom，the symmetryof the V-doped boron clusters decreases fromthe D2d symmetry of the B-4 0 cluster［35］ to the C2v symmetryof the structure Ⅰ of the VB-40 clusters. Comparedwith the Cs symmetry of the VB-20 drum cluster［22］，the VB-40 cage structure Ⅰ becomes higher C2vsymmetry as cluster size increases.

As we know，electron has much influence in thestructure of boron cluster. The neutral B40 is cage likewhile anion is quasi-planar［35］. When dotting a Vatom，does the electrons has great influence of B40cage？ Does the ground state structure of neutralVB40 change relative to VB-40？ Therefore，8 candidatestructures of neutral VB40 clusters are obtainedas shown in Fig. 4. The candidate structures of theneutral VB40 cluster are obtained by optimizing the anionicVB-40 clusters after removing an electron fromeach anionic cluster. For the VB40 clusters，the candidatestructures are classified into the same 3 types asanionic VB-40 clusters. The lowest energy structurepossesses endohedral structure with the metal atomresiding inside the cage，which has 2 hexagonal holesat the top and bottom and 4 heptagonal holes on thewaist. As observed in Fig. 1 and Fig. 4，the globalminimum of both neutral and anionic VB40 clustersare endohedral metalloborospherene with slight bondlength differences，which are well inherited in the configurationof neutral and anionic B40 clusters. Metalloborospherenesprovide indirect evidence for the robustnessof the borospherene structural motif. Andneutral VB40 clusters have similar structures and energyordering as their anionic counterparts.

3. 2. 1 Molecular orbitals

To explore the bondingproperties，the molecular orbitals and HOMO-LUMOgap of structure Ⅰ and structure Ⅱ of the VB-40 clustersare presented in Fig. 5，where blue and red parts correspondto positive and negative phase of the orbital，respectively. The energy gap between HOMO and LUMO is related to the thermodynamic stability of agiven compound［36］. A large HOMO?LUMO gap reflectsa high-energy cost for an electron excited from theHOMO to the LUMO，and also implies that the correspondingstructure is chemically inert. It can be seenthat the HOMO?LUMO gap is 1. 94 eV for structure Ⅰand 2. 44 eV for structure Ⅱ，which suggests that thenew found structure Ⅱ with a larger HOMO-LUMOgap presents a higher chemical stability. In addition，theHOMO?LUMO energy gap decreases from 2. 99 eV inB-40［35］ to 1. 94 eV（ structure Ⅰ） or 2. 44 eV（ structureⅡ） in VB-40，which indicates that vanadium doping reducesthe chemical stability of boron clusters. Comparedwith the HOMO?LUMO energy gap of the VB-2 0 clusters（2. 99 eV）［22］，the chemical stability of vanadiumdopedboron clusters decreases as the number of boronatoms increases. In order to characterize the bindingcapacity of V atom to the surrounding B atoms morequantitatively，the binding energies Eb of VB-4 0 clusterscalculated at the PBE0/def2-TZVP level are displayedin Tab. 1. The binding energies Eb of structure Ⅰ andstructure Ⅱ are 227. 2 and 225. 6 eV，respectively. ForB-40 boron clusters，the cage structure is 0. 1 eV higherin energy than the most stable quasi-planar structure［35］，which indicates that vanadium doping leads to a stronginteraction between V atom and the surrounding B atoms，stabilizing the existence of the cage clusters.

3. 2. 2 Wiberg bond order

Bond order is a veryimportant concept for the analysis of chemical bonds，which is a quantitative expression of the characteristicsof chemical bonds. Mayer bond order analysis［37，38］，Mulliken bond order analysis［39］，multi-centerbond order analysis［40，41］，Wiberg bond order analysis［42］，and fuzzy bond order［43，44］ are developed to analyzeand understand the bond property. Wiberg bondorder is a quantitative description of chemical bondsand is used to understand the nature of chemical reactionand predict the molecular reactivity and stability.For the same chemical bond，the higher the Wibergbond order value is，the shorter the bond strength is.Therefore，Wiberg bond order is used to analyze thebond formation in this work. The Wiberg bond orderof structure Ⅰ and structure Ⅱ of VB-40 clusters arelisted in Tab. 2，as well as their atomic numbersshown in photoelectron spectroscopy results.

According to Tab. 2，the B ? B bond lengths arein the range of 1. 584～1. 776 ?（ structure Ⅰ），andin the range of 1. 625～1. 780 ?（ structure Ⅱ），andthe corresponding Wiberg bond orders values arefrom 0. 587 to 0. 892 for structure Ⅰ and from 0. 513to 0. 771 for structure Ⅱ ，which suggests a covalentcharacter. The V-B distances （2. 182～2. 713 ? forstructure Ⅰ and 2. 051～2. 650 ? for structure Ⅱ ）are typical transition-metal-boron coordination bondlengths［45］ ，which are longer than the V-B bondlength（ 2. 003 ?） in VB diatomic molecule. The resultsindicate that the weak interactions exist betweenthe V and B atoms in VB-40 cluster，which can be supportedby the calculated Wiberg bond orders. The calculationresults suggest that the Wiberg bond ordersof V-B bonds are from 0. 279 to 0. 516 for structureⅠ and from 0. 204 to 0. 557 for structure Ⅱ ，respectively.The low V-B orders reveals the weak interactionsbetween the V and B atoms in VB-4 0 clusters，inagreement with the long V-B bond lengths. Particularly，for both of structure Ⅰ and structure Ⅱ ，theWiberg bond orders of V-B bond are lower thanthose of B-B bond，which means that the V-B interactionsare lower than the corresponding B-B interaction.Similar bond lengths and bond orders results canalso be found in smaller VB-20 cluster［22］. While comparedwith VB-20 cluster，the Wiberg bond orders ofV-B and B-B of VB-40 clusters are smaller，and thebond length of that are longer. This indicates that thechemical stability of vanadium-doped boron clustersdecreases with the increase of cluster size，which isconsistent with the results of molecular orbital analysisabove. Besides，for structure Ⅰ ，the Wiberg bondorders of B-B bond in B6 ring close to the V atomshow higher values than those in B6 ring far awayfrom the V atom，implying that the stability of B-Bbond in B6 ring close to the V atom are stronger. Forstructure Ⅱ ，the Wiberg bond orders of B-B bond inB5 ring are larger than those in B6 ring，so the B-B interactionsin B5 ring are stronger.

3. 2. 3 Electron localization function

Electron localizationfunction （ELF） is used to determine theprobability of electron pairing by Beck and Edgecombe［46］. ELF is within the range of 0～1. A largeELF value means that electrons are greatly localized，indicating that there is a covalent bond，a lone pair orinner shells of the atom involved. Specifically，ELF=0 indicates that there is no electron density betweenatomic orbitals；ELF=0. 5 shows the regions withbonding of a metallic character；ELF=1 correspondsto perfect localization，and means covalent bonds. Tovisualize the real space electron distribution，the ELFin different parts of the 2 lowest energy structure ofthe VB-40 clusters are displayed in Fig. 6. For bothstructure Ⅰ and structure Ⅱ，there is high electron localizationbetween B and B atoms in multiple boronrings，which means that a covalent B-B bond isformed. However，the ELF values between V and Batom possess relatively lower values. In addition，forstructure Ⅰ，the higher ELF values between B and Batom exists in B6 ring far away from the V atom com ?pared with B6 ring close to the V atom. While for structure Ⅱ ，the ELF values of B-B bond in B5 unitsshow higher values than those in B6 units. These resultsare consistent with the Wiberg bond ordersanalysis.

3. 2. 4 AdNDP analysis

To further elucidate thestability of the lowest energy VB-40 （endohedralmetalloborospherene structure Ⅰ ），we carry out theanalysis of its bonding pattern via adaptive naturaldensity partitioning （AdNDP）［47］ method，which isused to analyze the localized and delocalized multicenterbonds （labeled as nc-2e，namely an n-centertwo-electron bond）.

As can be seen from Fig. 7，each lobe in the 6frames represents a multicenter 2-electron bond，withthe occupation number （Nocc） given below eachframe. A total of 48 delocalized σ bonds and 15 delocalizedπ bonds are identified in the structure Ⅰ ofVB-40 cluster. Among these bonds，the 48 delocalizedσ bonds can be divided into 40 3c-2e σ bonds on the40 B3 triangles and 8 6c-2e σ bonds on 8 quasi-planarclose-packed B6 units on the surface of B40 cage. Becausethe central B3 triangles contribute to the 6c-2e σbonds significantly，the 48 σ bonds of structure Ⅰ canbe practically regarded as delocalized 3c-2e σ bonds.The remaining 15 π bonds are characterized as 4groups：6 5c-2e π bonds and 4 7c-2e π bonds at thetop and bottom of the cage， 4 6c-2e π bonds on thewaist，and 1 40c-2e π bond over entire surface of thecage. Both the σ and π bonds of the C2v VB-40 clusterare well inherited in the D2d B40 bonding characteristics［35］，and these bonding patterns are similar to thatof C2v Ca@B40 and C2v Na@B40［19，48］ as well，which togetherdemonstrates the chemical robustness of theB40 borospherene system. Hence，the valence electronsof cage structure Ⅰ are delocalized σ or πbonds，and there is no localized 2c-2e bond，which isdifferent from the bonding pattern of the 2D boronclusters［49，50］. This double （σ + π） delocalization ofthe electron clouds along the interwoven doublechains on the molecular surface cause a highly stablestructure（ structure Ⅰ） despite its intrinsic electrondeficiency.

3. 3 Simulated PES，IR，Raman and UV-Vis spec?tra of structure Ⅰ and Ⅱ of the VB-40 clusters

Photoelectron spectroscopy （PES） is an effectivetool for studying the electron structure of atomsand molecules using the photoelectric effect. PES directlydetects the energy levels of the valence electrons，which have a great influence on chemical bondingin clusters. Fig. 8 shows the simulated photoelec?tron spectra of the 2 lowest energy cage structure Ⅰand Ⅱ for the VB-40 clusters. As shown in Fig. 8，thespectra of structure Ⅰ and structure Ⅱ exhibit 6 wellspacedbands in the range of binding energies from 1to 6 eV. The characteristic peaks of structure Ⅰ areat 2. 54，3. 14，3. 50，4. 33，4. 69，and 4. 96 eV，respectively.While the simulated spectrum of structureⅡ has major peaks located at 2. 53，2. 80，3. 46，4. 12，5. 16，and 5. 51 eV. Interestingly，the first verticaldetachment energy （Evd1） values of structure Ⅰ（2. 54 eV） is very similar to that of structure Ⅱ（2. 53 eV）.

The Evd1 peak corresponds to the vertical detachmenttransition from the ground state of the anionicVB-40 cluster to that of the counterpart neutral VB40.While the other PES features originate from verticaldetachment transitions to the excited states of VB40，and the peaks with higher binding energy originatefrom detaching the electrons from lower molecular orbitals.Meanwhile，we also calculate the adiabatic detachmentenergy（ Ead），which represents the electronaffinity of corresponding neutral VB40，and the computedEad values are 2. 98 eV for the structure Ⅰ and3. 36 eV for the structure Ⅱ . The difference betweenEvd1 and Ead （0. 44 eV for the structure Ⅰ and0. 83 eV for the structure Ⅱ） can reflect the variationof the geometrical structure for the most stable structuresof the VB-40 and VB40 clusters. It can be seenthat there is a relatively obvious geometric change betweenthe studied anionic and neutral V-doped boronclusters.

Zhai et al.［35］ reported that B-40 has a cage clusterof all-boron fullerenes with extremely low electronbinding energy via experimental and theoretical efforts.At the PBE0 level，the Evd1 value from simulatedphotoelectron spectroscopy for the B-4 0 cage clusteris 2. 39 eV，which is in close agreement with theexperimental Evd1 value of 2. 62 eV. These studies［22，51，52］ suggest the reliability of the PBE0 functionalutilized in this work. In comparison with theB-40 cage cluster［35］，the Evd1 difference between theVB-40 cage cluster （2. 54 eV for structure Ⅰ and2. 53 eV for structure Ⅱ ） and the B-40 cage cluster（2. 39 eV） is small，which is ～0. 15 eV. This meansthat the addition of vanadium has little effect on theelectron transition from the ground state of anionicVB-40 to the ground state of the neutral one.

In contrast to the photoelectron spectrum of theVB-20 drum cluster［22］，the photoelectron spectra of theVB-40 cage cluster（ structure Ⅰ and structure Ⅱ） generallybecome more complex with less independentspectrum peaks. The Evd1 value of the VB-40 cluster isclose to that of the VB-20 cluster（ ～2. 82 eV）. Differently，there is a large energy gap between the firstand the second vertical detachment energy （Evd2）peaks in the photoelectron spectrum of VB-20 drum cluster，with a difference of ～1. 15 eV. However，thegap between Evd1 and Evd2 peaks of VB-40 cage clustersis much smaller，namely，0. 60 eV for structure Ⅰand 0. 27 eV for structure Ⅱ. This indicates that neutralVB40 cluster becomes less stable than neutralVB20 cluster with the increase of cluster size.

The simulated infrared（ IR），Raman，and UVVisspectra of the two lowest energy structures of theVB-40 clusters are displayed in Fig. 9. The IR and Ramanmodes of structure Ⅰ and Ⅱ could be used as diagnosticfingerprints of V-B bonding in spectroscopiccharacterizations，and provide theoretical basis for theidentification of the VB-40 clusters based nanomaterialsby experimental researchers. As far as we know，IR and Raman spectra of the VB-4 0 clusters have notyet been found，hence these two spectra of the twolowest energy VB-40 clusters are computationallysimulated at the PBE0/def2-TZVP level to facilitatetheir future experimental characterizations. As shownin Fig. 9，both the IR and Raman of structure Ⅰ andⅡ studied include active vibration modes and inactivevibration modes. Infrared inactive modes causeno changes in dipole moment and Raman inactivemodes cause no changes in polarizability. These inactivevibration characteristics are mainly due to thehigh C2v symmetry of structure Ⅰ and lower Cs symmetryof structure Ⅱ.

As shown in Fig. 9a，structure Ⅰ features 12major IR active peaks around 141 cm?1（A1），161 cm?1（B1），215 cm?1（A1），536 cm?1 （A1），636 cm?1（B2），739 cm?1（A1），752 cm?1（A1），873 cm?1（A1），1122 cm?1（A1），1152 cm?1（A1），1217 cm?1（B2） and 1252 cm?1（B1），respectively.Among them， the strongest infrared peak at1252 cm?1 is assigned to the breath stretch of all of Batoms with the central V atom almost silent. The infraredpeak at 161 cm?1 is basically caused by therocking vibrations between V atom and the coordinatedB atoms. For IR spectrum of structure Ⅱ inFig. 9d，the characteristic absorption bands appear at31 cm?1（A'），296 cm?1（A'），388 cm?1（A'），556 cm?1（A'），592 cm?1（A'），637 cm?1（A'），794 cm?1（A'），828 cm?1（A'），895 cm?1（A''），953 cm?1（ A''），1112 cm?1（ A'） and 1171 cm?1（ A''）.The most intense vibrational band is from 592 cm?1which corresponds to the obvious stretching of B atomsresiding at the center of the pentagons and at the2 vertices of the hollow pentagon. The peak at31 cm?1 is contributed to rocking vibration of the incagequadrilateral composed of V and B atoms.

The Raman spectra of structure Ⅰ and Ⅱ areshown in Fig. 9b and Fig. 9e. The characteristic Ramanpeaks of structure Ⅰ are at 177 cm?1（A1），206 cm?1（B2），216 cm?1（A1），262 cm?1（A2），325 cm?1（A1），475 cm?1（A1），507 cm?1（A1），527 cm?1（B1），602 cm?1 （A1），677 cm?1 （A1），752 cm?1 （A1），909 cm?1 （A2），1122 cm?1 （A1），1151 cm?1（ A1） and 1299 cm?1（ A1），with the strongestcharacteristic peak at 1299 cm?1. The most interestedmodes for the studied systems are radial breathingmodes（ RBMs） of the endohedral cage complex，which can be available for characterizing hollow structures.These RBMs of structure Ⅰ appear at 177cm?1，507 cm?1 and 621 cm?1，and they are assignedto the breathing vibration of the central V atom andthe B40 cage. For structure Ⅱ ，it has characteristicpeaks located at 235 cm?1（ A'），332 cm?1（ A''），362cm?1（ A'），385 cm?1（ A'），480 cm?1（ A'），513 cm?1（A'），617 cm?1（ A'），715 cm?1（ A'），740 cm?1（ A'），845 cm?1（ A'），983 cm?1（ A'） and 1080 cm?1（ A'）.Among these peaks，the strongest peak at 513 cm?1is exactly RBM which belongs to the breathing of thecentral V atom and the B40 cage. It's worth notingthat the Raman spectrum of structure Ⅱ has a distinctpeak at 31 cm?1，which is so significantly differentfrom the low frequency pattern of structure Ⅰ .This low frequency peak of structure Ⅱ correspondsto violent swing of quadrilateral hole consisting of Batom and V atom.

Compared with IR and Raman spectra of drumVB-20［22］，with the increase of cluster size，both IR andRaman have obvious redshifts for structure Ⅰ of theVB-40 clusters，with the IR redshift of 867 cm?1 andthe Raman redshift of 634 cm?1. While for the structureⅡ of the VB-40 clusters，the redshifts of the strongestIR and Raman peaks are less obvious（ 207 cm?1of IR and 152 cm?1 of Raman）. In contrast to the infraredand Raman spectra of B40，it is found that forthe structure Ⅰ of the VB-40 clusters only the additionof vanadium does not significantly change the IR andRaman absorption peak of the neutral host B40，producingsimilar high-resolution peaks. For the structureⅡ of the VB-40 clusters，the IR and Raman peakschange significantly. This may be because，in additionto the addition of vanadium，the cage-like structure ofstructure Ⅱ changes significantly as well compared to B40.

As is shown in Fig. 9c and Fig. 9f，the weakbands of structure Ⅰ at 531 nm originate from electronicexcitations from highest occupied molecular orbital（HOMO） of the monoanion to its high-lying unoccupiedmolecular orbitals， while the band at698 nm involve electronic transitions from HOMO-1，HOMO-2，HOMO-3. The strong UV bands around262 and 446 nm mainly originate from electronic transitionsfrom deep inner shells. For structure Ⅱ ，mostof the strong peaks （233，263，315，339，372，463 nm） involve the electrons transition from deep innershells of the monoanion to its high-lying unoccupiedmolecular orbitals，while the weak absorptionpeak at 753 nm involves electron transitions from thesecond highest occupied orbital（ HOMO-1）.

4 Conclusions

We have investigated structure，stability，spectra，and bonding of VB-4 0 clusters using the particleswarm optimization method combining with densityfunctional theory calculation. The main conclusionsare as follows.

（1） For the VB-4 0 clusters， endohedral metalloborospherenes（C2v structure Ⅰ and Cs structure Ⅱ）are predicted to be the 2 lowest energy structures.Structure Ⅰ with the V coordination number of 16 iswell inherited in the cage geometry of the D2d B40，while the new found cage geometry （structure Ⅱ ）has 20-fold coordination with the largest coordinationstate of the VB-4 0 clusters.

（2） Because of the addition of the vanadiumatom，the symmetry of the V-doped boron clustersdecreases from the D2d symmetry of the B-40 cluster tothe C2v symmetry of the VB-40 clusters（ structure Ⅰ）.Compared with the Cs symmetry of the VB-20 drumcluster，the VB-40 cage structure Ⅰ becomes higherC2v symmetry as cluster size increases.

（3） The chemical bond analysis for cage structureⅠ of VB-40 cluster indicates that the double（σ+π） delocalization of the electron clouds along theinterwoven double chains on the molecular surfacecause a highly stable structure despite its intrinsic electron deficiency.

（4） In comparison with the photoelectron spectroscopyof B-40 cluster，the VDE1 difference betweenthe VB-40 and B-40 cage cluster suggests that the additionof vanadium has little effect on the electron transitionfrom the ground state of anionic VB-40 to theground state of the neutral one. The VDE analysis betweenthe VB-40 cage cluster and VB-2 0 drum clustermeans that neutral VB40 cluster becomes less stablethan neutral VB20 cluster with the increase of clustersize.

（5） For Raman spectra，the most interestedmodes for the endohedral cage VB-40 cluster are radialbreathing modes（ RBMs），which can be available forcharacterizing hollow structures. These RBMs ofstructure Ⅰ and Ⅱ are assigned to the breathing vibrationof the central Ⅴ atom and the B40 cage.

References：

［1］ Lipscomb W. The boranes and their relatives ［J］. Science，1977， 199： 1047.

［2］ Albert B， Hillebrecht H. Boron： Elementary challengefor experimenters and theoreticians ［J］. AngewChem Int Ed， 2009， 48： 8640.

［3］ Douglas B， Ho S. Structure and chemistry of crystallinesolids［ M］. Berlin： Springer， 2006.

［4］ Oganov A， Chen J， Gatti C， et al. Ionic high-pressureform of elemental boron［ J］. Nature， 2009， 475： 863.

［5］ Jian T， Chen X， Li S， et al. Probing the structuresand bonding of size-selected boron and doped-boronclusters［ J］. Chem Soc Rev， 2019， 48： 3550.

［6］ Liu L， Moreno D， Osorio E， et al. Structure andbonding of IrB-12： Converting a rigid boron B12 plateletto a Wankel motor［ J］. RSC Adv， 2016， 6： 27177.

［7］ Guo J， Feng L， Wang Y， et al. Coaxial triple-layeredversus helical Be6 B-11 clusters： Dual structural fluxionalityand multifold aromaticity ［J］. Angew Chem IntEd， 2017， 56： 10174.

［8］ Wang Y， Feng L， Guo J， et al. Dynamic Mg2B8 cluster：A nanoscale compass ［J］. Chem Asian J， 2017，12： 2899.

［9］ Chen Q， Zhang S， Bai H， et al. Cage-Like B+41 andB+42： New chiral members of the borospherene fam ?ily［ J］. Angew Chem Int Ed， 2015， 54： 8160.

［10］ Li H， Jian T， Li W， et al. Competition between quasiplanarand cage-like structures in the B29 cluster： Photoelectronspectroscopy and ab initio calculations ［J］.Phys Chem Chem Phys， 2016， 18： 29147.

［11］ Zhai H， Kiran B，Li J， et al. Hydrocarbon analoguesof boron clusters-planarity，aromaticity and antiaromaticity［ J］. Nat Mater， 2003， 2： 827.

［12］ Li W， Chen Q， Tian W， et al. The B35 cluster with adouble-hexagonal vacancy：A new and more flexiblestructural motif for borophene ［J］. J Am Chem Soc，2014， 136： 12257.

［13］ Piazza Z， Hu H， Li W， et al. Planar hexagonal B36 asa potential basis for extended single-atom layer boronsheets［ J］. Nat Commun， 2014， 5： 3113.

［14］ Galeev T， Romanescu C， Li W， et al. Observation ofthe highest coordination number in planar species： DecacoordinatedTaB-10 and NbB-1 0 anions ［J］. AngewChem Int Ed， 2012， 51： 2101.

［15］ Romanescu C， Galeev T， Li W， et al. Aromaticmetal centered monocyclic boron rings：CoB-8 andRuB-9 ［ J］. Angew Chem Int Ed， 2011， 50： 9334.

［16］ Jian T， Li W I， Popov A， et al. Manganesecenteredtubular boron cluster MnB-16： A new class of transitionmetalmolecules［ J］. J Chem Phy， 2016， 144： 154310.

［17］ Popov I， Jian T， Lopez G， et al. Cobalt-centred boronmolecular drums with the highest coordination number inthe CoB-16 cluster［ J］. Nat Commun， 2015， 6： 8654.

［18］ Li C， Shen Z， Zhang J， et al. Analysis of the structures，stabilities and electronic properties of MB-16（M=V，Cr，Mn，F(xiàn)e，Co，Ni） clusters and assemblies［ J］. New J Chem， 2020， 44： 5109.

［19］ Bai H， Chen Q， Zhai H， et al. Endohedral and exohedralmetalloborospherenes： M@B40（M=Ca，Sr） andMamp;B40（M=Be，Mg） ［J］. Angew Chem Int Ed，2015， 54： 941.

［20］ Li W， Romanescu C， Piazza Z， et al. Geometrical requirementsfor transition-metal-centered aromatic boronwheels：The case of VB-10 ［J］. Phys Chem ChemPhys， 2012， 14： 13663.

［21］ Zhang L， Jia J， Wu H. Structural and electronic propertiesof V2Bn（ n =1～10） clusters ［J］. Chem Phys，2015， 459： 131.

［22］ Li C， Li H， Cui Y， et al. A density functional investigationon the structures，electronic，spectral and fluxionalproperties of VB-20 cluster ［J］. J Mol Liq， 2021，339： 116764.

［23］ Xia X， Hermann A， Kuang X， et al. Study of thestructural and electronic properties of neutral and chargedniobium-doped silicon clusters： Niobium encapsulated insilicon cages［ J］. J Phys Chem C， 2016， 120： 677.

［24］ Chen B， Sun W， Kuang X， et al. Modification of geometricand electronic structures of iron clusters by nitrogen：Fe-8 vs Fe8N? ［J］. J Phys Chem C， 2020，124： 3867.

［25］ Jin S， Chen B， Kuang X， et al. Structural and electronicproperties of medium-sized aluminum-doped boronclusters AlBn and their anions ［J］. J Phys ChemC， 2019， 123： 6276.

［26］ Chen B， Sun W， Kuang X， et al. Structural stabilityand evolution of medium-sized tantalum-doped boronclusters：A half-sandwich-structured TaB-12 cluster［ J］.Inorg Chem， 2018， 57： 343.

［27］ Wang Y， Lv J， Zhu L， et al. CALYPSO： A methodfor crystal structure prediction［ J］. Comput Phys Commun，2012， 183： 2063.

［28］ Wang Y， Lv J， Zhu L， et al. Crystal structure predictionvia particle-swarm optimization ［J］. Phy Rev B，2010， 82： 094116.

［29］ Niu Z， Tang M， Ge N. Structure，stability，infraredspectra，and bonding of OHm（H2O）7 （ m=0，±1） clusters：Ab initio study combining the particle swarm optimizationalgorithm ［J］. Phys Chem Chem Phys，2020， 22： 26487.

［30］ Frisch M， Trucks G， Schlegel H， et al. Gaussian 16，Revision C. 01 ［CP］. Wallingford CT： Gaussian Inc，2019.

［31］ Lu T， Chen F. Multiwfn： A multifunctional wavefunctionanalyzer［ J］. J Comput Chem， 2012， 33： 580.

［32］ Humphrey W， Dalke A， Schulten K. VMD-visual moleculardynamics［ J］. J Mol Graphics， 1996， 14： 33.

［33］ Li S， Zhang Z， Long Z， et al. Structures， stabilities andspectral properties of metalloborospherenes MB0/ -40 （M=Cu，Ag， and Au）［ J］. RSC Adv， 2017， 7： 38526.

［34］ An Y， Zhang M， Wu D， et al. Electronic transportproperties of the first all-boron fullerene B40 and itsmetallofullerene Sr@B40 ［J］. Phys Chem Chem Phys，2016， 18： 12024.

［35］ Zhai H， Zhao Y， Li W， et al. Observation of an allboronfullerene［ J］. Nat Chem， 2014， 6： 727.

［36］ Jin S， Sun W， Chen B， et al. Insights into the structuresand bonding of medium-sized cerium-doped boronclusters［ J］. J Phys Chem A， 2021， 125： 4126.

［37］ Mayer I. Charge， bond order and valence in the ab ini?tio SCF theory［ J］. Chem Phys Lett， 1983， 97： 270.

［38］ Mayer I. Improved definition of bond orders for correlatedwave functions［ J］. Chem Phys Lett， 2012， 544： 83.

［39］ Mulliken R. Electronic population analysis on LCAOMOmolecular wave functions，I ［J］. J Chem Phys，1995， 23： 1833.

［40］ Giambiagi M， Mundim K. Definition of a multicenterbond index［ J］. Struct Chem， 1990， 1： 423.

［41］ Matito E. An electronic aromaticity index for largerings［ J］. Phys Chem Chem Phys， 2016， 18： 11839.

［42］ Wiberg K. Application of the pople-santry-segalCNDO method to the cyclopropylcarbinyl and cyclobutylcation and to bicyclobutane ［J］. Tetrahedron，1968， 24： 1083.

［43］ Lu T， Chen F. Bond order analysis based on the laplacianof electron density in fuzzy overlap space ［J］. JPhys Chem A， 2013， 117： 3100.

［44］ Salvador M. Overlap populations，bond orders and valencesfor‘ fuzzy’ atoms ［J］. Phys Chem Phys Lett，2004， 383： 368.

［45］ Li H， Liu H， Lu X， et al. Cage-like Ta@Bqn complexes（n=23～28， q=?1～+3） in 18-electron configurationswith the highest coordination number oftwenty-eight［ J］. Nanoscale， 2018， 10： 7451.

［46］ Beck A. A simple measure of electron localization inatomic and molecular systems ［J］. J Chem Phys，1990， 92： 5397.

［47］ Zubarev D，Boldyrev A. Developing paradigms ofchemical bonding： Adaptive natural density partitioning［ J］. Phys Chem Chem Phys， 2008， 10： 5207.

［48］ Fa W， Chen S， Pande S， et al. Stability of metalencapsulatingboron fullerene B40 ［J］. J Phy Chem A，2015， 119： 11208.

［49］ Sergeeva A， Popov I， Piazza Z， et al. Understandingboron through size-selected clusters： Structure， chemicalbonding，and fluxionality［ J］. Accounts Chem Res，2014， 47： 1349.

［50］ Huang W， Sergeeva A， Zhai H， et al. A concentricplanar doubly π -aromatic B cluster ［J］. Nat Chem，2010， 2： 202.

［51］ He R，Zeng X. Electronic structures and electronicspectra of all-boron fullerene B40 ［J］. Chem Commun，2015， 51： 3185.

［52］ An W， Bulusu S， Gao Y， et al. Relative stability ofplanar versus double ring tubular structures of neutraland anionic boron cluster B20 and B-20 ［J］. Chem Phy，2006， 124： 15430.

（責(zé)任編輯： 于白茹）

基金項(xiàng)目： 四川省自然科學(xué)基金（2022NSFSC1243， 2022NSFSC1826）