Stability and Nonadiabatic Effects of the Endohedral Clusters X@Al12

Article pubs.acs.org/JPCC

Stability and Nonadiabatic Effects of the Endohedral Clusters X@Al12 (X = B, C, N, Al, Si, P) with 39, 40, and 41 Valence Electrons Bertha Molina,† Jorge R. Soto,*,† and Jorge J. Castro‡ †

Facultad de Ciencias, Universidad Nacional Autónoma de México, Apartado Postal 70-646, 04510 México D.F., México Departamento de Física, CINVESTAV del IPN, Apartado Postal 14-740, 07000 México D.F., México

‡

ABSTRACT: Metallic nanoclusters, with one electron or hole difference from closed shell, might end up in a degenerate state undergoing a Jahn−Teller distortion as a result of nonadiabatic effects, which are a manifestation of the electron−vibron coupling. Here, we report a theoretical study on the stability and nonadiabaticity of the neutral and charged endohedral clusters X@Al12 (X = B, C, N, Al, Si, P) around icosahedral symmetry, for 39, 40, and 41 valence electrons. The nonadiabatic effects are evaluated through the Jahn−Teller gain for the distorted cluster and their effect on the calculated electronic density of states is analyzed. For the 40 electron valence systems, we present the full vibrational spectra. Our results are discussed within the framework of the superatom model, and show that not all systems are well described by the spherical jellium model and that nonadiabaticity is better represented by ellipsoidal models. We present a detail discussion of the Al13−1 electron detachment process and show how, through a comparison with available experimental photoelectron spectroscopy data, the nonadiabaticity can be estimated.

1. INTRODUCTION The study of nanostructures has been a field of intense research, both theoretical and experimental for more than two decades. This has been basically motivated by the fact that the physical and chemical properties of atomic clusters are very different from their bulk phases, opening a wide field of novel behavior at nanoscale dimensions. One model that in spite of its simplicity has been successfully used for studying the electronic properties of many metal clusters is the jellium model. This is a simple electronic shell model, which was originally used to describe the electronic structure of alkali metal clusters,1−5 and has been generalized for nonalkali metal clusters.6 For the spherical jellium model, the valence electrons of the individual atoms move in a positive spherical potential, formed by the innermost electrons and the nuclei, producing a shell structure where the electrons are arranged in the states 1s21p61d102s21f142p6.... Hence, clusters containing 2, 8, 18, 20, 34, 40... electrons correspond to closed shell states and show high stability, high ionization potential, low electron affinity, and chemical inertness, similar to noble gases. One particular example where this model has been successfully applied is the gas phase of aluminum clusters.7,8 The valence electrons for the Al atom are 3s23p1, with an energy difference between the 3s and 3p states of 3.6 eV. Hence, for cluster sizes where the electron shell model holds, the valence electrons are delocalized implying that one of the 3s electrons should go to the 3p state as a result of the 3s and 3p states overlap.9 In bulk, Al is trivalent and has characteristics of an almost free electron metal. Within the jellium model, Al13 cluster behaves like a halogen atom, with an electron affinity comparable to Cl atom; Al14 © 2012 American Chemical Society

presents properties similar to an alkaline earth atom, and the anion Al13−1 being a closed shell cluster (with 40 valence electrons) mimics a noble gas atom.10 On the basis of those properties, which are reminiscent of the behavior of atoms in the periodic table, the idea of a new 3D periodic table formed by clusters, called superatoms has been proposed. This opens a myriad of possible applications of nanoclusters, with tailored physical and chemical characteristics, which are ultimately determined by their structure. Photoelectron spectra experiments of Al13−1 and theoretical analysis, based on density functional theory (DFT), have shown consistency with an icosahedral structure formed by a cage of 12 Al atoms on the surface with one Al atom at its inversion center as shown in Figure 1, with an electronic closed-shell structure with 40 electrons for Al13−1, and a quasi-icosahedral symmetry for Al13 as a consequence of a Jahn−Teller distortion.11−16 Theoretical and experimental work has demonstrated that several aluminum based doped clusters X@Al12 present higher stability when the dopant X is at the center of the icosahedral Al12 cage.17,18 The extra degree of freedom introduced by doping the Al12 cluster allows tuning their chemical and electronic properties, which can be used to construct specially tailored nanostructures. A case of particular interest is when the valence of the X atom couples with the valence of the cage structure to behave as a jellium-like cluster with super atom Received: January 13, 2012 Revised: March 27, 2012 Published: March 27, 2012 9290

dx.doi.org/10.1021/jp3004135 | J. Phys. Chem. C 2012, 116, 9290−9299

The Journal of Physical Chemistry C

Article

included in the symmetric product [Γ ⊗ Γ]= Γ2, with Γ the IR of the total electronic state.38 In this work, we report electronic and structural properties for the X@Al12±γ, (X = B, C, N, Al, Si, and P) clusters with γ being the necessary charge to complete 39, 40, or 41 valence electrons systems. We present a complete tabulation of the optimized geometries, electronic structures, density of states (DOS) and vibrational spectra for the 40 valence electrons endohedral clusters. For the 39 and 41 valence electrons clusters, the nonadiabatic effects are analyzed by estimating the Jahn−Teller stabilization energy and the possible distorted symmetries. We present a comparative analysis between our results for electron affinity and vertical detachment energy (VDE) with those from the photoelectron spectrum (PES) for the Al-centered cluster. We also compare the calculated electron affinities and vertical ionization potentials (vIPs) with available experimental data and reported theoretical calculations. Since we are interested in studying nonadiabatic properties, we restrict the analysis to those isomers with the dopant atom at the cage center. The work is based on allelectron full relativistic DFT calculations.

Figure 1. Schematization of the icosahedral structure optimized for the clusters with 40 valence electrons.

properties; being atoms in Groups IIIA, IV, and V of the Periodic Table some of the most studied.16−29 For the 40 valence electrons systems, which corresponds to the closed shell structure within the jellium model, the substitution of the Al central atom by the tetravalent atoms C and Si, as well as Al and B for the neutral and anionic clusters, respectively, results in a endohedral structure with Ih symmetry and noble gas superatom behavior;14,17−19,21−23,27,28,30 we should also expect to be the case for the cations of N and P.31 In the case of the clusters with 41 valence electrons, where there is one extra valence electron outside the jellium model closed shell, only P@Al12 and Si@Al12−1 have been reported as having icosahedral structure with an alkali superatom behavior;14,27,28 whereas the charging on the icosahedral C@ Al12 cluster to form C@Al12−1, shifts the C atom from the center to the surface of the cage.24,32 For N@Al12, we find in the literature controversial results concerning its structure and supertom properties; whereas some studies have found an icosahedral structure with a central N atom with superatom properties,20,33,34 others, report as a most stable structure, one that has the N atom at the surface of the cage, loosing therefore the superatom characteristics.25,35,36 For the other clusters Al13−2 and B@Al12−2, no references were found in the literature. The 39 valence electron system favors only Al13 and Si@ Al12+1 near icosahedral structures with magic cluster behavior with D3d reported symmetry.14,20 C@Al12+1 has a low symmetry Cs structure instead of an icosahedron.20 B@Al12 has been reported as consistent with superatom properties.37 For N@ Al12 + and P@Al12 +2, no references were found. Most of the 39 and 41 valence electrons clusters here studied possess an electronic structure with one-hole or one-electron in the highest occupied molecular orbital (HOMO) or lowest unoccupied molecular orbital (LUMO) with a degenerated irreducible representation (IR) in icosahedral symmetry. Hence, these structures are unstable under the Jahn−Teller effect because their total electronic states are degenerated. The Jahn−Teller theorem in its simplest form establishes that a symmetric nuclear configuration of a nonlinear molecule in a degenerated total electronic state (with exception of the 2-fold Kramer’s degeneracy) is unstable with respect to nuclear displacements, which remove the degeneracy. These displacements are in the direction of the Jahn−Teller (JT) active modes, which are the nontotally symmetric vibrational modes

2. COMPUTATIONAL DETAILS We performed, for the minimum electronic spin state, a geometry optimization calculation for endohedral aluminum clusters X@Al12±γ, X = B, C, N, Al Si, and P; γ indicates the system charge state corresponding to the X atom that yields clusters with 39, 40, and 41 valence electrons (each aluminum atom contributes with 3). We have optimized those clusters with 40 valence electrons constricting their symmetry to be icosahedral. For clusters with 39 and 41 valence electrons, we tried all possible distorted Jahn−Teller symmetries (D2h, D3d, D5d, and Th) as symmetry constrictions. For those clusters subject to Jahn−Teller distortion, we estimated the Jahn− Teller gain as the difference between the total energy for the high symmetry Ih cluster and the respective optimized distorted cluster. Since the adiabatic approximation is not valid in the higher symmetry cluster configuration, we found convergent self-consistent DFT states only for a single point calculation in a slightly distorted structure. This structure corresponds to that of the lowest energy selected among those that result from a distortion of 0.1 nj, where nj is the unitary vector that points in the jth Jahn−Teller distortion direction. The calculation was performed at a DFT level in the restricted spin−orbit zeroorder regular approximation (ZORA).39 The Perdew−Burke− Ernzerhof40 generalized gradient approximation (GGA) functional was used to the exchange and correlation terms. For all atoms, we used a standard Slater-type orbital all electron basis set, which can be described as a triple-ζ in the core, quadruple-ζ in the valence, with four sets of polarization functions (QZ4P). In order to improve the SCF convergence, an accuracy of 10−6 Hartree was selected, and for the optimization process, a gradient maximum limit of 10−5 Hartree/ Å was used. The geometry optimization was performed using a quasi Newton approach with the BFGS Hessian matrix update formula. In all cases, we carried out numerical second derivatives in order to obtain the Hessian and verify that the structure obtained after of the optimization is a minimum on the potential energy surface. The ADF (Amsterdam Density Functional) package41 was used for all calculations. 9291



Article

Table 1. Geometric and Electronic Parameters Obtained from the Geometry Optimization of the Clusters with 40 Valence Electronsa structure

elec. conf.

Eg

dc−v

dv−v

B@Al12−1 C@Al12 N@Al12+1 Al13−1 Si@Al12 P@Al12+1

1Ag21T1u62Ag21Hg102T1u61Gu81T2u62Hg03Ag0 1Ag21T1u62Ag21Hg102T1u61Gu81T2u63Ag02Hg0 1Ag21T1u62Ag21Hg102T1u61Gu81T2u63Ag02Hg0 1Ag21T1u62Ag21Hg101T2u62T1u61Gu82Hg01Gg0 1Ag21T1u62Ag21Hg102T1u61T2u61Gu82Hg03Ag0 1Ag21T1u62Ag21Hg102T1u61T2u61Gu82Hg03Ag0

2.19 2.02 1.35 1.87 2.00 2.09

2.54 2.54 2.56 2.66 2.63 2.64

2.67 2.67 2.69 2.79 2.77 2.77

a

Electronic configuration of the valence band (elec. conf.); HOMO−LUMO energy gap, Eg (eV); distance from the center to the vertex atoms, dc−v (Å); and vertex to vertex distance, dv−v (Å). All structures possess the Ih symmetry.

3. RESULTS AND DISCUSSION 3.1. Clusters with 40 Valence Electrons (Ih Symmetry). 3.1.1. Electronic Properties. Figure 1 shows the icosahedral structure obtained after of the optimization for all the 40 valence electron clusters. Table 1 displays the results of the calculation for the relevant distances, energy gaps, and electronic configurations for the Ih group. As can be seen in this Table, the largest distance between the central and vertex atoms dc−v was obtained for Al13−1, whereas the smallest corresponds to both B@Al12−1 and C@Al12. It is worth noticing that this follows the same increasing order of the corresponding central atom radii.42 Our calculated distances dv−v and dc−v differ at most by 0.03 Å from the corresponding values reported in the literature.14,21,28,29,37,43 With the exception of the N@Al12+1 cluster, the calculated HOMO−LUMO energy gaps (Eg in Table 1) are larger than those corresponding to C60 (1.57 eV) and Au20 (1.77 eV) clusters,44,45 and they agree with previously reported theoretical works within 0.15 eV.14,18,29 Since we are interested in the superatom behavior, we can express the electronic configuration in the decomposition of the angular momentum basis function, in the full rotational group, into the IRs of Ih: [s] → [A g] [p] → [T1u] [d] → [Hg]

(1)

Figure 2. DOS of the ground state structure for the icosahedral clusters. The spectra were shifted to set the highest occupied molecular orbitals in 0 eV.

In Figure 2 we show the result of a simulation of the DOS for the 40 valence electrons clusters, where we use a Lorentzian functions superposition with 0.06 eV of width. They present high, narrow, and well-defined peaks typically observed in structures with high symmetry. These peaks are representative of a shell structure and are labeled with their respective IR in both the Ih symmetry group and angular momentum basis. For comparison purposes, we have fixed the position of the HOMO at the origin. It should be noticed that in all cases the 1d and 1f shells are approximately at the same position (ca. −3 and 0 eV, respectively), whereas the rest of the shells are shifted toward the HOMO as a function of the atomic radius of the central atom for each period of the chemical table. It is worth commenting about the displacements shown for the 2s and 2p shells for those clusters whose central atom corresponds to the third period. From P to Al, the 2s shell has a progressive displacement toward the HOMO until an exchange with the 1d shell is observed for Al; the 2p shell presents the same behavior

until it exchanges with the 1f shell of the Al13−1, which presents a splitting of 0.3 eV, and the 2p shell lies on the middle of the splitting. This behavior is consistent with a mostly metallic bond type at the surface for Si and Al centered clusters, as recently found by Henry et al.46 However, the slight difference in the electron detachment energy reported might be connected with the difference in the position of the 2p shell for each cluster. In order to explore the possibility of whether the shift of the 2p shell is consistent with the size of the central atom, we optimized the 40 valence electrons icosahedral Mg@ Al12−2 cluster (Mg has the next large atomic radius in the third period after Al) and verified that it is a minimum on the potential energy surface. We observed an exchange of the 2p with the 1f shell, i.e., for this cluster, the 2p shell is localized at the HOMO. Obviously, this change in the orbital character of the HOMO modifies the chemical properties of this cluster compared with those of the clusters here studied. It should be point out that, as in the case of C and N, DFT calculations47

[f] → [T2u + Gu] [g] → [Hg + Gg ]

9292



Article

predict the Mg atom in Mg@Al12 at the cage. For those clusters whose central atoms correspond to the second period, from N to B, the 2s state also suffers a displacement toward the HOMO until a hybridization with the 1d shell is observed for B@Al13−1. We analyzed the shifts of the s/p states from the gross population density of states (GPDOS). The GPDOS is the weight factor of a basis function χμ (or a sum of such functions) in the molecular orbitals determined by means of a Mulliken population analysis. In Figures 3a,b and 4a,b, we show the

Figure 4. The GPDOS of the p orbitals of the central atom in X@Al12 with (a) X = B, C, and N and (b) X = Al, Si, and P.

increase of the Pauli repulsion with the cage electrons, being lower for P because of its smaller atomic radius. To better understand the metallic or covalent bonding behavior, we calculated the electron localization function (ELF), which permits to visualize the electronic shell structure of clusters in a real-space representation by distinguishing the electronic regions with metallic or covalent bonding.48 Metallic bonding is characterized by an ELF around 0.5 at any electronic density, while covalent bonding has an ELF near its maximum value (1.0), indicating that the electrons are highly localized; very low values of ELF indicate a poor electronic density. In Figure 5a,b, we show the contour plots of the ELF in two perpendicular planes crossing the central atom for the icosahedral P@Al12+1 cluster as an example. The plane in Figure 5a contains four aluminum atoms forming a parallelogram and the central dopant atom, while the plane in Figure 5b only contains the central atom. In both planes, we can see central regions where the ELF emulate the behavior of the doping atom with delocalized electron regions (s type: ELF = 0.5) and localized electron regions (p type: ELF around 0.75). At the cluster surface, the bonding with the central atom is mainly metallic in character in the plane (b), and there are electrons with an ELF around 0.75 distributed on the cage surface and electrons more localized on the aluminum atoms

Figure 3. The GPDOS of the s orbitals of the central atom in X@Al12 with (a) X = B, C, and N and (b) X = Al, Si, and P.

GPDOS corresponding to the s/p valence orbitals for the central atoms as a function of the energy. Basically, these GPDOS are composed of two s/p peaks and coincide with the 1s2s/1p2p states of the superatom model. This coincidence arises because the spherical potential in the jellium model is centered at the central atom. This fact confirms that the shifts of the 2s/2p states observed in Figure 2 are strongly related to the central atom electronic contributions. In all cases shown in Figures 3 and 4, the GPDOS contribution 1p/2p is greater than 1s/2s (except C and N where 1p < 1s). This indicates a greater hybridization between the p states of the central atom and the cage p states, and a greater Pauli repulsion between the cage s states and the central atom. Also, the GPDOS contribution 1p is less than 2p for B, C, and N; however, the GPDOS contribution 1p is greater than 2p for Al, Si, and P due to an 9293



Article

have nine different normal modes of vibration in icosahedral symmetry. The decomposition of its vibrational representation is Γ = a g + gg + 2hg + 2t1u + t 2u + g u + h u

(2)

In this γ-representation, two normal modes are infrared active (2t1u), three are Raman-active (ag + 2hg), and the remaining four are optically silent, all to first order. The calculated values for the frequencies of each complex are displayed in Table 2. The infrared intensities are shown in parentheses. It is worth noting that the sum of intensities of the two infrared modes is around 68 km/mol for B, C, and N and around 17 km/mol for Al, Si, and P. For these last complexes, the intensity of the 2t1u mode is very low. This behavior might be connected with properties of intensity sum rules.49 It should be noted that the infrared-active modes were calculated and reported previously by Charkin et al50 using the hybrid DFT B3LYP/6-31G* method. Our frequencies are consistently above 20 to 30 cm−1, with the exception of N, where the 1t1u mode is 50 cm−1 and the 2t1u is just 11 cm−1. In Figure 6 we show a few selected normal modes of vibration. The ag mode (unique nondegenerate mode) is the completely symmetric breathing mode and involves identical radial displacements for the 12 aluminum atoms of the cage. The endohedral cluster can be seen as one centered atom between two parallel pentagons with two atoms localized in front of each center as apexes. The hg(1) mode squashes the apex atoms along the 5-fold axis, and the hg(2) also squashes the pentagons along the same axis but in the opposite direction. The hu mode rotates the two pentagons in opposite form around the 5-fold axis. The t2u mode simultaneously compresses one pentagon and decompresses the other, perpendicular to the 5-fold axis. The infrared-active modes t1u(1) and t1u(2) are the basis of the three-dimensional (3D) vectorial representation (x,y,z). In the first mode, the normal modes are more intense at the apical cage atoms, while in the second it is at the center atom X. In order to better understand the bonding between the central atom and the Al12 cage, we did a vibrational mode analysis in a similar way as it is done for frustrated rotations in molecular endohedral fullerenes.51 In our case, we only have frustrated translational modes represented by the t1u modes, schematized in Figure 6. Unlike the frustrated rotational modes of the endohedral fullerenes, the t1u modes are in the highest part of the vibrational spectra (Table 2). For each period, we found a linear correlation between the frequency of one of these modes with [1/mred]1/2, where mred is the reduced mass of a hypothetical X-Al12 harmonic oscillator. The regression coefficients are r2 = 0.9886 and 0.9998 for the 1t1u mode (B, C, and N) and the 2t1u mode (Al, Si, and P) respectively. From this, we end up with two force constants for each of the corresponding harmonic oscillators: k1 = 7.48 N/cm (B, C, and N) and k2 = 27.31 N/cm. This shows that the bonding of the central atom and the cage, which is related to the force constants, is the same for atoms in each period, and increases with the period. The first is because, regardless of the nominal valence electrons of the central atom, the total charge is compensated in order to have the same total valence for the system. In the third period, the number of electrons of the central atom is greater, and since the cage radius is approximately the same, the electron density at the interior

Figure 5. ELF for the P@Al12+1 cluster. (a) Contour plot in the plane crossing five aluminum atoms including the central one. (b) Contour plot in the plane perpendicular to that in panel a, crossing only the central dopant atom. (c) Isosurface at ELF = 0.82.

forming protrusions that reach values of ELF up to 0.85 (see perpendicular plane (a)). In Figure 5c we show the localized electron regions with an ELF isosurface of 0.82. From this analysis, we conclude that the bonding between the cage and the central atom is metallic, which is well represented by the jellium model. On the cluster surface there are localized electrons, similar to some aluminum crystalline surfaces.48 A similar behavior is observed for the rest of the clusters. 3.1.2. Vibrational Analysis. According to group theory, the 40 valence electrons endohedral aluminum clusters X@Al12 9294



Article

Table 2. Frequencies (cm−1) for the 40-Electron Endohedral Clusters X@Al12 for Each IRa

a

cluster\IR

ag

gg

1hg

2hg

1t1u

2t1u

t2u

gu

hu

B@Al12−1 C@Al12 N@Al12+1 Al13−1 Si@Al12 P@Al12+1

315 313 304 288 292 287

244 243 236 197 199 201

239 213 191 261 258 246

353 362 358 299 306 312

283(26) 241(15) 155(37) 185(18) 201(13) 204(12)

357(45) 299(51) 290(31) 384(0.2) 358(2) 300(7)

165 127 94 213 202 183

312 322 314 230 247 256

145 137 129 99 89 89

The infrared intensities (km/mol) are in parentheses.

Table 3. Geometric and Electronic Parameters Obtained from the Geometry Optimization of the Clusters with 39 Valence Electronsa structure

EJT

Eg

dc−v

dv−v

D2h B@Al12

49.3

2.04

D3d C@ Al12+1 D3d N@ Al12+2 D3d Al13 D3d Si@ Al12+1 D3d P@ Al12+2

71.8

1.96

2.45, 2.54, 2.58 2.46, 2.59

2.60, 2.62, 2.65, 2.66, 2.67, 2.74 2.60, 2.65, 2.67, 2.71

105.9

1.23

2.47, 2.63

2.62, 2.67, 2.71, 2.75

187.8 92.4

1.61 1.84

2.65, 2.69 2.64, 2.67

2.75,2.75,2.86, 2.94 2.72, 2.74,2.83, 2.92

82.3

1.94

2.66, 2.67

2.74, 2.77,2.84, 2.93

a

The Jahn−Teller gain (EJT, meV), HOMO−LUMO energy gap (Eg, eV), distance from the center to the vertex atoms (dc−v, Å) and distance from the vertex to vertex atoms (dv−v, Å).

coupling are taken into account, we can have D5d-type or D3dtype minima with D3d-type or D5d-type maxima, respectively.52 We obtained D3d-type minima in C and N. For B, was achieved a deformed D2h symmetry, which disagrees with the quadratic approximation. For Al, Si, and P we have a G ⊗ (g + h) vibronic problem whose solution, in the linear coupling approximation, can have Th-type minima with D3d saddle points (g modes dominate) or D3d-type stationary points with D2h saddle points (h modes dominate),52 being the last one compatible with our results for the three clusters. In Table 3 we also show the EJT (Jahn− Teller stabilization energy), HOMO−LUMO energy gaps Eg, dv−v, and dc−v. Comparing the dc−v of these clusters with the corresponding dc−v of the icosahedral structure, we observe that the symmetry breaking from Ih toward D3d symmetry slightly contracts six of the dc−v distances and elongates the other six in such a way that this distortion can be compared with an oblate spheroid, which is in good agreement with the spheroidal or ellipsoidal jellium model.53 This model was proposed to describe the behavior of the electronic open-shell structures as an improvement to the jellium model, which fails when it is applied to this kind of system. Specifically, the spheroidal jellium model predicts that any molecular system with its last degenerate electronic shell, incomplete for only one electron, will suffer an aspherical distortion, resulting in an oblate spheroid. If the last degenerate electronic shell has only one electron, the aspherical distortion of the cluster will result in a prolate spheroid. From the DOS of the clusters with 39 valence electrons (Figure 7), we observed in all cases a diminution of the intensity of the peaks and a small shift toward more negative energy values compared with those of the icosahedral structure. Also clear is the splitting of the degenerate modes of the corresponding original icosahedral structure when going to the

Figure 6. Selected vibrational modes of the 40-electron endohedral cluster X@Al12.

of the cage increases with a corresponding decrease of the states available for the excitation of the normal mode. 3.2. Clusters with 39 Valence Electrons. For the 39 electron valence clusters, the hole symmetry for the total molecular wave function is given in Table 1: 2T2u for B, C, and N central atoms, and 2Gu for the remaining clusters. From the Jahn−Teller theorem we know that these clusters may suffer a distortion that breaks their icosahedral symmetry through a vibrational coupling. The normal modes that indicate the distortion direction would be included in the symmetric products [T2u]2 or [Gu]2 respectively. This yields the following JT-active modes: hg for B, C, and N, and (gg + 2hg) for Al, Si, and P. This fact limits the possible symmetry-breakings from Ih to D2h, D3d, or D5d for B, C, and N, and also Th for the remaining clusters. From our calculations, we found the predominant symmetry-breaking mode for each case (Table 3). For B, C, and N, the electron−vibron coupling problem is of the type (T2 ⊗ h); if the quadratic terms of the vibronic 9295



Article

3.3. Clusters with 41 Valence Electrons. From Table 1 we can observe the one-electron symmetry of the total molecular wave function, corresponding to the occupation of the LUMO for the unpaired last electron. This symmetry is 2Hg in all cases with exception of C and N. In these last cases, we have a nonspatially degenerate total electron symmetry 2Ag without the possibility of JT-breaking. Then, for B, Al, Si, and P we have an H ⊗ (g + h) vibronic problem with pentagonal D5d (h dominant) or trigonal D3d (g dominant) possible minima (mutually excluded) with D2h saddle points in the linear vibronic coupling approximation.52 Our results, shown in Table 4 confirm the validity of this approach, obtaining D5d as the final symmetry in all cases. Table 4. Geometric and Electronic Parameters Obtained from the Geometry Optimization of the Clusters with 41 Valence Electronsa structure

EJT

Eg

dc−v

dv−v

D5d B@Al12−2 Ih C@Al12−‑1

166.2 0.0 0.0 56.7 73.7 131.1

0.36 0.33 1.05 0.11 0.16 0.23

2.52, 2.79 2.55 2.56 2.65, 2.83 2.61, 2.83 2.60, 2.85

2.63, 2.69,2.77 2.68 2.69 2.77, 2.81,2.87 2.73, 2.78,2.84 2.72, 2.78, 2.85

Ih N@Al12 D5d Al13−2 D5d Si@Al12−1 D5d P@Al12

a The Jahn−Teller gain (EJT, meV), HOMO−LUMO energy gap (Eg, eV), distance from the center to the vertex atoms (dc−v, Å), and distance from the vertex to vertex atoms (dv−v, Å).

Figure 7. The DOS of the ground structure for the clusters with 39 valence electrons. All spectra were shifted to set all the highest occupied molecular orbitals in 0 eV.

As previously mentioned, C and N do not suffer any symmetry breaking and show a practically pristine 40-electron structure in both cases. The largest and smallest E JT corresponds for B and Al respectively, which is exactly the opposite behavior as for the 39-electron hole symmetry. The linear tendency of the EJT with the LUMO splitting is also observed as in the case of the HOMO in the 39-electron clusters. Comparing dc−v of the clusters with 41 valence electrons with the corresponding dc−v for the icosahedral structure, we note that the symmetry breaking from Ih toward D5d symmetry slightly contracts 10 dc−v distances and elongates the other two, resulting in a volume that can be compare with a prolate spheroid. Again, this is in good agreement with the spheroidal jellium model.53 As in the case of the structures with D3d symmetry, the splitting and the diminution of the intensity of the peaks (see Figure 8) is due to the symmetry breaking seen experimentally for the icosahedral isomers. In particular, the pristine Ih HOMO for B undergoes the splitting

deformed one, according to the decomposition given in eqs 3−5. It should be noted that this is in agreement with the spheroidal jellium model, which shows that the electronic spherical shells will split into subshells under shape deformation. In all cases the pristine Ih HOMO undergoes a splitting, which for B is [T2u]Ih → [B1u + B2u + B3u]D2h

(3)

while for C and N is [T2u]Ih → [A 2u + E u]D3d

(4)

and for Al, Si, and P [Gu]Ih → [A1u + A 2u + E u]D3d

(5)

[T2u]Ih → [A 2u + E 2u]D5d

Also, from our results for the DOS, we can see that the HOMO splitting shows a linear dependence with respect to the EJT, with different slopes for period. This fact is reminiscent of a metal−ligand model,54 indicating that the central atom is an electron donator to their coordinated atoms (aluminum cage). The 2Ag (2s) and 2T1u (2p) “itinerant” states have the same behavior as that in the case of the icosahedral clusters (compare Figures 2 and 7). All the HOMO−LUMO energy gaps decrease slightly with respect to those of the clusters with 40 valence electrons (see Table 3). The maximum change (260 meV) of the HOMO−LUMO energy gap is calculated for the Al13 structure, while the minimum (60 meV) is obtained for C@ Al12+1. Again, excluding the N@Al12+2 cluster, all HOMO− LUMO energy gaps are larger than that of C60.

(6)

whereas that for Al, Si, and P corresponds to [Gu]Ih → [E1g + E 2g]D5d

(7)

However, in these cases, the pristine Ih the LUMO splitting is more relevant. Thus, for the isomers with D5d symmetry, the LUMO of the respective icosahedral cluster undergoes the splitting [H u]Ih → [A1g + E1g + E 2g]D5d

(8)

−1

From the DOS of C@Al12 and N@Al12, is clear that the extra electron occupies the corresponding LUMO 3Ag (3s) of C@Al12 and N@Al12+1, with the new HOMO−LUMO gap 9296



Article

neutral at the geometry of the icosahedral anion E(neutral, anion Ih) cannot be defined. However, we can make an estimate by assuming a slightly distorted geometry toward the relaxed structure in order to obtain a self-consistent DFT electronic state. The DFT adiabatic detachment energy (ADE), being the energy difference between the anion and the neutral relaxed into its nearest local minimum (distorted symmetry), can be calculated due that both energies are adiabatically well-defined. In this approximation, the difference: ΔDE = VDE − ADE = E(neutral, anion Ih) − E(neutral, distorted symmetry)

(9)

is well-defined and, in absence of excitations or vibrational contributions, corresponds to the detachment relaxation energy, and should be slightly larger than the Jahn−Teller stabilization energy or Jahn−Teller gain (EJT) defined by E JT = E(neutral, neutral Ih) − E(neutral, distorted symmetry)

(10)

where E(neutral, neutral Ih) is the energy resulting from the Ih restricted optimization of the neutral cluster. This energy is slightly lower than E(neutral, anion Ih), and for this reason ΔDE > EJT. It is exemplified in Figure 9 for the Al13−1 case,

Figure 8. The DOS of the ground structure for the clusters with 41 valence electrons. All spectra were shifted to set all the highest occupied molecular orbitals in 0 eV.

equal to the LUMO−LUMO+1 of the closed shell structure. In all other cases, the lowest energy structure is obtained when the extra electron occupies the A1g nondegenerate state coming from the splitting of the LUMO with IR 1Hg (eq 8). Naturally, we got narrow HOMO−LUMO energy gaps (see Table 4) due to the small splitting obtained for Hg. Compared with N@Al12+1, a small change is obtained in the HOMO−LUMO energy gap of the N@Al12 (it diminishes 0.30 eV). Instead, for the C@Al12−1 cluster, a large reduction (1.69 eV) of its HOMO−LUMO energy gap is calculated compared with that of the C@Al12. A similar reduction is obtained for the structures with D5d symmetry. For C and N we have observed that the cluster with 42 valence electrons probably close shell in D5d symmetry filling the A1g state (see Table 1). If this is correct and the HOMO−LUMO energy gap is broad, these clusters could be magic clusters. In order to prove this hypothesis, we optimized the C@Al12−2 and N@Al12−1 with D5d symmetry. We obtained stability for C@Al12−2, even having a very small HOMO−LUMO energy gap (0.38 eV); whereas N@Al12−1, who has an electronic closed shell structure, presents three imaginary frequencies, therefore making it nonstable, i.e., subshell closure does not guarantee the stability of the cluster as might be expected from the spheroidal jellium model. 3.4. Electron Affinities, Adiabatic Ionization Potentials, and PES. When nonadiabatic effects are important, as in the case of detachment processes for the icosahedral B and Al center clusters here studied, the VDE, which is the energy difference between the anion and the neutral, at the geometry of the anion has no meaning since the DFT energy of the

Figure 9. Experimental photoelectron spectra of Al13− at 193 nm.25 Reproduced with kind permission from the European Physical Journal (EPJ). The vertical red lines indicate the calculated ADE and the VDE.

where we indicate with vertical red lines the positions of the calculated ADE and VDE superimposed on the experimental PES of Al13−1 at 193 nm.25 Since the ADE coincides with the electron affinity (EA), we aligned our ADE with the experimentally reported value of the EA (3.57 eV32). The calculated VDE, which is separated by approximately 0.2 eV of the ADE value, is in agreement within the experimental uncertainty with the electron binding energy (EBE) value corresponding to the highest intensity peak of the lower transition band, which is normally considered the experimental position of the VDE. This result confirms the above-mentioned statement that the ΔDE difference (eq 9) should be slightly larger than EJT (0.19 eV) (see Table 3). For the Boron case, the ΔDE is 0.07 eV, also slightly larger than the calculated EJT (0.05 eV), both values being on the order of their respective experimental and DFT resolution. The calculated ADE for Si@Al12 (1.64 eV) compares very well with the experimentally EA reported by Akutsu et al. (1.69 eV27). This value is in better agreement with the experimental value than the theoretical range previously reported in the literature (1.80−196 eV14). We have also calculated vIP for the Al, Si, and P neutral clusters, which can be compared with the experimental values available in the literature. The vIP is 9297



Article

(3) Beck, D. E. Solid State Commun. 1984, 49, 381. (4) Chou, M. Y.; Cleland, A.; Cohen, M. L. Solid State Commun. 1984, 52, 645. (5) de Heer, W. A.; Knight, W. D.; Chou, M. Y.; Cohen, M. L. Solid State Phys. 1987, 40, 93. (6) Brack, M. Rev. Mod. Phys. 1993, 65, 677. (7) Li, X.; Wu, H.; Wang, X.-B.; Wang, L.-S. Phys. Rev. Lett. 1998, 8, 1909. (8) Reveles, J. U.; Khanna, S. N.; Roach, P. J.; Castleman, A. W., Jr. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 18405. (9) Xi, L.; Hongbin, W.; Xue-Bin, W.; Lai-Sheng, W. Phys. Rev. Lett. 1998, 81, 1909. (10) (a) Castleman, A. W., Jr.; Khanna, S. N. J. Phys. Chem. C 2009, 113, 2664. (b) Bergeron, D. E.; Castleman, A. W., Jr.; Morisato, T.; Khanna, S. N. Science 2004, 304, 84. (11) Fowler, J. E.; Ugalde, J. M. Phys. Rev. A 1998, 58, 383. (12) Rao, B. R.; Khanna, S. N.; Jena, P. Phys. Rev. B 2000, 62, 4666. (13) Khanna, S. N.; Rao, B. K.; Jena, P. Phys. Rev. B 2002, 65, 125105. (14) Henry, D. J.; Varano, A.; Yarovsky, I. J. Phys. Chem. A 2008, 112, 9835. (15) Aguado, A.; López, J. M. J. Chem. Phys. 2009, 130, 064704. (16) Ma, L.; Issendorff, B. V.; Aguado, A. J. Chem. Phys. 2010, 132, 104303. (17) Khanna, S. N.; Jena, P. Phys. Rev. Lett. 1992, 69, 1664. (18) Gong, X. G.; Kumar, V. Phys. Rev. Lett. 1993, 70, 2078. (19) Kumar, V.; Sundararajan, V. Phys. Rev. B 1998, 57, 4939. (20) Nayak, S. K.; Khanna, S. N.; Jena, P. Phys. Rev. B 1998, 57, 3787. (21) Charkin, O. P.; Charkin, D. O.; Klimenko, N. M.; Mebel, A. M. Chem. Phys. Lett. 2002, 365, 494. (22) Li, G. F.; Peng, P. D.; Zhou, W.; Han, S. C. Comput. Mater. Sci. 2009, 47, 302. (23) Kumar, V.; Bhattacharjee, S.; Kawazoe, Y. Phys. Rev. B 2000, 61, 8541. (24) Li, S. F.; Gong, X. G. Phys. Rev. B 2004, 70, 075404. (25) Li, X.; Wang, L.-S. Eur. Phys. J. D 2005, 34, 9. (26) Wang, B.; Zhao, J.; Shi, D.; Chen, X.; Wang, G. Phys. Rev. A 2005, 72, 023204. (27) Akutsu, M.; Koyasu, K.; Atobe, J.; Hosoya, N.; Miyajima, K.; Mitsui, M.; Nakajima, A. J. Phys. Chem. A 2006, 110, 12073. (28) Li, S. F.; Gong, X. G. Phys. Rev. B 2006, 74, 045432. (29) Chen, G.; Kawazoe, Y. J. Chem. Phys. 2007, 126, 014703. (30) Nakajima, A.; Kishi, T.; Sugioka, T.; Kaya, K. Chem. Phys. Lett. 1991, 187, 239. (31) Xie, R.-H.; Bryant, G. W.; Zhao, J.; Kar, T.; Smith, V. H., Jr. Phys. Rev. B 2005, 71, 125422. (32) Li, X.; Wang, L.-S. Phys. Rev. B 2002, 65, 153404. (33) Wang, B.; Shi, D.; Chen, X.; Wang, G.; Zhao, J. Int. J. Mod. Phys. B 2005, 19, 2380. (34) Wang, B.; Zhao, J.; Shi, D.; Chen, X.; Wang, G. Phys. Rev. A 2005, 72, 023204. (35) Bai, Q.; Song, B.; Hou, J.; He, P. Phys. Lett. A 2008, 372, 4545. (36) Sharma, H.; Garg, I.; Dharamvir, K.; Jindal, V. K. J. Comput. and Theor. Nanos. 2010, l7, 2297. (37) Wang, L.; Zhao, J.; Zhou, Z.; Zhang, S. B.; Chen, Z. J. Comput. Chem. 2009, 30, 2509. (38) Bersuker, I. B. The Jahn−Teller Effect; Cambridge University Press: Cambridge, U.K., 2006. (39) (a) van Lenthe, E.; Baerends, E. J.; Snijders, J. G. J. Chem. Phys. 1993, 99, 4597. (b) van Lenthe, E.; Baerends, E. J.; Snijders, J. G. J. Chem. Phys. 1994, 101, 9783. (c) van Lenthe, E.; Snijders, J. G.; Baerends, E. J. J. Chem. Phys. 1996, 105, 6505. (40) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865. (41) ADF2009.01, SCM; Vrije Universiteit: Amsterdam, The Netherlands, 2009; www.scm.com (42) See the web element table at www.webelements.com. (43) Ashman, C.; Khanna, S. N.; Liu, F.; Jena, P.; Kaplan, T.; Mostoller, M. Phys. Rev. B 1997, 55, 15868.

calculated as the total energy difference between the cation and neutral clusters, where the cation is formed by ionizing the neutral clusters and carried out a single point calculation. The vIP values obtained for Al and Si (6.68 and 6.93 eV, respectively) are in good agreement with the corresponding experimental values (

Stability and Nonadiabatic Effects of the Endohedral Clusters X@Al12

Recommend Documents