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Structural, electronic and magnetic properties of binary transition metal aluminum clusters: absence of electronic shell structure
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 015006 (http://iopscience.iop.org/0953-8984/26/1/015006) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 015006 (8pp)
doi:10.1088/0953-8984/26/1/015006
Structural, electronic and magnetic properties of binary transition metal aluminum clusters: absence of electronic shell structure Vikas Chauhan1 , Akansha Singh1 , Chiranjib Majumder2 and Prasenjit Sen1 1 2
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India Chemistry Division, Bhabha Atomic Research Centre, Mumbai, India
Received 29 August 2013, in final form 17 October 2013 Published 25 November 2013 Abstract
Single Cr, Mn, Fe, Co and Ni doped Al clusters having up to 12 Al atoms are studied using density functional methods. The global minima of structure for all the clusters are identified, and their relative stability and electronic and magnetic properties are studied. FeAl4 and CoAl3 are found to have enhanced stability and aromatic behavior. In contrast to binary transition metal alkali and transition metal alkaline earth clusters, spherical shell models cannot describe the electronic structure of transition metal aluminum clusters. S Online supplementary data available from stacks.iop.org/JPhysCM/26/015006/mmedia (Some figures may appear in colour only in the online journal)
π -electrons are responsible for the aromaticity of the Al2− 4 unlike benzene-like molecules [9]. unit in MAl− 4 Another unusual feature of small Al clusters is an increase of ionization potential (IP) and electron affinity (EA) values. Early ab initio calculations found the ionization potential (IP) of the small Aln cluster (n < 6) to be higher than that of the Al atom [10]. This is rather unexpected if one considers the metal cluster as a classical metallic sphere. In the case of alkali clusters, the IP of a cluster always turns out to be lower than that of the atom. At small cluster sizes, electron affinity (EA) was found to rise sharply in these calculations. This was later confirmed in anion photoelectron spectroscopy (PES) experiments [11]. This was considered to be an indication of the lack of sp hybridization in the Al atoms at such small cluster sizes. We will return to this topic towards the end of the paper. Whereas pure Al clusters have been studied extensively, and are still being studied with the help of new experimental techniques [12], there have also been a few works on transition metal (TM)–Al clusters. In an old work Gong and Kumar studied magnetic properties of TM doped Al12 icosahedral clusters [13]. This study was confined to only
1. Introduction
Electronic shell models have played a central role in understanding the properties of alkali, alkaline earth and coinage metal clusters over the last three decades [1–4]. Clusters having 2, 8, 18, 20, 34, 40 . . . valence electrons are found to have completely filled electronic shells resulting in enhanced stability compared to neighboring sizes. When it comes to trivalent elements such as aluminum, the situation gets complicated. Experimental observations on Al clusters containing ∼7 atoms or more have been explained in terms of shell models. Anion PES experiments by Taylor et al [5] found signatures of shell closings at 20, 40, 58 and 70 electrons. A subsequent photoionization spectroscopy study by Schriver et al [6] found shell closings at 20, 40, 50, 68, 84, . . . electron numbers. Of these, shell closing at 50, however, could not be explained by spherical, spheroidal or ellipsoidal shell models [7]. In order to explain the stability of smaller Al clusters, the concept of aromaticity has been invoked. In their work on MAl− 4 (M = Li, Na, Cu) clusters, Li et al [8] introduced the idea of all-metal aromaticity for the first time. Later research showed that the σ -electrons rather than the 0953-8984/14/015006+08$33.00
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one size. More importantly, a thorough structure search was not undertaken. Hence many of the conclusions reached need further verification. Reddy et al studied the structural and magnetic properties of (FeAl)n clusters for n ≤ 6 [14]. They found an initial increase and then a decrease of magnetic moment with size. Menzes and Knickelbein measured the IP and reactivity of Co–Al clusters [15]. Pramann et al studied − the PES and reactivity of CoAl− n (n = 8–17) and Con Alm (n = 6, 8 10; m = 1, 2) clusters [16]. They found hybridization of Al s and p and TM d states in these clusters. CoAl− 12 was found to be particularly resistant to reactions. This was explained in terms of a 40 electron filled shell configuration. Harms et al − [17] explained the inertness of VAl− 6 and NbAl4 clusters in their reactions towards O2 in terms of electronic shell models. Lang et al studied the structure of V, Cr and Ti doped Al cluster cations over a large size range of 5–35 Al atoms [18]. Using the ability of the clusters to physisorb Ar atoms, it was concluded that the V and Cr atoms are completely encapsulated by an Al+ n cage at n = 16, and the Ti atom is encapsulated between n = 19 and 21. More recently, Wang et al studied single 3d TM doped Aln clusters for n = 1–7 and 12 using first-principles electronic structure calculations [19]. In addition to finding the ground state structures and spin multiplicities of these clusters, a major conclusion was that all TMAl3 clusters show enhanced stability. The magnetic properties of some of the TMAln clusters were explained using the shell model. For example, it was claimed that CoAl3 and CrAl4 are singlets because both these have an eighteen electron filled shell configuration. However, if CrAl4 is indeed an eighteen electron filled shell cluster, one would expect it to have enhanced stability, which was not found by these authors. Although the shell model was invoked to explain the properties of some of the clusters, a correspondence of the molecular orbitals (MOs) of these clusters with the shell orbitals was not established. In light of the fact that pure Al clusters at large sizes follow the shell model quite well, their claim that TM–Al clusters deviate from the spherical shell model for n > 4 was interesting and requires a closer inspection. The size range n = 8–11 was also not studied. In our recent works on TM-alkaline earth clusters we have shown that the interplay of crystal-field effect and Hund’s coupling can lead to subshell filling at unconventional electron counts, resulting in enhanced stability and large magnetic moments [20–22]. FeMg8 , FeCa8 , CrSr9 and MnSr10 were found to have enhanced stability though the first three have 24 and the last one has 27 valence electrons. In fact, these are all examples of magnetic superatoms. The crystal-field effect is expected to play a bigger role in Al clusters since the nuclear charge in these clusters is not screened as well as in alkali or alkaline earth clusters [6, 10]. The crystal-field effect has been observed in the reactivity of the CuAl22 cluster [23]. Therefore, it is interesting to research whether such interplay of crystal-field effect and Hund’s coupling leads to enhanced stability in TM–Al clusters. Since stable clusters can occur at unconventional electron counts, one has to perform first-principles calculations on a series of clusters to identify them. With this motivation, we decided to study the structural, electronic and magnetic properties of single Cr, Fe and
Mn doped Al clusters in the size range of 2–12 Al atoms. However, for reasons discussed below, we were led to study Co and Ni doped Al clusters as well over a limited size range. We identify the stable clusters and try to understand the origin of their stability in terms of electronic and geometric structures. The significant finding is that FeAl4 and CoAl3 clusters possess enhanced stability compared to other clusters. However, this is not due to shell filling within the electronic shell models. Rather, aromaticity plays a role in determining the stability of these clusters. Unlike in pure Al clusters, the electronic structure of TM–Al clusters does not correspond to the electronic shell models. The rest of the paper is organized as follows. In section 2 we present the theoretical methods that have been used to study the clusters. In section 3 the results obtained are presented, and attempts are made to rationalize these. Finally, we draw our conclusions in section 4. 2. Method
All of our first-principles electronic structure calculations were performed within the density functional theory (DFT) using two different approaches in different cases. In one approach, molecular orbitals were expressed as linear combinations of Gaussian type orbitals centered on the atomic nuclei. deMon2k [24] and Gaussian03 [25] codes were used for these calculations. The exchange–correlation effects were taken into account using the generalized gradient approximation (GGA) functional proposed by Perdew, Burke and Ernzerhof (PBE) [26]. In deMon2K these functionals were calculated through a numerical integration from the orbital density. Calculation of four-center integrals was avoided by a variational fitting of the Coulomb potential. The auxiliary density was expanded in primitive Hermite Gaussians using the GEN-A2 auxiliary function set. This contains s, p, d, f and g auxiliary functions, and adapts automatically to the chosen orbital basis set. For all the atoms we used double-ζ valence plus polarization basis sets in both deMo2K and Gaussian03. These have been optimized for the 3d TM atoms for GGA functionals by Calaminici et al (DZVP-GGA) [27]. In a second approach, a plane wave basis set was used along with PAW potentials in conjunction with the USPEX [28, 29] code for a global search of the lowest energy structure of the clusters as described in detail below. Unless specified otherwise, the results presented are obtained using the deMon2K code. Finding the ground state structure of a cluster is a challenging task. For clusters containing a few atoms one can use chemical intuition, or start from possible symmetric structures and relax them to the nearest minima using an optimization technique such as the conjugate gradient method. However, the number of local minima increases exponentially with cluster size, and one becomes less confident about having found the global minimum. Hence we adopt a dual approach here. In the first approach, that we prefer to call the educated guess, we construct a number of initial structures from our knowledge of structures of Al or other metal clusters. For a particular size Aln , known structures of Aln+1 are taken and different inequivalent Al atoms are 2
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Figure 1. Ground state structure and multiplicity of Cr-, Mn- and FeAln clusters for n = 2–12 .
the different numerical methods used in the two codes. Our analysis is based on the deMon2K results.
replaced by the TM atoms. We also construct other possible symmetric structures, and possible cage-like structures with the TM atom both inside and outside the cage. We also took structures found for TM-alkali and TM-alkaline earth clusters, replaced these metal atoms with Al and adjusted the bond lengths using the Avogadro code [30]. For all these initial structures we consider a number of possible spin states and relax each of these to the nearest local minimum without any symmetry constraints. We use deMon2K for these calculations. To ensure that the optimized structures are truly local minima, harmonic frequencies are calculated for each one of them. If any structure turns out to have imaginary frequencies, it is re-optimized by distorting it along the unstable vibrational modes. In the second approach we perform a direct global search for the minimum energy structure of these clusters based on an evolutionary algorithm using the USPEX code [28, 29]. For all of the USPEX calculations energies are calculated using the plane wave PAW approach. VASP [31] code is used for these calculations. More details about these calculations are given in supplementary information (available at stacks.iop.org/JPhysCM/26/015006/ mmedia). Whenever we find a minimum of structure in USPEX that is different from the lowest energy structure obtained in the educated guess approach, we further relax it in deMon2K and compare the energies of the two structures. It turns out that in most cases the two approaches produced the same lowest energy structures. For CrAl6 and FeAl7 the ground state structures obtained by USPEX were higher energy isomers in the educated guess approach, and vice versa. The energy difference between the two isomers in deMon2K was small for both of the clusters: 0.021 eV for CrAl6 and 0.21 eV for FeAl7 . This difference could be due to
3. Results and discussion 3.1. Ground state structure
The ground state structures and spin multiplicities of all the TMAln (for TM = Cr, Mn, Fe) clusters we have found are given in figure 1. The ground state as well as a few higher energy isomers of all TMAln clusters studied in this work are given as supplementary information (figures S1–S4 available at stacks.iop.org/JPhysCM/26/015006/mmedia). An important point to note in figure 1 is that the Cr and Mn atoms occupy exterior positions at all sizes. The Fe atom gets completely encapsulated for the first time in the Al cage at size 12. This is understandable from the fact that Cr and Mn atoms have larger atomic radii, compared with Fe. This is also consistent with the experiments of Lang et al on Cr–Al clusters [18]. We also observe that Cr and Mn doped clusters generally tend to have larger spins than the Fe doped clusters. MnAl5 is an interesting exception. It is a singlet though both MnAl4 and MnAl6 are sextets. In the FeAln series, FeAl4 and FeAl10 are both singlet, while others are doublet or triplet depending on the electron count. Only FeAl9 is a quartet and FeAl6 is a quintet. Our structures and multiplicities differ from those reported by Wang et al [19] in many cases. For CrAl4 , CrAl5 , FeAl4 and FeAl12 clusters, both ground state structures and spin multiplicities differ. For some clusters, while the structures we find are the same as those reported by Wang et al, the spin states are different. For example, MnAl4 , MnAl6 and FeAl5 turned out to have different spin states in the two 3
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Figure 3. HOMO–LUMO gaps of TMAl3 and TMAl4 clusters for
transition metals Cr–Ni.
rather small. There are no peaks at n = 6 or 9. Therefore, the stability at n = 6 and 9 arises purely due to structural reasons. In fact, as shown in figure 1, MnAl6 and MnAl9 have exactly the same ground state structures as CrAl6 and CrAl9 . There is a peak in the HOMO–LUMO gap at n = 5 perhaps because MnAl5 is a singlet. While the origin of the small peak in the HOMO–LUMO gap for MnAl3 requires further understanding, detailed analysis shows that this is not due to filling of the electronic shell within shell models. In summary we do find stable clusters with large spin moments in the CrAln and MnAln series, but these stabilities are not the results of shell filling, and the electronic structures of these clusters do not have any resemblance to shell orbitals. This point is elaborated later. The FeAln series presents an interesting picture. 12n has peaks at n = 4, 7 and 9. The HOMO–LUMO gap also has a pronounced peak at n = 4, the value of the HOMO–LUMO gap being 1.47 eV. This can be compared to the HOMO–LUMO of 1.87 eV in the Al− 13 in our calculations, which is known to be a very stable cluster that does not react with O2 [33]. The HOMO–LUMO gap has a tiny peak at n = 7, but the value is only ∼0.5 eV, and nothing pronounced at n = 9. Clearly, the enhanced stability of FeAl4 has an electronic origin. To establish the special stability of FeAl4 even further we plot the HOMO–LUMO gaps of TMAl4 clusters for TM = Cr–Ni in figure 3, and the spin excitation energies of FeAln clusters in figure 4. Spin excitation energy is defined as the energy difference between the ground state of a cluster and its next higher energy spin state, and measures the stability of a cluster with respect to spin excitations [22]. Clearly, FeAl4 has an enhanced stability as both of these quantities have the highest values in the respective series. An Mn atom has a 3d5 4s2 electronic configuration, i.e., five unfilled d orbitals in the minority spin channel, and MnAl5 turns out to be a singlet in the ground state. An Fe atom, with its four unfilled d orbitals, forms a singlet in FeAl4 which turns out to have enhanced stability. Therefore, it is interesting to ask whether CoAl3 also is singlet and whether it has enhanced stability. We would also like to point out that CrAl6 is not a singlet though Cr has six unfilled orbitals (five 3d and one 4s). Rather it has a large spin. The difference between CoAl3 , FeAl4 and MnAl5 on one hand, and CrAl6 on the other, is that in the former three the TM atom is bonded to all of the Al atoms, while in the latter it is bonded to only four
Figure 2. 12 and HOMO–LUMO gaps of Cr, Mn and FeAln
clusters as a function of the number of Al atoms in the cluster.
calculations. For CrAl6 and CrAl12 clusters, the spin states turned out to be the same in the two calculations while the ground state structures were different. 3.2. Stability
Having found their ground states we now study the relative stability of various TMAln clusters. We calculate the second order energy difference (12 ) and the gap between the highest occupied and the lowest unoccupied molecular orbitals (HOMO–LUMO gap) for this. 12n for a TMAln cluster is defined as 12n = E(n − 1) + E(n + 1) − 2E(n),
(1)
where E(n) is the total energy of a cluster containing one TM atom and n Al atoms. 12n measures the thermodynamic stability of a cluster of size n relative to sizes n − 1 and n + 1. The HOMO–LUMO gap, on the other hand, is a measure of kinetic stability, a larger gap indicating a less reactive species. Figure 2(a) shows the variation of 12n with the number of Al atoms for CrAln , MnAln and FeAln clusters. In figure 2(b) we show the variation of the HOMO–LUMO gap with n for these clusters. In the CrAln series 12n has small peaks at n = 3, 6 and 9. However, the HOMO–LUMO gap shows dips at all these sizes. We have shown that such peaks in 12n without any accompanying peaks in the HOMO–LUMO gap indicate enhanced stability due to geometric effects rather than electronic effects [32]. Thus we do not find any CrAln cluster in the size range studied here that has enhanced stability originating in electronic effects. MnAln clusters also have peaks in 12n at n = 3, 6 and 9. Although there is a small peak in the HOMO–LUMO gap at n = 3, the value is 4
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Table 1. Ground state spin multiplicity M(= 2S + 1), chemical
hardness η (in eV) and the nucleus-independent chemical shift (NICS, in ppm) for TMAl3 and TMAl4 clusters. TMAl3 TM
M
η
Cr Mn Fe Co Ni
4 3 2 1 2
2.35 2.52 2.41 2.86 2.38
TMAl4 NICS
M
η
−17.77 −10.23 −41.80 −126.28 −65.36
3 6 1 2 3
2.45 2.25 2.60 2.11 2.26
NICS −41.11 −29.66 −99.36 −12.88 −22.03
this quantity using the 1SCF method. We would like to point out here that it is the vertical IP and EA that are relevant for the calculation of η since the derivatives in equation (2) are to be taken at a constant external potential. The η values calculated using the 1SCF method for the TMAl3 and TMAl4 clusters are given in table 1. It is seen that CoAl3 has the highest η within the TMAl3 series, while FeAl4 has the highest η within the TMAl4 series. We have further given the hardness values of the CoAln and FeAln series in table 2 over a limited range of sizes. Again, CoAl3 and FeAl4 turn out to have the highest η, indicating their enhanced kinetic stability. Our results on the relative stability of TM–Al clusters differ from those reported in [19]. They found CrAl3 and FeAl3 to have enhanced stability. CrAl3 was found to have large 12 and HOMO–LUMO gap. FeAl3 was also found to have a peak in 12 and a large HOMO–LUMO gap. We do not have any such features for CrAl3 . In fact, CrAl3 has quite a low HOMO–LUMO gap compared to CrAl4 . On the other hand we find FeAl4 rather than FeAl3 to have enhanced stability. Having different structures and/or different spin states can cause such discrepancies in conclusions. In fact, as discussed before, the ground structure and spin multiplicity of CrAl4 , CrAl5 and FeAl5 and the multiplicity of FeAl5 differ in the two calculations. Having undertaken a global search, we are reasonably confident of having found the ground states of these clusters, and hence a correct picture of their relative stability.
Figure 4. Spin excitation energies of FeAln and CoAln clusters.
Figure 5. 12 (n) and HOMO–LUMO gaps of CoAln clusters.
of the Al atoms. In order to answer the question on CoAl3 , we plot 12n and the HOMO–LUMO gaps of CoAln clusters for the range n = 2–6 in figure 5. Interestingly, CoAl3 has the largest 12n and HOMO–LUMO gap. The HOMO–LUMO gap is 1.64 eV, even larger than that of FeAl4 . To further validate the enhanced stability of CoAl3 we also plot the HOMO–LUMO gaps of the TMAl3 clusters in figure 3 and the spin excitation energies of the CoAln clusters in figure 4. Again, CoAl3 turns out to have the largest spin excitation energy and HOMO–LUMO gap. Another measure of kinetic stability that has been used in the literature is chemical hardness. It is a measure of the difficulty in transferring charge to and from a chemical species [34]. The chemical hardness (η) of a cluster is defined as 1 ∂ 2E 1 ∂µ η= = . (2) 2 ∂N v 2 ∂N 2 v
3.3. Origin of stability
Having established the enhanced stability of FeAl4 and CoAl3 clusters, we now try to understand the origin of their stability. We have already mentioned that large values of both 12 and HOMO–LUMO gap indicate an electronic origin of stability. Indeed, both of these clusters are filled shell singlets with spin excitation energies of 0.4 eV (FeAl4 ) and 0.78 eV (CoAl3 ). The first question that emerges is whether these stabilities can be explained by spherical shell models. If the Al atom acts as a trivalent species in these clusters, then FeAl4 has 20 valence electrons while CoAl3 has 18 valence electrons. Both of these are shell filling numbers. They should nominally have 1S2 1P6 1D10 2S2 and 1S2 1P6 1D10 electronic configurations respectively. If shell filling does indeed play a role in their stability then we should be able to identify a clear correspondence of the MOs of these clusters with shell orbitals. We have shown this for many TM-alkali and
µ, E and N are the chemical potential, the total energy and the number of electrons in a cluster. All of the derivatives are taken at a constant external potential v. In the finite difference approximation, hardness can be expressed as IP − EA . (3) 2 According to the maximum hardness principle [35] a chemical species tries to maximize its hardness, and therefore a larger hardness value indicates higher stability. We have calculated η≈
5
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Figure 6. Molecular energy level diagrams and isosurfaces of (a) FeAl4 and (b) CoAl3 clusters. Table 2. Ground state spin multiplicity M(= 2S + 1), chemical hardness η (in eV) and the nucleus-independent chemical shift (NICS, in
ppm) for FeAln and CoAln clusters. Cluster
M
η
FeAl3 FeAl4 FeAl5 FeAl6 FeAl7
2 1 2 5 2
2.4 2.6 2.11 2.2 2.16
NICS −41.8 −99.36 9.06 −65.19 −51.11
TM-alkaline earth clusters [20, 21, 36]. Therefore, we plot the MO energy level diagrams and MO isosurfaces of these two clusters in figure 6. Clearly, not all MOs in FeAl4 can be identified with shell orbitals. The fourth orbital from below cannot be identified with any shell orbital of definite angular momentum character (S, P or D). Similarly HOMO and HOMO-1 do not correspond to any shell orbital. In CoAl3 the HOMO, one of the HOMO-1 and the two HOMO-2 orbitals do not bear any resemblance to shell orbitals of any definite angular momentum. Therefore, we conclude that the stability of these clusters cannot be explained in terms of shell filling within a spherical shell model. In view of the fact that the stability of many metal clusters has been associated with aromaticity, we now ask whether aromaticity plays a role in FeAl4 and CoAl3 clusters. Unfortunately, there is no unique definition in the literature of this very important concept. Various measures of aromaticity have been proposed: structural, energetic, reactivity, electronic and magnetic [37]. As a consequence, aromaticity is considered to be ‘multidimensional’. Although the concept of aromaticity was introduced in the context of planar molecules, the idea has since been extended to include non-planar molecules and clusters as well [38–40]. One of the most widely used (magnetic) measures of aromaticity is the nucleus-independent chemical shift
Cluster
M
η
CoAl2 CoAl3 CoAl4 CoAl5 CoAl6
2 1 2 1 4
2.77 2.86 2.11 2.28 2.11
NICS 36.37 −126.28 −12.88 31.54 −36.62
(NICS) [9]. The NICS value is the negative of the nuclear magnetic shielding tensor at the nucleus of an inert probe atom. This is calculated using the gauge-independent atomic orbital (GIAO) method [41] implemented in Gaussian03 code. A negative value of NICS indicates an aromatic character, while a positive NICS value indicates the anti-aromatic character of a given cluster [42]. Since it is a position dependent quantity, for planar molecules or clusters this quantity is usually calculated at the center of the plane ˚ away (NICS(0)) or at a certain distance (typically 1 or 2 A) from the center of the plane (NICS(1) or NICS(2)). For non-planar clusters, aromaticity has usually been calculated at the unweighted geometric center of the cluster [40]. We calculated NICS values for some of the FeAln and CoAln clusters using the Gaussian03 code. The NICS values at the center of the TMAl3 and TMAl4 clusters are given in table 1. While all of the values turn out to be negative, the CoAl3 and FeAl4 clusters turn out to have the most negative values. The NICS values for the FeAln and CoAln clusters are given in table 2. Again, FeAl4 has a large negative NICS. FeAl5 in fact has a positive NICS, indicating its anti-aromatic character. Among the CoAln clusters studied, only CoAl3 and CoAl4 have negative NICS values. CoAl2 and CoAl5 have positive NICS, indicating that these clusters are anti-aromatic. 6
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The frontier orbitals come from the p orbitals only. The p-derived orbitals are only 1/6th filled, and as a consequence, only the deepest bonding orbitals are occupied. This leads to an increase in IP and EA in small clusters [10]. Rao and Jena, based on their DFT calculations, argued that the sp hybridization takes place when the cluster has 7 Al atoms [45]. The issue of sp hybridization has also been addressed in anion PES experiments. Schriver et al linked the leveling off of the IP at n = 5 with the onset of sp hybridization [6]. The measured IP merges with the behavior of a charged metallic sphere between n = 25 and 45. This was claimed to be the size range in which sp hybridization is complete. In a recent work, Melko and Castleman [12], combining measurement of angular distribution of photoelectrons and ab initio theory, have argued that sp hybridization is present in clusters as small as Al3 , calling into question interpretations of earlier PES experiments. An important ingredient in their analysis is the DFT result for the contribution of atomic s and p orbitals to a particular MO in a cluster. Analyzing the MOs of TMAl3 and TMAl4 clusters we find that almost all of the frontier orbitals have contributions from Al s, p and TM d orbitals. For example, the HOMO of CoAl3 consists of 34% Co d and 65% Al orbitals. Of the Al contribution, 44% is s and 51% is p. The HOMO in FeAl4 has 62% contribution from the Fe d orbitals, and 37% contribution from the Al s and p orbitals. Out of the Al contribution, 39% is s and 56% is p. Thus sp hybridization is present in small TM–Al clusters just as in pure Al clusters. Overlap of these sp hybrids with the TM d orbitals gives rise to bonding between the Aln unit and the TM atoms.
Several structural measures of aromaticity have been introduced that include the harmonic oscillator model of aromaticity (HOMA), the bond alternation coefficient, the resonance energy etc [37]. The basic idea behind all of these is that the bond lengths are equalized in an aromatic cluster, and they tend to have high symmetry structures. This, however, does not mean that a given cluster has the highest attainable symmetry [38]. For example, C6 H6 has a planar D6h structure rather than a higher symmetry octahedral Oh structure. Here we find that bond lengths are equalized in FeAl4 and CoAl3 ˚ clusters. In FeAl4 all of the Al–Al bond lengths are 2.84 A. ˚ Two of the Fe–Al bond lengths are 2.23 A (to two Al atoms ˚ In CoAl3 , on opposite sides), and the other two are 2.35 A. ˚ the Co–Al bond lengths are all 2.24 A, while all of the Al–Al ˚ The reactivity measure of aromaticity bond lengths are 2.74 A. includes chemical hardness. We have already shown that FeAl4 and CoAl3 have the largest chemical hardness. From this discussion we conclude that aromaticity plays a role in determining the stability of the FeAl4 and CoAl3 clusters whereas there is no resemblance of the MOs of these clusters to the shell orbitals. However, the orbitals of the Al13 , Al− 13 and Al14 clusters do follow the shell picture quite nicely, each Al atom contributing three valence electrons. In particular, Al13 is one electron short of a 40 electron filled shell configuration due to which it behaves as a superhalogen [43]. Al− 13 has a 2 6 10 2 14 6 filled shell 1S 1P 1D 2S 1F 2P electron configuration as shown in supplementary figure S5 (available at stacks. iop.org/JPhysCM/26/015006/mmedia), and behaves similarly to an inert gas atom [33]. This raises two questions: first, whether TMAln clusters at larger sizes follow the shell model, and second, whether the lack of correspondence with the shell model at smaller sizes is due to the absence of sp hybridization, as a number of early studies on pure Al clusters claimed. If the shell model emerged at large sizes, MnAl11 with 40 electrons would have shown enhanced stability, which is clearly not the case. After a careful look at the MOs of the clusters studied in this work, we conclude that even at larger sizes the electronic structure of TMAln clusters cannot be described by the spherical shell models, unlike pure Al, TM-alkali and TM-alkaline earth clusters. The deep lying orbitals can be identified with 1S, 1P, 1D and 2S orbitals. However, the frontier orbitals do not bear any resemblance to shell orbitals in most cases. To illustrate the point more clearly we have shown MO energy level and isosurface plots for MnAl11 in supplementary figures S6 (available at stacks.iop. org/JPhysCM/26/015006/mmedia). The contrast with Al− 13 is obvious.
4. Conclusions
We have presented a comprehensive study of a class of TM doped Al clusters. Global minima of structures have been obtained through an evolutionary algorithm, and their electronic structure has been studied in detail. The stable clusters within the CrAln , MnAln , FeAln , CoAln , TMAl3 and TMAl4 series have been identified. The important points emerging out of this study are as follows. The electronic structure of these clusters cannot be described by the shell models at any size up to 12 Al atoms. FeAl4 and CoAl3 possess enhanced stability originating from aromaticity. This shows up in NICS values, symmetric structure and large chemical hardness. However, MnAl5 has a dip in the 12 value in the MnAln series as seen in figure 2, although it is a singlet with a larger HOMO–LUMO gap compared with its neighbors. This presents an interesting picture. NiAl2 also does not possess an enhanced stability as found in the 12 value (not presented here), Ni having two unfilled orbitals in the minority spin channel. We believe this work has clarified some of the issues determining the stability of TM–Al clusters that were left open in [19]. This has also thrown up some interesting questions, as follows. Why do TM–Al clusters behave so differently from TM-alkali and TM-alkaline earth clusters although pure Al clusters follow the electronic shell model quite well at sizes ≥ 7? Can this be explained in terms of the nature of the bonding, between the TM and Al atoms? Do larger TM–Al clusters, beyond those studied here, obey
3.4. sp hybridization
Finally we come to the issue of hybridization of s and p orbitals on the Al atoms. In an Al atom there is an energy gap of 3.6 eV between the 3s and the 3p orbitals [44]. Bulk Al, on the other hand, is a good free-electron metal with three valence electrons per atom. The obvious question is at what cluster size the sp hybridization takes place. It was argued that in small clusters the s and p orbitals remain separate. 7
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the shell picture? We hope that this work will encourage more studies, both theoretical and experimental, addressing these questions.
[24] K¨oster A M 2006 deMon2k, V. 2.3.6, The International deMon Developers Community, Cinvestav, M´exico; available at www.deMon-software.com [25] Frisch M J et al 2003 Gaussian 03, Revision D.1 (Wallingford, CT: Gaussian) [26] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865 [27] Calaminici P, Janetzko F, K¨oster A M, Mejia-Olvera R and Z´un˜ iga-Gutierrez B J 2007 J. Chem. Phys. 126 044108 [28] Oganov A R and Glass C W 2006 J. Chem. Phys. 124 244704 [29] Glass C W, Oganov A R and Hansen N 2006 Comput. Phys.Commun. 175 713 [30] Avogadro: an open-source molecular builder and visualization tool. Version 1.0.0. http://avogadro.openmolecules.net/ [31] Kresse G and Hafner J 1993 Phys. Rev. B 47 558 Kresse G and Hafner J 1994 Phys. Rev. B 49 14251 Kresse G and Furthm¨uller J 1996 Comput. Mater. Sci. 6 15 Kresse G and Furthm¨uller J 1996 Phys. Rev. B 54 11169 Kresse G 1993 Thesis Technische Universit¨at at Wien [32] Reveles J U, Sen P, Pradhan K, Roy D R and Khanna S N 2010 J. Phys. Chem. C 114 10739 [33] Reber A C, Khanna S N, Roach P J, Woodward W H and Castleman A W Jr 2007 J. Am. Chem. Soc. 129 16098 [34] Parr R G and Yang W 1989 Density Functional Theory of Atoms and Molecules (New York: Oxford University Press) [35] Chattaraj P K, Sarkar U and Roy D R 2006 Chem. Rev. 106 2065 Parr R G and Chattaraj P K 1991 J. Am. Chem. Soc. 113 1854 [36] Reveles J U, Clayborne P A, Arthur A C, Khanna S N, Pradhan K, Sen P and Pederson M 2009 Nature Chem. 1 310 [37] Chattaraj P K (ed) 2001 Aromaticity and Metal Clusters (Boca Raton, FL: CRC Press, Taylor and Francis) [38] H¨oltz T, Veszpr´emi T, Lievens P and Nguyen M T 2001 Aromaticity and Metal Clusters ed P K Chattaraj (Boca Raton, FL: CRC Press, Taylor and Francis) p 271 [39] Chi X X 2012 Proc. Comput. Sci. 9 348 [40] Clayborne P A, Gupta U, Reber A C, Melko J J, Khanna S N and Castleman A W Jr 2010 J. Chem. Phys. 133 134302 Furuse S, Koyasu K, Atobe J and Nakajima A 2008 J. Chem. Phys. 129 064311 Tai T B and Nguyen M T 2001 J. Phys. Chem. A 115 9993 Tai T B, Tam N M and Nguyen M T 2011 Chem. Phys. 388 1 [41] Ditchfield R D 1974 Mol. Phys. 27 789 [42] Schleyer P v R, Maerker C, Dransfeld A, Jiao H and Hommes N J R v E 1996 J. Am. Chem. Soc. 118 6317 [43] Bergeron D E, Castleman A W Jr, Morisato T and Khanna S N 2004 Science 304 84 [44] Li X, Wu H, Wang X B and Wang L S 1998 Phys. Rev. Lett. 81 1909 [45] Rao B K and Jena P 1999 J. Chem. Phys. 111 1890
Acknowledgment
All computation was done on the Cluster Computing Facility at HRI Allahabad (www.cluster.hri.res.in). References [1] Ekardt W 1984 Phys. Rev. B 29 1558 [2] Knight W D, Clemenger K, de Heer W A, Saunders W A, Chou M Y and Cohen M L 1984 Phys. Rev. Lett. 52 2141 [3] de Heer W A 1993 Rev. Mod. Phys. 65 611 [4] Sen P 2001 Aromaticity and Metal Clusters ed P K Chattaraj (Boca Raton, FL: CRC Press, Taylor and Francis) p 137 [5] Taylor K J, Pettiette C L, Craycraft M J, Chesnovsky O and Smalley R E 1998 Chem. Phys. Lett. 152 347 [6] Schriver K E, Persson J L, Honea E C and Whetten R L 1990 Phys. Rev. Lett. 64 2539 [7] Clemenger K 1985 Phys. Rev. B 32 1359 [8] Li X, Kuznetsov A E, Zhang H F, Boldyrev A I and Wang L S 2001 Science 291 859 [9] Bultinck P, Fias S, Mandado M and Ponec R 2001 Aromaticity and Metal Clusters ed P K Chattaraj (Boca Raton, FL: CRC Press, Taylor and Francis) p 245 [10] Upton T H 1986 Phys. Rev. Lett. 56 2168 [11] Gantef¨or G and Eberhardt W 1994 Chem. Phys. Lett. 217 600 [12] Melko J J and Castleman A W Jr 2013 Phys. Chem. Chem. Phys. 15 3173 [13] Gong X G and Kumar V 1994 Phys. Rev. B 50 17701 [14] Reddy B V, Khanna S N and Deevi S C 2001 Chem. Phys. Lett. 333 465 [15] Menezes W J C and Knickelbein M B 1993 Z. Phys. D 26 322 [16] Pramann A, Nakajima A and Kaya K 2001 J. Chem. Phys. 115 5404 [17] Harms A C, Leuchtner R E, Sigsworth S W and Castleman A W Jr 1990 J. Am. Chem. Soc. 112 5673 [18] Lang S M, Claes P, Neukermans S and Janssens E 2011 J. Am. Soc. Mass Spectrom. 22 1508 [19] Wang M, Huang X, Du Z and Li Y 2009 Chem. Phys. Lett. 480 258 [20] Medel V M, Reveles J U, Khanna S N, Chauhan V, Sen P and Castleman A W Jr 2011 Proc. Natl Acad. Sci 108 10062 [21] Chauhan V, Medel V M, Reveles J U, Khanna S N and Sen P 2012 Chem. Phys. Lett. 528 39 [22] Chauhan V and Sen P 2013 Chem. Phys. 417 37 [23] Roach P J, Woodward W H, Reber A C, Khanna S N and Castleman A W Jr 2010 Phys. Rev. B 81 195404
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Structural, electronic and magnetic properties of binary transition metal aluminum clusters: absence of electronic shell structure
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 015006 (http://iopscience.iop.org/0953-8984/26/1/015006) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 015006 (8pp)
doi:10.1088/0953-8984/26/1/015006
Structural, electronic and magnetic properties of binary transition metal aluminum clusters: absence of electronic shell structure Vikas Chauhan1 , Akansha Singh1 , Chiranjib Majumder2 and Prasenjit Sen1 1 2
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India Chemistry Division, Bhabha Atomic Research Centre, Mumbai, India
Received 29 August 2013, in final form 17 October 2013 Published 25 November 2013 Abstract
Single Cr, Mn, Fe, Co and Ni doped Al clusters having up to 12 Al atoms are studied using density functional methods. The global minima of structure for all the clusters are identified, and their relative stability and electronic and magnetic properties are studied. FeAl4 and CoAl3 are found to have enhanced stability and aromatic behavior. In contrast to binary transition metal alkali and transition metal alkaline earth clusters, spherical shell models cannot describe the electronic structure of transition metal aluminum clusters. S Online supplementary data available from stacks.iop.org/JPhysCM/26/015006/mmedia (Some figures may appear in colour only in the online journal)
π -electrons are responsible for the aromaticity of the Al2− 4 unlike benzene-like molecules [9]. unit in MAl− 4 Another unusual feature of small Al clusters is an increase of ionization potential (IP) and electron affinity (EA) values. Early ab initio calculations found the ionization potential (IP) of the small Aln cluster (n < 6) to be higher than that of the Al atom [10]. This is rather unexpected if one considers the metal cluster as a classical metallic sphere. In the case of alkali clusters, the IP of a cluster always turns out to be lower than that of the atom. At small cluster sizes, electron affinity (EA) was found to rise sharply in these calculations. This was later confirmed in anion photoelectron spectroscopy (PES) experiments [11]. This was considered to be an indication of the lack of sp hybridization in the Al atoms at such small cluster sizes. We will return to this topic towards the end of the paper. Whereas pure Al clusters have been studied extensively, and are still being studied with the help of new experimental techniques [12], there have also been a few works on transition metal (TM)–Al clusters. In an old work Gong and Kumar studied magnetic properties of TM doped Al12 icosahedral clusters [13]. This study was confined to only
1. Introduction
Electronic shell models have played a central role in understanding the properties of alkali, alkaline earth and coinage metal clusters over the last three decades [1–4]. Clusters having 2, 8, 18, 20, 34, 40 . . . valence electrons are found to have completely filled electronic shells resulting in enhanced stability compared to neighboring sizes. When it comes to trivalent elements such as aluminum, the situation gets complicated. Experimental observations on Al clusters containing ∼7 atoms or more have been explained in terms of shell models. Anion PES experiments by Taylor et al [5] found signatures of shell closings at 20, 40, 58 and 70 electrons. A subsequent photoionization spectroscopy study by Schriver et al [6] found shell closings at 20, 40, 50, 68, 84, . . . electron numbers. Of these, shell closing at 50, however, could not be explained by spherical, spheroidal or ellipsoidal shell models [7]. In order to explain the stability of smaller Al clusters, the concept of aromaticity has been invoked. In their work on MAl− 4 (M = Li, Na, Cu) clusters, Li et al [8] introduced the idea of all-metal aromaticity for the first time. Later research showed that the σ -electrons rather than the 0953-8984/14/015006+08$33.00
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c 2014 IOP Publishing Ltd Printed in the UK
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one size. More importantly, a thorough structure search was not undertaken. Hence many of the conclusions reached need further verification. Reddy et al studied the structural and magnetic properties of (FeAl)n clusters for n ≤ 6 [14]. They found an initial increase and then a decrease of magnetic moment with size. Menzes and Knickelbein measured the IP and reactivity of Co–Al clusters [15]. Pramann et al studied − the PES and reactivity of CoAl− n (n = 8–17) and Con Alm (n = 6, 8 10; m = 1, 2) clusters [16]. They found hybridization of Al s and p and TM d states in these clusters. CoAl− 12 was found to be particularly resistant to reactions. This was explained in terms of a 40 electron filled shell configuration. Harms et al − [17] explained the inertness of VAl− 6 and NbAl4 clusters in their reactions towards O2 in terms of electronic shell models. Lang et al studied the structure of V, Cr and Ti doped Al cluster cations over a large size range of 5–35 Al atoms [18]. Using the ability of the clusters to physisorb Ar atoms, it was concluded that the V and Cr atoms are completely encapsulated by an Al+ n cage at n = 16, and the Ti atom is encapsulated between n = 19 and 21. More recently, Wang et al studied single 3d TM doped Aln clusters for n = 1–7 and 12 using first-principles electronic structure calculations [19]. In addition to finding the ground state structures and spin multiplicities of these clusters, a major conclusion was that all TMAl3 clusters show enhanced stability. The magnetic properties of some of the TMAln clusters were explained using the shell model. For example, it was claimed that CoAl3 and CrAl4 are singlets because both these have an eighteen electron filled shell configuration. However, if CrAl4 is indeed an eighteen electron filled shell cluster, one would expect it to have enhanced stability, which was not found by these authors. Although the shell model was invoked to explain the properties of some of the clusters, a correspondence of the molecular orbitals (MOs) of these clusters with the shell orbitals was not established. In light of the fact that pure Al clusters at large sizes follow the shell model quite well, their claim that TM–Al clusters deviate from the spherical shell model for n > 4 was interesting and requires a closer inspection. The size range n = 8–11 was also not studied. In our recent works on TM-alkaline earth clusters we have shown that the interplay of crystal-field effect and Hund’s coupling can lead to subshell filling at unconventional electron counts, resulting in enhanced stability and large magnetic moments [20–22]. FeMg8 , FeCa8 , CrSr9 and MnSr10 were found to have enhanced stability though the first three have 24 and the last one has 27 valence electrons. In fact, these are all examples of magnetic superatoms. The crystal-field effect is expected to play a bigger role in Al clusters since the nuclear charge in these clusters is not screened as well as in alkali or alkaline earth clusters [6, 10]. The crystal-field effect has been observed in the reactivity of the CuAl22 cluster [23]. Therefore, it is interesting to research whether such interplay of crystal-field effect and Hund’s coupling leads to enhanced stability in TM–Al clusters. Since stable clusters can occur at unconventional electron counts, one has to perform first-principles calculations on a series of clusters to identify them. With this motivation, we decided to study the structural, electronic and magnetic properties of single Cr, Fe and
Mn doped Al clusters in the size range of 2–12 Al atoms. However, for reasons discussed below, we were led to study Co and Ni doped Al clusters as well over a limited size range. We identify the stable clusters and try to understand the origin of their stability in terms of electronic and geometric structures. The significant finding is that FeAl4 and CoAl3 clusters possess enhanced stability compared to other clusters. However, this is not due to shell filling within the electronic shell models. Rather, aromaticity plays a role in determining the stability of these clusters. Unlike in pure Al clusters, the electronic structure of TM–Al clusters does not correspond to the electronic shell models. The rest of the paper is organized as follows. In section 2 we present the theoretical methods that have been used to study the clusters. In section 3 the results obtained are presented, and attempts are made to rationalize these. Finally, we draw our conclusions in section 4. 2. Method
All of our first-principles electronic structure calculations were performed within the density functional theory (DFT) using two different approaches in different cases. In one approach, molecular orbitals were expressed as linear combinations of Gaussian type orbitals centered on the atomic nuclei. deMon2k [24] and Gaussian03 [25] codes were used for these calculations. The exchange–correlation effects were taken into account using the generalized gradient approximation (GGA) functional proposed by Perdew, Burke and Ernzerhof (PBE) [26]. In deMon2K these functionals were calculated through a numerical integration from the orbital density. Calculation of four-center integrals was avoided by a variational fitting of the Coulomb potential. The auxiliary density was expanded in primitive Hermite Gaussians using the GEN-A2 auxiliary function set. This contains s, p, d, f and g auxiliary functions, and adapts automatically to the chosen orbital basis set. For all the atoms we used double-ζ valence plus polarization basis sets in both deMo2K and Gaussian03. These have been optimized for the 3d TM atoms for GGA functionals by Calaminici et al (DZVP-GGA) [27]. In a second approach, a plane wave basis set was used along with PAW potentials in conjunction with the USPEX [28, 29] code for a global search of the lowest energy structure of the clusters as described in detail below. Unless specified otherwise, the results presented are obtained using the deMon2K code. Finding the ground state structure of a cluster is a challenging task. For clusters containing a few atoms one can use chemical intuition, or start from possible symmetric structures and relax them to the nearest minima using an optimization technique such as the conjugate gradient method. However, the number of local minima increases exponentially with cluster size, and one becomes less confident about having found the global minimum. Hence we adopt a dual approach here. In the first approach, that we prefer to call the educated guess, we construct a number of initial structures from our knowledge of structures of Al or other metal clusters. For a particular size Aln , known structures of Aln+1 are taken and different inequivalent Al atoms are 2
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Figure 1. Ground state structure and multiplicity of Cr-, Mn- and FeAln clusters for n = 2–12 .
the different numerical methods used in the two codes. Our analysis is based on the deMon2K results.
replaced by the TM atoms. We also construct other possible symmetric structures, and possible cage-like structures with the TM atom both inside and outside the cage. We also took structures found for TM-alkali and TM-alkaline earth clusters, replaced these metal atoms with Al and adjusted the bond lengths using the Avogadro code [30]. For all these initial structures we consider a number of possible spin states and relax each of these to the nearest local minimum without any symmetry constraints. We use deMon2K for these calculations. To ensure that the optimized structures are truly local minima, harmonic frequencies are calculated for each one of them. If any structure turns out to have imaginary frequencies, it is re-optimized by distorting it along the unstable vibrational modes. In the second approach we perform a direct global search for the minimum energy structure of these clusters based on an evolutionary algorithm using the USPEX code [28, 29]. For all of the USPEX calculations energies are calculated using the plane wave PAW approach. VASP [31] code is used for these calculations. More details about these calculations are given in supplementary information (available at stacks.iop.org/JPhysCM/26/015006/ mmedia). Whenever we find a minimum of structure in USPEX that is different from the lowest energy structure obtained in the educated guess approach, we further relax it in deMon2K and compare the energies of the two structures. It turns out that in most cases the two approaches produced the same lowest energy structures. For CrAl6 and FeAl7 the ground state structures obtained by USPEX were higher energy isomers in the educated guess approach, and vice versa. The energy difference between the two isomers in deMon2K was small for both of the clusters: 0.021 eV for CrAl6 and 0.21 eV for FeAl7 . This difference could be due to
3. Results and discussion 3.1. Ground state structure
The ground state structures and spin multiplicities of all the TMAln (for TM = Cr, Mn, Fe) clusters we have found are given in figure 1. The ground state as well as a few higher energy isomers of all TMAln clusters studied in this work are given as supplementary information (figures S1–S4 available at stacks.iop.org/JPhysCM/26/015006/mmedia). An important point to note in figure 1 is that the Cr and Mn atoms occupy exterior positions at all sizes. The Fe atom gets completely encapsulated for the first time in the Al cage at size 12. This is understandable from the fact that Cr and Mn atoms have larger atomic radii, compared with Fe. This is also consistent with the experiments of Lang et al on Cr–Al clusters [18]. We also observe that Cr and Mn doped clusters generally tend to have larger spins than the Fe doped clusters. MnAl5 is an interesting exception. It is a singlet though both MnAl4 and MnAl6 are sextets. In the FeAln series, FeAl4 and FeAl10 are both singlet, while others are doublet or triplet depending on the electron count. Only FeAl9 is a quartet and FeAl6 is a quintet. Our structures and multiplicities differ from those reported by Wang et al [19] in many cases. For CrAl4 , CrAl5 , FeAl4 and FeAl12 clusters, both ground state structures and spin multiplicities differ. For some clusters, while the structures we find are the same as those reported by Wang et al, the spin states are different. For example, MnAl4 , MnAl6 and FeAl5 turned out to have different spin states in the two 3
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Figure 3. HOMO–LUMO gaps of TMAl3 and TMAl4 clusters for
transition metals Cr–Ni.
rather small. There are no peaks at n = 6 or 9. Therefore, the stability at n = 6 and 9 arises purely due to structural reasons. In fact, as shown in figure 1, MnAl6 and MnAl9 have exactly the same ground state structures as CrAl6 and CrAl9 . There is a peak in the HOMO–LUMO gap at n = 5 perhaps because MnAl5 is a singlet. While the origin of the small peak in the HOMO–LUMO gap for MnAl3 requires further understanding, detailed analysis shows that this is not due to filling of the electronic shell within shell models. In summary we do find stable clusters with large spin moments in the CrAln and MnAln series, but these stabilities are not the results of shell filling, and the electronic structures of these clusters do not have any resemblance to shell orbitals. This point is elaborated later. The FeAln series presents an interesting picture. 12n has peaks at n = 4, 7 and 9. The HOMO–LUMO gap also has a pronounced peak at n = 4, the value of the HOMO–LUMO gap being 1.47 eV. This can be compared to the HOMO–LUMO of 1.87 eV in the Al− 13 in our calculations, which is known to be a very stable cluster that does not react with O2 [33]. The HOMO–LUMO gap has a tiny peak at n = 7, but the value is only ∼0.5 eV, and nothing pronounced at n = 9. Clearly, the enhanced stability of FeAl4 has an electronic origin. To establish the special stability of FeAl4 even further we plot the HOMO–LUMO gaps of TMAl4 clusters for TM = Cr–Ni in figure 3, and the spin excitation energies of FeAln clusters in figure 4. Spin excitation energy is defined as the energy difference between the ground state of a cluster and its next higher energy spin state, and measures the stability of a cluster with respect to spin excitations [22]. Clearly, FeAl4 has an enhanced stability as both of these quantities have the highest values in the respective series. An Mn atom has a 3d5 4s2 electronic configuration, i.e., five unfilled d orbitals in the minority spin channel, and MnAl5 turns out to be a singlet in the ground state. An Fe atom, with its four unfilled d orbitals, forms a singlet in FeAl4 which turns out to have enhanced stability. Therefore, it is interesting to ask whether CoAl3 also is singlet and whether it has enhanced stability. We would also like to point out that CrAl6 is not a singlet though Cr has six unfilled orbitals (five 3d and one 4s). Rather it has a large spin. The difference between CoAl3 , FeAl4 and MnAl5 on one hand, and CrAl6 on the other, is that in the former three the TM atom is bonded to all of the Al atoms, while in the latter it is bonded to only four
Figure 2. 12 and HOMO–LUMO gaps of Cr, Mn and FeAln
clusters as a function of the number of Al atoms in the cluster.
calculations. For CrAl6 and CrAl12 clusters, the spin states turned out to be the same in the two calculations while the ground state structures were different. 3.2. Stability
Having found their ground states we now study the relative stability of various TMAln clusters. We calculate the second order energy difference (12 ) and the gap between the highest occupied and the lowest unoccupied molecular orbitals (HOMO–LUMO gap) for this. 12n for a TMAln cluster is defined as 12n = E(n − 1) + E(n + 1) − 2E(n),
(1)
where E(n) is the total energy of a cluster containing one TM atom and n Al atoms. 12n measures the thermodynamic stability of a cluster of size n relative to sizes n − 1 and n + 1. The HOMO–LUMO gap, on the other hand, is a measure of kinetic stability, a larger gap indicating a less reactive species. Figure 2(a) shows the variation of 12n with the number of Al atoms for CrAln , MnAln and FeAln clusters. In figure 2(b) we show the variation of the HOMO–LUMO gap with n for these clusters. In the CrAln series 12n has small peaks at n = 3, 6 and 9. However, the HOMO–LUMO gap shows dips at all these sizes. We have shown that such peaks in 12n without any accompanying peaks in the HOMO–LUMO gap indicate enhanced stability due to geometric effects rather than electronic effects [32]. Thus we do not find any CrAln cluster in the size range studied here that has enhanced stability originating in electronic effects. MnAln clusters also have peaks in 12n at n = 3, 6 and 9. Although there is a small peak in the HOMO–LUMO gap at n = 3, the value is 4
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Table 1. Ground state spin multiplicity M(= 2S + 1), chemical
hardness η (in eV) and the nucleus-independent chemical shift (NICS, in ppm) for TMAl3 and TMAl4 clusters. TMAl3 TM
M
η
Cr Mn Fe Co Ni
4 3 2 1 2
2.35 2.52 2.41 2.86 2.38
TMAl4 NICS
M
η
−17.77 −10.23 −41.80 −126.28 −65.36
3 6 1 2 3
2.45 2.25 2.60 2.11 2.26
NICS −41.11 −29.66 −99.36 −12.88 −22.03
this quantity using the 1SCF method. We would like to point out here that it is the vertical IP and EA that are relevant for the calculation of η since the derivatives in equation (2) are to be taken at a constant external potential. The η values calculated using the 1SCF method for the TMAl3 and TMAl4 clusters are given in table 1. It is seen that CoAl3 has the highest η within the TMAl3 series, while FeAl4 has the highest η within the TMAl4 series. We have further given the hardness values of the CoAln and FeAln series in table 2 over a limited range of sizes. Again, CoAl3 and FeAl4 turn out to have the highest η, indicating their enhanced kinetic stability. Our results on the relative stability of TM–Al clusters differ from those reported in [19]. They found CrAl3 and FeAl3 to have enhanced stability. CrAl3 was found to have large 12 and HOMO–LUMO gap. FeAl3 was also found to have a peak in 12 and a large HOMO–LUMO gap. We do not have any such features for CrAl3 . In fact, CrAl3 has quite a low HOMO–LUMO gap compared to CrAl4 . On the other hand we find FeAl4 rather than FeAl3 to have enhanced stability. Having different structures and/or different spin states can cause such discrepancies in conclusions. In fact, as discussed before, the ground structure and spin multiplicity of CrAl4 , CrAl5 and FeAl5 and the multiplicity of FeAl5 differ in the two calculations. Having undertaken a global search, we are reasonably confident of having found the ground states of these clusters, and hence a correct picture of their relative stability.
Figure 4. Spin excitation energies of FeAln and CoAln clusters.
Figure 5. 12 (n) and HOMO–LUMO gaps of CoAln clusters.
of the Al atoms. In order to answer the question on CoAl3 , we plot 12n and the HOMO–LUMO gaps of CoAln clusters for the range n = 2–6 in figure 5. Interestingly, CoAl3 has the largest 12n and HOMO–LUMO gap. The HOMO–LUMO gap is 1.64 eV, even larger than that of FeAl4 . To further validate the enhanced stability of CoAl3 we also plot the HOMO–LUMO gaps of the TMAl3 clusters in figure 3 and the spin excitation energies of the CoAln clusters in figure 4. Again, CoAl3 turns out to have the largest spin excitation energy and HOMO–LUMO gap. Another measure of kinetic stability that has been used in the literature is chemical hardness. It is a measure of the difficulty in transferring charge to and from a chemical species [34]. The chemical hardness (η) of a cluster is defined as 1 ∂ 2E 1 ∂µ η= = . (2) 2 ∂N v 2 ∂N 2 v
3.3. Origin of stability
Having established the enhanced stability of FeAl4 and CoAl3 clusters, we now try to understand the origin of their stability. We have already mentioned that large values of both 12 and HOMO–LUMO gap indicate an electronic origin of stability. Indeed, both of these clusters are filled shell singlets with spin excitation energies of 0.4 eV (FeAl4 ) and 0.78 eV (CoAl3 ). The first question that emerges is whether these stabilities can be explained by spherical shell models. If the Al atom acts as a trivalent species in these clusters, then FeAl4 has 20 valence electrons while CoAl3 has 18 valence electrons. Both of these are shell filling numbers. They should nominally have 1S2 1P6 1D10 2S2 and 1S2 1P6 1D10 electronic configurations respectively. If shell filling does indeed play a role in their stability then we should be able to identify a clear correspondence of the MOs of these clusters with shell orbitals. We have shown this for many TM-alkali and
µ, E and N are the chemical potential, the total energy and the number of electrons in a cluster. All of the derivatives are taken at a constant external potential v. In the finite difference approximation, hardness can be expressed as IP − EA . (3) 2 According to the maximum hardness principle [35] a chemical species tries to maximize its hardness, and therefore a larger hardness value indicates higher stability. We have calculated η≈
5
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Figure 6. Molecular energy level diagrams and isosurfaces of (a) FeAl4 and (b) CoAl3 clusters. Table 2. Ground state spin multiplicity M(= 2S + 1), chemical hardness η (in eV) and the nucleus-independent chemical shift (NICS, in
ppm) for FeAln and CoAln clusters. Cluster
M
η
FeAl3 FeAl4 FeAl5 FeAl6 FeAl7
2 1 2 5 2
2.4 2.6 2.11 2.2 2.16
NICS −41.8 −99.36 9.06 −65.19 −51.11
TM-alkaline earth clusters [20, 21, 36]. Therefore, we plot the MO energy level diagrams and MO isosurfaces of these two clusters in figure 6. Clearly, not all MOs in FeAl4 can be identified with shell orbitals. The fourth orbital from below cannot be identified with any shell orbital of definite angular momentum character (S, P or D). Similarly HOMO and HOMO-1 do not correspond to any shell orbital. In CoAl3 the HOMO, one of the HOMO-1 and the two HOMO-2 orbitals do not bear any resemblance to shell orbitals of any definite angular momentum. Therefore, we conclude that the stability of these clusters cannot be explained in terms of shell filling within a spherical shell model. In view of the fact that the stability of many metal clusters has been associated with aromaticity, we now ask whether aromaticity plays a role in FeAl4 and CoAl3 clusters. Unfortunately, there is no unique definition in the literature of this very important concept. Various measures of aromaticity have been proposed: structural, energetic, reactivity, electronic and magnetic [37]. As a consequence, aromaticity is considered to be ‘multidimensional’. Although the concept of aromaticity was introduced in the context of planar molecules, the idea has since been extended to include non-planar molecules and clusters as well [38–40]. One of the most widely used (magnetic) measures of aromaticity is the nucleus-independent chemical shift
Cluster
M
η
CoAl2 CoAl3 CoAl4 CoAl5 CoAl6
2 1 2 1 4
2.77 2.86 2.11 2.28 2.11
NICS 36.37 −126.28 −12.88 31.54 −36.62
(NICS) [9]. The NICS value is the negative of the nuclear magnetic shielding tensor at the nucleus of an inert probe atom. This is calculated using the gauge-independent atomic orbital (GIAO) method [41] implemented in Gaussian03 code. A negative value of NICS indicates an aromatic character, while a positive NICS value indicates the anti-aromatic character of a given cluster [42]. Since it is a position dependent quantity, for planar molecules or clusters this quantity is usually calculated at the center of the plane ˚ away (NICS(0)) or at a certain distance (typically 1 or 2 A) from the center of the plane (NICS(1) or NICS(2)). For non-planar clusters, aromaticity has usually been calculated at the unweighted geometric center of the cluster [40]. We calculated NICS values for some of the FeAln and CoAln clusters using the Gaussian03 code. The NICS values at the center of the TMAl3 and TMAl4 clusters are given in table 1. While all of the values turn out to be negative, the CoAl3 and FeAl4 clusters turn out to have the most negative values. The NICS values for the FeAln and CoAln clusters are given in table 2. Again, FeAl4 has a large negative NICS. FeAl5 in fact has a positive NICS, indicating its anti-aromatic character. Among the CoAln clusters studied, only CoAl3 and CoAl4 have negative NICS values. CoAl2 and CoAl5 have positive NICS, indicating that these clusters are anti-aromatic. 6
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The frontier orbitals come from the p orbitals only. The p-derived orbitals are only 1/6th filled, and as a consequence, only the deepest bonding orbitals are occupied. This leads to an increase in IP and EA in small clusters [10]. Rao and Jena, based on their DFT calculations, argued that the sp hybridization takes place when the cluster has 7 Al atoms [45]. The issue of sp hybridization has also been addressed in anion PES experiments. Schriver et al linked the leveling off of the IP at n = 5 with the onset of sp hybridization [6]. The measured IP merges with the behavior of a charged metallic sphere between n = 25 and 45. This was claimed to be the size range in which sp hybridization is complete. In a recent work, Melko and Castleman [12], combining measurement of angular distribution of photoelectrons and ab initio theory, have argued that sp hybridization is present in clusters as small as Al3 , calling into question interpretations of earlier PES experiments. An important ingredient in their analysis is the DFT result for the contribution of atomic s and p orbitals to a particular MO in a cluster. Analyzing the MOs of TMAl3 and TMAl4 clusters we find that almost all of the frontier orbitals have contributions from Al s, p and TM d orbitals. For example, the HOMO of CoAl3 consists of 34% Co d and 65% Al orbitals. Of the Al contribution, 44% is s and 51% is p. The HOMO in FeAl4 has 62% contribution from the Fe d orbitals, and 37% contribution from the Al s and p orbitals. Out of the Al contribution, 39% is s and 56% is p. Thus sp hybridization is present in small TM–Al clusters just as in pure Al clusters. Overlap of these sp hybrids with the TM d orbitals gives rise to bonding between the Aln unit and the TM atoms.
Several structural measures of aromaticity have been introduced that include the harmonic oscillator model of aromaticity (HOMA), the bond alternation coefficient, the resonance energy etc [37]. The basic idea behind all of these is that the bond lengths are equalized in an aromatic cluster, and they tend to have high symmetry structures. This, however, does not mean that a given cluster has the highest attainable symmetry [38]. For example, C6 H6 has a planar D6h structure rather than a higher symmetry octahedral Oh structure. Here we find that bond lengths are equalized in FeAl4 and CoAl3 ˚ clusters. In FeAl4 all of the Al–Al bond lengths are 2.84 A. ˚ Two of the Fe–Al bond lengths are 2.23 A (to two Al atoms ˚ In CoAl3 , on opposite sides), and the other two are 2.35 A. ˚ the Co–Al bond lengths are all 2.24 A, while all of the Al–Al ˚ The reactivity measure of aromaticity bond lengths are 2.74 A. includes chemical hardness. We have already shown that FeAl4 and CoAl3 have the largest chemical hardness. From this discussion we conclude that aromaticity plays a role in determining the stability of the FeAl4 and CoAl3 clusters whereas there is no resemblance of the MOs of these clusters to the shell orbitals. However, the orbitals of the Al13 , Al− 13 and Al14 clusters do follow the shell picture quite nicely, each Al atom contributing three valence electrons. In particular, Al13 is one electron short of a 40 electron filled shell configuration due to which it behaves as a superhalogen [43]. Al− 13 has a 2 6 10 2 14 6 filled shell 1S 1P 1D 2S 1F 2P electron configuration as shown in supplementary figure S5 (available at stacks. iop.org/JPhysCM/26/015006/mmedia), and behaves similarly to an inert gas atom [33]. This raises two questions: first, whether TMAln clusters at larger sizes follow the shell model, and second, whether the lack of correspondence with the shell model at smaller sizes is due to the absence of sp hybridization, as a number of early studies on pure Al clusters claimed. If the shell model emerged at large sizes, MnAl11 with 40 electrons would have shown enhanced stability, which is clearly not the case. After a careful look at the MOs of the clusters studied in this work, we conclude that even at larger sizes the electronic structure of TMAln clusters cannot be described by the spherical shell models, unlike pure Al, TM-alkali and TM-alkaline earth clusters. The deep lying orbitals can be identified with 1S, 1P, 1D and 2S orbitals. However, the frontier orbitals do not bear any resemblance to shell orbitals in most cases. To illustrate the point more clearly we have shown MO energy level and isosurface plots for MnAl11 in supplementary figures S6 (available at stacks.iop. org/JPhysCM/26/015006/mmedia). The contrast with Al− 13 is obvious.
4. Conclusions
We have presented a comprehensive study of a class of TM doped Al clusters. Global minima of structures have been obtained through an evolutionary algorithm, and their electronic structure has been studied in detail. The stable clusters within the CrAln , MnAln , FeAln , CoAln , TMAl3 and TMAl4 series have been identified. The important points emerging out of this study are as follows. The electronic structure of these clusters cannot be described by the shell models at any size up to 12 Al atoms. FeAl4 and CoAl3 possess enhanced stability originating from aromaticity. This shows up in NICS values, symmetric structure and large chemical hardness. However, MnAl5 has a dip in the 12 value in the MnAln series as seen in figure 2, although it is a singlet with a larger HOMO–LUMO gap compared with its neighbors. This presents an interesting picture. NiAl2 also does not possess an enhanced stability as found in the 12 value (not presented here), Ni having two unfilled orbitals in the minority spin channel. We believe this work has clarified some of the issues determining the stability of TM–Al clusters that were left open in [19]. This has also thrown up some interesting questions, as follows. Why do TM–Al clusters behave so differently from TM-alkali and TM-alkaline earth clusters although pure Al clusters follow the electronic shell model quite well at sizes ≥ 7? Can this be explained in terms of the nature of the bonding, between the TM and Al atoms? Do larger TM–Al clusters, beyond those studied here, obey
3.4. sp hybridization
Finally we come to the issue of hybridization of s and p orbitals on the Al atoms. In an Al atom there is an energy gap of 3.6 eV between the 3s and the 3p orbitals [44]. Bulk Al, on the other hand, is a good free-electron metal with three valence electrons per atom. The obvious question is at what cluster size the sp hybridization takes place. It was argued that in small clusters the s and p orbitals remain separate. 7
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the shell picture? We hope that this work will encourage more studies, both theoretical and experimental, addressing these questions.
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Acknowledgment
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