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Article

Models of Molecular Structures of Hexa-Nuclear AlnFem Metal Clusters (n + m = 6): DFT Quantum-Chemical Design

by
Oleg V. Mikhailov
1,* and
Denis V. Chachkov
2
1
Department of Analytical Chemistry, Certification and Quality Management, Kazan National Research Technological University, K. Marx Street 68, 420015 Kazan, Russia
2
Kazan Department of Joint Supercomputer Center of Russian Academy of Sciences—Branch of Federal Scientific Center “Scientific Research Institute for System Analysis of the RAS”, Lobachevskii Street 2/31, 420111 Kazan, Russia
*
Author to whom correspondence should be addressed.
Materials 2021, 14(3), 597; https://doi.org/10.3390/ma14030597
Submission received: 8 December 2020 / Revised: 27 December 2020 / Accepted: 20 January 2021 / Published: 27 January 2021
(This article belongs to the Special Issue Superconducting and Quantum Metamaterials, Metacircuits, and Metadevices)
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Abstract

By using the density functional theory (DFT) method at the OPBE/QZVP level, key parameters of molecular structures of six-atomic (heterobi)nuclear metal clusters with an AlnFem composition (n + m = 6) (bond lengths, bond angles, and torsion (dihedral) angles) were calculated. It was found that each of these clusters exists in a large number of structural isomers that differ substantially in terms of their total energy. Furthermore, the molecular structures of these structural isomers significantly differ regarding the geometric parameters and geometric form. In addition, the most stable structural isomers of these metal clusters also differ rather considerably in terms of the geometric form.

1. Introduction

In previous studies [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33], quantum chemical calculations of metal clusters containing atoms of two variable p- and d-elements, known as (dd)- and (pd)heterobimetallic metal clusters, were carried out using the density functional theory (DFT) method. In the works [1,2,3,4,5,6,7,8,9,10,11,12,13,14], the objects of study were (dd)heterobimetallic metal clusters which included atoms of two different d-elements, in particular, (Cu, Fe) [1], (Pd, Fe) [2], (Pd, Ag) [3,4,5], (Pt, Cu) [6], (Au, Fe) [7], (Au, Pd) [8], (Au, Ag) [9], (Au, Ir) [12], and (Pd, Ir) [14]. Some of these metal clusters have been applied in various fields of science and technology [10,11,12,13,14]. The (pd)heterobimetallic metal clusters that include atoms of different metal categories, namely, p- and d-elements, are of greater interest than (dd)heterobimetallic ones, because, theoretically, it can be expected that they will demonstrate new properties that are not inherent to metal clusters containing metal atoms of only one category of chemical elements.
Among the most important p-elements is aluminum, which has a very wide industrial application; nevertheless, only a few (pd)metal clusters containing this p-element and any of the d-metals are described in the literature [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. On the other hand, the most important d-element is iron, which plays a key role in the iron and steel industry. Metal clusters containing aluminum and iron are of undoubted interest, if only because they are structural units of bimetallic alloys such as “alfer” and “alphenol”, containing 13% and 16% aluminum, respectively, and having a high magnetic permeability in weak magnetic fields and significant electrical resistance. In addition, both of them have a fairly high hardness, strength, and wear resistance, due to which the first of these alloys is used for the manufacturing of electroacoustic (magneto-strictive) transducers, and the second, in the production of cores for recording and reproducing heads of magnetic recording equipment. Based on the specifics of the phase diagram of the (aluminum–iron) system and the composition of the (AlFe) intermetallic compounds formed within it, it may be assumed that the key structural units of these compounds are tetra-, penta-, and hexa-nuclear (AlFe) metal clusters. However, quantum-chemical calculations (AlFe) of metal clusters containing four to six atoms have only been performed in a few works [23,24,25,26,27,28,29,30,31,32,33], and in all cases, the DFT method with the OPBE/TZVP basis set was used. In our opinion, hexa-nuclear aluminum–iron metal clusters with the general formula AlnFem, where the values of n and m vary from 1 to 5, are of the greatest interest here. Considering this, this article will be devoted to the presentation and systematization of the quantum-chemical calculation results of such compounds’ molecular structures using a more advanced method in the framework of density functional theory (DFT), specifically DFT OPBE/QZVP.

2. Results

The quantum chemical calculations carried out by using the DFT method at the OPBE/QZVP level showed that each of the hexa-nuclear aluminum–iron metal clusters with the stoichiometric composition AlnFem (n + m = 6), namely, Al5Fe, Al4Fe2, Al3Fe3, Al2Fe4, and AlFe5, exists in a very significant number of structural isomers, the number of which varies from 19 in the case of Al5Fe to 40 in the case of Al3Fe3. The information of these metal clusters in detail is presented in the Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9 and Table 10. As may be seen from the Table 1, Table 3, Table 5, Table 7, and Table 9, the relative total energies of these structural isomers for each of the metal clusters under examination also vary in a very wide range. The most energetically advantageous among these isomers for Al5Fe, Al4Fe2, Al3Fe3, Al2Fe4, and AlFe5 are shown in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5. The molecular structure parameters of these structural isomers are given in Table 2, Table 4, Table 6, Table 8, and Table 10, respectively. A complete assortment of molecular structures of all these metal clusters is presented in the Supplemental Materials. In terms of the numbering of metal clusters, the Arabic numeral in parentheses denotes the value of the spin multiplicity of the ground state (MS), and the Roman numeral is the ordinal number of the metal cluster with MS data in ascending relative energy.
NBO analysis data for the metal clusters indicated in Table 11 are presented in Table 12.

3. Discussion

As can be seen from the presented data, the Al5Fe metal cluster exists in 19 structural isomers, the Al4Fe2 metal cluster in 39 structural isomers, the Al3Fe3 metal cluster in 40 structural isomers, the Al2Fe4 metal cluster in 53 structural isomers, and the AlFe5 metal cluster in 23 structural isomers. There is no regularity in these numbers; there is only a tendency for them to first increase their number with an increase in the number of iron atoms (m) in the structural unit of the metal cluster (in the range m = 1–4), and then to decrease (in the range m = 4–5). For any of the considered metal clusters in the series Al5Fe–Al4Fe2–Al3Fe3–Al2Fe4–AlFe5, in general, a very significant variety of molecular structures is characteristic–from pseudo-octahedral [Al2Fe4 (3-V)] to strictly planar [Al4Fe2 (3-XIII)]. Both of these structures are unique and are not observed in any of the other AlnFem metal clusters (n + m = 6). For the overwhelming majority of the presented metal cluster structures, a pronounced asymmetry with an almost complete absence of any symmetrical elements is typical. Therefore, none of these metal clusters has a symmetry axis of the third or higher order, and only some of them have a plane of symmetry and a symmetry center (see Supplementary Materials). This fully applies to the most energetically stable (i.e., those with the minimum total energy values) metal clusters presented in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5. It is important to note that in pairs of so-called “inverted” metal clusters, where the numerical values of m and n for Al and Fe atoms are interchanged [(Al5Fe, AlFe5) and (Al4Fe2, Al2Fe4)], there is no noticeable similarity between the assortments of molecular structures in either quantitative or qualitative relations.
A remarkable feature of all the AlnFem metal clusters is the relatively small number of metal–metal bonds—no more than 8; at the same time, there is no apparent correlation between the total number of these bonds in the metal cluster and its energy stability (i.e., the value of its relative energy). It is very curious that the formation of Al–Al bonds in molecular structures of these metal clusters is much rarer than the formation of Fe–Fe bonds, and this phenomenon even takes place in the case of the Al4Fe2 metal cluster, where the number of aluminum atoms in each of the 39 molecular structures is twice as high as the number of iron atoms (in the most energetically stable Al4Fe2 metal cluster, namely Al4Fe2 (1-I), there is also no Al–Al bond (Figure 2)). Among the molecular structures of the Al2Fe4 metal cluster, there is not even one structure in which such a bond is present; the most energetically stable of them—Al2Fe4 (5-I)—is no exception (Figure 4). Moreover, Figure 2, Figure 3 and Figure 4 show that, among the five most stable structural isomers of each of the metal clusters Al2Fe4, Al3Fe3, and Al2Fe4, there is not a single one in which the molecular structure contains at least one Al–Al bond. Molecular structures with the absence of Fe–Fe bonds only occur in the case of the Al5Fe metal cluster (where such a bond cannot exist in principle) and the Al4Fe2 metal cluster, so they are in the minority compared to structures where the Fe–Fe bond is present (16 out of 39 molecular structures). Al–Fe bonds, as was expected, are present in each of the structural isomers of each metal cluster under consideration. It should be noted that the length of any of the metal–metal bonds in any of these metal clusters is more than 200 pm (see Table 2, Table 4, Table 6, Table 8, and Table 10). The Al–Al bonds are the longest on average, the Fe–Fe bonds are the shortest, and the Fe–Al bonds have an intermediate length between the lengths of bonds formed by two aluminum atoms and two iron atoms. This relationship seems to be real due to the atomic radii of Al (143 pm) and Fe (126 pm). As for the plane bond angles formed by metal–metal bonds, they are, as a rule, relatively small and almost always less than 90°; a similar situation occurs for planar non-bonded angles, as well as for dihedral (torsion) angles.
As can be seen from the data in Table 1, Table 3, Table 5, Table 7, and Table 9, the most energetically stable metal clusters Al5Fe (2-I) and Al4Fe2 (1-I) are low-spin, while the most energetically stable metal clusters Al3Fe3 (6-I) and Al2Fe4 (5-I) are high-spin; the AlFe5 (4-I) metal cluster occupies an intermediate position. This fact is quite understandable because the number of unpaired electrons in a neutral Al atom (the electronic configuration of the ground state 3s23p1) is significantly lower than the number of unpaired electrons in a neutral Fe (4s23d6) atom and with an increase in the number of Fe atoms (and, accordingly, a decrease number of Al atoms) in the molecular structure of the AlnFem metal cluster (n + m = 6), logically, there should be an increase in the total number of unpaired electrons and a higher probability of the realization of the high-spin ground state compared to the low-spin one. Such a prediction, however, is not fully justified when going from Al2Fe4 to AlFe5, because, contrary to our expectations, the ground state of the most stable AlFe5 metal cluster, which is AlFe5 (4-I), was not in a high-spin state with MS = 6, but a state with MS = 4. The reason for such a metamorphosis is apparently related to the fact that in the smallest homohexa-nuclear metal cluster Fe6, as the calculation shows, the ground state is the spin singlet (MS = 1), and the spin multiplicity of the ground state for AlFe5 should thus be intermediate, between those of Al2Fe4 and Fe6, which is confirmed by the calculation results. According to NBO analysis data, the charges on the Al and Fe atoms that make up the investigated metal clusters, on the whole, are relatively small and do not exceed 1.00 in absolute value. As should be expected, for the metal clusters presented in Table 11, the charges on the Al atoms are positive, and those on the Fe atoms are negative. This is also quite understandable since the electronegativity of Al on the Pauling scale (1.5) is lower than the electronegativity of Fe (1.8). An interesting exception is the AlFe5 (4-I) cluster, in which one of the Fe atoms, namely Fe2, has a positive charge (Table 12). As a rule, the magnitudes of the charges on different atoms of the same element are different, which is also quite obvious because of the asymmetry of most structural isomers of these metal clusters.
According to our calculation, the values of the Gibbs free energy of formation ΔfG0 (298 K), even for the most stable AlnFem metal clusters (n + m = 6), are positive (Table 11), which means that none of them can be obtained by direct interactions between metallic iron and metallic aluminum. However, the situation changes radically if gaseous aluminum and iron are taken as starting materials, i.e., the following reactions (1)–(5) are used:
5Al(gas) + Fe(gas) → Al5Fe(gas)
4Al(gas) + 2Fe(gas) → Al4Fe2(gas)
3Al(gas) + 3Fe(gas) → Al3Fe3(gas)
2Al(gas) + 4Fe(gas) → Al2Fe4(gas)
Al(gas) + 5Fe(gas) → AlFe5(gas)
each of which, according to the calculations, can be thermodynamically resolved and belongs to the number of chemical processes occurring with the so-called enthalpy factor (Table 13).
As can be seen from these values, for each of the reactions (1)–(5) in the gas phase, the values of both of these parameters are negative for any of the considered metal clusters; hence, all these reactions are thermodynamically allowed at relatively low temperatures and forbidden at high ones. It is noteworthy that all of these reactions are exothermic. Moreover, the thermal effect of each of them is quite significant. In the simplest version using the Gibbs–Helmholtz Equation (6) for the isobaric process,
ΔrG0 (T) = ΔrH0 (298 K) − TΔrS0 (298 K)
where ΔrH0 (298 K) and ΔrS0 (298 K) are the changes in the enthalpy and entropy as a result of a chemical process in standard conditions, respectively; T is the process temperature in K; and ΔrG0 (T) is the dependence of the Gibbs free energy on the temperature T). It is easy to find the temperature at which one or another of the Reactions (1)–(5) will not take place due to thermodynamic prohibition. Actually, this parameter is the temperature indicating the beginning of thermal destruction of a metal cluster (Ttd, K) or (ttd, °C) in the gas phase; the values of this parameter for each of the most energy-stable metal clusters, namely, Al5Fe (2-I), Al4Fe2 (1-I), Al3Fe3 (6-I), Al2Fe4 (5-I), and AlFe5 (4-I), are displayed in Table 14.
As can be seen from the given table, this temperature is very high and for each of the most stable AlnFem metal clusters (n + m = 6), it exceeds 2000 °C, so at least in this aggregation state, all of them are very resistant to thermal effects. These data also demonstrate that the most stable in this respect is Al3Fe3 (6-I), and the least stable is Al5Fe (2-I).
It seems appropriate to compare the results for the Al3Fe3 and Al2Fe4 metal clusters presented in this article and the results for the same metal clusters obtained using a simpler version of the DFT method, namely, DFT OPBE/TZVP (they are presented in a recently published review [33], as well as in earlier original publications [30,31,32]). According to these data, the Al3Fe3 metal cluster exists in 20 different structural isomers, whereas the Al2Fe4 metal cluster only exists in nine different structural isomers. These numbers of structural isomers are in sharp contrast to the analogous values for the data of metal clusters obtained using the DFT OPBE/QZVP method (40 and 53, respectively). It should be noted that for the Al4Fe2 metal cluster, DFT OPBE/TZVP gives practically the same number of possible structural isomers (38) as for the DFT OPBE/QZVP method (39); a similar coincidence can be observed for the Al5Fe metal cluster (any data on these two metal clusters, however, were not published by the authors of this article or any other researchers). Therefore, the difference in the number of possible structural isomers obtained by these two methods increases with an increase in the number of 3D-element atoms (in our case, Fe) in the structural unit of the AlnFem metal cluster. This difference is quite understandable because the DFT OPBE/QZVP method better takes into account the specifics of the electronic structure and wave functions of 3D-element atoms than DFT OPBE/TZVP, and the larger the fraction of Fe atoms in the structure, the more significant the difference in the data should be. It should be noted that, in this regard, the molecular structures of the most stable metal clusters Al3Fe3 and Al2Fe4, as well as the spin multiplicities of their ground states, obtained by the DFT OPBE/TZVP method, are similar to the analogous data obtained by the DFT OPBE/QZVP method and presented in this article.

4. Calculation Method

The quantum-chemical calculations of AlnFem metal clusters were carried out using the density functional method (DFT) combining the standard extended split-valence QZVP basis [34,35] and the OPBE functional [36,37]. The data of works [34,35,36,37,38,39,40,41] gave us reason to assert that the given method allows the most accurate estimation of the ratio between energies of the high-spin state and low-spin state to be obtained and, at the same time, rather reliably predicts the key geometric parameters of molecular structures for various compounds of 3p- and 3d-elements. To build quantum chemical models of the molecular structures of the metal clusters under examination, GAUSSIAN09 software was used [42]. As in [16,17,18,19,20,21,22], the accordance of the found stationary points to the energy minima was confirmed by the calculation of the second derivatives with respect to the atomic coordinates. Additionally, all equilibrium structures corresponding to the minima at the potential energy surface only revealed real positive frequency values. Parameters of the molecular structures for spin multiplicities (MS) higher than 1 were determined using the so-called unrestricted method (UOPBE), and those for MS = 1, using the so-called restricted method (ROPBE). Along with this, the unrestricted method in conjunction with the GUESS = Mix option was used for the cases when MS was equal to 1. Moreover, NBO analysis of these metal clusters (Natural Population, Natural Electron Configurations, and Natural Atomic Orbital Occupancies) was carried out according to the procedure described in the works [43,44]. NBO 3.0 version built-in GAUSSIAN09 was used. The energetically most favorable structure has always been checked with the STABLE = OPT procedure; in all cases, the wave function corresponding to it was stable. The standard thermodynamic parameters of the formation of metal clusters under study, and namely ΔfH0 (298 K), Sf0 (298 K), and ΔfG0 (298 K) for the metal clusters under examination, were calculated using the method described in [45].

5. Conclusions

As can be seen from the calculated data, each of the AlnFem hexa-nuclear metal clusters (n + m = 6) under examination has a rather significant number of structural isomers, which differ very significantly in terms of their structural and geometric parameters and relative total energies; the total number of such structural isomers increases from Al5Fe to Al2Fe4, while from Al2Fe4 to AlFe5, it decreases. It is also noteworthy that in the case of the so-called “inverted” metal clusters [(Al5Fe, AlFe5) and (Al4Fe2, Al2Fe4)], the number of structural isomers in AlFe5 and Al2Fe4 is noticeably higher than the number of structural isomers in Al5Fe and Al4Fe2, respectively (although, given the uniformity of their composition in each of these two pairs, the same number of these same isomers would be expected). As a rule, structural isomers of the analyzed metal clusters are either completely devoid of symmetry elements, or only have one plane of symmetry; at the same time, it is remarkable that, in most of these compounds, Al–Al chemical bonds are absent. For the spin multiplicity of the most stable metal clusters’ ground state in the series Al5Fe–Al4Fe2–Al3Fe3–Al2Fe4–AlFe5, the tendency to transition from the low-spin ground state in Al5Fe and Al4Fe2 to the high-spin number of unpaired atoms in Al3Fe3 и Al2Fe4 is quite clearly expressed. Taking into account the presence of a greater number of unpaired electrons in the Fe atom in comparison with that in the Al atom, this seems to be quite natural. The most stable AlnFem metal clusters (n + m = 6) are characterized by a very high thermal stability, although none of them can be directly obtained from metallic iron and aluminum (since, for any of them, the values of the standard Gibbs energies of formation ΔfG0 (298 K) > 0).
In conclusion, the (AlFe) metal clusters considered in the given article could be primarily used in the creation of new composite materials and alloys based on polymetallic nanoparticles in the future; it is quite possible that such alloys may have extraordinary magneto-chemical and physical-mechanical characteristics. Other possible areas of their application may be the doping of traditional alloys based on both non-ferrous and ferrous metals, metal complex catalysis, the creation of specific electrochemical systems, and semiconductor technology. It is also highly probable that they can be used as potential so-called quantum dots, the possibilities of technologies with the use of which are still far from exhausted.

Supplementary Materials

These data are available online at https://www.mdpi.com/1996-1944/14/3/597/s1. Figure S1: Molecular Structures of Hexanuclear Al5Fe Metal Clusters calculated by DFT OPBE/QZVP method, Figure S2: Molecular Structures of Hexanuclear Al4Fe2 Metal Clusters calculated by DFT OPBE/QZVP method, Figure S3: Molecular Structures of Hexanuclear Al3Fe3 Metal Clusters calculated by DFT OPBE/QZVP method, Figure S4: Molecular Structures of Hexanuclear Al2Fe4 Metal Clusters calculated by DFT OPBE/QZVP method, Figure S5: Molecular Structures of Hexanuclear AlFe5 Metal Clusters calculated by DFT OPBE/QZVP method.

Author Contributions

Conceptualization, O.V.M.; methodology, O.V.M. and D.V.C.; software, D.V.C.; validation, O.V.M. and D.V.C.; formal analysis, O.V.M. and D.V.C.; investigation, O.V.M. and D.V.C.; resources, D.V.C.; data curation, D.V.C.; writing—original draft preparation, O.V.M. and D.V.C.; writing—review and editing, O.V.M.; visualization, O.V.M. and D.V.C.; supervision, O.V.M.; project administration, O.V.M.; funding acquisition, D.V.C. All authors have read and agreed to the published version of the manuscript.

Funding

The present study was carried out with financial support in the framework of draft No. 4.5784.2017/8.9 to the competitive part of the state task of the Russian Federation in 2017–2019. Contribution of author D.V.C. was funded by the state assignment to the Federal State Institution “Scientific Research Institute for System Analysis of the Russian Academy of Sciences” for scientific research (http://www.jscc.ru).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Acknowledgments

All quantum-chemical calculations were performed at the Kazan Department of Joint Supercomputer Center of Russian Academy of Sciences–Branch of Federal Scientific Center “Scientific Research Institute for System Analysis of the RAS” that is acknowledged for technical support.

Conflicts of Interest

The authors declare that they have no conflict of interest, financial or otherwise.

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Materials 14 00597 g001aMaterials 14 00597 g001b
Figure 1. Molecular structures of the five most energetically stable Al5Fe metal clusters.
Figure 1. Molecular structures of the five most energetically stable Al5Fe metal clusters.
Materials 14 00597 g001aMaterials 14 00597 g001b
Materials 14 00597 g002
Figure 2. Molecular structures of the five most energetically stable Al4Fe2 metal clusters.
Figure 2. Molecular structures of the five most energetically stable Al4Fe2 metal clusters.
Materials 14 00597 g002
Materials 14 00597 g003
Figure 3. Molecular structures of the five most energetically stable Al3Fe3 metal clusters.
Figure 3. Molecular structures of the five most energetically stable Al3Fe3 metal clusters.
Materials 14 00597 g003
Materials 14 00597 g004
Figure 4. Molecular structures of the five most energetically stable Al2Fe4 metal clusters.
Figure 4. Molecular structures of the five most energetically stable Al2Fe4 metal clusters.
Materials 14 00597 g004
Materials 14 00597 g005aMaterials 14 00597 g005b
Figure 5. Molecular structures of the five most energetically stable AlFe5 metal clusters.
Figure 5. Molecular structures of the five most energetically stable AlFe5 metal clusters.
Materials 14 00597 g005aMaterials 14 00597 g005b
Table 1. Relative energies and spin multiplicities of the ground states of various structural isomers of metal clusters with an Al5Fe composition.
Table 1. Relative energies and spin multiplicities of the ground states of various structural isomers of metal clusters with an Al5Fe composition.
Structure
Designation		Relative Energy,
kJ/mol		Structure
Designation		Relative Energy,
kJ/mol
Metal clusters with MS = 2Metal clusters with MS = 4
Al5Fe (2-I)0.0Al5Fe (4-I)12.0
Al5Fe (2-II)4.4Al5Fe (4-II)14.0
Al5Fe (2-III)23.3Al5Fe (4-III)16.0
Al5Fe (2-IV)28.9Al5Fe (4-IV)21.1
Al5Fe (2-V)32.3Al5Fe (4-V)23.8
Al5Fe (2-VI)54.3Al5Fe (4-VI)61.9
Al5Fe (2-VII)62.7Al5Fe (4-VII)78.7
Al5Fe (2-VIII)104.4Al5Fe (4-VIII)105.2
Al5Fe (2-IX)125.1Al5Fe (4-IX)105.9
Al5Fe (2-X)126.3
Table 2. Key structural parameters of the most stable Al5Fe metal clusters.
Table 2. Key structural parameters of the most stable Al5Fe metal clusters.
Metal ClusterAl5Fe (2-I)Al5Fe (2-II)Al5Fe (4-I)Al5Fe (4-II)Al5Fe (4-III)
Distances between metal atoms, pm
Al1Al2273.2284.8266.7260.4274.8
Al1Al3260.1271.0263.0295.1271.0
Al2Al3443.9424.3421.2437.2377.5
Al1Fe1242.4238.0240.0240.5242.8
Al1Al4279.2251.4262.0277.2272.5
Al1Al5279.9303.3266.5262.9377.7
Al2Fe1395.2239.1362.2262.3242.8
Al2Al4256.2284.7254.7263.5272.4
Al2Al5273.2258.1259.9452.5271.2
Al3Fe1246.8442.1241.2247.4429.5
Al3Al4431.7270.8432.6260.0267.7
Al3Al5260.7259.3254.8255.6251.5
Fe1Al4244.3238.0241.2253.1237.5
Fe1Al5242.5415.0362.1251.6429.5
Al4Al5279.1303.2421.0422.2267.5
Plane angles between metal atoms, deg.
Al4Al1Al559.965.5105.3102.845.1
Fe1Al1Al455.358.157.158.054.5
Fe1Al2Al333.678.134.830.084.7
Al4Al2Al370.439.075.133.145.2
Al1Al2Al463.652.460.563.959.7
Al4Al1Al255.363.857.558.659.7
Al1Al2Al332.739.037.041.045.8
Al2Al3Fe162.332.059.132.034.3
Al3Fe1Al284.169.986.1118.161.1
Al5Al1Al259.251.958.3119.745.8
Fe1Al1Al554.899.491.159.884.7
Fe1Al1Al299.953.591.163.055.5
Al1Fe1Al242.973.347.462.268.9
Al3Al4Al535.353.334.734.756.0
Table 3. Relative energies and spin multiplicities of the ground states of various structural isomers of metal clusters with an Al4Fe2 composition.
Table 3. Relative energies and spin multiplicities of the ground states of various structural isomers of metal clusters with an Al4Fe2 composition.
Structure
Designation		Relative Energy,
kJ/mol		Structure
Designation		Relative Energy,
kJ/mol
Metal clusters with MS = 1Al4Fe2 (3-IV)57.2
Al4Fe2 (1-I)0.0Al4Fe2 (3-V)58.5
Al4Fe2 (1-II)93.4Al4Fe2 (3-VI)59.6
Al4Fe2 (1-III)95.0Al4Fe2 (3-VII)62.4
Al4Fe2 (1-IV)101.9Al4Fe2 (3-VIII)65.0
Al4Fe2 (1-V)103.3Al4Fe2 (3-IX)82.2
Al4Fe2 (1-VI)132.5Al4Fe2 (3-X)111.5
Al4Fe2 (1-VII)135.9Al4Fe2 (3-XI)218.4
Al4Fe2 (1-VIII)139.5Al4Fe2 (3-XII)229.7
Al4Fe2 (1-IX)147.7Al4Fe2 (3-XIII)246.8
Al4Fe2 (1-X)155.3Metal clusters with MS = 5
Al4Fe2 (1-XI)169.2Al4Fe2 (5-I)22.1
Al4Fe2 (1-XII)169.3Al4Fe2 (5-II)22.5
Al4Fe2 (1-XIII)183.4Al4Fe2 (5-III)30.8
Al4Fe2 (1-XIV)205.7Al4Fe2 (5-IV)32.2
Al4Fe2 (1-XV)211.8Al4Fe2 (5-V)32.8
Al4Fe2 (1-XVI)254.3Al4Fe2 (5-VI)34.3
Metal clusters with MS = 3Al4Fe2 (5-VII)51.0
Al4Fe2 (3-I)24.0Al4Fe2 (5-VIII)60.6
Al4Fe2 (3-II)25.0Al4Fe2 (5-IX)64.6
Al4Fe2 (3-III)40.0Al4Fe2 (5-X)69.9
Table 4. Key structural parameters of the most stable Al4Fe2 metal clusters.
Table 4. Key structural parameters of the most stable Al4Fe2 metal clusters.
Metal ClusterAl4Fe2 (1-I)Al4Fe2 (5-I)Al4Fe2 (5-II)Al4Fe2 (3-I)Al4Fe2 (3-II)
Distances between metal atoms, pm
Al1Al2256.6265.0288.8432.8261.9
Al1Al3362.9277.3335.9277.8275.6
Al2Al3256.6277.2443.2267.9275.8
Al1Fe1248.9244.1246.8241.2246.5
Al1Fe2248.9244.1246.8252.7246.5
Al1Al4256.6277.3443.2429.9275.6
Al2Fe1248.9244.1246.8250.4246.5
Al2Fe2248.9244.1246.8244.6246.5
Al2Al4362.9277.3336.3266.3275.6
Al3Fe1248.9245.3246.8238.0238.2
Al3Fe2248.9403.9246.8249.9399.8
Al3Al4256.6483.5288.8264.5483.8
Fe1Fe2340.8212.9217.5214.8213.1
Fe1Al4248.9403.9246.8393.5399.8
Fe2Al4248.9245.3246.8244.4238.2
Plane angles between metal atoms, deg.
Fe2Al1Al459.055.726.129.653.9
Fe1Al1Fe286.451.752.351.551.2
Fe1Al2Al359.055.726.154.553.9
Fe2Al2Al359.0101.426.158.299.8
Al1Al2Fe259.057.154.230.057.9
Fe2Al1Al259.057.154.229.057.9
Al1Al2Al390.061.549.338.361.6
Al2Al3Fe159.055.326.159.056.8
Al3Fe1Al262.169.0127.766.569.3
Al4Al1Al290.061.549.436.061.6
Fe1Al1Al459.0101.426.164.899.8
Fe1Al1Al259.057.154.228.957.9
Al1Fe1Al262.165.871.6125.364.2
Al3Fe2Al462.193.071.6119.895.3
Table 5. Relative energies and spin multiplicities of the ground states of various structural isomers of metal clusters of Al3Fe3.
Table 5. Relative energies and spin multiplicities of the ground states of various structural isomers of metal clusters of Al3Fe3.
Structure
Designation		Relative Energy,
kJ/mol		Structure
Designation		Relative Energy,
kJ/mol
Metal clusters with MS = 2Al3Fe3 (4-IV)102.4
Al3Fe3 (2-I)51.6Al3Fe3 (4-V)103.4
Al3Fe3 (2-II)72.4Al3Fe3 (4-VI)106.6
Al3Fe3 (2-III)82.2Al3Fe3 (4-VII)114.5
Al3Fe3 (2-IV)89.2Al3Fe3 (4-VIII)119.1
Al3Fe3 (2-V)94.6Al3Fe3 (4-IX)122.4
Al3Fe3 (2-VI)94.8Al3Fe3 (4-X)124.7
Al3Fe3 (2-VII)102.9Al3Fe3 (4-XI)129.6
Al3Fe3 (2-VIII)108.4Al3Fe3 (4-XII)147.2
Al3Fe3 (2-IX)114.8Al3Fe3 (4-XIII)162.5
Al3Fe3 (2-X)122.3Metal clusters with MS = 6
Al3Fe3 (2-XI)125.6Al3Fe3 (6-I)0.0
Al3Fe3 (2-XII)132.2Al3Fe3 (6-II)3.5
Al3Fe3 (2-XIII)134.6Al3Fe3 (6-III)32.7
Al3Fe3 (2-XIV)136.5Al3Fe3 (6-IV)57.4
Al3Fe3 (2-XV)137.1Al3Fe3 (6-V)61.0
Al3Fe3 (2-XVI)154.0Al3Fe3 (6-VI)65.9
Al3Fe3 (2-XVII)174.9Al3Fe3 (6-VII)67.6
Metal clusters with MS = 4Al3Fe3 (6-VIII)79.2
Al3Fe3 (4-I)49.1Al3Fe3 (6-IX)86.5
Al3Fe3 (4-II)81.2Al3Fe3 (6-X)99.6
Al3Fe3 (4-III)89.0
Table 6. Key structural parameters of the most stable Al3Fe3 metal clusters.
Table 6. Key structural parameters of the most stable Al3Fe3 metal clusters.
Metal ClusterAl3Fe3 (6-I)Al3Fe3 (6-II)Al3Fe3 (6-III)Al3Fe3 (4-I)Al3Fe3 (2-I)
Distances between metal atoms, pm
Al1Al2305.1270.6262.3259.8268.8
Al1Al3305.8270.7262.3258.4268.8
Al2Al3398.5368.5384.3372.3374.3
Al1Fe1236.0241.8271.4245.7241.0
Al1Fe2369.4354.8245.5346.1338.0
Al1Fe3236.0241.8245.5243.7243.8
Al2Fe1244.0239.7242.8251.0250.8
Al2Fe2242.3241.8389.6252.5236.8
Al2Fe3244.0239.7249.2242.8248.2
Al3Fe1244.0239.7242.8248.5250.8
Al3Fe2242.3241.8249.2246.8236.8
Al3Fe3244.0239.7389.6253.0248.2
Fe1Fe2222.3226.8219.1221.5235.4
Fe1Fe3270.9305.7219.1330.4328.7
Fe2Fe3222.3220.8253.4246.3222.9
Plane angles between metal atoms, deg.
Fe2Al1Fe335.039.256.745.441.2
Fe1Al1Fe235.039.249.839.644.2
Fe1Al2Al335.239.837.741.641.7
Fe2Al2Al334.740.437.641.237.8
Al1Al2Fe284.187.438.385.083.6
Fe2Al1Al240.742.9100.246.644.1
Al1Al2Al349.447.142.943.945.9
Al2Al3Fe135.239.837.742.141.7
Al3Fe1Al2109.5100.5105.696.496.5
Fe3Al1Al251.755.458.757.657.7
Fe1Al1Fe370.178.449.884.985.4
Fe1Al1Al251.755.454.159.558.6
Al1Fe1Al278.968.461.063.166.2
Al3Fe2Fe363.261.4107.661.865.3
Table 7. Relative energies and spin multiplicities of the ground states of various structural isomers of metal clusters of Al2Fe4.
Table 7. Relative energies and spin multiplicities of the ground states of various structural isomers of metal clusters of Al2Fe4.
Structure
Designation		Relative Energy,
kJ/mol		Structure
Designation		Relative Energy,
kJ/mol		Structure
Designation		Relative Energy,
kJ/mol
Metal clusters with MS = 1Al2Fe4 (1-XIX)359.2Al2Fe4 (3-IX)64.3
Al2Fe4 (1-I)15.3Al2Fe4 (1-XX)378.3Al2Fe4 (3-X)77.3
Al2Fe4 (1-II)15.9Al2Fe4 (1-XXI)378.7Al2Fe4 (3-XI)86.8
Al2Fe4 (1-III)16.1Al2Fe4 (1-XXII)394.4Al2Fe4 (3-XII)89.4
Al2Fe4 (1-IV)17.2Al2Fe4 (1-XXIII)394.6Al2Fe4 (3-XIII)91.2
Al2Fe4 (1-V)30.1Al2Fe4 (1-XXIV)402.1Al2Fe4 (3-XIV)121.1
Al2Fe4 (1-VI)41.9Al2Fe4 (1-XXV)403.0Al2Fe4 (3-XV)201.7
Al2Fe4 (1-VII)53.8Al2Fe4 (1-XXVI)403.3Metal clusters with MS = 5
Al2Fe4 (1-VIII)83.1Al2Fe4 (1-XXVII)409.8Al2Fe4 (5-I)0.0
Al2Fe4 (1-IX)129.3Al2Fe4 (1-XXVIII)426.0Al2Fe4 (5-II)16.8
Al2Fe4 (1-X)142.6Metal clusters with MS = 3Al2Fe4 (5-III)24.4
Al2Fe4 (1-XI)187.3Al2Fe4 (3-I)15.0Al2Fe4 (5-IV)29.0
Al2Fe4 (1-XII)231.1Al2Fe4 (3-II)15.7Al2Fe4 (5-V)41.2
Al2Fe4 (1-XIII)241.9Al2Fe4 (3-III)26.6Al2Fe4 (5-VI)45.2
Al2Fe4 (1-XIV)259.0Al2Fe4 (3-IV)27.6Al2Fe4 (5-VII)119.8
Al2Fe4 (1-XV)259.4Al2Fe4 (3-V)37.6Al2Fe4 (5-VIII)131.1
Al2Fe4 (1-XVI)330.8Al2Fe4 (3-VI)44.2Al2Fe4 (5-IX)204.2
Al2Fe4 (1-XVII)350.3Al2Fe4 (3-VII)56.8Al2Fe4 (5-X)212.1
Al2Fe4 (1-XVIII)352.4Al2Fe4 (3-VIII)57.5
Table 8. Key structural parameters of the most stable Al2Fe4 metal clusters.
Table 8. Key structural parameters of the most stable Al2Fe4 metal clusters.
Metal ClusterAl2Fe4 (5-I)Al2Fe4 (3-I)Al2Fe4 (1-I)Al2Fe4 (3-II)Al2Fe4 (1-II)
Distances between metal atoms, pm
Fe1Fe2333.1244.1217.0329.9218.5
Fe1Fe3246.1213.4252.9241.9379.5
Fe2Fe3226.5326.5335.6214.7254.4
Fe1Al1249.7249.2242.0246.6242.9
Fe1Al2249.7249.2350.2246.6242.9
Fe1Fe4246.1326.5248.5234.3395.9
Fe2Al1242.9247.6247.7248.7250.5
Fe2Al2242.9247.7252.6248.7250.5
Fe2Fe4226.5220.2253.0242.1389.9
Fe3Al1242.9249.2252.7248.7246.1
Fe3Al2348.7249.2247.8248.7246.1
Fe3Fe4231.8244.1217.0330.0216.7
Al1Al2270.1374.5262.6369.0277.5
Al1Fe4348.6247.7350.2246.6248.5
Al2Fe4242.8247.7242.0246.6248.4
Plane angles between metal atoms, deg.
Al2Fe1Fe458.648.743.761.636.8
Al1Fe1Al265.597.448.496.969.7
Al1Fe2Fe362.249.148.564.458.3
Al2Fe2Fe395.949.147.364.458.3
Fe1Fe2Al248.360.996.148.061.9
Al2Fe1Fe246.660.345.848.565.5
Fe1Fe2Fe347.640.848.947.1106.5
Fe2Fe3Al162.248.747.364.460.0
Fe3Al1Fe255.682.284.251.161.6
Fe4Fe1Fe242.842.465.447.272.4
Al1Fe1Fe489.448.791.161.636.8
Al1Fe1Fe246.660.365.048.565.5
Fe1Al1Fe285.158.952.683.552.6
Fe3Al2Fe441.558.952.683.552.0
Table 9. Relative energies and spin multiplicities of the ground states of various structural isomers of metal clusters of AlFe5.
Table 9. Relative energies and spin multiplicities of the ground states of various structural isomers of metal clusters of AlFe5.
Structure
Designation		Relative Energy,
kJ/mol		Structure
Designation		Relative Energy,
kJ/mol
Metal clusters with MS = 2AlFe5 (4-III)49.1
AlFe5 (2-I)18.7AlFe5 (4-IV)123.7
AlFe5 (2-II)57.6AlFe5 (4-V)131.5
AlFe5 (2-III)65.5AlFe5 (4-VI)132.2
AlFe5 (2-IV)69.7AlFe5 (4-VII)162.1
AlFe5 (2-V)123.0AlFe5 (4-VIII)252.0
AlFe5 (2-VI)139.1Metal clusters with MS = 6
AlFe5 (2-VII)177.4AlFe5 (6-I)33.8
AlFe5 (2-VIII)178.4AlFe5 (6-II)61.2
AlFe5 (2-IX)254.7AlFe5 (6-III)69.4
Metal clusters with MS = 4AlFe5 (6-IV)105.1
AlFe5 (4-I)0.0AlFe5 (6-V)161.9
AlFe5 (4-II)36.0AlFe5 (6-VI)189.8
Table 10. Key structural parameters of the most stable AlFe5 metal clusters.
Table 10. Key structural parameters of the most stable AlFe5 metal clusters.
Metal ClusterAlFe5 (4-I)AlFe5 (2-I)AlFe5 (6-I)AlFe5 (4-II)AlFe5 (4-III)
Distances between metal atoms, pm
Fe1Fe2223.6224.4233.1221.4251.4
Fe1Fe3252.1248.2249.8382.4251.5
Fe2Fe3252.8329.9335.1393.7342.7
Fe1Al1246.1353.0358.0219.5351.8
Fe1Fe4231.2248.3245.0233.1210.7
Fe1Fe5335.8224.4223.9258.7251.5
Fe2Al1350.9243.6250.5249.2243.7
Fe2Fe4223.5245.9238.1224.2252.6
Fe2Fe5252.7222.1228.8388.9225.1
Fe3Al1252.2252.1249.4247.4243.8
Fe3Fe4335.8218.0212.5255.7252.7
Fe3Fe5212.9245.7250.8211.7225.1
Al1Fe4246.1252.1244.8273.7254.4
Al1Fe5252.1243.6251.1252.1248.4
Fe4Fe5252.0329.9323.1254.0330.0
Plane angles between metal atoms, deg.
Fe4Fe1Fe548.688.487.061.990.7
Al1Fe1Fe462.045.643.069.045.8
Al1Fe2Fe345.949.447.837.445.4
Fe4Fe2Fe389.541.439.137.647.3
Fe1Fe2Fe462.363.562.763.149.4
Fe4Fe1Fe258.962.559.759.065.6
Fe1Fe2Fe363.648.848.270.647.1
Fe2Fe3Al188.047.248.137.745.3
Fe3Al1Fe246.083.484.2104.989.3
Fe5Fe1Fe248.859.360.1108.053.2
Al1Fe1Fe548.443.144.059.444.9
Al1Fe1Fe290.643.144.163.643.8
Fe1Al1Fe239.839.040.452.745.6
Fe3Fe4Fe539.348.150.949.142.9
Table 11. Standard thermodynamic parameters of formation for the most energetically stable (heterobi)hexa-nuclear metal clusters of AlnFem (n + m = 6).
Table 11. Standard thermodynamic parameters of formation for the most energetically stable (heterobi)hexa-nuclear metal clusters of AlnFem (n + m = 6).
Metal ClusterΔfH0 (298 K), kJ/molSf0 (298 K), J/mol KΔfG0 (298 K), kJ/mol
Al5Fe (2-I)805.0457.5718.9
Al4Fe2 (1-I)789.0445.8706.1
Al3Fe3 (6-I)807.5469.0717.4
Al2Fe4 (5-I)890.1480.3796.3
AlFe5 (4-I)927.6476.1834.7
Table 12. Charge distribution (in units of electron charge) of various Al and Fe atoms for the most stable AlnFem metal clusters (n + m = 6) according to NBO analysis data.
Table 12. Charge distribution (in units of electron charge) of various Al and Fe atoms for the most stable AlnFem metal clusters (n + m = 6) according to NBO analysis data.
Al5Fe (2-I)
Al1Al2Al3Al4Al5Fe1
+0.0719+0.1187+0.2723+0.2682+0.0716−0.8027
Al4Fe2 (1-I)
Al1Al2Al3Al4Fe1Fe2
+0.1861+0.1861+0.1861+0.1861−0.3722−0.3722
Al3Fe3 (6-I)
Al1Al2Al3Fe1Fe2Fe3
+0.5766+0.6292+0.6312−0.7980−0.2411−0.7979
Al2Fe4 (5-I)
Al1Al2Fe1Fe2Fe3Fe4
+0.3197+0.3197−0.1446−0.2312−0.1316−0.1320
AlFe5 (4-I)
Al1Fe1Fe2Fe3Fe4Fe5
+0.2554−0.1544+0.0823−0.0161−0.0782−0.0890
Table 13. Calculated values of the parameters ΔrH0 (298 K) and ΔrS0 (298 K) for the Reactions (1)–(5).
Table 13. Calculated values of the parameters ΔrH0 (298 K) and ΔrS0 (298 K) for the Reactions (1)–(5).
ParameterAl5FeAl4Fe2Al3Fe3Al2Fe4AlFe5
ΔrH0 (298 K), kJ−1257.1−1360.5−1429.4−1434.2−1484.1
ΔrS0 (298 K), J/K−545.1−572.7−565.5−570.1−590.2
Table 14. Temperatures of the beginning of thermal destruction (Ttd, K) and (ttd, °C) of the most energetically-stable metal clusters of AlnFem (n + m = 6).
Table 14. Temperatures of the beginning of thermal destruction (Ttd, K) and (ttd, °C) of the most energetically-stable metal clusters of AlnFem (n + m = 6).
Metal ClusterTtd, Kttd, °C
Al5Fe (2-I)2306.182008.02
Al4Fe2 (1-I)2375.582077.42
Al3Fe3 (6-I)2527.672229.51
Al2Fe4 (5-I)2515.572217.41
AlFe5 (4-I)2514.572216.41
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Mikhailov, O.V.; Chachkov, D.V. Models of Molecular Structures of Hexa-Nuclear AlnFem Metal Clusters (n + m = 6): DFT Quantum-Chemical Design. Materials 2021, 14, 597. https://doi.org/10.3390/ma14030597

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Mikhailov OV, Chachkov DV. Models of Molecular Structures of Hexa-Nuclear AlnFem Metal Clusters (n + m = 6): DFT Quantum-Chemical Design. Materials. 2021; 14(3):597. https://doi.org/10.3390/ma14030597

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Mikhailov, Oleg V., and Denis V. Chachkov. 2021. "Models of Molecular Structures of Hexa-Nuclear AlnFem Metal Clusters (n + m = 6): DFT Quantum-Chemical Design" Materials 14, no. 3: 597. https://doi.org/10.3390/ma14030597

APA Style

Mikhailov, O. V., & Chachkov, D. V. (2021). Models of Molecular Structures of Hexa-Nuclear AlnFem Metal Clusters (n + m = 6): DFT Quantum-Chemical Design. Materials, 14(3), 597. https://doi.org/10.3390/ma14030597

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