Quantum chemical study of small AlnBm clusters: Structure and physical properties

Chemical Physics 493 (2017) 61–76

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Quantum chemical study of small AlnBm clusters: Structure and physical properties Boris I. Loukhovitski, Alexander S. Sharipov, Alexander M. Starik ⇑ Central Institute of Aviation Motors, Scientific Educational Center ‘‘Physical and Chemical Kinetics and Combustion”, Moscow, Russia

a r t i c l e

i n f o

Article history: Received 7 December 2016 In final form 12 June 2017 Available online 13 June 2017 Keywords: Atomic clusters Aluminum Boron Structure Physical properties DFT

a b s t r a c t The structure and physical properties, including rotational constants, characteristic vibrational temperatures, collision diameter, dipole moment, static polarizability, the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), and formation enthalpy of the different isomeric forms of Aln Bm clusters with n þ m 6 7 are studied using density functional theory. The search of the structure of isomers has been carried employing multistep hierarchical algorithm. Temperature dependencies of thermodynamic functions, such as enthalpy, entropy, and specific heat capacity, have been determined both for the individual isomers and for the ensembles with equilibrium and frozen compositions for the each class of clusters taking into account the anharmonicity of cluster vibrations and the contribution of their excited electronic states. The prospects of the application of small Aln Bm clusters as the components of energetic materials are also considered. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction For past decades, small atomic clusters have attracted considerable attention of researchers in basic science due to their unique physical and chemical properties [1–9]. Such clusters play an important role in different areas of physics and in technical applications, especially, for the fabrication of nanomaterials with unique characteristics [10,4,11]. Nowadays, accurate characterization of the structure and determination of physical and chemical properties of small atomic clusters are in the focus of a number of researches [12–23]. So, pure aluminum Aln and boron Bm clusters were extensively studied in the past because of a broad range of their potential application [24–28,1,29–32]. However, clusters of mixed aluminum-boron composition (Aln Bm ) have not been investigated properly until now. Only a few researches addressed the analysis of structure and properties of certain neutral Aln Bm clusters of special composition: Al12B [33], AlBm (m ¼ 1::14) [34–36], Aln Bm (n þ m ¼ 13) [37]. Note that relatively greater attention was paid to the study of ionic Aln B m clusters [38–40,37,41–44]. Though the Al and B elements belong to the same group of the periodic table, and, as expected, should have, generally, similar properties, boron possesses some unique and distinguishing characteristics compared to the other elements of the group [45]. So, the structures, composed of aluminum and boron atoms, can exhi⇑ Corresponding author. E-mail addresses: [email protected] (B.I. Loukhovitski), [email protected] (A.S. Sharipov), [email protected] (A.M. Starik). http://dx.doi.org/10.1016/j.chemphys.2017.06.006 0301-0104/Ó 2017 Elsevier B.V. All rights reserved.

bit some unique features and can be applied to the design of new types of materials [41,42,35,36,46]. The other important issue in the application of small atomic clusters is the fabrication of high energy density propellants. The usage of such clusters as additives can improve the reactivity and energetic output of traditional organic fuels [47,48,20]. It should be emphasized that earlier different sorts of nano-sized particles, including aluminum [49–51] and boron [52–54] ones, were considered as possible additives enhancing the combustion of hydrocarbon fuels. It is also believed that the usage of combined aluminum-boron structures for the high energy density materials (HEDM) can be very promising [55–58]. Despite the great interest expressed to Aln Bm clusters, the available data on the physical characteristics of small Aln Bm structures are very fragmentary. The extensive analysis of the structure of different isomeric forms of such clusters, required for accurate calculation of their physical and thermodynamic properties (see, for example [18,16,22]), has not been conducted until now. Moreover, the data on the structural isomers of clusters and their properties are indispensable for the proper interpretation of experimental data for atomic clusters and nanoparticles, as far as, at experimental conditions, the concentrations of global minimum structures are frequently smaller than those of the high-lying isomers [1,59]. The present work is aimed at the systematic study of the structure of different isomeric forms of Aln Bm (n þ m 6 7) clusters and the determination of their physical and thermodynamic properties. An additional goal of this paper is the estimation of the combustion

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enthalpies of such clusters and analysis of their potentialities in the application for HEDM. 2. Methodology 2.1. Search and optimization of cluster structure The multistep heuristic algorithm, based on the sequence of semiempirical, Hartree–Fock and density functional theory (DFT) quantum chemical calculations, developed recently [16], was used for the search of possible isomeric forms of Aln Bm (n þ m 6 7) clusters of different multiplicity (the two lowest spin states were considered in each case). In order to reduce the computation time for the search of all isomeric forms of clusters with given composition, the basic algorithm was supplemented with the genetic type approach to generate initial geometries of the structures in the manner similar with that proposed in [60]. At the final stage of the optimization procedure, the B97-2 DFT functional [61] was applied, as far as it provided high accuracy in the predictions of atomization energy Eat both for Aln [60] and for Bm [17] clusters. Moreover, the usage of B97-2 DFT functional allows one to obtain accurate value of Eat for the ground state AlB molecule compared to that predicted by high level ab initio calculations (MRCI [62] and CCSD(T) [63] methods) and obtained using composite CBS-Q method [64] (see Table 1). In the present work, all quantum chemical calculations were performed with Firefly QC program package [65] that is partially based on the GAMESS(US) source code [66]. We looked for the structures with the isomerization energy T I (it specifies the energy difference at T ¼ 0 K between the given isomer and the lowest one) smaller than the cutoff criterion Ecut ¼ 5 eV. Notice that the applied value of Ecut ensures the proper predictions of thermodynamic properties of clusters up to the temperature T = 6000 K [17]. The structure of all clusters was optimized at the UB97-2/aug-cc-pvDZ level of theory. For each stationary point on the potential energy surface (PES), a vibrational frequency analysis at the same level of theory was performed. The structures, possessing low (x 6 50 cm1 ) or imaginary vibrational frequencies, were reoptimized employing tight convergence criteria (root-mean-square forces within 105 atomic units and root-

2.2. Thermodynamic properties Thermodynamic properties of Aln Bm clusters such as temperature-dependent enthalpy HðTÞ, entropy SðTÞ and specific heat capacity at constant pressure C p ðTÞ were calculated by means of standard statistical formalism [67] both for the individual isomers and for the given class of clusters with fixed n and m numbers. In the latter case, it was assumed that the whole ensemble of isomeric forms can be specified by Boltzmann distribution at the given gas temperature T. In doing so, the contribution of different isomeric forms and electronically excited states of clusters to the thermodynamic properties of cluster ensemble was estimated by applying the methodology reported in [16,17]. The total partition function Q tot , accounting for each isomer for given class of clusters, was expressed as following

Q tot ðTÞ ¼

! L X Ti Q itr Q irot Q ivib Q iel exp  I ; kb T i¼1

ð1Þ

where L is the number of isomers in the given group of clusters; kb is the Boltzmann constant, Q itr ; Q irot ; Q ivib and Q iel are the translational, rotational, vibrational, and electronic partition functions of ith isomer. The fraction of the ith isomeric form at the temperature T, in this case, can be written as

! Q itr Q irot Q ivib Q iel T iI : ci ðTÞ ¼ exp  Q tot ðTÞ kb T

ð2Þ

Translational and rotational partition functions were calculated in the same manner as it was usually accepted in cluster physics [68,5,18], when using classic and rigid-rotator approximations, respectively. A special approach was applied for the calculation of vibrational and electronic partition functions of isomers.

mean-square displacements within 4  105 atomic units). The structures, possessing imaginary vibrational frequencies even after reoptimization with tight criteria, were rejected from further consideration. This allowed us to select the stationary points that are actually local minima (all vibrational frequencies are real) at the chosen level of theory. Zero-point energy (ZPE) corrections, obtained from vibrational frequency analysis, were taken into account for energies of all isomers. Let us remark that the extrapolation of the electronic energy of critical points, revealed at the UB97-2/aug-cc-pvDZ level of theory, to the basis set limit was not performed, as far as such a procedure did not provide the systematic improvement of the accuracy for the atomization energy values both for Aln [60] and for Bm [17] clusters with respect to the reference ones. The same tendency takes place for the AlB molecule (see Table 1).

2.2.1. Contribution of cluster vibrations Nowadays, the harmonic oscillator (HO) approximation is frequently used for the estimations of thermodynamic properties of atomic clusters [68,13,59,18] as far as rigorous anharmonic vibrational analysis for all isomers of the given class of clusters, based on accurate PES analysis, is very computationally expensive. However, the neglect of the anharmonicity of cluster vibrations can potentially lead to the considerable errors in the predicted thermodynamic properties both at low and at high temperatures [69,60]. Therefore, in the present work, the vibrational partition function for each isomer was calculated with the usage of anharmonic oscillator (AHO) approximation in the manner proposed in [17]. In doing so, analogously with [69,60], the whole spectrum of cluster vibrational states was represented by the system of independent one-dimensional oscillators, specifying the individual vibrational modes. Each ith vibrational mode was specified by Morse potential with the depth of the potential well Emax;i . The total vibrational partition function of a cluster was calculated by direct summation of the relative populations of vibrational levels. The energy of the level with vibrational quantum number V for Morse oscillator is governed by the equations

Table 1 Atomization energy Eat of triplet AlB molecule calculated at different levels of theory.

   2 1 1  xie xie V þ Ei ðVÞ ¼ xie V þ ; 2 2

Method

Eat , eV

UB97-2/aug-cc-pvDZ UB97-2/aug-cc-pvTZ UB97-2/aug-cc-pvQZ CBS-Q MRCI/cc-pV5Z CCSD(T)/WMR

1.93 2.01 2.01 1.85 1.96 [62] 1.87 [63]

xie ¼

xie 4Emax;i

;

ð3Þ

ð4Þ

where xie is the frequency of normal vibrations for the ith oscillator and xie is the coefficient that specifies the anharmonicity of the given vibrational mode of cluster.

B.I. Loukhovitski et al. / Chemical Physics 493 (2017) 61–76

The potential well depth, for each mode of isomeric structure, was estimated using the simple approach that allowed one to avoid the extensive investigation of PES for the clusters under study. This approach is based on the suggestion of Zavitsas [70,71] that the bond dissociation energy of polyatomic molecule correlates both with the corresponding frequency of vibrational mode and with the reduced mass specified for this vibrational mode during the vibrational frequency analysis. In line with this supposition, the maximal energy of the ith mode Emax;i of Aln Bm cluster can be expressed as

Emax;i ¼

!b  i a Eat xe Mi ; n  1 xAlB M AlB e

ð5Þ

where n is the number of atoms in cluster, a and b are the adjustable parameters, xie and xAlB are the characteristic vibrational frequene

63

lowing values of the correction factors for ZPE and fundamental frequencies were recommended: kZP ¼ 0:9854; kFF ¼ 0:9606 [73]. Assuming that the account for the anharmonicity of vibrations leads to the decrease of ZPE by a factor equal to ð1  0:5xe Þ and fundamental frequencies by a factor of ð1  2xe Þ [72] [see Eq. (3)], the factor, compensating only the electron correlation effects, can be estimated as following: kel ¼ ð4kZP  kFF Þ=3 ¼ 0:9937. Precisely this value of scaling factor was used to adjust all computed harmonic frequencies. Because the harmonic frequency analysis, based on differentiation of PES in the point of energy minimum, can lead to notable systematic errors for the modes with low frequencies (x 6 100 cm1), the vibrational partition function for such modes was calculated at x ¼ 100 cm1 in accordance with the recommendations [29]. This allowed us to avoid the substantial overestimation of the vibrational partition function caused by the errors inherent in harmonic frequency analysis [29,74].

cies of the ith mode of cluster and AlB molecule, respectively, Mi and MAlB are their reduced masses. The values of a and b parameters were chosen on the basis of PES investigation along the trajectories of normal vibrations for the lowest energy isomers of the following clusters: Al2, Al3, Al4, B2, B3, B4, AlB, AlB2, Al2B, Al2B2. For stretching modes, the value of Emax was identified as the bond dissociation energy, whereas for bending or torsional modes of the polyatomic cluster, the values of isomerization or torsional barrier on the PES were taken as Emax;i . The estimates of Emax;i were performed at the UB97-2/augcc-pvDZ level of theory. The resulting values for a and b parameters were found equal to 1.66 and 2.52, respectively. Details and the results of these calculations can be found in the Supplementary material. Shown in Fig. 1 is the comparison of Emax;i values obtained directly from PES analysis and calculated with the usage of Eq. (5). One can see that the relationship (5) with the resulting values of a and b reproduces the maximal energy of the ith mode Emax;i with reasonable accuracy both for rigid (Emax;i > 1 eV) and for floppy (Emax;i < 0:1 eV) modes. A special attention was paid to the choice of scaling factor for the correction of computed harmonic vibrational frequencies. Such a factor is frequently used for the simultaneous compensation both of the incomplete treatment of the electron correlation and of the neglect of the anharmonicity of vibrations [72]. In practice, two types of scaling factor can be used: special correction factor for ZPE kZP and correction factor for fundamental vibrational frequencies kFF . However, as far as, in the present work, the anharmonicity of vibrations was treated explicitly, it is necessary to choose a special factor compensating only the effects of electron correlation (kel ). Earlier, for the UB97-2/aug-cc-pvDZ level of theory, the fol-

Fig. 1. The values of Emax;i obtained in the course of PES exploration (symbols) and calculated with the usage of Eq. (5) (curve).

2.2.2. Contribution of excited electronic states For the estimation of the electronic partition function, the timedependent density functional theory (TDDFT) [75] was applied. As was shown previously, TDDFT can provide reliable thermochemical properties of small clusters [68,17]. In doing so, for each structure under study, the energies of the electronic states of cluster with the same multiplicity as that for its ground electronic state were estimated based on single-point calculations at the TDDFT-UB97-2/ aug-cc-pvDZ level of theory. The excited electronic states with other multiplicities were taken into account implicitly at the optimization stage because the ground-state isomers of different multiplicities have been considered at optimization stage. Fig. 2 depicts the results of TDDFT calculations of the vertical excitation energy T e and the available data of other researchers for the lowest energy isomers of Al3 and B3 clusters. So, the data on the values of vertical excitation energy were obtained for Al3 both experimentally [76] and in the course of the MRCI calculations [77]. The T e values for B3 structures were estimated by using the configuration interaction approach [78] and CASSCF method [79]. One can see that the TDDFT allows one to obtain the energies of electronically excited states for aluminum and boron compounds with the accuracy comparable with that provided by much more computationally expensive multireference ab initio methods. A special attention was paid to the evaluation of a number of electronic states required for the proper calculation of electronic partition function. Following to [80,81], the number of electronic states, needed for the accurate evaluation of thermodynamic prop-

Fig. 2. The values of vertical excitation energy T e for the lowest energy isomers of Al3 and B3 clusters calculated at the TDDFT-UB97-2/aug-cc-pvDZ level of theory in the present work (lines) and reported by other researchers [78,79,76,77] (symbols).

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erties of individual species in the case of rarefied gas, can be bounded by the last state before the ionization threshold. In addition, there is no need to account for the electronically excited states lying higher than the cutoff criterion Ecut taken for the restriction of the number of considered isomers. When applying the frozen molecular orbital approximation, one can estimate the first ionization potential (IP) of the cluster as the energy of highest occupied molecular orbital EHOMO taken with the opposite sign (Koopmans theorem) [82]. Thus, the following energy cutoff crite-

with stoichiometric coefficients m1 ¼ 1, m2 ¼ 0:75ðn þ mÞ, m3 ¼ 0:5n, m4 ¼ 0:5 m in line with the equation

Qc ¼

1

m1 l1

2 X

m

0 i Df H 298;i



i¼1

4 X

m

!

0 i Df H298;i

;

ð7Þ

i¼3

rion Eel cut for the truncation of the set of electronically excited states was used:

where l1 is the molar mass of fuel. The value of Q c , determined in such a way, can be considered as an amount of energy released when the unit mass of a fuel (Aln Bm cluster) reacts completely with oxidizer (O2) to form higher oxides in liquid phase (so-called ”higher heating value”) [85].

Eel cut ¼ minfEcut ; EHOMO g:

3. Results and discussion

ð6Þ

It should be noted that the TDDFT methodology has a tendency to underestimate T e values upon approaching to the ionization threshold [83]. However, because the values of IP for the considered Aln Bm clusters, obtained through HOMO energy, are typically within the range of 4–6 eV, this factor can not affect substantially the accuracy of the calculations of cluster thermodynamic properties up to the temperature T = 6000 K. 2.3. Other cluster properties For all obtained isomers of Aln Bm clusters, the collision diameter, frequently used for the estimation of transport properties, were evaluated in line with the procedure proposed in [14]. The electric properties of clusters, such as dipole moment and dipole polarizability, were also estimated. The dipole moment was calculated as the expectation value of the electric dipole operator from the electronic wave function. For the estimation of static dipole polarizability, the analytical method [84], based on the coupled perturbed Hartree–Fock formalism, was applied. The difference between the energy of highest occupied molecular orbital EHOMO and the energy of lowest unoccupied molecular orbital ELUMO (Eg ) was also calculated. When the thermochemical properties of Aln Bm clusters have been calculated, it became possible to estimate the values of combustion enthalpy Q c of clusters. The magnitude of Q c characterizes the maximal amount of a heat that can be released during their combustion at standard initial conditions. It was calculated in the manner of [20] via the values of formation enthalpy Df H0298 of the reactants and products for the following global irreversible reaction

m1 Aln Bm þ m2 O2 ! m3 Al2 O3 ðlÞ þ m4 B2 O3 ðlÞ

3.1. Isomeric composition The isomers of Aln Bm clusters (n þ m ¼ 1 . . . 7) with different multiplicity values in the ground electronic states and those with the isomerization energy T I < 5 eV were identified during the calculations (in total, 916 isomers). Shown in Fig. 3a is the dependence of the number of found stable Aln Bm structures on the (n þ m) value. One can see that this dependence is very close to exponential one. This fact allows us to conclude that the revealed set of isomers is close to being complete, as far as, for the system of interacting atoms, the number of stable and distinguishing spatial arrangements must rise exponentially with the number of constituent units [86,87]. The number of isomeric forms, L, can be roughly approximated by the following formula: L ¼ 1:35 expð0:85ðn þ mÞÞ. Fig. 3b shows the dependence of the number of isomers for Aln Bm clusters of different size on their elemental composition [it is specified by the magnitude of ðn  mÞ]. As is seen, the maxima of a number of isomeric forms are achieved for the clusters with approximately the same numbers of aluminum and boron atoms. It is worth noting that the increase in the number of structural isomers with the growth of cluster size leads to a diminishing of the energy difference between adjacent isomeric forms upon passing from smaller systems to larger ones. This fact is illustrated in Fig. 4, where the energies of structural isomers, lying within T I < 5 eV, are depicted for Al2B, Al3B2, and Al4B3 clusters. One can see that the isomers of seven-atom clusters are distributed roughly uniformly, and the mean energy gap between adjacent isomers hDEi for such clusters can be as low as  0:05 eV that is comparable with the energy of thermal motion at normal conditions ( 0:03 eV). Therefore, in order to evaluate thermodynamic prop-

Fig. 3. Total number of obtained Aln Bm isomers with the isomerization energy T I < 5 eV versus number of atoms (n þ m) (a) and the number of isomers of different size versus (n  m) (b): symbols are the calculations; dashed curve is the exponential approximation.

B.I. Loukhovitski et al. / Chemical Physics 493 (2017) 61–76

Fig. 4. The energies of isomeric structures of Al2B, Al3B2, and Al4B3 clusters. The mean gap between energy levels hDEi is given in brackets.

erties for such clusters at the temperature T > hDEi=kb , it is feasible to employ continuous distribution of isomers over the energy [88]. 3.2. Structure and energy of clusters For each isomer, the electronic degeneracy (multiplicity) 2s þ 1, symmetry group, formation enthalpy Df H0298 , rotational constants A0 ; B0 ; C 0 , vibrational frequencies x1 , . . ., xn , dipole moment lD , energy gaps between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) Eg , static isotropic polarizability a, and collision diameter r were determined in accordance with the methodology described above (see Supplementary material). The obtained values and cluster structures for the lowest ground-state ones, along with their ionization potentials and electron affinities, are presented in Table 2. It is worth noting that the isomeric structures, found in the present work, are generally similar to those reported elsewhere for Aln [32,16], Bm [24,30,31,17] and AlBn [34,35] clusters. However, for some structures, minor discrepancies were detected. So, the structure of ground state AlB5 isomer, reported in [34,35], differs from that obtained in the present work: the structure, predicted there as the most stable one, is the first isomer of AlB5 cluster with T I ¼ 0:15 eV found in the present study. This discrepancy is associated with different DFT functionals applied in the course of the optimization in [34,35] (B3LYP) and in the present work (B97-2). It should be emphasized that the calculation of reliable thermochemical properties of Aln Bm clusters requires a sufficient confidence in the accuracy of predicted electronic energy values. As was mentioned in the Section 2.1, the methodology, applied in the present work, must describe properly the energies of pure Aln and Bm clusters. This fact can be proved by the comparison of the predictions, obtained in the present study, with the highly accurate calculations of other researchers. The values of atomization energy for the Al2, Al3, and Al4 clusters were estimated with high accuracy by Zhan et al. [27] using Feller-Peterson-Dixon method with complete basis set extrapolation of CCSD(T) predictions and additional corrections. Later, atomization energies of some Aln clusters were calculated by Schultz et al. [28] using a few composite methods (MC–CO/2, MCG3/3, and MC-QCISD/3) and by Kiohara et al. [32] with the usage of CCSD(T) method with a complete basis set. Atomization energies of Bm clusters were estimated by Tai et al. [31] by means of the composite G3B3 method. Shown in Fig. 5a are the values of binding energy Eb (atomization energy per atom) of aluminum and boron clusters, calculated in the present work, as well as those obtained elsewhere [27,28,31,32]. One can see that our predictions for Aln and Bm clusters coincide well with the accurate theoretical calculations of

65

other researchers. The applied methodology also describes properly the energies of boron clusters doped by aluminum atom. This is clearly seen from Fig. 5b which depicts the Eb values for AlBm clusters calculated in the present work and those obtained employing high-level MRCI [62] and DFT [34,35] approaches. Besides the energy values for the ground state structures, the applied methodology must reproduce properly the difference in electronic energies for the structures with distinct multiplicities. Note also that the possibility of such reproducing is a rather difficult task for conventional DFT functionals [89,22]. Table 3 lists the predicted differences between the energy values for the lowest electronic states with distinct multiplicities for some clusters and analogous differences obtained from the results of accurate multireference and coupled-cluster calculations reported elsewhere [79,77,89]. One can see that the DFT methodology, used in the present work, allows one to predict properly the energy gaps DEs between the electronic states of different spin values for considered clusters, even for the cases of the very small magnitude of the gap (as for Al6). It is interesting to analyze the dependence of binding energy on the cluster composition, as far as the magnitude of Eb is one of the simplest descriptors of cluster stability [90]. The binding energy values for the Aln Bm clusters, obtained by using UB97-2/aug-ccpvDZ level of theory, are shown in Fig. 6. One can see that the energy Eb increases with m number monotonically at fixed n. Thus, the clusters with higher boron atom content are, in general, energetically more stable with respect to the complete atomization, regardless of a number of aluminum atoms. This is because the bonding between boron atoms is much stronger than for Al-Al or Al-B pairs of atoms (as evidenced from Fig. 5 and from the data on the cohesive energy for bulk boron and metallic aluminum [91]). Notice that the structure of the lowest energy Aln Bm (with m > 2) clusters corresponds to the tightly coupled boron skeleton, surrounded by the aluminum environment (see Table 2). Consequently, the Eb value of Aln Bm cluster is mainly determined by strong bonds, delocalized over B nuclei [21] within the boron skeleton. Meanwhile, for clusters with fixed m values, the dependence Eb ðnÞ has a distinguishing behavior. So, for pure aluminum clusters (m ¼ 0), the value of Eb rises with the growth of n, whereas, for the structures with m > 3, the dependence Eb ðnÞ become negative. Note that the B7 cluster possesses the greatest value of Eb among all considered structures (Eb ¼ 4:26 eV), whereas, for the aluminum Al7 cluster, the binding energy is notably smaller (Eb ¼ 2:09 eV).

3.3. Electric properties The electric properties, such as dipole moment and static dipole polarizability, determine the optical and electrostatic characteristics of molecules and clusters [92,93] and play a dominant role in intermolecular interactions [94]. As is known, they are also of great importance in the estimation of the transport properties of clusters (see, for instance, [14]). In addition, polarizability, along with binding energy, is frequently considered as a descriptor of cluster stability [90,95]. However, the accurate calculations of electric properties of complex atomic systems are usually considered as challenging matter requiring, in certain cases, the application of high-level quantum chemical methods [26,96]. Hence, it would be interesting to assess the accuracy of the predictions of electric properties using the applied DFT-based approach. From the data, listed in Table 2, one can conclude that the majority of the lowest energy isomers of Aln Bm clusters are polar, though some of them (B6, AlB3, Al2B3) possess rather large dipole moment (lD > 3 D). Clusters with such great values of dipole moment can have an anomalously small gas phase diffusion coef-

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Table 2 Structures, symmetry, multiplicity, and properties of the lowest ground state isomers of Aln Bm clusters. Spec.

Sym.

2s + 1

Df H0298 , kJ/mol

lD , D

a, Å3

Eg , eV

VIP, eV

VEA, eV

r, Å

Al Al2 Al3

Kh D1h D3h

2 3 2

330.0 514.5 628.9

0.0 0.0 0.0

9.1 19.2 24.6

1.6 1.4 1.5

5.92 6.41 6.46

0.33 1.38 1.71

4.89 5.67 6.36

Al4

D2h

3

760.2

0.0

31.8

1.3

6.50

2.11

6.69

Al5

C 2v

2

843.5

0.5

39.8

1.7

6.51

2.12

7.10

Al6

D3d

1

891.4

0.0

41.6

1.7

6.69

2.58

7.64

Al7

C 3v

2

898.4

0.7

45.8

1.6

6.22

2.36

7.89

B B2 B3

Kh D1h D3h

2 3 2

565.0 873.2 869.7

0.0 0.0 0.0

3.2 10.2 7.8

2.5 2.2 2.7

8.48 9.55 9.60

0.25 2.15 2.53

3.89 4.37 4.81

B4

D2h

1

919.1

0.0

9.2

3.7

9.37

1.83

4.96

B5

C 2v

2

992.8

0.4

11.9

3.1

8.82

2.21

5.33

B6

C 5v

1

1093.6

3.0

12.4

3.8

8.81

2.26

5.86

B7

C 2v

2

1083.2

1.8

14.5

2.2

8.15

2.73

6.00

AlB

C 1v

3

709.1

1.8

18.2

1.8

7.61

1.60

5.35

AlB2

C 2v

2

835.8

2.6

13.1

2.2

8.22

2.15

5.39

AlB3

C 2v

1

917.5

3.1

16.8

2.6

7.76

2.35

5.82

AlB4

Cs

2

975.7

2.2

16.6

2.2

8.22

2.64

6.02

AlB5

Cs

1

1031.5

1.7

17.2

3.2

8.10

2.36

6.32

AlB6

Cs

2

1086.1

2.6

19.1

2.1

7.64

2.24

6.44

Al2B

C 2v

4

747.7

0.9

17.9

1.8

7.13

1.80

6.10

Al2B2

Cs

1

864.8

1.7

21.5

1.7

7.31

2.18

6.29

Al2B3

Cs

2

940.1

3.4

20.9

2.1

7.67

2.59

6.50

Al2B4

C 2v

1

1019.3

1.1

27.2

2.5

7.47

2.26

6.90

Al2B5

C1

2

1079.8

2.9

24.2

1.8

7.36

2.76

6.62

Al3B

C 3v

3

795.8

1.4

27.3

1.7

6.67

1.94

6.88

Al3B2

C 2v

2

911.8

0.6

28.8

1.7

6.99

2.34

6.89

Al3B3

Cs

1

986.1

1.6

27.2

2.5

7.18

2.17

7.09

Al3B4

C1

2

1036.3

1.0

26.8

1.9

7.14

2.56

7.16

Al4B

C 2v

2

871.8

1.4

35.1

1.8

6.80

2.27

6.89

struct.

67

B.I. Loukhovitski et al. / Chemical Physics 493 (2017) 61–76 Table 2 (continued) Spec.

Sym.

2s + 1

Df H0298 , kJ/mol

lD , D

a, Å3

Eg , eV

VIP, eV

VEA, eV

r, Å

Al4B2

Cs

1

926.4

1.3

31.9

2.6

6.98

2.16

7.07

Al4B3

Cs

2

996.8

0.3

30.7

1.9

6.94

2.60

7.24

Al5B

C 5v

1

933.9

0.0

38.3

1.9

6.53

2.22

7.30

Al5B2

C1

2

1004.8

1.3

36.6

1.8

6.50

2.45

7.52

Al6B

C1

2

970.0

0.7

41.3

1.8

6.41

2.25

7.62

struct.

Fig. 5. Values of binding energy Eb of Aln , Bm (a) and AlBm (b) clusters calculated in the present work (solid curves) and by other researchers with the usage of accurate quantum-chemical methods [62,27,28,34,31,35,32] (symbols).

Table 3 The difference in the electronic energy values DEs ¼ Eð2s2 þ 1Þ  Eð2s1 þ 1Þ in eV for the structures with distinct multiplicities. Molecule

2s1 þ 1

2s2 þ 1

This work

Other studies

B3 Al3 Al4 Al6

2 2 3 1

4 4 1 3

1.30 0.16 0.08 0.03

1.53 0.30 0.10 0.09

ficient and high reactivity. It should be emphasized that the available data on the dipole moment of aluminum-boron compounds are very scarce, and, therefore, it is very difficult to prove the validity of obtained values of lD for all Aln Bm clusters. However, there exists a nice coincidence between the values of dipole moment for the ground state AlB predicted in this work (1.76 D) and calculated by Gutsev et al. [63] applying CCSD(T) method (1.79 D). As to static polarizability, the respective literature data for the Aln Bm (n; m > 0) clusters are absent. To date, only the data for pure Aln and Bm clusters are available, and, therefore, it is reasonable to compare the predictions of the present work with the data reported elsewhere for such clusters. The experimental values of the polarizability of Al and Al2 species were obtained by Milani et al. [97] on the basis of molecular beam deflection in the inhomogeneous electric field. The analysis of possible reasons for the dis-

(CASSCF) [79] (MRCI) [77] [CCSD(T)] [89] [CCSD(T)] [89]

crepancies between existing experimental and theoretical data on the polarizability of Al atom was conducted by Fuentealba [98]. As well, Fleig performed highly accurate relativistic electronic structure calculations of the static dipole polarizability of Al atom [99]. The values of polarizability for the lowest-energy isomers of pure aluminum clusters, comprising up to 31 atoms, were estimated within the framework of density functional theory (PBE96 functional) by Alipour and Mohajeri [12]. Shown in Fig. 7 is the comparison of the values of polarizability per atom for Aln clusters, predicted in the present work, with the data reported by other researchers. One can see the existence of a reasonable agreement of our predictions both with the measurements and with the theoretical data on the polarizability of Aln clusters. However, there exists considerable distinction in the calculated value of the polarizability for an aluminum atom with

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Fig. 6. Binding energy Eb of the most stable Aln Bm clusters with different n and m values.

respect to that measured by Milani et al. [97]. So, accurate ab initio theoretical calculations [98,99] as well as the predictions of the 3

present work give the value of polarizability a(Al)=8–9 Å , whereas its measured magnitude is by a factor of 1.2–1.3 smaller. The nature of this discrepancy, apparently, results from the inaccuracies of experimental data [98]. Fig. 7 also depicts the values of polarizability per atom for Bm clusters, obtained with the methodology of the present work and in [26] with B3LYP functional, as well as the polarizability for the ground state boron atom calculated by using the accurate CCSD (T) and MRDCI methods [100]. It is seen the excellent agreement of the predictions of the present work with the data published elsewhere. As to the triplet B2 molecule, its abnormally large value of 3

polarizability per atom (5:11 Å ) compared to those for the B3 and B4 clusters, is a computational artifact associated with the application of single-reference DFT approach to the electronic system of multireference character [17]. Note that Bm clusters possess much smaller values of polarizability per atom compared with those for Aln ones (by a factor of 3). As was mentioned above, the polarizability is one of the possible descriptors of cluster stability. So, it would be interesting to compare the obtained values of polarizability with the binding energies of clusters specifying thermodynamic stability. From

Fig. 8a it follows that the polarizability per atom correlates with the binding energy of cluster rather well: clusters with higher Eb energy possess lower values of a=ðn þ mÞ. Note also that these results coincide with the conventional expectations [90,95]. The other quantity that is worth to be compared with the polarizability of clusters, is a HOMO-LUMO energy gap (Eg ) characterizing the chemical hardness of electronic structure [90,82]. The value of Eg also can be considered as a descriptor of stability [82]. In fact, there is no such a strong correlation between the Eg energy and the thermodynamic stability of cluster. However, generally, the more stable structures possess the larger HOMO-LUMO gaps [90]. The energy Eg also characterizes the metallic (eg., electroconductivity) [101] and optoelectronic [102] properties of clusters. The data on Eg and a=ðn þ mÞ values for the Aln Bm clusters are summarized in Fig. 8b. It is seen that there exists a correlation between HOMOLUMO energy gap and polarizability per atom: clusters with higher polarizability per atom have lower values of Eg . The AlB5 cluster possesses the greatest Eg value (3.17 eV) among the structures with mixed composition. This indicates on its potentially high stability. Let us remark that the similar conclusion can be made on the basis of the analysis of binding energy values [see Figs. 6 and 8a]. The other topic that should be discussed is the dependence of the polarizability of the cluster on its isomeric structure. Shown in Fig. 9a are the values of static polarizability for obtained isomers of some seven-atom clusters (Al7, Al5B2, Al2B5, and B7) as a function of their isomerization energy T I . One can see a notable growth of a value with increasing the isomerization energy, especially, for the clusters with a high content of aluminum atoms. Note that the tendency of a growth with the increase of T I energy was observed in the past for the other types of atomic clusters [103,17,22]. This correlation is due to the fact that the isomers with higher T I value have more loose structure [103], whereas the rise in the averaged bond length, in general, leads to the growth of polarizability [104,105]. The identical tendency was also observed for the electronically excited states of polyatomic molecules [106,107]. Thus, the change in the fractions of different isomeric forms with the temperature variation can lead to the change in the properties of the Boltzmann ensemble of clusters with given n and m numbers. Fig. 9b depicts the averaged static polarizabilities hai for the equilibrium ensembles of isomeric forms of Al7, Al5B2, Al2B5, and B7 clusters calculated in line with Eq. (2) for different temperatures. One can see that the temperature rise results in the notable increase of the value of averaged static polarizability due to the change in the fractions of different isomeric forms. The increase in hai value at T ¼ 6000 K achieves a factor of 1.2 compared to that at room temperature. It should be noted that for the ensemble of isomers of Al2B5 cluster, the nonmonotonic behavior of haiðTÞ dependence with the maximum at T ¼ 2600 K takes place. The observed distinction of the averaged static polarizability of the Boltzmann ensemble of isomeric forms of clusters from that of the lowest energy isomer can be treated as a special type of the contribution to the polarizability of polyatomic species along with the pure electronic polarizability and contributions of vibrational and rotational degrees of freedom, that were extensively studied in the past (see, in particular [108,105]). Such a type of the contribution to the polarizability of clusters, arising from the possibility of isomerization, was found to be of crucial importance for reproducing the experimental data for the complex atomic structures [103,109].

3.4. Ionization potential and electron affinity Fig. 7. Static dipole polarizability per atom for Aln and Bm clusters versus n and m numbers calculated in the present work (solid lines) and obtained by other researchers [97,100,26,98,99,12] (symbols).

The ionization potential and the electron affinity (EA) are important parameters, allowing for interpretation of photoelectron spectroscopy data. Besides, their combinations are used for esti-

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69

Fig. 8. The values of a=ðn þ mÞ vs. Eb and the values of Eg vs. a=ðn þ mÞ (symbols). Corresponding linear approximations are depicted by dashed lines.

Fig. 9. Static dipole polarizability of different isomeric forms of certain seven-atom clusters versus T I value (a) and dependence of averaged polarizability hai for the Boltzmann ensemble of isomeric forms of these clusters on temperature (b).

mating of such important properties as the chemical potential, the electronegativity, chemical hardness and softness [90,82]. The vertical ionization potential (VIP) is the energy difference between the neutral and ionic (cationic) clusters at the neutral equilibrium geometry. On the other hand, the energy difference between the ground state of the cation cluster and ground state of the neutral one is referred to as the adiabatic ionization potential (AIP). Thus, the VIP is always larger than the AIP and the energy difference between them is an indication of the energy gain due to structural relaxation. In this article, we calculated the VIPs of Aln Bm clusters for their lowest energy structures at the UB97-2/aug-cc-pvDZ level of theory. Corresponding values are given in Table 2 and in Fig. 10. The AIP values, calculated at the same level of theory are also presented in Fig. 10 alongside with the experimental data for IP of bare aluminum and boron clusters [110,112]. As expected, the AIP values are slightly lower than VIP ones, especially for large clusters. So, the energy gain by relaxation of the geometry of the ionized cluster lies in the range of 0.01–0.5 eV for the clusters studied. In addition, one can conclude that our calculated ionization potentials are in good agreement with experimental data both for aluminum [112] and boron clusters [110].

Fig. 10. The calculated values of VIP, AIP, VEA, AEA as well as the IP and EA estimates using the Koopmans theorem (lines) in comparison with the available measurements [110–114](symbols).

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As a whole, the IP decreases with increasing the number of Al atoms in a cluster. The same tendency is observed for the unsigned values of EHOMO (so-called Koopmans IPs [90]), also given in Fig. 10. However, the IP’s determined using the Koopmans theorem for the Kohn–Sham HOMO orbitals systematically underestimate the VIP and AIP values (by 1–2 eV). It is remarkable that the measure of this underestimation is a measure of the quality of the DFT functional employed, and that only at the limit of the exact functional, VIP ¼ EHOMO [115,83]. The electron affinity, the ability of an atom or molecule to hold an additional electron, is another important parameter to understand the electronic properties of clusters. As the case may be, the vertical or adiabatic electron affinities can be considered. The vertical electron affinity (VEA) is the difference in the energy between the ground state of the anionic cluster with the spin multiplicity that differs by 1 from the neutral cluster and the neutral cluster having the anionic geometry. The adiabatic electron affinity (AEA), on the other hand, is defined as the energy change when an electron is bound to the neutral clusters to form anionic clusters at their respective equilibrium geometries. The VEA and AEA values, calculated for Aln Bm clusters at the UB97-2/aug-cc-pvDZ level of theory, are also summarized in Fig. 10 along with the Koopmans EA and the experimental data for AE of aluminum [111] and boron clusters [113,114]. Similarly to IP, the AEA values are slightly lower than VEA ones, whereas the EA estimates, obtained from ELUMO , can overestimate them by 1–2 eV. Meanwhile, the calculated electron affinity values coincide well with the available measurements. Besides, as evidenced from the Table 2 and Fig. 10, the EA energy for larger Aln Bm clusters has a weak dependence on their elemental composition. So, for all clusters, composed at least of 5 atoms, VEA ¼ 2:4  0:3 eV.

3.5. Thermodynamic properties In order to estimate the thermodynamic properties for each isomer of considered clusters, the parameter of anharmonicity xie was calculated as it was described above [see Eq. (4)], and anharmonic effects for cluster vibrations were treated explicitly. Consequently, in the present study, there is no need to use the special factors for the correction of vibrational frequencies, applied usually to a wide set of molecular systems to allow for both the anharmonicity and the incomplete treatment of electronic correlation. However, the analysis, performed in the present work, made it possible to determine the scaling factors both for ZPE (kZP ) and for fundamental vibrational frequencies (kFF ) especially for the aluminum-boron clusters, when the UB97-2/aug-cc-pvDZ level of theory was used: kZP ¼ 0:990 and kFF ¼ 0:979. It should be emphasized that these values differ slightly from those recommended by Tantirungrotechai et al. [73] based on the analysis of the dataset for 122 small molecules (kZP ¼ 0:9854 and kFF ¼ 0:9606). Using the methodology described above, the temperaturedependent thermodynamic functions HðTÞ; SðTÞ and C p ðTÞ were calculated for each individual isomer of Aln Bm clusters. The corresponding data are presented in a polynomial form in Supplementary material. Let us estimate now the influence of the anharmonicity of cluster vibrations on the specific heat capacity as well as the contribution of electronically excited states of clusters to the C p ðTÞ value. Fig. 11 shows the temperature dependence of dimensionless heat capacity C p ðTÞ=R, where R is the universal gas constant, for the lowest energy isomers of Al2B, Al3B2, and Al4B3 clusters calculated by utilizing HO and AHO approximations for cluster vibrations and additionally with the account for the contribution of electronic states of these clusters. One can conclude that the account for the anharmonicity leads to the nonmonotonic behavior of specific heat capacity of clusters

Fig. 11. Dimensionless specific heat capacity C p ðTÞ=R for the lowest-energy isomers of some Aln Bm clusters calculated within the following models: (i) infinite HO (dotted curves), (ii) AHO (dashed curves), (iii) AHO with accounting for the excited electronic states of cluster (solid curves).

with respect to the calculations with HO approximation. So, at low temperatures, the value of C p ðTÞ, obtained with the AHO approximation, is somewhat higher than that predicted by using HO one (up to a factor 1.05–1.15). This is due to greater density of vibrational states compared to that for HO. However, the finiteness of the number of vibrational states for AHO, in contrast to infinite ones for HO, leads to the decrease of C p values with increasing T. The contribution of electronic states of clusters to specific heat capacity becomes notable only at higher temperatures (T > 3000 K), especially, for clusters of larger size (due to the higher density of electronically excited states). So, for the sevenatom Al4B3 cluster, at T ¼ 6000 K, it achieves 3:2R [50 electronic states within the criterion specified by Eq. (6)], whereas for the triatomic Al2B cluster it amounts to 2:1R only [26 states within the criterion specified by Eq. (6)]. It is evident that the existence of structural isomers can also contribute to the effective thermodynamic properties of the ensemble of clusters. In the case, when there exists thermodynamic equilibrium between different isomeric forms of the cluster, the thermodynamic functions can be evaluated by using the total partition function specified by Eq. (1). This case occurs when the characteristic time of temperature variation in gasdynamic process sgas is much greater than the characteristic time of isomerization reactions sis . The effect of the contribution of isomeric forms on the specific heat capacity, for this case, is illustrated in Fig. 12. As is seen, the isomeric forms can contribute substantially to the effective heat capacity of the Boltzmann ensemble of Aln Bm clusters. Such a contribution leads to the nonmonotonic behavior of C p ðTÞ dependence, typical for the equilibrium ensemble of clusters [116,5,59]. So, the C p ðTÞ dependence can have pronounced peaks (maxima) (see Fig. 12). The appearance of these peaks is associated with the additional reaction heat required for the transformation of the isomeric forms of clusters in course of the isomerization reactions. Shown in Fig. 13 is the variation of fractions of the most abundant isomers of certain Aln Bm clusters for the case of thermally equilibrium ensemble of isomeric structures. One can see that the pronounced peaks (maxima) for C p ðTÞ dependence are achieved at the temperatures for which the sharp growth in the fractions of high-energy isomers occurs. So, the appearance of the peak at T ¼ 1000 K for the Al3B2 cluster is caused by the intensive formation of IS(1), IS(3), and IS(4) isomers at this temperature, and such a peak for the Al4B3 cluster at T ¼ 1800 K is associated with the formation of IS(10), IS(11) and IS(13) isomers in the tem-

B.I. Loukhovitski et al. / Chemical Physics 493 (2017) 61–76

71

Fig. 12. Dimensionless effective heat capacity C p ðTÞ=R for the Al2B (a), Al3B2 (b), and Al4B3 (c) clusters calculated with AHO approximation and accounting for the contribution of excited electronic states for the lowest energy isomers (dotted curves) as well as for the equilibrium ensemble of isomeric forms (solid curves) and for the ensemble with frozen isomeric compositions at T  ¼ 2000 and 6000 K (dashed and dash-dotted curves, respectively).

perature range T ¼ 1500  2000 K. Note that the structural transformations of different isomeric forms can be considered as an analog of the second-order phase transitions for clusters and nanosystems [116,117], however, this question requires special analysis. It should be emphasized that the irregular behavior of the temperature dependence of the fractions of isomers with respect to their number is associated with the fact that the values of ci are defined not only by the isomerization energy T I , but also by its rotational, vibrational, and electronic partition functions [see Eq. (2)]. As illustrated in Fig. 13c, the most abundant isomer of Al4B3 cluster, at T > 2500 K, is IS(13) one, whereas the fractions of IS (1)-IS(12) isomers are substantially smaller at such temperatures. Let us consider now the case, when the rate of temperature variation is greater than the rate of isomerization reactions, i.e. sgas  sis , and thermodynamic equilibrium between different isomeric forms of the cluster is not established. In this case, the specific heat capacity of the ensemble of different isomers can be estimated by averaging over the frozen composition of isomeric structures with the fractions ci ðT  Þ [see Eq. (2)] determined for given temperature T  . The C p ðTÞ dependences for the ensembles of Al2B, Al3B2, and Al4B3 clusters, estimated in a such a way for T  ¼ 2000 and 6000 K, are depicted in Fig. 12. One can see that, in this case, the temperature dependence of effective heat capacity has no distinct peaks that are typical for the case when the isomerization processes have occurred. Nevertheless, the contribution of structural isomers to the specific heat capacity of the ensemble of isomeric forms is rather notable in this case too.

As to the case, when we have sgas  sis , the thermodynamic description of the ensemble of isomers with respect to the effective C p ðTÞ dependence becomes pointless. At such conditions, the kinetic model, treating the isomerization processes, should be applied. However, this approach can hardly be implemented, nowadays, due to a large number of possible isomeric forms and isomerization reactions that must be involved in the kinetic model. As a reasonable alternative, the molecular dynamic simulation, employing a realistic analytic many-body potential between atoms, can also be applied [118,119,59]. 3.6. Combustion enthalpy The combustion enthalpies of Aln Bm clusters can be evaluated by using the thermochemical data calculated above. The estimated values of Q c for the lowest energy Aln Bm isomers of clusters with n þ m ¼ 2 . . . 7 are depicted in Fig. 14. During the calculations, it was supposed that aluminum and boron oxides in combustion products formed in the liquid phase. The formation enthalpies for Al2O3(l) and B2O3(l), in line with [120], were taken equal to 1510 kJ/mol and 1250 kJ/mol, respectively. One can see that the higher values of combustion enthalpy per fuel mass unit are achieved for the clusters with a lower content of aluminum atoms. It is also seen that the magnitude of Q c for pure aluminum and boron clusters gradually decreases with increasing the number of atoms (n and m, respectively). Evidently, the Q c values should converge for Aln and Bm clusters to the combustion enthalpies for bulk aluminum (31.1 MJ/kg) and bulk boron (58 MJ/kg) [120]. Special

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Fig. 13. Fractions ci of the most abundant isomeric forms for the equilibrium ensembles of some Aln Bm clusters as a function of temperature.

analysis, based on the power function extrapolation, showed that Q c values for Aln and Bm clusters achieved the corresponding bulk limits at n  10 and m  15. Therefore, only rather small clusters (with the number of atoms no more than 10) can ensure the combustion enthalpy greater than that for bulk aluminum boride (if the lowest energy isomers are considered), and precisely such clusters are the most promising for the fabrication of HEDM. So, the lowest energy isomers of B5–B7 clusters have Q c ¼ 70:8—74:8 MJ=kg that is by a factor of 1.2–1.3 higher than the analogous value for bulk boron. It should be emphasized that some isomeric forms of Aln Bm clusters, possessing a high value of T I and, therefore, greater formation enthalpy compared to that for the lowest energy isomers, can potentially be more promising for the usage as the components of HEDM. However, it is reasonable to consider only those structures that can be stable with respect to the isomerization and dissociation processes. It is believed [47] that the long-term kinetic stability of highly energetic clusters at normal conditions can be ensured if the values of corresponding energy barriers on the PES for these structures are equal, at least, to  1 eV. In order to estimate the stability of revealed isomeric forms of Aln Bm clusters, the approach, proposed in [60], was applied. This approach is based on the analysis of the maximal vibrational energy of modes Emax;i of a cluster [see Eq. (5)]. So, let us assume that the Aln Bm isomers, that can be potentially interesting for the usage as the components of HEDM, must possess T I > 1 eV and Em;m ¼ minðEmax;i Þ P 0:8 eV, where Emax;i is the maximal vibrational energy of the ith mode of cluster. Shown in Fig. 15 are the Em;m –T I diagrams for the isomers of five- and seven-atom clusters.

Fig. 14. Combustion enthalpies Q c for the lowest energy isomers of Aln Bm clusters with different values of n þ m (given in brackets) as a function of n number as well as those for bulk boron and aluminum.

One can conclude that only five isomers possess needed T I and Em;m values. The selected isomers, satisfying the stability criteria, are listed in Table 4 along with the corresponding values of kinetic stability descriptor Em;m and combustion enthalpy Q c . Note that there are no structures promising for the usage in HEDM among the isomers of six-atom clusters. It is seen that the values of combustion enthalpy of potentially stable isomers of seven-atom Aln Bm

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Fig. 15. The values of Em;m vs. T I for different isomers of five-atom (a) and seven-atom (b) Aln Bm clusters. The structures, promising for the usage in HEDM, are depicted by arrows.

Table 4 The isomers of potential interest for using as components of HEDM, their isomerization energy T I , the minimal value of the maximal energy of vibrational modes Em;m as well as the values of Q c and DQ c . Spec.

T I , eV

Em;m , eV

Df H0298 , kJ/mol

Q c , MJ/kg

DQ c , MJ/kg

B5 IS(3) B7 IS(1) Al3B4 IS(51) Al4B3 IS(19) Al5B2 IS(15)

2.27 1.19 1.8 1.49 1.23

1.08 1.79 0.80 0.80 1.09

1211.6 1197.9 1210.0 1140.2 1123.3

78.8 72.3 47.8 42.8 39.2

4.0 1.5 1.4 1.0 0.8

clusters, possessing T I > 1 eV, is greater than those of the lowest energy isomers by  1  1:5 MJ/kg. Among the five-atom isomers with T I > 1 eV, only the third isomer IS(3) of B5 cluster is potentially stable, and the excess value of combustion enthalpy with respect to the lowest energy isomer DQ c is equal to 4 MJ/kg for this structure. It is worth noting that such small highly energetic clusters can possess much greater reactivity with respect to the oxidation process than that of the analogous bulk materials [121,122]. This allows one to consider them as very promising components of HEDM. 4. Conclusion The geometrical structure, symmetry and electronic energy of all possible isomeric forms of Aln Bm clusters of different multiplicity with ðn þ m 6 7Þ and the isomerization energy within T I < 5 eV were determined with the usage of density functional theory and the original heuristic multi-level algorithm developed for the global search and optimization of cluster structures. It allowed us to calculate cluster static isotropic polarizability, dipole moment, HOMO–LUMO gap, collision diameter and enthalpy of formation. The electronic properties such as ionization potential and electron affinity were additionally determined for the lowest energy isomers. The thermodynamic properties of Aln Bm clusters such as enthalpy, entropy, and specific heat capacity were also estimated. In doing so, we took into account the contribution of excited electronic states of clusters and used the anharmonic oscillator approximations for cluster vibrations. Good agreement of the predictions of the present work with the available reference data on

binding energy, the energy gap between the structures with different multiplicities as well as on the electric properties of clusters was achieved. The correlation between the polarizability of clusters and other stability descriptors, such as binding energy and HOMO–LUMO gap, was demonstrated for the lowest energy isomers of considered clusters. The tendency of the polarizability rise with increasing the isomerization energy for the high-lying isomeric forms of clusters was revealed. The thermodynamic functions both for the individual isomers and for the equilibrium ensemble of each class of clusters were estimated. It was found that the account for the anharmonicity of cluster vibrations results in the considerable corrections of the thermodynamic properties compared to those calculated with harmonic oscillator approximation practically at the whole considered temperature range (T ¼ 300—6000 K). It was also demonstrated that the effect of electronic degrees of freedom of clusters on their thermodynamic properties became notable at high temperatures (T > 3000 K), especially, for the clusters of large size. The contribution of structural isomers to the thermodynamic properties of the ensemble of clusters was analyzed both for the case of the existence of a thermodynamic equilibrium between different isomeric forms of clusters and for the case of the frozen composition of isomers. In the former case, the contribution of the isomeric forms to the specific heat capacity of the ensemble of isomeric structures with given composition was found to be rather substantial. It results in the nonmonotonic behavior of C p ðTÞ dependence with the distinctive peaks. In the latter case, the temperature dependence of effective specific heat capacity has not such peaks, though the contribution of structural isomers can be also notable.

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The prospects of the usage of small Aln Bm clusters as components of HEDM were analyzed. The computations revealed that the higher values of combustion enthalpy per fuel mass unit were achieved for the clusters with lower aluminum atom content. It was shown that the combustion enthalpies for pure aluminum and boron lowest-energy clusters gradually decrease with increasing the size of cluster approaching to those of bulk limits. Consequently, the application of only rather small clusters, comprising up to 10 atoms, is worthwhile for the fabrication of HEDM. Such clusters possess rather high combustion enthalpy that can be by a factor of 1.2–1.7 greater than the analogous values for bulk materials. Some high-energy isomeric forms of Aln Bm clusters, possessing larger formation enthalpy, can potentially be more promising for the usage as components of HEDM than the lowest energy isomers. However, only a few of them can be potentially stable with respect to isomerization process. The estimations showed that the excess value of combustion enthalpy with respect to that of the lowest energy isomers, for the potentially stable seven-atom clusters, can mount to  1—4 MJ=kg. Acknowledgments This work was supported by the Russian Foundation for Basic Research (projects Nos. 16-29-01098 and 17-01-00810). The authors also thank their esteemed colleague Ms. Liliya Pavlenko for the assistance with data processing during global optimization of cluster structures. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.chemphys.2017. 06.006. References [1] F. Baletto, R. Ferrando, Structural properties of nanoclusters: energetic, thermodynamic, and kinetic effects, Rev. Mod. Phys. 77 (2005) 371–421. [2] G. Zhao, Y. Lei, Z. Zeng, Absorption spectra of small silver clusters Agn (n = 4, 6, 8): a TDDFT study, Chem. Phys. 327 (2006) 261–268. [3] F.Y. Naumkin, Flat-structural motives in small alumino-carbon clusters, J. Phys. Chem. A 112 (2008) 4660–4668. [4] S.A. Claridge, A.W. Castleman Jr, S.N. Khanna, C.B. Murray, A. Sen, P.S. Weiss, Cluster-assembled materials, ACS Nano 3 (2009) 244–255. [5] A.S. Sharipov, B.I. Loukhovitski, A.M. Starik, Theoretical study of structure and physical properties of ðAl2 O3 Þn clusters, Phys. Scr. 88 (2013) 058307, 10pp. [6] B.J. Irving, F.Y. Naumkin, A computational study of ’Al-kanes’ and ’Al-kenes’, Phys. Chem. Chem. Phys. 16 (2014) 7697–7709. [7] S.E. Fioressi, R.C. Binning Jr, D.E. Bacelo, Structures and energetics of Ben Cn ðn ¼ 1  5) and Be2n Cn ðn ¼ 1  4Þ clusters, Chem. Phys. 443 (2014) 76–86. [8] B.H. Cogollo-Olivo, N. Seriani, J.A. Montoya, Unbiased structural search of small copper clusters within DFT, Chem. Phys. 461 (2015) 20–24. [9] H.-S. Hu, Y.-F. Zhao, J.R. Hammond, E.J. Bylaska, E. Apra, H.J. van Dam, J. Li, N. Govind, K. Kowalski, Theoretical studies of the global minima and polarizabilities of small lithium clusters, Chem. Phys. Lett. 644 (2016) 235– 242. [10] B.D. Leskiw, A.W. Castleman Jr, The interplay between the electronic structure and reactivity of aluminum clusters: model systems as building blocks for cluster assembled materials, Chem. Phys. Lett. 316 (2000) 31–36. [11] X.F. Zhou, A.R. Oganov, Z. Wang, I.A. Popov, A.I. Boldyrev, H.T. Wang, Twodimensional magnetic boron, Phys. Rev. B 93 (2016) 085406. [12] M. Alipour, A. Mohajeri, Computational insight into the static and dynamic polarizabilities of aluminum nanoclusters, J. Phys. Chem. A 114 (2010) 12709–12715. [13] Y. Dong, M. Springborg, Y. Pang, F.M. Morillon, Analyzing the properties of clusters: structural similarity and heat capacity, Comput. Theor. Chem. 1021 (2013) 16–25. [14] A.S. Sharipov, B.I. Loukhovitski, C.-J. Tsai, A.M. Starik, Theoretical evaluation of diffusion coefficients of ðAl2 O3 Þn clusters in different bath gases, Eur. Phys. J. D 68 (4) (2014) 99, http://dx.doi.org/10.1140/epjd/e2014-40831-2. [15] V.S. Baturin, S.V. Lepeshkin, N.L. Matsko, A.R. Oganov, Y.A. Uspenskii, Prediction of the atomic structure and stability for the ensemble of silicon nanoclusters passivated by hydrogen, Europhys. Lett. 106 (2014) 37002.

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