Physical and Thermodynamic Properties of AlnCm Clusters: Quantum

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Physical and Thermodynamic Properties of AlC Clusters: Quantum Chemical Study n

m

Boris I. Loukhovitski, Alexander S. Sharipov, and Alexander M. Starik J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp5108087 • Publication Date (Web): 28 Jan 2015 Downloaded from http://pubs.acs.org on February 1, 2015

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The Journal of Physical Chemistry

Physical and Thermodynamic Properties of AlnCm Clusters: Quantum Chemical Study Boris I. Loukhovitski, Alexander S. Sharipov and Alexander M. Starik* Central Institute of Aviation Motors, Moscow, Russia Scientific Educational Centre “Physical-Chemical Kinetics and Combustion”, Moscow, Russia

*Corresponding author. E-mail: [email protected] Keywords: aluminum carbide, clusters, isomers, thermochemistry, electric properties.

Abstract Geometrical structures and physical properties, such as rotational constants and characteristic vibrational temperatures, collision diameter, enthalpy of formation, dipole moment, static isotropic polarizability, magnetic moment of different forms of AlnCm clusters with n=0…5, m=0…5 have been studied with the usage of density functional theory. Different forms of clusters with the electronic energy up to 5 eV have been identified by using the original multi-step heuristic algorithm based on semiempirical calculations and density functional theory. Temperature dependencies of thermodynamic properties such as enthalpy, entropy and specific heat capacity were calculated both for the individual isomers and for the Boltzmann ensembles of each class of clusters.

1. Introduction For past years, great efforts of researchers addressed the studies of small cluster properties.1-9 Such clusters play an important role in astrophysics and are constituent in the atmosphere of different planets.10,11 Small clusters can form upon burning of hydrocarbon and metallized fuels12-15, in electric

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discharge16,17 during laser ablation.16,18,19 In addition, small clusters can be used as building blocks of new nanoscale materials with special properties.20,21 Non-stationary models of nanoparticle formation in complex plasma must involve the kinetics of clusterization that requires the knowledge of thermodynamic properties of small clusters. Considering the existence of different cluster isomeric forms, the actual problem is the reduction of kinetic nucleation mechanism. Such a reduction can be performed via combining some isomers into the groups isolated on the potential energy surfaces (PES) for given molecular system.22,23 This allows one to take into account only the processes with chosen ensembles of isomers. However, in this case, one needs to know the thermodynamic properties of such ensembles of clusters. Clusters composed of carbon and metal atoms of are considered as very important structures for different applications. So, such clusters can be applied for development of new class of materials for semiconductors, ceramics and for the fabrication of high energy density fuels.9,24-26 One of the most interesting clusters that can be synthesized via combustion of metallized fuels are the clusters composed of aluminum and carbon atoms. It should be emphasized that the smallest AlnCm clusters AlC, Al2C, Al2C2 were extensively investigated both theoretically and experimentally26-28. At the same time, previous works,4,26,29-30 addressed the analysis of more heavy clusters (n=2-4, m=2-8), were focused on the study of relatively symmetric structures or on AlnCm structures with fixed n/m ratio. However, even for such clusters, thermochemical properties were not calculated. Interest to the investigation of AlCn clusters (carbon cluster doped by Al atom) and AlnC clusters (aluminum cluster doped by C atom) were also appeared, because doped clusters can be utilized for synthesis of nanostructured and cluster-assembled materials.31-37 It should be noted that AlCn clusters come into play in astrochemistry because carbon and aluminum atoms are abundant in stellar atmospheres.33,38 In addition, aluminum carbide coating can be used for the protection of non-oxidized aluminum nanoparticles in energetic materials.39,40 However, until now there have been no data on such important physical properties of different forms of AlnCm clusters as collision diameter, static isotropic polarizability, dipole electric and magnetic 2 ACS Paragon Plus Environment

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moments that are important for the estimations of cluster reactivity and transport properties. Furthermore, it is unclear what are the contributions of the different isomeric forms of AlnCm structures to the thermodynamic and physical properties of the ensembles of such clusters. The present paper addresses the search and characterization of all possible isomeric forms of AlnCm (n=0..5, m=0..5) clusters with the electronic energy lower than the preselected cut-off criterion and calculation of their physical and thermodynamic properties.

2. Methodology Today, in order to obtain the information about the structure of clusters and their physical and thermodynamic characteristics, quantum-mechanical calculations, that are based on solving the Schrödinger steady-state equation for many-electron system, are used. The progress in numerical methods of solving Schrödinger equation makes it possible to choose the technique appropriate for the specific problem. Today, sophisticated post-Hartree-Fock methods41 or modern versions of density functional theory (DFT)42 are widely applied. Calculation of thermodynamic and physical properties of clusters implies the systematic search of possible isomeric forms of cluster with given atomic composition.6,43 The well-known Becke’s threeparameter hybrid functional (B3LYP)44 is one of the most popular methods to study AlnCm clusters.4,6,31,33,36 However, the strategies for locating the global and local minima on the PES based on generating all reasonable starting geometries and minimizing each of them to the nearest minimum with the usage of B3LYP functional, are computationally expensive even with the basis sets of moderate size. Note that simple estimates indicated that the number of minima on the PES correlating with stable cluster isomers rises exponentially with the number of atoms in clusters.43 In the present study, the following heuristic algorithm was developed for the systematic search of possible AlnCm isomers. At the first stage, atoms were placed in a random order in accordance with a randomly chosen way of the generation of starting geometry: C1 (no symmetry constraints), Cs, Ci, C2 symmetries, chain structure, cubic lattice or close packing of spheres. Note, that random alternation of 3 ACS Paragon Plus Environment


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types of symmetry constraints, including C1, allows one to accelerate the search of all possible isomeric forms substantially. The generated configuration was considered as suitable one for the initial approach of cluster structure, if the minimal distance between pairs of atoms was not smaller than 2min(Rci)/1.2 (1.27 Å), and maximal spacing of atom from the nearest one is not more than 2max(Rci)⋅1.2 (2.9 Å), where Rci is the covalent radius of atom of ith sort. Note that the limits of minimal and maximal distance between pairs of atoms were adjusted to obtain realistic initial configurations. At the second stage, the constructed configuration was optimized by means of semiempirical PM3 method.45 Obtained structure was considered as a suitable one for the next optimization step, if its energy does not exceed the minimal PM3 energy of obtained isomers by 10 eV. At the third stage, the single point calculations at the UHF/STO-3G level of theory41 were performed, that allowed us to exclude the structures optimized with PM3 method whose UHF energy exceeded the minimal HF one by 10 eV. At the fourth stage, the selected structures were re-optimized at the UHF/STO-6G level of theory. The obtained configurations with the energy higher than 6 eV as well as the duplicate structures were also excluded from further consideration. At the final stage, the selected configurations were re-optimized again at the UB3LYP/631+G(d) level of theory. AlnCm clusters with the electronic energy higher than 5 eV were excluded at this step, and it was supposed that all isomers of given composition were found. The procedure of the search of possible isomers was interrupted, when the number of randomly chosen configurations, used for the search of last 10 isomers, became 100 times higher than that used for the search of first 10 isomers. The applied optimization algorithm allows us to reduce the time necessary for the search of all isomers of given elemental composition as far as the “bad” structures were rejected at each stage of the level theory rise. The proposed procedure was tested earlier for the search of Al2O3 isomers reported recently,6 and all isomers were found successfully. Note that the developed hierarchical methodology comprising different levels of theory is not absolutely novel and its other versions were successfully applied for the search of clusters in the past.4, 46


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For each stationary point on the PES of AlnCm system, vibrational frequency analysis was performed at the same level of theory that as applied for final optimization. The calculated frequency values were scaled by a factor of 0.9648 in line with the recommendations of Merrick et al.47 to take into account an incomplete treatment of electron correlation as well as the use of truncated basis set and neglecting of anharmonicity of vibrations in the estimations. Refining the electronic energy values were performed via the single point calculations with UB3LYP functional at the extended 6-311+G(3df) basis set for geometry optimized at the UB3LYP/631+G(d) level of theory. The commonly applied notation for such calculations is UB3LYP/6311+G(3df)//UB3LYP/6-31+G(d). However, not long ago, a number of hybrid density functionals with perturbative second-order correlation were developed.48,49 These functionals allows one to achieve better predictive ability for energy values than the conventional DFT functionals. According to many realistic tests, the B2PLYP method can be regarded as the best general purpose one for thermochemical calculations.48,50 Therefore, precisely the UB2PLYP/6-311+G(3df)//UB3LYP/6-31+G(d) method was used for refining of energy values for the isomer of lowest energy in each group of clusters. In accordance with the recommendations of our recent work,8 the estimations of dipole moments µ and static polarizabilities α were performed by using the UB3LYP/6-311+G(2d) level of theory for the geometries optimized with 6-31+G(d) basis set (UB3LYP/6-311+G(2d)//UB3LYP/6-31+G(d)). The dipole moment was computed as the expectation value of the electric dipole operator for the electronic wave function. For the calculation of static dipole polarizabilities, at the first stage, the method developed by Cammi et al.51 was applied. Then, the values of static isotropic polarizability α were scaled by a factor of 1.11, that provided the best coincidence with the known experimental data obtained for 21 species including aluminum containing compounds.8 As well, the collision diameters of AlnCm clusters σ were estimated by using the procedure suggested by Sharipov et al.6 and based on the calculation of the dimensions of the aggregate of van der Waals spheres centered on all atoms. All DFT calculations were performed by using Firefly QC program package52 which is partially based on the GAMESS(US) source code53. ACS Paragon Plus Environment

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When the electronic degeneracy, electronic energy, rotational constants, vibrational frequencies were determined, the thermodynamic properties such as temperature-dependent enthalpy H(T), entropy S(T) and specific heat capacity Cp(T) can be calculated by means of standard statistical formalism through the partition function. However, the existence of different isomers of AlnCm clusters, especially possessing low electronic energy, must be taken into account when computing the thermodynamic properties for given class of clusters. The approach applied recently6 implies that the different isomers and electronic states of AlnCm with fixed n and m numbers are in thermal equilibrium, i.e. the ensemble of clusters with given n and m can be specified at given temperature T by Boltzmann distribution. Therefore, the temperature dependencies of enthalpy and entropy for Boltzmann ensemble of AlnCm clusters can be calculated with the use of total partition function Q accounting for each isomer for given class of clusters L  Ti  i i Q = ∑ Qtri Qrot Qvib × g i exp − e  , i =1  kT 

(1)

where L is the number of isomers in the given group of clusters; Qtri, Qroti and Qvibi are the translational, rotational and vibrational partition functions of ith isomer, gi and Tei are its statistical weight and electronic energy. The partition functions Qtri, Qroti and Qvibi were calculated in the same manner as in our recent work6 by using the harmonic oscillator and rigid rotator approximations. The thermodynamic properties of individual species, such as temperature-dependent enthalpy H(T), entropy S(T) and specific heat capacity Cp(T) was expressed through the partition function Q by means of standard statistical formalism.54 For the ensemble of clusters, the quantity x specifying some physical property should be averaged over all possible isomeric forms and in the case of thermal equilibrium can be expressed as

(

i i Qrot Qvib g i exp − Tei / T

L

x av (T ) = ∑ xi γ i , γ i = i

L

∑Q

j rot

(

)

Q g j exp − Te / T j vib

j

)

,

(2)

j

where xi is the value of this physical property for the ith isomer.


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3. Validation of applied methodology The isomers of AlnCm clusters (n=0..5, m=0..5) of different multiplicity in their ground electronic states with the electronic energy lower than 5 eV with respect to the most stable configurations (in total, 834 isomers) were found. Good correlation between isomeric structures, predicted at different stages of the optimization procedure, was observed. As an example of such correlation, the optimization history for the isomers of Al2C2 cluster is presented in Supporting Information. The number of stable AlnCm structures rises exponentially with the number of atoms in accordance with the conventional expectation43 (see fig. 1). So, for Al5C5 the total number of isomers within the range of 5 eV, respecting the lowest lying cluster, equals to 117. The number of isomeric forms N can be approximated by following formula N=0.95exp(0.5(n+m)). The difference in the energy between adjacent isomeric forms is approximately equal to 0.05 eV in this case, and this fact must be taken into account when computing thermochemical and thermodynamic properties of clusters. It should be emphasized that the calculation of reliable thermochemical properties of AlnCm clusters requires a special confidence in the accuracy of predicted electronic energy values. First of all, the methodology applied in the present work, must describe properly the energies of Cm and Aln clusters. Atomization energy ∆Eat of the carbon clusters were calculated by Karton et al.55 with the use of composite W4 method which was found to be very accurate to reproduce experimental data for Cm clusters.56 From fig. 2 one can see that the methodology, applied in the present study, allows one to describe the measurements of Gingerich et al.56 and to reproduce the predictions of Karton et al.55 Note that the applied UB3LYP/6-311+G(3df) and UB2PLYP/6-311+G(3df) methods provide almost identical results. It is worth noting that the ground state of C2 dimer, obtained in the present work, is triplet one, though the singlet state of C2 is commonly accepted as the lowest energy state.55 In fact, the singlettriplet energy gap is rather small (~0.1 eV55), and the calculations with the B3LYP functional, applied in the present work, gives the inverse arrangement of these electronic states. The calculations with the use of other popular DFT functionals demonstrate the same tendency. The use of composite computational methods allows one to reproduce the correct sign of this energy gap. However, the application of 7 ACS Paragon Plus Environment


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composite methods for the analysis of the structure of all isomeric forms of AlnCm clusters is unreasonable due to high computational costs. Nevertheless, the multiplicities of the rest ground state Cm species were determined correctly with the methodology used in the present work. The Aln clusters were studied by Ahlrichs and Elliott57 with the use of DFT approach with BP86 functional and SVP basis set. Values of atomization energy predicted there57 agree rather well with the experimental data for Al2 and Al3 clusters reported by Schultz et al.58 and NIST CCCB Database59 as well as with the results of high accurate multilevel MCG3 and MC-UT calculations for Al2, Al3, Al4 clusters.58 This is clearly seen from the data depicted in fig. 3. From the plots shown here one can also conclude that atomization energies of aluminum dimer and trimer estimated in the present work correlate rather well both with the experimental data58,59 and with the theoretical predictions of Schultz et al.58 However, for larger clusters, methodology applied in the present study slightly underestimates the values of the atomization energy of clusters in comparison with those obtained by Ahlrichs and Elliott57 and Schultz et al.58 Note that the B2PLYP method compared to the B3LYP one predicts the values of ∆Eat closer to those calculated by Ahlrichs and Elliott57 and by Schultz et al.58 Nevertheless, one can conclude that the B2PLYP approach makes it possible to estimate the atomization energy of small Aln clusters quite satisfactorily. From comparison of atomization energies of Cm and Aln clusters (see figures 2 and 3) it follows that Cn clusters are more stable than Aln ones. Earlier, the structures and energies of AlCm clusters were studied by Largo et al.33 at the B3LYP/6-311+G(d) level of theory. For the calculation of AlC atomization energy, the high-level multireference MRCI/aug-cc-pVQZ approach, assuming scalar relativistic corrections, was applied.60 Allendorf et al.61 estimated the thermochemical properties for quartet AlC on the basis of composite BAC-G2 method proved to be efficient for Al-containing compounds. The results of the calculations of AlCm clusters atomization energies along with the predictions of the present work are summarized in fig. 4. One can see that there is good agreement between the predictions with usage of these highly accurate quantum chemical approaches and the measurements of Thoma et al.62 The comparison


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suggests that the methodology applied in the present work can predict the atomization energies of AlCm clusters (m=1..5) very well. Ashman et al.63 determined the structures and energies of AlnC clusters on the basis of DFT approach (PBE functional). The structure and atomization energy of Al5C cluster were investigated by Gui-Fa et al.37 with the use of similar methodology. These data are also depicted in fig. 4. It is seen that B2PLYP method provides somewhat better agreement than B3LYP one with the calculations of Ashman et al.63 and Gui-Fa et al.37 for clusters with m>3. In addition, one can conclude that the values of the atomization energy of AlCm clusters are notably higher (approximately by a factor of 2) than those of AlnC clusters. Recently Irving and Naumkin30 calculated the dissociation energies of some AlnCm clusters at the PBE0/6-311(d,p) level of theory. In that work, the energy values for two different dissociation processes were computed. Firstly, upon dissociation one can expect that carbon fragment retains its structural integrity, whereas aluminum fragment is atomized in the course of this process: AlnCm=nAl+Cm (dissociation energy De). Secondly, the initial cluster can be decomposed into two separate aluminum and carbon clusters: AlnCm=Aln+Cm (dissociation energy De´). The comparison of our calculations with the data for De and De´ reported by Irving and Naumkin30 is presented in Table 1. One can observe reasonable agreement between the values of dissociation energy predicted by different methods.

4. Results and discussion For each isomer, the electronic degeneracy (multiplicity) 2s+1, formation enthalpy ∆fH2980, rotational constants A0, B0, C0, vibrational frequencies ω1,..,ωn, dipole moment µD, static isotropic polarizability α and collision diameter σ were determined in accordance with the methodology considered above. The obtained values for the ground state clusters along with cluster structures are given in Table 2. The appropriate data for the rest isomers of AlnCm clusters are presented as Supporting Information. Note that, for the calculations of formation enthalpy, the B2PLYP functional with


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perturbative second-order correlation were used, as far as this method proved to be more accurate than B3LYP one for reproducing atomization and dissociation energies of AlnCm clusters. It should be emphasized that the structures of the ground state AlC2, AlC3, AlC4, AlC5, Al2C2, Al2C4, Al3C, Al3C2, Al3C3, Al4C2, Al5C clusters, obtained in the present work, are generally similar to those reported elsewhere.4,26,31-33,36,38 However, for some structures, the minor discrepancies were detected. So, the structure of ground state Al5C2 isomer reported by Dong et al.26 is not the same as that obtained in the present work. In fact, the structure predicted by Dong et al.26 as the most stable one, is the second isomer of Al5C2 cluster revealed in the present study, though the energy gap between these structures is very small (0.09 eV). This discrepancy is associated with the different theoretical methods applied for the optimization procedure (MP2 by Dong et al.26 and B3LYP in the present study). It is remarkable that analyzing the energy of clusters one can gain an insight into their relative stability.36,57 One of the simplest stability descriptors is the cohesive energy Ecoh defined as the atomization energy per atom.64 The cohesive energy of the aluminum–carbon clusters AlnCm obtained upon UB2PLYP/6-311+G(3df)//UB3LYP/6-31+G(d) calculations is shown in fig. 5 as a function of n and m. One can see that the energy Ecoh increases with m number monotonically. The clusters with higher carbon atom content are energetically more stable with respect to the complete atomization. Meanwhile, for fixed m values, the dependence Ecoh(n) has a distinguishing behavior. So, for pure aluminum clusters (m=0), the value of Ecoh rises with the growth of n, whereas, for AlnC5 clusters, Ecoh decreases with the growth of aluminum atom number. For other m values, the dependence Ecoh(n) is non-monotonic. For example, for AlnC and AlnC2 structures, there exists minima at n=4 and 2, respectively. The electric properties of clusters such as dipole moment and static dipole polarizability are needed for understanding and modeling of different phenomena. Particularly, they specify the intermolecular interactions and transport properties of clusters.8 Polarizability also serves as a descriptor of stability and reactivity of clusters.64,65 Therefore, it would be interesting to assess the accuracy of the values of electric properties predicted by applied methodology. Figure 6 compares the results of the 10 ACS Paragon Plus Environment

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calculations of the dipole moment of AlCm (m=1..5) clusters performed in the present work with the data reported elsewhere.29,31,33,38 One can see that agreement is quite good. As it follows from Table 2, Cm and Aln clusters of the lowest energy possess zero dipole moment due to their symmetric structure, with the exception of Al5 one. However, some isomeric forms of these clusters have substantial values of dipole moment in the range of 1÷3.5 D (see Supporting Information). The majority of AlnCm clusters with the lowest energy (n, m>0) are polar (µ~0.5÷5 D), and only certain isomeric forms possess extremely large values of dipole moment up to 10÷15 D (see Supporting Information). Note, that high-lying isomeric forms of AlnCm clusters with such great dipole moment can have anomalously high reactivity and anomalous transport properties. The calculations of polarizability are usually considered to be more challenging to the accuracy of the applied level of theory. In view of the absence of corresponding data for the AlnCm (n, m>0) clusters, let us compare the predictions of the present work with the data for pure Cm and Aln clusters reported elsewhere. Static isotopic polarizability values of small Cm clusters were studied by using DFT methods by Fuentealba.66 Later, Abdurahman et al.2 estimated the polarizabilities of Cm clusters with the usage of single-reference and multireference methods. Parasuk et al.67 calculated the components of the static dipole polarizabilities of different electronic states of carbon dimer. The comparison of the calculations of the present work with the results of other researchers2,66,67 and with the data68 for Cm clusters is shown in fig. 7a. One can see that the methodology, applied in the present study, reproduces the polarizabilities of C1, C3, C4 and C5 structures rather well compared to the calculations of other researchers (the difference in the predictions does not exceed a factor of 1.05). As to triplet dimer C2, our calculations overestimate its polarizability somewhat greater. The main reason of this overestimation is the multireference character of C2 structure67, whereas the single-reference DFT approach, applied in the present work, can not provide very accurate results for C2 dimer.2 It should be emphasized that the data on the static polarizability of Aln clusters are very scarce. Experimental data on polarizability of Al and Al2 were reported by Milani et al.69 The discrepancy between calculated and measured values of Al atom polarizability was discussed previously.70 More 11 ACS Paragon Plus Environment


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accurate calculations of static dipole polarizability of Al were conducted by Fleig71 taking into account relativistic and electron correlation effects. With respect to larger clusters, the values of polarizability for the lowest-energy aluminum clusters containing from 3 to 31 atoms have been investigated systematically within the DFT approach by Alipour and Mohajeri.72 Shown in fig. 7b is the comparison of the calculations of polarizabilities of Aln clusters conducted in the present work with the data reported by other researchers. One can conclude that the methodology, applied in the present work, describes both the measurements of Milani et al.69 and the calculations of Fuentealba70 and Fleig71 for Al and Al2 rather well, whereas for Aln clusters with n>2 the present methodology predicts somewhat higher values of polarizability than those reported by Alipour and Mohajeri.72 This difference is caused, in a considerable extent, by the values of empirical scaling factor taken in the same manner as in our previous work8 and which was not implied by Alipour and Mohajeri.72 This factor allows one to take into account the finiteness of basis set and contribution of zero point vibrational motion of molecular system that can contribute notably to polarizability of polyatomic molecules.73 As it follows from figure 7 and Table 2, Cm clusters possess smaller values of polarizability compared to those for Aln clusters. For small pure aluminum and carbon clusters (n≤5, m≤5), roughly linear dependence of polarizability on the n and m numbers is observed. This means that the polarizability of these small clusters is roughly additive quantity. Note that, at larger m and n, this dependence deviates from linear one substantially.2,66,72 For small mixed AlnCm (n=0..5, m=0..5) clusters the dependence α(n, m) is more complex. The values of polarizability for the most stable isomers presented in Table 2 can be approximated by two-parameter dependence with the cross term:

α (n, m ) = 8.21n + 2.26m − 0.050n ⋅ m . This means that, even for small AlnCm clusters, the polarizability is not absolutely quantity. The other interesting issue is the dependence of polarizability on the structure of cluster. For example, figure 8 depicts the values of static polarizability calculated for obtained isomers of six-atomic clusters Al5C, Al3C3 and AlC5. One can see that certain isomers have anomalously large polarizability


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compared to other clusters with the same composition. As is seen from fig. 8, these isomers are near linear or contain linear fragments in their structures. This tendency is valid also for the isomeric forms of other clusters under study. Note that, in accordance with the maximum polarizability principle,64,65 the isomers having larger values of polarizability are considered to be unstable with respect to isomerization. The additional property, that can be discussed, is the existence of magnetism in AlnCm clusters. Since the contribution of orbital magnetic moment is usually small compared with the contribution of spin magnetic moment in clusters,74,75 the value of total spin magnetic moment is thought to be a reasonable estimate of the entire magnetic moment of AlnCm clusters. The total spin magnetic moment is usually computed as M = 2SµB = [nα − nβ]µB, where S is the total electron spin, nα and nβ are the numbers of the alpha and beta spin electrons, respectively, and µB is the Bohr magneton. From Table 2 one can see, that the majority of the obtained AlnCm clusters have, at least, one unpaired spin and thus, would possess non-zero magnetic moment. Consequently, they would undergo the deflection in a Stern– Gerlach inhomogeneous magnetic field. However, for such clusters as C3, C5, Al2C2, Al2C4, Al4C2, Al2Cm (m=1..5) the energetically preferred isomers are singlet ones. So, they will not deflect in a Stern– Gerlach field. The temperature-dependent thermodynamic functions H(T), S(T) and Cp(T) were calculated both for each individual cluster and for the Boltzmann ensembles of AlnCm clusters with given n and m. Fig. 9 depicts the dimensionless specific heat capacity for the lowest energy AlnCn clusters and the temperature-dependent effective specific heat capacity for the Boltzmann ensemble of AlnCn isomers. Data for thermodynamic properties of isomers are presented as Supporting Information. It is seen that there exists non-monotonic behavior of effective specific heat capacity of the ensembles of AlnCn clusters for n>1. The appearance of the maxima in Cp(T) dependence is associated with the contribution of structural isomers and isomers in the electronic states with higher electronic energies.6,76 It is worth noting that ignoring the contribution of isomers can lead to substantial


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underestimation of specific heat capacity of the whole ensemble of each class of clusters. The error in the magnitude of Cp can achieve the value of (5÷6)R, where R is the universal gas constant. It is interesting to estimate the validity of harmonic approximation used in the present work for the calculation of vibrational partition function. For this purpose, the temperature-dependent specific heat capacity for the lowest energy AlnCn clusters was also estimated with the usage of anharmonic oscillator approximation. In doing so, it was assumed that the energy of vibrational level of each mode in cluster depends on the vibrational quantum number V as following54

E (V ) = ω e (V + 1 / 2) − ω e xe (V + 1 / 2) , 2

(3)

where ωe is the frequency of normal vibrations and ωexe is the coefficient that specifies the anharmonicity of given vibrational mode in cluster. Assuming that E(V) dependence achieves its maximum with E=Emax at V=Vmax, in line with the work,77 one can obtain the following expressions for ωexe and Vmax

ω e xe =

ω e2 4 E max

, Vmax =

ω e −ω e xe . 2ω e xe

(4)

The value of Emax was evaluated via atomization energy of cluster in line with the expression Emax=∆Eat/(n-1), where n is the number of atoms in cluster. Note that, for diatomic molecule, the value

of Emax determined in this manner coincides with dissociation energy. In this case, the total vibrational partition function of cluster is calculated as Qvib = Π

M i =1

i Vmax

 E i (V ) − E i (0 )  , T 

∑ exp −

V =0

(5)

where M is the number of modes in cluster. The appropriate specific heat capacity was calculated in the same manner as for the case of harmonic oscillator approximation. From the plots shown in fig. 9 one can see that the effect of anharmonicity on heat capacity becomes substantial at T>3000 K. So, at T=3000 K, the neglect of unharmonicity of cluster vibrations leads to the underestimation of Cp value by 5-7%.


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The other interesting issue, which should be discussed, concerns the consequences of transitions between different isomeric forms and excited electronic state structures with temperature change in thermally equilibrium case. Note that the dependence of physical properties of the ensemble of clusters on the composition of isomers were discussed earlier.76, 78 The exploration of this dependence is closely related with the theory of second-order phase transition.79 Depicted in fig. 10a are the averaged static polarizabilities for the Boltzmann ensembles of isomeric forms of Al5C, Al3C3 and AlC5 clusters. One can see that the change in the composition of isomers with the temperature rise results in notable deviation of the value of averaged static polarizability. The most pronounced effect is observed for the ensemble of Al5C clusters, as far as the certain high-lying Al5C isomers have anomalously large polarizability (see fig. 8), and their contributions come into play at high temperatures. Another cluster property which can be of interest for the ensemble of cluster isomers is the mean distance to the center of mass RCM specified by the expression RCM =

1 n ∑ ri − rCM n i

where ri is the vector position of ith atom and rCM is the vector position of the center of mass.78 This value is widely used to characterize the size and compactness of a particle.76 The calculated values of

RCM for the Boltzmann ensembles of isomers of Al5C, Al3C3 and AlC5 clusters are shown in fig. 10b. One can see that the behavior of the dependencies depicted in fig. 10a and 10b are similar. The main reason of this similarity is the fact that high-lying isomers possessing larger value of polarizability have, in the most cases, near linear structure, and, hence, larger values of RCM. Therefore, precisely these isomers contribute considerably to both polarizability and mean distance to the center of mass. Note that the dependencies presented in fig. 10 are of qualitative significance only, as they were obtained by ignoring the effects caused by the nuclear motion in the course of vibrational and rotational excitation of clusters.80 On the basis of known thermodynamic properties of molecular clusters AlnCm, one can calculate equilibrium mole fractions of such clusters at different temperatures. The composition of cluster mixture ACS Paragon Plus Environment

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can be found in the course of the minimization of Gibbs free energy for given elemental composition in gas phase at fixed temperature and pressure.81,82 Figure 11 depicts the temperature dependencies of equilibrium mole fractions γi of different clusters for two cases: (1) only the lowest energy clusters are taken into account, (2) all obtained isomers are considered, and γi value of each class of AlnCm clusters is determined as a sum of mole fractions of all considered isomers. One can see that, at T1800 K, Al4C3 mole fraction decreases, whereby the mole fractions of smaller clusters such as Al2C2, Al2C4 and AlC2 rise. At very high temperature (T>4600 K), atomic aluminum and atomic carbon prevail. One can see that the difference in the composition of the mixture for the cases (1) and (2) is not very high for the most clusters, though the account for all isomeric forms is of crucial importance for the prediction of the abundance of some clusters (Al3C4 and Al4C4).

Concluding remarks The geometrical structures and physical properties of different forms of small AlnCm clusters with n=0…5, m=0…5 were found by applying original heuristic algorithm, based on density functional theory and semiempirical methods, for the systematic search of possible isomers. The obtained properties for AlnCm clusters such as atomization and dissociation energies, dipole moment and static dipole polarizability were validated against available data. It was shown that the calculation methodology chosen in the present work provides good agreement both with experimental data and with calculations of other researchers. It was found that the dipole moment and polarizability of some highlying isomers have anomalously large values respecting those for the lowest energy isomers. It was revealed that isomers with large polarizability have near linear structure or contain linear fragments. This fact must be taken into account when estimating the stability, reactivity and transport properties of AlnCm clusters. ACS Paragon Plus Environment

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On the basis of predicted physical properties of clusters, the static polarizability, mean distance to the center of mass and thermodynamic functions such as enthalpy, entropy and specific heat capacity both for the individual isomers and for the Boltzmann ensembles of each class of clusters were calculated. It was shown that the temperature dependence of the specific heat capacity of Boltzmann ensembles of AlnCm clusters demonstrates non-monotonic behavior. High-lying isomers can contribute considerably to both polarizability and mean distance to the center of mass of the ensembles of isomers of AlnCm clusters, and averaged values of these physical properties can change notably with temperature growth. The gas phase equilibrium thermodynamic calculations of AlnCm mole fractions, when elemental composition Al/C = 1/1, showed that, at atmospheric pressure, the most abundant cluster in the temperature range T=600…1800 K is aluminum carbide Al4C3, but, at higher temperature (T=20004000 K), Al and C atoms as well as small clusters Al2C2 and AlC2 dominate.

Supporting Information Table containing the geometrical configurations and properties of the predicted clusters and their isomers. This material is available free of charge via the Internet at http://pubs.acs.org.

Acknowledgements This work was supported by the Russian Foundation for Basic Research (grants 14-08-31247 and 12-0892008). The part concerning the development of special heuristic algorithm for the systematic search of possible isomers of AlnCm clusters and determination of their structure and electric properties such as dipole moment and static polarizability was supported by the grant of Russian Science Foundation (project 14-19-01128).


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160

number of isomers

140 120 100 80 60 40 20 0 0

2

4

6

8

10

n+m

Fig. 1. Number of AlnCm isomers with the electronic energy Te

Physical and Thermodynamic Properties of AlnCm Clusters: Quantum

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