1 Unusual electronic properties of sub -nano sized magnesium clusters Stanislav K. Ignatov 1 Artëm E. Masunov2

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Unusual electronic properties of sub-nanosized magnesium clusters

Stanislav K. Ignatov, 1* Artëm E. Masunov2

1 Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod 603950, Russia

2 NanoScience Technology Center, University of Central Florida, Orlando, Florida 32826, USA

* skignatov@gmail.com

ABSTRACT

The electronic parameters and, in particular, the isotropic electrostatic polarizability (IEP) of sub-

nanoscale magnesium clusters were studied in an expanded set of 1237 structurally unique isomers

found in the course of direct global DFT optimization of the structure of Mg2-Mg32 clusters at the

BP86/6-31G(d) level, as well as using global optimization based on DFT-calibrated MTP potential for

some larger structures. The calculation of the polarizability at the same DFT level reveals an unusual

property of the IEP – the dependence of the IEP of the most favorable isomers on the cluster nuclearity

n is linear with a high correlation coefficient, and its value for each n is close to the minimum value

among all found isomers of a given nuclearity. These features take place independently on the cluster

structure which allows hypothesizing that the energetic favorability of a cluster structure is connected

to their polarizability. A possible explanation of the observed dependence, its significance for quantum

chemistry, and the possibility of practical application are discussed.

2 
 Introduction 
Sub-nanosized metal clusters are polyatomic compounds of variable nuclearity (the number n 
of atoms in a cluster), occupying an intermediate position between atoms and nanoparticles, differ from 
the latter in a lower degree of crystalline ordering and in greater structural diversity. Their typical 
diameter is of 0.2–2 nm which corresponds to the nuclearity up to 150-200 atoms. When the size 
increases, the large crystalline domains appear in a particle which makes them similar to polycrystals, 
and  the  standard  theoretical  methods  for  the  description  crystalline  structures  can  be  applied  to 
evaluation of various physic-chemical properties (see, e.g., study of 1 where adsorption energy on Pt 
clusters went to the limiting values of crystal surface at n > 147). In contrast, the cluster structures at 
lower n (typically n=2-150) are poorly predictable, both at the levels of simple “chemical intuition” and 
standard metallic potentials which are used with high success to the description of bulk metal properties 
or the properties of metal surface. Among quantum chemical methods, only DFT methods can usually 
be used for the effective search for favorable structures of small clusters, although one cannot always 
be sure of the reliability of their predictions, since the benchmark high-level theories can often be 
applied in a limited range of nuclearities only. The same applies to the so-called novel “potentials of 
quantum accuracy” (GAP,2 SNAP,3 MTP,4 ACE 5), since they are usually calibrated by DFT results in 
a limited range of nuclearities. In this situation, the most reliable method for the sub-nano cluster 
structure prediction remains the direct DFT global optimization. At the same time, the properties of 
sub-nanosized metallic particles, both mono- and polyatomic ones, are of great interest because they 
frequently manifest higher activity 6,7 and selectivity 8 in catalysis 9-11, can serve as the base elements 
for the modern and further nanoelectronic,12,13 or spintronic14,15 devices or as a base for the novel 
nanodevices manifesting e.g., neuromorphic properties16. Previously, using the example of magnesium 
clusters, we demonstrated that the number of the cluster isomers which can be formed for the given n, 
although is high  and  quickly grows with  n,  nevertheless remains much  lesser than  the  number  of 

mathematically predicted number of connected graphs of the same number of vertices 17. This fact

allows to explore the complete set of isomers at once (at least for some n), trying to search for the

individual representatives with useful properties. In a recent study 18, we carried out such a search in

the extended set of isomers of clusters Mg2-Mg32 comprising 1237 isomeric structures located in the

direct DFT global optimization. The magnesium clusters are the convenient object for such studies

because they are simple for DFT calculations, their ground state is a singlet spin state, and many

reference data are known for them, both experimental and theoretical. In the study 18, we considered

mostly energetic and structural properties, which allowed us to locate the most stable representatives,

the energy distributions for a complete set of isomers as well as the distributions for separate

nuclearities. The analysis of these distributions allowed to predict that the size- and shape-selected

cluster formation can be performed for some n, in particular, for n=10 and 20. Also, some new tubular

structures and the new dependences between the structural features of clusters were established. In the

current study, we continue the investigation of the extended set of magnesium clusters, focusing now

on their electronic properties. Among these electronic properties, we find that the extended set of

isomers demonstrates quite unusual feature of their isotropic electrostatic polarizability (IEP): IEP of

the most favorable structures (and of only them) is linearly dependent on n, and, even more surprising,

these linearly dependent values are close in their magnitude to the minimum IEP values for the set of

isomers with the given n. This property is rather unusual because the most favorable isomers typically

have quite different geometry structure. Moreover, this property, if it will be common for other metals,

can be extremely useful since it gives the mean for the prediction how the random cluster structure is

close to the global minimum – the mean which opens new ways in cluster structure prediction. We also

show that the connection between the polarizability and the DFT energy of cluster cannot be simply

derived from any known models of polarizability although some models reproduce its linear

dependence (but do not “explain” it directly). The paper is organized as follows. In the next section, we

briefly describe the methodology of cluster generation and evaluation of their properties. In the

beginning of the Results and Discussion section, we describe the calculated cluster polarizability and

the found dependencies of it on different structural properties. We also discuss here some principles

and possible explanations of the found dependencies including the applications of some known models

of molecular polarizability. The Concluding remarks section contains a discussion of possible

implications of the found regularities and the directions that could be explored in order to rationalize

the patterns found.

Calculation details

Set of isomeric structures. Structures of magnesium clusters Mg2-Mg32 used in the analysis

were generated in the course of the direct DFT global optimization combined with graph generation

algorithm and manual construction of the highly symmetric structures as was described in detail in refs

17,18. In brief, the initial structures were generated from a complete set of connected graphs with n

vertices as well as using the evolutionary algorithm combined with taboo-search to avoid the generation

of similar structures. The geometries of generated structures were thoroughly optimized at the BP86/6-

31G(d) DFT level with higher optimization criteria (Tight stopping criteria and UltraFine DFT grid

implemented in Gausian16 19 program) and unique final structures were selected on the basis of two

different algorithms of similarity evaluation in order to establish the unique isomers. Such a procedure

continued until new unique structures ceased to appear. The Cartesian coordinates of 543 structures of

Mg2-Mg13 located by this method were reported earlier in Supporting Information for ref 17. In a course

of this work, about 9000 optimizations were carried out and about 820000 points of potential energy

surface (PES) were explored in total. All the located unique structures are the true local minima of PES

as was proven by the frequency calculations at the same theory level. The remaining 694 structures of

Mg14-Mg32 were located by the similar methodology, although without achieving the limit of all

5 
 
possible structures, with the DFT optimization using normal optimization criteria as was described in 
18. Their Cartesian coordinates were published as Supplementary Information for ref 18. 
DFT calculations. All energies, polarizabilities and other electronic properties were calculated 
at the BP86/6-31G(d) level of DFT theory which was proven to be the best level of theory describing 
the results of combined CCSD//MP2/cc-pVTZ level of theory for Mg2-Mg7 clusters.20,21 Our previous 
evaluation of the performance of this DFT level for Mg10 cluster shows that this theory level agrees 
well with  B3PW91/6-31G(d)  results  whereas PBE0/6-31G(d) results  are somewhat  worse.17  Also, 
comparison of BP86 results with the results of CCSD(T)/cc-pVQZ and MP2/cc-pVQZ shows that the 
bond lengths in small clusters Mg2-Mg4 are remarkable underestimated at the DFT level but they going 
better as nuclearity increases. At the same time, this DFT level better reproduces the cluster energies of 
CCSD(T) than it takes place in the case of MP2. All calculations were carried out using the Gaussian16 
19 program. 
Calculation  of  electronic  properties.  For the  complete  set  of  1237  structures  of  Mg2-Mg32 
obtained as described above, the additional single point calculation was carried out at the BP86/6-
31G(d) theory level with UltraFine DFT grid in order to calculate the  MO vectors of Kohn-Sham 
orbitals, HOMO/LUMO energies, the electrostatic polarizabilities, and other electronic properties. The 
obtained MO vectors were then  processed with  ChargeMol  program  22,23  in  order to calculate the 
Density Derived Electrostatic and Chemical (DDEC6) charges, bond orders, and the atom valences. 
23,24 Other electronic characteristics (Coulson’s, Mayer’s, Wiberg’s bond orders, charges and valences) 
were extracted from the calculation log files or calculated on the basis of MO vectors. Additionally, the 
values of coordination numbers, HOMO and LUMO energies, cluster energies, and, in some cases, 
NBO parameters were extracted and analyzed. DDEC6 method used here is the relatively new method 
for the bond order evaluation 23,24 which has significant advantages before the “classical” bond order 
descriptions including the Coulson’s, Mayer’s, Wiberg’s, and NBO bond orders and charges. Namely, 

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1Unusualelectronicpropertiesofsub-nanosizedmagnesiumclustersStanislavK.Ignatov,1*ArtëmE.Masunov21LobachevskyStateUniversityofNizhnyNovgorod,NizhnyNovgorod603950,Russia2NanoScienceTechnologyCenter,UniversityofCentralFlorida,Orlando,Florida32826,USA*skignatov@gmail.comABSTRACTTheelectronicparametersan...

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1 Unusual electronic properties of sub -nano sized magnesium clusters Stanislav K. Ignatov 1 Artëm E. Masunov2

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