The low lying isomers of the copper nonamer cluster, Cu9
Graphical abstract
A genetic algorithm approach is used to calculate electronic properties of the energetic lowest-lying isomers of the Cu9 cluster.
Introduction
The determination of the structure and properties of transition metal clusters has received considerable attention, both theoretically and experimentally, because of the importance of such clusters in nano-science and their potential use as catalysts. We are interested in calculating the global minimum structure, and the electric dipole moment and polarizability, of nano-sized metal clusters; the electric dipole moment and dipole polarizability are interesting properties to consider, as they provide a characterization of the electronic structure, and they can be probed by experiment. For neutral and ionized copper clusters in particular, there are many theoretical investigations of the geometrical and electronic structure [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. In this Letter we consider just the cluster. The motivation of this work is to test the efficiency of our recently developed genetic algorithm code by calculating the structures, energies, and the electric dipole moments and polarizabilities, of the low lying isomers using density functional methods with a large valence basis set for copper. Particular interest arises since there is a significant discrepancy between the experimental [17] and theoretical [8], [13] values of the polarizability for .
In a recent experimental beam deflection study of neutral clusters with , Knickelbein [17] observed a polarizability per atom () for of 16.2 ± 0.8 Å3, more than three times larger than the one predicted by density functional calculations (4.9 Å3) [8]. This discrepancy is surprising since the theoretical approach presented in [8] predicts polarizabilities for realistic, geometry-optimized alkali clusters that are similar to those experimentally measured, indicating that effects of the so-called electron spillout are accurately accounted for by the density functional theory (DFT) approach [17]. One expects Cu, Ag and Au clusters to have properties that are similar to those of the alkali metals since they have filled d-shells and electronic structures determined in large part by the half-filled bands of nearly free s electrons. The experimental polarizability values for the smaller Cu clusters in this size range lie significantly higher than the classical predictions for conducting ellipsoids, and convergence to the classical predictions is only achieved for [17]. However, as Knickelbein points out, within the weak-field limit (, where is the dipole moment, F is the electric field, is Boltzmann’s constant, and is the rotational temperature), the experimentally measured ‘effective’ polarizability represents an upper limit to the electronic polarizability as the permanent dipole moment contributes to the effective polarizability in a way that depends on temperature. For symmetric rotors with principal moments of inertia and , the effective polarizability is given by [18]where parameterizes the deviation from sphericity.
Yang and Jackson [13] calculated polarizabilities of copper clusters consisting of up to 32 atoms in all electron calculations using a small basis set and the generalized gradient approximation of Perdew–Burke–Ernzerhof (PBE). They also found a large discrepancy with experiment and confirmed the values from Ref. [8]. In addition they obtained good agreement with the predictions of the classical model already at cluster sizes as small as six atoms. Furthermore, they tested their DFT approach against wavefunction based theories, and tested basis set and electron correlation effects at different geometries for smaller copper clusters. They found that these effects are too small to account for the discrepancy between theory and experiment. They conclude, however, that according to Eq. (1), low rotational temperatures combined with non-negligible permanent dipole moments can in some cases explain the observed difference between theory and experiment.
Dipole moments and polarizabilities strongly depend on the geometric structure and the delocalization of the valence electrons. None of the aforementioned studies systematically searched for the ground state structures employing DFT or ab initio approaches. In the present work we searched for low lying isomers and subsequently refined the structures using an extensive valence basis set within a scalar relativistic pseudopotential approach. However, obtaining agreement with the experimental polarizability still proves somewhat elusive.
Section snippets
Global optimization and computational methods
The structures of the energetically lowest-lying isomers, including that of the global minimum, of the copper nonamer cluster were obtained using our recently developed genetic algorithm (GA) code Belphegor described in detail below. The main functionalities of Belphegor were inspired by Johnston’s review article on genetic algorithms [19] and Hartke’s work [20]. A GA can be divided into four essential steps: selection, mating (or crossover), mutation and evaluation. Fig. 1 depicts a flow chart
Polarizability of atomic copper
The most accurate evaluations of the isotropic static electric dipole polarizability published for the copper atom are 6.67 Å3[27], [28] and 6.89 Å3[29], where the former is obtained from relativistic pseudopotential QCISD(T) calculations and the latter from relativistic all electron CCSD(T) calculations within the framework of a Douglas–Kroll–Hess (DKH) transformation, respectively. In Fig. 3, calculated polarizabilities from a variety of different DFT functionals, comprising different exchange
Summary and conclusions
To the best of our knowledge, this is the first time that a systematic approach in the framework of DFT has been employed to find the global minimum of the copper cluster nonamer cluster. It is found that the isomers labeled , and lie energetically too close to adopt either of them as the global minimum. Also the polarizability and dipole moment are calculated using a rather large basis set. While previously reported polarizabilities per atom obtained from theory are confirmed,
Acknowledgements
This work was financed by a Marsden grant (MAU-313) through the Royal Society of New Zealand. We like to thank Sascha Schäfer and Robert Krawzcyk for useful discussions. PRB likes to thank Massey University for a visitor’s grant.
References (35)
- et al.
Chem. Phys. Lett.
(1995) - et al.
Chem. Phys.
(2005) Chem. Phys. Lett.
(1992)- et al.
J. Chem. Phys.
(1990) - et al.
Phys. Rev. Lett.
(1991) Phys. Rev. B
(1993)- et al.
Phys. Rev. B
(1994) - et al.
J. Chem. Phys.
(1996) - et al.
J. Chem. Phys.
(1998) - et al.
J. Chem. Phys.
(2000)