Metallic clusters have been the subject of intense interest in recent years [19,20]. Calcium is a very abundant metal that plays an important role in a variety of compounds, mechanisms, and processes. This element is of interest because of its potential use in excimer lasers, carbon-chemical engineering, and ion deposition. Despite its popular chemical usages, only a limited number of experimental observations for pure calcium clusters [21] have been reported. Calcium belongs to the group-IIA alkaline-earth metals with closed-shell electronic configuration, [Ar] 4s2. The bonding in the bulk quasi-metal is quite strong with a cohesive energy of 1.825 eV. In contrast, the Ca2 molecule presents a weakly bound ground state with dissociation energy of only 0.14 eV and strongly bound excited states [22]. The discrepancy in the bonding behavior at the two size limits, bulk and dimer suggests that a change in the bonding must take place as a function of cluster size.
One property of small metallic clusters that has received little attention is their vibrational spectra. Recently a detailed theoretical investigation of the harmonic vibrational frequencies of Rh2 through Rh6 using density functional methods and large basis sets [23] has been reported. For small calcium clusters there have been very few ab-initio calculations that include electron correlation effects or report vibrational frequencies: Ca2 [24–26], Ca3 [27], Ca4 [26-30], and Ca5 [26]. For larger calcium clusters there are no first principles calculations, although there have been attempts to
model the cluster geometries with the empirical potential of Murrell and Mottram (MM) [31-33] that has two and three body interactions. The results of the calcium cluster study in the thesis has been published by J. Mirick et al. [34]
In this chapter we perform an exhaustive all-electron
study within the DFT framework and the GGA of calcium clusters containing up to
13 atoms. Results are presented in Section 3.2 for the calcium dimer including
a thorough comparison between various calculation methods. The energetics,
vibrational frequencies, and thermodynamic properties in the harmonic
approximation for Ca3 through Ca13 are
described in Section 3.3. This section
also discusses several isomerization reaction paths and report several cluster
structures corresponding to saddles of the energy surface. A thermally induced structural transition
for Ca12 is highlighted in this section. The study will help us gain insight in to the
models and mechanisms that bring atoms together to form condensed matter.
The Kohn-Sham equations [2] were solved self consistently using the GGA representation of the correlation functional. The Becke’s three-parameter hybrid method [7] was used in the study of calcium clusters, as described in Chapter 2. The hybrid method includes the Hartree-Fock exchange and local and non-local terms to the correlation functional provided by Perdew-Wang [9,10]. A triple split valence basis set, 6-311g(d), was used in all calculations involving the calcium atom, i.e. calcium clusters and CaCO3 clusters (see Table 2.1). We found the basis set to be sufficiently large to give good results at a reasonable computational effort. Adding an additional diffuse function produced insignificant changes in both the energy and bond length of Ca2.
Based on the dimer results, all calculations reported in this study of calcium clusters were performed using the triple valence basis set with the addition of a d polarization function. Results in forthcoming sections do not include the Basis Set Superposition Error (BSSE) [35] correction. However, we did test for BSSE with calculations up to Ca4 and found an insignificant decrease in the total energy for these cluster sizes. Worth noting is that BSSE is important when the basis set is small, and our basis set is large.
One way to characterize the various methods and basis sets used in computational chemistry is to compare the results of the calculations with experiments. Experimental results of the interactions between calcium atoms are somewhat limited to the ionization potential of the calcium atom, the dissociation energy of the calcium dimer, and the equilibrium interatomic distance in the calcium dimer. Although calcium clusters have been obse