Binding Energy per Atom

STUDY OF THE STABILITY OF LITHIUM CLUSTERS 5.1. Binding energy per atom 

Due to the lack of experimental results, quantities like the dependence of the binding energy per atom with the number of atoms are useful for giving us hints as to the quality of our VB results. Let us study the anionic and neutral clusters separately. 

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Table 1. Calculated Average Bond Distance d, Number of Bonds Per Atom NB, Magnetic Moment Per Atom p, Binding Energy Per Atom Eb, and Spin Gaps Ai and A2 of Mnw Clusters.93... 

Table 2. Binding Energy Per Atom Eb, Distance D from Atoms to the Cluster Center, and Average Magnetic Moment Per Atom p for Octahedral Six-Atom Clusters. Data Collected from Zhang et al.107... 

TABLE 1. Results of GGA calculations for the binding energy per atom (eV) of the two lowest energy isomers, Xy and Xy, of noble metal clusters with i/=0, l (labels as in Fig. 1). Only geometries I, II, and III, are planar. 

Figure 2. Left equilibrium geometries of the two lowest energy isomeric states of Au clusters obtained using LDA or GGA scalar relativistic pseudo-potentials. The ground state is Au for GGA and Auj for LDA (except for n=6, which LDA structure is also Aue). Right difference in the binding energy per atom of the planar and 3D structures given in the left panel for neutral gold clusters with 6

Figure 5. Left panel lowest energy equilibrium structures of AunTM+ clusters with 2

An understanding of the structure of molecules requires a proper quantum mechanical description of the covalent bond that cannot be captured by the use of central pair potentials. We therefore extend our linear combination of atomic orbitals (LCAO) treatment of the s-valent dimer to three-, four-, five-, and six-atom molecules respectively. Following eqs (3.46) and (4.17), we write the binding energy per atom for an. -atom molecule as... 

We can understand the behaviour of the binding energy curves of monovalent sodium and other polyvalent metals by considering the metallic bond as arising from the immersion of an ionic lattice of empty core pseudopotentials into a free-electron gas as illustrated schematically in Fig. 5.15. We have seen that the pseudopotentials will only perturb the free-electron gas weakly so that, as a first approximation, we may assume that the free-electron gas remains uniformly distributed throughout the metal. Thus, the total binding energy per atom may be written as... 

This non-pairwise behaviour is most easily demonstrated by considering the coordination number dependence of the binding energy. It follows from eqs (5.68), (5.69), and (5.70) that the binding energy per atom of a lattice with coordination number may be written in the form... 

Fig. 5.17 The binding energy per atom U as a function of the coordination number for aluminium. The crosses correspond to LDA predictions, whereas the curve is a least-squares fit of the form of eqn (5.72). The lattice types considered are the linear chain ( = 2), graphite ( = 3), diamond ( = 4), two-dimensional square mesh ( = 4), square bilayer ( = 5), simple cubic (x = 6), triangular mesh (x - 6), vacancy lattice (x — 8) and face centred cubic (x = 12). (After Heine eta/. (1991).)...

The so-called band-structure contribution to the total binding energy per atom is given by subtracting this double-counting term, eqn (6.55), from Using eqn (6.26) we have... 

The total binding energy per atom of the NFE metal can then be written as the sum of the three terms ... 

Thus, the total binding energy per atom of a NFE metal can be expressed in a physically transparent form, as the sum of a volume-dependent contribution and a pair-potential contribution in a manner that is reminiscent of the semi-empirical embedded atom potential of eqn (5.68). It follows from eqs (6.59M6.72) that... 

The parabolic variation in the cohesive energy across the 4d series is driven by the d-bond contribution alone, as is clearly demonstrated by Fig. 7.11. The sizeable drop in the cohesive energy towards the middle of the 3d series is a free-atom phenomenon, resulting from the special stability of Cr and Mn atoms with their half-full d shells. The simplest model for describing the bonding of transition metals is, therefore, to write the binding energy per atom as... 

The binding energy per atom of the AB alloy may, therefore, be written very simply as... 

We have seen in the previous chapter that the total binding energy per atom of an elemental sp-valent system may be written within the approximation as the sum of three terms, namely... 

The semiclassical model always predicts the most compact cluster equilibrium structure because the binding energy per atom is a direct function of the number of bonds. The EH and CNDO methods, which include electron repulsion effects, predict that the charge, geometry, and energy levels are dependent on the type of site where the cluster is adsorbed. 

We will investigate the stability of the anionic lithium clusters in Section 5. A relevant quantity is the dependence of the binding energy per atom on the number of atoms, as well as the electron affinity of neutral Lin clusters. From the latter, we evaluate whether these neutral clusters are able to receive an extra electron and to form an anionic system. 

In Table 5, we compare our results with the literature through the binding energy per atom. 

Binding energy per atom Eb /n (kcal/mol) of the small anionic lithium clusters. 

The predictions of EH described previously for Ag clusters are confirmed by the CNDO-type calculation. The linear geometry is more stable than other geometries when 4d, 5s, and 5p orbitals are included in the calculation. The data calculated by CNDO in Table V for linear Ag particles confirm the behavior of IF, FA, BEjn, and AE with size found earlier by EH calculation. The binding energy per atom is greater for even- than for odd-size particles. The band gap... 

Lithium clusters have been a popular model for the calculation of metal properties because of their low atomic number. Lasarov and Markov (49) used a Hiickel procedure to determine the properties of a 48-atom Li crystal. They found a transition to metal properties with the binding energy per atom approaching 1.8 eV at 30 atoms. The ionization potential approached the bulk value since some electrons occupy antibonding molecular orbitals, as observed for Ag clusters. The calculated properties of the largest cluster were not those of a bulk metal. 

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