1. Cohesive Energies
Using UHF wavefunctions we calculated cohesive energies of the Cu10, Agθ, Ag8, Agio, Auιo, Liβ, Liχ0, Li14, and Nai0 symmetric ring clusters (low-spin) with respect to atomization (2) and dimerization (3). These are reported in Table 1, where the total cohesive energies have been divided by N. These cohesive energies are calculatedusing (i) the total energies of low-spin ring clusters at fixed values of the lattice constant (a), (ri') the total energies ofdiatomic molecules at their calculated equilibrium internuclear separations (Re), and (in) the total energies of the isolated
atoms. In all cases the UHF wavefunctions are optimized with no orbital symmetry
■restrictions. The UHF total energies are lower than the HF totalenergiesin all cases except for Auj and the isolated atoms, where the UHF and HF total energies are equal. Further details axe given in Appendices A-B.
The results given in Table 1 indicate that at the UHF level the symmetric ring clusters are all quite stable with respect to dissociation intoboth atoms and diatomic molecules. In contrast with HF, for UHF the cohesive energyper atom increases with increasing N.
Comparison of the UHF cohesive energiescalculated for the Afι0 ring clusters with the experimental cohesive energies for the three-dimensional bulk metals (given in Table2) indicates that the trend in the cohesive energy with respect to dimerization differs dramatically for theone-dimensional (Ag > Cu > Au) and three-dimensional (Au > Cu > Ag) noble metals. However, the experimental and calculatedatomiza
tion energies for the diatomic molecules both follow the trend Au > Cu > Ag (see Appendix B). This could indicate a fundamental difference in the bonding for the one-dimensional and three-dimensional systems.
2. Spin Density Waves
The UHF wavefunction contains a separate orbital for each valence electron, where the orbitals occupied with up-spin electrons (↑ or α) are allowed to overlap the orbitals occupied with down-spin electrons (‡ or β}. Foreach of the Cuιo,Agβ, Ag8, Ag10, Auιo, Liβ, Liιo, Li14, and Naio symmetricring clusters, the UHF ground state optimized without orbital symmetry restrictions is low spin and leads to valence orbitals having maximum absolute amplitudes centered at the bond midpoints, as shown in Figure 10 for Ag8.
By imposing orbital symmetry restrictions, we solved self-consistently for low- spin “excited” states having valence orbitals with maximum absolute amplitudes centered at the atoms (as shownin Figure 11 for Aga), leading to significantly higher total energies [for Ag8, the UHF energy for the atom-centered state is higher than that of thegroundstate (bond-centered) by 109.6 meV/atom; further details of these atom-centered (excited) UHF states are givenin Appendix D].
In each case, theα-spin orbitals and ∕3-spinorbitals optimizedwithout symmetry restrictions for the low-spin ground state break symmetry in such a manner that (i) the total valence α-spin density has maxima and minima on alternating sets of bond midpoints (periodicity 2α), (ii) the total valence ∕3-spin densityis phase-shifted fromthetotal total valenceα-spin density by a such that the maxima (and minima) of the valence α-spin and ∕3-spin densities axe staggered, leading to a spin density (defined by the difference between the α-spin density and the ∕3-spin density) with periodicity 2o, and (iti) the total valence electronic charge density (disregarding spin) is fully symmetric (with periodicity a). Hence, for each of the low-spin Cu10, Age, Aga, Agio, Auιo, Liβ, Liιo, Lii4, and Naio symmetric ring clusters, UHF leads to an antiferromagnetic description having a charge density with periodicity a and spin density with periodicity 2α. The local description of the ground state valence electronic structure in each case involves electrons centered at the bond midpoints with alternating spins, e.g., a β a β a β, etc.16
3. Peierls Instability-
In contrast with HF, the UHF description of the low-spin Age ring cluster leads to stabilitywith respect to the Peierls distortion (5α). The UHF total energy calculated without orbital symmetry restrictions increases quadratically as a function of δa as
shown in Figure 12.
At sufficiently large values of Sa, we anticipated the possibility that the valence orbitals could slide away from the bond midpoints (as shown in Figure 10) towards the atoms (as shown in Figure 11). However, this does not occur for Sa ≤ 0.30 Â even though these UHF wavefunctions are optimized without orbital symmetry restrictions, allowing complete freedom for the positions of maximum absolute or bital amplitudes. As a further test for sliding valence orbitals, we constructed a set of skewed “starting guess” orbitals for Sa = 0.30 Â having maximum absolute amplitudes centered at alternate positions midway between,the atoms and the bond midpoints. The UHF iterative self-consistent optimization of these skewed orbitals resulted in orbitals having maximum absolute amplitudes centered exactly at bond midpoints.
The UHF orbitals shown in Figure 10 implythat the cohesion of the symmetric ring cluster is due to two-center one-electronbonds, similar to the one-electronbonds ofthe diatomic molecular cations.16,17 Hence, the Peierls-distorted diatomic lattice is unfavorable because alternate one-electron bonds are stretched and compressed.