D. UHF Energy Bands
III. Discussion
The ab initio calculations indicate that the one-dimensional elemental metals com
posed of Cu, Ag, Au, Li and Na are stablewith respect to the Peierls distortion, and have large cohesive energies with respect to both atomization and dissociation into diatomicmolecules, as long as spinpolarization effects of the UHF wavefunctions are allowed (see Table 1). The UHF wavefunction for each of these systems leads to an antiferromagnetic (nonmetallic) ground state having a spin density wave, although ineach case the net electronic charge density of the groundstate (obtained by adding
the up-spin and down-spin densities) is fully symmetrical. This is consistent with the low-density (or large α) solution of the Hubbard hamiltonian.2’8’7’8 In each case, the local description ofthe valence electronic structure consists of electrons centered at the bond midpoints with alternating spins.
The HF wavefunctions (which do not allow spin polarizationeffects) give spurious results, such as charge density waves for the Aga ring cluster and negative cohesive energies with respect to atomization for both the Naio ring cluster (see Table 1) and the Na2 molecule (see Appendix B). TheHFwavefunctionsfor these one-dimensional metallic clusters lead to Peierls instabilities, and very small or negative cohesive energies with respect to dissociation into diatomicmolecules.15
Thus, the HF and UHF results calculated for the one-dimensional metals com
posed of Cu, Ag, Au, Li and Na are in absolute disagreement with one another.
Since the UHF total energies are all lower than the HF total energies, (as shownin Figure 15 for Age), the variational principle suggests that the UHF results are more likely to be correct. However, unlike HF, the low-spin UHF wavefunctions are “spin- contaminated,” i.e., they are not eigenfunctions of the many-electron spin operator S2 (see Appendices A.4 and A.5) and hence contain contributions from both the many-electron singlet (5 = 0) and higher spin states such as triplet (5 = 1), quintet (5 = 2), etc., up to high-spin (5 = 2V∕2, where N Is the number of atoms in the cluster). It is likely that errors due to spin contaminationdo not affect the diatomic molecules (M3) and the ring clusters(Mjv)in a consistent manner. For example, the Ags symmetric cluster has a UHF total energy 1.801 eV (225.1 meV/atom) lower than the HF total energy, whereas the Agj molecule has a UHF total energy only 0.020 eV (10.1 meV/atom) lower than the HF total energy (see Figure 15). Hence, the results of the ab initio calculations based on single-determinant wavefunctions
(HF and UHF) are somewhat inconclusive.
In order to resolve the disagreement between the HF and UHF results, we per formed ab initio total energy calculations for each of these systems with the mul tideterminant generalized valence bond (GVB) wavefunction,3° which is a multi
determinant generalization of UHF that is a proper eigenfunction of S2 (no spin contamination); details of these results are reported elsewhere.16 The GVB total en ergies are all lower than the UHF and HF total energies, as shown in Figure 15 for the Aga ring cluster. For Ag8, the GVB total energy calculations confirm the UHF result ofstability with respect to the Peierls distortion (see Figure 15). In general, for the ground electronic states of the one-dimensional Μχ ringclusters composed of Cu, Ag, Au, Li, and Na, the GVB results confirm that the HF-Peierls description is fundamentally incorrect and that the UHF-Hubbard description is basically correct except that (unlikeUHF) the GVB description of theantiferromagnetic ground state leads to fully symmetricalspin and charge densities (no spin-contamination or spin density wave).16
Thus, spin polarization is crucial for a proper single-determinant description of the valence electronic structures of these one-dimensional metals, especially for the antiferromagnetic ground states. However, both the spin contamination and the spin density waves resulting from UHF wavefunctions for these systems are due to an incomplete treatment of the electron correlation forced by the use of a single determinant wavefunction.lβ
The UHF calculations indicate that the trend in the cohesiveenergy with respect to dissociation into diatomic molecules differs dramatically for the one-dimensional (Ag > Cu > Au) and three-dimensional (Au > Cu > Ag) noble metals (see Tables 1-2). The experimental atomization energies for the diatomic molecules and bulk
metals both follow the trend Au > Cu > Ag (see Appendix B). However, the trend for the one-dimensional metals is consistent with both the atomic ∙s-to-p excitation energies [Ag (3.740 eV) < Cu (3.806 eV) < Au (4.947 eV)],31 and with the extent of p hybridization for the Mw ring clusters as revealed by the standard Mulliken19 orbital population analysis [Ag (42 % ) > Cu (40 %) > Au (18 %); see Appendix A].
The cohesive energies (Li > Na; see Tables 1-2), the atomic s-to-p excitation energies (Li, 1.848 eV; Na, 2.104 eV),31 and the extent of p hybridization (Li, 51 %;
Na, 35 %) are all consistent for Li and Na. Hence, it is clear that sp hybridization plays a crucial rolein the cohesion of these one-dimensional metals.
The participation of the low-lying p orbitals in the valence electronic structures of these one-dimensional metal clusters leads to singly-occupied UHF valence or bitals having maximum absolute amplitudes centered at bond midpoints (see Figure 10). The UHF up-spin and down-spin orbitals are staggered, leading to a local
ized descriptionwhere electrons are centeredat thebond midpoints with alternating spins (antiferromagnetism). This implies that the cohesionin these one-dimensional metals is due to two-center one-electron bonds, similar to the one-electron bonds of the diatomic molecular cations.18’17 Hence, the Peierls-distorted diatomic lattice is unfavorable because alternate one-electron bonds are stretched and compressed.
In terms ofenergy band theory,spin polarization effects resulting from the UHF wavefunction lead to a reduction ofthe Brillouinzone bya factor of two, resultingin a ground state having completely filled energy bands, explaining the lack of Peierls instability (which is predicted only for one-dimensional metals with partially filled energy bands).
The UHF calculations indicate that the usual half-filled band model for these systemsis fundamentallyincorrect due to spin polarization effects; hence,the Peierls
instability occurs for HF but not for UHF. A generalization of the one-dimensional Peierls instability1’32 has been used to explain the Hume-Rothery rules33 [these rules correlate particular alloy structures with particular valence-electron to atom ratios, e.g., the 7-brass structure occurs frequently for alloys with electron/atom ratios of approximately 21/13 (1.54 - 1.70)33,3* such as AggZng, Cu9Al4, etc]. The present results raise doubts concerning this energy band explanation of the stability of the Hume-Rothery phases, sincespin polarization effects are neglected.