Thesis by

Here we present results of extensive ab initio totalenergycalculations for various one-dimensional ring clusters composed of copper, silver, gold, lithium and sodium. Hence, UHF leads to stable one-dimensional structures (no Peierls instability) for Cu, Ag, Au,Li, and Na.

Energy Band Theory — Non-Interacting Electrons

The Peierls Instability

Peierls1 showed that the application ofband theory to symmetric one-dimensional metals with partially occupied valence bands leads to instability with respect to geometric distortions. Thus, for a one-dimensional metal with a partially occupied valence band, Peierls showed that it is always possible to find a distortion lowering the energies of the one-electronstates below the Fermi level (and raising theenergies of the one-electron states above the Fermilevel).

Hartree-Fock

Cohesive Energies

Results for the series Liβ, Liχo, Liχ4, and the series Aga, Aga, Agio indicate that the HF cohesive energies decreasewith increasing N. Both the HF and UHF cohesive energies for the one-dimensional metal clusters (Table 1) are much smaller than the respectiveexperimental values for the three-dimensional bulkmetals (Table 2).3βι37∙28.

Charge Density Waves

For low-spinAg8, the lowest energyHF wavefunction leads to a charge density wave having maximaand minima centered about alternate atoms (see Figure 8a). This excited charge density wave HF state(Figure8b) was optimizedself-consistentlybyimposing D4h orbital symmetry restrictions, leading to four orbitals that are D4∣1 symmetry combinations of four two-center, two-electronbonds (similar to the bond of Ag2).

Peierls Instability

Hence, Peierls instability results for the low-spin states having charge density wave maxima centered at bond midpoints. For Sa ≥ 0.10 Â, the charge density wave maxima and minima are located at the exact centers of alternate bond midpoints (as in Figure 8b).

Unrestricted Hartree-Fock

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. In contrast with HF, the UHF description of the low-spin Age ring cluster leads to stabilitywith respect to the Peierls distortion (5α).

UHF Energy Bands

The Symmetric Cluster

The Peierls-Distorted Cluster

The Peierls distortionstabilizes the α-spin valence orbitals since theyhave nodes bisecting the expandedbond midpoints but destabilizes the β- spinvalenceorbitals since they have nodes bisecting the compressed bond midpoints. Thenet result is that the one-dimensionalmetalcomposedof monovalent atoms does not lead to a Peierls instability (ifspinpolarization effects are allowed).

Discussion

Thenet result is that the one-dimensionalmetalcomposedof monovalent atoms does not lead to a Peierls instability (ifspinpolarization effects are allowed). the up-spin and down-spin densities) is fully symmetrical. 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.

Summary

• Basis Sets and Effective Potentials
• Energy Bands and Atomic Orbital Populations
• The Frozen Core Approximation
• Details of the Valence Electron Wavefunctions
• UHF Spin Contamination
• The Energy Expression for Complex Bloch Orbitals
• Symmetric Ring Clusters
• Peierls-Distorted Ring Clusters
• Comparison with GVB-PP

We evaluated the spin contaminations ofthe UHF wavefunctions for the various Mχ ring clusters and Λf2 diatomic molecules, by calculating the “average” spin. the results are given in Table 9). This leads to total energies (Figure 16, dashed curve) much higher than those calculated withoutthe orbital symmetry restriction (Figure 16, solid curve),e.g., the energy differencebetween these two states is 109.6, meV/atom for δa = 0 and 69.0 meV/atomfor δa = 0.30 Â. Both of these triplet states are lower in energy than the singlet state for the symmetric ring cluster (Sa = 0), but for ∣iα∣ > 0.13 Â the energy ofthe singlet drops belowthat of thetriplet due to the Peierls instability for the singlet.

The valence band is half filled in the HF description of the low-spin state of the one-dimensional metal,i.e.5 the one-electron states with ∣⅛∣ < 7r∕2α and one of the two degenerate ⅛ = ±7r∕2α states are double-occupied.

Charge Density Waves, Spin Density Waves,

Qualitative aspects of the bonding

Each of the bond pairs for the Mu ring cluster is similar to the bond pair ofMa (compare Figures 5b and 1 for Ag). Pauling, The Nature of the Chemical Bond (3rd edition, Cornell University Press, Ithaca, New York, 1960); L. Valence orbitals for each of the Cuχo, Agio, Auιo, Liio, and Nai0 ring clusters (each orbital contains one electron; overlaps are given in Tables 3 and 5).

For each of the Λfy ring clusters, the GVB total energies for the allowed magne­.

The GVB Many-Electron Wavefunction

Results

In the process of exploring the bonding in various metal clusters,14’24 we performed extensive ab initio calculations for various one-dimensional ring and chain clusters of lithium atoms up to N = 12, where N is the number of atoms in the cluster, and have extrapolated various results to infinite N,24 These studies show that the cohesive properties of the ring clusters converge rather quickly, and that the Ms and Mw ring clusters are qualitatively correct and fairly acpurate as models for the infinite chain (eachis periodic in one dimension). Here we examine ring clusters composed of Cu, Ag, Au, Li, and Na, with lattice constants (a) for the undistorted (symmetric) clusters equal to the nearest- neighbor distancesfor the bulkmetals.25 The cohesive properties of the one-dimensional alkali and noblemetals are dominated by the valence sp electrons,7 and sp hybridiza­.

GVB-PP

• Charge Density Waves
• Peierls Instability

The GVB-PP ground state wavefunction of eachof these systems leads to a charge density wave — the electronic charge density does not have the same periodicity as the lattice. For each case, these two distinct charge density wave states are each doubly degenerate(for the symmetricringcluster)in the sense that “translating” the valence orbitals by a (or rotating by 2τr∕2V) results in anew charge density wave state with the same energy (for finite N these many-electronstates are not orthogonal).

Full GV

• Charge and Spin Densities
• Peierls Distortion
• Discussion

In addition, for the Ag8 symmetric ring cluster we carried out GVB-CI(SCF) calculations restricted so that the final (converged) orbitals would be atom-centered, leading to a 1B2g excited state (Figure 9b).2β The orbitals optimized in this fashion (Figure 9b) also result in a fully-symmetrical (tDah) electronic charge distribution, in contrast to the GVB-PP skewed orbitals for the Ag8 atom-centered state (Figure 5b, leading to a charge density wave). For ∣tfα∣ > 0.30 Â, the converged GVB-CI(SCF) wavefunction results in a charge density wave having C,4⅛symmetry [the GVB-PP charge density wave has C4⅛ symmetryfor all ∖Saj ≠ 0].

Restricted Hartree-Fock

Cuιo, Agβ, Ag8, Agio, Auιo, Liβ, Lii0, Lii4, and Nawsymmetricring clusters, thelocal HF description of each electron pair involves an atom-centered orbital [⅛≈,(↑)^(i)].τ,24 Expanding this atom-centeredorbital φ as a sum of two adjacent orthogonal bond- centered orbitals. The GVB total energy is always lower than the HF total energy (as shown in Figure 12 for the Aga- ring) because electron correlation is included in GVB but excluded in HF.

Unrestricted Hartree-Fock

For the one-dimensional ring clusters composed of Cu, Ag, Au, Li, and Na, these GVB orbitals {φi} are centered at adjacent bond midpoints and overlap (<S12 ≠ 0; . see Figure 3 and Table 3). Hence, the neglect of electron correlation is responsible for the spurious HF results. atoms in the cluster).7 As a result of this “spin contamination,” the lack of Peierls instability for UHF does not imply a lackof Peierls instabilityfor the exact ground state (which would be aproperly described singlet state).

Generalized Valence Bond

• Summary
• The GVB-CI(SCF) Wavefunction
• The Hartree Localization Method
• The Wannier Localization Method
• Basis Sets and Effective Potentials
• GVB-CI(SCF) Wavefunctions
• GVB-CI(SCF) Hartree Localized Orbitals
• UHF Hartree Localized Orbitals
• Cu2, Aga, Au 2 , Li2 , and Na2 ,
• Cu‡, Agj", Au‡, Lij^, and Na‡

For M‡, the UHF nearest-neighbor overlap integrals are 1-10% larger than the respective GVB-PP values (see Table 5). For Cu‡ and Ag‡, the errors in the GVB De values (in comparison to experi­ ment) aremainlydueto the neglect of core-core and core-valence electron correlation effects.44 For Li‡, almost all of thediscrepancy between the listed GVB results and the experimental results can be removed with basis set improvements (such as opti­ . mizing thep basis scale factor and adding a set of dfunctions).14’46. For the Mχ ring clusters (with monovalent Λf), GVB-CI(SCF) includes all configurations within N orbitals (see Appendix A).

Localized orbitals are shown for UHF (see Appendix B.3), GVB-PP (the two orbitals ofa bond pair are shown), and full GVB (see Appendix B.2). a) The Ma symmetric ring cluster as a model of the undistorted one­.

Magnon Dispersion, Spin Density Waves,

Results

Hence, with respect tothe Peierls instability, N = 4i ring clusters are expected to be better models ofthe one-dimensional metal than N = 4λ + 2 ring clusters, although the distinction between N = 4i and N = 4i + 2 is expected to vanish in the limit as N approaches infinity.35 .. the isolated atoms where the core orbitals are doubly-occupied and the valence s orbital is singly-occupied).]33 Further details are given in Appendices A-B. For the small distortion ofSa = 0.04 Â, the Peierls distortion causes the charge density wave maxima to slide towards the compressed bond midpoints and the charge density wave minima to slide towards the expanded bond midpoints, resulting in a net stabilization of the total energy. However, unlike HF, the UHF wavefunctions are “spin-contaminated.” The low-spin UHF wavefunction is not an eigenfunction of the many-electron spin operator (S2) and hence contains contribu­.

The GVB wavefunctions lead to the following conclusions concerningthe ground electronic state of each of the Cu, Ag, Au, Li, and Na one-dimensional metal clus­ ters. t) The HF-Peierls description of the valence electronic structure (half-filled band) is fundamentallyincorrect. Since both Φ and χ are products of one-electron functions, the UHF wavefunction contains a single Slater determinant, isan eigenfunction ofSz but is not aneigenfunction of S2 unless the down-spin orbitals are identical to the up-spin orbitals (this leads to the restricted Hartree-Fock wavefunction, denoted simply HF herein).7 For the general casewhere Φubf ≠ ΦjJ^f is “spin-contaminated”, e.g., Φ^^rcontains a mixture of spins ∣Ms∣ ≤ S ≤ 7V∕2.τ The UHF orbitals are optimized without restrictions with respect to overlap or symmetry, resultingin a spin density wavefor the ground state of each Λf1o ring cluster.7. The HF wavefunction leads to Peierls instability and lack of cohesion with respect to dissociation into diatomic molecules.7 The lack of cohesion in the HF description of the Ag8 ground state is due to the inaccuracy ofdescribing one-electronbonds with doubly-occupied orbitals.

Introduction

GVB study of these systems is presented in full detail elsewhere.13 We show that the UHF magnon spectra are consistent with a nearest-neighbor Ising model.14’15 However, HF leads to incorrect results (such as a ferromagnetic ground state for LItf). Hence, the same frozen core approximation as utilized previously (including the closed-shell d1° electrons ofthe noble metal rings)10’11 is expected to be reasonably accurate for magnetizations 0 ≤ μ < 1.1°.

The High-Spin State

• Ab Initio Results
• Ising and Generalized Ising Models

Restricted Hartree-Fock

• HF Results Without Orbital Symmetry Restrictions
• HF Results With Orbital Symmetry Restrictions

Weoptimized HF wavefunctions for all allowed magnetizations 0 ≤ μ of each of the Cujo, Age, Aga, Agio, Auιo, Lie, Liu, Liι<, and Nau ring clusters. The HF magnon spectra calculated by imposing orbital symmetry restrictions are tabulated in Table 1 for each of the nine Λfχ ring clusters.

HF Charge Density Waves

• Intermediate Magnetizations (0 < μ < 1)
• Low Spin (μ = 0)

The key measure of the ionic character is the average numberof doubly-occupied Wannier orbitals per configuration, (J). However, for each of the Aga, Lixo, L1l4, and Na10 rings, the HF μ = 0 state contains achargedensitywave— with maxima and minima centered at alternate atoms10’11 — but no spin density wave.

Spin Density Waves and Spin Polarization

The honeycomb valence electronic lattice leads to the classical model of anti­ ferromagnetism — where the two sublattices consist of the {B} and {N} sites of graphitelike BN. The wurzite lattice leads to the classical model of antiferromagnetism ■— where the two sublattices consist of the {Zn} and {S} sites of hexagonal ZnS.

Ising Models for Simple Metals

• Planar Close-Packed Metals
• Hexagonal Close-Packed Metals
• Face Centered Cubic Metals
• The Frozen Core Approximation
• Hartree-Fock Wavefunctions
• The Hartree-Fock Energy Expression
• Two Electrons
• Ring Clusters
• HF Without Symmetry Restrictions
• Djv⅛ symmetry-restricted HF
• The Wavefunctions
• Energy Analysis for Mw
• Approximate Energy Expression for
• Failure of Simple Energy Band Theory
• Detailed Results for Cu 10 and Liχo
• One-Electron Energy Bands
• HF States Involving 5d Excitations
• UHF Results and the Ising Models
• HF Results and Symmetry Breaking Effects

Magnon Dispersion, Spin Density Waves,

Details of the Calculations

Results and Discussion

Ab Initio (GVB) Magnon Spectra

The Heisenberg Model

Comparison of GVB with Hartree-Fock

• UHF
• HF
• Summary
• The GVB Energy Expression
• Two Electrons
• Mu Ring Clusters
• Exact HM Solutions
• Normal UHF (Ising)
• Spin Projected UHF
• Simple Valence Bond
• Resonating Valence Bond

New Concepts of Metallic Bonding Based

Generalized Valence Bond Studies of Metallic Bonding: Naked Clus