UHF

Chapter 4. Magnon Dispersion, Spin Density Waves,

C. Comparison of GVB with Hartree-Fock

1. UHF

differencebetween UHF and HF (∖Ebf(c^ — Eubf∖", see Figure 20) increases mono- tonically with decreasing magnetization. Hence, for these systems, spin polarization is crucial for a HartreeFock (single-determinant) descriptionof thevalenceelectronic structure.

In each case, the local valence electronic structure of the UHF ground state consists of singly->occupiedorbitals centered at the bond midpoints with alternating spins (see Figures 2 and 5), e.g.,

≠ι(α)≠a(∕3)≠3(α)≠√∕3) ∙ ∙ ∙ ≠JV-ι(α)⅛(∕3)

(for even N). This leads to a spin density wave, although in each case the net electronic charge density (obtained by adding the up-spin and down-spin densities) isfully symmetrical.

For each magnetization, localized orbitals {φi} similar to those for μ = 0 can be obtained by separate Hartree localizations11 of the up-spin canonical UHF orbitals

{⅛5<s 1 ≤ i ≤ A} and the down-spin canonical UHF orbitals {⅛5t∙, A+ 1 ≤ i ≤ N}

(the UHF total energy is invariant to this orbital transformation).11 In terms of the

{≠i}, for UHF, the magnetization isincreased byflipping down-spin (∕3) electrons to up-spin (a), as shown in Figure 5 for Miq.

The UHF energyincreases monotonically with increasing magnetization, because the nearest-neighbor exchange interaction between adjacent parallel spins is anti­ bonding (U = —l)s as opposed to the nonbonding (U = 0) interaction between adjacent antiparallel spins (see Appendix B).

The UHF Hartree orbitals exhibit only minor variations asa function of μforeach of the Cu10, Agβ, Age, Agio, Au10, Liβ, Liio, Lii4, and Na10 ring clusters (see Figure 2 and Table 2). As a result, the UHF total energies for the allowed magnetizations can be interpolated accurately with a generalized Ising Model

¾m = Em ~μNJ(l + μNS2).

This generalized Ising energy expression includes effects due to the nearest-neighbor overlap (5) to first order and results from the solutions of the nearest-neighbor Ising Hamiltonian14’15

Hising = J∑ ∣ + 2silsl∙+li (8)

i

including only nearest-neighbor exchange interactions J where s<i is the z com­ ponent of spin for lattice position i. This generalized Ising model reduces to the usual simple Ising model in the limit as 5 approaches zero (such as for a lattice of singly-occupied d or ∕ orbitals).

The reason for the successof the generalized Ising model in fitting the UHF total

energies is that (i) the UHF wavefunctions for the various magnetizations of Mχ can all be described fairly wellin terms of a single set of N equivalent well-localized non-orthogonal orbitals {φj} (see Appendices B-C), and (w) and S decrease ex­ ponentially with increasing distance;11’13 hence, non-nearest-neighbor overlaps and exchange interactions are negligible.

C. Generalized Valence Bond

The optimum wavefunction allowing an independent particle description of the elec­

tronic structure isthe generalizedvalence bond wavefunction (GVB).2°,21 Both GVB and UHF include a separate spatial orbital for each valence electron, where the orbitals are energy-optimized without any restrictions. For UHF, the orbitals are optimized for a fixed many-electronspinfunction (%) that is a simple product of one- electron spin functions (χusp = aAßB}. However, unlike UHF, for GVB, both the valence orbitals andthe many-electron spinfunction areoptimized simultaneously.20,11 Hence, for μ < 1, GVB always leads to a lower total energy than UHF.

We have optimized GVB wavefunctions for all allowed magnetizations of each of theCuιo, Agβ, Agβ, Agio, Aui0,Liβ, Lii0, and Nai0 ring clusters. Fulldetails of these results are presented elsewhere.13

In agreement with UHF, GVB leads to a μ = 0 ground state in each case. Also in agreement with UHF, GVB leads tolocalized orbitals centered at bond midpoints that exhibit only minor variations as a function ofμ, However, unlike UHF, in each case GVB leads to fully symmetrical charge and spin densities for all magnetizations 0 ≤ μ ≤ 1∙ Therefore,for these systems, the spin density waves resulting from UHF are dueto anincomplete treatment of the electron correlation forced by the use of a restricted (product) spin function.11’13

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

tizations arein excellent agreement with the exact solutions of the nearest-neighbor Heisenberg Hamiltonian15,22

hheis = t7Ç I +2⅛ ∙ 7i+1. (9)

i

The Heisenberg Hamiltonian leads to the energy expression ' ⅛ = ⅛‰0 + (‰μ)JVJ∙

where Ε^μ=0 is the Ising energy for μ = 0, and (Ujv,μ) is average nearest-neighbor exchange coefficient (see Appendix B). The Ising model is an approximate solution of (9)leading to (t⅛,μ) = μ∙ Therefore, the exact solutions of (9) lead to (Ujf,μ} ≤ μ∙

As an example, for an M2 antiferromagnetic chain, the exact μ = 0 solution is simply (aß —∕3α)∕√z2, leading to U = 1 (bonding), and theUHF-Ising μ = 0 solution is aβ, leading to U = 0 (nonbonding). For μ = 1, the exact solution and the Ising solution are both aa, leading to U = — 1 (antibonding).

Forthe Mnring, the Ising solution for μ = 0 leads to (Uj(^_0) = 0. However, for Afχo, the exact solution leads to (Z7jv,μ=o) =0∙403 (bonding), and for M00, the exact solutionleads to (∑7jv,μ=o) = 2In 2 — 1 = 0.386.23

The UHF-Ising description of the electronic structures ofthese systems can be useful as an approximation to the more exact GVB-Heisenbergdescription.

De

Ising

Models for Simple Metals

Based on the success ofthe Ising model in approximating the UHF results for the one-dimensional metals composed of Cu, Ag, Au, Li, and Na, it is likely that the Ising model couldproveusefulfor describing the valence electronic structures oftwo- dimensional metals (metal surfaces) and three-dimensional metals (bulk metals).

The classical model of antiferromagnetism is based on dividing the principal lat­

tice into two sublattices {A} and {5} — occupied by up-spin and down-spin elec­

trons, respectively — such that all nearest neighbors of sublattice {A} belong to sublattice {B} and vice versa.15 However, this two-sublattice model of antiferromag­ netism is impossible if the principal lattice contains triangles.15

The UHF results for the rings are consistent with this classical model of antiferromagnetism; however, since the valence orbitals are centered at the bond midpoints, the principal lattice for describing the valence electrons is the lattice of bond midpoints — not the atomic lattice.

ingeneral, for a n-dimensional atomic lattice, we expect that the “characteristic” localizedvalence orbital is composed of sphybrid orbitals from n +1 adjacent atoms (for metallic systems). This expectation is based on UHF and generalized valence bond studies ofsmall lithium clusters containing up to thirteen atoms.24 Hence, the characteristic localized valence orbitals are centered at bond midpoints, triangular facesand tetrahedral hollows for ID metals, 2D metals, and 3Dmetals,respectively.24 Therefore, for 2D and 3D metals, the lattice appropriate for simulating the valence

electronic structure — based on interstices — need not have the same coordination as the atomic lattice!

For 2D and 3D close-packed metals, we show in each case that the valence elec­

tronic lattice (based on orbitals centered at interstices) leads to the classical model of antiferromagnetism. Hence,the common belief that the classical model of antifer­

romagnetism cannot applyto the valence electronsof close-packed metals — because their atomic lattices contain triangles — is incorrect.

For a planar close-packed metal, each atom has six nearest neighbors. If this lattice is filled with Cu2+ (3d9) ions, then the atomic lattice is correct for describing the electronic structure of the hole states.

However, ifthis lattice is filled with Cu+ (3d10) ions andone valence electron per ion is added,the honeycomb (hexagonal) lattice definedby the triangular hollows — where eachlatticepoint has three nearest neighbors — should be used for describing the valence electronic structure.

For transition metals, such as Ni (3d94∙s1), the valence electronic lattice would interact with the atomic lattice, since the d electrons are atom-centered.

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.

Since there aretwo triangular hollows peratom, for monovalent metal atoms, this leads to a half-filled valenceband for both the up-spin and the down-spin electrons, leading to a two-dimensional metal. For divalent simple metals (Be, Mn, Ca, etc.

and Zn, Cd, Hg), the up-spin and down-spin valence bands are filled, leading to two-dimensional insulators.

2o Hexagonal Close-Packed Metals

For hep metals, the “valence electronic lattice” consists of the centers of the tetra­ hedral hollows. For ⅛cp, each tetrahedral hollow shares a face (three edges) with another tetrahedral hollow, and the three remaining edges are shared with three other tetrahedral hollows. Hence, for hep the lattice of tetrahedral hollows corre-

sponds to the wurzite lattice (such as for hexagonal ZnS, where each lattice point has four neighbors).

The wurzite lattice leads to the classical model of antiferromagnetism ■— where the two sublattices consist of the {Zn} and {S} sites of hexagonal ZnS.

3. Face Centered Cubic Metals

For feemetals, the electronic latticealso consists of the centers of tetrahedralhollows.

For fee, each tetrahedral hollow shares an edge with six other tetrahedral hollows;

hence, for fee the lattice oftetrahedral hollows corresponds to the rock salt lattice (NaCl).

The rock salt lattice leads to the classicalmodelof antiferromagnetism — where the two sublattices consist ofthe {Na} and {C1} sites ofNaCl.

For both fee and hep,there are two tetrahedral hollows per atom. Hence, in each case the Ising model leadsto a half-filledenergy band for monovalent metalatoms and a completely filled energy band for divalent metal atoms. This isconsistent with the observation that the monovalent metals tend to have higher electrical conductivities than thedivalent metals.

IV. Summary

We optimized UHF and HF wavefunctions for all allowedmagnetizations of each of the Cuιo, Agβs Ag8, Agio, Auι0, Liβ, Lii0, Liι∙t, and Nai0 ring clusters. The results for these one-dimensional metals—and speculationsfor two-dimensional metalsand three-dimenstional metals — are summarized as follows.

Restricted Hartree-Fock leads to spurious results, such as (i) a charge density

wave for each of the Ag8, Li10, Li14, and Nai0 μ = 0 (low-spin) symmetric ring clusters,10,11 (it) charge and spin density waves for all intermediate magnetizations (0 < μ < 1) of each ring cluster, and (tit) ground states for Ag8, Naχo, and Li1o having 0.25, 0.6, and 1.0 unpaired electrons per atom, respectively. These spurious resultsindicate a fundamental flawinHF for these systems — the double occupation of valenceorbitals for describing magnetizations μ < 1.

For each Mpr ring cluster, unrestricted Hartree-Fock leads to an antiferromagnetic (μ = 0) ground state that does not have a charge density wave. However, for each magnetization μ < 1 (including μ = 0), UHF leads to a spin density wave (orbital localization). The UHFtotalenergyincreases monotonically with increasing magnetizationas long as spin densitywaves are allowed. The UHFmagnon spectrum for each of the various Mjf rings is described accurately in terms of a generalized Ising modelincludingonly effectivenearest-neighbor exchange interactions.

However, for these systems, the spin density waves resulting from the UHF- Ising description are due to an incomplete treatment of the electron correlation.11,13 Hence, generalized valence bond calculations11’13 lead to lower total energies and fully symmetrical charge and spin densities for all magnetizations 0 ≤ μ ≤ 1.

For these one-dimensional ring clusters, the UHF-Ising model is consistent with the classical two-sublattice model of antiferromagnetism,ιs where the principallattice is divided into two sublattices A and B — occupied by the up-spin and down-spin electrons, respectively— such that all nearest neighbors of one sublattice belong to the other sublattice. However, for these rings, the valence orbitalsarecentered at the bond midpoints; hence, the principal latticefor describing the valence electrons is the latticeofbond midpoints— not the atomic lattice.

For close-packed metals, the valence orbitals are centered at interstitial hollows

— triangular hollowsfortwo-dimensional metalsand tetrahedral hollows for three di­

mensional metals.34 For three cases (planar close-packed,feeand hep), weshow that the lattice defined by the interstitial hollows is consistent with the two-sublattice model of antiferromagnetism. Each of these three close-packed metals contains two interstitial hollows per atom, leading (for monovalent metal atoms) to metallic prop­ erties resultingfrom a half-filled valenceenergyband. Therefore, the common belief that the classical (two sublattice) model of antiferromagnetism cannot apply to the valence electrons of close-packed metals — because their atomiclattices containtri­

angles —- is incorrect.

Appendix A. Details of the Calculations

1.

The Frozen

Core

Approximation

The frozen core approximation can be described briefly as follows (more detail is given elsewhere).10

First, both the coreorbitals and the valenceorbitals are optimized simultaneously for the valence high-spin state (chosensincethe HF and UHF valencewavefunctions are identical for high-spin). The core orbitals optimized for the high-spin state are then used to construct the valence hamiltonian

« = sSosx + ∑ι(∙) + ∑-

i=l i>,∙ rij

i(i) _ -lv,1 + v(fi ) +vc°∞

Vcore = ∑(2Jc-⅛c) c

where (t) E%-ore includes the nuclear repulsion energy and all one-electron and two-electron energy terms involving only the core electrons {the [Ar]3d10, [Kr]4d10,