Chapter 2. Charge Density Waves, Spin Density Waves,
B. Full GV
2. Charge and Spin Densities
The GVB-CI(SCF) many-electron wavefunction for the ground state of each of the Cu10, Agβ, Aga, Agio, Au10, Liβ, Li10, and Nai0 symmetric ring clusters leads to a fully-symmetrical (Djv⅛) electronic charge distribution (no spin density wave or charge density wave; see Figures 3 and 9a, and see Appendix B for further details).
For each system, each localizedvalenceorbital φi optimizedforthe groundstate is symmetrically centered at abond midpoint and is composed primarily of sp hybrids from the two adjacent atoms. Orbitals φi and φj are related by a rotation through (j — ») bond midpoints and hence are equivalent. The high symmetry of the {φi}
leadsto afully-symmetrical (Djvfc) electronic charge distribution (as shownin Figure 3 for Cuιo, Agio, Auio, Liι0, and Nai0 and in Figure 9a for Ag8). Consequently, each GVB orbital has equal overlaps with the two adjacent orbitals (and with the two next-nearest neighbor orbitals, etc., see Appendix B).
The overlapsof adjacent GVB orbitals are compared with the GVB-PPbondpair overlaps in Table 3. The GVB-PP orbitals are also centered at the bond midpoints but each GVB-PP orbital has a non-zero overlap with one adjacent orbital and is orthogonal to the other adjacent orbital. Hence, the GVB-PP orbitals are skewed (leading to a charge density wave, as shown in Figure 3 for Cuι0, Agio, Au1o, Liιo, and Nai0 and inFigure5a for Ag8). GVB-CI(SCF)corrects this GVB-PP deficiency.
Since the GVB ground state for each Mn symmetric ring cluster (S = 0, A = B = AΓ∕2)
‰,s=o = Λ [Φλγ Xλγ,s=o] (10)
= ΦlΦ2Φs ,∙∙<i>N
contains a fully symmetrical spatial orbital product Φ7∖r, the many-electron sym-
metry is in each case determined by the symmetry of the spin function χπts=o (in combination with √4). The spin function XNts=o can be written24 as a “resonance combination” of two primary spin couplings
Æ=o = {aβ - ßa}Ä/VN
Xn*s=q = la(aß-ßa)A~1ß-ß(aß-ßa)A~1aJ/VN
Xn,s=o = c1 [⅛ - (-l)xχ⅞‰0] + ∙ ∙ ∙ (11) and numeroussecondary spin couplings [there are (A+1)~1 linearly independent spin eigenfunctions for the 7V-electron singlet]. Here χ⅞⅞≈0 and χ⅞⅞=0 are the two perfect-pairing spin functions involved inthetwodegenerate GVB-PPcharge-density wave states of Λfjy.
The ground state of the Aga symmetric ring cluster (bond-centered orbitals, Fig ure 9a) has 1Big symmetry, whereas the ground state of eachof the Cuι0, Agβ, Ag10, Auιo, Lie, Liχo, and Na10 symmetric ring clusters has 1Alg symmetry.28
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). Details of the Ag8 atom-centered state are also given in Table 3.
The energy splitting between the atom-centered state (1^2β) a∏d the bond- centered state (1-H⅛) is 1.043 eV (130.4 meV/atom) for the Ag8 symmetric cluster.
This is a measure of the very strong preference of the valence orbitals for centering about the bond midpoints (as opposed to centering about the atoms). The Ag8
atom-centered state (Figure 9b) is unstable with respect to diatomic molecules (9) by 0.102 eV (12.7 meV/atom). The Agg bond-centered state (Figure 9a) is stable with respect to diatomic molecules (9) by 0.941 eV (117.7 meV/atom).
These results show again that two-center one-electron bonding, similar to the bonding ofM‡ and linear M‡ (compareFigures 2, 5a, and 9a), plays a crucial role in the cohesion of the Mχ ring clusters.
8. Peierls Distortion
Again, we focus ourattentionon the Agg ring cluster as the test forPeierls instability.26 The Agg ring isstable with respect to the Peierls distortion for both GVB andGVB- PP. Hence, the total energy of the Ag8 ground state increases quadratically as a function of the Peierls distortion (Sa°, see Figure 10).
Although alternate bond midpoints are compressed and expanded for Sa ≠ 0, all the nuclei are equivalent by symmetry (Z>4∕l). We tested for charge density waves in theGVB-CI(SCF) wavefunction byusing the GVB-PP skewed orbitals as “starting- guess” orbitals (shown in Figure 7 for Sa = 0.00, 0.10, 0.20 and 0.30 Â) and then solving iteratively for the optimum (self-consistent) orbitals (shown in Figure 11 for Sa = 0.00, 0.10, 0.20 and 0.30 Â). The converged GVB-CI(SCF) wavefunction results in a fully symmetrical charge density for ∣5α∣ < 0.20 Â. 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].
Hence, for Sa= 0.30 Â theoptimum GVB orbitals are slightly skewed. However, the GVB-CI(SCF) energy calculatedwithJ94⅛ orbital symmetryrestrictions (Figure 10, dashed line) at ∣5α∣ = 0.30 Ä is only 0.0049 eV (0.61 meV/atom) higher than that calculated without orbital symmetry restrictions (Figure 10, solid line).
The optimum GVB orbitals (shownin Figure 11) imply thatthePeierls-distorted diatomiclattice is unfavorable because alternateone-electronbonds are stretched and compressed. As 15α∣ increases, orbitals centeredat expanded bond midpointstend to delocalize somewhat overadjacent compressedbond midpoints, and orbitals centered at compressed bondmidpoints tend tocontract,resultinginincreased gradients along the compressed bond axes.
V. Discussion
Theseabinitiocalculations [GVB-PP and (full) GVB] indicate that the one-dimensional elemental metals composed of Cu, Ag, Au, Li, and Na have large cohesive energies with respect to both atomization and dissociation into diatomic molecules, and are stable with respect to the Peierls distortion (see Table 2 and Figure 12).
For each of the Cu10, Agβ, Age, Agio, Au10, Liβ, Li10, and Na10 symmetric ring clusters, the GVB ground state wavefunction consists of singly-occupied valence or
bitals centered at the bond midpoints, forming one-electron bonds similar to the bonds in the respective M‡ and linear M‡ molecules (compare Figures 2, 3, and 9a). Adjacent bond-centered orbitals overlap, leading to antiferτomagnetic (singlet) ground states having fully symmetricalelectronic charge densities (no charge density or spin density waves for full GVB).
The cohesion in each of these one-dimensionalmetalsis dominated bythese two- center one-electron bonds; hence, the Peierls distortion stretching and compressing alternate one-electron bonds is unfavorable.
For a correct description of the cohesion due to these one-electron bonds, the wavefunction must include the configuration occupying eachlocalized nonorthogonal
valence orbital with only one valence electron. Hence,the wavefunction mustinclude at least one orbital for each valence electron, as is the case for unrestricted Hartree- Fock (UHF, spin polarized), GVB-PP, and GVB, eachincluding exactly one orbital per valence electron.