D. UHF Energy Bands
IV. Summary
6. The Energy Expression for Complex Bloch Orbitals
The energy expression for the configuration
⅛,(1‰(2) c√l)∕3(2) can be written in terms of the {≠m} as
e = I [(≠J⅞l≠m) + (≠-m∣⅛l≠-m) + (≠m<∣Λ∣≠m-) + (≠-m<l⅛l≠-m')]
+ I [Ojû’Oj + (≠-m≠-m^Γ2Ι≠m-≠m') + (≠∏8≠mkΓ2Ι≠-m<‰') + (≠-m≠-mkΓ21∣≠-m^≠-mθ] ‘
This energy expression is simply the average of the energy expressions for each of the configurations
≠m(1) ≠ra∙(2) α(l) /3(2)
≠-m(l) ‰,.(2) α(l) /3(2)
≠m(1) ≠-ra.(2) α(l) /3(2)
≠-m(1) ≠-m∙(2) α(l) /3(2).
Energy expressions forsingle-determinant wavefunctions written in terms of the {≠m} can always be expressed in terms of the {V,m}j leading to an “averagefield” descrip tion in terms of the {≠to}.
Appendix B. Results- for the Diatomic Molecules
We calculated the optimum internuclear separations (Re), dissociation energies (L>β), and force constants (⅛e) forhomonucleardiatomic moleculescomposedof Cu, Ag, Au, Li, and Na at the HF and UHF levels. The frozen core approximation of Appendix A.3, whichis fairlyaccurate forthe Mjf ring cluster calculations,is substantially less accurate for diatomic molecules (see Table 8);43 hence, we optimized both the core orbitals and the valence orbitals for all diatomicmolecules.
These HF and UHF results are compared with results obtained both from ex periment and from calculations45’46 including treatments of the electron correlation effects that are more complete than that of HF or UHF (see Table 10). The errors in the HF values of J2e, ke and De for the various metal dimers in comparison with experiment are as follows. Re too large by 8-10 % (Cu,Ag,Au), 4-5 % (Li,Na); ke too small by 42-45 % (Cu,Ag), 23 % (Au), 5-8 % (Li,Na); De too small by 77 % (Cu,Ag), 68 % (Au), 87 % (Li), 104 % (Na).
Note that although the HF value of De for Na2 is negative (for our basis set, the energy of Na2 is higher than twice the energy of the isolated Na atom), the HF values of Re and fce for Na2 are rather accurate. Due to ionic terms in the HF wavefunction (which are present inthe HF descriptionofthe dimer but absent inthe HF description of the isolated atom), HF rarely gives accurate bond energies, and occasionally gives negative bond energies. The HF De value for Na2 in thelimit ofa complete basis set (e.g., numerical HF) is approximately 0.014 eV (too small by 98
%).47
For Cu2, our HF results (Re = 2.44 Â, De = 0.463 eV, ⅛e = 4.44 eV/Â2) calculated with an effective potential (22-electron wavefunction) and an inflexible d-basis are in very good agreement with “all-electron” HF results (Re = 2.42 Â, De = 0.54 eV, ⅛⅛ = 4.05 eV/Â2) calculated with all 58 electrons included in the wavefunction and a flexible (triple-^) d-basis.45 Similar agreement has been reported for Ag2 between the effective potential and all-electron HF results.48
These results indicate that our basis sets are sufficiently flexible for the HF de scription, and that the effective potentials are fairly accurate. Hence, the errors in the various HF results (in comparison to experiment) are mainly due to the neglect of electron correlation effects (inherent to the single-determinant form of the HF
wavefunction).
TheUHFwavefunctions lead to somewhat more accurate De valuesin comparison to HF. However, UHF leads to values of Re and ke that are less accurate than the corresponding HF values (especially for Na2). The reason for the decreased accuracy for the J2β and ⅛e values is that any improvement in De afforded by UHF is at the expense of mixing in triplet character (see Appendix A.5). The lone exception of these trends is Au2, where the HF and UHF results are identical.
A reasonably accurate description of the potential energy well (Re, ke and De) for each of these metal dimers requires an accurate description of valence electron correlation effects.45,48,48 The noble metal dimers also require correlation effects in
volving the subvalence dw electrons as well as relativistic effects (especially for Ag and Au).45,4β The effective potentialsused in this study includerelativistic effects for Ag and Au but not for Cu.
Appendix C. Details of the UHF Energy Bands
For the UHF descriptionof the Ags symmetricring cluster, the valence energy band of the high-spin state (Figure 3, band width 23) splits into upper and lower energy bands for the low-spin state (Figure 13, band widths 231 and 232, respectively) when spin polarization effects are allowed. This results in an antiferromagnetic insulator since the energy gap (ΔW) between the upper and lower energy bands is at the Fermi level. This is in qualitative agreement with the Hubbard hamiltonian2,β
Hbub = B ∑[ctτci+lι∣ + ct+uci,i] + U 52 ni,τni,i
8=1 8=1
B -2zh⅛l (Cl)
¾ = (ui∖h∖uj)
U = J"1 (C2)
j⅛ = (ωiωi∣r⅛%∙ωj>
Ήΐ = ci,Tc<,ι n<i = ⅛cu
where B is the band width without correlation (the band width for U = 0), z is the coordinationnumber (z — 2 for one-dimensional metals), <⅞ιβ∙ (c∣σ) are operators for creating (annihilating) an electron with spin σ in the localized Wannier3 orbital u⅛, and U is the intra-atomic coulomb energy. U > 0 tends to prevent two electrons from occupying the same localized orbital ω<.
Inthis appendix we assess for Age the quantitative agreement ofthe UHF energy bands withthose obtained with the Hubbard model. To do this we obtainedHubbard parameters (B and Z∕) directly from the ab initio calculations by two methods.
(t) Equations (Cl) and (C2) [using the high-spin {u⅛}] result in the values U — 8.001 eVs B = 5.433 eV, and B∣U = 0.679 for Aga. The value B = 5.433 eV obtained from thenearest-neighbor one-electron integral h"2 = —1.358 eV is in very good agreement with the high-spin valence band width B = 5.524 eV obtained from the orbital energies [hence including all one-electron interactions and two-electron interactions with (IV — 1) valence electrons;21 see Figure 3]. Values of B and U for Cuio, Age, Aga, Agio, Auio, Lie, Liio, Lil4, and Na10 obtained by this method are given in Table 11.
(m) “Effective” values of B and TÂ are obtained by satisfying the relations2’8 ΔWr = U -0.5(Bi + B2)
= U-B-2B ∑(-l)i [√l+p(2W∕βp - j(2W∕S)]
(C3)
[J1(u) is the Bessel function]8 where ΔW, Bχ and S2 are taken directly from the UHF energy bands (see Figure 13). For Ag8, the UHF values ΔW = 2.893 eV, B1 = 2.626 eV, and B2 = 1.187 eVlead to Ueff = 4.800 eV, Beff = 2.273 eV5 and βEFF∣llEFF = 0.474).
The values 1Ä and B are much larger than Ueff and Beff, respectively. The values U = 8.001 eV and B = 5.433 eV lead to a Hubbard band gap ΔW = 3.802 eV (C3) 31% larger than the UHF band gap ΔW = 2.893 eV for Ag8.
The value Beff = 2.273 eV is much closer to the average of the widths of the upper (Si) andlower (S3) energy bands3 (Bave = 1.907 eV) than it is to S = 5.433 eV. Indeed, Si and S2
Si ≈ — 2z (φ-lUPPER B2 ≈ — 2z (φ:EOWER
l⅛3 ,∣i∣⅛
UPPER
LOWER
,LOWER
are based on next-nearest neighbor hopping integrals where {ΦYpper} and {Φeower} are sets of localized nonorthogonal orbitals obtained Horn separate localizations (Fourier transformations) of the up-spin and down-spin canonical orbitals for the upper and lower energy bands, respectively (see Figure 10).lβ
For Λf8, the two-electron coulomb energy for the covalent configuration ω√↑ )u>3(J,)ω3 ( ↑ )ω4( i)ω5(↑ )u>β(i W ↑ W‡)
can be simplified as
eCOUL = 8jω + g jcu + g j« + 4j«
duetothe cyclical nature of the {ωf}. For Ma therearetwodifferent types oflocalized ionic configurations occuring with equal weights:
ωi(↑ >ι(i)ω3(↑ )ω4(J,)ω5(↑)ωβ(i)ω7(↑)ω8(J.)
= jω + lj^ + 8 j« + 8 + 4 j«
where the doubly-occupied orbital and hole are adjacent (u>1 and u⅛), and ωι(↑)ω2(∣)ω3(↑)ω1(∣)ω5(↑)ω6(j,)ω7(↑)ω8(^)
Ecovl = J1ω1 + 8J1ω2 + 8J1ω3 +7 J" + 4 J1ω5
where the doubly-occupied orbitaland hole are third-nearest neighbors (u>ι and ω4).
Hence, for AΓ8 a more accurate definition ofU is
⅛ = J1ω1-0.5(J⅛+Λω4)∙
For Ag8, the values J"1 = 8.001 eV, J"2 = 4.536 eV, and J%i = 2.028 eV lead to the value ⅜ = 4.719 eV, in very good agreement with Ueff = 4.800 eV.
The Ιίχ values for Mχ symmetric ring clusters (even N) N/2
un = jr1-2jv-1∑¾
j=l.
Urn Un = J“
N→∞ 11
are hence expected to converge as N~1. Hence, Equation (C2) does not accurately define U for a finite cluster. Since Beff is expected to decrease with increasing N for Mff clusters composed of Cu, Ag, Au, Li, and Nalβ (the {u⅛}, {^fPP£Ä}, and
{Φi°wεii} are all centered at the bond midpointsand the distances between adjacent bond midpoints and next-nearest neighbor bond midpointsincreases with increasing N), the net result is that Beff∕Ueff is expectedto decrease with increasing N and ΔW is expected to increase with increasing AT.