Comparison with GVB-PP

D. UHF Energy Bands

IV. Summary

3. Comparison with GVB-PP

Total energies of Ag8 as a function of the Peierls distortion (£a) for both the bond- centered state and the atom-centered state calculated with perfect-pairing gener­ alized valence bond wavefunctions (GVB-PP)49 are also given in Figure 16. The GVB-PP wavefunction is a restricted form of the full GVB wavefunction. Hence, GVB-PP leads to higher total energies than full GVB (compare Figures 15 and 16;

further details ofthese GVB-PP results arepresentedelsewhere).16 For both Ag8 and Aga, the GVB-PP total energies are lower thanthe UHF total energies. In addition, theGVB-PP wavefunction is aneigenfunction of S2 while theUHF wavefunctionis not (see Appendix A and Tables 9 and 12).

For the bond-centered (ground) state of Ag8, GVB-PP and UHF both lead to a positivecohesiveenergywithrespect to dissociation into Ag2 molecules, and stability with respect to the Peierls distortion.

However, for the atom-centered (excited) state of Ag8, GVB-PP leads a nega­ tive cohesive energy with respect to dissociation into Aga molecules, and a Peierls instability (in disagreement with UHF).

Further details of the GVB-PP results for the Cu10, Ag6, Ag8, Ag10, Au10, Liβ, Liχo, Li14, and Naχ0 ring clusters are presented elsewhere.16

Appendix E. HF Results for the Ag8 Triplet State

The triplet state of Ag8 described by thevalence configuration

≠o(U)≠-ι(∏)≠ι(U)≠.3(↑)^2(↑) (El)

leads to fully symmetrical electronic spin, and charge densities; however, optimizing the orbitals at the HF level without orbital symmetry restrictions

⅛Ι(ΠMU)φ3(U)φ4(↑)9,5(↑)

leads to a triplet state having both a spin density wave and a charge density wave with a total energy lower than that of (El) by 0.041 eV for the symmetric ring cluster.

The HF total energies for these two states as a function of the Peierls distortion (Sa) are shownin Figure 17, where the dashed curve shows the results obtained with Dih orbital symmetry restrictions. Figure 17 also includes the HF total energy for the lowest-energy low-spin state (singlet, dotted curve). Neither triplet state leads to a Peierlsinstability, although they both are unstable with respect to the limit of four low-spin diatomic molecules. 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 HF orbitals optimized without symmetry restrictions for the triplet state are shown in Figure 18 for Sa = 0.00, 0.10, 0.20, and 0.30 Â. Although the triplet spin state having spin and chargedensity waves is stable with respect to the Peierls distortion, the spin and charge density waves slide as a function of Sa in such a manner that the two singly-occupiedorbitals localize about a pair of adjacent atoms forming a compressed bond for Sa = 0.30 Â.

Appendix F. Detailed Energy Data for Ag§

In this appendix we use the energy expression

βTOTAL

E one

ττ∣CORE

&N N

Σ⅛ 8=1

ONE , &TWO

hij

= (

w

(

i

)I¾

i

)I<

pj

(

i

)>

+ + E

= ∑⅛-∑¾- Σ K‘i

i>3 i>3 i>j>A

ja = MiM2)∣-⅛K1M2))

^12

Kij = (φi(l)vJj∙(2)∣-∣^∙(l)⅛5i(2))

**12

based on the valence electron hamiltonian of Appendix A.4 where (i) Ecore in­ cludes the nuclear repulsion energy and all one-electron and two-electron energy terms involving only the core electrons {the [Ar]3d10, [Kr]4d10, [Xe]4∕145dlθ, Is2, and ls32s32pβ shellsfor Cu, Ag, Au, Li, and Na,3β respectively (see SectionA.3)}, (m) Eone Is the total “one-electron” (valence) contribution to the total energy includ­

ing the electron-nuclear attraction and kinetic energy of the valence electrons, and all two-electron interactions between core electrons and valence electrons, and (iii) Eτwo is the total two-electron (valence) contribution to the total energy where Jjj∙

and Kij are the valence two-electron integrals (coulomb and exchange,respectively).

Calculated values of Ecore, Eone, Etwo, and Etotal are given in Table 13 as afunction ofthe Peierls distortion (5α) for selected states of the Age ring cluster where the average nearest-neighbor internuclear separation is a = 2.889 Â. Table 13 also includes similar data for low-spin Age calculated with the GVB wavefunction.

The total energyof low-spinAg8 as a function ofSa is also presentedinFigure 15 for HF, UHF and GVB. The total energies for these wavefunctions were all optimized

using the same basis sets and frozen core hamiltonian (see Appendix A); further details of the GVB calculations are given elsewhere.16

In all cases, Ecore increases quadratically as a function of δa. For the low-spin HF wavefunctionoptimized with D<⅛ orbital symmetryrestrictions, the variation of Eone with δa is approximately linear, leading to the Peierls instability (the variation of Eτwo with δa is negligiblefor this case). Likewise, the Peierls instability for the low-spin HF wavefunctionoptimized with Cs orbital symmetryrestrictions is driven by Eone, For the high-spin HF wavefunction and the low-spin UHF and GVB wavefunctions the variations of Eone and Eτwo with δa are quadratic but the variations cancelone another, i.e., in each case the variation ofEone + Eτwo with δa is negligible. Hence, the symmetric ring is stable for these cases due to Ecore.

Appendix G. Results for Hydrogen Ring Clusters

Since the stable form of hydrogen is the diatomic molecule, we tested our model calculations by performing analogous calculations for ring clusters composed of hy­

drogen. Extensive total energy calculations for one-dimensional arrays of hydrogen atoms are presented elsewhere.50

In order to model the anticipated Peierlsinstability, we chose the Hg ring cluster with an average internuclearseparation (α = 1.483Â) equal to twice the experimental bond length of the Ha molecule51 [the Huzinaga53 (5s∕3s) unsealed basis set was used]. The results of these calculations are given in Table 14. The low-spin HF wavefunction leads to acharge densitywavefor the symmetric cluster. The low-spin UHF wavefunction leads to a spin densitywave (but no charge density wave). The low-spin GVB wavefunctionleads tofully symmetrical spin and charge densities. All

[10] Low-Dimensional Cooperative Phenomena. The Possibilityof High-Temperature Superconductivity, editedby H. J. Keller (Plenum, New York, 1974); Chemistry andPhysics of One-Dimensional Metals, edited by H. J. Keller (Plenum, New York, 1977); One-Dimensional Conductors, edited by H. G. Schuster (Springer- Verlag, New York, 1975); Extended Linear Chain Compounds, edited by J. S.

Miller (Plenum, New York, 1982, 1982, 1983), Vols. 1-3.

[11] L. F. Dahl and D. L. Wampler, Acta Crystall. 15, 903 (1962). Also see M.-H.

Whangbo and M. J. Foshee, Inorg. Chem. 20, 113 (1981).

[12] Z. G. Soos, in Low-Dimensional Cooperative Phenomena. The Possibility of High-Temperature Superconductivity, edited byH.J. Keller (Plenum, New York, 1974).

[13] C. Castiglioni, G. Zerbi, and M. Gussoni, SolidState Commun. 56, 863 (1985);

K. A. Chao and S. Stafstroem, Mol. Cryst. Liq. Cryst. 118, 45 (1985); I. Bo- zovic, Phys. Rev. B 32, 8136 (1985); M. Kertesz and R. Hoffmann, SolidState

Commun. 47, 97 (1983).

[14] Thecalculationsfor Li and Na are “abinitio” in the traditional definition in that the total energy expressions (Φ∣7f]Φ)∕(Φ∣Φ) include all terms resulting from the true Hamiltonian (7f ) [within the usual Born-Oppenheimer approximation (frozen nuclei)] and no additionalterms. The only approximations utilized in ab initio calculations arein the many-electron wavefunction (Φ). “Exact” calcula­

tions would require Φ toinclude essentially an infinite numberof configurations within an infinite basis set and are generally unattainable; the configuration and basis set expansions aretruncatedin actual applications. Both the HF and

UHF wavefunctions include only a single configuration. Localdensity functional calculations (such as ∑α) are not ab initio in this traditional definition since the exchangeintegrals are replaced by a local “exchange-correlation” potential obtained from calculations on the free electron gas. The calculations for Cu, Ag and Au are ab initio for eleven electrons per atom (dlθs1), where “ab ini­ tio” effective potentials model all effects of the remaining core electrons. See Appendix A and Reference [36] for further details.

[15] W. Fδrner and M. Seel, J. Chem. Phys. 87, 443 (1987).

[16] M. H. McAdon and W. A. Goddard III, J. Phys. Chem., accepted for publica­ tion; M. H. McAdon and W. A. Goddard III, unpublished.

[17] M. H. McAdon and W. A. Goddard III, J. Phys. Chem. 91, 2607 (1987); M.

H. McAdon and W. A. Goddard III, J. Non-Cryst. Solids 75, 149 (1985); M.

H. McAdonand W. A. Goddard III, Phys. Rev. Lett. 55, 2563 (1985).

[18] J. Donohue, The Structure of the Elements (Wiley, New York, 1974).

[19] R. S. Mulliken, J. Chem. Phys. 23, 1833 (1955).

[20] C. A. Coulson, B. O’Leary, and R. B. Mallion, Hückel Theory for Organic Chemists (Academic, New York, 1978); L. Salem, The Molecular OrbitalTheory of Conjugated Systems (Benjamin, NewYork, 1966).

[21] T. Koopmans, Physica 1, 104 (1933).

[22] Orbital energies from the valence electron high-spin stateprovide energy levels that are consistent for the entire valence energy band — see Reference [29].

These energy levels are used in the qualitative description of other valence many-electron spin states.

[23] Actually, the lowest energy HF states for the Cui0, Agβ, Agi0, and Au10 rings are low-spin, but the lowest energy HF states for the Ag8, Nai0, Liβ, Liχ0, and Liχ4 rings contain 0.25, 0.6, 1.0, 1.0, and 1.0 unpaired electrons per atom, respectively. The lowest energy UHF state for each of these nine clusters is low-spin (M. H. McAdon and W. A. Goddard III, unpublishedresults).

[24] H. A. Jahn and E. Teller, Proc. Roy. Soc. London A 161, 220 (1937). Also see R. Englman, The Jahn-Teller Effect in Molecules and Crystals (Wiley, New York, 1972).

[25] This is consistent with the HF results of Reference [15] leading to Peierls in­

stability for Liio and Li1< but apparently not for Lie.

[26] R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelley and D.

D. Wagman, Selected Values of the Thermodynamic Properties ofthe Elements (American Society for Metals, Metals Park, Ohio, 1973).

[27] M. D. Morse, Chem. Rev. 86, 1049 (1986).

[28] M. M. Kappes, M. Schär and E. Schumacher, J. Phys. Chem. 89, 1499 (1985).

[29] W. J. Hunt and W. A. Goddard III, Chem. Phys. Lett. 3, 414 (1969).

[30] W. A. Goddard III, T. H. Dunning, Jr, W. J. Hunt, and P. J. Hay, Accts.

Chem. Res. 6, 368 (1973); W. A. Goddard III and L. B. Harding, Ann. Rev.

Phys. Chem. 29, 363 (1978).

[31] C. E. Moore, Atomic Energy Levels (Nat. Stand. Rev. Data Ser., Nat. Bur.

Stand. (U.S.), 1971). Values of atomic excitation energies are obtained weight­

ing the variousJ values for eachconfigurationbythe 2J+1 degeneracy, e.g.,for the noble metal atoms the s-to-p excitation energyinvolves the d10s1 (J = 0) and d10p1 (J = 3/2 and J = 1/2) atomic states.

[32] H. Jones, Proc. Phys. Soc. London, A 49, 250 (1937); N. F. Mott and H.

Jones, Properties of Metals and Alloys (Oxford Univ. Press, London, 1936); H.

Jones, The Theory ofBrillouin Zones and ElectronicStates in Crystals (North- Holland, Amsterdam, 1960).

[33] W. Hume-Rothery and G. V. Raynor, The Structure of Metals and Alloys, 4th Ed. (Institute ofMetals, London, 1962).

[34] C. Barrett and T. B. Massalski, Structure ofMetals, Crystallographic Methods, Principles and Data, 3rd (revised) Ed. (Pergamon Press, New York, 1980).

[35] T. H. Upton and W. A. Goddard III, UHF program, unpublished; R. A. Bair, W. A. Goddard III, A. F. Voter, A. K. Rappé, L. G. Yaffe, F. W. Bobrowicz, W. R. Wadt, P. J. Hay and W. J. Hunt, GVB2p5 program, unpublished; see R.

A. Bair, Ph. D. Thesis, California Instituteof Technology, 1980; also see F. W.

Bobrowicz and W. A. GoddardIII, in: Modem Theoretical Chemistry: Methods ofElectronic Structure Theory,editedbyH. F. Schaefer III (Plenum, NewYork, 1977) Vol. III, Chap. 4. The UHF and HF calculations were performed using the UHF and GVB2p5 programs, respectively.

[36] P. J. Hay and W. R. Wadt, J. Chem. Phys. 82, 270, 1985.

[37] T. H. Dunning, Jr., and P. J. Hay, in: Modem Theoretical Chemistry: Methods of Electronic Structure Theory,editedbyH. F. Schaefer III (Plenum, New York, 1977), Vol. III, Chap. 1.

[38] S. Huzinaga, J. Andzelm, M. Klobukowski, E. Radzio-Andzelm, Y. Sakai, and H. Tatewaki, Gaussian Basis Sets for Molecular Calculations (Elsevier, New York, 1984).

[39] J. Demuynck, M.-M. Rohmer, A. Strich and A. Veillard, J. Chem. Phys. 75, 3443 (1981); C. Bachmann, J. Demuynck, and A. Veillard, Faraday Symp.

Chem. Soc. 14, 170 (1980).

[40] H. Tatewaki, E. Miyoshi and T. Nakamura, J. Chem. Phys. 76, 5073 (1982).

[41] E. Miyoshi, H. Tatewaki, and T. Nakamura, J. Chem. Phys. 78, 815 (1983).

[42] L. G. Yaffe, LTR.AN program, unpublished; see L. G. Yaffe and W. A. God­ dard III, Phys. Rev. A. 13, 1682 (1976). Also see W. J. Hunt, T. H. Dunning, Jr., and W. A. Goddard III, Chem. Phys. Lett. 3, 606 (1969).

[43] The electronic structure of the M2 molecule differs markedly from that of the Mn ring cluster for M = Cu, Ag, Au, Li, and Na; hence, it is not surprising that the frozen core approximation ■— using core orbitals optimized for the high-spin valence state to describe other valence states — does not treat M2 and Mjf consistently. For example, at β = 2.889 Â, high-spin Ag10 is unbound with respect to the separatedatomlimit by 86.3 meV/atom, whereas high-spin Aga is unbound with respect to the separated atomlimit by 157.7 meV/atom.

[44] A. T. Amos and G. G. Hall, Proc. Roy. Soc. London A 263, 483 (1961).

[45] S. R. Langhoff, C. W. Bauschlicher, Jr., S. P. Walch, and B. C. Laskowski, J.

Chem. Phys. 85, 7211 (1986); S. P. Walch, C. W. Bauschlicher, Jr. and S. R.

Langhoff, J. Chem. Phys. 85, 5900 (1986).

[46] D. D. Konowalow and M. L. Olson, J. Chem. Phys. 71, 450 (1979); D. D.

Konowalow, M. E. Rosenkrantz and M. L. Olson, J. Chem. Phys. 72, 2612 (1980).

[47] R. L. Martin and E. R. Davidson, Mol. Phys. 35, 1713 (1978).

[48] P. J. Hay and R. L. Martin, J. Chem. Phys. 83, 5174 (.1985); R. L. Martin, J.

Chem. Phys. 86, 5027 (1987).

[49] W. J. Hunt, P. J. Hay, and W. A. Goddard III, J. Chem. Phys. 57, 738 (1972).

[50] R. D. Poshusta and D. J. Klein, Phys. Rev. Lett. 48, 1555 (1982); M. Seel, P.

S. Bagus and J. Ladik, J. Chem. Phys. 77, 3123 (1982); A. Karpfen, Chem.

Phys. Lett. 61, 363 (1979); M. Kertesz, J. Koller, and A. Azman, Phys. Rev.

B 14, 76 (1976); S. P. Ionov, I. I. Amelin, V. S. Lyubimov, G. V. Ionova, and E. F. Makarov, PhysicaStatus Solidi B 77, 441 (1976).

[51] K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure: IV.

Constants ofDiatomic Molecules (Van Nostrand Reinhold, New York, 1979).

[52] S. Huzinaga, J. Chem. Phys. 42, 1293 (1965).

Table 1. Cohesiveenergies for low-spin ring clusters,αl

system α

(Â)

atomization6) energy (meV/atom)

dimerization c) energy (meV/atom)

HF UHF HF UHF

Agβ 2.889 246.4 352.8 57.6 153.8

Ags 2.889 145.7 370.8 -43.1 171.8

Agio 2.889 191.3 379.8 2.4 180.8

Liβ 3.014 60.4 361.7 -10.3 269.1

Liιo 3.014 15.0 398.2 -55.8 305.6

Liι⅛ 3.014 9.6 408.9 -61.1 316.3

Cuιo 2.556 239.7 388.7 8.1 152.6

Agio 2.889 191.3 379.8 2.4 180.8

Auιo 2.884 374.0 511.5 1.1 138.6

Liιo 3.014 15.0 398.2 -55.8 305.6

Naιo 3.659 -72.9 177.6 -56.3 136.5

a) Results for the lowest energy low-spin states (HF, 5 = 0; UHF, Ms = 0) calculated without orbital symmetry restrictions. A frozen core approximation was in effect. See Appendix A for further details.

δ) The total atomization energy Mpr → N M divided by N atoms.

c) The total dimerization energy Ms → N/2 M? divided by N atoms. In calculating the dimerization energies, the bond length of the diatomic molecule is optimized (see Table 10).

Table 2. Experimentalcohesiveenergies for bulk three-dimensional metals.

system

atomization energy βl (eV∕ atom)

dimerization energy (eV/atom)

Cu 3.49 2.47

Ag 2.94 2.11

Au 3.82 2.66

Li 1.64 1.11

Na 1.11 0.74

α) Reference [26].

⅛) Cohesion with respect to diatomic molecules, e.g., Af(,) → 1/2 Afg, References [26,27,28].

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element basis set °1

ground state·

configuration

energy 61 (hartrees)

Cu (3s,2p,5d)∕(2s,2p,ld) [Ar]3d104s1 -50.43769

Ag (3s,3p,4d)∕(2s,2p,ld) [JΓr]4d105s1 -37.40668

Au (3s,3p,3d)∕(2s,2p,ld) [Xe]4∕145dlθ6s1 -33.38745

Li (θs,4p)∕(3s,2p) ls32∙s1 -7.43174

Na (lls,7p)∕(4s,3p) ls32s32p63s1 -161.79530

α) The basis sets are composed of contracted gaussian-type functions and contain the smallest possible number of functions to describe the core orbitals and twice the minimum number of functions to describe the valence s and low-lying p orbitals.

Calculations for Cu, Ag and Au utilize basis sets and ab initio effective potentials from Reference [36]. Calculations for Li and Na are all-electron ab initio, with basis sets from Reference [37] and Reference [38], respectively.

δ) For each case the energy of the isolated atom is the same for HF and UHF.

Table6.AtomicorbitalpopulationsfortheMioringclusters,HF“1

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Au2 2.672 0.000 0.000

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Naa 3.620 0.644 0.446

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Agio 2.889 2.789 1.243

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Au10 2.884 2.343 1.110

Liιo 3.014 3.973 1.555

Naιo 3.659 3.541 1.447

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