Lecture 27 March 11, 2010 Metals, Hypervalent systems, cuprates Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy William A. Goddard, III, wag@wag.caltech.edu 316 Beckman Institute, x3093 Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology Teaching Assistants: Wei-Guang Liu <wgliu@wag.caltech.edu> Ted Yu <tedhyu@wag.caltech.edu> Ch120a-Goddard-L27 © copyright 2010 William A. Goddard III, all rights reserved 1 Remaining Course schedule Wednesday, Mar. 10 no class (wag at conference in Chicago) Thursday Mar. 11, 2pm, L27 TODAY (make up for Mar. 10) Last lecture Final available Friday March 12 Final due back Friday March 19 Ch120a-Goddard-L27 © copyright 2010 William A. Goddard III, all rights reserved 2 Last time Ch120a-Goddard-L27 © copyright 2010 William A. Goddard III, all rights reserved 3 The heme group The net charge of the Fe-heme is zero. The VB structure shown is one of several, all of which lead to two neutral N and two negative N. Thus we consider that the Fe is Fe2+ with a d6 configuration Each N has a doubly occupied sp2 s orbital pointing at it. 4 Energies of the five Fe2+ d orbitals x2-y2 z2=2z2-x2-y2 yz xz xy 5 Exchange stabilizations 6 Ferrous FeII x2-y2 destabilized by heme N lone pairs z2 destabilized by 5th ligand imidazole or 6th ligand CO y x 7 Out of plane motion of Fe – 4 coordinate 8 Add axial base N-N Nonbonded interactions push Fe out of plane is antibonding 9 Free atom to 4 coord to 5 coord Net effect due to five N ligands is to squish the q, t, and s states by a factor of 3 This makes all three available as possible ground states depending 10 on the 6th ligand Bonding of O2 with O to form ozone O2 has available a ps orbital for a s bond to a ps orbital of the O atom And the 3 electron p system for a p bond to a pp orbital of the O atom 11 Bond O2 to Mb Simple VB structures get S=1 or triplet state In fact MbO2 is singlet Why? 12 change in exchange terms when Bond O2 to Mb O2ps O2pp Assume perfect 10 K 7 Kdd dd VB spin pairing 5*4/2 up spin 4*3/2 Then get 4 cases + Thus average Kdd is down spin 2*1/2 (10+7+7+6)/4 =7.5 7 Kdd 6 Kdd 4*3/2 + 2*1/2 3*2/2 + 3*2/2 13 Bonding O2 to Mb Exchange loss on bonding O2 14 Modified exchange energy for q state But expected t binding to be 2*22 = 44 kcal/mol stronger than q What happened? Binding to q would have DH = -33 + 44 = + 11 kcal/mol Instead the q state retains the high spin pairing so that there is no exchange loss, but now the coupling of Fe to O2 does not gain the full VB strength, leading to bond of only 8kcal/mol instead of 33 15 Bond CO to Mb H2O and N2 do not bond strongly enough to promote the Fe to an excited state, thus get S=2 16 compare bonding of CO and O2 to Mb 17 New material 18 Consider Co-heme Co Doublet S=1/2 d Quartet S=3/2 q Exchange stabilizations Up spin down spin total S=1/2 4*3/2 3*2/2 =9 S=3/2 5*4/2 2*1/2 = 11 1 2 -44 kcal/mol 44 kcal/mol Co stabilization for high spin 44 kcal/mol; Fe 88 kcal/mol Result: Co-heme is low spin d state 19 Bond O2 to Co-heme Co Co Cannot make, xz doubly occupied Before bond Co After bond Co before bond S=1/2 with net spin on Co After bond S=1/2 with net on O2 p 20 Exchange stabilizations for ferric Fe, FeIII doublet S=1/2 d quartet S=3/2 q 1 Sextet S=5/2 s 1 Exchange stabilizations Up spin down spin total S=1/2 3*2/2 2*1/2 = 4 S=3/2 4*3/2 0 = 6 S=5/2 5*4/2 0 = 10 1 2Kdd = 44 kcal/mol 6Kdd = 132 kcal/mol FeIII stabilization for high spin 132 kcal/mol; FeII 88 kcal/mol Result: FeIII cannot bind O2, must reduce to FeII first 21 Hypervalent compounds It was quite a surprize to most chemists in 1962 when Neil Bartlett reported the formation of a compound involving XeF bonds. But this was quickly followed by the synthesis of XeF4 (from Xe and F2 at high temperature and XeF2 in 1962 and later XeF6. Indeed Pauling had predicted in 1933 that XeF6 would be stable, but no one tried to make it. Later compounds such as ClF3 and ClF5 were synthesized These compounds violate simple octet rules and are call hypervalent 22 Noble gas dimers Recall that there is no chemical bonding in He2, Ne2 etc This is explained in VB theory as due to repulsive Pauli repulsion from the overlap of doubly occupied orbitals It is explained in MO theory as due to filled bonding and antibonding orbitals (sg)2(su)2 23 Noble gas dimer positive ions On the other hand the positive ions are strongly bound This is explained in MO theory as due to one less antibonding electron than bonding, leading to a three electron bond for He2+ of 55 kcal/mol, the same strength as the one electron bond of H2+ (sg)2(su)1 The VB explanation is less Using (sg) = L+R and (su)=L-R straightforward. We consider that there are Leads to (with negative sign two equivalent VB structures neither of which leads to much bonding, but superimposing them leads to 24 resonance stabilization Re-examine the bonding of HeH Why not describe HeH as (sg)2(su)1 where (sg) = L+R and (su)=L-R Would this lead to bonding? The answer is no, as easily seen with the VB form, where the right structure is 23.6-0.7=23.9 eV above the left. Thus the energy for the (sg)2(su)1 state would be +12.0 – 2.5 = 9.5 eV unbound at R=∞ Adding in ionic stabilization lowers the energy by 14.4/2.0 = 7.2 eV (overestimate because of shielding) , still unbound by 2.3 eV He H He+ IP=+24.6 eV H- EA = 0.7 eV25 Examine the bonding of XeF Consider the energy to form the charge transfer complex Xe Xe+ The energy to form Xe+ F- can be estimated from Using IP(Xe)=12.13eV, EA(F)=3.40eV, and R(IF)=1.98 A, we get E(Xe+ F-) = 1.45eV (unbound) Thus there is no covalent bond for XeF, which has a weak bond of ~ 0.1 eV and a long bond 26 Examine the bonding in XeF2 The energy to form Xe+F- is +1.45 eV Now consider, the impact of putting a 2nd F on the back side of the Xe+ Xe+ Since Xe+ has a singly occupied pz orbital pointing directly at this 2nd F, we can now form a covalent bond to it How strong would the bond be? Probably the same as for IF, which is 2.88 eV. Thus we expect F--Xe+F- to have a bond strength of ~2.88 – 1.45 = 1.43 eV! Of course for FXeF we can also form an equivalent bond for F-Xe+--F. Thus we get a resonance, which we estimate below We will denote this 3 center – 4 electron charge transfer bond as FXeF 27 Stability of XeF2 Ignoring resonance we predict that XeF2 is stable by 1.43 eV. In fact the experimental bond energy is 2.69 eV suggesting that the resonance energy is ~ 1.3 eV. The XeF2 molecule is stable by 2.7 eV with respect to Xe + F2 But to assess whether one could make and store XeF2, say in a bottle, we have to consider other modes of decomposition. The most likely might be that light or surfaces might generate F atoms, which could then decompose XeF2 by the chain reaction XeF2 + F {XeF + F2} Xe + F2 + F Since the bond energy of F2 is 1.6 eV, this reaction is endothermic by 2.7-1.6 = 1.1 eV, suggesting the XeF2 is relatively stable. Indeed it is used with F2 to synthesize XeF4 and XeF6. 28 XeF4 Putting 2 additional F to overlap the Xe py pair leads to the square planar structure, which allows 3 center – 4 electron charge transfer bonds in both the x and y directions. The VB analysis indicates that the stability for XeF4 relative to XeF2 should be ~ 2.7 eV, but maybe a bit weaker due to the increased IP of the Xe due to the first hypervalent bond and because of some possible F---F steric interactions. There is a report that the bond energy is 6 eV, which seems too high, compared to our estimate of 5.4 eV. 29 XeF6 Since XeF4 still has a pz pair, we can form a third hypervalent bond in this direction to obtain an octahedral XeF6 molecule. Indeed XeF6 is stable with this structure Here we expect a stability a little less than 8.1 eV. Pauling in 1933 suggested that XeF6 would be stabile, 30 years in advance of the experiments. He also suggested that XeF8 is stable. However this prediction is wrong 30 Estimated stability of other Nobel gas fluorides (eV) Using the same method as for XeF2, we can estimate the binding energies for the other Noble metals. KrF2 is predicted to be stable by 0.7 eV, which makes it susceptible to decomposition by F radicals 1.3 1.3 1.3 1.3 1.3 1.3 -2.9 -5.3 -0.1 1.0 2.7 3.9 RnF2 is quite stable, by 3.6 eV, but I do not know if it 31 has been observed XeCl2 Since EA(Cl)=3.615 eV R(XeCl+)=2.32A De(XeCl+)=2.15eV, We estimate that XeCl2 is stable by 1.14 eV with respect to Xe + Cl2. However since the bond energy of Cl2 is 2.48 eV, the energy of the chain decomposition process is exothermic by 2.48-1.14=1.34 eV, suggesting at most a small barrier Thus XeCl2 would be difficult to observe 32 Halogen Fluorides, ClFn The IP of ClF is 12.66 eV comparable to the IP of 12.13 for Xe. This suggests that the px and py pairs of Cl could be used to form hypervalent bonds leading to ClF3 and ClF5. Stability of ClF3 relative to ClF + 2F We estimate that ClF3 is stable by 2.8 eV. Indeed the experiment energy for ClF3 ClF + 2F is 2.6 eV, quite similar to XeF2. Thus ClF3 is endothermic by 2.6 -1.6 = 1.0 eV 33 Geometry of ClF3 34 ClHF2 We estimate that Is stable to ClH + 2F by 2.7 eV This is stable with respect to ClH + F2 by 1.1 ev But D(HF) = 5.87 eV, D(HCl)=4.43 eV, D(ClF) = 2.62 eV Thus F2ClH ClF + HF is exothermic by 1.4 eV F2ClH has not been observed 35 ClF5 36 BrFn and IFn 37 SFn 38 SF6 The VB rationalization for octahedral SF6 would be to assume that S is promoted from (3s)2(3p)4 to (3s)0(3p)6 which would lead to 3 hypervalent bonds in the x, y, and z directions. With an “empty” 3s orbital, the EA for SF6 would be very high 39 PFn The VB view is that the PF3 was distorted into a planar geometry, leading the 3s lone pair to become a 3pz pair, which can then form a hypervalent bond to two additional F atoms to form PF5 40 Donor-acceptor bonds to oxygen 41 Ozone, O3 The simple VB description of ozone is, where the terminal pp electrons are not doing much This is analogous to the s system in the covalent description of XeF2. Thus we can look at the p system of ozone as hypervalent, leading to charge transfer to form 42 Diazomethane leading to 43 Application of hypervalent concepts Origin of reactivity in the hypervalent reagent o-iodoxybenzoic acid (IBX) Hypervalent O-I-O linear bond Enhancing 2-iodoxybenzoic acid reactivity by exploiting a hypervalent twist Su JT, Goddard WA; J. Am. Chem. Soc., 127 (41): 14146-14147 (2005) Ch120a-Goddard-L27 © copyright 2010 William A. Goddard III, all rights reserved 44 Hypervalent iodine assumes many metallic personalities Hypervalent I alternative O Oxidations O O OH I CrO3/H2SO4 OAc Radical cyclizations SnBu3Cl I OAc OH Electrophilic alkene activation CC bond formation HgCl2 I OTs O I Pd(OAc)2 this remarkable chemistry of iodine can be understood in terms of hypervalent concepts Martin, J. C. organo-nonmetallic chemistry – Science 1983 221(4610):509-514 Ch120a-Goddard-L27 © copyright 2010 William A. Goddard III, all rights reserved 45 High Temperature Superconductors Cuprates Chiral plaquette polaron theory of cuprate superconductivity; Tahir-Kheli J, Goddard WA Phys. Rev. B 76 (1): Art. No. 014514 (2007) The Chiral Plaquette Polaron Paradigm (CPPP) for high temperature cuprate superconductors; Tahir-Kheli J, Goddard WA; Chem. Phys. Lett. (4-6) 153-165 (2009) Plaquette model of the phase diagram, thermopower, and neutron resonance peak of cuprate superconductors; Jamil Tahir-Kheli and William A. Goddard III, J. Phys. Chem. Lett, submitted Jamil 46 Tahir-Kheli Superconducting Tc; A Story of Punctuated Evolution All Serendipity Discovery is made. Then all combinations tried. Then stagnation until next discovery. Theory has never successfully predicted a new higher temperature material. Embarrassing state for Theorists. To ensure progress we need to learn the fundamental mechanism in terms of the atomistic interactions Metal Era A15 Metal Alloy Era Today Cuprate Era Theoretical Limit (Anderson) 2020 BCS Theory (1957) 47 Short history of superconductivity 4.15 K 1911 Hg (Heike Kamerlingh Onnes, Leiden U, Netherlands) 3.69 K 1913 Tin (Onnes) 7.26 K 1913 Lead (Onnes) 9.2 K 1930 Niobium (Meissner, Berlin) Metals Era 1.14 K 1933 Aluminum 16.1 K 1941 NbN (Ascherman, Frederick, Justi, and Kramer, Berlin) 17.1 K 1953 V3Si (Hardy and Hulm, U Chicago) A15 Metal 18.1 K 1954 Nb3Sn (Matthias, Gebelle, Geller, and Corenzwit, Bell Labs) Alloy Era 9.8 K 1962 Nb0.6Ti0.4 (First commercial wire, Westinghouse) 23.2 K 1973 Nb3Ge (Gavaler, Janocho, and Jones, Westinghouse) 30 K 1986 (LaBa)2CuO4 (Müller and Bednorz, IBM Rüschlikon, Switzerland) 92 K 1987 YBa2Cu3O7 (Wu, Ashburn, and Torng (Alabama), Hor, Meng, Gao, Huang, Wang, and Chu (Houston)) The Cuprate Era 105 K 1988 Bi2Sr2CaCu2O8 (Maeda, Tanaka, Fukutomi, Asano, Tsukuba Laboratory) 120 K 1988 Tl2Ba2Ca2Cu3O10 (Hermann and Sheng, U. Arkansas) 133 K 1993 HgBa2Ca2Cu3O8 (Schilling, Cantoni, Guo, Ott, Zurich, Switzerland) 138 K 1994 (Hg0.8Tl0.2)Ba2Ca2Cu3O8.33 (Dai, Chakoumakos, (ORNL) Sun, Wong, (U Kansas) Xin, Lu (Midwest Superconductivity Inc.), Goldfarb, NIST) after a 15 year drought, the next generation is due soon, what will it be? 48 Fundamental Goals in Our Research on Cuprate Superconductivity Determine the fundamental mechanism in order to have a sound basis for designing improved systems. Criterion for any proposed mechanism of superconductivity: Does it explain the unusual properties of the normal and superconducting state for cuprates? There is no precedent for a theory of superconductivity that actually predicts new materials. Indeed we know of no case of a theorist successfully predicting a new improved superconducting material! Bernt Matthias always claimed that before trying new compositions for superconductors he would ask his Bell Labs theorists what to try and then he would always do just the opposite. 49 Perovskites Perovskite (CaTiO3) first described in the 1830s by the geologist Gustav Rose, who named it after the famous Russian mineralogist Count Lev Aleksevich von Perovski crystal lattice appears cubic, but it is actually orthorhombic in symmetry due to a slight distortion of the structure. Characteristic chemical formula of a perovskite ceramic: ABO3, A atom +2 charge. 12 coordinate at the corners of a cube. B atom +4 charge. Octahedron of O ions on the faces of that cube centered on a B ions at the center of the cube. Together A and B form an FCC structure 50 (La0.85Z0.15)2CuO4: Tc = 38K (Z=Ba), 35K (Z=Sr) 1986 first cuprate superconductor, (LaBa)2CuO4 (Müller and Bednorz) Nobel Prize Isolated CuO2 sheets with apical O on both sides of Cu to form an elongated octahedron Structure type: 0201 Crystal system: Tetragonal Lattice constants: a = 3.7873 Å c = 13.2883 Å Space group: I4/mmm Atomic positions: La,Ba at (0, 0, 0.3606) Cu at (0, 0, 0) O1 at (0, 1/2, 0) O2 at (0, 0, 0.1828) CuO6 octahedra 51 YBa2Cu3O7–d Tc=92K (d=0.07) Per formula unit: two CuO2 sheets (five coordinate pyramid) one CuO chain (four coordinate square) Structure type: 1212C Crystal system: Orthorhombic Lattice constants: a = 3.8227 Å b = 3.8872 Å c = 11.6802 Å Space group: Pmmm Atomic positions: Y at (1/2,1/2,1/2) Ba at (1/2,1/2,0.1843) Cu1 at (0,0,0) Cu2 at (0, 0, 0.3556) O1 at (0, 1/2, 0) O2 at (1/2,0,0.3779) O3 at (0,1/2,0.379) O4 at (0, 0,0.159) 1987: Alabama: Wu, Ashburn, and Torng Houston: Hor, Meng, Gao, Huang, Wang, Chu 52 Tc depends strongly on the number of CuO2 layers: Bi2Sr2Can-1CunO4+2n single sheet CuO2 Tc= 10 K double sheet CuO2 Tc= 85 K Triple sheet CuO2 Tc= 110 K a = 3.85 Å c = 26.8 Å a = 3.85 Å c = 30.9 Å a = 3.85 Å c = 36.5 Å 53 Dependence of Tc on layers is not monotonic TlBa2Can-1CunO2n+3 n= 2 Tc= 103 K n= 3 Tc= 123 K n= 4 Tc= 112 K n= 5 Tc= 107K CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 Double sheet CuO2 a = 3.86 Å c = 12.75 Å Triple sheet CuO2 a = 3.84 Å c = 15.87 Å 4 sheet CuO2 5 sheet CuO2 a = 3.85 Å c = 19.15 Å a = 3.85 Å 54 c = 22.25 Å Reining Champion since 1994: Tc=138K (Hg0.8Tl0.2)Ba2Ca2Cu3O8.33 This has the same structure as TlBa2Ca2Cu3O9 n= 3 Tc= 123 K 1994 Dai, Chakoumakos (ORNL) Sun, Wong (U Kansas) Xin, Lu (Midwest Superconductivity Inc.), Goldfarb (NIST) CuO2 CuO2 CuO2 a = 3.84 Å c = 15.87 Å Triple sheet CuO2 55 Isolated layer can be great CuO2 Tl2Ba2Can-1CunO2n+4 CuO2 n= 1 Tc= 95 K CuO2 n= 4 Tc= 112 K CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 single sheet CuO2 CuO2 a = 3.86 Å c = 23.14 Å a = 3.85CuO Å 2 c = 41.98 Å 4 sheet CuO56 2 (Ba,Sr)CuO2 Tc=90K single sheet CuO2 Structure type: 02"∞ -1"∞ Crystal system: Tetragonal Lattice constants: a = 3.93 Å c = 3.47 Å Space group: P4/mmm Atomic positions: Cu at (0,0,0) O at (0,1/2,0) Ba,Sr at (1/2,1/2,1/2) 57 Some cuprates lead to electron doping not holes (Nd,Ce)2CuO4d Tc =24K For d=0 2 Nd (+3) and 1 Cu (+2) lead to 8 holes 4 O (2) lead to 8 electrons, get insulator Dope with Ce (+4) leading to an extra electron CuO2 single sheet CuO2 Structure type: 0201T ' Crystal system: Tetragonal Lattice constants: a = 3.95 Å c = 12.07 Å Space group: I4/mmm Atomic positions: Nd,Ce at (0,0,0.3513) Cu at(0,0,0) O1 at (0, 1/2, 0) O2 at (0, 1/2,1/4) CuO2 CuO2 58 Our Goal Explain which systems lead to high Tc and which do not Explain how the number of layers and the location of holes and electrons affects the Tc Use this information to design new structures with higher Tc 59 Structural Characteristics of HighTc Superconductors: Start with Undoped Antiferromagnet Cu O O Cu O Cu O Cu O Cu O Cu O Cu O Cu Cu O O O O Cu O O O O Cu O O O Cu O O O Cu Cu Cu O O Cu 2D CuO2 square lattice (xy plane) Oxidation state of Cu: CuII or d9. (xy)2(xz)2(yz)2(z2)2(x2-y2)1 d9 hole is 3d (x2–y2). Oxidation state of O: O2– or p6. (pσ)2 (pp)2 (ppz)2. Cu – O bond = 1.90 – 1.95 Å Cu can have 5th or 6th apical O (2.4 Å) to form an octahedron or half-octahedron Undoped system antiferromagnetic with TNeel = 325 K for La2CuO4. Describe states as a Heisenberg AntiFerromagnet with Jdd = 0.13 eV for La2CuO4: H dd J dd Si S j Superexchange Jdd AF coupling 60 Superexchange coupling of two Cu d9 sites exactly the same as the hypervalent XeF2 Two Cu d9 separated by 4Å leads to no bonding (ground state singlet and excited triplet separated by 0.0001 eV) With O in-between get strong bonding (the singlet is stabilized by Jdd = 0.13 eV = 1500K for LaCuO4) Explanation: a small amount of charge transfer from O to right Cu Cu(x2-y2)1-O(px)2-Cu(x2-y2)1 Cu(x2-y2)1-O(px)1-Cu(x2-y2)2 allows bonding of the O to the left Cu, but only for the singlet state The explanation is referred to as superexchange. bond bond No direct bonding 61 267-14 Characteristic of High Tc Superconductors: Doping The undoped La2CuO4 is an insulator (band gap = 2.0 eV) La2CuO4 (Undoped): La3+, Sr2+, O2–, Cu2+ Thus cation holes = 3*2 + 2 = 8 and anion electrons = 4*2 = 8 Cu2+ d9 local spin antiferromagnetic coupling To get a metal requires doping to put holes in the valence band Doping (oxidation) La2-xSrxCuO4: Assuming 4 O2- requires 8 cation holes. But La2-xSrx 6-x holes, thus must have x Cu3+ and 1 – x Cu2+ Second possibility: assume that excitation from Cu2+ to Cu3+ is too high, then must have hole on O2– leading to O– This leads to x O– and (4 – x) O2– per formula unit. YBa2Cu3O7: Assume that all 7 O are O2– Must have 14 cation holes: since Y3+ +2 Ba2+ leads to +7, then we must have 1 Cu3+ and 2 Cu2+ The second possibility is that all Cu are Cu2+ requiring that there be 1 O– and 6 O2– 62 Essential characteristic of all cuprate superconductors is oxidation (doping) Typical phase diagram La2-xSrxCuO4 Superconductor: 0.05 < x < 0.32 Spin Glass: 0.02 < x < 0.05 Antiferromagnetic: 0 < x < 0.02 Minimum doping to obtain superconductivity, x > 0.05. Optimum doping for highest Tc=35K at x ~ 0.15. Maximum doping above which the superconductivity disappears and the 63 system becomes a normal metal. Summary: Central Characteristics of cuprate superconductors, square CuO2 lattice, 16% holes CuO2 plane Cu O O Cu O Cu O Cu O O pσ O O Cu Cu O O O O Cu O O O Cu Cu O Cu O Cu O pπ O Cu O Cu O O La2CuO4 (Undoped): La3+, Sr2+, O2–, Cu2+ Cu d9 Cu2+ spin, with antiferromagnetic coupling Cu O O Cu Doping (oxidation) La2-xSrxCuO4: Hole x Cu3+ and 1 – x Cu2+, Or Hole x O– and 4 – x O2– YBa2Cu3O7: Y3+, Ba2+, O2– 1 Cu3+ and 2 Cu2+, Or Y3+, Ba2+, Cu2+ 1 O– and 6 O2– Where are the Doped Holes? CuIII or d8: Anderson, Science 235, 1196 (1987), but CuII CuIII IP = 36.83 eV O pσ: Emery, Phys. Rev. Lett. 58, 2794 (1987). O pπ: Goddard et al., Science 239, 896, 899 (1988). O pσ: Freeman et al. (1987), Mattheiss (1987), Pickett (1989). All wrong: based on simple QM (LDA) or clusters (Cu3O8) 64 Which is right: ps or pp holes? Goddard et al. carried out GVB calculations on Cu3O10 + 998 point charges and found pp holes (found similar E for ps) undoped Electronic Structure and Valence Bond Band Structure of Cuprate Superconducting Materials; Y. Guo, J-M. Langlois, and W. A. Goddard III Science 239, 896 (1988) The Magnon Pairing Mechanism of Superconductivity in Cuprate Ceramics G. Chen and W. A. Goddard III; Science 239, 899 (1988) doped 65 pp holes Goddard et al showed if the ground state has pp holes there is an attractive pair that leads to triplet P-wave Cooper pairs, and hence superconductivity. The Superconducting Properties of Copper Oxide High Temperature Superconductors; G. Chen, J-M. Langlois, Y. Guo, and W. A. Goddard III; Proc. Nat. Acad. Sci. USA 86, 3447 (1989) The Quantum Chemistry View of High Temperature Superconductors; W. A. Goddard III, Y. Guo, G. Chen, H. Ding, J-M. Langlois, and G. Lang; In High Temperature Superconductivity Proc. 39th Scottish Universities Summer School in Physics, St. Andrews, Scotland, D.P. Tunstall, W. Barford, and P. Osborne Editors, 1991 However experiment shows that the systems are singlet D-wave. Thus, pp holes does not correct provide an explanation of superconductivity in cuprates. 66 ps holes Emery and most physicists assumed ps holes on the oxygen. Simplifying to t–J model, calculations with on-site Coulomb repulsion suggest that if the system leads to a superconductor it should be singlet D-wave. Thus most physicists believe that ps provides the basis for a correct explanation of superconductivity. Goddard believed that if ps holes were correct, then it would lead to strong bonding to the singly occupied dx2-y2 orbitals on the adjacent Cu atoms, leading to a distortions that localize the state. This would cause a barrier to hopping to adjacent sites and hence would not be superconducting 67 Current canonical HighTc Hamiltonian Ops holeThe t–J Model Undoped Cu d9 hole is 3d x2–y2. O 2pσ doubly occupied. Heisenberg AF with Jdd = 0.13 eV for La2CuO4. H dd J dd Si S j Doped Doping creates hole in O pσ that bonds with x2–y2 to form a bonded singlet (doubly occupied hole). Singlet hole hopping through lattice prefers adjacent sites are same spin, this frustrates the normal AF coupling of d9 spins. J t J Because of Coulomb repulsion cannot have doubly occupied holes. 68 Summary of the t–J model Coulomb repulsion of singlet holes leads to singlet d-wave Cooper pairing. d-wave is observed in phase sensitive Josephson tunneling, in NMR spin relaxation (no Hebel-Slichter coherence peak), and in the temperature dependence of the penetration depth (λ~T2). t-J predicts an ARPES (angle-resolved photoemission) pseudogap which may have the right qualitative dependence. The t–J model has difficulty explaining most of the normal state properties (linear T resistivity, non-standard Drude relaxation, temperature dependent Hall effect, mid-IR optical absorption, and neutron ω/T scaling). 69 Universal Superconducting Tc Curve (What we must explain to have a credible theory) ≈ 0.16 Superconducting Phase ≈ 0.05 ≈ 0.27 Where do these three special doping values come from? 70 Basis for all theories of cuprate superconductors LDA Band calculations of La2CuO4 LDA and PBE lead to a half filled ps-dx2-y2 band; predicting that La2CuO4 is metallic! This is Fundamentally Wrong Experimental Band Gap is 2 eV Empty Fermi Energy (0,p) Un-Occupied Un-Occupied Occupied Occupied G(0,0) (p,0) Occupied Occupied Un-Occupied ps-dx2-y2 band (p,p) Un-Occupied LDA: Freeman 1987, Mattheiss 1987, Pickett (1989) Occupied 71 G2(MAD) H-Ne TM He2 Ne2 (H2O)2 0.0109(2.377) 0.0231(2.595) 0.391(2.710) Units: eV Method DHf IP EA PA Etot HF 6.47 1.036 1.158 0.15 4.49 G2 or best ab initio 0.07a 0.053a 0.057a 0.05a 1.59a WeDE(R want to do QM onDcuprate DE(Re) e(RO..O) e) superconductors. Which 1.09 Unbound Unbound 0.161(3.048) DFT functional Is most accurate? 0.19b 0.0011(2.993)c 0.0043(3.125)c 0.218(2.912)d 3.94a 0.665 0.749 0.27 6.67 0.54e BP86 0.88a 0.175 0.212 0.05 0.19 BLYP 0.31a 0.187 0.106 0.08 0.19 Validation of accuracy of DFT for 148 molecules (the G2 0.194(2.889) Set) with very 0.46 Unbound Unbound accurate experimental data 0.37e Unbound Unbound 0.181(2.952) BPW91 0.34a 0.163 0.094 0.05 0.16 0.60 PW91PW91 0.77 0.164 0.141 0.06 0.35 0.52 OLYP mPWPWf 0.20 0.65 0.185 0.161 0.133 0.122 1.38 0.05 0.16 0.38 PBEPBEg 0.74g 0.156 0.101 0.06 1.25 0.34 XLYPk 0.33 0.186 0.117 0.09 0.95 0.24 0.18 0.94 0.139 0.207 0.107 0.247 1.13 0.07 0.08 0.72 Unbound B3P86 m 0.78a 0.636 0.593 0.03 2.80 0.34 Unbound B3LYP n 0.13a 0.168 0.103 0.06 0.38 e 0.25LDA Unbound (3.9 eV B3PW91 o 0.15a 0.161 0.100 0.03 0.24 0.38 PW1PW p 0.23 0.160 0.114 0.04 0.30 0.30 0.0066(2.660) 0.0095(3.003) 0.227(2.884) mPW1PW q 0.17 0.160 0.118 0.04 0.16 0.31 0.0020(3.052) 0.0023(3.254) 0.199(2.898) PBE1PBE r 0.21g 0.162 0.126 0.04 1.09 B3LYP and X3LYP are most generally 0.30 0.0018(2.818) 0.0026(3.118) 0.216(2.896) X3LYP s 0.12 0.154 0.087 0.07 0.11 0.22 accurate DFT methods 0.0010(2.726) 0.0028(2.904) 0.216(2.908) Exptl. 0.00 0.000 0.000 0.00 0.00 0.00 0.0010(2.970)h 0.0036(3.091)h 0.236i(2.948 j) LDA (SVWN) GGA DExc Unbound Unbound 0.156(2.946) DH f = Heat of Formation (298K) 0.0100(2.645) 0.0137(3.016) 0.235(2.886) IP = Ionization Potential 0.0052(2.823) 0.0076(3.178) 0.194(2.911) EA = Electron affinity 0.0032(2.752) 0.0048(3.097) PA = Proton Affinity 0.222(2.899) 0.0010(2.805) 0.192(2.953) Etot = total0.0030(3.126) atomic energy Hybrid Methods O3LYP BH&HLYPl Data isUnbound the mean 0.214(2.905) average deviation from experiment Unbound 0.206(2.878) Unbound 0.198(2.926) error) and HF (6.5 eV error) Unbound Unbound 0.175(2.923) Useless for thermochemistry 72 U-B3LYP calculations of La2CuO4 U-B3LYP leads to an insulator (2eV band gap) with a doubled unit cell (one with up-spin Cu and the other down-spin) ky Empty Fermi Energy Band gap Occupied LDA 0.0 PBE 0.0 PW91 0.0 Hartree-Fock 17.0 B3LYP (unrestricted) 2.0 Experiment 2.0 Freeman et al. 1987, Mattheiss 1987, Pickett 1989 Tahir-Kheli and Goddard, 2006 Tahir-Kheli and Goddard, 2006 Harrison et al. 1999 Perry, Tahir-Kheli, Goddard Phys. Rev. B 63,144510(2001) 73 (Ginder et al. 1988) B3LYP leads to excellent band gaps for La2CuO4 whereas LDA predicts a metal! LDA 0.0 PBE 0.0 PW91 0.0 Hartree-Fock 17.0 B3LYP (unrestricted) 2.0 Experiment 2.0 (Freeman et al. 1987, Mattheiss 1987, Pickett 1989) Tahir-Kheli and Goddard, 2006 Tahir-Kheli and Goddard, 2006 (Harrison et al. 1999) (Perry, Tahir-Kheli, and Goddard 2001) (Ginder et al. 1988) Conclusion #1: To describe the states of the La2CuO4 antiferromagnet we must include some amount of true (Hartree-Fock) exchange. Plane wave based periodic codes do not allow this (Castep, CPMD, VASP, Vienna, Siesta). LDA or GGA is not sufficient (e.g. PBE, PW91, BLYP). Crystal does allow B3LYP and X3LYP. Conclusion #2: Unrestricted B3LYP (U-B3LYP) DFT gives an excellent description of the band gap. Hence, B3LYP should be useful for describing doped cuprates (both hole type and electron type). 74 B3LYP Works Well for Crystals with Transition Metals Too Expt. B3LYP Si 3.5 3.8 Diamond 5.5 5.8 GaAs 1.4 1.5 ZnO 3.4 3.2 Al2O3 9.0 8.5 Cr2O3 3.3 3.4 MgO 7.8 7.3 MnO 3.6 3.8 NiO 3.8 3.9 TiO2 3.0 3.4 FeS2 1.0 2.0 ZnS 3.7 3.5 Si B3LYP band structure 1.2eV 3.5eV Points are Expt., GW, and QMC Direct (Vertical) Bandgaps in eV B3LYP accurate for transition metals and band structures J. Muscat, A. Wander, and N. M. Harrison, Chem. Phys. Lett. 342, 397 (2001). 75 Undoped x=0.0 Band gap 2eV Density of States for Explicitly Doped La2–xSrxCuO4 using UB3LYP Use superlattice (2√2 x 2√2 x 1) with 8 primitive cells La15SrCu8O32 This allows antiferromagnetic coupling of Cu atoms. Allows hole to localize (but not forced to) HOMO LUMO The down-spin states show a clear localized hole with Opz-Cudz2 character Note exactly 0.125 doping leads to ordered supercell with small gap. Band gap Real system disordered holes + d9 spins x=0.125 Becomes conductor for x>0.06 Doped UB3LYP: Perry, Tahir-Kheli, and Goddard, Phys Rev B 65, 144501 (2002) 76 Now use B3LYP for La2–xSrxCuO4 B3LYP leads to a hole along the SrO-Cu Apical axis (z) the apical Polaron Use superlattices with 8 primitive cells La15SrCu8O32 allowing antiferromagnetic coupling of Cu atoms. Allows hole to localize (but not forced to) 0.26 Å Find extra hole localized on apical Cu and O atoms below the Sr site. 2/O pz Cu z This state has Cu character on the Cu and hole Opz character on the bridging O atom. dz2 Because dz2 not doubly occupied, the top O – Cu bond goes from 2.40 to 2.14 Å and O – Cu bond below Cu decreases 0.11 Å to 2.29 Å. The spin of the Cu dz2 and Cu dx2-y2 on the same atom are the same (d8 high spin) The singlet state is ~2.0 eV higher. Bottom line: the hole is NOT in the Cu-O plane (as assumed in ALL previous attempts to explain superconductivity of cuprates) Sr Perry, Tahir-Kheli, and Goddard; Phys. Rev. B 65, 144501 (2002)]. 0.11Å 77 Nature of the new hole induced by LaSr: the Apical Polaron Two holes Mulliken Populations 4 Planar O1 in CuO6 near Sr O2 O1 One hole O2’ 2 O1 per hole Located on apical Cu–O just below the Sr. We call this the Apical Polaron. 78 2nd LaSr polaron from Ab-Initio DFT: the Plaquette Polaron The Plaquette Polaron state is localized on Apical O pz + the four-site Cu plaquette above the Sr. It has Cu z2 hole de-localized apical O pz, Cu dz2, and planar O pσ character over the plane of four Cu atoms. over plaquette The Plaquette Polaron state is calculated to for low doping be 0.053 eV per 8 formula units above the 0.09 Å apical polaron state this is 0.1 Å 0.007 eV = 0.2 kcal/mol per Cu in the La0.875Sr0.125CuO4 cell. The apical O below the Sr shifts up 0.1 Å to a Cu – O bond distance of 2.50 Å (seen in Sr XAFS) leading to a plaquette state. The apical O below the plaquette Cu distance optimizes to a Cu – O bond distance of 2.29 Å. Sr Assumption: LaSr Doping leads to Plaquette Polarons. 79 The Plaquette Polaron States Consider 4 Cu-O dz2 orbitals and the 4 Ops orbitals. For the undoped system, there are 16 electrons in these 8 orbitals In the Plaquette Polaron, one electron is removed. This leads to a hole in either the Px or Py orbital (degenerate). 80 Real orbital view of Plaquette Polarons get two degerate states Main hole character Cu dz2 O pz = ps Sr Cu dz2 O pz = ps Sr 81 Coupling of plaquette spin to neighboring d9 antiferromagnetic lattice (Ising) The Px plaquette is compatable with the left antiferromagnetic coupling of the d9 regions while the Py plaquette is compatable with the right antiferromagnetic coupling of the d9 regions. This Ising-like description is over-simplified. Must find ground state of Heisenberg system, including the Plaquette spin H dd J dd Si S j 82 The Chiral Coupling Term J CH S z 2 ( S1 S2 ) J CH S z 2 ( S2 S1 ) Chiral coupling twists the spins into a right or left-handed system. 83 An unusual degeneracy: complex linear combinations (e+=Px + i Py and e-=Px - i Py) It is natural to assume the hole wavefunction is Px or Py or some real linear combination of the two. The interaction of the polaron with the background d9 spins chooses the complex combinations. This implies each polaron has a clockwise or counter-clockwire current. This result is surprising because we usually think of spins as Ising spins rather as 3D quantum spins. -i -1 Px+iPy 1 i i -1 3D Quantum character of spins leads to Chiral Polarons. Px-iPy 1 -i 84 We obtain 3 types of Electrons 1. “Undoped” Cu AF d9 sites 2. 4-site polarons (out of plane) • Two types of polarons a) Surface polarons (neighboring) AF d9 sites b) Interior polarons (surrounded by other polarons) 3. x2-y2/pσ band electrons inside the percolating polaron swath (the “Doped” Cu sites) x2-y2/pσ band Surface Polaron Interior Polaron d9 AF 85 Total Spin Hamiltonian Time-Reversed Chiral Polarons J CH S z 2 ( S1 S2 ) The total spin Hamiltonian is, Htot = Hdd + Hpd + HCH. d9)–(AF d9) (AF spin coupling (AF d9)–(polaron) spin coupling Chiral coupling | Px iPy ; s | Px iPy ; s J CH J CH , Sz Sz . HCH is invariant under polaron time-reversal. Hdd is polaron time-reversal invariant. Hpd is not invariant. JpdSz•Sd – JpdSz•Sd Hpd splits the energy between time-reversed chiral polarons. The energy difference is on the order of Jpd ~ (8/4)Jdd=2Jdd ~ 0.28 eV. The energy difference between polarons with the same spin but different chiralities is on the order of 4JCH ~ 1.1 eV. The energy splitting between time-reversed polarons is largest for low doping because there are more d9 spins to induce the splitting. 86 Isolated Plaquette Polaron in a d9 sea. Dopant The Plaquette is pinned down by the Sr dopant. The Cu d9 sea leads to an insulator, just as for undoped 87 As increase doping, weaken AF coupling among Cud9 states. Above 5% get conductor. 1. “Undoped” Cu AF d9 sites 2. 4-site polarons (out of plane) • Two types of polarons a) Surface polarons (neighboring) AF d9 sites b) Interior polarons (surrounded by other polarons) 3. x2-y2/pσ band electrons inside the percolating polaron swath (the “Doped” Cu sites) x2-y2/pσ band Surface Polaron Interior Polaron d9 AF 88 Superconducting Pairing only on Surface Plaquettes 89 Assume Optimal Tc Maximum Surface Polarons per Volume experiment 1000 x 1000 lattice 200 ensembles ≈ 0.16 Sp prediction Percolation threshold Becomes metallic Surface area of pairing leads to correct optimal doping 90 Predict maximum Tc • Given our Plaquette Theory of Cuprate Superconductors – Find a way to calculate Tc for different plaquette arrangements as a function of doping • The hope is to predict dopant configurations that could lead to increased Tc – Here, we show some math for the gap equations in the hope that there will be interest in the group to attack this large computational problem 91 The Goal • Compute Tc for different arrangements of dopings – Presumably, will find the prior argument of Tc peak near 0.16 to be correct with random dopings • The question is, is there a non-random doping distribution that can lead to a higher Tc prediction? – How much higher? – Increasing surface area by punching holes into metallic swath should increase Tc 92 D-Wave Pairing Local singlet Cooper pairing within a plaquette. The intermediate state of the plaquette is the time-reversed partner (P↓ P’↑). Only coupling that leads to pairing is spin-exchange coupling with x2y2/pσ electron. Sign of the wavefunction (from Pauli principle part), −i −1 P 1 i (r↑, r’↓, P↓) (r↓, r’↓, P’↑) (r↓, r’↑, P↓) (+) sign i −1 Spin-exchange matrix element part, If r and r’ on same diagonal, then (+) sign. If r and r’ along Cu-O bond, then (−) sign because P and P’ are time-reversed! One more (−) sign due to denominator in second-order perturbation (Eground–EI). Net (+) coupling attractive singlet + P’ −i 1 P↓ r’↑ r↓ − P’↑ (−) repulsive for singlet P, P’ energy splitting ~ Jdd maps to Debye energy in BCS Tc. D-Wave − + r↑ P↓ r’↓ 93 Nature of Pairing −V −V −V −V −V −V 94 Gap Equation in BCS Superconductors k k pairs with k k Scatters into (k ,k ) Matrix element V kk k k D k D s wave k D d k x2- y2 95 BCS Ground State Wavefunction 1 H ( k )c c Vk 'k ck' ck ' ck ck 2 k 'k ks ks ks | G (uk v c c k k ' k ' )|0 uk2 vk2 1 D k Vkk ' G | ck ck | G k' H HF ( 1 ( k )c c D k ck' ck ' Dk ck ck 2 k 'k ks ks ks ) 96 Solution (Min F=E-TS) k k Dk u k vk 2 Ek 1 k u 1 2 Ek 2 k Ek k2 D2k 1 k v 1 2 Ek 2 k k ' D k Vkk ' (uk 'vk ' ) tanh 2kT Dk ' k ' D k Vkk ' tanh 2 Ek ' 2kT 97 Going beyond the BCS Gap equation: the Bogolubov-De Gennes Equations In essence, rewrite the BCS pairing equations in Real-Space. This was originally developed to address the questions of non-uniform magnetic fields in type-II superconductors and also impurities (magnetic and non-magnetic). Leads to Ginzburg-Landau phenomenological theory of superconductivity in complex magnetic fields and disorder. −V −V −V −V −V −V 98 The Bogolubov-De Gennes Equations Allows us to have a pairing in real-space, V(r’,r) that leads to a gap that is a function of position Δ(r’,r). We can incorporate a spatially varying pairing at the interface between the d9 spins and the metallic swath and determine Tc as a function of doping by solving the real-space gap equation. If this leads to correct Tc(x) curve, it shows that the strong coupling Eliashberg formulation is unnecessary. -V neighboring pair attraction 99 B-dG Equations H c cRs t c cR 's Rs R Rs RR ' Rs RR ' 1 VRR'cR cR ' cR ' cR 2 RR' D RR' (VRR' ) cRcR ' D R 'R −V −V −V E u H u D v n n R 0 n RR ' R ' n RR ' R ' R' −V −V −V E v H v D u n n R 0 n RR ' R ' n RR ' R ' R' D RR' 1 n n En n n (VRR' ) u R 'vR u R vR ' tanh 2 2kT 100 Computational Approaches • 2D need 1000 x 1000 lattice – Could start from 0.25 doping of plaquettes where there is no surface (Δ=0) and lower doping (Periodic Boundary Conditions) • Only need to get down to ~ 0.16 doping • If need 3D lattice, then could do 100 x 100 x 100, but this may be too discrete for band structure • Is there a Greens function approach? 101 Estimate of Maximum Tc Chemical Physics Letters 472 (2009) 153–165 To estimate Tc, use the formula from BCS theory Tc = 1.13 ħωD exp(-1/N(0)V) ħωD is Debye energy, N(0) is the density of states at the Fermi level, and V is the strength of the attractive coupling. In CPPP, the Debye energy is replaced by the scale of the energy splitting between opposite chirality plaquettes. For a plaquette surrounded on all four sides by d9 spins get ~ 2Jdd = 0.26 eV ~ 3000K. Expect range from Jdd/2 for one-side with d9 spin neighbors to 3Jdd/2 for case with three-side interfacing d9 spin neighbors Assume exponential term is ~ 1/10 as for A15 superconductors (Tc ~ 23K) Expect that Maximum Tc for a cuprate superconductor is in range of 0.05Jdd to 0.15Jdd or 150K to 450K. Current maximum of 138K may be 0.05Jdd case. Expect that Tc of ~ 300K might be attainable.. Using 100x100 supercell, self-consistent calculations for 100 random 16% doping cases we adjusted the d9-plaquette coupling to give gap Tc ~138K, then we chose specific doping patterns and calculate Tc. We have found cases with Tc > 200K. We expect to 102 predict optimum doping structure to have Tc > 200K. May be a challenge to synthesize. The Three Assumptions of the Chiral Plaquette Polaron Model Assumption 1: A polaron hole due to doping will be in a chiral combination of Px’ and Py’. Each chiral polaron has an orbital symmetry and a spin. Leads to neutron incommensurability and Hall effect. Assumption 2: A band is formed by Cu x2–y2/O pσ on the polaron sites when the polaron plaquettes percolate through the crystal. Leads to ARPES background. d9 Assumption 3a: Interaction of the undoped AF spins with the chiral polarons breaks the energy degeneracy between time-reversed chiral polarons. This leads to the D-wave superconducting pairing. 1 (Px ' iPy ' ) 2 E (Px ' iPy ' ;s ) E (Px ' iPy ' ; s ). Assumption 3b: Since the environment of each polaron is different, the distribution of energy splittings between the polaron states is uniform. Yields neutron scaling and linear resistivity. 103 Cuprate Superconductivity Puzzles Must all be explained by any correct theory Exp. Couples to Electron Charge Linear Resistivity ρ ~ T Drude scattering 1/τ ~ max(ω,T) Excess Mid-IR absorption Low temperature resistivity ~ log(T) Negative Magnetoresistance low T Superconductivity “Semi-conducting” c-axis resistivity Phase transition to superconductivity Hall Effect ~ 1/T (expect ~ constant) 2 2 Dx –y Gap Symmetry Hall Effect RH ~ const for field Evolution of Tc with doping in CuO2 plane. Co-existence of magnetism and Photoemission Pseudogap superconductivity Photoemission Background Large A successful theory must explain experiments from each category. Previous theories leave many of the very puzzling properties unexplained. The chiral plaquette paradigm based on out-of-plane holes explains all of these Exp. Couples to Electron Spin Neutron spin incommensurability Neutron spin ω/T scaling (expect ω/Jdd or ω/EF) Cu, O different NMR relaxations Chiral plaquette polaron theory of cuprate superconductivity Tahir-Kheli, Goddard; Phys. Rev. B 76: 014514 (2007) Explains each of these phenomena 104 Bonding in metallic solids Most of the systems discussed so far in this course have been covalent, with the number of bonds to an atom related to the number of valence electrons. Thus we have discussed the bonding of molecules such as CH4, benzene, O2, and Ozone. The solids with covalent bonding, such as diamond, silicon, GaAs, are generally insulators or semiconductors We also considered covalent bonds to metals such as FeH+, (PH3)2Pt(CH3)2, (bpym)Pt(Cl)(CH3), The Grubbs Ru catalysts We have also discussed the bonding in ionic materials such as (NaCl)n, NaCl crystal, and BaTiO3, where the atoms are best modeled as ions with the bonding dominated by electrostatics Next we consider the bonding in bulk metals, such as iron, Pt, Li, etc. where there is little connection between the number of bonds 105 and the number of valence electrons. Elementary ideas about metals and insulators The first attempts to develop quantum theory started with the Bohr model H atom with electrons in orbits around the nucleus. With Schrodinger QM came the idea that the electrons were in distinct orbitals (s, p, d..), leading to a universal Aufbau diagram which is filled with 2 electrons in each of the lowest orbitals For example: O (1s)2(2s)2(2p)4 106 Bringing atoms together to form the solid As we bring atoms together to form the solid, the levels broaden into energy bands, which may overlap . Thus for Cu we obtain Energy Fermi energy (HOMO and LUMO Thus Cu does not have a band gap at ordinary distances Density states 107 Metals vs inulators 108 conductivity For systems with a band gap, there is no current until excite an electron from the occupied valence band to the empty conduction band The population of electrons in the conduction band and holes in the valence bond is proportional to exp(-Egap/RT). Thus conductivity incresses with T (resistivity decreases) 109 The elements leading to metallic binding There is not yet a conceptual description for metals of a quality comparable to that for non-metals. However there are some trends, as will be described 110 Body centered cubic (bcc), A2 A2 111 Face-centered cubic (fcc), A1 112 Alternative view of fcc 113 Closest packing layer 114 Stacking of 2 closest packed layers 115 Hexagonal closest packed (hcp) structure, A3 116 Cubic closest packing 117 Double hcp The hexagonal lanthanides mostly exhibit a packing of closest packed layers in the sequence ABAC ABAC ABAC This is called the double hcp structure 118 mis fcc hcp b cc Structures of elemental metals some correlation of structure with number of valence electrons 119 Binding in metals Li has the bcc structure with 8 nearest neighbor atoms, but there is only one valence electron per atom. Similarly fcc and hcp have 12 nearest neighbor atoms, but Al with fcc has only three valence electrons per atom while Mg with hcp has only 2. Clearly the bonding is very different than covalent One model (Pauling) resonating valence bonds One problem is energetics: Li2 bond energy = 24 kcal/mol 12 kcal/mol per valence electron Cohesive energy of Li (energy to atomize the crystal is 37.7 kcal/mol per valence electron. Too much to explain with resonance New paradigm: Interstitial Electron Model (IEM). Each valence electron localizes in a tetrahedron between four Li nuclei. Bonding like in Li2+, which is 33.7 kcal/mol per valence electron120 GVB orbitals of ring M10 molecules Get 10 valence electrons each localized in a bond midpoint note H10 is very different, get orbital localized on atom, not bond midpoint R=2 a0 Calculations treated all 11 valence electrons of Cu, Ag, Au using effective core potential. All electrons for H and Li 121 122 Bonding in alkalis 123 The bonding in column 11 Get trend similar to alkalis 124 Geometries of Li4 clusters For H4, the electrons are in 1s orbitals centered on each atom Thus spin pair across sides. Orthogonalization cases distortion to rectangle For Li4, the electrons are in orbitals centered on each bond midpoint Thus spin pair between bond midpoint. Orthogonalization cases distortion to rhombus 125 Geometries of Li6 cluster For H6, the electrons are in 1s orbitals centered on each atom Thus spin pair across sides. Orthogonalization cases distortion to D3h hexagone For Li6, the electrons are in orbitals centered on each bond midpoint Thus spin pair between bond midpoint. Orthogonalization cases distortion to triangular structure 126 Geometries of Li8 cluster For Li8, the electrons are in orbitals centered on each bond midpoint Thus spin pair between bond midpoint. Orthogonalization cases distortion to out-of-plane D2d structure 127 Li10 get closest packed structure 128 Li two dimensional Electrons localize into triangular interstitial regions Closest packed structure has 2 triangles per electron One occupied and one empty Spin pair adjacent triangles but leave others empty to avoid Pauli Repulsion Calculation periodic cell with 8 electrons or 4 GVB pairs with 129 overlap = 0.52 Crystalline properties of B column 130 Binding of CH3 to Pt clusters 131 Binding of alkyl CH3-xMex to (111) surfaces Prefers on-top site Decreased binding with increasing x due to steric interactions with 132 other atoms of Pt (111) surface Binding of alkylidene CH2-xMex to (111) surfaces Prefers bridge binding site Decreased binding with increasing x due to steric interactions with 133 other atoms of Pt (111) surface Binding of alkylidyne CH2-xMex to (111) surfaces 134 Average bond strength in CHx/M8 cluster 135 Comparison of bonding energies of CHx and C2Hx 136 Energy barriers for CH4 dehydrogenation on Pt 137 Energy barriers for benzene dehydrogenation on Pt 138 Geometries and Energetics of ethyl binding to M8 139 Geometries and Energetics of ethylene binding to M8 140 Geometries and Energetics of vinyl binding to M8 141 Geometries and Energetics of ethylidene binding to M8 142 Geometries and Energetics of vinylidene binding to M8 143 Geometries and Energetics of dicarbond binding to M8 144 Geometries and Energetics of ethynel binding to M8 145 Geometries and Energetics of acetylene binding to M8 Confirmed experimentally by Wilson Ho 146 Heats of formation for C2Hx and CHx species for Pt Most stable is to form CHad 147 Energetics for C2Hx on Pt 148 149 150 151

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