Lecture 27 March 9, 2011 Cuprates, metals

1© copyright 2011 William A. Goddard III, all rights reservedCh120a-Goddard-L27

Nature of the Chemical Bond with applications to catalysis, materials

science, nanotechnology, surface science, bioinorganic chemistry, and energy

Lecture 27 March 9, 2011Cuprates, metals

William A. Goddard, III, [email protected] Beckman Institute, x3093

Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics,

California Institute of Technology

Teaching Assistants: Wei-Guang Liu <[email protected]>Caitlin Scott <[email protected]>


Last time


Examine the bonding of XeF

Xe Xe+

The energy to form Xe+ F- can be estimated from

Consider the energy to form the charge transfer complex

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


Examine the bonding in XeF2

The energy to form Xe+F- is +1.45 eVNow 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 itHow 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


Estimate stability of XeF2 (eV)

XeF2 is stable with respect to the free atoms by 2.7 eV

Bond energy F2 is 1.6 eV.

Thus stability of XeF2 with respect to Xe + F2 is 1.1 eV

1.3

2.7

Energy form F Xe+ F- at R=∞

F-Xe+ covalent bond length (from IF)

Energy form F Xe+ F- at R=Re

F-Xe+ covalent bond energy (from IF)

Net bond strength of F--Xe+ F-

Resonance due to F- Xe+--F

Net bond strength of XeF2


Stability of gas of XeF2

The XeF2 molecule is stable by 1.1 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 XeF2 is used with F2 to synthesize XeF4 and XeF6.


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.

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.


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

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

Here we expect a stability a little less than 8.1 eV.


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

2.71.0 3.9-5.3-2.9 -0.1

RnF2 is quite stable, by 3.6 eV, but I do not know if it has been observed


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.

We estimate that ClF3 is stable by 2.8 eV.

Stability of ClF3 relative to ClF + 2F

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


Geometry of ClF3


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


ClF5


SFn


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


Donor-acceptor bonds to oxygen


Ozone, O3

The simple VB description of ozone is, where the terminal p 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


Diazomethane

leading to


Origin of reactivity in the hypervalentreagent o-iodoxybenzoic acid (IBX)

Hypervalent O-I-O linear bond

Application of hypervalent concepts

Enhancing 2-iodoxybenzoic acid reactivity by exploiting a hypervalent twist Su JT, Goddard WA; J. Am. Chem. Soc., 127 (41): 14146-14147 (2005)


Hypervalent iodine assumes many metallic personalities

IOH

O O

O

I

OAc

OAc

I

OH

OTs

I

O

Oxidations

Radicalcyclizations

CC bondformation

Electrophilicalkene activation

CrO3/H2SO4

Pd(OAc)2

SnBu3Cl

HgCl2

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

Hypervalent I alternative


New material

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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 electrostaticsNext we consider the bonding in bulk metals, such as iron, Pt, Li, etc. where there is little connection between the number of bonds and the number of valence electrons.

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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 orbitalsFor example:O (1s)2(2s)2(2p)4

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Bringing atoms together to form the solidAs we bring atoms together to form the solid, the levels broaden into energy bands, which may overlap . Thus for Cu we obtain

Energy

Density states

Fermi energy (HOMO and LUMO

Thus Cu does not have a

band gap at ordinary

distances

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Metals vs inulators

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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)

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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

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Body centered cubic (bcc), A2

A2

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Face-centered cubic (fcc), A1

30

Alternative view of fcc

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Closest packing layer

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Stacking of 2 closest packed layers

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Hexagonal closest packed (hcp) structure,

A3

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Cubic closest packing

35

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

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bcc

hcp fcc

mis

Structures of elemental metals

some correlation of structure with number of

valence electrons

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Binding in metalsLi 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 electron

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GVB orbitals of ring M10 moleculesGet 10 valence electrons each localized in a bond midpoint

Calculations treated all 11 valence electrons of Cu, Ag, Au using effective core potential.

All electrons for H and Li

R=2 a0

note H10 is very different, get orbital localized on

atom, not bond midpoint

39

40

Bonding in alkalis

41

The bonding in column 11

Get trend similar to alkalis

42

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 midpointThus spin pair between bond midpoint. Orthogonalization cases distortion to rhombus

43

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 hexagoneFor Li6, the electrons are in orbitals centered on each bond midpointThus spin pair between bond midpoint. Orthogonalization cases distortion to triangular structure

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Geometries of Li8 cluster

For Li8, the electrons are in orbitals centered on each bond midpointThus spin pair between bond midpoint. Orthogonalization cases distortion to out-of-plane D2d structure

45

Li10 get closest packed structure

46

Li two dimensional

Electrons localize into triangular interstitial regions

Closest packed structure has 2 triangles per electronOne occupied and one emptySpin pair adjacent triangles but leave others empty to avoid Pauli RepulsionCalculation periodic cell with 8 electrons or 4 GVB pairs with overlap = 0.52

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Crystalline properties of B column

48

Binding of CH3 to Pt clusters

49

Binding of alkyl CH3-xMex to (111) surfaces

Prefers on-top site

Decreased binding with increasing x due to steric interactions with other atoms of Pt (111) surface

\

50

Binding of alkylidene CH2-xMex to (111) surfaces

Prefers bridge binding site

Decreased binding with increasing x due to steric interactions with other atoms of Pt (111) surface

51

Binding of alkylidyne CH2-xMex to (111) surfaces

52

Average bond strength in CHx/M8 cluster

53

Comparison of bonding energies of CHx and C2Hx

54

Energy barriers for CH4 dehydrogenation on Pt

55

Energy barriers for benzene dehydrogenation on Pt

56

Geometries and Energetics of ethyl binding to M8

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Geometries and Energetics of ethylene binding to M8

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Geometries and Energetics of vinyl binding to M8

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Geometries and Energetics of ethylidene binding to M8

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Geometries and Energetics of vinylidene binding to M8

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Geometries and Energetics of dicarbond binding to M8

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Geometries and Energetics of ethynel binding to M8

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Geometries and Energetics of acetylene binding to M8

Confirmed experimentally by Wilson Ho

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Heats of formation for C2Hx and CHx species for Pt

Most stable is to form CHad

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Energetics for C2Hx on Pt

66

67

68

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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 Tahir-Kheli

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Superconducting Tc; A Story of Punctuated Evolution All Serendipity

Today

MetalEra

A15 Metal AlloyEra

CuprateEra

2020

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

BCS Theory (1957)

Theoretical Limit(Anderson)

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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) 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) 18.1 K 1954 Nb3Sn (Matthias, Gebelle, Geller, and Corenzwit, Bell Labs) 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)) 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)

Metals Era

A15 Metal Alloy Era

The Cuprate Era

Short history of superconductivity

after a 15 year drought, the next generation is due soon, what will it be?

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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?

Fundamental Goals in Our Research on Cuprate Superconductivity

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.

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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

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(La0.85Z0.15)2CuO4: Tc = 38K (Z=Ba), 35K (Z=Sr)

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

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

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YBa2Cu3O7– Tc=92K (=0.07)

Structure type: 1212CCrystal system: Orthorhombic

Lattice constants: a = 3.8227 Åb = 3.8872 Åc = 11.6802 Å

Space group: PmmmAtomic 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

Per formula unit: two CuO2 sheets (five coordinate pyramid)one CuO chain (four coordinate square)

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Tc depends strongly on the number of CuO2 layers: Bi2Sr2Can-1CunO4+2n

a = 3.85 Åc = 26.8 Å

a = 3.85 Åc = 30.9 Å

a = 3.85 Åc = 36.5 Å

Triple sheet CuO2

Tc= 110 K

single sheet CuO2

Tc= 10 K

double sheet CuO2 Tc= 85 K

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Dependence of Tc on layers is not monotonic TlBa2Can-1CunO2n+3

CuO2

CuO2

n= 2 Tc= 103 K n= 3 Tc= 123 K n= 4 Tc= 112 K n= 5 Tc= 107K

Double sheet CuO2Triple sheet CuO2 4 sheet CuO2 5 sheet CuO2

a = 3.86 Åc = 12.75 Å

a = 3.84 Åc = 15.87 Å

a = 3.85 Åc = 19.15 Å

a = 3.85 Åc = 22.25 Å

CuO2

CuO2CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

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Reining Champion since 1994: Tc=138K (Hg0.8Tl0.2)Ba2Ca2Cu3O8.33

This has the same structure

asTlBa2Ca2Cu3O9

n= 3 Tc= 123 K

CuO2

a = 3.84 Åc = 15.87 Å

Triple sheet CuO2

CuO2

CuO2

1994 Dai, Chakoumakos (ORNL)

Sun, Wong (U Kansas) Xin, Lu (Midwest

Superconductivity Inc.), Goldfarb (NIST)

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Isolated layer can be greatTl2Ba2Can-1CunO2n+4

n= 1 Tc= 95 K n= 4 Tc= 112 K

CuO2

CuO2

single sheet CuO2 4 sheet CuO2

a = 3.86 Åc = 23.14 Å

a = 3.85 Åc = 41.98 Å

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

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(Ba,Sr)CuO2 Tc=90K

Structure type: 02"∞ -1"∞Crystal system: Tetragonal

Lattice constants: a = 3.93 Åc = 3.47 Å

Space group: P4/mmmAtomic positions:

Cu at (0,0,0)O at (0,1/2,0)

Ba,Sr at (1/2,1/2,1/2)

single sheet CuO2

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Some cuprates lead to electron doping not holes (Nd,Ce)2CuO4 Tc =24K

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)

For =0 2 Nd (+3) and 1 Cu (+2) lead to 8 holes 4 O (2) lead to 8 electrons, get insulatorDope with Ce (+4) leading to an extra electron

CuO2

single sheet CuO2

CuO2

CuO2

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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

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Structural Characteristics of HighTc Superconductors: Start with Undoped Antiferromagnet

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

Undoped system antiferromagnetic withTNeel = 325 K for La2CuO4.Describe states as a Heisenberg AntiFerromagnetwith Jdd = 0.13 eV for La2CuO4:

jidddd SSJH

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 (p)2 (pz)2.

Cu – O bond = 1.90 – 1.95 ÅCu can have 5th or 6th apical O (2.Å) to form an octahedron or half-octahedron

Superexchange Jdd AF coupling

84

Superexchange coupling of two Cu d9 sitesexactly the same as the hypervalent XeF2

267-14

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 CuCu(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 stateThe explanation is referred to as superexchange.

No direct bonding

bond

bond

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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 = 8Cu2+ d9 local spin antiferromagnetic couplingTo get a metal requires doping to put holes in the valence band

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–

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.

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Essential characteristic of all cuprate superconductors is oxidation (doping)

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 system becomes a normal metal.Antiferromagnetic: 0 < x < 0.02

Spin Glass: 0.02 < x < 0.05

Superconductor: 0.05 < x < 0.32

Typical phase diagram

La2-xSrxCuO4

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Summary: Central Characteristics of cuprate superconductors, square CuO2 lattice, 16% holes

CuO2 plane La2CuO4 (Undoped): La3+, Sr2+, O2–, Cu2+

d9 Cu2+ spin, with antiferromagnetic coupling

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 eVO 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).

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

Doping (oxidation) La2-xSrxCuO4:Hole x Cu3+ and 1 – x Cu2+, OrHole x O– and 4 – x O2–

pσ

pπ

All wrong: based on simple QM (LDA) or clusters (Cu3O8)

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Goddard et al. carried out GVB calculations on Cu3O10 + 998 point charges and found p holes (found similar E for p)

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)

Which is right: p or pholes?

undoped

doped

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pholes

Goddard et al showed if the ground state has p 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, p holes does not correct provide an explanation of superconductivity in cuprates.

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p holes

Emery and most physicists assumed p 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 p provides the basis for a correct explanation of superconductivity.

Goddard believed that if p 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

91

Current canonical HighTc Hamiltonian Op holeThe t–J Model

Cu d9 hole is 3d x2–y2. O 2pσ doubly occupied.

Heisenberg AF with Jdd = 0.13 eV for La2CuO4.

Undoped

jidddd SSJH

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.Doped

J

tJ

Because of Coulomb repulsion cannot have doubly occupied holes.

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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).

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Universal Superconducting Tc Curve(What we must explain to have a credible theory)

Where do these three special doping values come from?

≈ 0.05

≈ 0.16

≈ 0.27

SuperconductingPhase

94

Basis for all theories of cuprate superconductors LDA Band calculations of La2CuO4

LDA and PBE lead to a half filled p-dx2-y2 band; predicting that La2CuO4 is metallic!

This is Fundamentally Wrong

Experimental Band Gap is 2 eV

LDA: Freeman 1987, Mattheiss 1987, Pickett (1989)

(

Occupied

Un-Occupied

Occupied

OccupiedOccupied

Un-Occupied

Un-Occupied

Un-Occupied

(

(

Fermi Energy

Occupied

Empty

p-dx2-y2 band

95

Validation of accuracy of DFT for 148 molecules (the G2 Set) with very

accurate experimental data

Hf = Heat of Formation (298K)IP = Ionization PotentialEA = Electron affinityPA = Proton AffinityEtot = total atomic energy

Units: eV

Data is the mean average deviation from experiment

O3LYP 0.18 0.139 0.107 1.13

OLYP 0.20 0.185 0.133 1.38

LDA (3.9 eV error) and HF (6.5 eV error) Useless for thermochemistry

B3LYP and X3LYP are most generally accurate DFT methods

We want to do QM on cuprate superconductors. Which

DFT functional Is most accurate?

96

Occupied

U-B3LYP calculations of La2CuO4

Fermi Energy

EmptyU-B3LYP leads to an insulator (2eV band gap) with a doubled unit cell (one with up-spin Cu and the other down-spin)

Band gapky

LDA 0.0 Freeman et al. 1987, Mattheiss 1987, Pickett 1989PBE 0.0 Tahir-Kheli and Goddard, 2006PW91 0.0 Tahir-Kheli and Goddard, 2006Hartree-Fock 17.0 Harrison et al. 1999B3LYP (unrestricted) 2.0 Perry, Tahir-Kheli, Goddard Phys. Rev. B 63,144510(2001)Experiment 2.0 (Ginder et al. 1988)

97

B3LYP leads to excellent band gaps for La2CuO4 whereas LDA predicts a metal!

LDA 0.0 (Freeman et al. 1987, Mattheiss 1987, Pickett 1989)PBE 0.0 Tahir-Kheli and Goddard, 2006PW91 0.0 Tahir-Kheli and Goddard, 2006Hartree-Fock 17.0 (Harrison et al. 1999)B3LYP (unrestricted) 2.0 (Perry, Tahir-Kheli, and Goddard 2001)Experiment 2.0 (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).

98

B3LYP Works Well for Crystals with Transition Metals Too

J. Muscat, A. Wander, and N. M. Harrison, Chem. Phys. Lett. 342, 397 (2001).

3.7

1.0

3.0

3.8

3.6

7.8

3.3

9.0

3.4

1.4

5.5

3.5

Expt.

3.5ZnS

2.0FeS2

3.4TiO2

3.9NiO

3.8MnO

7.3MgO

3.4Cr2O3

8.5Al2O3

3.2ZnO

1.5GaAs

5.8Diamond

3.8Si

B3LYP

Direct (Vertical) Bandgaps in eV

Si B3LYP band structure

Points are Expt., GW, and QMC

B3LYP accurate for transition metals and band structures

1.2eV

3.5eV

99

Density of States for Explicitly Doped

La2–xSrxCuO4 using UB3LYP

Doped UB3LYP: Perry, Tahir-Kheli, and Goddard, Phys Rev B 65, 144501 (2002)

The down-spin states show a clear localized hole with Opz-Cudz2 character

Undopedx=0.0

x=0.125

HOMO LUMO

Band gap

Note exactly 0.125 doping leads to ordered supercell with small gap.

Real system disordered holes + d9 spins

Becomes conductor for x>0.06

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)

Band gap 2eV

100

Find extra hole localized on apical Cu and O atoms below the Sr site.

Sr

0.11Å

0.26 Å

Cu z2/O pz hole

B3LYP leads to a hole along the Sr-O-Cu Apical axis (z) the apical

Polaron

Bottom line: the hole is NOT in the Cu-O plane (as assumed in ALL previous attempts to explain superconductivity of cuprates)

This state has Cu dz2 character on the Cu and Opz character on the bridging O atom.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.

Perry, Tahir-Kheli, and Goddard; Phys. Rev. B 65, 144501 (2002)].

Now use B3LYP for La2–xSrxCuO4

Use superlattices with 8 primitive cells La15SrCu8O32

allowing antiferromagnetic coupling of Cu atoms. Allows hole to localize (but not forced to)

101

Nature of the new hole induced by LaSr: the Apical Polaron

Mulliken Populations4 Planar O1 in CuO6 near Sr

On

e h

ole

Tw

o h

oles

2 O1 per hole

Located on apical Cu–O just below the Sr. We call this the Apical Polaron.

O1

O2

O2’

102

The Plaquette Polaron state is localized on the four-site Cu plaquette above the Sr. It has apical O pz, Cu dz2, and planar O pσ character over the plane of four Cu atoms.The Plaquette Polaron state is calculated to be 0.053 eV per 8 formula units above the apical polaron state this is 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.

2nd LaSr polaron from Ab-Initio DFT: the Plaquette Polaron

Sr

Apical O pz + Cu z2 hole

de-localizedover plaquettefor low doping

0.09 Å0.1 Å

Assumption: LaSr Doping leads to Plaquette Polarons.

103

The Plaquette Polaron StatesConsider 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).

104

Real orbital view of Plaquette Polaronsget two degerate states

Sr Sr

Cu dz2

O pz = p

Main hole character

Cu dz2

O pz = p

105

Coupling of plaquette spin to neighboring d9 antiferromagnetic lattice (Ising)

jidddd SSJH

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

106

The Chiral Coupling Term

)( 212 SSSJ zCH )( 122 SSSJ zCH

Chiral coupling twists the spins into a right or left-handed system.

107

We obtain 3 types of Electrons

1. “Undoped” Cu AF d9 sites2. 4-site polarons (out of plane)

• Two types of polaronsa) 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)

SurfacePolaron

InteriorPolaron d9 AFx2-y2/pσ

band

108

Total Spin HamiltonianTime-Reversed Chiral Polarons

)( 212 SSSJ zCH

.

,

;| ;|

zz

CHCH

yxyx

SS

JJ

iPPiPP

The total spin Hamiltonian is,

Htot = Hdd + Hpd + HCH.

(AF d9)–(AF d9) spin coupling

(AF d9)–(polaron) spin coupling

Chiral coupling

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.

109

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

110

As increase doping, weaken AF coupling among Cud9 states. Above 5% get conductor.

1. “Undoped” Cu AF d9 sites2. 4-site polarons (out of plane)

• Two types of polaronsa) 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)

SurfacePolaron

InteriorPolaron d9 AFx2-y2/pσ

band

111

Superconducting Pairing only on Surface Plaquettes

112

Assume Optimal Tc Maximum Surface Polarons per Volume

1000 x 1000 lattice200 ensembles

≈ 0.16

Surface area of pairing leads to correct optimal doping

Sp

Percolation threshold

Becomes metallic

prediction

experiment

113

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

114

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

115

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.

1 i

−1

−1

1 −i

−i

i

P

P’

P↓

P’↑

P↓

r↑ r’↓

r↓ r’↑

Sign of the wavefunction (from Pauli principle part),

(r↑, r’↓, P↓) (r↓, r’↓, P’↑) (r↓, r’↑, P↓) (+) signSpin-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(−) repulsive for singlet

−

−

+

+

D-Wave

P, P’ energy splitting ~ Jdd maps to Debye energy in BCS Tc.

116

Nature of Pairing

−V−V −V

−V−V −V

−V−V −V

−V−V −V

117

Gap Equation in BCS Superconductors

kk withpairs

k

k

k

k

kkV

kk

element Matrix

),( into Scatters

k

wavesk y2-x2

d k

118

BCS Ground State Wavefunction

kk

kkkkkkk

kkk ccccVccH'

'''2

1)(

0|)(|

'' kkkkG ccvu

GkkGk

kkk ccV ||'

'

kk

kkkkkkk

kkkHF ccccccH'

''2

1)(

122 kk vu

119

Solution (Min F=E-TS)

kk22kkkE

k

kkk E

vu2

k

kk E

u

12

12

k

kk E

v

12

12

kTvuV k

kkkkk 2tanh)( '

'''

kTE

V k

k

kkkk 2

tanh2

'

'

''

120

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

−V−V −V

−V−V −V

121

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

122

B-dG Equations

'

''''

'' 2

1

RRRRRRRR

RRRRRR

RRRR ccccVcctccH

−V−V −V

−V−V −V

−V−V −V

−V−V −V

RRRRRRRR ccV '''' )(

'

'''0

'R

nRRR

nRRR

nRn vuHuE

kT

EvuvuV nn

RnR

nR

nRRRRR 2

tanh2

1)( ''''

'

'''0

'R

nRRR

nRRR

nRn uvHvE

123

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?

124

Estimate of Maximum TcChemical 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 predict optimum doping structure to have Tc > 200K. May be a challenge to synthesize.

125

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.

Assumption 3a: Interaction of the d9 undoped AF spins with the chiral polarons breaks the energy degeneracy between time-reversed chiral polarons. This leads to the D-wave superconducting pairing.

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.

''21

yx iPP

. ;

;

''

''

yx

yx

iPPE

iPPE

126

Cuprate Superconductivity Puzzles Must all be explained by any correct theory

Exp. Couples to Electron SpinNeutron spin incommensurability Neutron spin ω/T scaling (expect ω/Jdd or ω/EF)Cu, O different NMR relaxations

Exp. Couples to Electron ChargeLinear Resistivity ρ ~ TDrude scattering 1/τ ~ max(ω,T)Excess Mid-IR absorptionLow temperature resistivity ~ log(T)Negative Magnetoresistance low T“Semi-conducting” c-axis resistivityHall Effect ~ 1/T (expect ~ constant)Hall Effect RH ~ const for field in CuO2 plane.Photoemission PseudogapPhotoemission Background Large

SuperconductivityPhase transition to superconductivityDx2–y2 Gap SymmetryEvolution of Tc with dopingCo-existence of magnetism and superconductivity

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

Chiral plaquette polaron theory of cuprate superconductivity Tahir-Kheli, Goddard; Phys. Rev. B 76: 014514 (2007) Explains each of these phenomena

127

Summary Ch120a

Fundamental concept of QM KE = (Ћ2/2me)<(Φ Φ>

Keeps atom from collapsing (best PE = e2/Rave

Leads to bond formation: decrease gradient for sum of two orbitals

H atom states 1s, 2s, 2p, 3s, 3p, 3d corrected for shielding leads to Aufbau principle atoms 1s<<2s<2p<<3s<3p<3d ~ 4s<4p etc

Get covalent bond when have unpaired spins on bond atoms

Start with ground config of atoms and find maximum spin pairs

Two bonds to p orbitals 90 degree bond angle (but HH repulsion H2O is 104.5°

Bonds to Be,B,C columns: 2s pair 2s ±2p hybrids

Transition metals start with Ground config of atoms Pt (6s)(5d)9

Explain structures, reactions, properties organometallics, solids, etc

Lecture 27 March 9, 2011 Cuprates, metals

Documents

occupied valence band

covalent bonding

valence electrongvb

bonding of molecules

number of bonds

band gap

li2 bond energy

energy bands