Molecular Orbital Theory Hand-Outs: 19 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Molecular Orbital Theory (“Chemists”) Tight-Binding Theory (“Physicists”) Atomic Orbital Basis; Construct Symmetry-Adapted Linear Combinations of AO’s; Hamiltonian (Energy Operator) has total symmetry of point group of the molecule; Diagonalize Hamiltonian matrix for each IR to obtain eigenvalues (energies) and eigenvectors (orbital coefficients); Outcomes: MO energy diagram (HOMO, LUMO); orbital coefficients (population analysis) Construct Symmetry-Adapted Linear Combinations of AO’s with respect to translational symmetry (wavevector k); Hamiltonian (Energy Operator) has total symmetry of space group of the solid; Diagonalize Hamiltonian matrix at each k for each IR to obtain eigenvalues (energies) and eigenvectors (orbital coefficients); Outcomes: density of states (Fermi level, valence and conduction bands), energy dispersion, En(k), and COOP/COHP curves (population analysis)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 20 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR + 

IV. Electronic Structure and Chemical Bonding Hand-Outs: 20 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR +  Symmetry Adapted Linear Combination of Basis Functions (SALCs): (Bloch)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 20 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR +  Symmetry Adapted Linear Combination of Basis Functions (SALCs): k = 0: eikma = e0 = 1

IV. Electronic Structure and Chemical Bonding Hand-Outs: 20 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR +  Symmetry Adapted Linear Combination of Basis Functions (SALCs): k = /2a: eikma = emi/2 = (i)m (Real part)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 20 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Atomic Orbital Basis: 1s AO at each H atom (1 AO/atom) OR +  Symmetry Adapted Linear Combination of Basis Functions (SALCs): k = /a: eikma = emi = (1)m

IV. Electronic Structure and Chemical Bonding Hand-Outs: 21 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix

IV. Electronic Structure and Chemical Bonding Hand-Outs: 20 Hand-Outs: 21 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix Hückel Approximation: Ignore interactions beyond first nearest neighbors “Coulomb” integral = AO Energy “Resonance” integral

IV. Electronic Structure and Chemical Bonding Hand-Outs: 21 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Chain of H atoms; lattice constant a; 1 H atom per unit cell… N (large) = Periodic Boundary Conditions. Hamiltonian (Energy) Matrix: 1 H atom/unit cell = 1 1s AO/unit cell… 11 matrix Hückel Approximation: Ignore interactions beyond first nearest neighbors “Coulomb” integral = AO Energy “Resonance” integral (NOTE: E(k) = E(k), so we limit k to 0  k  /a)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 21 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Outcomes: Band Structure Density of States Crystal Orbital Overlap Population Bandwidth Antibonding Orbitals Fermi Level for H Chain Bonding Orbitals

IV. Electronic Structure and Chemical Bonding Hand-Outs: 21 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Outcomes: Comparison of Band Structure and DOS Curve Band Structure Density of States Crystal Orbital Overlap Population Bandwidth Antibonding Orbitals Fermi Level for H Chain Bonding Orbitals k

IV. Electronic Structure and Chemical Bonding Hand-Outs: 22 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Bandwidth Band Center /a

IV. Electronic Structure and Chemical Bonding Hand-Outs: 22 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 -Bandwidth -Bandwidth Band Center /a

IV. Electronic Structure and Chemical Bonding Hand-Outs: 22 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3

IV. Electronic Structure and Chemical Bonding Hand-Outs: 23 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Band Crossings: Band centers vs. Bandwidths p  s > |  |’s p-Band

IV. Electronic Structure and Chemical Bonding Hand-Outs: 23 IV. Electronic Structure and Chemical Bonding Tight-Binding Model J.K. Burdett, Chemical Bonding in Solids, Ch. 1-3 Band Crossings: Band centers vs. Bandwidths p  s > |  |’s p  s < |  |’s

IV. Electronic Structure and Chemical Bonding Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 1 H atom / unit cell 1 1s AO / unit cell 2 a 2 H atoms / unit cell 2 1s AOs / unit cell 2 H atoms / unit cell 2 1s AOs / unit cell

IV. Electronic Structure and Chemical Bonding Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 1 H atom / unit cell 1 1s AO / unit cell 2 a 2 H atoms / unit cell 2 1s AOs / unit cell   2 H atoms / unit cell 2 1s AOs / unit cell 2 1 2 Energy Matrix (Hamiltonian Matrix):

IV. Electronic Structure and Chemical Bonding Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 2 a 1 = 2   No Distortion 2 1 2 Half-filled Band is unstable with respect to a Peierls Distortion: Electronically-driven

IV. Electronic Structure and Chemical Bonding Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 2 a 1 = 2   2 1 2 “Band Folding”

IV. Electronic Structure and Chemical Bonding Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 Polyacetylene Metallic

IV. Electronic Structure and Chemical Bonding Hand-Outs: 24 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 Polyacetylene Metallic Semiconducting

IV. Electronic Structure and Chemical Bonding Hand-Outs: 25 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 -Bands 11 valence e 10 valence e

IV. Electronic Structure and Chemical Bonding Hand-Outs: 25 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 4 orbitals (BC *) -Bands 11 valence e 10 valence e 10 orbitals (BC , ) 2 orbitals (C 2s)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 25 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 YBC -Bands 11 valence e 10 valence e

IV. Electronic Structure and Chemical Bonding Hand-Outs: 25 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 ThBC -Bands 11 valence e 10 valence e

IV. Electronic Structure and Chemical Bonding Hand-Outs: 26 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 NbI4 High Temperatures Low Temperatures

IV. Electronic Structure and Chemical Bonding Hand-Outs: 26 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 NbI4 High Temperatures Low Temperatures (33 valence electrons)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 26 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 2 NbI4 High Temperatures Low Temperatures kF = /2a kF = /2a (33 valence electrons)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 27 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (a) Oxidation or Reduction Polyacetylene (2x)+ (Br)2x

IV. Electronic Structure and Chemical Bonding Hand-Outs: 27 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (b) Chemical Substitutions

IV. Electronic Structure and Chemical Bonding Hand-Outs: 28 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (b) Chemical Substitutions: Charge Density Waves (static or dynamic) Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X) + (Pt3+) Susceptible to a Peierls Distortion Pt 5dz2 Br 4p Br 4s

IV. Electronic Structure and Chemical Bonding Hand-Outs: 28 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (b) Chemical Substitutions: Charge Density Waves (static or dynamic) Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X) + (Pt3+) Susceptible to a Peierls Distortion Pt 5dz2 Br 4p Br 4s Pt-Br Bond length alternation does not change the qualitative picture!

IV. Electronic Structure and Chemical Bonding Hand-Outs: 28 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (b) Chemical Substitutions: Charge Density Waves (static or dynamic) (Pt4+) (Pt2+) Wolfram’s Red Salt: [Pt(NH3)4Br]+ (X) + (Pt3+) Pt 5dz2 Br 4p Br 4s

IV. Electronic Structure and Chemical Bonding Hand-Outs: 27 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (c) Interactions between Chains: Polysulfur nitride (SN)x

IV. Electronic Structure and Chemical Bonding Hand-Outs: 27 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (c) Interactions between Chains: Polysulfur nitride (SN)x

IV. Electronic Structure and Chemical Bonding Hand-Outs: 27 IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (c) Interactions between Chains: Polysulfur nitride (SN)x “Less than 1/2-filled” “More than 1/2-filled”

IV. Electronic Structure and Chemical Bonding Peierls Distortion J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Preventing Peierls Distortions (d) Applying Pressure: Near-neighbor repulsive energy vs. orbital overlap (e) Increasing Temperature: Fermi-Dirac Distribution f(Fermi-Dirac) = [1+exp(EEF)/kT]1 EF

IV. Electronic Structure and Chemical Bonding R. Hoffmann, Solids and Surfaces: A Chemist’s View of Bonding in Extended Structures, 1988. Summarizes material published in these review articles: “The meeting of solid state chemistry and physics,” Angewandte Chemie 1987, 99, 871-906. “The close ties between organometallic chemistry, surface science, and the solid state,” Pure and Applied Chemistry 1986, 58, 481-94. “A chemical and theoretical way to look at bonding on surfaces,” Reviews of Modern Physics 1988, 60, 601-28.

IV. Electronic Structure and Chemical Bonding Hand-Outs: 29 IV. Electronic Structure and Chemical Bonding Square Lattice J.K. Burdett, Chemical Bonding in Solids, Ch. 3 Real Space: H atoms at lattice points Reciprocal Space: Brillouin Zone ky y kx x (0, /a) (0, 0) (/a, /a) (Only nearest neighbor interactions:  )

IV. Electronic Structure and Chemical Bonding Hand-Outs: 29 IV. Electronic Structure and Chemical Bonding Square Lattice J.K. Burdett, Chemical Bonding in Solids, Ch. 3 Wavefunctions M X 

IV. Electronic Structure and Chemical Bonding Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands J.K. Burdett, Chemical Bonding in Solids, Ch. 3 y x a2 G (2) (1) a1 a2* K M a1* G: (0, 0) M: (1/2, 0) K: (1/3, 1/3)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands J.K. Burdett, Chemical Bonding in Solids, Ch. 3 G K M DOS Curve COOP Curve p-Antibonding “Zero-Gap Semiconductor” p-Bonding

IV. Electronic Structure and Chemical Bonding Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at  (0, 0)? G -Antibonding K M -Bonding

IV. Electronic Structure and Chemical Bonding Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at  (0, 0)? Totally Antibonding G K M Totally Bonding

IV. Electronic Structure and Chemical Bonding Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at  (0, 0)? Totally Antibonding G K M Totally Bonding

IV. Electronic Structure and Chemical Bonding Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at M (1/2, 0)? G -Antibonding K M -Bonding

IV. Electronic Structure and Chemical Bonding Hand-Outs: 30 IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What do the Wavefunctions Look Like at M (1/2, 0)? G K M

IV. Electronic Structure and Chemical Bonding Graphite: -Bands – What is the Advantage of Reciprocal Space? Graphite C6 C13 C24

IV. Electronic Structure and Chemical Bonding Hand-Outs: 31 IV. Electronic Structure and Chemical Bonding Graphite: Valence s and p Bands DOS Curve C-C COOP Curve -Bands Optimized C-C Bonding at EF 2pxpy “Poor” Metal 2pz  (“sp2”) 2s M G K M

IV. Electronic Structure and Chemical Bonding Hand-Outs: 31 IV. Electronic Structure and Chemical Bonding Boron Nitride: Valence s and p Bands – Electronegativity Effects DOS B-N COOP Nonmetallic “N 2p” B-N Bonding “N 2s” B-N Bonding

IV. Electronic Structure and Chemical Bonding Hand-Outs: 32 IV. Electronic Structure and Chemical Bonding MgB2 and AlB2: Valence Bands B: 63 Nets Integrated COHP Mg or Al DOS B-B COHP AlB2 MgB2 Mg or Al 3s, 3p AOs Some Mg-B or Al-B Bonding

IV. Electronic Structure and Chemical Bonding Hand-Outs: 32 IV. Electronic Structure and Chemical Bonding MgB2 and AlB2: Energy Bands s Band below EF in AlB2 -Bands at EF in MgB2

IV. Electronic Structure and Chemical Bonding Hand-Outs: 33 IV. Electronic Structure and Chemical Bonding Tight-Binding Model: Si (Integrated DOS = # Valence Electrons) (Integrated ICOHP) Si-Si Antibonding “sp3” Si-Si Bonding “sp3” 3s

IV. Electronic Structure and Chemical Bonding Hand-Outs: 34 IV. Electronic Structure and Chemical Bonding Tight-Binding Model: Main Group Metals Valence s, p only Free-Electron Metal Nearly Free-Electron Metals Semi-Metals

IV. Electronic Structure and Chemical Bonding Hand-Outs: 35 IV. Electronic Structure and Chemical Bonding Atomic Orbital Energies A.Herman, Modelling Simul. Mater. Sci. Eng., 2004, 12, 21-32. Hartree-Fock Valence Orbital Energies

IV. Electronic Structure and Chemical Bonding Hand-Outs: 36 IV. Electronic Structure and Chemical Bonding How are Bands Positioned in the DOS? NaCl Structures (Semimetallic) (Semiconducting) (Insulating) CaO ScN TiC

IV. Electronic Structure and Chemical Bonding Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 Re 5d (t2g) (3 orbs.) EF (WO3) O 2p (9 orbs.)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 yz  (0, 0, 0)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 yz R (1/2, 1/2, 1/2)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 Re 5d (t2g) (3 orbs.) EF (WO3) O 2p (9 orbs.)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 yz  (0, 0, 0)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 37 IV. Electronic Structure and Chemical Bonding What Controls Band Dispersion? ReO3 yz R (1/2, 1/2, 1/2)

IV. Electronic Structure and Chemical Bonding Hand-Outs: 38 IV. Electronic Structure and Chemical Bonding Populating Antibonding States: Distortions Inorg. Chem. 1993, 32, 1476-1487 t2g Band d2 d3; d5 d6

IV. Electronic Structure and Chemical Bonding Hand-Outs: 39 IV. Electronic Structure and Chemical Bonding NbO: Metal-Metal Bonding J.K. Burdett, Chemical Bonding in Solids, Ch. 4 3 “NbO” per unit cell 33 e Nb-Nb 24 e O 2s + 2p Nb-O

IV. Electronic Structure and Chemical Bonding Hand-Outs: 38 IV. Electronic Structure and Chemical Bonding NbO: Metal-Metal Bonding J.K. Burdett, Chemical Bonding in Solids, Ch. 4 NbO in “NaCl-type” 3 “NbO” per unit cell 33 e 11 e Nb-Nb Nb-Nb 24 e 8 e O 2s + 2p Nb-O O 2s + 2p Nb-O

IV. Electronic Structure and Chemical Bonding Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Electron-Electron Interactions: TB Theory predicts NiO to be a metal – it is an insulator! E = 0 “Higher Potential Energy” Spin-Pairing Energy “Higher Kinetic Energy” Ligand-Field Splitting

IV. Electronic Structure and Chemical Bonding Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Electron-Electron Interactions: E = 0 “Higher Potential Energy” Spin-Pairing Energy “Higher Kinetic Energy” Ligand-Field Splitting EHS  ELS = 22P = 2(P) High-Spin:  < P Low-Spin:  > P

(Independent Electrons) Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 H2 Molecule A B Energy ( > 0) EIE = 2() (Independent Electrons) A

IV. Electronic Structure and Chemical Bonding Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 H2 Molecule Molecular Orbital Approach (Hund-Mulliken; “Delocalized”) A B MO(1,2) = ½ (A1A2 + A1B2 + B1A2 + B1B2) Energy “Covalent” “Ionic” “Ionic” contribution is too large; Poorly describes H-H dissociation ( > 0) EIE = 2() (Independent Electrons) EMO = 2() + U/2 A

IV. Electronic Structure and Chemical Bonding Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 H2 Molecule Valence Bond Approach (Heitler-London; “Localized”) A B VB(1,2) = (A1B2 + B1A2) / 2 Energy “Ionic” contribution is too small; Describes H-H dissociation well ( > 0) EVB = 2 EIE = 2() (Independent Electrons) 0th Order – neglecting 2-electron Coulomb and Exchange Terms A

IV. Electronic Structure and Chemical Bonding Hand-Outs: 40 IV. Electronic Structure and Chemical Bonding Hubbard Model J.K. Burdett, Chemical Bonding in Solids, Ch. 5 Energy If U/ is small: If U/ is large: “Microstates” “Configuration Interaction”