Lecture 6. Bonds to Bands. But for most problems we use same approximation methods: 6. The Tight-Binding Approximation

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1 6. The Tight-Bidig Approximatio or from Bods to Bads Basic cocepts i quatum chemistry LCAO ad molecular orbital theory The tight bidig model of solids bads i 1,, ad 3 dimesios Refereces: 1. Marder, Chapters 8, pp Kittel, Chapter 9, pp Ashcroft ad Mermi, Chapter 8 4. R. Hoffma, Solids ad Surfaces: A chemists view of bodig i exteded structures" VCH, 1988, pp 1-55, P.A. Cox, The Electroic Structure ad Chemistry of Solids, Oxford, 1987, Chpts. 1, (skim), 3 (esp. 45-6), ad 4 (esp. Lecture 79-88). 5 1 Bods to Bads Forces i solids Covalet (e.g., Si, C ) Ioic (e.g., NaCl, MgO ) Metallic (e.g., Cu, Na, Al ) Molecular (weak) (e.g., N, bezee) Forces betwee atoms (chemistry) Covalet Polar covalet Ioic Weak (Lodo/dispersio, dipole) Lecture 5 But for most problems we use same approximatio methods: Two mai approaches i electroic structure calculatios: Build up from atom: atomic orbitals + Ifiite solid dow: plae waves + Two methods used i quatum mechaics are variatioal ad perturbatio theory methods Perturbatio theory: H = H o + λ H 1 + λ H + H o Ψ o = E o Ψ o (ofte kow) 3 1

2 Variatioal Method Variatioal Method Use matrix otatios Ψ = Ψ> Η Ψ> = Ε Ψ> (time idep. Sch. Eq.) For statioary stats, if Ψ is ormalized, well-behaved fuctio, it ca be show that < Ψ * Η Ψ> = < Ψ * Ε Ψ> = Ε <Ψ * Ψ> (variatioal itegral) Miimize variatioal itegral to get groud state Ψs Ψ* H Ψ E = (overlap itegral) Ψ* Ψ 4 Liear Combiatio of Atomic Orbitals (LCAO) Best to start to solve Sch. eq. by choosig good ψ s! ψ = Ay ψ ca be expaded from set of orthoormal fuctios φ (satisfyig appropriate boudary coditios) c φ I quatum physics/chemistry ofte use hydroge-like atomic orbitals φ (s, p, d ) to study molecules (molecular ψ s) This approach is called LCAO liear combiatio of atomic orbitals 5 LCAO Geeral problem is to miimize variatioal itegral by fidig coefficiets c that make variatioal itegral statioary: c = 0! E HΨ = EΨ H ( H E) c φ = E c φ = 0 c ( Hφ Eφ ) = 0 * * * c( φmhφdv φm c[ H m Eδ m ] = 0 c[ H ES ] = 0 m m c φ * Eφ dv ) = 0 if φ m orthoormal, if ot 6

3 Molecular system H + Most simple molecular system H + (molecular io with oe electro) Hˆ elec h e e e e = m ria rib r r1 ki Elect.-uc. attract. ucl repulsio fro H electroic or rep. larger systems For trial fuctios choose simple 1s atomic orbitals ψ = c φ + c φ S S i H H Ai = φ = φ = φ = 1 ISA * ISA * ISB * ISB H φ H φ φ ISA Bi ISB ISA ISA = S = H = H BA BB BA 7 Bods to Bads Determiat becomes: H E H ES H ES BB H E = 0 ( H (1 S E)( H ) E + (H E) ( H ES BB H ) = 0 ) E + ( H H ) = 0 Solve quadratic eqs ad get solutios: H H S + H H S E = (1 S )(1 + S ) H H S H + H S E = (1 S )(1 + S ) H + H E1 = ( + ) 1+ S H H E = (-) 1 S + sol' sol' Eergy diagram: β α 1 ψ 1 = ( φisa + φisb ) 1 ψ = ( φisa φisb) E E 1 8 Defiitios ψ 1 is called bodig orbital ψ is called ati-bodig orbital Remember: each orbital ca have electros For + electro system must add e-e repulsio + screeig of core effective potetial e.g., Hydroge 9 3

4 Nitroge 10 Correlatio Diagrams Homouclear diatomics Heterouclear diatomics 11 Some symmetry cosideratios Stregth of bod due to: - overlap of wavefuctios i space - iv. prop. to eergy differece of o-iteractig orbitals - electro + core repulsios 1 4

5 Exteded Hückel Theory - EHT Quatum chemical approximatio techique (semi-empirical) Study oly valece (i.e., bodig ) orbitals Solve the problem with LCAO variatioal method, but usig ideas from perturbatio theory The iitial state orbitals are atomic Slater type orbitals STOs X = Ar i 1 exp 0 Y ( θ, ϕ) The iitial state eergies, H ii are take from experimets or some other calculatios 5r lm 13 Exteded Hückel Theory - EHT Ituitive idea is that the stregth of a bod, as represeted by the offdiagoal matrix elemets Hij, should be proportioal to the extet of overlap (Sij) ad the mea eergy of the iteractig orbitals(hii+hjj)/ H Hii + H = K S jj ij K is a empirical parameter betwee 1 ad Problems: - H ij is ot exact - o charge self cosistecy - H ij icreases with overlap causig system to show miimum eergy whe atoms collapse - o core-core repulsio ij But EHT does usually show qualitative correct treds i a easily iterpretable maer 14 Orbitals ad Bads i 1D 1. Equally spaced H atoms. Isomorphic p-system of a o-bod-alteratig delocalized polyee 3. Chai of Pt II square-plaar complexes 15 5

6 16 Bad Dispersio or bad width The bad structure of a chai of hydroge atoms spaced 3,, ad 1 Å apart. The eergy of a isolated H atom is eV 17 Desity of States (DOS) Cocept E(k) curve, the bad structure, has a simple cosie curve shape DOS(E) is proportioal to the iverse of the slope of E(K) vs k; the flatter the bad, the greater the desity of states at that eergy 18 6

7 s vs p orbitals For s orbitals, the iteractio itegral is egative (lowerig eergy) Total eergy spa from -β to +β total width is 4β, which is proportioal to the degree of iteractig betwee eighbourig atoms Geeral tred: strogly overlappig orbitals give large values of β, ad wide bads, where as cotracted atomic orbitals that overlap poorly give rise to arrow bads Eergy as a fuctio of k for bads of s Liear combiatio p σ orbitals for k = 0 ad pσ orbitals i a liear chai (atibodig) ad k= ±π/a (bodig) 19 Examples (a) E(k) curve showig the allowed k values for a chai with N = 8 atoms; (b) Orbital eergies for eight-atom chai, showig clusterig at the top ad bottom of the bad; (c) Desity of states for a chai with very large N X-ray photoelectro spectrum of the log chai alkae C 36 H 74, showig the desity of states i the s bad. (J.J.Pireau et.al.,phys.rev.a,14 (1976), 133.) 0 Acroyms of curret experimetal methods for bad structure studies AES EELS EXAFS IPS STM(S) Vis(UV) abs Auger electro spectroscopy Electro eergy loss spectroscopy Exteded x-ray absorptio fie structure Iverse Photoemissio Spectroscopy Scaig Tuelig Microscopy (Spectroscopy) Vis./ UV absorptio Photo (electro) i, electro out, ier shells Electro i/out; Coductio electros Photo i; filled bads Electro i, photo out; Empty Levels Electros i, coductio ad valece bads Photo i; bad gap, defects UPS XPS XAS Ultraviolet photoelectro spectroscopy X-ray photoelectro spectroscopy X-ray absorptio Photo i, electro out, filled bads Photo i, electro out, filled bads Photo i, empty levels Kittel, p modified 1 7

8 Spectroscopic techiques (a) optical absorptio i the visible /UV rage; (b) photoelectro spectroscopy; (c) iverse photoelectro spectroscopy; (d) X-ray absorptio; (e) X-ray emissio. Electros with eergies above the vacuum level ca eter or leave the solid; i techiques (b) ad (c) the scale shows the kietic eergy measured i a vacuum outside the solid. Crystal Orbital Overlap Populatio (COOP) Better tha DOS for determiig extet of bodig ad ati-bodig iteractios 3 p ad d orbitals i liear chai p ad d orbitals (i additio to s) for liear chai 4 8

9 A polymer - liear array of PtH 4 molecules Moomer-moomer separatio ~ 3 Å The major overlap betwee d z ad p z orbitals 5 Bad structure ad DOS for PtH 4 - The DOS curves are broadeed so that the two-peaked shape of the xy peak i the DOS is ot resolved 6 Bucklig of chai Bucklig of 1D chai to miimize E (periodicity chages - doubles!) 7 9

10 Eergy miimizatio for 1D chai - Peierls istability Solid-state chemistry aalog of Jah-Teller effect 8 Peierls Distortio Qualitative electroic structure ad eergy miimizatio of cyclobutadiee Symmetry breakig 9 LCAO theory for D array of s orbitals The simplest possible D crystal cosists of atoms of the same kid arraged o a square lattice, with spacig a. I the LCAO approach, we eed to fid crystal orbitals i the form: ψ = c r, sχ r, s r, s where the atomic orbital coefficiets c r,s are determied by the periodicity of the lattice cr, s = exp( irkxa + iskya) (a) The bad structure ad (b) DOS of a square lattice of Lecture H atoms

11 D array of p orbitals p z orbitals will give a bad structure similar to that of the s orbitals (topology of the iteractio of these orbitals is similar) Each crystal orbital ca be characterized by σ or π bodig preset E.g., at Γ the x ad y combiatios are σ atibodig ad π bodig 31 Qualitative Bad Structure Schematic bad structure of a plaar square lattice of atoms bearig s ad p orbitals The s ad p levels have a large separatio that the s ad p bad do ot overlap 3 The electroic structure of graphite P z orbitals for differet wave-vectors i graphite