5.04 Principles of Inorganic Chemistry II
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1 MIT OpenCourseWre Principles of Inorgnic Chemistry II Fll 2008 For informtion bout citing these mterils or our Terms of Use, visit:
2 5.04, Principles of Inorgnic Chemistry II Prof. Dniel G. Nocer Lecture 9: Bnd Theory in Solids The LCAO method for cyclic systems provides convenient strting point for the development of the electronic structure of solids. At very lrge N, s the circumference of the circle pproches, the cyclic problem converges to liner one, Qulittively, from MO energy level perspective, Prof. Dniel G. Nocer Pge 1 of 10
3 More quntittively, in moving from cyclic to liner systems, insted of describing orbitl (tom) positions ngulrly, the position of n tom is described by m, where m is the number of the tom in the rry nd is the distnce between toms. Thus, the θ of the N-cyclic derivtion becomes m, ψ j = e i jθ φ m m ψ k = e ik(m) φ m m E j = α + 2β cos 2 j E k = α + 2β cos 2 j N N multiplied by / = α + 2β cosk 2 j where k= N A few words bout k. It is: mesure of the number of nodes n index of wvefunction nd ccordingly symmetry of wvefunction quntum number for given ψ k mesure of length, relted to wvelength λ 1 from DeBroglie s reltion, λ = h, therefore k is lso wve vector tht mesures momentum p Returning to the foregoing discussion, note tht k prmetriclly depends on. Since is lttice prmeter of the unit cell, there re s mny k s s there re unit cells in the crystl. In the liner cse, the unit cell is the distnce between djcent toms: there re n toms n unit cells or in other terms there re s mny k s s toms in the 1-D chin. Let s determine the energy vlues of limits, k = 0 nd k = : t k = 0: ψ 0 = e i0(m) φ m = φ m m m t k = : ψ = e φ m = e im φ m = ( 1) m φ m n m m i (m) Prof. Dniel G. Nocer Pge 2 of 10
4 The energies for these bnd structures t the limits of k re: E 0 = α + 2β cos(0) = α + 2β E = α + 2β cos( ) = α 2β Note tht k is quntized; so there re finite number of vlues between α+2β nd α 2β but for very lrge number (~10 23 toms) between the limits of k. Thus, the energy is continuous nd smoothly vrying function between these limits. The rnge k or k is unique becuse the function repets itself outside these limits. This unique rnge of k vlues is clled the Brillouin zone. The first Brillouin zone is plotted bove from 0 to (symmetric reflection from to 0). With given number of e s in the solid, the levels will be filled to certin energy clled the Fermi level, which corresponds to certin vlue of k (= k F ). In the k bove exmple, there re more electrons thn there re orbitls, so k F >. If 2 ech tomic orbitl contributed 1e to the system, then E F would occur for k F =. 2 Prof. Dniel G. Nocer Pge 3 of 10
5 The symmetry of the individul tomic orbitls determines much bout bnd structure. Consider p-orbitls overlpping in liner rry (vs the 1s orbitls of the bove tretment). Anlyzing limiting forms: t k = 0: ψ 0 = φ 1 + φ 2 + φ 3 + φ 4 + NOTE: This is the highest + energy orbitl with N nodes t k = : ψ = φ 1 φ 2 + φ 3 φ 4 + NOTE: Lowest energy + orbitl with N nodes The energy bnd is opposite of tht for the sσ orbitl LCAO becuse the (+) LCAO for pσ orbitl is ntibonding. Thus, molecules re esily relted to solids vi Hückel theory. Not surprisingly, there is lnguge of chemistry describing the electronic structure of molecules tht is relted to the lnguge of physics describing the electronic structure of solids. Below re some of the terms tht chemists nd physicists use to describe similr phenomen in molecules nd solids: Prof. Dniel G. Nocer Pge 4 of 10
6 LCAO MO tight-binding model moleculr orbitl crystl or bnd orbitl Jhn Teller distortion Peierls distortion high or intermedite spin mgnetic low spin non mgnetic Bnd Width or Dispersion Wht determines the width or dispersion of bnd? As for the HOMO-LUMO gp in molecule, the overlp of neighboring orbitls determines the energy dispersion of bnd the greter the overlp, the greter the dispersion. Note how the bnd dispersion of liner chin of H toms vries s the 1s orbitls of the H toms re spced 1, 2, 3 Å prt (E of n isolted H tom is 13.6 ev): Prof. Dniel G. Nocer Pge 5 of 10
7 Density of Sttes The totl energy of the system is, E totl = N 2 kf k F 2E(k)dk = 2N normliztion constnt 0 k F E(k)dk or in other words, it is the re under the curve to k F. Another useful quntity is the number of orbitls between E(k) + de(k), clled the density of sttes (DOS). For 1-D system, inversely proportionl to slope DOS 1 of E(k) curve t given k. Since n(e(k)) E( k) k slope t k = 0 nd k = / pproches 0, n(e(k)) is lrge A plot of the bove equtions is, DOS In the bove DOS digrm, no energy gp seprtes the filled nd empty bnds, i.e. there is continuous density of sttes this property is chrcteristic of metl. If n energy gp between filled nd empty orbitls is present nd it cn be thermlly surmounted, then it is semiconductor; n energy gp tht cnnot be surmounted is n insultor. A 1-D Exmple Argubly the best known 1-D system in inorgnic chemistry is K 2 Pt(CN) 4 nd its prtilly oxidized compound (e.g. K 2 Pt(CN) 4 Br 0.3 ). Prof. Dniel G. Nocer Pge 6 of 10
8 Norml pltinocynide, K 2 Pt(CN) 4 : n insultor, ple yellow compound Prtilly oxidized pltinocynide, K 2 Pt(CN) 4 Br 0.3 3H 2 O: long chin: conductor ( Ω 1 cm 1 ) copper-colored with metllic luster to chin: n insultor ( Ω 1 cm 1 ), yellow compound Note: d(pt-pt) = 2.78 Å in Pt metl To explin these disprte properties of the 1-D compounds, consider the moleculr subunit Pt(CN) 4 2- : for d 8 system vlence bonds completely filled the energy gp between the M-Lσ derived vlence bnd nd the M-Lσ* derived conduction bnd is too lrge to be thermlly populted, thus n insultor Prof. Dniel G. Nocer Pge 7 of 10
9 The dispersion of the bnds is due to the different overlps of the dσ, d nd dδ orbitls. Bnd structure (or first Brillouin zones) derived from the frontier MO s is: For prtilly oxidized system, the σ bond derived from d z 2 should be prtilly filled nd thus metllic, but it is not, prtilly oxidized K 2 Pt(CN) 4 Br x is semiconductor. To explin this nomly, consider how the bnd structure is perturbed upon prtil oxidtion: Prof. Dniel G. Nocer Pge 8 of 10
10 Isolting on the d z 2 bnd in K 2 Pt(CN) 4, the Pt toms re evenly spced with lttice dimension (I). Upon oxidtion, the Pt chin cn distort to give lttice dimension 2 (II). In the cse of the K 2 Pt(CN) 4 Br 0.3 3H 2 O the distortion is rottion of Pt subunits nd formtion of dimers within chin, thus the unit cell dimension is pinned to every other Pt tom. For K 2 Pt(CN) 4 (I): For K 2 Pt(CN) 4 Br 0.3 3H 2 O (II): Thus Brillouin digrm distorts Color code red: for prtilly occupied d z 2 bnd with no distortion boundry condition for K 2 Pt(CN) 4 boundry condition for K 2 Pt(CN) 4 Br 0.3 3H 2 O Color code qu mrine: 2 for prtilly occupied d z bnd with distortion or pictorilly Prof. Dniel G. Nocer Pge 9 of 10
11 The Brillouin digrm bove is for distorted K 2 Pt(CN) 4 but filled d z 2 orbitl. Hence both qu mrine bnds re filled nd the mteril tht thus possesses the electronic structure shown below is insulting, despite the distortion Upon prtil oxidtion of K 2 Pt(CN) 4 to produce K 2 Pt(CN) 4 Br 0.3 3H 2 O, the d z 2 bnd is now prtilly filled. Hence conductive mteril results. The conductivity is long the Pt-Pt xis, explining the electricl nisotropy of the mteril Prof. Dniel G. Nocer Pge 10 of 10
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