5.4紧束缚模型（tight-bindingmodel)、定性说明、微扰计算三、具体例子参考：黄昆书4.5节p189阎守胜书3.3节p75和近自由电子近似认为原子实对电子的作用很弱相反，本节，我们假定原子实对电子的束缚作用很强，因此，当电子距某个原子实比较近时，电子的运动主要受该原子势场的影响，受其它原子势场的影响很弱。因此固体中电子的行为同孤立原子中电子的行为更为相似。这时可将孤立原子看成零级近似，而将其他原子势场的影响看成小的微扰，由此可以给出电子的原子能级和晶体能带之间的相互联系。这种方法称为紧束缚近似 (Tight Binding Approximation)
5.4 紧束缚模型 (tight-binding model) 一、定性说明 二、微扰计算 三、具体例子 参考：黄昆书4.5节p189 阎守胜书3.3节p75
一.定性说明：FIGURE24-13(a)TWOXBfinite potential wells B and (Ca)Eigenfunction associated with an没有相互作用的双势阱(b)clectron in well B with the groundXcstate energy, (c) Eigenfunctionassociated with an clectron in wel G(c)with the ground state energy.x,=a(xB+x)对称本征波函数FIGURE24-14(a)Symmetriceigentunction representing anclcctron that can be found with equlprobability in the two wellsofFig.2413 with the ground stateenergy (0)Probabiliry density associated with反对称本征波函数the symmeric cigenfunction in (a)(6)XA=a(XB-X)FIGURE24-15Antisymmetrecigentunction representing anelectron that can be found withequal probability in the two wellsof Fig, 24-13 with thc groundstate cnergy. (b) Probability densinyassociated with the antisymmetnccigenfunction of (e)
一 . 定性说明： 没有相互作用的双势阱 xB xC xs=a(xB+xC) 对称本征波函数 xs2 反对称本征波函数 xA=a(xB-xC) xA2
势阱靠近并产生相互作用对称本征波函数（成键态）XS=a(XB+XC)(b)(a)FIGURE 24-16When the two wells are very close together, the symmetric cigenfunction ofFig,24-14 looks like the ground state eigenfunction fora finite well of width 2a (see Fig.24-12a)反对称本征波函数（反键态）(b)FIGURE 24-17When the wells are very closetogether,the antisymmetric cigenfunction ofFig.24-15looks like thefirst excited state ofa finitewell of width2a (see Fig,24-12b)
势阱靠近并产生相互作用 对称本征波函数（成键态） 反对称本征波函数（反键态）
紧束缚模型：体系波函数是原子波函数的线性组合-Linearcombinationofatomicorbitals(LCAO)首先忽略电子之间相互作用，其次采用单电子近似V?+VA,+VBiJV,=6,VHy,2m则猜测波函数为两个原子的线性组合：, =C,[PA(r,)+ A,PB(r))=±1若两个波函数相等，可以解出JyHydr2C?(H.. +Hab)V+=C+(PA+PB)Jyiy.drV_ =C(PA-PB)[y'Hy_dr= 2C2(Hg - Hab)Jy'y.dr
紧束缚模型：体系波函数是原子波函数的线性组合 - Linear combination of atomic orbitals (LCAO) 首先忽略电子之间相互作用，其次采用单电子近似 2 i VAi VBi i i i m H ] 2 [ 2 则猜测波函数为两个原子的线性组合： ( ) ( ) i i A i i B i C r r 则猜测波函数为两个原子的线性组合： ( ) ( ) i i A i i B i C r r 若两个波函数相等，可以解出 1 ( ) C A B 2 ( ) 2 * * C Haa Hab dr H dr ( ) C A B 2 ( ) 2 * C H H b H dr dr 2 ( ) * C Haa H a b dr
Hydr=2C?(Ha。+Hab）成键轨道，对称波函数Jyy.drBonding orbitalJy'Hy_dr= 2C?(H..-H.b)反键轨道，反对称波函数-Jyy.drAnti-bondingorbitalHaa =[,HPadr =HPedr 80Hab=Jo,Hodr<0H表征了电子受到两个原子核的库仑相互作用，与两个原子波函数重叠成正比，这个波函数的重叠也被称为重叠积分，表征了共价键中相互作用的强弱。两个电子同时占据成键轨道，因此能量得以下降原子轨道线形组合方法（LCAO）是化学学科常用说法，其物理思想本质和固体物理中的紧束缚方法（tight-binding)方法是完全一致的
2 ( ) 2 * * C Haa Hab d H dr 成键轨道，对称波函数 2 ( ) 2 * C H H H dr dr Bonding orbital 2 ( ) 反键轨道 反对称波 数 2 * C Haa Hab dr 反键轨道，反对称波函数 Anti-bonding orbital * 0 * * H H dr H dr aa A A B B 0 * H H dr ab A B Hab 表征了电子受到两个原子核的库仑相互作用，与两个原子波函数重叠成正 比，这个波函数的重叠也被称为重叠积分，表征了共价键中相互作用的强 弱。 两个电子同时占据成键轨道，因此能量得以下降 原子轨道线形组合方法（LCAO）是化学学科常用说法，其物理思想本质 和固体物理中的紧束缚方法（tight-binding)方法是完全一致的
常用原子轨道基组形式：Slater基组与Gaussian基组d(r)=e"αr2d(r)=e"αrGaussian-type1sorbitalSlater-type 1s orbital
常用原子轨道基组形式：Slater基组与Gaussian基组
Slater基组双原子成键与反键轨道.+)FIGURE 24-18Radial part of the ground state eigenfiunctions of rwo isolated hydrogen atomsas a function of the distance of the electrons from the respective nuclei.Ns=am+Xc)(a)a(Xg 3c)(b)FIGURE24-20(a)Probabilitydensity associated with the symmetriccigenfunction of Fig. 24-19a, (6)Probability density associated withthe antisymmetric cigenfunction of(b)Fig. 24-196. Note that x2 is large imthe region berween the two nuclei soFIGURE24-19 (a)Symmetrictherefore the electron represented bycombination of the ground stateX spends onnsiderahle rime hetweencigenfunctionsof the twoindividuaboth nuclei. Xa does not have thishydrogen atoms of Fig, 24-18. (h)featureAntisymmetric combination of thesame two cigenfunctions
Slater基组双原子成键与反键轨道
六个原子轨道的线性组合6th leve(Highest E)5thleFIGURE 24-21Six possible2nd 1combinations of theground statecigenfunctions of six hydrogenatoms,each corresponding to.aIst levedifferent energystate,Ar the bottom(Lowest E)of the figure the six individualeigenfunctions are sketched. Thesketches above them represent sixpossible ways of adding them cithersymmetrically or antisymmetrically
六个原子轨道的线性组合
Splitting of 1s State of Six Atomsls stateInteratomicseparationFIGURE24-22 Splittingofthe Is state of six hydrogen atomsintoaband of sixenergylevels astheseparationbetween theatomsdecreases. Note the increase inthe bandwidth with decreasinginteratomicseparation
Splitting of 1s State of Six Atoms Splitting of 1s State of Six Atoms
Splitting of Atomic Levels in SodiumTight-binding模型Na原子形成晶体的原理3s3sband2pband202sband28FIGURE24-23Expectedsplittingofthefirstfouratomiclevel1sbandofsodiumintofourenergybandsi18SolidasodiumcrystalAtom
Splitting of Atomic Levels in Sodium Splitting of Atomic Levels in Sodium Tight-binding 模型Na原 体的原 子形成晶体的原 理