《分子电子结构》研究生课程教学资源（Electronic Structure of Molecules）Lecture 2-1 is a brief review of quantum mechanics

幻灯片2LasttimeCourse overviewModelA1.What is a model?2.Nature?3.Attitudeto use4.Howto evaluate?2Electronicstructurecalculation1.Whatiselectronicstructure calculation?2.Challenges3.Askrelevantquestions4.Advantages vs.disadvantages5.Status quo6.Environment7.ResourcesGaussian16andGaussview63.我们先来回顾一下上次课程的主要内容。1、模型与模拟信息简化和损失-》模型：模型是能够自动将输入转换为输出的“黑匣子”模型的关键是确定描述因子，建立描述因子与所关心性质间的关系，即机理、算法；模型好坏的评价；模型不能滥用；2、计算化学的根本问题预测化学反应要求的化学精度（1kcal/mol）：基团的性质无法迁移，因为性质的叠加会带来巨大的误差，远超化学精度；【还是有人在这条路上尝试，divideandconquer;QM/MM;FragmentMolecularOrbital Method 总能量（Etotal）计算准确（万分之一或者更高），但是反应看能量变化（△E），其精度就很难达到化学精度；计算资源消耗随所计算体系的大小的增加呈指数增加，对于HF或者DFT至少N^3。3、介绍了高斯和GV。安装是否有问题？

幻灯片 2 Course overview 1. Model 1. What is a model? 2. Nature? 3. Attitude to use 4. How to evaluate? 2. Electronic structure calculation 1. What is electronic structure calculation? 2. Challenges 3. Ask relevant questions 4. Advantages vs. disadvantages 5. Status quo 6. Environment 7. Resources 3. Gaussian 16 and Gaussview 6 Last time 2 我们先来回顾一下上次课程的主要内容。 1、模型与模拟 信息简化和损失-》模型； 模型是能够自动将输入转换为输出的“黑匣子” ； 模型的关键是确定描述因子，建立描述因子与所关心性质间的关系，即机理、 算法； 模型好坏的评价； 模型不能滥用； 2、计算化学的根本问题 预测化学反应要求的化学精度（1kcal/mol）； 基团的性质无法迁移，因为性质的叠加会带来巨大的误差，远超化学精度；【还 是有人在这条路上尝试，divide and conquer; QM/MM; Fragment Molecular Orbital Method 】 总能量（Etotal）计算准确（万分之一或者更高），但是反应看能量变化（ΔE）， 其精度就很难达到化学精度； 计算资源消耗随所计算体系的大小的增加呈指数增加，对于HF或者DFT至少 N^3。 3、介绍了高斯和GV。安装是否有问题？

幻灯片3Computational chemistrytryTheoryQuantummechanicClassicalmechanicsAccuracyHighLowStationaryQuantummecics(QM)Molecularmecchanics (MM)stateElectronicstructurecalculationNewtonianahinitiomoleculardvnamicMolecular dynamics (MD)mechanics(AIMD)/first-principlesMDSystemsizeSmallLargeMolecular (intramolecular)Macroscopic (intermolecular)PropertiesHybridsQM/MMReaxFF (QM-basedforcefield+ MD)Semi-empiricalmethodsCoarse-grained methods (e.g-SimplificationDensityfunctional-basedtightDPD)bindinemethodElectronicstructurecalculationInabiggerpicture.上节课更重要的是给出了计算化学的大体分类，如表。名称：电子结构计算、量子化学；分子模拟对于【稳态stationarystate/staticstate】可以采用QM或者MM方法处理。基于解不含时的薛定方程的QM的方法，即【电子结构计算】，主要是处理稳态（包括过渡态），也叫【量子化学QC】，这是本课程主要讲述的范围。【分子力学】也可以处理稳态，但不能处理过渡态。分子力学后面是力场(forcefield)如果对原子核的运动，采用牛顿力学描述，包含时间，QM方法变为AIMDQM/MM还是静力学，处理稳态；ReactiveFF反应力场，处理反应。DPDdissipativeparticledynamics耗散粒子动力学

幻灯片 3 Computational chemistry 3 Molecular simulation/ molecular modelling Electronic structure calculation/quantum chemistry Name Theory Quantummechanics Classical mechanics Accuracy High Low Quantum mechanics (QM)/ Molecular mechanics (MM) Electronic structure calculation Stationary state ab initio molecular dynamics Molecular dynamics (MD) (AIMD)/ first-principles MD Newtonian mechanics System size Small Large Properties Molecular (intramolecular) Macroscopic (intermolecular) ReaxFF (QM-based force field + MD) Hybrids QM/MM Coarse-grained methods (e.g. DPD) Semi-empirical methods; Density functional-based tight binding methods Simplification Electronic structure calculation In a bigger picture. 上节课更重要的是给出了计算 化学的大体分类，如表。 名称：电子结构计算、量子化学；分子模拟 对于【稳态stationary state/static state】可以采用QM或者MM方法处理。 基于解不含时的薛定谔方程的QM的方法，即【电子结构计算】，主要是处理稳 态（包括过渡态），也叫【量子化学QC】，这是本课程主要讲述的范围。 【分子力学】也可以处理稳态，但不能处理过渡态。分子力学后面是力场 （force field） 如果对原子核的运动，采用牛顿力学描述，包含时间，QM方法变为AIMD QM/MM还是静力学，处理稳态；Reactive FF反应力场，处理反应。 DPD dissipative particle dynamics耗散粒子动力学

幻灯片4ContentsBrief review of quantummechanics1.1.Describe wave properties of anelectron;2.Quantum free particle model and its various derivatives;本课内容1、电子的波动性，粒子在势阱中的模型和其变种2、继续学习高斯，计算单点能

幻灯片 4 Contents 4 1. Brief review of quantum mechanics 1. Describe wave properties of an electron; 2. Quantum free particle model and its various derivatives; 本课内容 1、电子的波动性，粒子在势阱中的模型和其变种 2、继续学习高斯，计算单点能

幻灯片5Matterwave:wave-particle dualityE=hyhe-For a photon:E=mc2p=mcPlanck's constalde Broglie Wavelength (1924)h=6.626×10-34ls77278The pilot-wave modelCarFoat10009.1×10-31m (kg)100km/hr0.0102.7 × 10-24p (kg m/s)2.8 ×1042.4 × 10-382.4 × 10-10[入 (m)Too small to detect.Comparableto sizeof atomRemarkClassical object!Must accountforwavepropertiesof an electron!eneDiffraction Nature1999,401,680FlPhthalocvalaninederivatives (CaaH2sF2aNgOg)shNanotechnology2012,Z.29Z.物质波，波粒二象性Louis-VictordeBroglie德布罗意公爵，Langevin的学生，其工作直接引发薛定调方程。还提出了几个重要的思想（wavepacket），有兴趣的同学可以看看他的传记。其对波动方程的解释不同于主流的哥本哈根解释即Born几率解释，他提出了thepilot-wave model, de Broglie-Bohm theory, and the causal interpretation of quantummechanics，(https://plato.stanford.edu/entries/gm-bohm/)。可能对现在研究热门的量子纠缠的理解有帮助。"It is the simplest exampleof what isoftencalleda hidden variables interpretation ofquantummechanics.InBohmianmechanicsasystemofparticlesisdescribedinpartby its wavefunction,evolving,as usual, according to Schrodinger's equation.However,thewavefunctionprovidesonlyapartial descriptionofthesystem.Thisdescriptioniscompletedbythespecificationoftheactualpositionsoftheparticles.Thelatterevolveaccordingtothe“guidingequation",whichexpressesthevelocitiesof the particles in terms of the wavefunction.Thus, in Bohmian mechanics theconfigurationofa system of particles evolves via a deterministicmotionchoreographed by thewave function. In particular, when a particle is sent intoa two-slit apparatus, the slit through which it passes and its location upon arrival on thephotographicplatearecompletelydeterminedbyitsinitial positionandwavefunction."1、其工作受到Planck和Einstein的影响【创新来自新思想的组合！】v=nu。2、宏观物体和微观粒子的主要差异是质量，大概10^34，速度差异宏观物体慢10^5，差异最终由质量主导，约为10^283，最大显示衍射和干涉性质（diffractionandinterfere）的分子weshowhowa

幻灯片 5 Matter wave: wave-particle duality de Broglie Wavelength (1924) Planck's constant For a photon: Car Electron 9.1 ×10−31 m (kg) 1000 v 100 km/hr 0.01 C 2.7 ×10−24 2.8 ×104 p (kg m/s) 2.4 ×10−10 2.4 ×10−38 λ (m) Comparable to size of atom. Must account for wave properties of an electron! Too small to detect. Classical object! Remark 5 The pilot-wave model Phthalocyanine derivatives (C48H26F24N8O8 ) show quantum interference, Nature Nanotechnology 2012, 7, 297. Fullerene Diffraction Nature 1999, 401, 680. 物质波，波粒二象性 Louis-Victor de Broglie 德布罗意公爵，Langevin的学生，其工作直接引发薛定谔 方程。 还提出了几个重要的思想（wave packet），有兴趣的同学可以看看他的传记。其 对波动方程的解释不同于主流的哥本哈根解释即Born几率解释，他提出了 the pilot-wave model, de Broglie-Bohm theory, and the causal interpretation of quantum mechanics, (https://plato.stanford.edu/entries/qm-bohm/)。可能对现在研究热门 的量子纠缠的理解有帮助。 “It is the simplest example of what is often called a hidden variables interpretation of quantum mechanics. In Bohmian mechanics a system of particles is described in part by its wave function, evolving, as usual, according to Schrödinger’s equation. However, the wave function provides only a partial description of the system. This description is completed by the specification of the actual positions of the particles. The latter evolve according to the “guiding equation”, which expresses the velocities of the particles in terms of the wave function. Thus, in Bohmian mechanics the configuration of a system of particles evolves via a deterministic motion choreographed by the wave function. In particular, when a particle is sent into a two￾slit apparatus, the slit through which it passes and its location upon arrival on the photographic plate are completely determined by its initial position and wave function.” 1、其工作受到Planck和Einstein的影响【创新来自新思想的组合！】ν = nu。 2、宏观物体和微观粒子的主要差异是质量，大概10^34，速度差异宏观物体慢 10^5，差异最终由质量主导，约为10^28 3，最大显示衍射和干涉性质（diffraction and interfere）的分子 we show how a

combination of nanofabrication and nano-imaging allows us to record the full two-dimensional build-up of quantum interference patterns in real time forphthalocyanine（酞菁https://pubchem.ncbi.nlm.nih.gov/compound/phthalocyanine#section=3D-Conformer))moleculesandforderivativesofphthalocyaninemolecules,whichhave masses of514AMUand1,298AMUrespectively.4，思考问题：时间与空间是否是连续的还是有最小单位？

combination of nanofabrication and nano-imaging allows us to record the full two￾dimensional build-up of quantum interference patterns in real time for phthalocyanine （酞菁 https://pubchem.ncbi.nlm.nih.gov/compound/phthalocyanine#section=3D￾Conformer） molecules and for derivatives of phthalocyanine molecules, which have masses of 514 AMU and 1,298 AMUrespectively. 4，思考问题：时间与空间是否是连续的还是有最小单位？

幻灯片6Describe wave properties of an electronSchrodinger equation(1926)What Is Life?0-H业ynot aphysicalohserExpressed as differentialequationV(r.r)+V(r.t)甲(r.t)Single particle,non-relativisticP(r.r)Steady-st-independentV(r)allowsthe separatiorw(r)+(r)(r)=E(r)of variables r and TKinetic energy + Potential energy = Total EnergyQuantumharmonicoscillatorAstationarystateisnotmathematicallyconstant甲(r,t)=(r)e%The probability that the particle is at location xiindependentoftim(,)=e/@(2,0)e//(2,0)2=(2,)薛定的波函数。1926年连发4篇文章，单手建立了波动力学，课本的经典描述。1、波动力学描述氢原子；2、量子谐振子，刚性转子，双原子分子；3、证明波动力学与矩阵力学的等价；4、含时波函数，引进复数波函数减少求导次数。但是现在实际上计算中常用的是矩阵力学的表示方式，波动力学的表达多用来教学、解析模型。除了薛定谔的猫以外，最著名的可能是他的一本书ErwinSchrodinger(1944),“What Is Life?:The Physical Aspect of the LivingCell"【http://www.whatislife.ie/downloads/What-is-Life.pdf]:1、其中的生命就是非周期性的晶体（aperiodiccrystal)；2、基因以化学键和分子构型存在；3、基因突变=量子跃迁；4、生命=负熵；5、引发许多物理学家，如JamesD.WatsonFrancisCrick转向生物研究，导致生物物理的发展。学生有LinusPauling【1954化学奖，1962和平奖；后又有Linux的作者linusTorvalds的父母仰慕Pauling】，FelixBloch【1952诺贝尔物理奖】等人。"had an unconventional personal life"1、薛定谔在德布罗意的物质波的思路上将粒子的状态方程以波函数的形式进行了描述。含时波函数表达简洁，左边是波函数psi对时间求导，h-bar是约化

幻灯片 6 = Describe wave properties of an electron Schrödinger equation (1926) Expressed as differential equation: Kinetic energy + Potential energy = Total Energy Steady-state, or time-independent: V(r) allows the separation of variables r and T Single particle, non-relativistic: A stationary state is not mathematically constant The probability that the particle is at location x is independent of time Quantum harmonic oscillator 6 What Is Life? Ψ not a physical observable! 薛定谔的波函数。1926年连发4篇文章，单手建立了波动力学，课本的经典描 述。 1、波动力学描述氢原子； 2、量子谐振子，刚性转子，双原子分子； 3、证明波动力学与矩阵力学的等价； 4、含时波函数，引进复数波函数减少求导次数。 但是现在实际上计算中常用的是矩阵力学的表示方式，波动力学的表达多用来 教学、解析模型。 除了薛定谔的猫以外，最著名的可能是他的一本书Erwin Schrödinger (1944), “What Is Life? : The Physical Aspect of the Living Cell“【http://www.whatislife.ie/downloads/What-is-Life.pdf】： 1、其中的生命就是非周期性的晶体（aperiodic crystal）； 2、基因以化学键和分子构型存在； 3、基因突变=量子跃迁； 4、生命=负熵； 5、引发许多物理学家，如 James D. Watson Francis Crick转向生物研究，导致生 物物理的发展。 学生有Linus Pauling【1954化学奖，1962和平奖；后又有Linux的作者linus Torvalds的父母仰慕Pauling】，Felix Bloch【1952诺贝尔物理奖】等人。 "had an unconventional personal life" 1、薛定谔在德布罗意的物质波的思路上将粒子的状态方程以波函数的形式进行 了描述。含时波函数表达简洁，左边是波函数psi对时间求导，h-bar是约化

planck常数，=h/2pi。i是虚数，右边是哈密顿，能量算符作用于波函数psi上。2、对于一个粒子，且不考虑相对论效应，即其运动速度远小于光速时，含时波函数可以展开为。左侧没有变化，波函数是位置r和时间t的函数。右边的哈密顿算符展开为动能和势能两项。动能项包括波函数对位置的二价导数，√2istheLaplacian拉普拉斯算符，势能项简单的是势能与波函数的乘积。3、如果粒子处于稳态，那么就可以用不含时波函数进行描述。4、【稳态或者不含时波函数不代表波函数是常数，它也随时间进行相位变化。】5、不变或者稳定的是几率，在空间某点的粒子出现几率=稳态波函数的模的平方，这个值不随时间变化。例如量子谐振子的几个状态，如果是稳态，则它的波函数的实部和虚部都随时间变化，而谐振子空间出现几率不变。如果不是稳态，则其几率会随时间变化。-6、为何动能项前面有个负号？一句话解释：波函数含有exp(i·p·r/h)部分，两次求导会产生-1.https://physics.stackexchange.com/questions/9557/why-is-there-a-minus-sign-in-this-wave-equation-derivationWhyisthereaminussigninthiswaveequationderivation(Px=-i-h·o/oxor-inkinetic part ofthe schrodingerequation）?Px=hk==-ih-0/0x, E= p^2/2m = -(h^2/2m)-0^2/0x^2The relative sign is not justa convention.OnceyoudecidethatEisrepresentedbyiho/ot,中(t) =-iEt/hand (r) =i·k·x/h or 中=Φ(r) (t) = exp[i (p*:x -Et)/h)there must be a minus sign in the formula for p, namely p=hk=-iho/oxp. Or viceversa.The second derivative of (r)to rwill producea-1,whichwill cancel the-1 infrontof kinetic term, which is produced by p.The relativeminussign inp,Ep,Emaybecome invisible if you onlyact withsecondderivatives -squared momentum, squared energy-but it is visible if you act with firstpowers of the operators.AevendeeperexplanationDe Broglie's wave-or a waveassociated witha particle-is proportional toexp[i (px -Et)/h]TherelativesignbetweenpxandE·tindeBroglie'sformulaaboveisphysicallynecessarybecauseonlyEt-pxisthecorrectLorentzianinnerproductofthevectors(E,p)and(t,x):therelativeminussigncomesfromtheoppositesignsofspaceandtimeinthesignatureofspacetime

planck常数，=h/2pi。i是虚数，右边是哈密顿，能量算符作用于波函数psi上。 2、对于一个粒子，且不考虑相对论效应，即其运动速度远小于光速时，含时波 函数可以展开为。左侧没有变化，波函数是位置r和时间t的函数。右边的哈密 顿算符展开为动能和势能两项。动能项包括波函数对位置的二价导数，∇ 2 is the Laplacian拉普拉斯算符，势能项简单的是势能与波函数的乘积。 3、如果粒子处于稳态，那么就可以用不含时波函数进行描述。 4、【稳态或者不含时波函数不代表波函数是常数，它也随时间进行相位变 化。 】 5、不变或者稳定的是几率，在空间某点的粒子出现几率=稳态波函数的模的平 方，这个值不随时间变化。 例如量子谐振子的几个状态，如果是稳态，则它的波函数的实部和虚部都随时 间变化，而谐振子空间出现几率不变。如果不是稳态，则其几率会随时间变 化。 6、为何动能项前面有个负号？一句话解释：波函数含有exp(i⋅p⋅r/ℏ)部分，两次 求导会产生-1. https://physics.stackexchange.com/questions/9557/why-is-there-a-minus-sign-in￾this-wave-equation-derivation Why is there a minus sign in this wave equation derivation （Px = -i·ℏ·∂/∂x or – in kinetic part of the schrodinger equation）? Px =ℏk== -iℏ·∂/∂x, E = p^2/2m = - (ℏ^2/2m)·∂^2/∂x^2 The relative sign is not just a convention. Once you decide that E is represented by iℏ∂/∂t, ψ(t) = -iEt/ℏ and ψ(r) = i⋅k⋅x/ℏ or ψ=ψ(r) ψ(t) = exp[i (p⃗ ⋅x⃗ −Et)/ℏ] there must be a minus sign in the formula for p, namely p=ℏk=−iℏ∂/∂xp. Or vice versa. The second derivative of ψ(r) to r will produce a -1, which will cancel the -1 in front of kinetic term, which is produced by p. The relative minus sign in p, E·p, E may become invisible if you only act with second derivatives - squared momentum, squared energy - but it is visible if you act with first powers of the operators. A even deeper explanation De Broglie's wave - or a wave associated with a particle - is proportional to exp[i (p⃗ ⋅x⃗ −Et)/ℏ] The relative sign between p⃗ ⋅x⃗ and E⋅t in de Broglie's formula above is physically necessary because only Et−p⃗ ⋅x⃗ is the correct Lorentzian inner product of the vectors (E,p⃗) and (t,x⃗ ): the relative minus sign comes from the opposite signs of space and time in the signature of spacetime

幻灯片7Wavefunctions allowed inquantummechanics1.Single-valued;2.Continuous;Nowhere infinite (exception:3.入Dirac delta-function)aPiecewise continuous firstderivatives;5Square-integrable(exception:plane wave)1, 3, 5 required by Borninterpretation;2 and 4 required by kineticoperator (2nd derivative of w)品优波函数Physicallyallowed各种书本上具体的要求不尽相同。Functionsthat aresingle-valued,continuous,nowhereinfinite,andhavepiecewise continuous first derivatives will be referred to as acceptable functions.Inmostcases,thereisonemoregeneral restrictionsquare-integrable orquadraticallyintegrable(exceptionplanewave).it maybe infinite overan infinitesimal range for sucha function is square-integrable (it corresponds to a Dirac delta-function).Kronecker delta oij =1 i=j; =0j.Itmusthaveacontinuousfirstderivative,exceptat ill-behaved regionsofthepotential.Noticeg and h haveas coordinate

幻灯片 7 Wave functions allowed in quantum mechanics 7 1. Single-valued; 2. Continuous; 3. Nowhere infinite (exception: Dirac delta-function); 4. Piecewise continuous first derivatives; 5. Square-integrable (exception: plane wave) 1, 3, 5 required by Born interpretation; 2 and 4 required by kinetic operator (2nd derivative of Ψ) 品优波函数Physically allowed 各种书本上具体的要求不尽相同。 • Functions that are single-valued, continuous, nowhere infinite, and have piecewise continuous first derivatives will be referred to as acceptable functions. In most cases, there is one more general restriction square-integrable or quadratically integrable (exception plane wave). • it may be infinite over an infinitesimal range for such a function is square￾integrable (it corresponds to a Dirac delta-function). Kronecker delta σij = 1 i=j; =0 i≠j. • It must have a continuous first derivative, except at ill-behaved regions of the potential. • Notice g and h have θ as coordinate

幻灯片8Kineticenergyvs.potentialenergyWAN(alSincewieely.whichthat 入isV = O.E=TFor hieher Th is mshorter (Since is periodic for a free particle, A is defined.) (b) As Vincreases from left to right, becomes less wiggly. (c)-(d) is most wigglywhere V is lowest and T is greatest.动能vs势能1、对psi的二阶导数反应了psi的曲率，即曲线斜率的变化率。the second derivative of ywith respect to a given direction isa measureof therateofchangeofslope（i.e.,the【curvature曲率】，or【“wiggliness”波动性、摆动性】)of in that direction.2、势能为零，动能随着总能量增加而增加，波函数变得更加波动频繁。波长变短。Hence, we see that a more wiggly wavefunction leads, through the Schrodingerequation,toahigherkineticenergy.This isinaccordwiththespirit ofdeBroglie'srelation, since a shorter wavelength function is a more wiggly function. But theSchrodinger equation is more generally applicable because we can take secondderivativesofanyacceptablefunction,whereaswavelengthisdefinedonlyforperiodicfunctions.3、b图，注意纵坐标变了，总能量守恒，势能增加，动能下降，【右侧】波函数的波动性下降。Since E is a constant, the solutions ofthe Schrodinger equation must bemore wigglyin regions where Vis low and less wiggly where Vis high. Examples for some one-dimensional cases are shown in Fig.4、C-d图，总能守恒，势能最低处，动能最大，波函数波动剧烈；反之亦然。dcorrespondstoquantumharmonicoscillator

幻灯片 8 Kinetic energy vs. potential energy 8 (a) Since V = 0, E = T . For higher T , ψ is more wiggly, which means that λ is shorter. (Since ψ is periodic for a free particle, λ is defined.) (b) As V increases from left to right, ψ becomes less wiggly. (c)–(d) ψ is most wiggly where V is lowest and T is greatest. 动能vs势能 1、对psi的二阶导数反应了psi的曲率，即曲线斜率的变化率。 the second derivative of ψ with respect to a given direction is a measure of the rate of change of slope (i.e., the 【curvature曲率】, or【“wiggliness” 波动性、摆动 性】) of ψ in that direction. 2、势能为零，动能随着总能量增加而增加，波函数变得更加波动频繁。波长变 短。 Hence, we see that a more wiggly wavefunction leads, through the Schrodinger equation, to a higher kinetic energy. This is in accord with the spirit of de Broglie’s relation, since a shorter wavelength function is a more wiggly function. But the Schrodinger equation is more generally applicable because we can take second derivatives of any acceptable function, whereas wavelength is defined only for periodic functions. 3、b图，注意纵坐标变了，总能量守恒，势能增加，动能下降，【右侧】波函数 的波动性下降。 Since E is a constant, the solutions of the Schrodinger equation must be more wiggly in regions where V is low and less wiggly where V is high. Examples for some one￾dimensional cases are shown in Fig. 4、c-d图，总能守恒，势能最低处，动能最大，波函数波动剧烈；反之亦然。 d corresponds to quantum harmonic oscillator

幻灯片9Quantization,energylevels anddegeneracy:ananalogyofstablestatesofachairnormalpositioneoon the sideThedegreeofdegeneracyequalstwoinclinedaonthesupportDifferentdeCy ofenergylevels lead todensityofstates (DOS);Using k as x axis leads to band structureArranging x axis according to reaction leads to reaction pathway量子化-》能级-》简并态我们先来通过一个宏观的例子，来说明几个概念：构型，能级，稳态。如图所示的一把普通靠背椅放在地上，它有多少种放置方式？其中有几个稳定的放置方式？1、每一个椅子与地面的相对位置和角度的摆放，我们叫做一个构型configuration，构型理论上有无穷多个。2、每个构型都有自己的势能，由椅子重心离地面的高度决定，我们把每一个势能叫做一个能级，energylevel。3、我们稍微想一下就可以知道，不是每个构型都能够稳定存在，而且稳定的构型的数量很少。如图所示，只有五种稳定构型，我们把他们叫做稳态【stable/steady/staionarystates】。其中能量最低的是靠背贴地的放置构型，也叫基态groundstate，其他能量高的状态我们称为激发态，excitedstates。侧放的椅子有两种构型，它们的能级完全相同，它们被叫做简并态，degeneratestates。我们坐在椅子上时，椅子的稳态反而是能量最高的一个激发态。4、如果将椅子从一个稳态放置为另一个稳态，多数情况下椅子需要经过能量相对更高的不稳定构型，即过渡态。5、如果我们将构型以某种方式组织，形成横坐标，以其势能为纵坐标，我们可以得到所谓的椅子的势能面。每一个构型，不论稳定与否都是势能面上的一个点

幻灯片 9 Quantization, energy levels and degeneracy: an analogy of stable states of a chair on the support Potential energy of the chair inclined The degree of degeneracy equals two on the side normal position 9 • Different degrees of degeneracy of energy levels lead to density of states (DOS); • Using k as x axis leads to band structure • Arranging x axis according to reaction leads to reaction pathway 量子化-》能级-》简并态 我们先来通过一个宏观的例子，来说明几个概念：构型，能级，稳态。如图所 示的一把普通靠背椅放在地上，它有多少种放置方式？其中有几个稳定的放置 方式？ 1、每一个椅子与地面的相对位置和角度的摆放，我们叫做一个构型 configuration，构型理论上有无穷多个。 2、每个构型都有自己的势能，由椅子重心离地面的高度决定，我们把每一个势 能叫做一个能级，energy level。 3、我们稍微想一下就可以知道，不是每个构型都能够稳定存在，而且稳定的构 型的数量很少。如图所示，只有五种稳定构型，我们把他们叫做稳态 【stable/steady/staionary states】。其中能量最低的是靠背贴地的放置构型，也 叫基态ground state，其他能量高的状态我们称为激发态，excited states。侧放 的椅子有两种构型，它们的能级完全相同，它们被叫做简并态，degenerate states。我们坐在椅子上时，椅子的稳态反而是能量最高的一个激发态。 4、如果将椅子从一个稳态放置为另一个稳态，多数情况下椅子需要经过能量相 对更高的不稳定构型，即过渡态。 5、如果我们将构型以某种方式组织，形成横坐标，以其势能为纵坐标，我们可 以得到所谓的椅子的势能面。每一个构型，不论稳定与否都是势能面上的一个 点