《分子电子结构》研究生课程教学资源（Electronic Structure of Molecules）ESM-2-revQM

LasttimeCourse overview1. Model1.What is a model?2.Nature?3.Attitude to use4.How to evaluate?2.Electronic structurecalculation1.Whatiselectronic structurecalculation?2.Challenges3.Askrelevantquestions4.Advantagesvs.disadvantages5.Status quo6.Environment7.Resources3.Gaussian16andGaussview62

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

Computational chemistryNameElectronic structureMolecular simulationcalculation/quantum chemistrymolecularmodellingTheoryQuantummechanicsClassical mechanicsHighLowAccuracyStationaryQuantummechanics(QM)/Molecularmechanics (MM)stateElectronic structure calculationNewtonianab initiomoleculardynamicsMoleculardynamics (MD)mechanics(AIMD)/first-principlesMDSmallSystemsizeLargePropertiesMacroscopic(intermolecular)Molecular (intramolecular)HybridsQM/MMReaxFF(QM-basedforcefield+ MD)SimplificationCoarse-grained methods (e.g.Semi-empiricalmethodsDPD)Densityfunctional-basedtightbinding methods3

Computational chemistry 3 Name Electronic structure calculation/quantum chemistry Molecular simulation/ molecular modelling Theory Quantum mechanics Classical mechanics Accuracy High Low Stationary state Quantum mechanics (QM)/ Electronic structure calculation Molecular mechanics (MM) Newtonian mechanics ab initio molecular dynamics (AIMD)/ first-principles MD Molecular dynamics (MD) System size Small Large Properties Molecular (intramolecular) Macroscopic (intermolecular) Hybrids QM/MM ReaxFF (QM-based force field + MD) Simplification Semi-empirical methods; Density functional-based tight binding methods Coarse-grained methods (e.g. DPD)

Contents1.Brief review of quantum mechanics1. Describe wave properties ofanelectron;2. Quantum free particle model and its various derivatives;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;

Matterwave:wave-particledualityhcE=hvc = AvFor a photon:E:=pc-E=mc2p= mcPlanck's constanthhde Broglie Wavelength (1924)h = 6.626 × 10-34 Js1pmvThe pilot-wave modelCarElectron9.1 X 10-31m (kg)1000v100 km/hr0.01 C2.7 X 10-242.8 X 104p (kg m/s)2.4 X 10-102.4 X 10-38入 (m)RemarkToo small to detect.Comparabletosizeofatom.Classical object!Mustaccountforwaveproperties ofan electron!FullereneDiffractionNature1999,401,680Phthalocyanine derivatives (C4gH26F24NgOg) show quantum interference, NatureNanotechnology 2012,Z, 2975

Matter wave: wave-particle duality de Broglie Wavelength (1924) Planck's constant For a photon: Car Electron m (kg) 1000 9.1 × 10−31 v 100 km/hr 0.01 C p (kg m/s) 2.8 × 104 2.7 × 10−24 λ (m) 2.4 × 10−38 2.4 × 10−10 Remark Too small to detect. Classical object! Comparable to size of atom. Must account for wave properties of an electron! 5 𝐸 = ℎ𝜈 𝐸 = 𝑚𝑐 2 𝑐 = 𝜆𝜈 𝑝 = 𝑚𝑐 𝐸 = ℎ𝑐 𝜆 = 𝑝𝑐 λ = ℎ 𝑝 = ℎ 𝑚𝑣 ℎ = 6.626 × 10−34 Js The pilot-wave model Phthalocyanine derivatives (C48H26F24N8O8 ) show quantum interference, Nature Nanotechnology 2012, 7, 297. Fullerene Diffraction Nature 1999, 401, 680

DescribewavepropertiesofanelectronWhat Is Life?Schrodingerequation(1926)a=HUY not a physical observable!访Expressedasdifferentialeguation:at记-v?y(r,t)+V(r,t) Y(r,t)Single particle, non-relativistic:2matSteady-state,ortime-independentV(r)allowstheseparation-y(r)+V(r)y(r)= Ey(r)of variablesrandT2mKinetic energy+Potential energy=Total EnergyQuantumharmonicoscillatorotoAstationarystateisnotmathematicallyconstantY(r,t)=y(r)e"%pTheprobabilitythattheparticleisat locationxisindependentoftime(r,t)2 =e-at/h(r,0)2 = e-it/(r,0)2 =(a,0)vp6

= 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!

Wavefunctionsallowed inquantummechanicsY(x)P(x)no1.Single-valued;nob)+a)2.Continuous;P(x)3.Nowhereinfinite(exception:Dirac delta-function);4.Piecewise continuousfirstP(x.yPoxvederivatives;5. SSquare-integrable(exceptionplane wave)vesY(0)(0)1, 3, 5 required by Bornh)00g)interpretation;2 and 4 required by kinetic(x)P(x)nooperator (2nd derivative of )yesi

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)

Kineticenergyvs.potentialenergyIV-O(c)(a(b)(d)(a)SinceV=O,E=T.ForhigherT,ismorewiggly,which meansthat入isshorter. (Since is periodic fora free particle,入is defined.)(b) As Vincreases from left to right, becomes less wiggly. (c)-(d) is most wigglywhereVislowestandTisgreatest

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

Quantization,energylevelsanddegeneracy:ananalogyof stable statesofa chaireaEsnormalpositionnon the sideThedegreeofdegeneracyequalstwoinclinedE1onthesupportEoDifferent degrees of degeneracy of energy levels lead to densityof states (DOS);UsingkasxaxisleadstobandstructureArranging x axis according to reaction leads to reaction pathway9

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

The origin of quantization:boundary conditions2k2h2d2(z)Quantumfreeparticley(x)= eikxEb(a)E,≥02mdr22mkany real number,no restrictionParticle-on-a-ringAs assumed byde Broglie(a)Periodicboundarycondition(0)=(0+2元)2元R = n入1h2n2ine山En =?2mR2V2元Mismatch-contrarytoassumption3-DRepreseritation2-DRegresentation(b)nDe Broglie Waves around a Closed Orbit±2(a)Anintegralnumberofwavelengthsonthe circumference.(b)Notan integralnumber.+Combinedegeneratecomplexwavefunctionstoformvisualizable (localized)real wavefunctions+attheexpenseoftheknowledgeofmomentum,oinwhichcasepositionisunknown(*w=const)10

The origin of quantization: boundary conditions De Broglie Waves around a Closed Orbit (a) An integral number of wavelengths on the circumference. (b) Not an integral number. Quantum free particle 10 ψ(x)= e ikx k any real number, no restriction Particle-on-a-ring Periodic boundary condition n Combine degenerate complex wavefunctions to form visualizable (localized) real wavefunctions, at the expense of the knowledge of momentum, in which case position is unknown (Ψ*Ψ=const) 𝜓 = 1 2𝜋 𝑒 𝑖𝑛𝜃 𝜓(𝜃) = 𝜓(𝜃 + 2𝜋) 𝐸𝑛 = ℏ 2𝑛 2 2𝑚𝑅2 2𝜋𝑅 = 𝑛λ 𝐸𝑛 = ℏ 2𝑘 2 2𝑚 ≥ 0

Benzeneandits元orbitalsdelocalized6p-orbitalsUsingabsorptionspectra,effectiveradiusofbenzeneBENZENEcan be fitted807OX8CPOpeayonticedothitaCyclicpolyynesC18 ring synthesizedELECIRONShttps://en.wikipedia.org/wiki/Benzene11

Benzene and its π orbitals https://en.wikipedia.org/wiki/Benzene 11 Cyclic polyynes C18 ring synthesized Using absorption spectra, effective radius of benzene can be fitted