Part IlI Symmetry and BondingChapter7NormalModes（简正模/简正振动模式）Prof.DrXinLu（吕鑫）Email:xinlu@xmu.edu.cnhttp:/ /pcoss.xmu.edu.cn/xlv/index.htmlhttp:/ /pcoss.xmu.edu.cn/xlv/courses/theochem/index.html15:31
Part III Symmetry and Bonding Chapter 7 Normal Modes (简正模/简正振动模式) Prof. Dr. Xin Lu （吕鑫） Email: xinlu@xmu.edu.cn http://pcoss.xmu.edu.cn/xlv/index.html http://pcoss.xmu.edu.cn/xlv/courses/theochem/index.html 15:31
7. Normal modes (简正模This section is devoted to using symmetry considerations to help understand the vibrations ofmolecules and spectrathat arisedueto transitions betweenthe associated energy levels.The vibrations of a molecule can be separated into contributions from a/afinitenumber of special vibrations called normal modesEachnormal modehasasetofenergylevels,andthetransitionsbetweentheselevelsgiverisetoinfra-red spectraofthetype.WateINFRAREDSPECTRUMe.g., three normal modes of H,O and theirfundamental transitionsDueseannejaA,V,3652cm-tA,Vz1595cm-B,V,3756cm-nber (cm-1)(https://webbook.nist.gov/chemistry)15:31
7. Normal modes (简正模) • This section is devoted to using symmetry considerations to help understand the vibrations of molecules and spectra that arise due to transitions between the associated energy levels. • Each normal mode has a set of energy levels, and the transitions between these levels give rise to infra-red spectra of the type. e.g., three normal modes of H2O and their fundamental transitions • The vibrations of a molecule can be separated into contributions from a finite number of special vibrations called normal modes. 15:31 (https://webbook.nist.gov/chemistry)
7. Normal modesHere we will show ideas abouti)howto classifynormal modes according to symmetryii) how to predict which modes give rise to infra-red spectra and vibrationalRamanscattering. We will use the symmetry arguments to explain the occurrence of more complexfeatures ofinfra-red spectra,such as overtones and combination bands15:31
7. Normal modes Here we will show ideas about i) how to classify normal modes according to symmetry, ii) how to predict which modes give rise to infra-red spectra and vibrational Raman scattering. • We will use the symmetry arguments to explain the occurrence of more complex features of infra-red spectra, such as overtones and combination bands. 15:31
7.1 Normal mode analysis.Vibrations involvethephysicaldisplacement of atomsfromtheirequilibriumpositionsTo analyse the symmetry of vibrations, we simply imagine a basis which consists of an x, y and zdisplacementvector attachedtoeach atom inthemolecule.3NFor the ith normal mode (vibration) of an N-atom molecule, Q: =cijqj(g:各原子位移基失)defineitsnormalcoordinateQ（简正坐标）asj=1 Example, H,O (C2v), basis (x, y and z displacement vectors on each atom).0.Q.yoozoJzEC2C2v1111Air2..zH211A2-1-1RzxyB111-1-1RxXzHi.ZB2-11Rx1-1yzyBasis(9vectors)→a9-Drep.!.To simplify the problem, we first separate the displacement vectors into groups which are mappedonto one another(!!!!!!!!) by the operations of the point group与之前对原子轨道做对称性分类相似！
7.1 Normal mode analysis • Vibrations involve the physical displacement of atoms from their equilibrium positions. • To simplify the problem, we first separate the displacement vectors into groups which are mapped onto one another(!!!!!!!!) by the operations of the point group. Basis (9 vectors ) a 9-D rep.! 与之前对原子轨道做 对称性分类相似！ • Example, H2O (C2v), basis (x, y and z displacement vectors on each atom). For the ith normal mode (vibration) of an N-atom molecule, define its normal coordinate Qi (简正坐标) as 𝑸𝒊 = 𝒋=𝟏 𝟑𝑵 𝒄𝒊𝒋𝒒𝒋 (q: 各原子位移基矢) • To analyse the symmetry of vibrations, we simply imagine a basis which consists of an x, y and z displacement vector attached to each atom in the molecule
7.1 Normal mode analysisOXZEC2orzC2vO.-0,xyo111x2: y2:z2A11Z中IN11-1-1RzA2xyH2,XHaO,2H1.xB111-1-1Ryxzx11B2-1-1RxyyzHi,zHa.22020I=A, @B,(H,x, H2,x)IRVector(s)-202=A, ④B20I2(Hy, Hy)O,xB, (from the table)O.yB2SALCFull set(3N)3A, ④ 3B,@ A,④ 2B,0,2AlTranslations (x,y,z)B, B2, A, (from the table)A, B(Hj,x, H2x)Rotations(R,R,R)B2, B, A2 (from the table)(Hi-,H2-) A, @BI2A, 甲 B,Vibrations (3N-6)(Hiy, H2y) A, @B,Total 3A, ④ 3B,④ A,@ 2B3N-6normalmodesfornon-linearmolecules15:3T
7.1 Normal mode analysis (H 1 2 0 2 0 = A1 B1 1 ,x, H2 ,x) (H1 ,y, H2 O, ,y) 2 2 0 -2 0 = A2 B2 x O,y O,z (H1 ,x, H2 ,x) A1 B1 (H1 ,z, H2 ,z) A1 B1 (H1 ,y, H2 ,y) A2 B2 3A1 3B1 A2 2B2 • 3N-6 normal modes for non-linear molecules. Vector(s) IR Full set （3N) 3A1 3B1 A2 2B2 Translations (x,y,z) B1 , B2 , A1 (from the table) Rotations (Rx ,Ry ,Rz ) B2 , B1 , A2 (from the table) Vibrations （3N-6) 2A1 B1 B1 (from the table) B2 A1 Total SALC 15:31