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
一7.1.1 Form of the normal modesEx.32·In a normal mode,thecentre of mass hastoremainfixed.Accordingly,the atomshavetomoveinways which balance oneanotherout and in additiontheamount by which each atom moves willbeaffectedbyitsmass.(lowermass→largerdisplacement)3N.However, it is rather tedious to derive the form of the normal modesQi =Cijqjina basis of (x,y,z)displacements evenfor simplemolecules!一.Alternatively,use internal displacements to derive the forms of normal modes-two rules(i)there is 1 stretching vibration per bondInternalcoordinates（内坐标）bondlengths,bondangles(ii)wemusttreatsymmetry-related atomstogetherdihedralangles? H,O has two stretching modes and one angle bending modeA1V,3652cm-11Va1595cm-B, v, 3756 cm-115:31
7.1.1 Form of the normal modes Ex.32 • H2O has ？ stretching modes and ？ angle bending mode. • Alternatively, use internal displacements to derive the forms of normal modes—two rules (i) there is 1 stretching vibration per bond (ii) we must treat symmetry-related atoms together • In a normal mode, the centre of mass has to remain fixed. Accordingly, the atoms have to move in ways which balance one another out and in addition the amount by which each atom moves will be affected by its mass. (lower mass larger displacement) two one Internal coordinates(内坐标): bond lengths, bond angles, dihedral angles v3 15:31 • However, it is rather tedious to derive the form of the normal modes in a basis of (x,y,z) displacements even for simple molecules! 𝑸𝒊 = 𝒋=𝟏 𝟑𝑵 𝒄𝒊𝒋𝒒𝒋
7.1.1 Form of the normal modesExample: H,OUsinginternal(coordinate)displacements!XyO8NHH2 T2: First use the two O-H bond stretches (r,,r2) as a basisTo-zoJzEC2C2v(ri+r2)The A, stretching (--like):xy2221111~Symmetric (in-phase) stretchingAiZ11A2-1-1RzxyThe B, stretching (x-like):(-ri+r2)BiRy1-1-11xXzB21-1-1Rx1yyz~anti-symmetric (out-of-phase) stretching2020r(2)=A, @Bir(α)1111Usethe H-O-H angleαbending as a basis=A1The angle bending transforms as A, IRA;v,3652cmA,vg1595cm1B,v,3756cm*1The A, bending & symmetric stretching further mix!Neitherpurelybendingnor purely stretching15
7.1.1 Form of the normal modes Example: H2O • First use the two O-H bond stretches (𝒓𝟏,𝒓𝟐) as a basis. O H1 H2 𝑟 1 𝑟 2 𝚪 (𝟐𝒓) 2 0 2 0 = A1 B1 x z y The A1 stretching (z-like): (r 1+ r 2 ) The B1 stretching (x-like): (–r 1+r 2 ) ~Symmetric (in-phase) stretching ~anti-symmetric (out-of-phase) stretching 𝚪 (𝜶) • Use the H-O-H angle bending as a basis. 1 1 1 1 = A1 The angle bending transforms as A1 IR. Using internal (coordinate) displacements! The A1 bending & symmetric stretching further mix! Neither purely bending nor purely stretching. 15:31
7.1.2 Normal modes of HExample: intersteller molecule H (point group D3h)x,1ED3h2C33C22S330vChy.12.1Z.1(b.11111A'11X2+y2;22A2111x,2x,2-1R21-1b.3E'20-12-10(x2 - y2,2xy)(x,y)11A"1-1-1-13y,2@z,2z.3E11-11-1-1ZZ.3z,2a.2a.32-10-210(Rx,R,)(xz, yz)b,2. In a general axis system: (z, I z,2 z,3), (x, I x, 2 x, 3, y, I y,2 y,3) a 6-D rep.!.Inalocalaxissystem:(z, 1 z,2 z,3), (a, 1 a,2 a,3), and (b, 1 b,2 b,3) → all 3-D reps.!Radial displacementsTangentialdisplacements15:31
7.1.2 Normal modes of 𝑯𝟑 + • Example: intersteller molecule 𝑯𝟑 + (point group D3h). • In a general axis system: (z,1 z,2 z,3), (x,1 x,2 x,3, y,1 y,2 y,3) • In a local axis system: (z,1 z,2 z,3), (a,1 a,2 a,3), and (b,1 b,2 b,3) all 3-D reps.! a 6-D rep.! Radial displacements Tangential displacements 15:31
7.1.2 Normal modes of Ha.1ASANGD3hE2C33C22S330vChz.1(-b.1x2 +y2;z211111Ai1A211-111Rz-1b,3E'22-10-10(x? - y2,2xy)(x,y)1A"1-11-1-1OZ2A211-1-1-11z7Ka.2a.3Ei20-20-11(Rx,R,)(xz, yz)b,20-3 01)(3A,"@E"(z, 1 z,2 z,3)Q:How does its three00DEE13(a, I a,2 a,3)31)normal modes look like?E2br-b2-b3030-1-11,E(b, 1 b,2 b,3)3Ist approx.:E'b,-b3Sym. ringTotalA'④A,"@2E'④A,"④E"A,:a,+a,ta3breathing-translations (x,y,2)E'@A,"E'2ar-az-aAsym. ringE"-rotations (R,R,R.)④A,E'breathingaz-a3VibrationsE15:31
7.1.2 Normal modes of 𝑯𝟑 + (z,1 z,2 z,3) ( 3 0 -1 -3 0 1）A2 E (a,1 a,2 a,3) ( 3 0 1 3 0 1) A1 E (b,1 b,2 b,3) ( 3 0 -1 3 0 -1) A2 E Total A1 A2 2E A2 E –translations (x,y,z) E A2 –rotations (Rx ,Ry ,Rz ) A2 E Vibrations A1 E Q: How does its three normal modes look like? A1 : a1+a2+a3 E y 2a1 -a2 -a3 E x a2 -a3 1 st approx.: x y E y 2b1 -b2 -b3 E x b2 -b3 15:31 Sym. ring breathing Asym. ring breathing
7.1.3 X-H stretching analysis.Onaccountofthelowmassofthehydrogenatom,itisoftenthecasethatparticularnormalmodes are dominated by X-H stretching motions. Therefore it is practically useful to make a symmetry analysis using a basis consisting ofonly X-H stretches, but nota general set of (x,yz) displacements on each atom:Of course, such an approach will only reveal the symmetries of those normal modesinvolvingtheX-H stretches.: Example: the C-H stretches of ethene (point group D2)15:31
7.1.3 X–H stretching analysis • On account of the low mass of the hydrogen atom, it is often the case that particular normal modes are dominated by X–H stretching motions. • Therefore it is practically useful to make a symmetry analysis using a basis consisting of only X–H stretches, but not a general set of (x,y,z) displacements on each atom. • Of course, such an approach will only reveal the symmetries of those normal modes involving the X–H stretches. • Example: the C–H stretches of ethene (point group D2h). 𝒓𝟒 𝒓𝟏 𝒓𝟐 𝒓𝟑 15:31