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