7.1.3 X-H stretching analysisC2D2hEChCO-1zgyzo-ty12:2:72Ag1111日B1g-1-1RxyCB2gR,xzB3gRxyzAuBuB2u1y=A,④ Big④ Bzu④ B3uB31x0001FTikexylikexlikeyA,Totally symmetricBig, like xyBzu, like yB3u, like x(fi+t2-r3-1)15:31(1+T2+Tg+T)(T1-T2+T3-T)(1-12 -13+r)
7.1.3 X–H stretching analysis 4 0 0 0 0 4 0 0 = Ag B1g B2u B3u Ag , Totally symmetric B2u , like y B3u B1g , like xy , like x 𝒓𝟒 𝒓𝟏 𝒓𝟐 𝒓𝟑 (𝒓𝟏– 𝒓𝟐 – 𝒓𝟑 + 𝒓𝟒 ) (𝒓𝟏 + 𝒓𝟐 + 𝒓𝟑+ 𝒓𝟒 ) (𝒓𝟏 + 𝒓𝟐 – 𝒓𝟑 – 𝒓𝟒 (𝒓 ) 𝟏 - 𝒓𝟐 + 𝒓𝟑 - 𝒓𝟒 ) 15:31
Ex.33.Thesepictures arisingfrom combination of internal displacements are only approximationsto the real normal modes.(In reality, the carbon atoms would also need to move by smallamounts in order to ensure that the centre of mass remainfixed.). In the next two sections, we will see how a symmetry analysis helps us to determinewhetherornotaparticularnormal modewill giverisetoabsorptions intheinfra-red orvibrational Raman scattering(i.e.whether ornotamode isinfra-red orRaman active). We will start out by looking at the symmetry ofthe harmonic oscillatorwavefunctions,whichareafirstapproximationtothevibrationalwavefunctions ofthemolecule,andthenmoveontodiscussthe selectionrulesfortransitionsbetweenthem15:31
• These pictures arising from combination of internal displacements are only approximations to the real normal modes. （In reality, the carbon atoms would also need to move by small amounts in order to ensure that the centre of mass remain fixed.） • In the next two sections, we will see how a symmetry analysis helps us to determine whether or not a particular normal mode will give rise to absorptions in the infra-red or vibrational Raman scattering (i.e. whether or not a mode is ‘infra-red or Raman active’). • We will start out by looking at the symmetry of the harmonic oscillator wavefunctions, which are a first approximation to the vibrational wavefunctions of the molecule, and then move on to discuss the selection rules for transitions between them. Ex.33 15:31
7.2 Symmetry of the vibrational wavefunctions:If we assumethatthe vibrations areharmonic,eachnormal modehas associated witha setof energy levels: (o,is the vibrational frequency ofthe ith normal mode)EyhwVi = 0, 1, 2 ...2.Thenormal modes canbeexcited independentlyof oneanotherso,forexample,wecanhave the first normal mode in the v,= 1 level, the second in the ground state (v,=0), the thirdin the v,= 3 level and so on.33..22..*Thesetofenergylevels3112availableforH,O1V, = 0V3=00energyzeromode 1, Amode 2, Amode3,B,15:31
7.2 Symmetry of the vibrational wavefunctions • If we assume that the vibrations are harmonic, each normal mode has associated with a set of energy levels: (ωi is the vibrational frequency of the ith normal mode). 𝑬𝒗𝒊 = 𝒗𝒊 + 𝟏 𝟐 ℏ𝝎𝒊 𝒗𝒊 = 𝟎, 𝟏, 𝟐 . • The normal modes can be excited independently of one another so, for example, we can have the first normal mode in the v1= 1 level, the second in the ground state (v2=0), the third in the v3= 3 level and so on. The set of energy levels available for H2O. 15:31 B1
7.2 Symmetry of the vibrational wavefunctionsHermitepolynomials.Foradiatomic,theharmonicoscillatorEv = H,(@) exp(-q2 /2)Vwavefunctionsdependonlyonthe120exp(-q)displacement x, where x=(r-r)3125127122g exp(-q2)1In terms of the scaled coordinate q(ac x), the2(4q2-2) exp(-q2)formofthefirstfewwavefunctionsandtheirenergies (unit in hw)aretabulated here3(8q-12q) exp(-q2)Normal coordinate Q, in the place of g for complex molecules!: In more complex molecules, a normal mode involves several atoms changing theirpositions, but we can define a single normal coordinate Q,(筒正坐标) to describe themotionof ithnormal mode· Key point: I(Q)= r(i)15:31
7.2 Symmetry of the vibrational wavefunctions • For a diatomic, the harmonic oscillator wavefunctions depend only on the displacement x, where x=(r-re ). In terms of the scaled coordinate q( x), the form of the first few wavefunctions and their energies (unit in ℏ𝝎) are tabulated here. • In more complex molecules, a normal mode involves several atoms changing their positions, but we can define a single normal coordinate Qi (简正坐标) to describe the motion of ith normal mode. Normal coordinate Qi in the place of q for complex molecules! • Key point: (𝑸𝒊 ) = (𝒊) = 𝑯𝒗 𝒒 exp(−𝑞 2/2) 15:31 2q (4q2 – 2) (8q3 – 12q) Hv (q) Hermite polynomials
7.2.1 Symmetry of the ground state vibrational wavefunction. For a non-degenerate normal mode, its ground-state wavefunction is o= exp (-; Q?)Q; (as a basis) transforms as a particular 1-D IR, i.e.,Rg, =x(R)g, with x(R)= +1 or -1(effect of symmetry operation R on Q;>RQ? = [x(R),}?2 = (+1)Q? (valid for arbitrary R of the very point group!)> Q? transforms as the totally symmetric IR, so does o= exp (-Q?)The ground-state wavefunction always transforms as the totally symmetric IR: For degenerate normal modes, the conclusion remains the same and the statement abovetherefore appliesto all normal modes15:31
7.2.1 Symmetry of the ground state vibrational wavefunction • For a non-degenerate normal mode, its ground-state wavefunction is ψ0= exp (– 𝟏 𝟐 𝑸𝒊 𝟐 ). Qi (as a basis) transforms as a particular 1-D IR, i.e., 𝑹 Qi =𝜒(𝑹 )Qi with 𝜒(𝑹 )= +1 or –1 (effect of symmetry operation 𝑹 on Qi ) 𝑹 𝑸𝒊 𝟐 = [𝜒(𝑹 )Qi ] 2 = (+1)𝑸𝒊 𝟐 (valid for arbitrary 𝑹 of the very point group!) 𝑸𝒊 𝟐 transforms as the totally symmetric IR, so does ψ0= exp (− 𝟏 𝟐 𝑸𝒊 𝟐 ). The ground-state wavefunction always transforms as the totally symmetric IR. • For degenerate normal modes, the conclusion remains the same and the statement above therefore applies to all normal modes. 15:31