7.2.2 Symmetry of excited states: non-degenerate normal modesKey point: r(Qi) = r(i)StateWavefunctionSymmetry of vo= exp (-Q?)V,=0The first excited state transforms asrot. symthesameIRasthenormal mode41= 2;exp (-1Q2)y,=1i @Pot. sym = Ti2 = (4Q2-2) exp (-, Q2)Prot. sym ot. sym. =Irot. sym.V; =23 = (8Q:-12Q) exp (-,Q:) Ti) @rot sym = ri)V, =3For non-degenerate normal modes,vibrational wavefunctions with v=O,2,4,:..(evenv)transform as the totally symmetric IR, and those with odd have the same IR as doesthe normal mode.15:31
State 7.2.2 Symmetry of excited states: non-degenerate normal modes ψ1 = Qi exp (– 𝟏 𝟐 𝑸𝒊 𝟐 ) (i) tot. sym. = (i) ψ2 = (4𝑸𝒊 𝟐 –2) exp (– 𝟏 𝟐 𝑸𝒊 𝟐 ) ψ3 = (8𝑸𝒊 𝟑 –12Qi ) exp (– 𝟏 𝟐 𝑸𝒊 𝟐 ) vi =0 ψ0 = exp (– 𝟏 𝟐 𝑸𝒊 𝟐 ) vi =1 tot. sym. vi =2 Wavefunction Symmetry of 𝝍𝒗𝒊 vi =3 tot. sym.tot. sym. =tot. sym. (i) tot. sym. = (i) For non-degenerate normal modes, vibrational wavefunctions with v = 0, 2, 4, . . . (even v) transform as the totally symmetric IR, and those with odd v have the same IR as does the normal mode. Key point: 𝚪 (𝑸𝒊) = 𝚪 (𝒊) The first excited state transforms as the same IR as the normal mode. 15:31
7.2.2 Symmetry of excited states:non-degenerate normal modesFor non-degenerate normal modes, vibrational wavefunctions with v= O, 2, 4, : ..(even v)transform as the totally symmetric IR, and those with odd v have the same IR as does thenormal mode.E3B3-2A2.3B,24TAV1=0AV3=0A0energyzeromode1mode2mode3B,AANormal Modesof H,O15:31
7.2.2 Symmetry of excited states: non-degenerate normal modes E Normal Modes of H2O For non-degenerate normal modes, vibrational wavefunctions with v = 0, 2, 4, . . . (even v) transform as the totally symmetric IR, and those with odd v have the same IR as does the normal mode. 15:31