Chapter 7 Symmetry and Spectroscopy Molecular Vibrations p. 1 -

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1 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p - 7 Symmetry and Spectroscopy Molecular Vbratons 7 Bases for molecular vbratons We nvestgate a molecule consstng of N atoms, whch has 3N degrees of freedom Tang the translatons (3) and rotatons (3 for non-lnear, for lnear molecules) nto account, we obtan 3N-6 (5) vbratonal degrees of freedom for the non-lnear (lnear) case In order to descrbe those, we choose an arbtrary set of bass vectors n order to descrbe the 3N dsplacements from the equlbrum poston (the choce s arbrtrary, but a choce havng smple transformaton propertes mght be advantageous) Example: BF 3 or CO 3 - degrees of freedom: 3 translatons, 3 rotatons, 6 vbratons dsplacement bass d: 7 Normal modes We are nterested n the collectve vbratonal modes of the molecules These collectve modes wth dentcal frequency and phase are denoted as normal modes As a pctoral representaton of the modes, we ndcate the set maxmal dsplacements, eg:

2 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p - As we wll see later, normal modes can be classfed wth respect to the rreps of the pont group of the molecule In analogy to the strategy followed n chapter 5 for the electronc structure of molecules, we can transform the dsplacement bass of dmenson 3N nto a symmetry adapted set of dsplacement coordnates D: x r r y r D αd wth d z x M In the next step we transform the bass D nto the bass of normal modes r r βd In analogy to chapter 5, the matrx β s blocdagonal, e the normal coordnates are lnear combnatons of the SALCs of dsplacements belongng to one rrep of the pont group of the molecule (Wgner s theorem) As an example we classfy the normal vbratons of CO 3 -, BF 3 (D 3h ): (7 Character table D 3h) Note: 3, 4 and 5, 6 are transformed nto lnear combnatons of each other by symmetry operatons of the group They belong to dmensonal rreps

3 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p 3-73 Symmetry of normal modes We would le to classfy the normal modes of the molecules wth respect to the rreps whch they belong to For ths purpose, we analyse the 3N dmensonal representaton of the dsplacements wth respect to the rreps of the group Example: BF 3 (C 3v ), dmensonal dsplacement bass E: all dagonal elements Γ (E) χ ( E ) σ h : x, y, dsplacements are retaned, z dsplacement changes sgn χ σ ) 4 ( h C : dsplacements of atoms, whch change poston do not contrbute to Γ (off dagonal elements) χ ( C )

4 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p 4 - C 3 : z s nvarant ( χ ), for x,y : x' cos0 sn0 x y' sn0 cos0 y ( χ ) χ ( C ) 0 3 S 3 : analogous to C3, but z -z ( χ ) χ ( C ) 3 σ v : y, z, dsplacements at two postons are nvarant, x dsplacement changes sgn χ σ ) ( v D 3h E C 3 3C σ h S 3 3σ v χ 0 4 Γ A ' + A ' + 3 E' + A '' + E'' Symmetry of translatons and rotatons from character table: Γ Tx, Ty E' ; Γ A '' Tz Γ Rx, Ry E'' ; Γ ' A Rz (7 Classfy translatons and rotatons n D 3h) The remanng vbratonal normal modes are: Γ A' + E' A '' vb +

5 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p 5 - (73 Excercse: classfy the vbratonal modes of SF 6 wth respect to ther symmetry, substtute one F by Cl) 74 Analyss of normal modes We assume that we have determned the complete potental energy surface of an N atom molecule V(r,r,r 3N ) and expand V nto a Taylor seres at the equlbrum geometry: V V0 { equlbrum energy 3 V V V + + r + rr rr r + r, r r 6,, r r r for expanson at equlbrum poston force constants (harmonc) anharmonctes We start from the equaton of moton for the molecule m r& r and use the ansatz m r ω r 0 0 r ωt r e whch yelds 0 Introducng mass weghted coordnates R m r ths equaton s smplfed to / r r R or Rω KR m m 44 / / K R ω 0 0 As usual we solve the set of 3N lnear equatons by determng the egenvalues egenvectors Ω r By transformng to a new bass r r ΩR wth Ω ( Ω, Ω Ω ) r r,, bass of normal coordnates, the matrx of force constant κ ΩKΩ becomes dagonal: r 3N ω and, e the v r ω κ or 0ω κ 0 κ 0 Thus we obtan a set of 3N decoupled degees of moton (vbratons, rotatons, translatons) If we express the total energy E T + V mr& +, r r n terms of the new coordnates, we obtan

6 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p 6 - E & + κ & + κ & + In a quantum mechancal descrpton we can descrbe the Hamltonan as a smple sum over harmonc oscllators κ Hˆ Hˆ wth Hˆ h + κ wth the well nown solutons φ v ( ) / ω ω h N v H v / e h ; Ev ( v + ) hω ; ω ; v,,3, ; / κ (H (x): Hermte polynomals) Consequently, the vbratonal wavefuncton of the molecules s ( ) vb φ wth v vb E E v (74 Example: normal modes of C H ) 75 Symmetry of vbratonal wavefunctons: fundamental modes We consder a molecules wth N vbratonal modes The exctaton from the vbratonal ground state ( v 0 ; N ) to a state wth a sngly excted normal mode ( v ; v 0;,, + N ) s referred to as a fundamental mode of the molecule Symmetry of the ground state: vb Ne ω h Effect of symmetry operaton Rˆ on (a) non degenerate modes: ± Rˆ vb Ne ω h ( ± ) vb (a) n degenerate modes:

7 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p 7 - α wth α α δ (orthonormalzaton) Rˆ vb Ne ω h Ne ω h α Ne ω h vb Thus, the vbratonal ground state s nvarant under all symmetry operatons, e Γ vb, 0 A Symmetry of the excted state: vb N e ω h Symmetry of wavefuncton s equal to symmetry of, e Γ vb, Γ 76 Symmetry of vbratonal wavefunctons: combnaton modes The exctaton from the vbratonal ground state ( v 0 ; N ) to a state wth several normal modes excted by one vbratonal quantum each ( v ; v ) s referred to as a combnaton mode of the molecule Symmetry of the excted state: l vb N e l ω h Symmetry of wavefuncton s equal to the drect product of the representatons of and, e Γ vb,, l Γ Γ l (75 Excercse: symmetry of the combnaton modes of BF 3) 77 Symmetry of vbratonal wavefunctons: overtones The exctaton from the vbratonal ground state ( v 0 ; N ) to a state wth several vbratonal quanta n one normal mode ( v > ) s referred to as a overtone of the mode Symmetry of the excted state: (a) non degenerate case

8 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p 8 - vb v ( γ / ) NUH wth U e ω h and ω γ h Hermte polynomals: H ( z) 0 ( z) z H H ( z) 4z ( z) 8z 3 z H 3 H 4 ( z) 6z 48z 4 + Note: even for even v, odd for odd v Therefore Rˆ vb Γ vb ( R) vb for v for v n n + or Γ vb A Γ for v for v n n + (a) degenerate case Smlar as for the problem of several electrons occupyng the same set of degenerate orbtals (chapter 6), quantum statstcs restrcts the number of vbratonal states f several vbratonal quanta occupy the same degenerate mode Thus the problem becomes sgnfcantly more complcated We consder the case of a twofold degenerate vbratonal mode (, v ;, v ): vbratonal quantum: (v ; v 0) (v 0; v ) -fold degenerate vbratonal quanta: (v ; v 0) (v ; v ) (v 0; v ) 3-fold degenerate 3 vbratonal quanta: (v 3; v 0) (v ; v ) (v ; v ) (v 0; v 3) 4-fold degenerate n vbratonal quanta: n+ fold degenerate

9 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p 9 - (analogous: n vbratonal quanta n three-fold degenerate mode: ½(v+)(v+) vbratonal wavefunctons; see textboos) As an example we consder the wavefunctons for v +v 3: 3/ 3 / ( 8γ γ ) 3,0 NU a a / ( 4γ )( γ ), NU a b / ( γ a )( 4γ ), NU b 3/ 3 / ( 8γ γ ) 0,3 NU b b We consder lnear combnatons a, b, for whch the transformaton matrx of a gven symmetry operaton R s dagonal (ths s always possble for sngle operaton, for other symmetry operatons the transformaton s of course not dagonal because the functons belong to a two dmensonal representaton): a b ' Γ ' Γ aa bb a b As the cubc and the lnear terms of the wavefunctons must transform ndetcally, we only consder the cubc terms: ,0 : a ' Γaa a, a ' b' aa : Γ Γ bb a b, : a ' b ' ΓaaΓbb ab ,3 : b' Γbb b Therefore, the character of the 4 dmensonal representaton s χ 3 3 ( R ) R + R R + R R + R 3 a a b a b b 3 Analogous: χ ( R ) R + R R + R and χ ( R ) + a a b b n n n Repeated applcaton of R: χ ( R ) R + R a b R a R b

10 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p 0 - In general, one can show that the general recurson formula s vald (see Wlson, Decus, v Cross): χ ( R) ( χ( R) χ ( R) + χ ( R ) v v (analogous for trply degenerate normal coordnates: v ( R) χ( R) χ ( R) + ( χ( R ) [ χ( R) ] ) χ ( R) + χ( R ) χv v v 3, see Wlson, Decus, Cross) Wth ths nformaton t s possble to construct the characters of an -fold excted vbratonal states correspondng to a twofold degenerate normal coordnate and analyse the representaton (notaton (Γ) n ) whch they span n terms of the rreps of the group (smlar formulae exst for hgher degenerate normal coordnate, see Wlson, Decus, Cross) (76 Exercse: What s the symmetry of the frst and second overtone of an E vbratonal mode of CH 4?) The ntensty of overtones and combnaton bands s relatvely low n most cases In some cases these wea transtons become mportant We assume that there s a sngly excted state v and a st overtone v, v wth smlar vbratonal energy If both states have the, v 0 0 same symmetry (or contan a component wth dentcal symmetry), a couplng between both state s possble, e matrx elements ˆ v, v 0 H v 0, v can be nonzero The egenstates of the coupled modes are mxed vbratonal wave functons ' c v, v 0 + c v 0, v The effect s denoted as Ferm resonance and leads to an ntensty borrowng of the wea combnaton band or overtone from the fundamental as well as to a repulson between both states (compare secton 56) (77 Example: Ferm resonance n CO ) 78 Symmetry of vbratonal wavefunctons: arbtrary excted states

11 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p - Accordng to sectons 75 to 78 the symmetry of an arbtrary vbratonal state wth the vbratonal quantum numbers v, v, v 3, correspondng to the normal modes of symmetry Γ, Γ, Γ 3, s: Γ vb v v v3 ( Γ ) ( Γ ) ( Γ ) 3 Note: We have dscussed the symmetry of vbratonal wavefunctons n terms of the harmonc approxmaton However, the results are not affected, f we tae anharmonctes nto account, as the dfferent parts of the Taylor expanson of the Hamltonan (secton 74) must be totally symmetrc (A ) wthn the group of the molecule If we thus express the anharmonc wave functon as a sum over harmonc functons anh a, only those functons contrbute, whch have the same symmetry 79 Selecton Rules Electrcal dpole transtons are descrbed by matrx elements f the type f µˆ where µˆ are the components of the vector of the dpole operator (wth x,y,z) We have seen that the ntegrand must be totally symmetrc (A ) for the ntegral to be non zero Ths mmedately leads to the dpole selecton rule: A Γvb, f Γµ Γ vb, Note: We consder the components of the dpole moment operator x qx; µ ˆ y q y; z µ ˆ µ ˆ q z () A symmetry operaton exchanges equvalent atoms wth dentcal charge

12 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p - () The operator contans the sums x ; y; z only, whch reman unaffected by a change of ndces Therefore, t s suffcent to nvestgate the transformaton propertes of the functons x, y, z, whch span the same representaton as the components of the dpole operator Typcally, these are tabulated n the character table Remar concernng IR reflecton absorpton spectroscopy (IRAS) at metallc surfaces: For metallc surfaces the parallel component of the electrc feld s effcently screened by the conducton electron Thus the electrc feld vector must be nearly perpendcular to the surface, and only the electrc feld component perpendcular to the surface contrbutes to the feld strength at the surface Smlarly, dpole felds of the oscllatng molecules are screened by the conducton electrons Ths resultng feld s normally constructed by assumng an (magnary) mage charge nsde the metal The mage charge leads to the effect that perpendcular dpole moments are amplfed, whereas parallel dpoles are extngushed Thus the relevant matrx element for IR absorpton at metallc surfaces s: f µˆ z If we assume exctaton from the ground state ( Γ A ) and tae nto account that µˆ or z s z symmetrc wth respect to any possble symmetry operaton of a surface, e Γ µ A, we obtan the so called metal surface selecton rule (MSSR) as: z Γ f A Raman transtons are descrbed by matrx elements f the type f αˆ where αˆ are the components of the tensor of the delectrc polarzablty (wth,x,y,z) Analogous to the prevous case, we obtan the selecton rule:

13 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p 3 - A Γvb, f Γα Γ vb, As we wll dscuss n the next secton, the polarzablty s a symmetrc tensor of ran connectng the electrc feld and the nduced dpole moment: α r µ α α 3 α α α 3 α α χ v E We wll see that upon a symmetry operaton descrbed by the transformaton matrx R ths type of tensor transforms accordng to a ' R R lαl l eg a xx ' R xx α + R α + R α + R R α + R R α + R xx xy yy xz zz xx xy xy xx xz xz xy R xz α yz Transformaton of functon x for comparson: ( x' ) ( R x + R y + R z) R xx + R yy + R zz + R R xy + R R xz + R R yz xx xy xz xx xy xz xx xy xx xz Thus, the components of the tensor show the same transformaton behavour as the quadratc forms x, y, z, xy, xz, yz xy xz Typcally, these functons are tabulated n the character table But how are they determned? In prncple there are two methods: () Applcaton of the symmetry operaton to the bass (x, y, z, xy, xz, yz), determnaton of the representaton matrces and of ther characters, analyss n terms of the rreps of the symmetry group () Analogous to the method appled n the symmetry analyss of overtones (secton 77)

14 Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p 4 - wavefunctons of st overtone of 3fold deg mode: a, b, c, a b, a c, b c quadratc functons of three coordnates: x, y, z, xy, xz, yz Example: Symmetry of the quadratc forms n T d : Γ xyz T R E 8C 3 3C 6S 4 6σ d R E C 3 E C E χ(r) (E) χ(r ) Recurson formula for trply degenate modes (see secton 77): χ ( R) χ( R) χ( R) + ( χ ( R ) [ χ( R) ] ) χ ( R) + χ( R ) 3 [( χ( R) ) + χ ( R )] 0 χ (R) A + E + T (78 Exercse: CO 3 - : Symmetry of normal modes, IR / Raman actvty of fundamentals and st overtones) Note: from the IR and Raman selecton rules t s mmedately evdent that f the nverson s element of the symmetry group of the molecule, a IR actve transton s Raman nactve and vce versa (excluson prncple) Moreover all st overtones of centrosymmetrc molecules are IR nactve