MOLECULAR VIBRATIONS

Transcription

1 MOLECULAR VIBRATIONS Here we wsh to vestgate molecular vbratos ad draw a smlarty betwee the theory of molecular vbratos ad Hückel theory. 1. Smple Harmoc Oscllator Recall that the eergy of a oe-dmesoal harmoc oscllator s gve by the sum of the ketc ad potetal eergy. Suppose a partcle of mass m s movg a potetal gve by V (x) = 1 kx oe-dmeso, the, the Hamltoa s gve by H = 1 mẋ + 1 kx (1) where the frst term s the ketc eergy ad the secod the potetal eergy. Classcally, the partcle obeys Newto s Law, whch states that: mẍ = dv () dx gvg rse to the followg equato of moto upo substtutg the form of the potetal V (x) = 1 kx : mẍ = kx (3) whch solves to gve oscllatory solutos havg the form of s, cos or expoetal. We could choose the cose form: x(t) = A cos(ωt + φ) (4) where A ad φ are costats determed by tal codtos (sce Eq. (3) s a secod-order dfferetal equato, we expect two costats here) ad ω s the agular frequecy gve by k ω = (5) m Usg Eq. (5), we could wrte the Hamltoa as H = 1 mẋ + 1 mω x (6) 1

2 . Datomc molecule Now, cosder a datomc molecule havg atoms wth masses m 1 ad m teractg through a harmoc potetal descrbed wth a force costat k. Assumg that the moto of the two ucle s cofed to oe dmeso, the Hamltoa of the problem becomes H = 1 m x + 1 k(x 1 x ) (7) where x 1 ad x are the dsplacemet coordates relatve to the equlbrum geometry of the molecule (F. 1). Fgure 1. Dsplacemet coordates of a datomc molecule. We frst troduce a mass-weghted coordate: q = m x (8) to remove ay explct mass depedece from Eq. (7): H = 1 q + 1 ( k q1 q ) m1 m = + 1 ( k q k 1 q 1 q + k ) q m 1 m1 m m = 1 q + 1 q K j q j (9) where K s the dyamcal or Hessa matrx gve, ths case, by k k m K = 1 m1 m k k m1 m m j We ca fd the egevalues ad egevectors of the dyamcal matrx K by solvg The solutos ca be wrtte as j (10) K λi = 0 (11) K j j = λ (1) c Xglog Zhag 017

3 where the superscrpt refers to the th ormalsed egevector, ad λ the correspodg egevalue. Sce the matrx K s Hermta (deed K s symmetrc sce all ts compoets are real), ther egevectors are orthogoal, we ca also ormalsed them to gve orthoormal egevectors: c (m) = δ m (13) Now we wsh to fd ormal mode coordates, Q, terms of (a lear combato of) scaled atomc dsplacemets q, such that these ormal modes are decoupled from each other ad ca be aalysed separately. Expressed more formally, we wsh to be able to wrte the Hamltoa Eq. (9) a dagoal form: H = ( 1 Q + 1 ω Q ). (14) Sce the Hamltoa Eq. (9) cotas the atomc dsplacemets q, we cosder the followg coordate trasform: q = Q (15).e., wrtg the scaled coordates as a lear combato of the ormal mode coordates, where the coeffcets s the th compoet of the th ormalsed egevector from Eq. (1). Substtutg ths to the Hamltoa of the system Eq. (9), we have H = 1 q + 1 q K j q j j = 1 [ ] [ ] Q c (m) Q m + 1 [ ] [ ] Q K j c (m) Q m m j m = 1 [ ] Q c (m) Q m + 1 K j c (m) Q m m m j = 1 Q δ m Q m + 1 λ m δ m Q m m m = 1 Q + 1 λ Q (16) where achevg the secod last le, we used the equatos from Eq. (1) ad (13). Ths ow deed has the dagoal form of Eq. (14) whch we detfy the egevalues from solvg the dyamcal matrx as λ = ω (17) c Xglog Zhag 017 3

4 At ths stage, after solvg for the egevalues ad the egevectors of the dyamcal matrx, we are fally able to wrte the ormal mode coordates as a lear combato of the (scaled) atomc dsplacemets. We have the coordate trasform Eq. (15), left multply by traspose of aother egevector ad usg the orthoormalty codto of Eq. (13), we have ] c (m) q = Q = c (m) Q = [ c (m) Q = δ m Q = Q m q (free relabellg of m ) (18) Now, the above equato wrte the ormal mode coordates as a lear combato of the mass-scaled atomc dsplacemets so that each ormal mode s decoupled from the other ormal modes (Ref Eq. 16). To solve the system of molecular motos for the datomc system, oe s the requred to solve the dagoalsato of the dyamcal matrx Eq. (10). Ths s left to the readers as a smple exercse. 3. Geeralsato to polyatomcs Cosder a molecule cosstg of N atoms. Classcally, the Hamltoa for molecular vbratos has the geeral form H = 1 3N m Ṙ + V (R) (19) where R s the posto vector for atom, m s the mass of the atom wth compoets R ( = 1,, 3N). The postos of all ucle ca be wrtte as The quatum expresso s gve stead by Ĥ = 1 R {R 1, R,, R N } (0) 1 ˆp + V (R) = 1 1 m m R + V (R) (1) However, we wll ote that usg the mass-weghted coordate Eq. (8), we ca get the dyamcal matrx usg the potetal term V (R) aloe. Suppose the molecule s at equlbrum geometry, deoted by R (0) {R (0) 1, R (0),, R (0) N }, () we ca defe ay uclear motos relatve to the equlbrum geometry by R = R (0) + x (3) c Xglog Zhag 017 4

5 where x s a collecto of 3 coordate dsplacemets for ucleus : x = (x 1, x, x 3 ). (4) The codto for equlbrum geometry for a molecule s gve by ( ) V = 0 (5) x where the subscrpt 0 deotes that the partal dervatves are evaluated at equlbrum geometry. For small-ampltude dsplacemets aroud the equlbrum, we ca Taylor-expad the potetal eergy as ) V (R) = V ( V j x x j 0 x x j + cubc terms x x j x k + (6) 0 where V 0 = V (R (0) ). Note that the lear term vashes due to equlbrum codto. I the harmoc approxmato, the Taylor expaso above s trucated after the quadratc terms. Comparg the quadratc term above wth the potetal eergy term (the secod term) Eq. (9), ad usg the mass-weghted coordates Eq. (8), we ca see that 1 the dyamcal matrx s gve by ( 1 ) V K j = (7) m m j x x j 0 ( ) V = (8) q q j We ow have a geeral method for solvg for ormal modes of a system wth a potetal eergy V. We ca costruct the dyamcal matrx from t usg Eq. (8). We the solve for ts egevalues ad egevectors to gve the (square) of the vbratoal frequeces (Eq. 17) ad the ormal mode coordates (Eq. 18). Ths s deed the same approach we use solvg the Hückel secular equatos. Of course, symmetry could be used to smplfy the solvg of these matrces. 4. Parallels betwee Hückel theory ad molecular vbratos aalyss Although the two areas appear to be very dfferet, actual fact, the ways to solve these problems are very smlar. We ote the followg resemblace betwee Hückel theory ad molecular vbratos aalyss: 0 1 you could do a chage of varable dfferetato by usg cha rule c Xglog Zhag 017 5

6 Hückel Theory atomc orbtals, φ molecular orbtals, ψ Hückel matrx, H j orbtal eerges, ɛ symmetry orbtals, θ Γ mass-weghted coordates, q ormal mode coordates, Q dyamcal matrx, K j squared frequeces, ω symmetry-adapted lear combato of dsplacemets, Q Γ Table 1. Close resemblace betwee Hückel theory ad molecular vbratos. c Xglog Zhag 017 6