L 1 = L G 1 F-matrix: too many F ij s even at quadratic-only level

Transcription

1 5.76 Lctur #6 //94 Pag of 8 pag Lctur #6: Polyatomic Vibration III: -Vctor and H O Lat tim: I got tuck on L G L mut b L L L G F-matrix: too many F ij vn at quadratic-only lvl It obviou! Intrnal coordinat: typ N 6 indpndnt on contraint * tranlation * rotation -vctor St * dirction of fatt incra t * magnitud rulting from limit diplacmnt in optimum dirction N St( { ρ } ) t ρ rigid tranlation ρ ε r for all contraint contraint If th normal diplacmnt ar built from t vctor that atify th contraint, thn, for infinitimal diplacmnt from quilibrium, thr i no rotation. For larg diplacmnt, or for mall diplacmnt away from a non-quilibrium configuration, thr i a mall vibrational angular momntum. Thi dfinition of vibration mbd a pcific partitioning btwn rotation and vibration. TODAY: G from t t Exampl of t. valnc bond trtch r. valnc angl bnd φ 0 no cntr of ma tranlation ρ dω R d Ω S ( Ω ) t d dω t R rigid rotation by d r Ω ( ) R 0 t ECKART (minimiz vibrational angular momntum) G matrix uing diagram and tabl from WDC pag 04 and 05 H O FG handout

2 5.76 Lctur #6 //94 Pag of 8 pag In 07 March 994 Lctur I dfind G DD rcall S B ξ q Μ / ξ B DM / BM / D G DD BM / ( M / ) B BM B G N * tt Bti Bt i i mi dfinition of S B ξ S t ξ i 0 S t ξ i 0 N t m N G tt t t m ( S ) ( S ) 0 t 0 Thi way to driv G i convnint * locally dfind t. Eay to comput t * Each S t involv mall numbr of t (only th involvd atom). * Small numbr of topological ca for intrnal diplacmnt. All analyzd in WDC, pag Vctor Mthod. WDC pag * tart with all atom at quilibrium poition; * dirction of t i dirction of th atom mut mov to yild maximum incra in St; * magnitud of t i incra in St that rult from unit diplacmnt of atom in optimal dirction; * mut vrify or impo th 6 contraint. t 0 0 R t t.

3 5.76 Lctur #6 //94 Pag of 8 pag Svral poibl typ of intrnal diplacmnt.. Bond Strtch. Valnc angl bnd. angl btwn a bond and a plan (non-planar A A tat of HCO) dfind by bond 4. torion tran-bnt xcitd A A u tat of HCCH. Bond Strtch S t r t t only nonzro Atom Atom Ar th contraint atifid? r vctor (vn in a long linar chain)! ˆ ˆ t t + t + 0! R ( ˆ) ( ˆ t R + R ) R R ˆ ( ) unit diplacmnt S t ({ ξ } ) ˆ ( ρ ρ ) t t t t (diplacmnt of all othr atom hav no ffct on r ) Th ar th vctor rprntation of S r. R R R R R R 0! cntr of ma

4 5.76 Lctur #6 //94 Pag 4 of 8 pag. Valnc Angl Bnd S t φ ² φ r φ r t t Exactly atom ar involvd. nonzro t. How to mov ach atom to incra φ by maximum amount? tan² φ ² φ t r How to dfin a UNIT VECTOR pointing in corrct dirction? ˆ ˆ ˆ ˆ Rcall ê ê t in φ right hand rul to plan, up out of board in φ Rul for vctor tripl product ˆ ˆ t t ˆ ˆ ˆ ˆ ˆ ˆ ( ) ( ) co φˆ in φ ˆ in φ co φ Now, how much do unit diplacmnt of atom in ˆ dirction incra S t? t tan ² S t ²S t unit r r t co φˆ ˆ ˆ t t t r in φ thi i a vctor of pcifid lngth and dirction

5 5.76 Lctur #6 //94 Pag 5 of 8 pag imilarly for atom t coφê ê r in φ now for th hard on: atom! Eay way: impo contraint t 0 ( ) t t+ t ( co φ ) ˆ + ( co φ) r r r r ˆ rr in φ Hard way: mov atom unit in optimal dirction, thn tranlat dformd molcul rigidly back to put atom back at it original poition. Thi vidntly lav atom and diplacd by t and t rpctivly. mov atom tranlat ditortd tructur back to put atom in original location t t t ( ) t t+ t Thi obviouly atifi t 0 It i hardr to how that it alo atifi 0 R t. Grind out th algbra! ( Non-Lctur on nxt pag)

6 5.76 Lctur #6 //94 Pag 6 of 8 pag Altrnativ dfinition of S θ a a linar diplacmnt rathr than an angular diplacmnt i poibl..g. r φ, r φ, or (r r ) / φ. Thn S φ would hav dimnion of lngth and all bnding forc contant would hav am unit a trtching on. Th drivation of S φ would follow am path, but ach t gt multiplid by th rlvant lngth factor, r or r or (r r ) /. Proof that t atify Eckart Condition? R + R + R t 0 t t t t t co φˆ ˆ r in φ NON-LECTURE co φˆ ˆ r in φ ( co φ ) ˆ + ( co φ) r r r r ˆ rr in φ R R R R R + R R R + R? ( R R) + ( R R) + R R ( + + ) ( R + R ) 0 t t t? 0 t t t t t 0 R ê 0 R ŝ 0 0? ê R ê R r in φ r in φ R ê r ê ê r ê ê R ê r ê ê QED

7 5.76 Lctur #6 //94 Pag 7 of 8 pag G-Matrix G N tt t t m Could comput dirctly from WDC, pag 05. t, but air to u diagram from WDC pag 04 and tabl on DIAGRAMS an atom involvd in both t and t diagonal trtch-trtch off-diagonal trtch-adjacnt trtch G rr # of atom common to both t and t trtch diagonal bnd-bnd G φφ off-diagonal bnd-intrnal trtch G rφ tc. TABLE G rr µ + µ G φφ µ m µ cφ cφ coφ ρ µ +ρ µ +ρ ( +ρ ρ ρ cφ)µ ρ ij r ij ( ) G rφ ρ µ φ φ in φ 4 ci bnt actyln ² r ² r ²r ² r ² r4 ² r²t

8 5.76 Lctur #6 //94 Pag 8 of 8 pag G: 0 G rφ G rφ G matrix ha 6 7 indpndnt lmnt G rτ 4 G φφ 5 G φφ 6 4 G ττ S J. C. Dciu Journal of Phyical Chmitry 6 05 (948)! for torion and out of plan bnd ditortion!