3. Prior Distribution

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Imogene Clarke
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1 004 Reactin Dynamics, (c) A. Miyshi 3 3. Prir Distributin - statistically equal distributin in final states 3. Rtatinal distributin and rtatinal sum Phtdissciatin f HCN at 5 nm (465 cm ): HCN + hν (465 cm ) H + CN( 0) + 3 cm H + CN( ) + 5 cm * rel. translatin + CN rtatin Prbability f finding CN in a specific rratinal state ρ [ ] ( J ) g( J ) ρ ε ( ) ε ( J ) trans rt channel-0 channel-, (3..) Rtatinal distributin f CN ia channel- ( J + ) ρ trans [ 5 BJ ( J + ) ] ( J + ) [ J ( J + ) ] / (J max 4, fig. 3.) Prbability f finding CN in a ratinal state (indistinctie f rt. states) Rtatinal sum: ρ [ ] ( ) g( J ) ρ ε ( ) ε ( J ), [ ε ( J ) ε ( ) J trans (summatin integratin) ε ( ) (D) ρ ρ ε ( ) rt [ rt ] ρtrans ε ( ) 0 ε 0 ( ) c (D) rt ctrans rt rt [ ε ] [ ε ( ) ε ] / rt rt dε dε rt rt * ] (3..) c 3 (D) rt ctrans Branching rati f channel-0 : channel- (3) 3/ : (5) 3/ ~ 4.66 : 3/ [ ε ( )] Prblem-3. [OPTION] Derie an equatin similar t 3..3 fr the frmatin f an atm and a nn-linear mlecule. (3..3) [Rtatinal sum] Rtatinal sum fr tw fragments frmatin: rt sum rt sum n + r ρ C (3..4) C r nt nr, i Γ Γ mr ( n ) i r, i ctrans crt nt + nr Γ i ε ( i ) : nergy in translatin and rtatin (*: Ttal excess energy) rt sum i n t : Translatinal degree f freedm ( 3), f freedm ( m n r, i ) i r m (3..5) m r : Number f rtatrs, n r : Ttal rtatinal degree

2 004 Reactin Dynamics, (c) A. Miyshi 3 fr integer n : Γ( n ) ( n )!, Γ () Γ( ) / / fr half integer n : Γ ( / ) π, ( 3/ ) Γ( / ) ( / ) π / Γ,... Prblem-3. ) Calculate the prir ratinal energy distributin f OH frmed frm O( D) + H OH + H ( H 0K 8. kj ml * ) at 98 K. (ttal excess energy) H 0K +. 5RT, (. freq.) ~ ν (OH) 3568 cm. ) [OPTION] Calculate the prir rtatinal distributin OH in its 0 state frmed in the abe reactin. (rt. cnst.) B(OH) 8.5 cm. 3. Vibratinal sum Branching prbability fr a reactin channel ρ ( ) ρ sum rt sum (ratinal sum) O( D) + HD H + OD channel-a D + OH channel-b H 0K [O( D) + H OH + H] 8. kj ml Vibratinal frequencies (cm ) - OH: 3568, OD: 63, H : 46, HD: 3633 Rtatinal cnstants (cm ) - OH: 8.5, OD: 9.87 (after ZP & thermal 98 K) *(a) 595, *(b) 5484 (cm ) 3/ µ H + OD µ D + OH Crt sum : Crt sum ( b) : ~ :. 38 BOD BOH max (a) 6 [579 cm ], max (b) 4 [47 cm ] ρ a : ρ b ρ a, : ρ b, Branching rati ( ) ( ) ( ) ( ) 6 0 3/ sum sum rt sum rt sum [ 63 ] 3/ 4 : C ( b) [ ( b) 3568] 3/ :. 07 Crt sum rt sum cf.) summatin integratin cl Crt 3/ Crt sum 5/ ρ sum cl [ cl ] d [ cl ] h ε ε (3..) ν 5 hν OD sum ( OD) 0 0 ( ) * * * * nte) cl + ZP ( OD) 768 ( b) + ZP( OH) cl ( b) Crt sum Crt sum ( b) ρ sum cl : ρ sum cl ( b) : :.09 hν( OD) hν( OH)

3 004 Reactin Dynamics, (c) A. Miyshi 3 3 [Vibratinal sum] General frmula by replacement f summatin by integratin: Γ( r + ) ( l + n + ) ( l n ) C ρ sum cl (3..) Γ l r ( n +) / r Prblem-3.3 l rt sum r + n cl r hν i i, n : ratinal degree f freedm, ν i : ratinal frequency cl : excess energy measured frm the classical rigin O( D) + CH 4 OH + CH 3 ( H 0K 8.3 kj ml ): Prir ratinal distributin f OH (98 K) Mtin ther than OH-ratin (3..), hν OH 3568 cm ZP(CH 3 ) 76. kj ml * cl ( ) [kj ml ] + 3RT cm cf. * 8.3 [kj ml ] + 3RT 5860 cm max 4 [47 cm ], l r (3 + + ) / 3, n 6, l r + n 9 9 Vibratinal distributin ( ) ( ) cl cl hν OH (fig. 3.) Calculate the prir ratinal distributin f OH frmed by O( D) + C H 6 OH + C H 5 ( H 0K 0.7 kj ml, ZP(C H 5 ) 48.5 kj ml ) similarly t abe, and cmpare it with the cases fr O( D) + H and O( D) + CH Degeneracy f the electrnic states [Atms] (except fr the excited states f rare gas atms) (spectrum) S+ Term : [L] J S : electrn spin q. n. L : electrn rbital angular mmentum q. n. J: ttal angular mmentum q. n. J L S, L S +, K, L + S [L]: symblic representatin f L - S, P, D, F, G, H,... (fr L 0,,, 3, 4, 5,...) S + L + g LS Ttal degeneracy: ( )( ) g J Degeneracy f the fine-structure state: J +

4 004 Reactin Dynamics, (c) A. Miyshi 3 4 term L S g LS D g J J fine str. term S 0 / / P 3 P / 6 9 / P / 4 3/ P 3/ 0 3 P P 5 3 P branching rati HCl + hν H + Cl( P 3/ ) (a) 4 ρ trans (a) H + Cl( P / ) (b) ρ trans (b) [Linear mlecules] (spectrum) S Term : + ( + [ Λ] ) Λ+Σ Λ: prjectin f L t the mlecular axis Σ: prjectin f S t the mlecular axis [Λ]: symblic representatin f Λ - Σ, Π,, Φ,... (fr Λ 0,,, 3,...) (+ ): parity (+ r, nly fr Σ states) Ttal degeneracy: g Λ S ( S + ) g Λ ; g Λ ( Λ 0) r ( Λ 0) Degeneracy f the fine-structure state: g ( ) term Λ S g Λ S Λ Λ +Σ + g Λ Λ+Σ 0 fine str. term Σ 0 / ±/ Π 3 Π / 4 6 / Π / 3/ Π 3/ 0 3 Π 0 3 Π 3 Π branching rati NH 3 H + NH(X 3 Σ ) (X) 3 ρ sum(x) H + NH(a ) (a) ρ sum (a)

5 004 Reactin Dynamics, (c) A. Miyshi 3 5 [Nn-linear mlecules] (spectrum) S+ Term : Γ Γ: symmetry species f the electrnic state... A, A', A, B,, F, etc. gγ S S + gγ; gγ,,3,4,5, K fr Γ [ A, B],, T ( F), G, H, nn-degenerates (A, B) and f spherical grup: n angular mmentum f cylindrical grup, T[F], G, H,... f spherical grup: (nn-integer) angular mmentum Degeneracy f the fine-structure state: g Γ Ttal degeneracy: ( ) K term S g Γ S gγ T A / fine str. term / 4 / 3/ branching rati ~ CH 3 OH H + CH 3 O ( X ) (X) 4 ρ sum (X) ~ H + CH 3 O A A ) (A) ρ sum(a) ( 3.4 Number f ptical ismers and rtatinal symmetry number [Number f ptical ismers] branching rati CFCl Br + hν Cl( P 3/ ) + CFClBr (a) ρ sum (a) Br( P 3/ ) + CFCl (b) ρ sum (b) channel-a tw ptical ismers fr CFClBr ( tw reactin pathway inersin dubling)

6 004 Reactin Dynamics, (c) A. Miyshi 3 6 [Rtatinal distributin and nuclear spin statistics] Nuclear spin I : g I I + Resultant ttal nuclear spin f a mlecule T: g T T + Bse/Fermi particle beys Bse/Fermi statistics (symmetric/asymmetric t permutatin) H ( 0) Rtatinal state distributin - H nucleus : I / ( prtn / Fermi particle) ψ tt shuld be Asymmetric - ψ elec ( Σ + g ) : Sym., ψ ( 0) : Sym. rth-h para-h T ( ) 0 ( ) ψn.s Sym. Asym. ψrt Asym. Sym. J, 3,...(dd) 0,,...(een) g T g J 3 (J + ) (J + ) N r D ( 0) Rtatinal state distributin - N r D nucleus : I (Bse particle) ψ tt shuld be Symmetric rth-n /D para-n /D T 0( ) r ( ) ( ) ψn.s Sym. Asym. ψrt Sym. Asym. J 0,,...(een), 3,...(dd) g T g J (+5) (J + ) 3 (J + ) σ either in sym. r asym. rtatinal state, rtatinal density f states is / Prblem-3.4

Born-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated.

Born-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated. Rtatinal transitins Vibratinal transitins Electrnic transitins Brn-Oppenheimer Apprximatin is the assumptin that the electrnic mtin and the nuclear mtin in mlecules can be separated. This leads t mlecular