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1 CH CH CH 3 CH SFG 1 1 SFG SFG GF 8 4 CH 3 C 3v 31 CH CH c CH c µ c α c α cc a b CH α aa = α bb α aa = r α cc µ c α cc α aa CH r CH 1 ( µ c / r CH ) 0 ( α/ r CH ) 0 β 0 = ( α cc / r CH ) 0 ( µ c / r CH ) 0 rβ 0 = ( α aa / r CH ) 0 ( µ c / r CH ) 0 (0.1a) (0.1b) CH CH 3 CH (1) CH HCH high-frequency isolation () 1 CH CH SFG CH CH CH c β ccc = G CH β 0 (Q CH ) v+1,v β aac = β bbc = G CH rβ 0 (Q CH ) v+1,v (0.) G CH = 1 m C + 1 m H = m C + m H m C m H (Q CH ) v+1,v [< v + 1 Q CH v >] = h v + 1 πcω CH CH - 1

2 r = α aa /α cc, (c.f., depolarization ratio ρ = (1 + r 1 r ) ) vinyl-ch CH CH HCH C H c HCH a c HCH HCH α a 1 (1 + r) + (1 r) cosα β ccc = G a1 [ cos(α /)]β 0 (Q a1 ) v+1,v (1 + r) (1 r) cosα β aac = G a1 [ cos(α / )]β 0 (Q a1 ) v+1,v (0.3) β bbc = G a1 [r cos(α /)]β 0 (Q a1 ) v+1,v G a1 = 1+ cosα m C + 1 m H = m C + m H (1+ cosα) m C m H =(3m C + m H )/(3m C m H ) = 1 m H m C for methylene group (α = ), (m C + m H )/(m C m H ) = 1 m H m C for vinyl CH (α = 10 ). (Q a1 ) v+1,v [< v + 1 Q a1 v >] = h πcω a1 (v + 1) (c.f., depolarization ratio ρ = b r 1 r ) β caa = G b1 [ 1 r sinα sin(α /)]β 0(Q b1 ) v+1,v β aca = G b1 [ 1 r (0.4) sinα sin(α /)]β (Q 0 b1 ) v+1,v G b1 = 1 cosα + 1 = m C + m H (1 cosα ) w m C m H m C m H =(3m C + 4m H )/(3m C m H ) = 1 m H m C for methylene group (α = ), (m C + 3m H )/(m C m H ) = 1 m H m C for vinyl CH (α = 10 ). (Q b1 ) v+1,v [< v + 1 Q b1 v >] = h πcω b1 (v + 1) IR (dµ a1 /dr CH ) /(dµ b1/dr CH ) G a1 cos (α/)/g b1 sin (α/) Raman CH -

3 I a1 /I b1 G a1 [30(1 + r) + 7(1 - r) ]/[14G b1 (1 - r) ] (1 + r) /(1 - r) CH 3 C 3 C H 3 c 3 HCX a b HCX HCX τ a 1 β ccc = 3G sym [ r + (1 r) cos τ β aac = β bbc = 3G sym [ r + (1 r)sin τ G sym = cosτ ]β 0 (Q a1 ) v+1,v cosτ ]β 0 (Q a1 ) v+1,v (0.5) 1+ cosτ m C + 1 m H = m C + m H (1 + cosτ ) m C m H = 1 m H (Q sym ) v+1,v [< v + 1 Q sym v >] = (Q sym ) v,v+1 [< v Q sym v + 1 >] = (c.f., depolarization ratio ρ = 3 4 h πcω sym (v + 1) r 1 r ) 1 m C e β caa = β cbb = β aca = β bcb = 3G deg [ 1 r sin τ cosτ ]β 0 (Q deg ) v+1,v β aaa = β bba = β abb = β bab = 3 G deg [1 r sin3 τ ]β 0 (Q deg ) v+1,v (0.6) G deg = 1 cosτ m C + 1 m H = m C + m H (1 cosτ ) m C m H = 1 m H (Q deg ) v+1,v [ < v + 1 Q deg v >] = (Q deg ) v 1,v [< v 1 Q deg v >] = 1 m C h πcω deg (v + 1)(v + ) h πcω deg v(v +1) l v + 1 v v = 1 v v + 1 E = A 1 + E (v = ), E 3 = E (v = 3) IR (dµ sym /dr CH ) /(dµ deg /dr CH ) G sym cos τ/g deg sin τ Raman I sym /I deg G sym [15(1 + r) + (1/4)(1 - r) (1-3cos τ) ]/[(63/)G deg (1 - r) sin τ(1 + 3cos τ)] (1 + r) /(1 - r) (v = 1-0) η vib = β 0 (Q vib ) 1,0 CH - 3

4 η sym = β 0 (Q sym ) 1,0 η deg = β 0 (Q deg ) 1,0 = ω sym /ω deg 1.95 η sym η a1 = β 0 (Q a1 ) 1,0 = ω sym /ω a η sym η b1 = β 0 (Q b1 ) 1,0 = ω sym /ω b η sym SFG β aac β aac * CH 3 CH β aac = β aac * β aac = β aac * β ccc = 0.5 β aac * β ccc = β aac * β aaa = 1.49 β aac * β aaa = β aac * β caa = β aac * β caa = β aac * =CH CH CH β aac β + aac CH CH C=C c a =CH CH β aac = β aac + β ccc = 0.33 β aac + β aaa = β caa = 1.04 β aac + + β aac = β aac + β acc = β cac = β cca = β aac + β ccc = β aac + β aaa = β aac + β aca = β caa = β aac 1 SFG CH (dµ/dr) 0 r = µ CH r (dα/dr) 0 r = a 0 r r CH c (mass adjusted) HCH HCC CC CO potential energy distribution = PES CH CH 3 CH CH CH [(dq/dt) + λq ] Q CH = 1 G CH r CH (1.1) CH - 4

5 G CH G CH = (m C + m H )/m C m H = 1/m H + 1/m C (1.) V K V = K( r) = KG CH Q CH (1.3) F CH = K λ CH = F CH G CH [ν (cm -1 ) = (λ CH )/πc] CH CH r 1 r Q a1 Q b1 Q a1 = r 1 + r 1 G a1 (1.4a) Q b1 = r 1 r 1 G b1 (1.4b) G a1 G b1 HCH α G a1 = [m C + m H (1 + cosα)]/m C m H = 1/m H + (1 + cosα)/m C G b1 = [m C + m H (1 - cosα)]/m C m H = 1/m H + (1 - cosα)/m C (1.5a) (1.5b) V T d cosα = -1/3 V = K[( r 1 ) + ( r ) ] + K' r 1 r = (K + K' + 4F/3)G a1 Q a1 + (K - K' - 0.F/3)G b1 Q b1 (1.6) F Shimanouti field H H repulsion term F F '' = -0.1F Td λ a1 = (K + K' + 4F/3)G a1 λ b1 = (K - K' - 0.F/3)G b1 CH 3 3 CH r 1 r r 3 Q sym Q deg,a Q deg,b ac H 1 Q sym = r 1 + r + r G sym (1.7a) Q deg,a = r 1 r r G deg (1.7b) Q deg,b = r r 3 1 G deg (1.7c) G sym = [m C + m H (1 + cosτ)]/m C m H = 1/m H + (1 + cosτ)/m C G deg = [m C + m H (1 - cosτ)]/m C m H = 1/m H + (1 - cosτ)/m C (1.8a) (1.8b) τ HCH T d cosτ = -1/3 CH - 5

6 V V = K[( r 1 ) + ( r ) + ( r 3 ) ] + K'[ r 1 r + r r 3 + r 1 r 3 ] = (K + K' +.03F)G sym Q sym + (K - K' F)G deg Q deg (1.9) λ sym = (K + K' +.03F)G sym λ deg = (K - K' F)G deg CH CH 3 CH CH CH CH CH [h/(4π cω)] 1/ Q µ vib = (dµ CH /dr) 0 ( r) = (dµ CH /dr) 0 G CH Q CH (1.10) CH r 1 r [ r 1 sin(α/), 0, r 1 cos(α/)] [ - r sin(α/), 0, r cos(α/)] Q a1 Q b1 c a e a e c a c µ vib = (dµ CH /dr) 0 ( r 1 + r ) = (dµ CH /dr) 0 [e a ( r 1 - r )sin(α/) + e c ( r 1 + r )cos(α/)] = (dµ CH /dr) 0 [ G a1 cos(α/)q a1 + G b1 sin(α/)q b1 ] (1.11) µ/ Q a1 = (dµ CH /dr) 0 G a1 cos(α/) µ/ Q b1 = (dµ CH /dr) 0 G b1 sin(α/) (1.1) [cos(α/) = 1 /3, sin(α/) = / 3 for methylene] and [cos(α/) = -1/, sin(α/) = 3 / for vinyl CH )] CH 3 µ vib = (dµ CH /dr) 0 ( r 1 + r + r 3 ) = (dµ CH /dr) 0 [e a ( r 1 - r - r 3 )sinτ + e b ( r - r 3 )sinτ + e c ( r 1 + r + r 3 )cosτ] = (dµ CH /dr) 0 3G sym cosτq sym + 6G edeg / sinτ[q deg,a + Q deg,b ]} (1.13) CH - 6

7 µ/ Q sym = (dµ CH /dr) 0 3G sym cosτ µ/ Q deg = (dµ CH /dr) 0 6G edeg / sinτ (1.14) (cosτ = -1/3, sinτ = 8 /3) CH (c ) CH C H ζ ξ η (dα/dr) 0 r = α 0 r (dα ζζ /dr) 0 r = α 0 r (dα ξξ /dr) 0 r = (dα ηη /dr) 0 r = rα 0 r (1.15) CH (ξ, η, ζ) (a, b, c) (1) (E a, E b, E c ) (E ξ, E η, E ζ ) () (P ξ, P η, P ζ ) (3) (P a, P b, P c ) P a E a (4) r i α aa 1 CH C H c a b (a, b, c) (ξ, η, ζ) CH E c E a E b CH P c = (dα zz /dr) 0 re c = a 0 E c r = a 0 G CH Q CH E c P a = (dα xx /dr) 0 re a = ra 0 E a r = ra 0 G CH Q CH E a P b = (dα hh /dr) 0 re b = ra 0 E b r = ra 0 G CH Q CH E b a 0 = (dα zz /dr) 0 ζ ξ H (a) (1.1) Q CH = (1/ G CH ) r (c) α C ξ 1 H 1 CH - 7 ζ 1

8 α cc = a 0 G CH Q CH α aa = α bb = ra 0 G CH Q CH (1.16) CH CH HCH ξζ η CH HCH α C H c HCH H 1 H a E a E b E c CH ξ η ζ E ξ (1) = E a cosα/ + E c sinα/ E η (1) = E b E ζ (1) = -E a sinα/ + E c cosα/ E ξ () = E a cosα/ - E c sinα/ E η () = E b E ζ () = E a sinα/ + E c cosα/ (1.17) CH(1) CH() P ξ (1) = (dα ξξ /dr) 0 r 1 E ξ (1) = ra 0 r 1 E ξ (1), P η (1) = (dα ηη /dr) 0 r 1 E η (1) = ra 0 r 1 E η (1), P ζ (1) = (dα ζζ /dr) 0 r 1 E ζ (1) = a 0 r 1 E ζ (1), P ξ () = (dα ξξ /dr) 0 r E ξ () = ra 0 r E ξ (), P η () = (dα ηη /dr) 0 r E η () = ra 0 r E η (), P ζ () = (dα ζζ /dr) 0 r E ζ () = a 0 r E ζ () (1.18) CH(1) CH() (ξ, η, ζ) (a, b, c) η α/ CH(1) CH() (a, b, c) P a = P ξ (1)cosα/ + P ξ ()cosα/ - P ζ (1)sinα/ + P ζ ()sinα/ P b = P η (1) + P η () P c = P ξ (1)sinα/ - P ξ ()sinα/ + P ζ (1)cosα/ + P ζ ()cosα/ (1.9) (1.17) (1.18) P a = a o {[(1 - cosα) + r(1 + cosα)]( r 1 + r )/ E a [(1 - r)sinα]( r 1 - r )/ E c } P b = ra o ( r 1 + r )E b P c = a o {[(1 + cosα) + r(1 - cosα)]( r 1 + r )/E c [(1 - r)sinα]( r 1 - r )/ E a } (1.0) (1.4) CH - 8

9 P a = a o {[(1 - cosα) + r(1 + cosα)] G a1 Q a1 E a [(1 - r)sinα] G b1 Q b1 E c } P b = ra o (G a1 )Q a1 E b P c = a o {[(1 + cosα) + r(1 - cosα)] G a1 Q a1 E c [(1 - r)sinα] G b1 Q b1 E a } (1.1) a 1 α aa = a o [(1 - cosα) + r(1 + cosα)] G a1 Q a1 α bb = ra o (G a1 )Q a1 α cc = a o [(1 + cosα) + r(1 - cosα) G a1 Q a1 (1.) b 1 α a c = a o [(1 - r)sinα] G b1 Q b1 α ca = a o [(1 - r)sinα] G b1 Q b1 (1.3) CH(1) CH() (ξ, η, ξ) (a,b,c) b -α/ +α/ b φ c' θ b" χ (a, b, c) (a", b", c") (χ, θ, φ) (0, -α/, 0) (0, +α/, 0) (ξ, η, ξ) (a,b,c) (0, +α/, 0) (0, -α/, 0) V (V a, V b, V c ) (V ξ, V η, V ζ ) V ξ V η V ζ [CH(1)] cos(α / ) 0 sin(α /) = sin(α / ) 0 cos(α / ) V a V b V c V ξ V η V ζ [CH()] cos(α /) 0 - sin(α / ) = sin(α / ) 0 cos(α / ) V a V b V c (1.4a) V a V b V c cos(α / ) 0 - sin(α /) = sin(α /) 0 cos(α / ) V ξ V η V ζ V a V b V c cos(α / ) 0 sin(α / ) = sin(α / ) 0 cos(α /) V ξ V η V ζ (1.4b) (1.4a) (1.4b) V ξ (1) = U(-α/)V a, V ξ () = U(+α/)V a (1.5a) V a = U(+α/)V ξ (1), V a = U(-α/)V ξ () (1.5b) (1.17) ~ (1.1) (1.17) E ξ (1) = U(-α/)E a, E ξ () = U(+α/)E a (1.6) (1.18) (dα ξξ /dr) 0 r 1 (dα ξξ /dr) 0 r α ξξ CH - 9

10 P ξ (1,) = P ξ (1,) α ξξ 0 0 E P η (1,) = ξ (1,) 0 α ηη 0 E η (1,) P ζ (1,) 0 0 α ζζ E ζ (1,) = α ξ E ξ (1,) (1.7) CH (a, b, c) (1.19) P a (1,) = U(±α/)P ξ (1,) = U(±α/) α ξ E ξ (1,) (1.6) P a (1) = U(+α/)α ξ U(-α/)E a (1) P a () = U(-α/)α ξ U(+α/)E a () (1.8) U(+α/)α ξ U(-α/) U(-α/)α ξ U(+α/) CH(1) CH() α ξ α a (1.4a) (1.4b) α aa (1,) = [(5cosα + 7)/1]α ξξ + [(cosα - 1)/1]α ηη - [(cosα - 1)/]α ζζ α bb (1,) = [( -cosα + 1)/1]α ξξ + [(cosα + 11)/1]α ηη α cc (1,) = -[(cosα - 1)/3]α ξξ - [(cosα - 1)/6]α ηη + [(cosα + 1)/]α ζζ α ca (1) = α ac (1) = -α ca () = -α ac () = (sinα/)(α ζζ - α ξξ ) P a (1) P a () CH r 1 r (1.15) α aa = {[(5cosα + 7)/1]α ξξ + [(cosα - 1)/1]α ηη - [(cosα - 1)/]α ζζ }[ r 1 + r ] = a 0 [r(cosα + 1)/ - (cosα - 1)/][ r 1 + r ] α bb = {[(-cosα + 1)/1]α ξξ + [(cosα +11)/1]α ηη }[ r 1 + r ] = a 0 r[ r 1 + r ] α cc = { -[(cosα - 1)/3]α ξξ - [(cosα - 1)/6]α ηη + [(cosα + 1)/]α ζζ }[ r 1 + r ] = a 0 [ -r(cosα - 1)/ + (cosα + 1)/][ r 1 + r ] α ca = α ac = (sinα/)(α ζζ - α ξξ )[ r 1 - r ] = a 0 [(1 - r)sinα/][ r 1 - r ] (1.) CH 3 C 3 CH(1) ξζ CH ξ CH η 10 c C 3 C H 3 a XCH(1) C 3 H 1 C CH - 10

11 H(1) H() H(3) CH(1) CH() CH(3) (ξ, η, ζ) (a,b,c) (0, π + τ, 0) (0, π + τ, π/3) (0, π + τ, -π/3) (ξ, η, ζ) (a,b,c) (0, π - τ, 0) (-π/3, π - τ, 0) (π/3, π - τ, 0) (CH(1)) (ξ, η, ζ) (a,b,c) -cosτ 0 sinτ U(0, π + τ, 0) = sinτ 0 -cosτ (1.9a) (ξ, η, ζ) (a,b,c) -cosτ 0 -sinτ U 1 (0, π + τ, 0) = U(0, π τ, 0) = sinτ 0 -cosτ (1.30a) (CH()) (ξ, η, ζ) (a,b,c) -cosτ 0 sinτ -1 / - 3 / 0 U(0, π + τ, - π /3) = / -1 / 0 - sinτ 0 -cosτ /cosτ 3 / cosτ sinτ = 3 / -1 / 0 1 /sinτ 3 / sinτ -cosτ (1.9b) (ξ, η, ζ) (a,b,c) -1 / 3 / 0 -cosτ 0 -sinτ U 1 (0, π + τ, - π /3) = U (π / 3, π τ, 0) = - 3 / -1 / sinτ 0 -cosτ 1/ cosτ 3 / 1/ sinτ = 3 /cosτ -1 / 3 / sinτ sinτ 0 -cosτ (1.30b) (CH(3)) (ξ, η, ζ) (a,b,c) CH - 11

12 -1 / 3 / 0 -cosτ 0 sinτ U(0, π + τ, π / 3) = - 3 / -1 / sinτ 0 -cosτ 1/ cosτ - 3 / -1 / sinτ = 3 /sinτ -1 / - 3 / sinτ -sinτ 0 -cosτ (1.9c) (ξ, η, ζ) (a,b,c) -1 / - 3 / 0 -cosτ 0 -sinτ U 1 (0, π + τ, π / 3) = U( π / 3, π τ, 0) = 3 / -1 / sinτ 0 -cosτ 1 /cosτ - 3 / 1/ sinτ = - 3 /cosτ -1 / - 3 / sinτ sinτ 0 -cosτ (1.30c) α a = U -1 α ξ U (CH(1)) α ξξ cos τ + α ζζ sin τ 0 ( α ξξ + α ζζ )sinτ cosτ α a (1)= 0 α ηη 0 ( α ξξ + α ζζ ) sinτ cosτ 0 α ξξ sin τ + α ζζ cos τ (1.31a) (CH()) α a ()= 1 4 α ξξ cos τ + 3α ηη + α ζζ sin τ 3(α ξξ cos τ α ηη + α ζζ sin τ ) (α ξξ α ζζ )sinτ cosτ 3(α ξξ cos τ α ηη + α ζζ sin τ ) 3α ξξ cos τ + α ηη + 3α ζζ sin τ 3(α ξξ α ζζ ) sinτ cosτ (α ξξ α ζζ )sin τ cosτ 3(α ξξ α ζζ ) sinτ cosτ α ξξ sin τ + α ζζ cos τ (CH(3)) α a (3)= 1 4 (1.31b) α ξξ cos τ + 3α ηη + α ζζ sin τ 3(α ξξ cos τ α ηη + α ζζ sin τ ) (α ξξ α ζζ )sinτ cosτ 3(α ξξ cos τ α ηη + α ζζ sin τ ) 3α ξξ cos τ + α ηη + 3α ζζ sin τ 3(α ξξ α ζζ ) sinτ cosτ (α ξξ α ζζ )sin τ cosτ 3(α ξξ α ζζ ) sinτ cosτ α ξξ sin τ + α ζζ cos τ (1.31c) CH α aa = (1/)[α ξξ cos τ + α ηη + α ζζ sin τ][ r 1 + r + r 3 ] + (1/4)(α ξξ cos τ - α ηη + α ζζ sin τ)[ r 1 - r - r 3 ] CH - 1

13 α bb = (1/)[α ξξ cos τ + α ηη + α ζζ sin τ][ r 1 + r + r 3 ] - (1/4)(α ξξ cos τ - α ηη + α ζζ sin τ)[ r 1 - r - r 3 ] α cc = [α ξξ sin τ + α ζζ cos τ][ r 1 + r + r 3 ] α ab = α ba = ( 3/4)(α ξξ cos τ - α ηη + α ζζ sin τ)[ r - r 3 ] α bc = α cb = ( 3/)(α ξξ - α ζζ )sinτcosτ[ r - r 3 ] α ac = α ca = (-1/)(α ξξ - α ζζ )sinτcosτ[ r 1 - r - r 3 ] (1.3) (1.7) α aa = ( 3/)[α ξξ cos τ + α ηη + α ζζ sin τ] G sym Q sym + ( 6/4)(α ξξ cos τ - α ηη + α ζζ sin τ) G deb Q deg,a α bb = ( 3/)[α ξξ cos τ + α ηη + α ζζ sin τ] G sym Q sym - ( 6/4)(α ξξ cos τ - α ηη + α ζζ sin τ) G deb Q deg,a α cc = 3[α ξξ sin τ + α ζζ cos τ] G sym Q sym α ab = α ba = ( 6/4)(α ξξ cos τ - α ηη + α ζζ sin τ) G deb Q deg,b α bc = α cb = ( 6/)(α ξξ - α ζζ )sinτcosτ G deb Q deg,b α ac = α ca = (- 6/)(α ξξ - α ζζ )sinτcosτ G deb Q deg,a (1.33) CH α ξξ = α ηη = rα ζζ (dα ζζ /dr) 0 = a 0 α aa = a 0 {( 3/)[(1 - cos τ) + r(1 + cos τ)] G sym Q sym + ( 6/4)(1 - cos τ)(1 - r) G deb Q deg,a } α bb = a 0 {( 3/) [(1 - cos τ) + r(1 + cos τ)] G sym Q sym - ( 6/4)(1 - cos τ)(1 - r) G deb Q deg,a } α cc = a 0 { 3[cos τ + r(1 - cos τ)] G sym Q sym } α ab = α ba = a 0 {( 6/4)(1 - cos τ)(1 - r) G deb Q deg,b } α bc = α cb = a 0 {(- 6/)(1 - r)sinτcosτ G deb Q deg,b } α ac = α ca = a 0 {( 6/)(1 - r)sinτcosτ G deb Q deg,a } (1.34) SFG SFG β ijk = α ij µ k (1.35) b 0 = (dα ζζ /dr) 0 (dµ CH /dr) o CH β aac = β bbc = rb 0 G CH Q CH β ccc = b 0 G CH Q CH (1.36a) (1.36b) CH (1.11) 3 Q a1 c a 1 (1-35) k c Q b1 CH - 13

14 a b 1 (1.35) k a a 1 β aac = b 0 [(1 - cosα) + r(1 + cosα)]cosα/ G a1 Q a1 β bbc = rb 0 cosα/ G a1 Q a1 β ccc = b 0 [(1 + cosα) + r(1 - cosα)]cosα/ G a1 Q a1 (1.37a) (1.37b) (1.37c) b 1 β aca = β caa = -b 0 (1 - r)sinα/sinα G b1 Q b1 (1.38) CH 3 (1.13) 3 Q sym c Q deg,a Q deg,b a b a 1 β aac = β bbc = b 0 (3/)[(1 - cos τ) + r(1 + cos τ)]cosτg sym Q sym β ccc = b 0 (3)[cos τ + r(1 - cos τ)]cosτg sym Q sym e β aaa = -β bba = b 0 (3/4)(1 - cos τ)sinτ(1 - r)g deg Q deg,a β caa = β aca = -b 0 (3/)(1 - r)sin τcosτg deb Q deg,a β abb = β bab = -b 0 (3/4)(1 - cos τ)sinτ(1 - r)g deg Q deg,b β bcb = β cbb = -b 0 (3/)(1 - r)sin τcosτg deb Q deg,b (1.39a) (1.39b) (1.40a) (1.40b) (1.40c) (1.40d) CH SFG α ξξ α ηη q Q q = (λ /h ) 1/4 Q, q = (πcω /h)q (Q sym ) v+1,v [< v + 1 Q sym v >] = (Q sym ) v,v+1 [< v Q sym v + 1 >] = h 4πcω sym (v + 1) (.1) CH - 14

15 (Q deg ) v+1,v [ < v + 1 Q deg v >] = (Q deg ) v 1,v [< v 1 Q deg v >] = h 8πcω deg (v + 1)(v + ) h 8πcω deg (v + 1)v (.) (1.4) (1) l l = v, v -, v - 4, 1 or 0,, -(v - ), -v q deg,a q deg,b <v + 1, l - 1 q deg,a v, l> = <v, l q deg,a v + 1, l - 1> = (1/) [(v - l + )/] <v + 1, l - 1 q deg,b v, l> = -<v, l q deg,b v + 1, l - 1> = (i/) [(v - l + )/] <v + 1, l + 1 q deg,a v, l> = <v, l q deg,a v + 1, l + 1> = (-1/) [(v + l + )/] <v + 1, l + 1 q deg,b v, l> = -<v, l q deg,b v + 1, l + 1> = (i/) [(v + l + )/] v l v + 1 l <v + 1 q deg,a v><v q deg,a v + 1> = <v + 1 q deg,b v><v q deg,b v + 1> = (1/4)[(v - l + )/ + (v + l + )/] = (v + )/4 <v - 1 q deg,a v><v q deg,a v - 1> = <v - 1 q deg,b v><v q deg,b v - 1> = (1/4)[(v - l)/ + (v + l)/] = v/4 (.) l v + 1 SFG q deg,a q deg,b () l cm -1 l v + 1 (1-4) (1) "high-frequency isolation" CH CH () CH CH (3) CH 1 C CH 1 CH G D 13 C CH - 15

16 Avogadro constant: , m H = kg/mol = kg, 1/m H = kg -1 m D = kg/mol = 3.345x10-7 kg, 1/m D = kg -1 m C = kg/mol=19.97x10-7 kg, 1/m C = kg -1 m 13C = kg/mol=1.593x10-7 kg, 1/m H = kg -1 CH 3 G sym = m + m C H (1 + cosτ ) = m C m H m H 3m C = (3.157) [6.13, (3.144)] 10 6 kg -1 G deg = m C + m H (1 cosτ ) m C m H = 1 m H + 4 3m C = (3.659) [6.595, (3.607)] 10 6 kg -1 (.3) CH ( ) G a1 = m + m C H (1+ cosτ ) = 1 + m C m H m H 3m C = 6.31 (3.34) [6.86, (3.98)] 10 6 kg -1 G b1 = m C + m H (1 cosτ ) m C m H = 1 m H + 4 3m C = (3.659) [6.595, (3.607)] 10 6 kg -1 (.4) CH ( ) G a1 = m + m C H (1+ cosα ) = m C m H m H m C = 6.8 (3.41) [6.09, (3.1)] 10 6 kg -1 G b1 = m C + m H (1 cosα) m C m H = 1 m H + 3 m C = (3.743) [6.67, (3.684)] 10 6 kg -1 (.5) CH G CH = m C + m H m C m H = 1 m H + 1 m C = (3.49) [6.440, (3.453)] 10 6 kg -1 (.6) 3-1 (1) F CH - 16

17 CH 3 F sym = K CH + k + F - F'/3 F deg = K CH - k + F'/3 (3.1) CH (alkyl) F a1 = K CH + k + 4F/3 F b1 = K CH - k + F'/3 (3.) CH (vinyl) F a1 = K CH + k + 3F/ F b1 = K CH - k + F'/ (3.3) CH F CH = K CH (3.4) 3 CH -CH 3 ~ 960, , 853 cm -1, -CH - ~ 930 ~ 850 cm -1 -CH< ~ 890 cm -1 -CH= 3040 ~ 3010 cm -1 CH = 3095 ~ 3075 cm ~ 965 cm -1 CH 3300 cm -1 -CHO 900 ~ 700 cm -1 ( ) CH(aromatic) ~ 3030 cm -1 ((md/å = 10 N/m) K CH = 4. F H H = 0.44 (in (-CH -CH -) n K CH = 4.4 F H H = 0.1 (CH 4 ) K CH = 3.46 (in HC0 - ) K CH = F H H = 0.03 (in CH 3 C0 - ) Urey-Bradley-Shimanouti k = 0 F' = -0.1F R. G. Snyder, J. Chem. Phys. 47, 1316 (1967) n- valence force field ν sym = 870 cm -1 ν deg = 950 cm -1 ν a1 = 845 cm -1 ν b1 = 915 cm -1 K CH = ± F H H = 0.03 ± 0.00 K CH = ± F H H = ± CH CH 3 K CH = F H H = CH K CH = F H H = K CH = 4.6 F H H = 0.03 md/å (CH 3 ) ν sym = 839 cm -1, ν deg = 933 cm -1 (CH ) alkyl ν a1 = 871 cm -1, ν b1 = 933 cm -1 (CH) vinyl ν a1 = 853 cm -1, ν deg = 95 cm -1 (CH) ν CH = 960 cm -1 (a) (b) () CH (dµ/dr CH ) 0 CH, CH CH 3 CH - 17

18 <v + 1, l - 1 q deg,a v, l> (1.10) ~ (1.14) (1.41) (1.4) v = 1-0 CH I CH = (1/)G CH /ω CH (3.5) CH I a1 = G a1 /ω a1 cos α/ = (1/3)G a1 /ω a1 I b1 = G b1 /ω b1 sin α/ = (/3)G b1 /ω b1 I a1 /I b1 ~ (3.6) CH 3 I sym = (3/)G sym /ω sym cos τ = (1/6)G sym /ω sym I deg = (3/4)G deg /ω deg sin τ = (4/3)G deg /ω deg I sym /I deg ~ (3.7) CH =CH I CH /I b1 /I a1 ~ G CH /3G b1 /G a1 ~ 1.0/1.56/ a = (1/3)(α aa + α bb + α cc ) γ = (1/4)[(α aa - α bb - α cc ) + 3(α aa - α bb ) + 3(α ab + α ba ) + 3(α bc + α cb ) + 3(ϖ ca + α ac ) ] (4.1) 90 I par a = (45a + 4γ )/45 I perp = γ /15 (4.) 45a + 7γ I perp /I para ρ ρ = I perp /I para = 3γ /(45a + 4γ ) (4.3) CH - 18

19 0 1/3 a = 0 3/4 45a = 5[α cc + α cc (α aa + α bb ) + (α aa + α bb ) ] γ = α cc - α cc (α aa + α bb ) + (1/4)(α aa + α bb ) + (3/4)(α aa - α bb ) (4.4) 45a + 7γ = 1α cc + 3α cc (α aa + α bb ) + (7/4)(α aa + α bb ) + (1/4)(α aa - α bb ) (4.5a) α ρ = cc α cc (α aa + α bb ) + (α aa + α bb ) α aa α bb α cc + ( /3)α cc (α aa + α bb ) + (α aa + α bb ) + ( /3)α aa α bb (4.5b) (4.5a) (4.5b) CH a = (1/9)a 0 (1 + r) G CH Q CH γ = a 0 (1 - r) G CH Q CH I CH Raman = (3/45)a 0 (4 + r + 9r )G CH Q CH ρ CH = (4.6) 1 5r( + r) (4.7) 3(1 r) CH a 1 mode b 1 mode a = (/9)a 0 (1 + r) G a1 Q a1 γ = (1/4)a 0 (1 - r) (1 + 3cos α)g a1 Q a1 I a1 Raman = (1/6)a 0 [60(1 + r) + 14(1 - r) ]G a1 Q a1 ρ a1 = 1 (1 r) 10 30(1+ r) + 4(1 r) (4.9) a = 0 γ = (3/4)a 0 (1 - r) sin αg b1 Q b1 I b1 Raman = (14/3)a 0 (1 - r) G b1 Q b1 (4.8) (4.10) ρ b1 = 3/4 (4.11) Raman I a1 Raman = G a1 I b1 G b (1+ r) 7(1 r) (4.1) CH 3 symmetric mode a = (1/3)a 0 (1 + r) G sym Q sym γ = (3/4)a 0 (1 - r) (1-3cos τ) G sym Q sym I sym Raman = a 0 [15(1 + r) + (1/4)(1 - r) (1-3cos τ) ]G sym Q sym CH - 19 (4.13) ρ sym = 3 (1 r) 45 45(1 + r) + 4(1 r) (4.14)

20 degenerate mode a = 0 γ = (9/4)a 0 (1 - r) sin τ(1 + 3cos τ)g deg Q deg (4.15) Raman I deg = (63/4)a 0 (1 - r) sin τ(1 + 3cos τ)g deg Q deg ρ deg = 3/4 (4.16) Raman I sym Raman = I deg 63 G sym G deg 15(1+ r) + (1/ 4)(1 r) (1 3cos τ ) (1 r) (1 + 3cos τ ) sin (4.17) τ r CH r (1-61) r = -1.0 ρ = 0.57 r = -0.5 ρ = 0.75 r = 0.0 ρ = 0.33 r = 0.5 ρ = 0.04 r = -0.9 ρ = 0.61 r = -0.4 ρ = 0.73 r = 0.1 ρ = 0.3 r = 0.6 ρ = 0.0 r = -0.8 ρ = 0.66 r = -0.3 ρ = 0.66 r = 0. ρ = 0.16 r = 0.7 ρ = 0.01 r = -0.7 ρ = 0.68 r = -0. ρ = 0.57 r = 0.3 ρ = 0.10 r = 0.8 ρ = r = -0.6 ρ = 0.74 r = -0.1 ρ = 0.38 r = 0.4 ρ = 0.06 r = 0.9 ρ = maximum at r = -0.5, grradual decrease converging to ρ = 0.15 (ρ = 0.5 at r = -3.0). minimum (ρ = 0.00) at r = 1.0, converging to r = 0.15 at infinite r (ρ = 0.05 at r = 3.0). HCOONa(aq) ρ = 0.3 ± 0.04 r = 0.10 ± 0.05 CHCl 3 CHBr 3 CHF 3 CHBrCl ρ ~ 0. HCCl CCl 3 ρ ~ 0.3 r = 0.05 ~ 0.15 CH IR I a1 /I b1 = 0.49 r I a1 /I b1 ρ a1 r I a1 /I b1 ρ a1 r I a1 /I b1 ρ a1 r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = CH - 0

21 (cm -1 ) IR int Raman int ρ r from ρ a1 CH Cl 984 ~ 45 ~ 41 ~ 0.1 ~ ~ 65 ~ 3 ~ 0.8 CH Br 987 ~ 15 ~ < 0.1 ~ ~85 ~ 4 ~ 0.8 CH I 967 ~50 ~ 14 ~ 0. ~ ~90 ~ 0.5 ~ 0.9 CH BrCl 987 w ~ 70 ~ 0.05 ~ w ~ 5 ~ 0.35 CH ClCCN 987 vs ~ 4 < 0.1 ~ vs ~ 35 ~ 0.6 CH 3 IR I sym /I deg = 0.1 r I sym /I deg ρ sym r I sym /I deg ρ sym r I sym /I deg ρ sym r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r r = ~ 0.1 ~ -0. SFG (cm -1 ) IR int Raman int ρ r from ρ a1 M ~ 6 < 0.1 S ~ 1 ~ 0.1 ~ 13 ~ 0.3 S ~ 11 ~ 0.3 ~ 0. CH 3 CH Br CH 3 CHCl CH 3 CHBr CH 3 F CH 3 Cl CH 3 I ~ 5 P ~ 55 < 0.05 ~ ~ 75 ~ 10 ~ 0.7 P ~ 0.5 P ~ 9 ~ 0.05 ~ dp ~ 1.5 P s ~ 6 ~ 0.1 s ~ 41 ~ 0.1 m ~ 7 ~ 0. ~ 9.5 ~ 0.1 s ~ 3.5 ~ ~ 44 < ~ ~ 0.9 CH - 1 ~ 0.16

22 3051 s CH 3 OH 835 ~ 80 ~ ~ 55 ~ 0.1 CH 3 COCl 938 ~ 0.05 CH 3 COBr 93 w ~ 80 < w ~ 55 ~0.75 CH 3 CN 946 ~ 44 < CH 3 COCCl ~ 3 ~ 0.8 ~ 15 < 0.1 ~ 3 ~ 3 r ~ 0.1 r ~ 0.35 r ~ 0.5 CH IR I CH /I a1 /I b1 = 0.4/0.3/1.0 r I CH /I b1 I a1 /I b1 r CH ρ a1 r I CH /I b1 I a1 /I b1 r CH ρ a1 r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = r = (cm -1 ) IR int Raman int ρ r from ρ a1 H C=CCl 3036 ~ 41 ~ 37 < 0.1 ~ H C=CHCN ~ 3 ~ 48 ~ 3.5 P ~ 19 ~ 0.1 s ~ 3 ~ 0.4 ~ 4 dp ~ 1 P ~ 0.15 ~ (as CH) 1 1 r CH -

23 T V T = (1/)µ[d( r)/dt] V = (1/)K rr ( r) µ = m 1 m /(m 1 + m ) K rr (A.1) r mass-adjusted coordinate Q = (A.1) T = (1/)[dQ/dt] V = (1/)λQ λ = K rr /µ µ r (A.) T E v = hν(v + 1/), v = 0, 1,, ν(s -1 ) = (1/π) K rr /µ q = (4π λ/h ) 1/4 Q = (4π λ/h ) 1/4 µ r (A.3) q <v q v - 1> = <v - 1 q v> = [v/] 1/ (A.4) <v = 1 q v = 0> = [1/] 1/, <v = q v = 1> = [/] 1/ <v q v> = <v q v - 1><v - 1 q v> + <v q v + 1><v + 1 q v> = v + 1/ <v + q v> = <v + q v + 1><v + 1 q v> = [(v + )(v + 1)/] 1/ (A.5) µ α.4,.5 µ = µ e + (dµ/d r) 0 r = µ e + (dµ/dq) 0 q + (A.6) α = α 0 + (dα/d r) 0 r = α 0 + (dα/dq) 0 q + (A.7) µ e α 0 (A.5) H' = K rrr ( r) 3 + K rrrr ( r) 4 + CH - 3

24 3 K rrr v v ± 1 v ± 3 4 K rrrr v v (diagonal) v ± v ± 4 K rrr A: (1) X T(X) T(X) = (1/) m i [(dx i /dt) + (dy i /dt) + (dz i /dt) ] (B.1) m i i ( amu) X i, Y i, Z i i U(R) B: () X' (1) (X 0, Y 0, Z 0 ) () i (X i ', Y i ', Z i ') (X i - X 0, Y i - Y 0, Z i - Z 0 ) A (B.1) T = (1/){M(dX 0 /dt) + (dy 0 /dt) + (dz 0 /dt) + m i [(dx i '/dt) + (dy i '/dt) + (dz i '/dt) ]} M = m i m i X i ' = m i Y i ' = m i Z i ' = 0 () (X i ', Y i ', Z i ') () CH - 4

25 U(X') T(X') T(X') = (1/) m i [(dx i '/dt) + (dy i '/dt) + (dz i '/dt) ] (B.) C: r (a, b, c) r (a, b, c) ab H O b O b H a b ab initio DFT D, E (χ, θ, φ) X' X' r r U(χ, θ, φ) r = U(χ, θ, φ)x' (B.3) X' r (χ, θ, φ) (χ, θ, φ) r Eckart Eckart a b X' r Eckart N r 3N 3N - 6 3N - 5 X' r ω dx'/dt = ω r + dr/dt dr/dt (ω r) = ω (r dr/dt) T = M(dX 0 /dt) + (dy 0 /dt) + (dz 0 /dt) + i m i [(da i /dt) + (db i /dt) + (dc i /dt) ]] + (I aa ω a + I bb ω b + I cc ω c ) - (I ab ω a ω b + (I bc ω b ω c + (I ca ω c ω a ) + (Ω a ω a + Ω b ω b + Ω c ω c ) I aa = i m i [b i + c i ], I bb = i m i [c i + a i ], I cc = i m i [a i + b i ] I ab = i m i a i b i, I bc = i m i b i c i, I ca = i m i c i a i CH - 5

26 Ω a = - i m i [(db i /dt)c i - b i (dc i /dt)], Ω b = - i m i [(dc i /dt)a i - c i (da i /dt)], Ω c = - i m i [(da i /dt)b i - a i (db i /dt)] r 0 dr r r = r 0 + dr 0 = i m i da i = i m i db i = i m i dc i x 0 0 = i m i x 0 i y 0 i = i m i y 0 i z 0 i = i m i z 0 i x 0 i Eckart Eckart 0 = i m i [a 0 i d(db i /dt) - b 0 i d(da i /dt)] = i m i [b 0 i d(dc i /dt) -c 0 i d(db i /dt)] = i m i [c 0 i d(da i /dt) - a 0 i d(dc i /dt)] (m i )a i = s l (a) is Q s, (m i )b i = s l (b) is Q s, (m i )c i = s l (c) is Q s i [(l (a) is ) + (l (b) is ) + (l (c) is ) ] = 1, 0 = i l (a) is l (a) Is' = i l (b) is l (b) is' = i l (c) (c) is l is' s (dq s /dt) V 0 V 0 = s (λ s /)Q s λ s = (πcω s ) Q s = [(h/π) /λ s ] 1/4 q s q s H vib (0) = [(h/4π) s λ s ] 1/ [q s + (p s π/h) ] p s q s D: R CH HCH R 7.6 (1) X i -X j distance r ij () X i -X j - X k angle bending, deformation, φ ijk (3) θ (4) dihedral angle between X i -X j -X k and X j -X k -X l planes τ ijkl CH - 6

27 U vib T vib r ij T vib = T(R) U vib = K ii r ij + K ij r ij r jk C r R r R E: S CH CH 3 redundancy 360 (H i CH j ) (redundant coordinate) BF 3 CH - 7

28 F: Q i X i V = ij f ij X i X j X 1, X,, X n Q T = i (dq i /dt), V = i λ i Q i (dq i /dt) + λ i Q i = 0 Q i Eyring & Kimball, "Quantum Chemistry" (Wiley & Sons) d 3 GF C ~ E P R X R R; x, R, S, Q T = ij P R ip R jg R ij U = ij X R ix R jf R ij G R ij G R F R ij F R G R F R Λ λ i ν i = (λ i ) 1/ /πc G R F R L R Λ (G R F R )L R = L R Λ (L R ) -1 (G R F R )L R = Λ λ i Λ (ν i = (λ i ) 1/ /πc G R F R - Eλ i = 0 ν i P R i = T X R i T = i atom [(P i,a ) +(P i,b ) +(P ic ) ](1/m i ) CH - 8

29 G x 1/ m /m /m G x = /m N /m N / m N 7 µ i i 1/m i ρ ij i j 1/r 0 ij r 1 G( r 1, r 1 ) = µ 1 + µ φ 10 G( φ 10, φ 10 ) = ρ 01 m 1 + ρ 0 m + µ 0 (ρ 01 + ρ 0 - ρ 01 ρ 0 cosφ 10 ) ,, 3 G( θ 1 sinφ 03, θ 1 sinφ 03 ) = µ 1 ρ 01 sin φ 03 + µ ρ 0 sin φ µ 3 ρ 03 sin φ 10 + µ 0 (ρ 01 sinφ 03 + ρ 0 sinφ ρ 03 sinφ 10 ) r 01 r 0 G( r 01, r 0 ) = µ 0 cosφ 10 r 01 φ 10 G( r 01, φ 10 ) = -µ 0 ρ 0 sinφ 10 r 01 φ 03 G( r 01, φ 03 ) = -µ 0 [(u 30 - w 03 )cosφ 10 + (u 03 - w 30 )cosφ 301 ] u ijk = ρ ij /sinφ ijk, w ijk = ρ ij cosφ ijk /sinφ ijk F (valence force field) UB (Urey-Bradley) (general force field) CH - 9

30 UB UB Urey-Bradley-Shimanouti field 144 (brute force field) GF 4 CH 3 C 3v CH CH (CH(1) ) CH (CH() CH(3) ) C 3v C s (CH(1) CH()H(3) ) (CH()H(3) ) Q sym Q deg,a SFG CH 3 Q deg,b CH()H(3) Q sym Q deg,a IRAS SFG S = 0 1/ 1/ 0 1 / - 1/ r CH1 r CH r CH3 G F µ C + µ H µ C cosτ µ C cosτ G r = µ C cosτ µ C + µh µ C cosτ µ C cosτ µ C cosτ µ c + µh µ C = 1/m C, µ H = 1/m H, τ HCH angle CH - 30

31 K 1 + s F + t F' s F t F' s F t F' F r = s F t F' K + s F + t F' s F t F' s F t F' s F t F' K + s F + t F' s = r CH (1-cosτ)/q HH, t = r CH sinτ/q HH, q HH = r CH (1-cosτ), F = -0.1F K: C-H stretching force constant ( = 4.6 md/å), F H H repulsion constant ( < 0.3 md/å) G S = µ C + µ H µ C cosτ 0 µ C cosτ µ C + µ H + µ C cosτ µ C + µ H µ C cosτ K 1 + s F + t F' (s F t F' ) 0 F S = (s F t F') K + s F K + t F' (G S F S ) 1,1 = (µ C + µ H )(K 1 + s F + t F') + µ C cosτ (s F t F') (G S F S ),1 = [(µ C + µ H )(s F t F') + µ C cosτ (K 1 + s F)] (G S F S ) 1, = [(µ C + µ H )(s F t F') + µ C cosτ (K + s F)] (G S F S ), = (µ C + µ H )(K + s F) + µ C cosτ (K + 4s F t F') (G S F S ) 3,3 = [(µ C + µ H ) µ C cosτ ](K + t F') G s F s λ 1, = (1/){[(G s F s ) 1,1 + (G s F s ), ] ± D} D = [(G s F s ) 1,1 - (G s F s ), ] + 4(G s F s ) 1, (G s F s ),1 K 1 = K + K D = 9[(µ C + µ H )(s F - t F ) + µ C cosτ(k + s F)] -[(µ C + µ H )(s F - t F ) + µ C cosτ(k + s F)][(µ C + µ H ) - 4µ C cosτ] K + (µ C + µ H ) K {3[(µ C + µ H )(s F - t F ) + µ C cosτ(k + s F)] - (1/3)[(µ C + µ H ) - 4µ C cos] K } + (8/9)[(µ C + µ H ) + (µ C + µ H ) µ C cosτ - (µ C cosτ) ] K D 3[(µ C + µ H )(s F - t F ) + µ C cosτ(k + s F)] - (1/3)[(µ C + µ H ) - 4µ C cosτ] K + (4/7)[(µ C + µ H )/ µ C cosτ]( K/K )(µ C + µ H ) K CH - 31

32 λ 1 (CH) = (µ C + µ H + µ C cosτ)[k + (1/3) K + 3s F - t F ] + (/7)[(µ C + µ H )/µ C cosτ]( K/K )(µ C + µ H ) K λ (CH, s ) = (µ C + µ H - µ C cosτ)[k + (/3) K + t F ] - (/7)[(µ C + µ H )/µ C cosτ]( K/K )(µ C + µ H ) K λ 3 (CH, a ) = (µ C + µ H - µ C cosτ)[k + t F ] ~ (1/3)( K/K )ν deg ~ (1/6)( K/K )ν sym T d cosτ = -1/3 s = / 3 t = 1 /3 λ 1 (CH) (µ H + (1/3)µ C )[K + (1/3) K + F] - (/9)[(µ C + µ H )/µ C ]( K/K )(µ C + µ H ) K λ (CH, s ) (µ H +(4/3)µ C )[K + (/3) K] + (/9)[(µ C + µ H )/µ C ]( K/K )(µ C + µ H ) K λ 3 (CH, a ) = (µ H + (4/3)µ C )K L Q S L S = LQ GFL = LΛ, LL t = G t GFL = LΛ E = GF L 1 = (E 1 E ) / (E 1 E ) / 4 + E 1 E 1 E 1 L L 1 = (E 1 E ) / + (E 1 E ) / 4 + E 1 E 1 E 1 L 11 LL t = G L 11 + L 1 = G 1, L 1 + L = G, L 11 L 1 + L 1 L = G 1 L 11 = (E 1 E 1 ) G 1 [(E 1 E ) / (E 1 E ) / 4 + E 1 E 1 ] G E 1 (E 1 E ) (E 1 E ) / 4 + E 1 E 1 L = (E 1 E 1 ) G 1 E 1 + G [(E 1 E ) / (E 1 E ) / 4 + E 1 E 1 ] (E 1 E ) (E 1 E ) /4 + E 1 E 1 K 1 = K >> F L 1 - L L 1 L 11 L 11 L 1/3 L 11 1/ 3 L -1/ 3 Q 1 (1 3)[ r 1 + r + r 3 ] = Q sym CH - 3

33 Q (1 6)[ r 1 - r - r 3 ] = Q deg,a C 3v SFG CH 3 CH - 33