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1 (t) t ˆ() = tr{ ˆ ˆ ρ()} t () ˆ i () exp ˆ eq ( ) ˆ i ρ t H τ dτ ρ exp Hˆ = ( τ) dτ () t ˆ ρ eq H ˆ β = / kt B { ˆ } eq ˆ ρ = exp[ βhˆ ]/ tr exp[ βh ] (3) H ˆ ( τ ) H ˆ - 5 -

2 ˆ( ρ t) () Â Hˆ( τ ) = Hˆ ( ) ˆ f τ B (4) B f ( τ ) ˆ ρ() t = ˆ ρ eq i t i ˆ ( )exp ( ) ˆ i exp ˆ eq ˆ i exp ˆ + dτ f τ H t τ B Hτ ρ Ht 0 i t i ( )exp ˆ eq i ˆ exp ˆ ( ) ˆ i dτ f τ H exp ˆ t ρ H t τ B Hτ + 0 (5) eq { ˆ ρ } { ˆ ρ } () t ˆ() t ˆ tr ˆ () t tr ˆ i t i ˆ ˆ i exp exp ˆ i exp ˆ ˆ i exp ˆ eq = dtr τ Ht ˆ Ht Hτ B Hτ ρ 0 (6) i t i exp ˆ ˆ i exp ˆ i dtr τ H exp ˆ τ B Hτ Ht ˆ i ˆ eq 0 exp ˆ H t ρ + i t = dτ f( τ) ˆ ( t), Bˆ ( τ) + 0 i t X () t dτ f( t τ) ˆ ( τ), Bˆ (7) 0 f ( τ ) Bˆ Â - 6 -

3 H( τ ) = H f( τ) Bˆ g( τ ) Bˆ (8) ˆB ˆB Â t τ Xt () = dτ dτ f( τ) g( τ ) t ˆ(), Bˆ( τ), Bˆ ( τ ) 0 0 (9) t τ = dτ dτ f( t τ) g( t τ τ ) ˆ ( τ), Bˆ( τ ), Bˆ 0 0 f ( τ ) Bˆ = g( τ ) Bˆ (4) (8) h( ) Bˆ τ i t τ τ Xt () = dτ dτ dτ ft ( τ) gt ( τ τ ) ht ( τ τ τ ) ˆ ( τ), Bˆ( τ ), Bˆ ( τ ), Bˆ, (0) i = ˆ ˆ () K ( t) ( t), B () = ˆ ˆ ˆ () K ( t, t) ( t ), B( t), B i ˆ ˆ ˆ ˆ (3) K ( t, t, t3) = ( t 3 3), B( t), B( t), B () (3) 3. CO HF H ˆ q H pˆ = + U( qˆ ) (4) m 0-7 -

4 C H O F NO µ ( q) Hˆ( τ) = Hˆ E( τ ) µ ( qˆ ) (5) µ ( qˆ ) µ ( qˆ) = µ + µ qˆ+ µ qˆ + (6) 0 µ 0 (4)() µ E( τ) qˆ f( τ) qˆ (5) 0 5 ec f ( τ ) (5) cm - CO - 8 -

5 ( ˆ ) P () ( ) IR t < µ q τ > (7) (5)(7) i t P () ( ) ( ˆ( )), ( ˆ IR t = dτe t τ µ q τ µ q(0) ) 0 (8) Et () = E0δ () t (6) i P 0 [ ˆ ˆ IR () t = µ E q( τ), q(0) ] (9) (9) pˆ Hˆ ˆ = + m0ω 0q (0) m 0 CO - 9 -

6 µ E0 PIR () t = in 0t m ω 0 0 ( ω ) m ω j j µ E 0 PIR () t = dωsir ( ω)in( ωt) 0 S IR () () ( ω) = δω ( ω j ) (3) m ω j j j S ( ω) PIR () t () SIR ( ω) IR. α( q) µ ( q) = α( q) E( t) q ˆ ˆ H( τ) = H ( ) ( ˆ E τ α q) (4) 0000[cm - ]

7 xyz α ( q) CO xxy E(t) α ( qet ) ( )

8 (5)(4) α( qˆ) α α qˆ α qˆ = (5) E () t ( ˆ ) P () t = E () t α q() t (6) Raman T (4) i t P () () ( ) ( ˆ( )), ( ˆ Raman t = ET t dτe t τ α q τ α q(0) ) 0 (7) N N+ (8)(7) (0) ()-(3) T CO - 3 -

9 αee 0 T () t PRaman () t = dωsraman( ω)in( ωt) (8) S Raman ( ω) = δω ( ω j ) (9) m ω j j j (3)(9) E(,) r t = E()exp[ t ikr Ω i t] + cc.. (30) r k E() t Et () = Eexp[ δ t] /δ c.c. (7) (30) ( ) E( r, t) = E ( t) exp[ ik r iω t] + exp[ ik r iω t] + E ( t)exp[ ik r iω t] + cc.. (3) T T T

10 k = ( k ± k± kt ) k = ( k + k kt ) Ω = ( Ω + Ω Ω ) T Ω (N+) ( ) E ( r, t) = E ( t) exp[ ik r iω t] + exp[ ik r iω t] + cc.. (3) Raman j j j j j j k j = k j k j Ω=Ω j Ωj N E ( r, t) = E ( t)exp[ ik r iω t] + cc.. (33) IR j j j j k j k j Ω j Ω j

11 9 0 (8)(9) 3 4. (7)-(9)(6)-(7) (5) t t PRaman () t = E T () t dt dte( t t) E ( t t t) 0 0 (34) α( qt ˆ( + t) ), α( qt ˆ( ) ), α( qˆ(0) ) 0 350cm - CN

12 (3) i t t t PIR () t = dt dt dt3e3( t t 3 ) E( t t t) E( t t t + t3) (35) µ ( qt ˆ( + t + t3) ), µ ( qt ˆ( + t) ), µ ( qt ˆ( ) ), µ ( qˆ(0) ) R ( t, t,, t ) = (N+ ) Raman ( N ) IR N N i α( qt ˆ( + + t )), ( ˆ( ) ),, ( ˆ( ) ), ( ˆ N α qt+ + tn α qt α q) R ( t, t,, t ) = N N i µ ( qt ˆ( + + t )), ( ˆ( ) ),, ( ˆ( ) ), ( ˆ N µ qt+ + tn µ qt µ q) (6)(5) (6) { } eq ( ˆ( τ) ) ( ˆ( τ )) = ( ˆ( τ )) ˆ ρ ( ˆ( τ) ) Bq q tr q Bq qt () = qt q () t= qt t t t t q(0) = q i q (0) = qi = dq dq δ( q q ) D[ q( τ)] D[ q ( τ)] q( τ ) Bq ( τ) e ρ ( q, q ; β)e ( ) ( ) i i dτl( q, q; τ) dτl( q, q ; τ) eq i i (36) (37) (38) (N+)

13 Lqqt (, ; ) τ > τ qt () = qt q () t= qt B( qˆ( τ )) ( qˆ( τ )) = B dqt Dq [ ( τ)] Dq [ ( τ)] i J( τ ) i J ( τ) q(0) = q i q (0) = qi i i dτ( L( q, q; τ) + J( τ) q) dτ( L( q, q ; τ) + J ( τ) q ) e ρeq ( qi, q i; β)e (39) B G( J(), t J () t ) i J ( τ ) i J( τ ) J( τ) = J ( τ) = 0 J( τ) = J ( τ) = 0 G(J(t), J (t)) ( q) B( q) Jt () Jt () = 0 { ( ˆ ) ( ˆ eq ( τ) ( τ )) ˆ ρ ( ˆ )} tr q B q B q = (40) B B G( J(), t J () t ) i J( τ) i J( τ ) i J (0) J= J = 0 m 0 ω 0 i G J t J t d d J J m 0 0 0ω 0 t t ( ( ), ( )) = exp τ τ ( ( τ ) ( τ )) βω0 ( ω τ τ ) ( J τ J τ ) i ( ω τ τ )( J τ J τ ) co 0( ) coth ( ) + ( ) in 0( ) ( ) ( ) (4) 3 (9) N

14 (3) iµ RIR ( t) = q( t), q(0) 0 0 [ ] iµ = + G J t i J( t) i J(0) i J (0) µ = in m ω ( ω t ) 0 ( (), J () t ) J= J = 0 (4) ()(6) < [[ qt ( + t), qt ( )], q(0)] >α 3 α α 3 (5) RRaman ( t, t) [ α( t+ t), α( t) ], α(0) αα = in ( ω0t) in ( ω0t) + in ( ω0( t+ t) ) m 0ω0 (43) (5) (3) i RIR ( t,0, t3) 3 [ µ ( t+ t3), µ ( t) ], µ ( t), µ (0) µµ = in ( ω0 t ) in ( ω03 t ) m 3 3 0ω0 (44) t = 0 µ 4 µ µ µ 3 µ ()(8) (3) α RRaman( t) = dωs( ω)in( ωt) (5) RRaman( t, t) = αα dωs ( ω)in ( ωt) in( ωt) + in ( ω( t+ t) ) (45) (46) µµ R t t = d S t t (3) 3 IR (,0, 3) ω ( ω)in ω 3 in ω 3 ( ) ( ) 5 0 E () t = E δ ( t T), E () t = E δ( t T T ), E () t = E δ( t T T ) (48) (7)(34)(35) RRaman( (3) T ) R (5) Raman ( T, T) (3) R ( T,0, T ) J ( ω) ω 0 / IR 3 (47)

15 ( ω ω 0) e S( ω) = (49) mω ω 0 = 38.5[cm ] 3 a = ω0 3ω 0 5ω 0 0ω 0 (b)(c) = ω0 (8)(46)(47) T = T αα < [[ qt ( + T), q( T)], q] > α q ( T) (c). 3 (a)()(b) (c)

16 0 350cm [cm - ] 350 [cm - ] q p m ω Fq ( ) cx j j x j p j m ω j j ( p ) j cfq ( ) p j ( ) j ( ω j) j (50) m j m j m j ( ω j) H = + U q + + m x ( ) ( )/ ( j j j) c F q m ω Fq ( ) = q U ( q) = mω q (5) q β = / kt B J tq() t G( J(), t J () t ) G( J(), t J () t ) (5) i t t G( J(), t J ()) t = exp dτ dτ ( J( τ ) J ( τ )) 0 0 (53) ( S( τ τ) ( J( τ) J ( τ) ) ic( τ τ) ( J( τ) J ( τ) ))

17 ωγ ( iω) βω S ( t) = dω coth co( ω t) + πm ( ω ω ) ω γ ( iω) (54) ωγ ( iω) C () t = dω in( ωt) πm ( ω ω ) + ω γ ( iω) (55) c j γ ( ω) = δ( ω ω j ) (56) j m jω j γ ( ω) = γ (3) Raman α R ( t ) = C ( t ) (57) 3 (5) αα RRaman( t, t) = C ( t) C ( t) + C ( t+ t) [ ] (3) IR [ ] (58) µµ R ( t,0, t ) = C ( t ) C ( t ) (59) γ t C() t = e in t ω γ 4 m ω γ 4 4 ω = 38.7[cm ] (/ ω = 86[f]) = 0[cm ] T T (58) T Γ T (59)T T 3 T T 3 ( + D ) () t γ (60) - 4 -

18 3(a) 4a 5 6 7, (a) (50) ()(b)(c) - 4 -

19 5. q q (4) i () ˆ (), ˆ i ˆ ˆ t t H H () t t ρ = ρ ρ (6) B ˆ, ˆ B ˆ ˆ (6) (6) ˆ i ( ) exp ˆ i i ρ (0)exp exp ˆ t = Ht ρ Ht Ht ρ(0) (6) eq eq ( ˆ ρ ˆ ρ ) i ˆ ˆ ˆ i ˆ i (), exp ˆ ˆ i t B exp ˆ = tr Ht B B Ht ˆ i exp ˆ i ( ˆ eq = tr ˆ H t B ρ ) t = 0 (63) (64) i ˆ ρ (0) B ˆ eq = ˆ ρ (65) (6) t Â

20 ˆ ρ () ˆ ˆ t ρ() t (66) ρ ˆ () t ( q q ) ρ (, ; ) ˆ qq t < q ρ ( t) q > 9 px i x x W( p, r; t) = dxe ρ r, r ; t π + (67) (6) W( p, r; t) = Lˆ W( p, r; t) (68) t ˆ p LW ( prt,;) W( prt,;) dpv ( p p, rw ) ( p,;) rt m r π (69) px x x V( p, r) dxin U r+ U r 0 ( qˆ ) (66) ˆ ˆ ˆ ˆ dp ρ W ( p p, r) W ( p, r) π i ˆ ˆ ˆ dp B ˆ ρ XW B XB( p p, rw ) ( p, r) π (70) B ( qˆ ) (65) ˆq (7) (7) ipx x pr (, ) i dxe r+ (73) px x x XB ( p, r) dxin B r+ B r i { ˆ eq ( )} t ˆ( ˆ ˆ ˆ ˆ ), B tr exp Lt = XBW (74) (75)

21 5. α α < [[ qt ( + t), qt ( )], q(0)] >α 3 j > ˆq eq jω qˆ j+ >< j + j >< j ˆ ρ = j > e < j qt ˆ( ) qt ˆ( ) qˆ j >< j tr{} qˆ( t) = exp[ iht ˆ / ] qˆexp[ iht ˆ / ] α α, Uq ( ) q q (50) γ ( iω) = γ 3,4,5,6 ρ ( qq, ; t) tr{ ρ( qxq,,, x ; t) } = B 7 W( prt, ; ) = LW ˆ (, ; ) ˆ prt+γw( prt, ; ) (76) t ˆ β Γ η p + (77) p m p  ˆB (75)(65) (68)(76) q ( ) = Bq ( ) = α( q) µ ( q) 8 { ( ˆ ) eq α ( α )} R (3) () ˆ exp ˆ ˆ ˆ Raman t = tr L Γ t X W (78)

22 IR ˆ { ( ˆ ˆ) ˆ { ( ˆ ˆ) ( ˆ ˆ eq α α α )}} ˆ ( ˆ ˆ) ˆ ˆ { { { ( ˆ ˆ) ( ˆ ˆ eq µ µ µ µ )}} } R ( t, t ) = tr exp L Γ t X exp L Γ t X W (5) Raman R ( t,0, t ) = tr exp L Γ t X X exp L Γ t X W (3) IR (79) (80) ˆ, X ˆ α µ, α µ (66)(65) IR(79)t ˆX µ ˆX α (78) (76) () IR ˆ eq W α( q) µ ( q) Xα ( pr, ) X µ ( pr, ) (7) p 0 t α( q) µ ( q) α ( p, r) µ ( pr, ) (7) p p r tr t RRaman( (3) t) R ( t ) (80) α( q) µ ( q) Xα ( pr, ) X µ ( pr, ) (7) p t t + t α( q) µ ( q) α ( p, r) µ ( p, r) (7) p p r tr t t RRaman( (5) t, t)

23 R ( t, t ) () IR (80) { [ ]} Uq ( ) = U exp aq (8) e ω 0 = U ( q)/ m ω [ cm ] = U e = [cm ] a = Γ= 0[cm ] Γ= 00[cm ] α( q) α q α q = + () α = α = 0 () α = α = T=50, 300, 450 [K] πω0 / kt B

24 q m ω 0 / q p / mω p 5 300[K](a) 5(b)α( qˆ ) T = 0[ p] α( qˆ) = α ˆ q 5(e)T = 0. [ p] (d)(b) 5(f) ( p, r) α

25 7 (a)(a )(b)(b ) α( q) α q α q = + α = α = 0 (c)(c ) α = α = (a) (57)(a)(a )300[K] (b)-(c ) - 7(b) 7(b ) (c) (c) αα < >q q [ q ( t), q] 7 (a)(b) (c)

26 α α q 7(c) (b)α α 7(c ) α 3 < [[ qt ( ), qt ( )], q(0)] > [K]4. 8 α α 9(a) 7(b) α 3 8(a)T qt () α 3 < [[ qt ( ), qt ( )], q(0)] > 9(b) 9(a) α α 8(a) 0 7(b )(c ) 8 (a) (b)

27 9 (a) (b) 8 0 (a) (b) 8-5 -

28 5.3 9, 0 5. qq qq ζγω J ( ω) = π γ + ω (8) a + aj b + j b j (a+a + )(b j +b + j ) (a + b j +ab + j ) J ( ω) ηω T T / T = / T - 5 -

29 (a+a + ) (b j +b j + )(aa + +aa + )(b j +b j + ) / T < / T + / T' (50) Fq ( ) = q (8) coth( β ω/ ) / βω β = / kt B 0 (0) ˆ (0) () W ( prt, ; ) = LW ( prt, ; ) + r W ( prt, ; ) t p t ( ˆ γ ) (, ; ) = + (, ; ) + (, ; ) p W () p r t L W () p r t r W () p r t m + ζ r p+ W p r t β p (0) (, ; ) (83) (84)

30 t ( ˆ γ ) (, ; ) = + (, ; ) + (, ; ) p m ( n ) + nζ r p+ W ( p, r; t) β p ( n ) ( n ) ( n+ W p r t L n W p r t r W ) p r t 9 ω Nγ ω ( ˆ γ ) ( N ) ( N m (, ; ) ) (, ; ) 4 ( N W p r t = L + N W p r t + ζr r p+ W ) ( p, r; t) t p β p m ( N ) + Nζ r p+ W ( p, r; t) β p (8) 3,4,5 W ( n) W (0) W () 3 W () 0,,, (85) (86)

31 5.5. (83)-(86)N 5. ζ ' ζ /mω 0 3 3(a)4(a) (a) ζ ' 3(a) 4(a) (aa + +aa + )(b j +b + j ) 3(b)(c) 3(a) δ ζ γ < [[ qt ( ), qt ( )], q(0)] >α α 4(a)(a )() 4 3 4(b) 4(a)(a ) j >< j j >< j± T T 4(b)(b ) γ 3(b )T = T

32 3(b) 4(b.) 4 (a) (a)(c ) (b)

33 5 5(a) 5 4 (a) (a)(c ) (b)

34 (c) 5(c ) T = T 3 (c) (c) < [ qt ( ), qt ( )], qt ( )], q(0)]]] >α 4 α 4 5(c ) α αα α 4 α. CO ( p ) H = m m m + gq (,, q, ) p j cq j mωq m j ( ω j) x + + j + j j j ( ω j) gq (,, q, ) 3 4 gnh( q,, q, ) = gq + gq (88) 3! 4! gfermi ( q,, q, ) = g qq + g q q (89) 3! 3! (87)

35 gdd ( q,, q, ) = g q q (90)!! () 3 α( q,, q, ) = α q + α q q + (9) ( ) ( ) R ( t, t ) α q ( t + t ), α q ( t ), α q (0) ({ }) { } ( ) ({ }) (5) Raman (9) i R ( t,0, t ) µ q ( t + t ), µ q ( t ), µ q ( t ), µ q (0) ({ }) { } ( ) ({ }) ({ }) (3) IR (93) R ( t, t ) α q ( t + t ), µ q ( t ), µ q (0) ({ }) { } ( ) ({ }) (3) IR Raman 4,5 Wright 6 (94) (5) iω T iωt (5) ω ω 0 0 R (, ) dt dt e e R ( T, T ) = (95) (5) [ + ] () () α ( α ) g q( t t), q( t), q (0) α( q ) = α q ( ) () () [ + ] ( α ) α g q( t t), q( t), q(0)

36 < > α 3 7 α 4 [ + ] () () ( α ) ( α ) g q ( t t3), q ( t), q( t), q(0) (9) [ + ] () () ( α ) α q( t t), q( t), q(0) 7 R R ( ω, ω ) = R ( ω, ω ) + R ( ω, ω ) (96) (5) (5) (5) P nh ( i ω i ω ωω ) ( ) Ω Ω +ΓΓ Γ Γ ( ) ( ) ( ) 4 n (5) α α α P ( ω, ω) = n n n n (97) ' ζζ MM n= Γ +Ω n iγω ω ( Γ n +Ωn iγnω ω) ( ) ( ) ( ) ( ) 4 (5) g α α α n R ( nh ω, ω) = ( ) ( Fn( ω, ω) Fn( ω, ω) ) 4MM M ζζ ζ n= (98) ( m) Γ ( ) ( ) 0 Ω n Γ iω +Ω n Γm iω F ( ω, ω ) = m ( )( i ) ( ) i ( m) Ω0n( ΩnΩ n +ΓΓm iγmω iγω ωω ) m ( )( i ) ( ) i n ( ) Γ 0 +Ω0n Γ +Ωn Γω ω Γ m + Ωn Γmω ω + ( ) Γ 0 +Ω0n Γ +Ω n Γω ω Γ m + Ωn Γmω ω (99) Γ= γ ζ = Ω γ (00), /4 Γ Ω Ω γ ζ ζ γ ζ ζ Γ Ω Ω Γ γ 3 Ω 3 Ω 3 + γ ζ ζ + ζ Γ Ω Ω γ + γ ζ ζ ζ ( ) ( ) (0)

37 Ω = 58[cm ] Ω = 368[cm ] γ = 5 [cm ] γ = [cm ] ζ Ω (a) ( ) g = 0 (b) α = 0 g 33 = g 33 = g 33 = g ( ) (c) α = g 0 g3 = g3 = g3 = g g 6 (97) 6(a) ( ( ) ) ( α α ) (58) ( ω, ω ) = ( ± ζ, ζ ) ( ±,0) ( ± ζ, ± ζ ) ζ 6 ( ζ, ζ ) ( ζ, ζ ζ ) 6(b)(c) ( Ω, Ω ) ( Ω, Ω ) g [ q( t + t), q( t) ], q (0) g [ q ( t + t), q ( t) ], q(0) q ( t t ), q ( t ) q ( t + t ), q ( t ) [ + ] [ ] 6 (a) (b) qq (c) qq 7-6 -

38 8 I( ω, ω ) = I ( ω, ω ) + I ( ω, ω ) (0) nh D D I = F + F (03) () () nh( ω, ω) ( ω, ω3) ( ω, ω3) () () () ID D( ω, ω) = F ( ω, ω3) + F ( ω, ω3) + F ( ω, ω3) (04) () () () + F ( ω, ω3) + F ( ω, ω3) + F ( ω, ω3) Γ jkl =Γ j +Γ k +Γ l ζ jkl = ε j ζ j + ε k ζ k + ε l ζ l ε 4 = εεε 3 4 ( ijkl) F ( ω, ω3) = 8 i εε ε3=± MiM jmkmlζζ i jζkζl (05) ε( Γ jkl iω3) ( Γ j iω)( Γi iω3) + εζiζ j + ζ jkl ( Γ j iω) ζi ε( Γi iω3) ζ j Γ i ( iω3 ζi) Γ j ( iω ζ j) Γ jkl i( ω3 ζ jkl) =± Ω = 3[cm ] Ω = 374[cm ] Γ =Γ = 36[cm ] g = g = g =

39 - 63 -

40 (b )(c ) 30,3 (46) { B ˆ ˆ} i B ˆ, ˆ, (06)

41 eq ( ) i i R (3) () ( ()), ( ) ( ()) ( ), ˆ Raman t = α q t α q = tr α q t α q ρ (07) ( q() t ) α( q) ( ) eq ( q) ρ α( q) (3) eq α ρ RRaman () t tr( α( q() t ){ α( q), ρ }) = tr α( q() t ) q p p q α q H eq α q q eq = tr α( q() t ) β ρ = βtr α( q() t ) ρ t q p q = β α (34) (5) RRaman ( t, t) = α ( t+ t) α ( t) α kt B α + α α kt B ( q ) ( q ) ( q) { } ( q( t t )) ( q( t )), ( q) ( ) eq (08) (09) q() t q() t H H q() t q() t d q(0) p(0) (0) (0) qp pp q p = (0) dt p() t p() t H H p() t p() t (0) (0) (0) (0) p p qq qp p p,,

42 , Fleming Tokmakoff,, Duppen Steffen

43 Miller, Miller,55 Fleming Blank,57MIT Tokmakoff Miller 9 Miller T T k k j k = k5 + ( k k) ( k3 k4) 9 k Miller

44 zzzzz T = 0[ f] T = 0 Fleming Tokmakoff α

45 DOVE (8) Et ( τ ) = E0 in ω( t τ) t 6,60,6,6,63,64,65,66 Wright Hochtraer Hamm Hochtraer N-methylacetamide-D (CH 3 CONDCH 3 ) 70 acetylproline-nh,7 Hamm D O la 3 Tokmakoff Rh(CO) (C 5 H 7 O ) Dlott

46 3 DO N-methylacetamide-D (a) D O cm - (b)

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