Nonequilibrium initial conditions of a Brownian oscillator system observed by two-dimensional spectroscopy

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1 JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 5 1 AUGUST 001 Nonequlbrum ntal condtons of a Brownan oscllator system observed by two-dmensonal spectroscopy Yoko Suzuk and Yoshtaka Tanmura Insttute for Molecular Scence, Myodaj, Okazak , Japan Receved 1 February 001; accepted 6 Aprl 001 We study effects of a nonequlbrum ntal condton of a Brownan oscllator system upon two-, three-, and four-tme correlaton functons of an oscllator coordnate as a subject of multdmensonal spectroscopy. A nonequlbrum ntal condton s set by a dsplacement of a Gaussan wave packet n an oscllator potental. Such stuaton may be found n a vbratonal moton of molecules after a sudden bond breakng between a fragmental molecule and a targetng vbratonal system or a movement of wave packet n an electronc excted state potental surface created by a laser pump pulse. Multtme correlaton functons of oscllator coordnates for a nonequlbrum ntal condton are calculated analytcally wth the use of generatng functonal from a path ntegral approach. Two-, three-, and four-tme correlaton functons of oscllator coordnates correspond to the thrd-, ffth-, and seventh-order Raman sgnals or the frst-, second-, and thrd-order nfrared sgnals. We plotted these correlaton functons as a sgnal n multdmensonal spectroscopy. The profle of the sgnal depends on the ntal poston and momentum of the wave packet n the ffth- and seventh-order Raman or the second and thrd order nfrared measurement, whch makes t possble to measure the dynamcs of the wave packet drectly n the phase space by optcal means. 001 Amercan Insttute of Physcs. DOI: / I. INTRODUCTION The vbratonal mode of molecules n condensed phases has been studed n many expermental and theoretcal works. Femtosecond nonlnear optcal spectroscopes are powerful tools to obtan nformaton about a varety of dynamc processes, ncludng such mportant processes as mcroscopc dynamcs, ntermolecular couplngs, and tme scales of solvent evoluton that modulate the energy of a transton. However, snce vbratonal lnes from these processes are often broadened and also appear n smlar postons, t s not easy to dstngush them from lnear spectroscopy. Ths dffculty can be overcome by hgher-order nonlnear optcal processes nvolvng many laser nteractons. Two-dmensonal Raman spectroscopy and twodmensonal nfrared spectroscopy are such examples. 1, Many expermental efforts along ths lne of research have been made to probe nhomogenety of lquds and nter- and ntramolecular vbratonal moton. 3 7 The D nformaton content of these tme doman experments can also be obtaned from a frequency doman experment, and also demonstrated that vbratonal nteractons n lquds can be observed. 8 1 It s obvous that hgher-order spectroscopy can contan many tme ntervals and these can be used to separate the mechansm of dynamcal processes from the others. Theores so far developed are to access varous dynamcal nformaton for nstance the degree of nhomogeneous broadenng, 1,13 15 the anharmoncty of potentals and the nonlnearty of polarzablty, 16 1 the couplng mechansm between dfferent vbratonal modes 6 and the structural nformaton of large molecules. 7,8 In ths paper, we am to demonstrate that multdmensonal spectroscopy s useful not only to nvestgate the targetng dynamcal processes but also to elucdate nformaton about a dfference for an ntal dstrbuton of vbratonal modes. For the purposes of ths work, we consder a Gaussan wave packet n the harmonc vbratonal mode, whose center s shfted from the equlbrum poston, as an ntal condton. Such ntal condton may arse from a sudden bond breakng between a fragmental molecule and a targetng vbratonal system. A possble example s the reacton drven coherence n MbNO the pump pulse creates the reactant excted state (MbNO*), whch rapdly decays to MbNO Fg. 1a. 9 One may also fnd smlar stuaton n a movement of a wave packet created n an electronc excted state by a laser pump pulse Fg. 1b. Dsplacement and movement of the wave packet s usually observed by the tme-dependent emsson or absorpton spectrum. For example, n a dsplaced oscllator case, such effects can be seen by the so called dynamcal Stokes shft. In some case, however, such measurements are very dffcult, snce the emsson or absorpton spectrum s often broadened and featureless from a convoluton of all the dynamcal and statc nformaton wthn t. One-dmensonal 1D spectroscopy does not allow unque extracton of nformaton for supermposed dynamcal tme scales. Multdmensonal spectroscopy, whch measures the magntude of a dpole moment or a nonlnear polarzaton as a functon of the two ndependent coherence evoluton perods, can provde more nformaton about the molecular structure and dynamcs than 1D spectroscopy. Here we demonstrate a possblty to use multdmensonal spectroscopy to probe the nature of the ntal dstrbuton. For ths purpose, we employ a Brownan oscllator model for /001/115(5)/67/15/$ Amercan Insttute of Physcs

2 68 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Y. Suzuk and Y. Tanmura FIG. 1. Schematc vew of a creaton of nonequlbrm ntal condton at tt I. Example a shows the case of a sudden bond breakng between a fragmental molecule and a targetng vbratonal system at tt I. In ths stuaton, the reactant excted state rapdly decays to the ground state and then the wave packet s created n the ground state at tt I. Example b shows the stuaton n whch a wave packet s created n an excted state by a resonant pulse at tt I. a molecular vbratonal mode and ncorporate the dsplacement of the ntal dstrbuton nto the Brownan moton theory usng nonequlbrum generatng functonal calculated from a path ntegral approach. We reformulate optcal response functons expressed by tme correlaton functons of the molecular polarzablty or the dpole moment and obtan ther analytcal expressons whch are the observable of multdmensonal Raman or nfrared spectroscopy. In Sec. II, the (N1)th order Raman and Nth order nfrared sgnals are descrbed to arbtrary order n terms of response functons whch are expressed by multtme correlaton functons of polarzablty or dpole moment. In Sec. III, we show any order of response functon can be expressed by a generatng functonal whose calculatonal detals are shown n Appendx A. We then calculated the response functons analytcally wth the use of the dagrammatcal rule descrbed n Appendx B. The numercal results are presented n Sec. IV and fnally conclusons are gven n Sec. V. II. RESPONSE FUNCTIONS FOR HIGHER-ORDER OPTICAL PROCESSES We consder a molecular system n the condensed phase whch s subjected to laser pulses. If the system s descrbed by a sngle oscllaton mode specfed by ts coordnate Qˆ and momentum Pˆ, the total Hamltonan of the system and the bath s wrtten as Ĥ Pˆ M 1 M Qˆ pˆ m m qˆ c Qˆ m..1 Here, denotes the oscllator frequency. The coordnate, conjugated momentum, mass, and frequency of an th oscllator are gven by qˆ, pˆ, m, and, respectvely. The nteracton between the system and the th oscllator s assumed to be Ĥ SB c qˆ Qˆ. The term c Qˆ /(m )sa counter term whch cancels the unphyscal dvergence from FIG.. Pulse confguraton for a the ffth- and b the seventh-order offresonant Raman experments. The nonequlbrum ntal condton s created at tt I. Then the movement of the wave packet s detected by a followng sequence of pulses,.e., two or three pars of pulses are appled to the system, whch followed by the last probe pulse. Here the frst par of pulses nteract wth the system at t. In ths paper, the temporal profles of pulses E 1, E, E 3, and E T are assumed to be mpulsve, and are gven by a Eq..5 and b Eq..6. the couplng to the bath degrees of freedom. The summaton over goes to nfnty n order to descrbe the dsspaton on the molecular system. We consder optcal measurements the molecular system s nteractng wth a laser feld, E(t). For off-resonant Raman spectroscopy, n whch resonance arses from a par of laser pulses through Raman exctaton processes, the effectve Hamltonan s gven by Ĥ Raman ĤE tqˆ,. (Qˆ ) s the coordnate dependent Raman polarzablty. For resonant IR spectroscopy, the Hamltonan ncludng laser nteracton s gven by Ĥ IR ĤEtQˆ,.3 (Qˆ ) s the coordnate dependent dpole moment. As both Raman polarzablty and dpole moment can be expanded as (Qˆ ) 0 1 Qˆ Qˆ /, or (Qˆ ) 0 1 Qˆ Qˆ /, the optcal responses of Raman and resonant IR are formally dentcal besdes the fact that the (N 1)th-order Raman spectroscopy corresponds to the Nth order IR spectroscopy. Therefore, hereafter we do not dstngush between the Nth-order IR and (N1)th-order Raman processes and present only the results for Raman spectroscopy. Notce however that the even-order of IR response sgnals vansh for sotropc materal. In the N1th order off-resonant Raman experment, the sgnal contrbutons are from Raman exctaton that occurs whle Raman pulse pars are temporally overlapped. Thus each nteracton occurs wth a tme concdent pulse par. Also, the polarzaton detected s temporally overlapped wth the probe pulse. The pulse confguratons for thrd-, ffth-, and seventh-order processes are descrbed as see Fg. E 1 tt, E T ttt 1,.4

3 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Two-dmensonal spectroscopy 69 E 1 tt, E ttt 1, E T ttt 1 T, E 1 tt, E ttt 1, E 3 ttt 1 T, E T ttt 1 T T Here we have assumed that laser pulses are mpulsve. The pump pulses and the probe pulse have been wrtten as E j (t) ( j1,,...,n) and E T (t), respectvely. The Raman sgnals for the optcally heterodyned detecton are expressed by the response functons as I (N1) T 1,T,...,T N ;R (N1) T N,...,T,T 1 ;,.7 whch are the N tme correlaton functons of the polarzablty operator (Qˆ ) gven n terms of R (N1) T N,...,T,T 1 ; N ˆ T 1 T T N, ˆ T 1 T T N1 ],...],ˆ T 1 ],ˆ ],.8 ˆ (t) represents the Hesenberg operator gven by ˆ (t)e (/)Ĥt (Qˆ )e (/)Ĥt and means the expectaton value of defned by Tr I /Tr I n whch I mples an ntal densty matrx. Notce that I s chosen as the nonequlbrum densty matrx n the present study, so that correlaton functons are not statonary,.e., ˆ (t t),ˆ (t)ˆ (t),ˆ (0). III. GENERATING FUNCTIONAL IN NONEQUILIBRIUM PROCESS A generatng functonal s defned as a functonal of the external force whch s obtaned from the densty matrx by tracng over all degrees of freedom of the total system. It s convenent to calculate the hgher-order response functons whch are derved by performng the functonal dfferentaton n terms of the external force. To calculate the generatng functonal, we need to trace out the system and bath degrees of freedom. The path ntegral method s sutable to carry out such procedure. In ths secton, we demonstrate how to apply the generatng functonal formalsm to the calculaton of the hgher-order response functon. Let us ntroduce the sources J(t) and K(t) coupled to Qˆ and (Qˆ ), Ĥ J,K tĥjtqˆ KtQˆ. The generatng functonal WJ,K s then defned by 3.1 exp WJ,K TrÛ J1,K 1,t I ˆ IÛ J,K,t I dq I dq I dq I dq I Q I q I ˆ IQ I q I Q I q I Û J,K,t I Û J1,K 1,t I Q I q I, Û J,K,t I T exp dtĥj,k t, and Û J,K s the adjont of Û J,K. The symbol T mples the tme orderng operator. The real tme paths are represented by suffx 1, and the sources J 1 and K 1 are for the lefthand sde tme evoluton kernel, as J and K are the rght. The matrx ˆ I s an arbtrary densty operator at the ntal tme t I whch does not need to be an equlbrum dstrbuton. In the second lne of Eq. 3., we have nserted the completeness relaton for the bass Q I,q I : 1 dq I dq I Q I,q I Q I,q I. The (N1)th order response functon s gven by the dfferentaton of the generatng functonal WJ,K as follows: R (N1) T N,...,T,T 1 ; T 1 T T N G (N1) R t 0,t 1,t,...,t N N N1 WJ,K N G (N1) R t 0,t 1,t,...,t N, 3.4 K t 0 K t 1 K t K t N, K0,J0 3.5 we set t T 1 T T N for 0,1,...,N 1, t N, and K t K 1tK t, K tk 1 tk t. 3.6 From Eq. 3.5, we can systematcally derve the hgherorder response functons, once we obtan the generatng functonal. We assume that the system and the bath are ntally factorzed. Consequently the factor Q I q I ˆ IQ I q I n Eq. 3. can be expressed as the product of the system part and the bath part, Q I q I ˆ IQ I q I Q I ˆ Q I(S) I q I ˆ q I(B) I. 3.7 We further assume the bath s n the equlbrum state, ˆ I(B) expĥ B, 3.8 Tr B expĥ B Ĥ B pˆ m qˆ m, 3.9

4 70 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Y. Suzuk and Y. Tanmura S K, dt K t t t K t 1 t 1 4 t and Tr B denotes the trace over the bath coordnates. As mentoned n Sec. I, we consder the case the ntal state of the system s set by the dsplaced Gaussan wave packet gven by Q I ˆ Q IS I a 1/ e a(q I Q 0 ) a(q I Q 0 ), 3.10 Q 0 s the dsplacement from the bottom of the harmonc potental and 1/a s the wdth of the ntal wave packet. We ntroduce the contour path for tme ntegraton to wrte varous formulas n a compact way The contour tme ntegral C dt runs from C 1 to C defned by C 1 :t I and C : t I return path see Fg. 3. The path ntegral method allows us to obtan the analytcal expresson of the generatng functonal WJ,K of the Brownan moton model even for the strong system bath couplng and the heat bath wth a fnte correlaton tme. The detaled dervaton of WJ,K wth the ad of the path ntegral s shown n Appendx A. By usng the notaton J t J 1tJ t, J tj 1 tj t, 3.11 the generatng functonal for (Qˆ ) 1 Qˆ Qˆ / s wrtten as exp exp WJ,K WJK0 exp 1J,K;/ S exp S, 3.1 K,0,J0 S 1 J,K;/ dt dt K () tt 1 FIG. 3. Contour paths C 1, C. t J t 1 K t t J t 1 K t K () t,t t J t 1 K t t J t 1 K t, 3.13 Here we ntroduced the functon J (t) as J tj t dtd ()1 ttq 0 cos t, 3.15 D () tttt sn tt, 3.16 M and the renormalzed frequency s defned by m The propagator for the total Hamltonan, K (), s expressed n the Laplace representaton form as M M c K () z 0 dttk () tte z(tt) Mz c m K () z,z 0 dt 0 dt e z(tt I ) e z(tt I ) K () t,t a M 4a zz z 1 z, 3.18 c G () z,z K() zk () z The spectral dstrbuton functon, I(), s formally defned by I() (c /m )( ), whch descrbes the character of the heat bath. In the followng, we consder the Ohmc dsspaton, I()M, the constant corresponds to the strength of the dampng. Wth the ad of Eq. 3.18, the propagator K () (tt) s wrtten as 1 K () tttt M exp tt sntt, 3.0 /4. Settng JK0, the tme evoluton of the expectaton value of Qˆ n the Ohmc dsspaton case s calculated from Eqs. 3.1, 3.15, and 3.0 as

5 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Two-dmensonal spectroscopy 71 Q (0) tqˆ t JK0 W dtk () t,tj t cos 0, sn 0, J tjk0 K () t,tj tjk0 ) Q 0 tt I e /(tt I cos t, ttt I The above equaton descrbes a damped oscllator of the wave packet n the harmonc potental started from the poston Q (0) (t I )Q 0. The hgher-order response functons are derved by usng the Feynman rule gven n Appendx B that s derved from the generatng functonal WJ,K. The Feynman role provdes the way of generalzng Brownan dynamcs, for example, to take nto account the anharmoncty of potental. In accordance wth Eqs. 3.4, B, and B3, the 1 N1 - and 1 N -terms of the (N1)th-order response functons (N 1,),.e., the thrd- and ffth-order response functons, for T 0(1,...,N) and t I 0 are gven by R (3) T 1 ; 1 Q (0) T 1 K () T 1 1 Q (0), 3.5 R (5) T 1,T ; 1 Q (0) T 1 T K () T K () T 1 1 Q (0) 1 Q (0) T 1 K () T K () T 1 T 1 Q (0). 3.6 Usng Eq. 3.1, the response functons are expressed by the ntal dsplacement Q 0 whch corresponds to the ampltude of oscllaton and the phase of the wave packet oscllator at the tme,. Note that f Q 0 s replaced by Q 0, the phase n Eqs. 3.5 and 3.6 becomes by (),.e., the negatve dsplacement leads to the phase shft of the sgnal. Physcally one can more easly understand the effects of nonequlbrum ntal condton by ntroducng the poston and momentum at the tme t nstead of Q 0 and (). We also ntroduce Q Q (0) () and P M(dQ (0) ()/d), mples the tme when the frst pump pulses nteract wth the system. From Eq. 3.1, we have Q (0) ttq e / T cost M K() T P K () T, 3.7 for T0 and t I 0. Wth the use of Eq. 3.7, we have R (3) T 1 ; 1 K () T 1 1 Q e / T 1 cost 1 M K() T 1 1 P K () T 1 K () T 1, 3.8 R (5) T,T 1 ; 1 K () T 1 K () T K () T 1 T K () T 1 Q MK () T 1 T K () T 1 K () T 1e /(T 1 T ) cost 1 T K () T 1 K () T 1e / T 1 cost 1 K () T 1 T K () T P K () T 1 T K () T 1 K () T. 3.9 In the same manner, we express the seventh-order response as a functon of Q and P as follows. The seventh-order Raman response functon s temporally three-dmensonal but up to now only temporally two-dmensonal seventhorder experments have been performed. In the Raman echo the second propagaton tme T s zero as n Raman pump probe experment the tme varable T 1 s zero. When we set T 1 to be zero, the response functon s expressed as R (7) T 3,T,0; 1 Q K () T K () T 3 K () T T 3, 3.30 wth the use of Eqs. 3.4 and B4. Ths ndcates that the sgnal n terms of T and T 3 does not depend on P.Ifwe set T to zero, K () (T ) vanshes n Eq. B4 and the seventh-order response functon s reduced to R (7) T 3,0,T 1 ; 1 3 Q e / T 1 cost 1 1 K () T 1 K () T 3 M K() T 1 1 P K () T 1 K () T 1 K () T

6 7 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Y. Suzuk and Y. Tanmura From Eqs. 3.8 and 3.31, we fnd that R (7) (T 3,0,T 1 ;) R (3) (T 1 ;)(K () (T 3 )). The 1 N1 -terms n R (N1) are ndependent of the wave packet moton and are the contrbuton for the equlbrum ntal condton,.e., ˆ I e Ĥ /Tre Ĥ. On the other hand, the 1 N -terms n R (N1) are contrbuton of the wave packet moton at tme and are functons of Q and P. In Eqs. 3.8, 3.9, and 3.31, the response functons do not depend on the ntal wdth of the wave packet and the temperature. As we show n Eqs. B, B3, and B4, sgnals depend on these hgher than the terms proportonal to N1. IV. NUMERICAL RESULT In ths paper, we consder the Raman experment for a nonequlbrum ntal condton set at tt I by the Gaussan wave packet wth the dsplacement Q 0, and the observaton s started at t by the rradaton of the par of pump pulses to the system. From a dfferent pont of vew, ths stuaton can be regarded as the thrd-, ffth-, and seventh-order Raman experments that provde the observaton of the wave packet whose ntal coordnate and ntal momentum are gven by Q and P at t. In ths secton the thrd-, ffth-, and seventh-order response functons of off-resonant Raman process are numercally calculated for the Ohmc dsspaton model for dfferent coordnate Q and momentum P. We set 1000cm 1 whch s the typcal value for molecular vbratonal moton and /0.1 underdamped case. Takng, Q and P vansh as can be seen from Eq. 3.1 and P dq /d. Then the (N1)th-order response functons approach to the equlbrum ones whose leadng contrbuton s gven by the 1 N1 -order term of R (N1), as seen from Eqs. 3.8, 3.9, and In order to see the roles of Q and P that characterze the state at t, we plot R (N1) NE T N,...,T,T 1 ;Q,P R (N1) T N,...,T,T 1 ; R (N1) T N,...,T,T 1 ;. 4.1 Hereafter, we employ the dmensonless coordnate and momentum defned by Q Q / 1 and P P / ( 1 M). We frst plot the magnary part of the Fourer transform of the thrd-order Raman response functon, R (3) ; 0 dt1 e T 1R (3) T 1 ;. 4. As a reference, n Fg. 4, we present the sgnal Im R (3) (;). Fgure 5 shows the magnary part of R (3) NE (;Q,P ) for /0.1 for a 0.01Q 0.01 wth P 0 and b 0.01P 0.01 wth Q 0. To understand the poston of the resonant peak, t s helpful to use energy level dagrams. For the thrd order Raman spectroscopy, we show some representatve dagrams n Fg. 6. In these dagrams, tme runs horzontally from the left to the rght. The vbratonal states are denoted as v FIG. 4. Plot of the spectral densty of the thrd-order Raman response Im R (3) (; ), for 0.1. wth v0,1,.... From Qˆ ââ, â and â are annhlaton and creaton operators (â vv1v1, âv vv1), we have j k dagrams for j k Qˆ j( T 1 ),Qˆ k(), whch consst of j arrows at the tme t T 1 and k arrows at t. The upward and downward arrows stand for the transton v v1 created by â and v v1 by â, respectvely. Frst, we should notce that the dsplaced Gaussan wave packet that we observe at t nvolves the off-dagonal elements n the energy level representaton,.e., ˆ v,w vw vw wth vw 0 for v w. Here ˆ s a densty matrx at the tme t defned as ˆ e Ĥ(t I )/ ˆ Ie Ĥ(t I )/. The exstence of the offdagonal elements can be understood n the followng way. Any state at t n the present study can be expressed n the phase space as e a(qq 0 cos ) [P/(M)Q 0 sn ], 4.3 s the phase determned by the rato of Q and P. Ths state can be generated from the Gaussan wave packet 0 e a[q (P/M) ] by the untary transformaton Dˆ e Q 0 (M sn Qˆ cos Pˆ ) as ˆ Dˆ ˆ 0Dˆ, snce Dˆ Qˆ Dˆ Qˆ Q 0 cos and Dˆ Pˆ Dˆ Pˆ MQ 0 sn. Thus n the energy level representaton, we have ˆ n,m e (nm) l 0 l,l D (0) m,l D (0) n,l *nm, 4.4 D (0) m,l mdˆ l 0, because Dˆ e M/(e â e â) and ˆ 0 l 0 l,l ll. The above equaton clearly ndcates the exstence of the off-dagonal elements. Fgure 6a shows the dagram for the term 1 Qˆ ( T 1 ),Qˆ (). The laser nteracton wth the lnear polarzablty, 1 Qˆ 1 (ââ ), changes the vbratonal state of the system from v to v1. Note that Qˆ (T 1 ),Qˆ () s the c-number correspondng to the functon K () (T 1 ). Fgures 6b and 6c show the dagrams for 1 Qˆ ( T 1 ),Qˆ () and 1 Qˆ (T 1 ),Qˆ (), respectvely. The laser nteracton wth the nonlnear polarzablty Qˆ (ââ ) changes the vbratonal state from v to v, as 1 Qˆ changes the state from v to v1.

7 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Two-dmensonal spectroscopy 73 FIG. 5. Three-dmensonal profle of the spectral densty of the thrd-order Raman response, Im R (3) NE (;), n the underdamped case (0.1) for a 0.01Q 0.01, P 0 andb Q 0, 0.01P The graph n the rght s the spectral densty for a (Q,P )(0.01,0) dotted lne, (Q,P )(0.005,0) dashed lne, and (Q,P )(0.01,0) sold lne, and b (Q,P )(0,0.01) dotted lne, (Q,P )(0,0.005) dashed lne, and (Q,P )(0,0.01) sold lne. Because of the off-dagonal elements of wave packet at t, the dagrams wth the dfferent ntal vbratonal state and the fnal vbratonal state can contrbute to the sgnal. It ncreases lnearly wth Q or P due to the contrbuton of the off-dagonal element of the state, v,v1. For negatve Q and P, the sgnal has the opposte sgn, whch can be seen from Eq. 4.4 by settng. For fxed P 0n Fg. 5a, the spectrum shows the two peaks at and wth wdth / and, whle for fxed Q 0n Fg. 5b, the spectrum does not show the clear peak and the spectral lne changes the sgn at 0 and. These features can be explaned clearly by usng the dagram Fgs. 6b and 6c. Wth the ad of the relaton Qˆ ( T 1 ),Qˆ () Qˆ ()Qˆ ( T 1 ),Qˆ () Qˆ ( T 1 ), Qˆ ()Qˆ (), the dagram Fg. 6b can be dvded nto two parts denoted by the dashed crcle and dotted one. The contrbuton of the dashed part that represents Qˆ ( T 1 ),Qˆ () s the same as Fg. 6a. The contrbuton of the dotted part gves the factor Qˆ ()Q whch s related to v,v1. Then Fg. 6b leads the Q dependence and contrbutes to Fg. 5a. The frequency of the sgnal oscllaton derved from Fg. 6b s due to the transton v v1 at the tme tt 1. The sgnal dependence on the dsplacement Q and the momentum P can be understood from the dagram Fg. 6c as follows. In Fg. 6c, the process can be dvded nto the two parts due to the relaton Qˆ (T 1 ),Qˆ ()Qˆ (T 1 )Qˆ (T 1 ), Qˆ (). The contrbuton encrcled by dashed lne s the same as Fg. 6a, as the contrbuton encrcled by dotted lne s the same as Q T1. By usng Qˆ (T) e ĤT/ Qˆ ()e ĤT/ cos()qˆ ()M sn()pˆ (), we fnd that ths contrbuton depends on both Q and P. Therefore Fg. 6c leads to the sgnal n Fg. 5. Fgures 5a and 5b correspond to the magnary and real parts of R (3) (), respectvely. Ths s because the matrx elements nvolved n Fg. 6c are v,v1 and, from Eq. 4.4, we have (a) the relaton v,v1 e / (b) v,v1 for the element n the case a and b that correspond to 0 and /. The sgnals n Fg. 5 show the oscllaton wth the frequency due to the transton v v at the tme T 1 usng â (t)e t â. Ther lne shapes are expressed by the superposton of the two sgnals whose lne shapes are gven by the replacement of, n R (3) (;) by0, and,, respectvely. Fgures 7 and 8 llustrate the ffth-order Raman sgnal FIG. 6. Examples of the energy level dagrams assocated wth the thrdorder response functon are shown. The vbratonal states are denoted as v, v1, and v. a represents the 1 Qˆ (T 1 ),Qˆ (). b and c correspond to 1 Qˆ (T 1 ),Qˆ () and 1 Qˆ (T 1 ),Qˆ (). The contrbuton of the dashed part that represents Qˆ (T 1 ),Qˆ () s the same as the dagram a. The contrbuton of the dotted part gves the factor Qˆ ()Q.

8 74 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Y. Suzuk and Y. Tanmura FIG. 7. Contour plot of the equlbrum part of the ffth-order Raman response, I (5) (T 1,T ;)R (5) (T,T 1 ; ), n the underdamped case ( 0.1). Dashed contours are negatve. FIG. 8. Contour plot of the ffth-order Raman response R (5) NE (T,T 1 ) n the underdamped case (0.1) for a (Q,P )(0,0.01), b (Q,P ) (0.01,0), and c (Q,P )(510 3, ). Dashed contours are negatve. FIG. 9. Examples of the energy level dagrams assocated wth the ffthorder response functon are shown. The vbratonal states are denoted as v, v1, v, and v3. a and b correspond to 1 Qˆ,Qˆ,Qˆ and 1 Qˆ,Qˆ,Qˆ. c and d are 1 Qˆ,Qˆ,Qˆ, 1 Qˆ,Qˆ,Qˆ, respectvely. The contrbuton of the dashed parts n c and d are the same as the dagrams a and b, as the dotted parts n c and d gve the factor Qˆ ()Q. Usng Eq. 4.5, the term 1 Qˆ,Qˆ,Qˆ s dvded n the two parts expressed by e and f. I (5) (T 1,T ;)R (5) (T,T 1 ;) calculated from Eq. 3.9 for the dampng constant /0.1. Fgure 7 s the equlbrum part R (5) (T,T 1 ;), as Fgs. 8a, 8b, and 8c depct the nonequlbrum part R (5) NE (T,T 1 ;Q,P ) for a Q 0, P 0.01, b Q 0.01, P 0, and c Q , P Ther phase () correspond to a ()/, b ()0, and c ()/3, respectvely. The sgnal R (5) (T,T 1,) vanshes at T 0 n all the cases Fgs. 7 and 8a 8c as can be seen from the defnton, Eq..8. The ffth-order response functon s dagrammatcally expressed n Fg. 9. In the dagrams for j k l Qˆ j(t 1 T ),Qˆ k(t 1 ),Qˆ l(), j, k, and l, arrows are depcted at the tme tt 1 T, T 1, and, respectvely. The 1 3 -terms are canceled out by each other because of the commutaton relaton n Qˆ,Qˆ,Qˆ. The leadng order terms are therefore 1 Qˆ,Qˆ,Qˆ and 1 Qˆ,Qˆ,Qˆ and are shown n Fgs. 9a and 9b, respectvely. These dagrams lead the sgnal n Fg. 7. Accordng to the commutaton relaton Qˆ (T 1 T ), Qˆ ( T 1 ),Qˆ () Qˆ (T 1 T ),Qˆ (T 1 )Qˆ (T 1 T ),Qˆ (), Fg. 9a s represented by the product of two parts, a-1 and a-, whch are assocated wth the thrd order response functon Qˆ (T 1 T ),Qˆ (T 1 )K () (T ) and Qˆ (T 1 T ),Qˆ ()K () (T 1 T ), respectvely; t shows the oscllaton wth the frequency along T drecton. In the same manner, Fg. 9b s represented by the product of two processes b-1 and b- that are assocated wth K () (T 1 ) and K () (T ), respectvely; t shows the oscllaton wth the frequency n both the T 1 and T drecton. Therefore we have the sgnals wth the frequency n the T 1 drecton and the frequency and n the T drecton n the 1 -order. The frequences n the 1 -order terms can be understood wth the use of the dagrams Fgs. 9c and 9d whch correspond to 1 Qˆ,Qˆ,Qˆ and 1 Qˆ,Qˆ,Qˆ, respectvely. These dagrams have the same frequences as Fgs. 9a and 9b and are ndependent of the momentum P, snce they can be dvded nto the dashed crcle of Fg. 9c, that of Fg. 9d, and the dotted crcles of Fgs. 9c and 9d, whch correspond to the dagrams Fgs. 9a, 9b, and Q, respectvely. The dependence of P s derved from the remanng term 1 Qˆ,Qˆ,Qˆ. Usng the commutaton relaton, the

9 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Two-dmensonal spectroscopy 75 term 1 Qˆ,Qˆ,Qˆ s rewrtten n two terms as follows, 1 Qˆ T 1 T,Qˆ T 1,Qˆ 1 Qˆ T1 T Qˆ T1 T,Qˆ T1 Qˆ T1,Qˆ 1 Qˆ T1 T, Qˆ Qˆ T1 T,Qˆ T1 Qˆ T The dagrams Fg. 9e and 9f correspond to the frst and the second term of Eq From Eq. 3.7, Q T1 T and Q T1 are expressed n terms of Q and P, whch means that the sgnals depend on both the poston and the momentum at the tme. The product of e-1, e-, and e-3 n Fg. 9e or that of f-1, f-, and f-3 n Fg. 9f shows the oscllaton wth frequency along T 1 and T. From the above dscusson, we can understand the profle of the sgnals for dfferent parameters. In Fg. 7, the sgnal oscllates wth the frequency n the T 1 drecton and and n the T drecton as dscussed n Fgs. 9a and 9b. In Fg. 8a, the sgnal oscllates wth the frequency n the T 1 and the T drecton, whch s attrbuted to the zero quantum transtons v v at the tme tt 1 and the two quantum transton v v at the tme tt 1 T shown n Fgs. 9e and 9f. In Fg. 8a, the sgnal s symmetrc wth respect to T 1 and T axs snce the dagrams Fgs. 9e and 9f cast nto the form P K () (T 1 T )K () (T 1 )K () (T ) wth the use of the relatons Q T1 T P K () (T 1 T ) and Q T1 P K () (T 1 ) for Q 0, whch s derved from Eq The sgnal n Fg. 8b ncludes varous components correspondng to the dagrams n Fgs. 9c, 9d, 9e, and 9f that lead the oscllatons wth the frequency and n the T 1 and T drecton wth the dfferent weght dependng on a condton at t and as a consequence, the sgnal s asymmetrc. In Fg. 8c, the sgnal conssts of the P and Q contrbuton whch are depcted n Fgs. 8a and 8b wth the rato sn () tocos(). A profle of any sgnal n the present model can be characterzed by the phase (). Thus, by obtanng a proper () to smulate expermental data, we can trace the moton of the wave packet at t moved from the ntal state at tt I. The man advantage of the present method s that, by measurng the sgnal for dfferent, we can drectly trace the tme evoluton of the wave packet n the phase space,.e., we can obtan the momentum and the coordnate of the wave packet at once. Note that, although the same argument can be appled to the thrd-order response, the hgher-order response that leads the two-dmensonal profle reveals the more crtcal nformaton. Fnally, we plot the seventh-order sgnal, I (7) (T 1,T 3 ;) R (7) (T 3,T 0,T 1 ;) calculated from Eq for / 0.1. The equlbrum part I (7) (T 1,T 3 ;) s gven n Fg. 10. The nonequlbrum parts, I (7) NE (T 1,T 3 ;Q,P ) R (7) NE (T 3,0,T 1 ;Q,P ), for a Q 0, P 0.01, b Q 0.01, P 0, and c Q 510 3, P whch correspond to the phase a ()/, b () 0, and c ()/3 are gven n Fgs. 11a, 11b, and FIG. 10. Contour plot of the equlbrum part of the seventh-order Raman response, I (7) (T 1,T 3 ;)R (7) (T 3,T 1 ; ), n the underdamped case (0.1). Dashed contours are negatve. 11c, respectvely. Fgures 1a 1e represent the dagrams correspondng to 1 Qˆ,Qˆ,Qˆ,Qˆ, 1 Qˆ,Qˆ,Qˆ,Qˆ, 1 3 Qˆ,Qˆ,Qˆ,Qˆ, 1 3 Qˆ,Qˆ,Qˆ,Qˆ, and 1 3 Qˆ,Qˆ,Qˆ,Qˆ, respectvely. Each of them s expressed by the product of crcled parts, whch lead to the thrd-order correlaton functon K () and Q and P. Usng the smlar manpulaton as n the ffth-order case, the dagrams n Fgs. 1a 1d lead to the oscllaton wth frequency n the T 1 drecton FIG. 11. Contour plot of the seventh-order Raman response R (7) NE (T 3,T 1 )n the underdamped case (0.1) for a (Q,P )(0,0.01), b (Q,P ) (0.01,0), and c (Q,P )(510 3, ). Dashed contours are negatve.

10 76 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Y. Suzuk and Y. Tanmura FIG. 1. Examples of the energy level dagrams assocated wth the seventh-order response functon at T 0 are shown. The vbratonal states are denoted as v, v1, v, v3, and v4. a and b correspond to 1 Qˆ,Qˆ,Qˆ,Qˆ and 1 Qˆ,Qˆ,Qˆ,Qˆ. c, d, and e correspond to 1 3 Qˆ,Qˆ,Qˆ,Qˆ, 1 3 Qˆ, Qˆ,Qˆ,Qˆ, and 1 3 Qˆ,Qˆ,Qˆ,Qˆ, respectvely. The contrbuton of the dashed parts n c and d are the equvalent to the dagrams a and b, as the dotted parts n c and d gve the factor Q. and n the T 3 drecton, and the dagram n Fg. 1e leads to the oscllaton wth the frequency both n the T 1 and T 3 drecton. Then we observe the oscllaton n the T 1 drecton and n the T drecton n Fg. 10, snce only the dagrams n Fgs. 1a and 1b contrbute to the equlbrum sgnal. In the case Fg. 11a, we observe the oscllaton both T 1 and T 3 drecton, snce the process correspondng to Fg. 1e contrbutes to the sgnal. In the case Fg. 11b, the dagram Fgs. 1c 1e contrbute to the sgnal. Hence the oscllaton n the T 1 drecton and that n the T 3 drecton becomes by the superposton of the oscllatons wth the frequences and. The sgnal n Fg. 11c s gven by the lnear combnaton of the sgnals n Fgs. 11a and 11b wth the rato sn () tocos(). Ths stuaton s same as the ffth-order case and we can use the seventhorder experment to see the dynamcs of the wave packet n the phase space, as well. V. CONCLUSION In ths paper, we derved the generatng functonal for a Brownan oscllator system whose ntal state s descrbed by dsplaced Gaussan wave packet from the path ntegral approach. The generatng functonal allows for the calculaton of the thrd-, ffth-, seventh-order Raman response of a harmonc oscllator wth coordnate dependence of the polarzablty. To demonstrate effects of the nonequlbrum ntal condton, we plot the Raman response for the dfferent dsplacement and momentum of the wave packet at the tme t when the frst pump pulses nteract wth the system. Any state at tme can be expressed by a Gaussan wave packet centered at P /(M)A sn and Q A cos, A and are the ampltude and the phase n the phase space and are gven by AQ 0 and for an oscllator wth frequency. Due to the off-dagonal elements of the state at t, the sgnals depend on the wave packet moton and show the mode wth the frequency whch does not appear n the equlbrum case. In the thrd order response, the sgnal decays wth decreasng the dsplacement Q for the postve dsplacement (0) and t ncreases wth decreasng Q for the negatve dsplacement (). Consequently, the tme evoluton of the sgnal mples whether the wave packet s dsplaced ntally toward decreasng or ncreasng bond length. In the ffth- seventh- order response, the component of the sgnal whch s proportonal to P s symmetry wth respect to T 1 and T T 1 and T 3. On the other hand, the component whch s proportonal to Q s asymmetrc. These propertes can be explaned wth the help of the energy level dagrams. In the sgnal, the rato of a Q contrbuton to a P /(M) contrbuton s cos to sn. Thus, by lookng for the phase to smulate expermental data, we can trace the moton of the wave packet at tme t moved from the ntal state at tme t0. The man advantage of the present method s that we can obtan the nformaton about not only the poston but also the momentum of the wave packet. In the present studes, we restrcted our analyss to the order of 1 N1 and 1 N, so the response functons do not depend on the ntal wdth of the wave packet and the temperature as stated n Sec. III; the dependence of the temperature and the wdth of the wave packet appear n the order of N1. Such effects as well as the effects of the anharmoncty of an oscllaton mode may be studed from the equaton of moton approach. 17,33,34 ACKNOWLEDGMENTS The authors thank professor Hro-o Hamaguch mentonng about a possblty to use D spectroscopy to detect the phase of the wave packet moton. The present nvestgatons were supported by the Grant-n-Ad on Prorty Area of Chemcal Reacton Dynamcs n Condensed Phases , the Grant-n-Ad for Scentfc Research B APPENDIX A: DERIVATION OF THE GENERATING FUNCTIONAL In ths Appendx, we derve the generatng functonal WJ,K defned by Eq. 3. for the ntal state Eqs By ntegratng over the bath coordnate q I and q I, WJ,K s gven by exp WJ,K 1 snh exp 1 dtdt C1 C c C Jt (m G, ) C t,t C Jtexp W SJ,K, A1

11 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Two-dmensonal spectroscopy exp W SJ,K VQ;KsKsQ, 77 A7 dq I dq I Q I I (S) Q I Q I Û S t F,t I Û S1 t F,t I Q I, A Û S t,tt exp t t ds Pˆ M 1 M Qˆ G (m,) C t,t m J sqˆ K sqˆ 1,, 1 snh C ttcos tt A3 C ttcos tt. A4 Here s gven n Eq Notce that we take the fnal tme t F to be set nfnty n the end of ths Appendx. The functonal dfferentaton / C J(t) means /J 1 (t) and /J (t) for tc 1 and tc, respectvely. The factor Q I Û S (t F,t I )Û S1 (t F,t I )Q I s calculated as follows. Insertng the completeness relaton 1 dq F Q F Q F at the fnal tme t F, we have Q I Û S t F,t I Û S1 t F,t I Q I dq F Q I Û S t F,t I Q F Q F Û S1 t F,t I Q I. A5 The tme evoluton kernel wth the sources J 1,K 1, Q F Û S1 (t F,t I )Q I s gven by see, for example, Ref. 35 Q F Û S1 t F,t I Q I M sn T1/ exp t I t FdsV J 1 1s s ;K exp 1 t I t Fdt t FdtJ1 tt,tj 1 t Q I t Fdt sn tf t J 1 t sn T Q F t Fdt sn t M J 1 t Q I Q F sn T sn T M cos T Q F Q I sn, T A6 t,t 1 M tt sn t sn tf t sn T tt sn t sn tf t sn T. A8 The kernel of the return path wth the source J and K, denoted by Q I Û S (t I,t F )Q F, s gven by replacng J 1 and K 1 wth J and K for the complex conjugate of Eq. A6. By ntegratng over Q F, Eq. A8 s expressed as dq F Q I Û J t F,t I Q F Q F Û J1 t F,t I Q I A9 exp t Fds V V J 1 s 1s ;K J s s ;K t FdtD () tt I Q I Q I exp Q IQ I t Fdt cos t J t t Fdt t FdtJ td () ttj t, A10 J s defned by Eq Choosng the dsplaced Gaussan wave packet 3.10 as the ntal state, W S J,K s calculated by ntegratng over (Q I Q I ) and (Q I Q I )/ as exp W SJ,K exp t Fds V V J 1 s 1s ;K J s s ;K exp t Fdt t FdtJ td () ttj t Q 0 cos tt I tt t Fdt t FdtJ td () t,tj t. A11 Here D () and D () are the propagators wthout couplng to the bath and are gven by Eq and D () t,t ad () tt I D () tt I 1 4a cos t cos t, A1

12 78 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Y. Suzuk and Y. Tanmura 1exp C1 C exp t FdtJ t tj t dtjtt0 t0 from the rght-hand sde of Eq. A1, and usng the relaton e F 1 [/J] e F [J] e dsj(s)(s) 0 e F [J/] e F 1 [] 0, we obtan the followng result: exp WJ,K 1 snh D () tt exp t Fdt t Fdt t t J t t J Q 0 cos tt I tt A13 t Fdt t Fdt t t J D t,t () t t J exp t Fdt t Fdt tc G () t,t t tc G () tt t exp t FdsV1 s;k 1 sv s;k, A14 s0 G () (m ttg, ) (m, ) 1 11 t,tg1 t,ttt sn m tt, A15 G () t,t 1 4 G (m, ) (m, ) (m, ) (m, ) 11 t,tg1 t,tg1 t,tg t,t m coth cos tt. A16 Equaton A14 leads to the Feynman rule descrbed n Fg. 13. Usng ths rule, WJ,K can be cast nto the followng form: exp WJ,K exp 1 t Fdt t snh Fdt t t J K tt () t t J t t J 1 t t J K t,t () exp t FdsV1 s;k 1 sv s;k. A17 s0 Here K () and K () are the propagators for the system ncludng the effects of the bath. By usng the fact that K () (t t) s causal, we can fnd the graphcal expresson Fg. 14 from the Feynman rule Fg. 13. The algebrac expresson of Fg. 14 s represented as K () ttd () tt t Fds t FdsD () ts t Fds t Fdst Fdst FdsD () ts c G () ss D () st c G () ss D () ss c G () ss D () st. A18

13 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Two-dmensonal spectroscopy 79 Summng up the rght-hand sde of Fg. 14, we have the followng relaton: K () ttd () tt t Fds t FdsD () ts c G () ss K () st. A19 Usng the Laplace transformaton, K () n t F s represented as K () zd () zd () z c G () z K () z, A0 D () (z) and G () (z) are gven by D () z 1 M 1 z, G () z 1 m 1 z. A1 A Ths leads the expresson n Eq Next, K () (t,t) can be obtaned n the same manner. The algebrac expresson of K () s K () t,td () t,t t Fds t FdsD () t,s c G () ss K () st t Fds t FdsD () ts c G () ss K () s,t t Fds t FdsD () ts c G () s,s K () st, FIG. 13. Feynman rule. A3 FIG. 14. Dagrammatcal expresson of a propagator K (). G () (tt)g () (tt), K () (tt) K () (tt) and another propagators D () (t,t) and G () (t,t) are gven by Eqs. A1 and A16, respectvely. Now, we take the fnal tme t F. Usng the frst lne of Eq and substtutng Eqs. A1 and A16 nto Eq. A3, we obtan the propagator K () as n Eq Applyng the relaton, e F[/] e dsk(s)(s) e dsk(s)(s) e F[(/)K], to Eq. A17, we arrve at the result gven n Eq APPENDIX B: DERIVATION OF N TIME CORRELATION FUNCTION BY USE OF FEYNMAN RULES A4 In ths Appendx, we present the Feynman rules that lead the N tme correlaton functons, G (N1) R, defned by Eq From Eqs. 3.5 and 3.14, we have followng rules: 1 Prepare N1 whte crcles correspondng to k0, k1,..., and kn (k 1,...; 0,1,...,N). The whte crcle correspondng to k from whch k lnes emerge shall be called the external pont; Prepare black crcles correspondng to sources J that (0) are treated as one-pont vertces. A lne emerges from ths black crcle; 3 Attach a tme valuable t 0 to the external pont correspondng to k0. We call t an external pont labeled t 0 ; 4 Attach a tme valuable t to the external pont correspondng to k. We call t an external pont labeled t ; 5 Attach a tme valuable t to a black crcle correspondng to J ; (0) 6 Prepare lnes correspondng to propagators, K () ; 7 Attach the ndex or to each lne from an external pont or a black crcle as n Fg. 15. The factors on the rght of the graphs mply attachments to the graphs. 8 Usng the propagators, external ponts, and one-pont vertces, draw all connected dagrams whch are topologcally dstnct. Note here that the dagram whch contans the propagators connectng the ndces and are excluded; 9 Carry out the ntegraton over all nternal tme from t I to ; 10 Multply the contrbuton of each dagram by (/) N1 /S, S s the symmetry factor. The symmetry factor S s defned as the order of the permutaton group of the nternal lnes and vertces leavng the dagram unchanged when the external lnes are fxed.

14 80 J. Chem. Phys., Vol. 115, No. 5, 1 August 001 Y. Suzuk and Y. Tanmura FIG. 15. Feynman rule for response functons. As an example, let us consder the second-order correlaton functon, G R () (t 0,t 1 ). In accordance wth the above rules, G R () (t 0,t 1 ) s dagrammatcally gven by Fg. 16a after ntroducng the expresson 1 dsk () tsj (0) s 1 Q (0) t, B1 the last equaton s obtaned wth the use of the second lne of Eq Followng to the rules, we can wrte down the analytcal expresson n the form FIG. 16. Dagrammatcal representaton of a second-, b thrd-, and c ffth-order response functons expressed by the terms Eq. B1 and Fg. 15. G R () t 0,t 1 1 Q (0) t 0 K () t 0 t 1 1 Q (0) t 1 K() t 0,t 1 K () t 0 t 1. B In the same way, the thrd-order correlaton functon, G (3) R (t 0,t 1,t ), and the fourth order correlaton functon, G (4) R (t 0,t 1,t,t 3 ), are dagrammatcally expressed as n Fgs. 16b and 16c, respectvely. Then the analytcal expresson for these dagrams are gven by G (3) R t 0,t 1,t 1 Q (0) t 0 K () t 0 t 1 K () t 1 t 1 Q (0) t t 1 t 1 Q (0) t 1 K () t 1 t 0 K () t 0 t 1 Q (0) t 3 K () t 0,t 1 K () t 0 t K () t t 1 K () t 0,t K () t 0 t 1 K () t 1 t, B3 3 G (4) R t 0,t 1,t,t 3 1 Q (0) t 0 K () t 0 t 1 K () t 1 t K () t t 3 1 Q (0) t 3 5 terms that are all permutaton of t 1,t,t 3 1 Q (0) t 1 K () t 1 t 0 K () t 0 t K () t t 3 1 Q (0) t 3 fve terms that are all permutaton of t 1,t,t 3 4 K () t 0 t 1 K () t 1 t K () t t 3 K () t 3,t 0 5 terms that are all permutaton of t 1,t,t 3. B4

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