Dissipative dynamics in a curve-crossing system. The N2 molecule

5 August1994

CHEMICAL PHVSICS LElTmS

ELSEVIER

ChemicalPhysicsLetters225 ( 1994)511-518

Dissipative dynamics in a curve-crossing system. The N2 molecule 0. Kiihn, V. May Institut ftir Physikund WIPbei der Max-Plan&AG Halbleitertheorie,Humboldt-Universitiit zu Berlin, Hausvogteiplatz5-7, D-10117 Berlin, Germany

Received10May 1994

Density matrix theory is applied to the study of environmentaleffects on the dynamicsof vibrational degreesof freedom in coupled Potential energy surfaces.For this purpose we formulate the theory in a diabatic state representationand incorporate the environment to second order of perturbation theory. As an examplewe study the dynamicson the coupled b’ ‘z-c’ 5: Potentialenergy surfacesof the Nz moleculewhich is supposedto be immobilizedin a solid state matrix. The resultingdependence.of the vibrationaldynamicson the couplingstrengthto the matrix degreesof freedom is discussedin the context of a pump probe experimentalconfiguration.It is found that with increasingcouplingstrengthoscillatorystructuresin the probe spectrum are stronglysuppressed. 1. Introdnctlon

Investigations of ultrafast dynamical phenomena in molecular systems have attracted considerable attention during the last years ‘. With the development of sophisticated experimental techniques it became possible to extract information on the time scale of the vibrational motion of the constituent atoms. Nowadays the theoretical description of these phenomena is well elaborated. For the treatment of the purely coherent dynamics there exist several techniques. These are based on different schemes for propagating the wavefunction as a solution of the time-dependent Schriidinger equation for a particular Hamiltonian (see e.g. Ref. [ 21). Besides measurements in the gaseous phase dissolved molecules have also been studied [ 3 1. Here it is necessary to ’ For a recentreviewon theoreticaland experimentalresults,see Ref. [ 11.

include dissipation into the theoretical description and thus change to the description of a partly coherent motion or incoherent motion. This situation is most favorably dealt with using the density matrix method. Depending on the problem at hand the density matrix can be treated in a coordinate representation [ 41, Wigner representation [ 5 ] or in a state representation [ 6,7]. In this Letter, we use the density matrix in the state representation for the investigation of dissipative vibrational dynamics in diatomic molecules (rotational degrees of freedom are not considered). Energy dissipation and vibrational relaxation will be of major concern here, because the molecule is assumed to be embedded into a solid state matrix or any other kind of environment. Interaction with this surrounding will cause a dynamical behavior different from that in the coherent regime. Section 2 gives a brief outline of the theoretical ingredients which are used in the numerical simulation of Section 3. We will dis-

0009-2614/94/$07.000 1994ElsevierScience B.V. Allrightsreserved

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0. K&a, V.May /Chemical Physics Letters 225 (1994) 511-518

512

cuss our results in the light of an experimental scheme corresponding to a time-resolved pump probe measurement in the limit of impulsive excitation [ 8 1. As an illustrative example we have chosen the N2 molecule which has been considered by Broeckhove and co-workers [ 9 ] for the gaseous phase. Although experimental results for pump probe spectra are not available for Nz dissolved in a solute or isolated in a solid state matrix it is interesting to compare our theoretical calculations with results obtained previously [91. 2.Theoretical approach We start our discussion with the assumption that the Schriidinger equation for the electronic degrees of freedom of the considered molecular system plus the environment has been solved. The resulting adiabatic potential energy surfaces (PES) depend on the nuclear coordinates of the molecule (the interatomic distance R in the present case of a diatomic molecule) as well as of the environment Z= {Z,>. Changing to a diabatic representation which refers to the different minima of the original PES the potential energy part of the Hamiltonian matrix contains offdiagonal elements instead of the kinetic energy part. Since we are interested in the curve-crossing problem we concentrate on two excited diabatic PES in the following. These two diabatic potential energy surfaces are coupled through the interstate coupling function, V,,,(R, Z), which in general depends on the interatomic distance and all degrees of freedom (DOF) of the environment. Here we take the approximation that the interstate coupling is constant in the region of curve-crossing where the wavefunction overlap has its maximum, i.e. we put V12(R, Z) k: VIZ,which is treated as a parameter *. Using this approximation we can write the Hamiltonian matrix for the two coupled diabatic states as fl*

v*

H&0=-2,

O (

+

~I’,(R-RP) (

v2,

0 v2 >

VI,

V,(R-R$)

>’

(1)

Compare the discussion of a R-dependent Gaussian coupling function for the N2 molecule by Broeclchoveet al. in Ref. [ 91. 2

The R”, are the equilibrium positions of the mth diabatic PES V,,,(R) and p is the reduced mass. In general these molecular parameters are modified by the environment. To account for this effect would amount to a quantum chemical calculation for a specific system. This is, however, beyond the scope of the present Letter. In Section 3 we will consider the N2 system using the parameters of the free molecule. The environment then enters only via its fluctuations as will be outlined subsequently. First, we have to specify the Hamiltonian for the environmental DOF, HE(Z), which is done by assuming the environment is a macroscopic set of independent harmonic oscillators in thermodynamic equilibrium, i.e.

For the coupling between the interatomic coordinate and the environmental DOF we take the following bilinear expression HsE(R,

Z) = c k,Z, e

(3)

We have chosen this well established type of interaction Hamiltonian (see e.g. Ref. [ 10 ] ) for numerical convenience although our approach is capable of the description of more general functional forms for this system environment coupling (SEC) [ 111. It should be mentioned that the choice of the Hamiltonian, Eqs. ( 1)-( 3) can be justified by a Taylor expansion of the diabatic PES with respect to both types of coordinates. The quantity of interest for the description of the ultrafast dynamics and related optical properties is the reduced density matrix (RDM ) for the single vibrational DOF, R. There exist a number of different techniques to derive the equations of motion for the RDM (see e.g. Refs. [ 6,121). Here we use an approach which starts with a representation of the density matrix in appropriately defined product states of the system and the environment [ 111. These states are given as the solution of the following stationary Schrijdinger equation

0. K&n, V.May /Chemical Physics Letters 225 (I 994) 51 l-518

-gv+

vm(R-R:)+HE(Z)

=&&AR

Z) *

>

&JR, Z) (4)

The index (Ycomprises the quantum numbers for the diabatic state, the vibrational states of the molecule and of the environment (Q = m, M, &!) . Introducing the timedependent statistical operator for the total system, W(t), the full density matrix can be written as Pa,r=trs+E (W(t)]ar)(p]). The RDM is obtained from this expression by taking the trace with respect to the environmental DOF M =trS+E(

W(t)

1 mM)

< nNi

) .

(5)

For low energetic vibrational excitation we can approximate the diabatic PES by parabolas which have, in general, different curvature. This assumption is not suitable for the description of highly excited vibrational states or any dissociation process of the molecule. To describe such situations a full quantum chemical calculation which, however, offers the basis states for the representation of the RDM only numerically, is necessary. Thus we will proceed with V,(R-Re,)

= Vt++&(R-R”,)2.

513

(6)

This parabolic potential form has the advantage that the expansion of the RDM will be carried out in simple oscillator eigenfunctions, x,,& R - RC, ) , and certain matrix elements of the Hamiltonian can be obtained analytically. The time evolution of the RDM follows from the Liouville-von Neumann equation of motion. In order to obtain a closed form expression which can be treated by standard numerical methods we must invoke certain approximations. When we partitioned the coordinates we implicitly assumed that the environmental DOF represent only a weak perturbation to the dynamics of the relevant system. Thus it is justified to include the SEC perturbatively to second order with respect to the coupling strength ke This can be done if one utilizes the projection operator formalism [ 111. After some lengthy algebra the equation of motion becomes

-&

(J’wppcf’i+v)+Rpv.

(7)

First, we notice the appearance of vibrational dephasing rates which include the inverse relaxational lifetimes for the state x,& 1 -=z,,

2;

m

[MY(%Y)+(M+l)y(-c-Ql,

(8)

with c,,,= p~,/fi. ‘The damping function for the vibrational levels, y(w ) , is given by the Fourier-transformed autocorrelation function of the environmental contributions to the SEC Hamiltonian Eq. ( 3 ) . It can be expressed in terms of the spectral density of the environment, a(o), as y(w)=2xa(o) [l+ n (0) ] ( n (co) is the Bose-Einstein distribution). For the present purpose we do not consider particular models for this spectral function. Instead we assume a(w) to be constant in the frequency range under consideration. This constant is treated as a parameter. For the actual numeric, however, it will be convenient to relate y(w) to a single PES (m = 1) which is done by introducing the SEC strength parameter G= xa ( w1) /c, (for details see Ref. [ 111). Thus we havey(+w,,2)=f[1+n(+ol,2)]o,G/w,,2. This damping function also enters the second dissipative part of the equation of motion, R,,. Without giving the detailed structure (see Ref. [ 111) we remark that R,, contains contributions which connect pllywith different elements of the RDM, such asp,,,,. Finally, we come to the second term on the righthand side of Eq. (7) which is responsible for the intersite transfer. Here we introduced the interstate coupling matrix which is in the Condon approximation Vco= V,, (mMI nN) . The Franck-Condon overlap integral (mMI nN), can be expressed in terms of a recursion relation [ 111. The static coupling, Pm,, will be an adjustable parameter of the present theory. .3. Discussion of the dynamics and related pump probe spectra This section is aimed at the demonstration of the power of the present approach in the investigation of dissipative dynamics particular in curve-crossing

0. Kiihn, V. May/Chemical

514

Table 1 Parameters for the diabatic PES used for the N2 molecule (after Ref. [ 131) Electronic state

m

x rz+ c”c: b”I;Y+ ”

0

1 2

VE) (103cm-‘) 0.0 104.52 104.48

orn

Xl

(cm-‘)

(A)

2360.0 2160.0 750.0

1.094 1.12 1.45

problems and related ultrafast optical properties. To achieve this goal we have chosen the Nz molecule as an example. According to the calculations in Ref. [ 91 we take Vi as the PES corresponding to the excited electronic state with symmetry b’ ‘Z,’ , and V, as corresponding to the state c’ ‘Z: . The parameters we use are summarized in Table 1. If we assume the molecule to be excited under conditions for which the limit of impulsive excitation holds, i.e. the duration of the laser pulse is short compared to any other time scale involved, the vibrational wavefunction is placed onto an excited PES without any change of its shape. (In the diagram of the PES versus R this appears as a vertical shift of the wavefunction with respect to the energy axis.) In the present case we provide excitation conditions in such a manner that only the PES Vi will

Physics Letters 225 (1994) 511-518

be excited. If the molecule was in its ground state before the excitation we have as an initial condition for the equation of motion Eq. ( 7 ) (see e.g. Ref. [ 111) .

The assumption of impulsive excitation by the pump beam avoids the incorporation of density matrix elements depending on the ground-state electronic quantum number. Thus the dynamics are reduced to a motion on the excited PES alone. For the actual parameters the wave packet is energetically located below the curve-crossing region. Looking at the occupation probabilities for the vibrational levels of the PES one finds that almost 85% of the probability is in the 1m= 1, M=O) state (compare Fig. 2a). This is in agreement with the findings of Stahel et al. [ 13 ] and Broeckhove and co-workers [91. Following the dissipative dynamics is most easily done if one looks at the occupation probability of the electronic states defined through P,(t)=

&P??amnM*

(9)

Additionally, it can be shown that the probe spec-

1.0

P(t) ;;

Fig. 1. Time evolution of the total occupation probability of the diabatic states (Eq. (9) ) for different SEC strengths: (a) G=O, (b) G=O.OOlor and (c) G=O.Ol w,. The staticcouplingis taken to be V,2=850cm-‘hereandthroughout theremainingfgures. The initial condition is such that PI (0) = 1 (upper curves in the panels). Note that in the limit of impulsive excitation these curves are proportional to a possible probe spectrum as discussed in the text.

0. Kiihn, Y. May /Chemical Physics Letters 225 (1994) 511-518

(b) i

t [fsl Fig. 2. Time evolution of the occupation probability of the vibrational levels of (a) the PES V,and (b) the PES V2for G=O.Ol o, ((-) M=O; (---)M=l; (---) M=2; (...) M=3; (-.A) M=4). The respective vibrational relaxation times are approximately r,,,,=M-‘245 fsand ~~=M-‘85 fs.

trum for a transition from the PES V1,Zto a higher PES is proportional to Pi I2( T) in the impulsive excitation limit [ 141. Here, r is the time delay between the pump and the probe pulse. Fig. 1 shows P,(t) for three different situations. Fig. la corresponds to the coherent limit, i.e. G=O, which was studied previously in Ref. [ 9 1. Although in this earlier work more realistic Morse potentials were used for the propagation of the wavefunctions our results show all characteristic features discussed in this context. In particular, we find oscillations on two different time scales. The longer one (approximately 100 fs) has its origin in the interference between the original wave packet which moves on the PES Vi and parts of it which were transferred to the

515

PES VZ.The short time scale, on the other hand, reflects the successive entering of the crossing region by the original wave packet. The vibrational periods corresponding to the PES Vr and V, are T, = 15 fs and T2= 44 fs, respectively. The effect of the SEC is considered in Fig. 1b (G=O.OOl or) and Fig. lc (G=O.Ol or). The corresponding vibrational lifetimes rW for the excited vibrational states in Fig. lc are rlM=M-‘245 fs and r,=iV-‘85 fs. The topology of the PES is such that for the low temperatures under consideration ( T= 4.2 K) the thermodynamic equilibrium state is one in which the right PES is preferably populated. This tendency is clearly visible in Figs. lb and lc where the occupation probability PI (t) decreases in favor of P2( t). This goes along with a smearing of the oscillatory structure of the curves with increasing G. The latter is a consequence of the intricate interplay between interstate transfer and intrastate relaxation. The dynamics of this relaxation process can be further visualized looking at the occupation probabilities of the vibrational states P-(t) =~-,~(t) which are shown in Fig. 2a (m= 1) and Fig. 2b (m=2) for G=O.Ol wl. The overall trend is a decrease of PIM and a population of the lowest vibrational levels in the PES V,. Note that the vibrational energies are such that 30~ x ol. This leads to a pop ulation of higher states in the PES V, which in addition have a larger wavefunction overlap with the respective states in the PES V,. Due to the SEC this population of the PES V2relaxes towards the ground states as is clearly indicated in Fig. 2b. A convenient measure for the action of the environment on the dynamics of the system is the entropy. In general the entropy is defined with respect to the density operator of the total system. This, however, does not provide any valuable information because of its constancy. On the other hand, one can define a subsystem entropy using the reduced density operator [ 15,161. Since the environment was assumed to consist of an infinite number of harmonic oscillators the thus defined entropy of the relevant system is a monotonic increasing function of time. In the present context, however, it is more enlightening to use the Shannon entropy [ 15,161 defined with respect to the occupation probabilities of the diabatic PES

516

0. K&I, K May /Chemical Physics Letters 225 (1994) 511-518

Fig. 3. Time evolution of the Shannon entropy, Eq. (lo), of the electronic subsystem S,(t) for: (a) G=O, (b) G=O.OOl o, and (c) G=O.Ol w,.

The difference to the entropy of the total relevant system becomes clear from Figs. 3a-3c. Here, S,(t) is plotted for the parameters of Fig. 1. Since we start in a pure state with respect to the electronic populationwehaveS,(O)=O (note,thatxlnxl,,O=O).The following temporal evolution of S,(t) reflects more or less the behavior of P,,,(t). Comparing the three panels of Fig. 3 we can conclude that already for G = 0 (Fig. 3a) the entropy is growing to an average value with respect to the two diabatic states. Around this average the entropy is oscillating with well developed recurrences which can be interpreted as a partial rephasing of the original state. This behavior has the same origin as the oscillations of P, ( t ) discussed before. Switching on the SEC leads to a damping of these oscillations (Fig. 3b) and, in particular, of the recurrences since constructive interference is hardly to establish now. For high enough G we notice a rephasing of the electronic population again after passing a global maximum, i.e. the entropy is manifestly decreasing in the given time interval. This confirms the statement that in the thermodynamic limit Pz x 1 and P, w 0, or in other words, the electronic subsystem is moving towards a more or less pure state again.

Finally, we show the time evolution of the initial wave packet in Fig. 4 for G= 0.01 o,. In doing so we must calculate the probability distribution in coordinate space which is given by the diagonal elements of the respective RDM P(R t) =p(R, R 0 = &‘N~Lu(R-R:)

xWmlV(R-R”m)P-,mN(t).

(11)

This figure nicely comprises all features of dissipative dynamics, discussed so far from a global viewpoint. Looking at Fig. 4a we notice that the initial wave packet oscillates with decreasing amplitude in the left PES. Whenever it hits the crossing region part of it is transferred to the higher excited states of the right PES (nodal patterns for short times). The subsequent relaxation leads, finally, to an accumulation of probability in the right PES as was found in Fig. 2. In order to give an account of the energy dissipation into the environment we have plotted in Fig. 4b the probability distribution P( R, t) for selected times at the energetic position of the internal energy of the system which is

&t(t)= +2 c

MN

,C, [VCm+fiWn(M+f)lPmA4,mM(t) ~mf,IN

RebLM,2N(f)

1.

(12)

0. Kiihn, V.May /Chemical Physics Letters 225 (1994) 511-518

517

0.5

P(RJ)

104 -

0.8

1.0

1.2

1.4 R 14

1.6

1.8

Fig. 4. Time-dependent dissipative wave packet dynamics in coordinate space according to Eq. ( 11) for (a) G= 0.0 lo,. In panel (b ) we show the wave packet for t = 0 fs and t = 500 fs (from top) at their respective internal energies (Eq. ( 12 ) ). Note that the vertical lines indicate the positions of the vibrational levels in the respective diabatic PES.

In summary, we have given a theoretical formulation for the microscopic description of dissipative dynamics as well as of the related pump probe spectra in curve-crossing systems. The approach we have used is based on the density matrix theory in the state representation of the diabatic PES. The inclusion of system environment interaction to second order of perturbation theory led to a dramatic change in the vibrational dynamics and thus in the respective probe spectra which was discussed for

the Nz molecule. In doing so we could extend previous results of Broeckhove and co-workers for the gas phase [ 91 to the more general situation of Nz in an arbitrary environment (e.g. a rare-gas matrix). For the sake of simplicity we restricted ourselves to parabolic potential energy surfaces only. While the overall features of the coherent dynamics in the Morse potentials could be reproduced here, it is desirable to extend the approach to more realistic potentials. This will be done in a forthcoming paper.

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0. Kiihn, V. May/Chemical

One of us (OK) gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft.

References [ 1] J. Manz and A.W. Castleman Jr., eds., Femtosecond

Chemistry: The Berlin Conference, J. Phys. Chem. 97, No. 48 (1993). [2] J. Broeckhove and L. Lathouwers, eds., Time-dependent molecular dynamics (Plenum Press, New York, 1992). [3] N.F. Seherer,L.D. Ziegler and G.R. Fleming J. Chem. Phys. 96 (1992) 5544; N.F. Scherer, D.M. Jonas and G.R. Fleming, J. Chem. Phys. 99 (1993) 153. [4] M. Berman and R. Kosloff, Computer Phys. Commun. 63 (1991) I. [ 5] Y.J. Yan, L.E. Fried and S. Mukamel, J. Phys. Chem. 93 (1989) 8149. [ 61 S.H. Lin, R. Alden, R. Islampour, H. Ma and A.A. Villaeys, Density matrix method and femtosecond processes (World Scientific, Singapore, 199 1).

Physics Letters 225 (1994) 511-518 [ 71 V. May and M. Schreiber, Phys. Rev. A 45 ( 1992) 2868.

[ 81 S. Mukamel, Ann. Rev. Phys. Chem. 41 ( 1990) 647. [9] J. Broeckhove, B. Feyen, L. Lathouwers, F. Arickx and P. Van Leuven, Chem. Phys. Letters 174 ( 1990) 504; J. Broeckhove, B. Feyen, L. Lathouwers and P. Van Leuven, in: Time-dependent molecular dynamics, eds. J. Broeckhove and L. Lathouwers (Plenum Press, New York, 1992 ) . [lo] J.N. Onuchic, D.N. Beratran and J.J. Hoptield, J. Phys. Chem. 90 (1986) 3707. [ 1110. Ktihn, V. May and M. Schreiber, J. Chem. Phys., submitted for publication. [ 121 K. Blum, Density matrix theory and applications (Plenum Press, New York, 1981). [ 131 D. Stahel, M. Leoni and K. Dressler, J. Chem. Phys. 79 (1983) 2541. [ 141 G. Stock and W. Domcke, J. Opt. Sot. Am. B 7 ( 1990) 1970. [ 151 E. Fick and G. Sauermann, The quantum statistic of dynamic processes (Springer, Berlin, 1990). [ 161 S.J.D. Phoenix and P.L. Knight, Ann. Phys. (NY) 186 (1988) 381.