Hydration shell effects in the relaxation dynamics of photoexcited Fe-II complexes in water
P. Nalbach, A.J.A. Achner, M. Frey, M. Grosser, W. Gawelda, A. Galler, T. Assefa, C. Bressler, M. Thorwart
aa r X i v : . [ phy s i c s . a t m - c l u s ] A p r Hydration shell effects in the relaxation dynamics of photoexcited Fe-IIcomplexes in water
P. Nalbach , , A. J. A. Achner , , M. Frey , , M. Grosser , , W. Gawelda , , A. Galler , , T. Assefa , , C.Bressler , , and M. Thorwart , I. Institut f¨ur Theoretische Physik, Universit¨at Hamburg, Jungiusstraße 9, 20355 Hamburg,Germany The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg,Germany European XFEL GmbH, Notkestraße 85, 22607 Hamburg, Germany (Dated: 19 July 2018)
We study the relaxation dynamics of photoexcited Fe-II complexes dissolved in water and identify the relax-ation pathway which the molecular complex follows in presence of a hydration shell of bound water at theinterface between the complex and the solvent. Starting from a low-spin state, the photoexcited complex canreach the high-spin state via a cascade of different possible transitions involving electronic as well as vibra-tional relaxation processes. By numerically exact path integral calculations for the relaxational dynamics of acontinuous solvent model, we find that the vibrational life times of the intermittent states are of the order ofa few ps. Since the electronic rearrangement in the complex occurs on the time scale of about 100 fs, we findthat the complex first rearranges itself in a high-spin and highly excited vibrational state, before it relaxesits energy to the solvent via vibrational relaxation transitions. By this, the relaxation pathway can be clearlyidentified. We find that the life time of the vibrational states increases with the size of the complex (within aspherical model), but decreases with the thickness of the hydration shell, indicating that the hydration shellacts as an additional source of fluctuations.
I. INTRODUCTION
When a photoexcited molecule is placed in a polariz-able solvent, it will relax its energy in presence of poten-tially strong interactions with its bath, i.e., its nearestneighbor solvent molecules. This interplay manifests it-self already in the properties of the steady state by theobserved Stokes shift between the absorption and emis-sion energies of the solute, which typically reflect therearrangement of the caging solvent around the excitedsolute . A pioneering femtosecond transient absorptionlaser study of photoexcited NO in solid Ne and Ar raregas matrices was capable of extracting mechanistic move-ments of the caging rare gas atoms in combination withmodel calculations , but in liquid media this connec-tion to the actual solvent movements in response to thecreation of an excited state dipole moment is inherentlydifficult to observe experimentally. Quantum chemicalcalculations have meanwhile advanced and now permitsimulating the dynamic response inside a box containingthe excited molecule itself and a certain number of mov-ing solvent molecules. In this way, simple ions , but alsomore complex molecules, such as aqueous [Fe(bpy) ] could be treated . In a recent experiment, Haldrup etal. have attempted to tackle this phenomenon exploit-ing combined x-ray spectroscopies and scattering tools .This picosecond time-resolved experiment used x-ray ab-sorption spectroscopy to unravel the electronic changesvisible around the Fe K absorption edge. They occurconcomitant to the geometric structural changes alreadyextracted from the extended x-ray absorption fine struc-ture (EXAFS) region . The latter monitors the molec-ular changes around the central Fe atom. While these studies only shed light on the excited molecular dynam-ics within itself, the recent study combined x-ray emissionspectroscopy (XES) with x-ray diffuse scattering (XDS)to obtain a picture of the internal electronic and struc-tural dynamics (via XES) simultaneously with the ge-ometric structural changes in the caging solvent shell.One surprising result from this experimental campaignhas yielded information about a density increase rightafter photoexcitation (i.e., within the 100 ps time resolu-tion of that study), which was fully in line with the MDsimulations of Ref. 7. They calculated a change in thesolvation shell between the low spin (LS) ground and highspin (HS) excited state, which resulted in the expulsion ofon average two water molecules from the solvation shellinto the bulk solvent. This showed up in the XDS dataas a density increase in the transient XDS pattern, andeven the quantitative analysis extracted an average den-sity increase due to about two water molecules expelledinto the bulk solvent per photoexcited [Fe(bpy) ] .This success has triggered the current theoreticalstudy: If it is becoming possible to experimentally gainnew insight into guest-host interactions in disorderedsystems like aqueous solutions, would it be possible toeventually understand the influence of guest-host inter-actions on the dynamic processes occurring within thesolute? Indeed, aqueous [Fe(bpy) ] is an ideal modelsystem for several reasons. Internally, it undergoes sev-eral ultrafast transition processes involving correlationswithin the 3 d orbitals: after photoexcitation from the A ground state into its singlet excited metal-to-ligandcharge transfer state ( MLCT), it rapidly undergoes anintersystem crossing into the triplet manifold ( MLCT)within about 30 fs , and leaves the MLCT manifold in120 fs . Femtosecond XAS studies observed the ap-pearance of the finally accessed HS T state in lessthan 250 fs , which was also confirmed by an ultra-fast optical-UV transient absorption study . A very re-cent femtosecond XES study revealed the existence ofa metal-centered intermediate electronic state on the flybefore the system settles into the HS state . This elec-tronic and spin-switching process sequence starts fromthe LS ground state which is formed by six paired elec-trons in the lower t g level. Then, the cascade proceedsto the HS excited state. There, the six electrons aredistributed via t g e g and both e g electrons with parallelspins to two of the four t g electrons. Overall, S = 2in the HS state (against S = 0 in the ground state) re-sults. Such a transition is very common in Fe-II basedspin crossover (SCO) compounds, but little is understoodabout both the internal dynamic processes involved aswell as about the possible influence of the solvent on thisrapid spin-switching scheme. Indeed, the initially excitedMLCT manifold should interact with the caging solventmolecules, but currently little is known about the actualdynamic processes. This mystery motivates the calcula-tions performed in this work.Here we investigate the energy relaxation dynamics inphotoexcited aqueous [Fe(bpy) ] theoretically in orderto provide a new view of the short-time guest-host in-teractions in this complex sequence of relaxation. Thewater molecules close to the compound are polarized anda hydration shell of bound water is formed. On the onehand, this hydration shell may shield the complex frompolarization fluctuations provided by the bulk water. Onthe other hand, it may also act as an additional sourceof polarization fluctuations and thus enhance the relax-ation process. To be specific here, we consider the case of[Fe(bpy) ] in water . The set of states which are in-volved in the cascade of transitions from the LS to the HSstate is schematically shown in Fig. 1. Also, several in-termediate vibronic states of the complex are relevant .An initial photoexcitation (green solid arrow) brings theFe-II complex from the ground state of the LS configu-ration into an excited vibronic state of a configuration ofthe metal-to-ligand-charge-transfer (MLCT) state. Thephotoexcitation at 400 nm provides an energy of about3 . ∼ − . More precisely, a state onthe MLCT manifold is initially excited, but rapidly un-dergoes an intersystem crossing into the triplet manifold( MLCT) within about 30 fs . The two manifolds aresimilar in their vibrational frequencies and correspondto the skeleton mode of bpy in the MLCT configura-tion. This mode has a rather high vibrational frequencyof Ω MLCT = 1607 cm − and its vibrational ground statehas an energy of about 18000 cm − . Hence, the photoex-citation populates mostly the vibrational state | i MLCT with a quantum number ν MLCT = 4.The relaxation out of this state can now occur via twoalternative relaxation pathways. Elements of these path-ways are known, but the path which is eventually cho-sen by the system is not fully understood in detail up to present. On the one hand, the relaxation can proceed viaenergetically lower-lying vibrational states on the MLCTmanifold, i.e., following | i MLCT → | i MLCT → . . . (theblue path, see the sequence of blue arrows in Fig. 1).In fact, the available MLCT states form a broad mani-fold of metal-centered states . From the MLCT groundstate, the energy could be transferred to a vibrationallyexcited state of one of the metal-centered triplet states( / T). In the T state, the Fe-N bond length increases,such that the Fe-II complex expands by about 0 . ofΩ T ∼
250 cm − which corresponds to a vibrational modeof the Fe-N bond. It is experimentally well-establishedthat the transfer from the MLCT manifold to the inter-mediate T states occurs in about 120 fs. The systemwould reach the vibrational ground state of the T con-figuration via a sequence of vibrational relaxation steps.From the T-vibrational ground state, the energy wouldbe transferred to a vibrational excited state of the HSconfiguration. Its HS vibrational ground state has anenergy of ∼ − . The vibrational energy gapis again determined by a vibrational mode of the Fe-Nbond and is estimated as Ω HS ∼
150 cm − . It is es-tablished that the transfer from the T to the HS stateoccurs within 70 fs . Along with this occurs anotherrearrangement of the compound which results in an ef-fective growth of the molecule (and thus somewhat thecaging cluster) of 0 . .The second possible pathway (the red path, see the se-quence of red arrows in Fig. 1) would start in a highly ex-cited vibrational state on the MLCT manifold as before.Without performing a vibrational relaxation transitionwithin the MLCT manifold, it directly yields to a highlyexcited vibrational state on the T manifold within 120fs and continues again without a vibrational relaxationtransition to another highly excited vibrational state ofthe HS configuration within 70 fs. From there, the com-plex relaxes into the HS ground state via vibrational tran-sitions and removal of the corresponding energy into thehydration shell and bulk water within 960 fs .The final HS to LS relaxation occurs in 665 ps .Both scenarios would allow the system to reach theHS electronic manifold within roughly 200 fs via severalintermediate states. The initial energy is intermittentlystored within molecular vibrations but finally transferredout of the complex into the solvent environment. The Fe-II complex expands, since the Fe-N bond lengths increase,when the compound is excited from the LS to HS state.Here, we assume that Fe-N stretching and bending modesare dominant.What is unknown from the experimental perspective, isthe vibrational life times of the intermediate vibrationalelectronic states (blue and red question marks in Fig.1). For instance, if the highly excited vibrational stateson the MLCT manifold live long enough such that thetransfer to the T-manifold can occur within 120 fs, the FIG. 1. Sketch of the energies of the LS, MLCT, T and the HSstate (not to scale). The details are given in the text. The twopossible relaxation pathways are indicated by the sequencesof the red and blue arrows. The unknown life time of thevibrational states is indicated by the blue question mark andis determined in this work. system would most likely choose the red pathway. Onthe other hand, if the highly excited vibrational stateson the MLCT manifold rapidly relax within 120 fs to theMLCT ground state, the system would prefer to followthe blue relaxation pathway.To decide this question from a theoretical point ofview, we follow a simplified model description which isaccurate enough such that a clear qualitative answer fol-lows. For this, we establish a model of a quantum me-chanical two-state system which describes a bath-inducedvibrational relaxation from an excited vibrational stateto the ground state on a generic manifold. We therebymodel the environmental polarization fluctuations in-cluding the effects of a hydration shell in terms of a re-fined Onsager model combined with a Debye relaxationpicture . A crucial aspect here is that we include thebulk solvent and the hydration shell on the same foot-ing in terms of a continuum description of environmentalGaussian modes. This model allows us easily to modifythe radius of the solvated complex (taken as a sphere inthis work) and the thickness of the surrounding hydrationshell. Within this simplified model, we determine the en-ergy relaxation rate for several representative vibrationalmodes including the Fe-N stretching and bending modesin dependence of the Fe-N bond length and the hydra-tion shell thickness. Technically, we use numerically ex-act real-time path integral simulations on the basis of afluctuational spectrum which is highly structured and farfrom being Ohmic. Such a “slow” bath reflects the similarphysical time scales on which the vibrational relaxationtransitions within a vibrational manifold and the polar-ization fluctuations of the surrounding water occur. The a (cid:1) ε C ε ( (cid:3) ) s a (cid:0) ε (cid:2) b ε ( (cid:4) ) s ε ( (cid:5) ) bw Model 1 Model 2
FIG. 2. Sketch of continuum dielectric models for thecomplex-bound-water-solvent system, see text for details. a denotes the radius of the inner sphere, while b refers to theradius of the outer sphere. ε s ( ω ) is the frequency-dependentcomplex dielectric function of the continuum bulk watermodes. ε bw ( ω ) is the frequency-dependent complex dielec-tric function associated with the bound water shell. ε c = 1 isthe dielectric constant of the vacuum inside the cavity. highly non-Ohmic form (see below) of the bath spectraldensities a priori calls for the use of an advanced theoret-ical method beyond the standard Markov-approximateddynamical Redfield equations.We find vibrational energy relaxation times on genericmanifolds in the range of 2 − II. MODEL
To determine the life time of the excited vibrationalstates, we formulate a minimal model in form of a quan-tum two-level system which is immersed in its solventenvironment (model 1) and is, in addition, surroundedby a hydration shell (model 2). After expansion of theFe-II complex, the stretching and bending modes in-volving the Fe-N bond change their respective vibrationalfrequency. We investigate their relaxation dynamics in-dependently and use the spin-boson Hamiltonian as aminimal model, i.e., H = ~ Ω2 σ z + ~ σ x X j c j ( b j + b † j ) + X j ~ ω j b † j b j . (1)Here, the Pauli matrix σ z contains the ground state | g i and the excited state | e i between which we investigate therelaxation transitions. The two states are separated inenergy by the vibrational frequency Ω. The bath modesproduce Gaussian fluctuations stemming from harmonicoscillators with frequencies ω j , the corresponding cre-ation and destruction operators of the bath modes aredenoted as b † j and b j . The fluctuations induce transi-tions in the system via the Pauli matrix σ x . They can becharacterized by a single function , the spectral density J ( ω ) = 2 π X j c j δ ( ω − ω j ) . (2)It provides the spectral weight contained in the fluctua-tions at frequency ω which are provided by a Gaussianbath at thermal equilibrium at a given fixed temperature T = 1 / ( k B β ). The correlation function of the quantumbath fluctuations ξ ( t ) is given by ( t > h ξ ( t ) ξ (0) i = 1 π Z ∞ dωJ ( ω ) (cid:20) coth ~ ωβ ωt − i sin ωt (cid:21) . (3)This quantity determines the relaxation and dephasingrates . In this work, we consider several representativeFe-N stretching and bending modes with the frequenciesΩ = 60, 120, 150, and 250 cm − . Moreover, we use a con-tinuum description of the solvent (bulk) water and thehydration shell following Gilmore and McKenzie . Thekey quantity to characterize the environment, i.e., thespectral distribution J ( ω ) of the fluctuations, is deter-mined in terms of the standard Onsager model of polar-ization fluctuations of the solvent water molecules. Theirrelaxation properties are described within a Debye relax-ation picture . In this approach, the spectral density isrelated to the continuum dielectric function ε ( ω ) of thehost material.To be more specific, we consider two differentsituations , see Fig. 2: In model 1, we assume thatthe complex with its vibrational mode is placed insidea vacuum spherical cavity of radius r a with a dielectricconstant ε c = 1. This is situated in a continuum of bulkwater modes with a frequency-dependent complex dielec-tric function ε s ( ω ). In model 2, we add to model 1 anouter sphere with radius r b > r a . The shell formed bythe two spheres describes the bound water or hydrationshell in terms of a second frequency-dependent complexdielectric function ε bw ( ω ). This model allows us to deter-mine the relaxation rates also for varying the radii r a and r b independently. Throughout this work, we set T = 300K. A. Model 1: Bulk water
Following Gilmore and McKenzie , one can calcu-late the reaction field by solving Maxwell’s equation forthe particular geometry shown in Fig. 2. This yields thespectral density J ( ω ) = (∆ µ ) πε r a Im ε s ( ω ) − ε s ( ω ) + 1= (∆ µ ) πε r a ε s , − ε s , ∞ )(2 ε s , + 1)(2 ε s , ∞ + 1) ωτ s ω τ + 1 , (4)with the respective transition dipole moment ∆ µ of thevibration, ε s , being the static dielectric constant of thebulk solvent, ε s , ∞ being the high-frequency dielectricconstant of the bulk solvent, and τ s = 2 ε s , ∞ + 12 ε s , + 1 τ D , s (5)and τ D , s is the Debye relaxation time of the solvent. Forwater, we have ε s , = 78 . , ε s , ∞ = 4 . τ D , s = 8 . a and we thus collect all constants in a prefactor.We arrive at J ( ω ) = α a ωω τ + 1 , (6)with α = 12 π ~ (∆ µ ) πε a ε s , − ε s , ∞ )(2 ε s , + 1)(2 ε s , ∞ + 1) τ s , (7)where a is the typical length scale of the problem andwhere the now dimensionless radius a = r a /a is mea-sured in units of a . We fix this to a = 1 ˚A throughoutthis work. The spectrum is purely Ohmic with a cut-offfrequency given by ω c , s = 1 /τ s . For our considerations,we fix the dipole moment to a typical value of ∆ µ = 1 D= 3 × − Cm. Collecting all parameters yields α ≈ J ( ω ) of model 1 for thecase a = 6 (corresponding to r a = 6 ˚A). Maximal spec-tral weight is observed at roughly 70 cm − . Hence, it isclear that the resulting bath correlation times are com-parable to or exceed internal system periods. This alsoprevents us from using a standard Markov approximationa priori, since a correlated and non-Markovian dynamicscan in principle be expected (see below). B. Model 2: Bulk water plus hydration shell
We also include the hydration shell of bound water anddo this by a second sphere with outer radius r b = ba with b being the corresponding dimensionless number.We assume that the hydration shell is thin relative tothe radius of the inner sphere and may then perform a ω [cm -1 ] J ( ω ) [ c m - ] a = 6a=6, (b-a)/a = 0.15 FIG. 3. Spectral densities for model 1 (black line, circles) andmodel 2 (red line, squares) for water and for a cavity radiusof r a = 6 ˚A and a relative shell thickness of ( b − a ) /a = 0 . Taylor expansion in the relative shell thickness ( b − a ) /a .The resulting spectral density is J ( ω ) = J ( ω ) + J bw ( ω ) , (8)with J bw ( ω ) = (∆ µ ) πε r a ε s ( ω ) + 1) (cid:18) ε s , | ε bw ( ω ) | (cid:19) Im ε bw ( ω ) , (9)where ε bw ( ω ) is the complex dielectric function of thebound water layer. Within the Debye relaxation model,we find J bw ( ω ) = α bw a b − aa ωω τ + 1 (10)with α bw = 12 π ~ µ ) πε a ( ε , + ε , )( ε bw , − ε bw , ∞ ) ε , (2 ε s , + 1) τ bw . (11)Here, we have the static dielectric constant ε bw , andthe high-frequency dielectric constant ε bw , ∞ of the boundwater layer. From generic considerations , one may in-fer that the relaxation time of the bound water shell isone order of magnitude large than the solvent relaxationtime, i.e., we set τ bw = 10 τ s . Likewise, we know that ε s , ≫ ε s , ∞ . Moreover, ε bw , ≫ ε bw , ∞ and ε s , ≫ ε bw , .Hence, we may use this and set ε bw , = 1 to obtain α bw = 12 π ~ µ ) πε a τ bw . (12)For the parameters mentioned, we find α bw ≈ J ( ω ) for these parameters and for a = 6and ( b − a ) /a = 0 .
15. Again, maximal spectral weightis observed at roughly 70 cm − . In general, the spectral time [ps] popu l a ti on d i ff e r e n ce (b-a)/a FIG. 4. Time evolution of the population difference P ( t ) forΩ = 150 cm − for T = 300 K for different values ( b − a ) /a ofthe shell thickness. weight of model 2 is higher than of model 1. This alreadyindicates that within the continuum approach, the boundwater shell acts as an additional source of fluctuationsand not as a spectral filter for the continuous bulk modes.Hence, the calculated relaxation times for model 2 willbe faster than for model 1.Moreover, it is clear that the vibrational life times onthe MLCT manifold are much larger since there the spec-tral weight of the solvent environmental modes aroundthe frequency of Ω MLCT = 1607 cm − is strongly sup-pressed (in fact, we do not consider the vibrational re-laxation around this frequency in this work). III. REAL-TIME DYNAMICS OF THE RELAXATIONTRANSITIONS
To investigate the quantum relaxation dynamics of thetwo vibrational states under the influence of environmen-tal fluctuations, we employ the numerically exact quasi-adiabatic propagator path-integral (QUAPI) schemewhich we have extended to allow treatment of multiplebaths . Specifically, QUAPI is able to treat highly struc-tured and non-Markovian baths efficiently . It deter-mines the time dependent statistical operator ρ ( t ) whichis obtained after the harmonic bath modes have been in-tegrated over. We briefly summarize here the main ideasof this well-established method and refer to the litera-ture for further details. The algorithm is based on asymmetric Trotter splitting of the short-time propagator K ( t k +1 , t k ) for the full Hamiltonian Eq. (1) into a partdepending on the system Hamiltonian alone and a partinvolving the bath and the coupling term. The short-time propagator gives the time evolution over a Trottertime slice δt . This splitting in discrete time steps is exactin the limit δt →
0, i.e., when the discrete time evolu-tion approaches the limit of a continuous evolution. For
5 5.5 6 6.5 7a 0 0.1 0.2 0.3 ( b - a ) / a FIG. 5. Relaxation time (color scale in ps) of the excitedvibrational state for varying radius r a = aa with a = 1 ˚Aand for varying relative shell thickness ( b − a ) /a for Ω =150cm − . Model 1 with no hydration shell is contained via thecut along the line ( b − a ) /a = 0. any finite time slicing, it introduces a finite Trotter errorwhich has to be eliminated by choosing δt small enoughsuch that convergence is achieved. On the other side, theenvironmental degrees of freedom generate correlationsbeing non-local in time. We want to avoid any Marko-vian approximation at this point and take these correla-tions into account on an exact footing. We may, how-ever, use the fact that for any finite temperature, thesecorrelations decay exponentially quickly on a time scaledenoted as the memory time scale. The QUAPI schemenow defines an object called the reduced density tensor.It corresponds to an extended quantum statistical oper-ator of the system which is nonlocal in time since it liveson this memory time window. By this, one can establishan iteration scheme by disentangling the dynamics in or-der to extract the time evolution of this object. All cor-relations are fully included over the finite memory time τ mem = Kδt , but are neglected for times beyond τ mem .To obtain numerically exact results, we have to increaseaccordingly the memory parameter K until convergenceis found. The two strategies to achieve convergence, i.e.,minimize δt but maximize τ mem = Kδt , are naturallycounter-current, but nevertheless convergent results canbe obtained in a wide range of parameters, including thecases presented in this work.
IV. RESULTS
At first, we consider modes with a vibrational fre-quency of Ω =150 cm − . We determine the difference P ( t ) = h σ z i t = tr[ ρ ( t ) σ z ] of the populations of theground and the excited states. We start out from theinitial preparation of the excited state, i.e., ρ (0) = | e ih e | .Fig. 4 shows examples of the relaxation dynamics for theenvironmental models 1 and 2 for different values of the a τ r e l [ p s ] (b-a)/a FIG. 6. Solid lines: Cut through the 2D plot of Fig. 5 alongthe lines ( b − a ) /a = 0 , . , . .
3. The dashed linesindicate the results of the vibrational life times calculatedwithin a Born-Markov approximation (see text). shell thickness ( b − a ) /a . We mainly observe exponentialrelaxation on a time scale of a few picoseconds. For anincreasing shell thickness, a tendency towards a decayingoscillatory dynamics appears. A pronounced oscillationwith a period of ∼
250 fs develops for the largest thick-ness considered.To quantify the decay in terms of life times of the ex-cited state, we extract from the time evolution the corre-sponding rate by a fit to an exponential. Fig. 5 shows therelaxation time in ps (colour scale) as a function of theradius a of the complex varying it between 5 to 7 ˚A andthe relative shell thickness ( b − a ) /a varying it between 0to 30%, which is consistent with the numerical findingsof Ref. 7. The plot shows results of both, models 1 and2 (model 1 corresponds to the line with ( b − a ) /a = 0).The data for ( b − a ) /a = 0 , . , . . /r a . This reflects theassumption that the effective transition dipole sits in thecenter of the sphere and an increasing complex pushesthe solvent fluctuations further away. This reduces theirstrength due to the distance dependence of the dipolarcoupling. Moreover, the life times decrease with increas-ing hydration shell thickness. Thus, the hydration shelldoes not act as a shield from bulk solvent fluctuationsbut acts as an additional source of fluctuations instead.Fig. 6 also shows the results of the vibrational life timescalculated within a Born-Markov approximation . Theinverse life time or the relaxation rate can be obtainedafter expanding the transition rates in a master equationapproach up to lowest order in the system-bath inter-action, together with a Markovian approximation of thebath-induced correlations. This corresponds effectively FIG. 7. Vibrational life times (color scale in ps) for Ω = 60cm − (a), 120 cm − (b), 150 cm − (c) and 250 cm − (d) formodels 1 and 2 for T = 300 K. to only including single-phonon transitions in the bath.The inverse vibrational life time then follows as τ − = J (Ω) coth ~ Ω β . (13)As is shown by the dashed lines in Fig. 6, significantdeviations from the exact life times occur and the ap-proximated life times are overestimated by up to 10%.Next, we show the results for the calculated lifetimes for other vibrational frequencies, i.e., for Ω =60 , , − in Fig. 7. These valuesspan the regime of the vibrational frequencies for theFe-N stretching and bending modes in the LS and HSstate . Note that the frequencies are comparable to orlarger than the frequency for which the maximal spec-tral weight in the environmental fluctuation spectrumoccurs. Hence, the energy relaxation dynamics occurs inthe regime in which non-Markovian multi-phonon tran-sitions already are noticeable . We note that for largervalues of Ω, no convergent results have been achieved,which is a further strong indication of non-Markovianbehavior. V. CONCLUSIONS
We observe that under the assumption of equalstrengths of the coupling to the environmental fluctu-ations, all Fe-N stretching and bending modes in the LSand HS state exhibit quite similar vibrational life timeson the order of 5 ps. The vibrational energy gap has beenmodified from 60 to 250 cm − and all cases show similarresults. An increased radius of the complex results in alarger life time since the fluctuating solvent molecules aremoved further outside. A finite hydration shell thicknessreduces the vibrational life times noticeably.Our results indicate that all vibrational modes con-tribute similarly to the energy relaxation after initial pho- toexcitation. At the same time, all vibrational modes livetoo long in order to relax the energy already in the MLCTor the T state (assuming here the vibrational modes be-ing identical to the modes in the LS state). Hence, theenergy after the photoexcitation is first rapidly transferedfrom a highly excited vibrational MLCT state to a highlyexcited vibrational T state and then further to a highlyexcited vibrational HS state within about less than 200fs. Only then, the full excess energy is dissipated whilethe electronic subsystem is in the HS state. Hence, thesystem follows the “red relaxation pathway” sketched inFig. 1.Energy redistribution within more molecular vibra-tional states is not included in our simplified model.Assuming the excess energy initially equally distributedamong the Fe-N stretching and bending modes , eachmode gets roughly an excitation energy of 440 cm − .This implies that roughly two excitations of the modeΩ = 250 cm − and up to three or four excitations ofthe mode with Ω = 120 cm − and Ω = 150 cm − oc-cur. Thus, the total equilibration time of the complexafter photoexcitation roughly follows as three times 5 pswhich yields a value of 15 ps. These results could beexperimentally verified by ultrafast spectroscopy of theintermediate MLCT and T states. VI. ACKNOWLEDGMENTS
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