Biomolecular electron transfer under high hydrostatic pressure
Mart Tars a, Aleksandr Ellervee a, Michael R. Wasielewski b,c, Arvi Freiberg a,*a Institute of Physics, Riia 142, EE2400 Tartu, Estonia
b Chemistry Di6ision, Argonne National Laboratory, Argonne, IL 60439, USAc Department of Chemistry, Northwestern Uni6ersity, E6anston, IL 60208, USA
Abstract
The dependence of the photoinduced electron transfer rate on hydrostatic pressure up to 8 kbar was studied at 295K in a bridged Zn-porphyrin donor and pyromellitimide acceptor supermolecule dissolved in toluene. A picosecondfluorescence emission kinetics of the donor, limited by the electron transfer rate, was detected by using synchroscanstreak camera. The experiment was complemented with model calculations based on modified classical andsemi-classical nonadiabatic electron transfer theory. A peculiar asymmetric inverted parabola-like dependence of theelectron transfer rate on pressure was observed. The dependence was successfully reproduced by nonadiabatic theoryin the high-temperature limit assuming that the reorganisation free energy or both the reorganisation free energy andthe reaction driving force (linearly) changed with pressure. The reaction driving force dependence on pressure alonefailed to explain the asymmetry, suggesting that the electron transfer was accompanied with vibration frequencychanges. It was inferred that the effective frequency in the product state should be larger than in the reactant state.The usage of the nonadiabatic theory is well justified due to the fulfilment of the inequalities V�kBT and V��tL�−1
Keywords: High pressure; Electron transfer; Porphyrins; Donor-acceptor complexes; Picosecond time-resolved fluores-cence
1. Introduction
Electron transfer (ET) reactions play a majorrole in chemistry and biology, particularly in pho-tosynthesis. Both for cognitive reasons and fordevelopment of practical molecular electronicsdevices based on these processes, the underlyingphysical mechanisms should be understood in de-
tail. To gain proper understanding of the ETreactions, a wide range of equilibrium and dy-namical effects need to be characterised.
There are two commonly used experimentalapproaches to study various aspects of ET reac-tions in solutions: by changing the chemical com-position of the solvent and by varying thetemperature. Although important results havebeen achieved this way, the quantitative under-standing of the ET kinetics and dynamics is stillrather limited. Moreover, the above methods are
M. Tars et al. / Spectrochimica Acta Part A 54 (1998) 1177–11891178
subject to serious inherent limitations. Tempera-ture causes simultaneous changes in volume andthermal energy that are difficult to separate.Changes in the solvent chemical composition arediscontinuous and also alter many physical prop-erties (density, viscosity, dielectric susceptibility)of the bulk solvent at the same time. This moti-vates the search for complementary experimentalapproaches.
Pressure as a universal thermodynamic variableoffers an attractive alternative. Continuous andcontrolled tuning of the properties of molecularmatter can be achieved over a wide range withoutchanging the structure of matter and chemicalnature of molecules. Unfortunately, as with tem-perature, the response of matter to increasingpressure is complex. In a broad sense, pressureeffects can be divided into equilibrium or staticeffects and dynamic effects. The most importantstatic effect of pressure is an alteration of the freeenergy surfaces, thus changing the reaction driv-ing force, reorganisation free energy, and barriersbetween the reactant and product states. Dynamiceffects are due to modification of the system,comprising of the reactant, product, and the sur-rounding solvent, relaxation properties underpressure.
The field is still relatively new. Earlier ET stud-ies under high hydrostatic pressure of which weare aware of have considered photosynthetic reac-tion centre proteins [1–5] and some other donor-acceptor compounds [6–9]. Both increase anddecrease of the ET rate have been observed indifferent samples. However, the underlying reac-tion mechanisms remain mostly unexplored.
The aim of the present work is 2-fold. Firstly,we present our results that show that externalpressure may be effectively used to study andcontrol intramolecular ET kinetics. Secondly, wedevelop a conceptual framework to understandthe effect of pressure on ET processes in molecu-lar systems. A preliminary report of this work hasbeen presented in the XIIth International Bio-physics Congress (11–16 Aug. 1996, Amsterdam)[10,11].
The donor-acceptor complex under study con-sists of the zinc meso-tripentylmonophenylpor-phyrin (ZnP) electron donor and the
pyromellitimide (PM) electron acceptor in a solu-tion at room temperature. The donor molecule iscovalently bridged via a phenyl side-ring to theacceptor molecule (see the structure in Fig. 1).The so-formed supermolecule has a center-to-cen-ter donor-acceptor distance of about 1.5 nm andedge-to-edge distance is about 0.5 nm. The por-phyrin-based donor–acceptor complex was cho-sen because of the central role of the porphyrinring system in photosynthesis, oxygen transport,and in biological oxidation and reduction pro-cesses. As it will be shown below (Section 4), theemission spectrum of the ZnP–PM complex in the600–700 nm region looks like a typical Q-typefluorescence emission spectrum of metallopor-phyrins [12–15]. This implies only a weak pertur-bation of the ZnP donor valence electrons by thepresence of the PM acceptor. At the same time,the ZnP–PM emission intensity is stronglyquenched and its lifetime considerably shortenedas compared to the closed shell metal (Zn, Mg)metalloporphyrins. For the latter, fluorescenceemission decay in the nanosecond time range hasbeen recorded in various solvents [12,15].Nanosecond lifetimes have also been observedunder kilobar pressures [13,14]. In contrast, weobserve only about 70 ps excited state lifetime atnormal pressure. This lifetime agrees well with themeasured time constant for formation of theZnP+ –PM− charge transfer state by transientabsorption (S. Millen and M.R. Wasielewski, un-published). Under these circumstances one canwrite
Fig. 1. Structure of the ZnP–PM supermolecule.
M. Tars et al. / Spectrochimica Acta Part A 54 (1998) 1177–1189 1179
tobs= (kR+kNR+kET)−1:kET−1 (1)
where tobs is the observed fluorescence emissionlifetime and kR, kNR, and kET are, respectively, theradiative, nonradiative intersystem crossing andinternal conversion, and nonradiative ET deacti-vation rates of the S1 excited state. Eq. (1) allowsa straightforward experimental determination ofthe ET rate and its pressure dependence by mea-suring fluorescence emission kinetics with picosec-ond time resolution at various pressures.
The rest of the paper is organised as follows. InSection 2, the theoretical framework we shall dis-cuss the experimental high pressure ET data ispresented. In Section 3, the experimental ap-proach used is described. Section 4 presents theresults of the spectral and kinetic measurementsunder pressure. Section 5 deals with the interpre-tation of the experimental data. Various ET mod-els will be considered and compared toexperimental results. Section 6 contains conclud-ing remarks.
2. Theoretical background
By analogy with chemical reactions, the ther-mally activated ET rate constant at the high tem-perature limit (kBT\hn, for all relevantvibrational frequencies n) can be expressed by theArrhenius law [16]:
kET= �V �2(4p3/h2ErkBT)1/2 exp(−G+/kBT) (2)
where G+, the free energy of activation, is givenby
G+ = (Er+Eg)2/4Er (3)
Where h is the Planck’s constant; kB is the Boltz-mann’s constant; T is the temperature; Eg is thefree energy gap, i.e. the energy difference betweenthe minima of the free energy parabolas for theproducts and reactants (the driving force); Er isthe reorganisation free energy defined as shown inschematic Fig. 2; and V is the electronic couplingmatrix element between the donor and acceptorstates.
Eq. (2) represents the physical situation in thenonadiabatic weak coupling limit, which is true in
Fig. 2. Single configuration co-ordinate model to explain theintramolecular electron transfer.
many cases. In this limit, the ET (and not thenuclear relaxation rate) is the rate-limiting stepproceeding slowly at the crossing point of thereactant and product potential energy surfaces.The ET rate explicitly depends only upon solventenergetics and not upon solvent dynamics.
Another extreme is the adiabatic process beingalmost independent of the electronic couplingstrength. In this view, the electronic coupling re-quired for ET is strong enough, so that whatlimits the reaction rate is the time required toachieve the proper nuclear configuration for anisoenergetic transfer to occur. In this limit, theaverage solvation time �tL� determines the over-all ET rate, kET8�tL�−1.
According to [17–19], the distinction betweenadiabatic and nonadiabatic reactions can be madeby comparing the electronic coupling energy Veither with the thermal energy, kBT, or with theaverage solvation rate, �tL�−1. Note that �tL�−1
is a measure of the energy uncertainty of thereaction coordinate. The reaction is adiabatic, ifV�kBT or V\�tL�−1 and nonadiabatic, ifV�kBT or VB�tL�−1.
To the best of our knowledge, existing ETtheories do not consider pressure effects. Yet, a
M. Tars et al. / Spectrochimica Acta Part A 54 (1998) 1177–11891180
general discussion in [20] based on a theory ofoptical transitions under pressure [21–25] shouldbe mentioned.
The effects of hydrostatic pressure can be de-scribed by adding the pressure-volume work term,px, to the reactant and product free energy sur-faces. (By analogy with [24], a single configurationcoordinate model is treated. The configurationcoordinate x can be related to the most significantmolecular vibration mode or a linear combinationof the modes.) What this means is that the partic-ular molecular state i has a characteristic volumeVi and a pressure-dependent total energy, Ei=Ei(0)+pVi. The pVi term, different for differentstates, represents the work that the state has toperform against (isotropic) environment upon ex-pansion from zero volume [26]. In our simplifiedmodel the characteristic volume of the states isconsidered alike and the corresponding potentialsUR(x) and UP(x) (see Fig. 2) are taken harmonic.Under these conditions, we have:
UR(x)= (1/2)kRx2+px (4)
UP(x)= (1/2)kP(x−x0)2+Eg+px (5)
where x0 is the displacement of the free energyminima along the configuration coordinate uponelectronic transition, kR and kP are the forceconstants that characterise the steepness of thefree energy parabolas, respectively. The force con-stants kR and kP may be different. Note that Eg,the free energy gap in Eq. (5), is negative. It isdefined as the energy difference, UP(x0)−UR(0)B0, between the minima of parabolas for the final(product) and initial (reactant) states.
The following expression for the pressure de-pendence of the free energy gap results from theconfiguration coordinate diagram, Fig. 2:
Eg(p)=Eg(0)+x0p+1/2(kR−1−kP
−1)p2 (6)
where Eg(0) is the zero-pressure value. Eq. (6)contains both linear and quadratic pressure cou-pling terms. The former part is due to horizontaldisplacement of the potential energy minima atET. The latter term becomes imperative when theET is accompanied by essential change of thevibrational frequencies. The quadratic couplingterm is absent, if the force constants remain unal-tered at ET.
From Fig. 2, it is also possible to get theexpression for the pressure change of x0:
x0(p)=x0(0)+ (kR−1−kP
−1)p (7)
where x0(0) is the zero-pressure displacement. Theeffective nuclear displacement following ET isseen to be pressure-sensitive only if the reactantand product potentials are characterised by differ-ent force constants. The force constants them-selves, and thus vibrational frequencies, areconsidered independent of pressure, similar toRef. [24].
The reorganisation free energy that character-ises the coupling strength between the electronicand nuclear degrees of freedom at ET, Er (see Fig.2), is defined via x0 and kP as
ER=x02kP/2 (8)
Er=0, if x0=0. From Eqs. (7) and (8) one gets
Er(p)=kP/2[x02(0)+2x0(0) (kR
−1−kP−1)p
+ (kR−1−kP
−1)2p2] (9)
Note that as long as kR"kP, Er(p)"0 at p\0,even if x0(0)=0.
3. Experimental
3.1. Sample handling
The ZnP–PM molecule was synthesised accord-ing to the published procedures [26]. The sub-stance was dissolved in toluene (2×10−5 M)without further purification. Toluene was chosenas a solvent in order to minimise the slow viscos-ity-dependent solvation that limits the ET rate inmore polar solvents (for a recent review, see [27]).
3.2. Pressure equipment
The solution under study was contained in aspecial sample cell having two windows about 2mm apart with one window fixed and anothertranslationally movable like a piston to adapt thepressure variations. The construction of the sam-ple cell prevented direct contact of the samplewith the pressure transmitting medium. The sam-
M. Tars et al. / Spectrochimica Acta Part A 54 (1998) 1177–1189 1181
ple cell was mounted in a simple home-madepiston-cylinder type high-pressure optical cell.Three 10 mm thick sapphire windows of 12 mmdiameter cover the light port of 4.5 mm apertureand 30° cone angle in window plugs. A portablehydraulic press was exploited to generate pressureup to about 10 kbar inside the cell by pushingdown the piston. The pressure was measured byusing a manganin gauge. In this work, pressuresup to 8 kbar were applied. As a pressure transmit-ter various liquids may be utilised. Here, a mix-ture of glycerol with water was used. Allmeasurements have been performed at room tem-perature (295 K).
Enough time was left between the pressurechange and the measurement to reach the equi-librium temperature and pressure. Reversibility ofthe pressure-induced effects at a fixed temperaturewas checked by comparison of the data recordedalong two experimental paths: in the course ofcompression and decompression.
3.3. Spectral and kinetic measurements
The excitation source was the Coherent 700styryl-9M mode-locked dye laser synchronouslypumped at 76 MHz by the Coherent Antares 76SNd: YAG laser. The output of the dye laser wasfrequency-doubled by using the LiIO3 crystal. Thefluorescence kinetics was recorded through a sub-tractive-dispersion double grating monochroma-tor (LOMO, Russian Federation) with 2 nmspectral bandwidth by using the HamamatsuC1587 Temporal Photometer in synchroscanmode. The streak camera was linked via a vidiconto the OSA 500 optical multichannel analyser(B&M Spektronik, Germany). The temporal re-sponse function of the whole instrument wasabout 15 ps (FWHM). A polariser at the magicangle (54.7°) to exclude the rotational diffusioneffects on the fluorescence decay kinetics at roomtemperature was routinely used. The steady-statefluorescence emission spectra were recorded usinga 1 nm bandwidth double grating monochroma-tor. Care was taken to avoid sample degradationor multiple excitation effects due to excess lightintensity. No degradation was observed at excita-tion intensities 520 mW cm−2. Absorption spec-
tra were measured by a Beckman Acta MVIIspectrophotometer (applying 1 nm bandwidth)equipped with a special holder for placing thehigh-pressure cell. Absorption spectra were runbefore and after each set of kinetics experimentsto verify that the sample was not degraded duringthe measurement.
3.4. Data handling
The data were transferred to IBM Pentium PCand handled by utilising the Spectra Solve soft-ware (LasTek, Australia). The measured spectrawere corrected for the spectral sensitivity of theset-up. The peak energies and bandwidths of thesteady-state spectra were obtained by fitting to theGaussian lineshape. Global as well as single-curveanalysis of the fluorescence emission kinetics wasperformed. Lifetimes were calculated using aleast-squares fitting algorithm assuming multiex-ponential kinetics
I(t)=%Ai exp(− t/ti) (10)
and taking into account the finite instrument re-sponse function. In Eq. (10), the sum is over thenumber of kinetics components, with the initialamplitude Ai, used in the fitting procedure.
The fluorescence emission spectra of the ZnP–PM complex measured at 0.3 and 3.2 kbar isshown in Fig. 3a. Excitation (excitation band-width about 10 nm) was into the Soret absorptionband at about 424 nm (here and below, if notindicated otherwise, the specific numbers aregiven at ambient pressure, 1 atm). Excitationwavelength was adjusted when pressure waschanged in order to always hit the peak of theabsorption band. The spectrum in Fig. 3(a) con-sists of two bands: the Qx(0, 0) origin band, whichpeaks at 606.2 nm and its Qx(0, 1) vibronic satel-lite, at 656.1 nm. As it was mentioned in Section
M. Tars et al. / Spectrochimica Acta Part A 54 (1998) 1177–11891182
Fig. 3. (a) Room-temperature emission spectra of the ZnP–PM complex at 0.3 and 3.2 kbar. The spectra are normalised to the samemaximal intensity. (b) Emission band positions (dots) as a function of pressure and corresponding linear fitting functions (solidlines): n [Qx(0, 0)]=1649696 to 2493 p ; n [Qx(0, 1)]=1524295 to 1792 p. Parameters of fitting functions are represented incm−1 for the band position and in kbar units for the pressure.
1, the shape of the ZnP–PM fluorescence emis-sion spectrum looks like a typical metallopor-phyrin spectrum, as if only the ZnP molecule inthe ZnP–PM complex was locally excited. Like inabsorption spectrum (data not shown), the bandsin the fluorescence emission spectrum suffer alinear red-shift when pressure is raised (Fig. 3 (b)).The common nature of the (0,0 ) band in theemission and absorption spectra is validated bythe same pressure-induced shift at the approxi-mate rate of −2493 cm−1 kbar−1. The rate ofthe shift is similar for both emission bands, imply-ing a relatively weak dependence of the in-tramolecular vibrational frequencies on pressure.This is in agreement with the data obtained onother metalloporphyrins [13,14], There is alsosome intensity redistribution between the vibronicbands preserving the general shape of the emis-sion spectrum. The decrease in intensity of the
Qx(0, 1) band with respect to the Qx(0, 0) bandprobably provides an evidence of a decreas-ing intramolecular vibronic coupling with com-pression.
4.2. Pressure dependence of the fluorescenceemission kinetics
The pressure-dependent fluorescence emissionkinetics recorded at the maximum of the Qx(0, 0)band are demonstrated in Fig. 4. Apart from asmall (1–2%) nondecaying background, the ob-served kinetics at all pressures can be fitted to twodecaying exponentials with the major fast and theminor slow components. At 1 atm, the followingdecay constants were observed: t1=7393 ps andt2=15309120 ps with A1/A2:5.5 (910%) (seeEq. (12)). The kinetics and amplitude ratio de-tected at five different wavelengths over the emis-
M. Tars et al. / Spectrochimica Acta Part A 54 (1998) 1177–1189 1183
sion band (at 605, 630, 657, 670, and 685 nm) didnot reveal any essential wavelength dependence ofthese parameters.
The pressure dependence of t1 and t2 is ratherunusual. The t1 value, equal to about 73 ps at 1atm, shortens down to the minimum value ofabout 47 ps at 2 kbar and starts to increasegradually at still higher pressures (Fig. 5b). Yet,this increase is much slower than the initial de-crease and the normal-pressure value of t1 isrecovered only at about 8 kbar. The long lifetime(t2), equal to about 1530 ps at 1 atm, also turnsinitially shorter under compression, but this short-ening stops at about 2 kbar by attaining a valueof 7009150 ps (Fig. 5a). At higher pressure this
time does not almost change any more. The am-plitude ratio A1/A2 increases from about 5.5 at 1atm to 25–30 at 3 kbar and then gradually de-creases to about 7–8 at 8 kbar. It should beemphasised that all described pressure effects arefully reversible in the studied 1 atm–8 kbar pres-sure range.
5. Discussion
5.1. E6aluation of the experimental data
The most impressive experimental result of thisstudy which needs an explanation is the observa-tion of the inverted parabola-like dependence oft1
−1 on applied pressure (Fig. 5). However, beforeto do that, let us first consider a possible origin ofthe experimentally observed two lifetime compo-nents, t1 and t2.
The simplest assumption is that the system isheterogeneous and the two components belong totwo different species. The fact that the ratio A1/A2
is basically spectrally independent implies that thespectral distribution of the A1 and A2 componentsis similar. The latter strongly indicates that alsothe A2 component has to be related to ZnP. Forexample, it may be due to ZnP–PM complexeswith inhibited ET function, traces of free ZnPmolecules or its derivatives. The fluorescenceemission kinetics of ZnP and its pressure depen-dence has not been measured yet. The availabledata about related Zn tetraphenylporphyrinmolecules [11,12] do not contradict the given in-terpretation. There is also an alternative explana-tion to the slow emission component possible asdue to recombination luminescence. This assumeselectron back transition from the state with sepa-rated charges by restoring ZnP–PM excited sin-glet state from which delayed emission takesplace. We have dismissed this version on thegrounds of large energy barrier for the back trans-fer. According to [26], the free energy gap betweenthe actual states at normal pressure is about 1600cm−1, much too large for any substantial ther-mally activated back ET to occur at roomtemperature.
Fig. 4. Normalised fluorescence emission kinetics of the ZnP–PM complex recorded at the Qx(0, 0) emission band maximumunder various pressures at room temperature. The curves arenormalised to the same maximal intensity. The instrumentresponse function is shown by the thin solid line.
M. Tars et al. / Spectrochimica Acta Part A 54 (1998) 1177–11891184
Fig. 5. Emission decay constants as a function of pressure. Solid lines are drawn as a guide for the eye. (a) Slow component, t2. (b)Fast component, t1.
If at this point, the source of the long lifetimecomponent is still open, the fast component canquite safely be attributed to the ET in properlyfunctioning ZnP–PM complex. As we have al-ready explained above, in this complex, the in-tramolecular ET is the major process that limitsthe lifetime of the ZnP donor S1 photoexcitedstate.
In the following discussion, we shall assumethat t2
−1 is a measure of the nonradiative decayrate of the ZnP–PM excited state when thedonor-acceptor ET is blocked (t2
−1=kNR). Underthe conditions that kR�kNR, kET, and t1
−1=tobs−1
(see Eq. (1)), the ET rate constant kET can then beevaluated as
kET(p)=t1−1(p)−t2
−1(p) (11)
If t2 is due to some impurity, then a less preciseequation
kET(p):t1−1(p) (12)
should be used, because of unknown value of kNR.In Fig. 6, the filled data points are calculated
according to Eq. (11) and the open ones, accord-ing to Eq. (12). For the sake of completeness,model calculations have been performed for bothvariants with qualitatively very similar results.
Therefore, in the following, only the fitting resultsbased on Eq. (11) will be presented.
5.2. Model calculations
In Section 5.2.1and Section 5.2.2, we shallmodel the pressure dependence of the ET ratebased on the weak coupling approximation, Eq.(2). In section Section 5.2.3, a semi-classicalmodel is introduced in which the solvent is treatedclassically, but high-frequency intramolecular vi-brations of the donor and acceptor are quantified.As the simplest assumption, the model parameters(free energy gap, reorganisation energy, anddonor-acceptor separation) are supposed to belinear functions of pressure. A common [28,29]expression for the distance dependence of the ETrate, assuming an exponential decrease of theelectronic coupling matrix element with thedonor-acceptor separation R, is used:
V=V0 exp[−b(R−R0)/2] (13)
Here R0 (:0.5 nm) is the value of R at p=1 atmand V0 is the electronic coupling matrix elementat 1 atm. b is the coefficient describing the contri-bution made by intervening medium in propagat-ing electronic wavefunctions. For covalently
M. Tars et al. / Spectrochimica Acta Part A 54 (1998) 1177–1189 1185
linked donor-acceptor compounds, b:7 nm−1
[28,29].
5.2.1. Reaction dri6ing force is pressure-dependentLet us first consider the simplest model where
both Eg and R depend linearly on pressure and Er
is a constant. Modifying Eqs. (2) and (13), we get:
kET(p)
=' 4p3
h2ErkBT�V0�2 exp(−bap)
exp�− (Eg(0)+xp+Er)2
4ErkBT�
. (14)
Here x=DEg/p is the rate of the free energy gapchange with pressure and a= (R−R0)/p (for sim-plicity, the normal pressure value is taken zero, sothat Dp:p). For R0:0.5 nm, a:−0.005 nmkbar−1, if the linear compressibility is 0.01 kbar−1,a value rather typical for the condensed molecularmatter [30,31].
Figure 6 and Table 1 show the result of theleast-squares fit with a as a fixed parameter. As
Table 1The fitting curve parameters in Fig. 6 based on Eq. (14)
seen, the modified Marcus formula, Eq. (14), thattakes into account only linear pressure depen-dence of the free energy gap, successfully repro-duces the inverted parabola-like pressuredependence of the ET rate constant. Fig. 5 alsoshows that pressure dependence of the electroniccoupling matrix element due to the pressure varia-tion of R in Eq. (13) has only a minor influenceon the overall ET rate. It is important to note thatthe least-squares fitting parameters Eg(0) andEr(0) in the first and second column of Table 1look acceptable. From [26], it is known thatEg(0):−1600 cm−1 and Er(0)5800 cm−1. Thex value exceeds several times the red-shift rate ofthe optical spectra. However, this does not seemunreasonable, considering an increased polarity ofthe charge separated state. The positive sign of x
means that −Eg is getting smaller along pressureincrease. If −a is taken nonphysically large (lastcolumn of Table 1), the other parameters becomeunreasonable too.
The fulfilment of the criterion V�kBT (V0:5cm−1 from Table 1, V:6 cm−1, as calculatedfrom Eq. (13) for a:−0.005 nm kbar−1 andp=8 kbar, whereas kBT:205 cm−1 at 295 K)suggests that the studied reaction can indeed beconsidered as nonadiabatic.
To double check we need to know the relax-ation rate of the solvent polarisation fluctuations.Although the experimental data for toluene arenot available, an estimate can be made based onFig. 4 of Ref. [19]. It follows that toluene is a fastrelaxing solvent with �tL�−1:3×1012 s−1 cor-responding to the energy uncertainty of 100 cm−1.Since the dielectric constant of toluene is knownto change by less than 20% over the
Fig. 6. Plots of experimental emission decay rates t1−1 (open
circles) and electron transfer rate constants kET (filled circles,calculated according to Eq. (11)) against pressure in the ZnP–PM complex at room temperature. Different curves are least-squares fits of Eq. (14) to the experimental data. Fitparameters are collected in Table 1.
M. Tars et al. / Spectrochimica Acta Part A 54 (1998) 1177–11891186
relevant pressure range [32], the same approxi-mate value, 100 cm−1, should be characteristicfor toluene at pressures used. So the inequalityV��tL�−1 is fulfilled too, and this provides an-other evidence for the nonadiabatic ET.
Despite the success in mastering the main trendin the pressure dependence of the ET rate, Eq.(14) completely fails when the asymmetry of thisdependence is considered. The fitting curves arefairly symmetric with respect to their maximum,whereas the experimental data show a substantialasymmetry (see Fig. 6). Therefore, further elabo-ration of the model is needed.
5.2.2. Reorganisation free energy ispressure-dependent
Here we assume that Eg is constant whereas Er
and R depend linearly on pressure. According toEq. (9), Er is pressure dependent, if the forceconstants or what is the same, vibration frequen-cies, characterising the reactant and product stateschange at ET (kR"kP). Note that the Marcus ETmodel [16] considers only the case of identicalreactant and product force constants. We have:
kET(p)
=' 4p3
h2(Er(0)+dp)kBT�V0�2 exp(−bap)
exp�− (Eg+Er(0)+dp)2
4(Er(0)+dp)kBT�
, (15)
where d is the rate of the reorganisation energychange with pressure, DEr/p.
The result of the least-squares fit of Eq. (15) tothe experimental data with different parametersets collected into Table 2 is shown in Fig. 7.
Fig. 7. Modelling of the ET rate as a function of appliedpressure in the ZnP–PM complex when the reorganisation freeenergy is pressure dependent. The fitting formula is Eq. (15)and the fit parameters are given in Table 2.
One can conclude that this model reproducesquite successfully both the inverted parabola-likecurve as well as its asymmetry. The best fit isobtained when the free energy gap Eg is taken asa free parameter (see dotted curve in Fig. 7 andthe second column in Table 2). However, in thiscase, the iterated Eg value is too small relative toits assumed value, about −1600 cm−1. The posi-tive d implies that Er is increasing with pressure.It follows from Eq. (9) that in this case kP\kR,meaning that the characteristic vibration in theproduct state has higher frequency compared tothe vibration in the reactant state. Optical spectrarender some support to this conclusion. The in-tramolecular vibration frequency to which theelectronic transition is mainly coupled is about1254 cm−1 in the ground electronic state andabout 1234 cm−1 in the lowest singlet excitedstate. It can be shown that much bigger frequencychanges should accompany ET in order to obtainthe d value in Table 2 (if the vibronic couplingstrength is kept in reasonable limits). An attrac-tive alternative seems to be a combination of thetwo models discussed assuming that the free en-
Table 2The fitting curve parameters in Fig. 7 based on Eq. (15)
M. Tars et al. / Spectrochimica Acta Part A 54 (1998) 1177–1189 1187
ergy gap and solvent reorganisation energy bothchange with pressure. In the linear approxima-tion, this possibility has been already brieflyanalysed in [11]. Note in Eq. (6) that if the forceconstants are different, kR"kP, the free energygap dependence on pressure is actually nonlinear.This adds an additional fitting parameter, whichmakes regression too flexible and the fitting resultrather ambiguous.
5.2.3. Reaction dri6ing force ispressure-dependent. A semi-classical model
The classical model in Eq. (15) is valid underthe condition that kBT�hn for all participatingvibrations. However, like radiative electronictransitions in our system, the nonradiative ETmay be coupled to high-frequency intramolecularmodes for which the above condition is not fulfi-lled. Moreover, the classical theory underesti-mates the ET rates in the so-called Marcusinverted region, which is characterised by thefollowing inequality: −Eg\Er. This is the situa-tion expected in our system where Eg(0):−1600cm−1 and Er(0)5800 cm−1 [26]. To address thisissue an extended model, which treats the solventand coupled donor and acceptor low-frequencyvibrations classically and the high-frequency in-tramolecular modes quantum mechanically,should be applied as follows:
k(p)
=' 4p3
h2lSkBT�V0�2 exp(−bap) %!exp(−S)Sm
m !
exp�− (Eg(0)+xp+ls+mhn)2
4lskBTn"
(16)
Like in Section 5.2.1, we again assume a con-stant reorganisation energy, but Eg and R linearlydependent on pressure. The total reorganisationenergy Er is now divided into two parts, theinner-cell (high-frequency intramolecular) vibra-tional reorganisation energy (ln) and the outer-cell (solvent plus low-frequency intramolecular)reorganisation energy (lS): Er=ln+lS. For sim-plicity, only a single high-frequency mode (with afrequency not dependent on pressure) is consid-
Fig. 8. A semi-classical modelling of the ET rate in theZnP–PM complex assuming that the reaction driving force ispressure dependent. The fitting formula is Eq. (16) and the fitparameters are given in Table 3.
ered. The (linear) dimensionless electron-vibrationcoupling strength of this mode is defined as S=ln/hn. Eq. (16) describes the case when the donoris in its vibrational ground state before ET.
In Fig. 8 the fitting result for the two cases ofdifferent high-frequency vibrations is presented.The outcome by using the 454 cm−1 vibration isclearly inadequate. A more satisfactory result isgiven testing with the 1200 cm−1 vibration. Alsothe above-mentioned asymmetry of the experi-mental data is well reproduced. Yet the fittingparameters gain rather unanticipated values. Asseen in Table 3, −Eg(0)BEr (Er=ln+lS=1692
Table 3The fitting curve parameters in Fig. 8 based on Eq. (16)
M. Tars et al. / Spectrochimica Acta Part A 54 (1998) 1177–11891188
cm−1; Eg(0)= −503 cm−1) which suggests thatET proceeds in normal rather than in invertedregion. Besides, in Section 5.2.1 we concluded thatthe driving force decreases with pressure. Here areversed tendency is observed. More work isneeded to resolve this controversy. Tentatively webelieve that the contribution of high-frequencyvibrations to the ET rate in ZnP–PM complex issmall and the classical theory more or less ade-quately describes the process.
6. Concluding remarks
In this work, an intramolecular ET kinetics wasstudied as a function of external high hydrostaticpressure. A combination of the ultrafast kineticspectroscopy with the high pressure technique hasthe potential to bring new insights into the natureof couplings that govern ET in molecular systems.The given experimental approach was furthercomplemented with model calculations. Expres-sions that describe the pressure dependence of ETwere worked out based on the classical and semi-classical nonadiabatic ET theory. Besides thestandard approximations, a quadratic electron-vi-brational coupling approximation was taken intoaccount, meaning that the force constants (orvibrational frequencies) were changed at ET. Areasonable agreement between the theory and ex-periment has been demonstrated taking into ac-count that the reaction driving force orreorganisation free energy is a function of pres-sure. The application of the nonadiabatic ETmodel to the present system is well-justified bear-ing in mind consistent model parameter valuesgained.
Sketching out a framework of understandinghow the molecular matter responds to externalpressure may be considered the main positiveresult of the work. Unfortunately, selecting out aunique interpretation (or set of parameters) iscurrently impossible because of too many parame-ters of the model and of limited scope and accu-racy of the experiment. More work should bedone along these lines in future.
Acknowledgements
It is a pleasure to thank Drs S.H. Lin and A.Laisaar for careful reading of the initial version ofthe manuscript and for many useful correctionsand Mr A. Krisciunas for the effective managerialsupport. The work was supported by the ArgonneNational Laboratory Contract (REP) No. 94-62JC-033 and by the Estonian Science Founda-tion grant No. 2271. M.R. Wasielewski issupported by the Division of Chemical Sciences,Office of Basic Energy Sciences, US DOE undercontract W-31-109-ENG-38.
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