J. Phys. Chem. 1995,99, 2469-2476
2469
Models for Electron Transfer with Vibrational State Resolution Kenneth G. Spears Department of Chemistry, Northwestem University, Evanston, Illinois 60208 Received: August 2, I994@
We use model calculations to study how electron-transfer rates in solution depend on populations of specific vibrational levels. The models are for nonadiabatic electron transfer with specific quantum populations in either a 2000-cm-' or a 430-cm- ' vibrational mode, characteristic of inorganic complexes in which we have previously demonstrated quantum effects. We find a major increase in the electron-transfer rate at small and large energy gaps when one or more quanta are in vibrations having large geometry changes during the electron transfer, and the effects are greater for high-frequency modes. The rates for different energy gaps, the ratio of rates for different quantum numbers, and the populations of vibrational energy after the electron transfer were shown to be useful for testing specific molecular models of electron transfer. We also modified standard nonadiabatic models to provide rate predictions when excess energy is communicating among a subset of vibrational modes by intramolecular vibrational redistribution. This modification converts excess energy to an effective vibrational temperature, which is used in temperature-dependent Franck-Condon factors. These models show that an order of magnitude increase of rate is possible for excess energies of 2000-6000 cm-' but only for electron-transfer rates slower than the maximum. At the maximum rate, excess energy slows the rate. The various models demonstrate that quantum effects for electron transfer in solution can be large, and therefore, such experiments can be used to provide a new molecular level test of electron-transfer mechanisms.
I. Introduction The earliest theoretical models of electron transfer (ET) were dominated by the powerful classical models of Marcus,' but the quantum aspects, in particular the understanding developed through work on radiationless transitions,2-6 were soon applied to this area by several group^.^-'^ The theoretical and experimental effort on electron transfer has proceeded on many fronts for the last 20 years, and we only note a few review articles here.I3-l5 It is surprising, especially when considering the close relationship of ET theory to radiationless transition theory, that the same experimental appro ache^'^,'^ successful in the late 1960s have not been applied to ET phenomena. In particular, there have been no measurements of electron transfer as a function of vibrational state, which might be useful in testing the vibrational part of electron-transfer models. In analogy to the evolution in radiationless transitions, such experimental tests of theory might probe molecular details of not only vibrational surfaces but also of the nature of coupling matrix elements. However, the reason for this situation persisting for over 20 years is clear; only a molecule selected such that intramolecular vibrational redistribution (IVR) and vibrational relaxation (VR) are slower than electron transfer can be studied with vibrational resolution, and then it may only be for one or two special vibrations. Recently, we have demonstrated that vibrationally resolved rates of electron transfer can be measured for at least one vibrational mode.I8 In addition, we have measured the vibrational populations after the electron transfer. The molecular system was carefully selected to match our experimental capability in picosecond infrared spectroscopy, but it is likely that other types of molecules may also show such behavior. As molecular systems are created to achieve device goals, ultrafast electron transfer is becoming more common, and therefore, the @
Abstract published in Advance ACS Abstracts, February 1, 1995.
0022-3654/95/2099-2469$09.00/0
consequences of vibrational population, even if only transient, may become practically important for design considerations.The importance of such measurements to refine our molecular level understanding of electron transfer seems obvious when compared to the molecular details now understood for radiationless transitions, details that have been helpful in molecular design. The particular solution experiments that we have doneIs use optical excitation in the charge-transfer band of an ion pair to prepare a neutral pair in specific vibrational states. One component of this ion pair is a metal carbonyl anion, Co(CO)4-, which has large vibrational distortion and an infrared active CO stretching mode that is involved in the exothermic electron transfer which retums the neutral pair back to an ion pair. The CO stretching mode has very slow IVR and VR, and this mode retains population while energy in other modes can cool by VR, thereby achieving quantum populations in a single vibration. Spontaneous electron transfer from the neutral pair with quantum resolution is studied through vibrational resolution of the anharmonic CO stretching modes via picosecond IR spectroscopy. We are in the process of making more measurements on this and related systems, as well as performing spectroscopic analyses and computing molecular potential energy surfaces. A great deal of work needs to be done before precise tests of electron-transfer theory can be made in these molecules. Interestingly, despite many investigations of quantum effects in ET,'9-21 there have been no predictions of trends in electrontransfer rates as a function of vibrational quantum number in a single vibration. Recently, a model for gas-phase electron transfer was done that included some quantum effects.22 However, in this case, it was assumed that an initial optical excitation populated many vibrational modes, all having ET in parallel. Therefore, at different energies, the excitation populates different distributions of vibrational quanta in these modes, and as the mode content varies, the quantum differences between modes are revealed as fluctuating ET rates. This work also 0 1995 American Chemical Society
2470 J. Phys. Chem., Vol. 99, No. 9, 1995
considered IVR effects for isolated molecules and predicted an increase in ET rates as a function of increasing energy. The goal of this paper is to study how vibrationally resolved electron-transfer rates depend on basic molecular parameters. This paper is our first effort to guide molecular and experimental designs, and we only treat the most obvious aspects of the problem. Section I1 discusses the mathematical model used in our treatment. Section I11 has the results and discussion, with subsections on high-frequency modes, midfrequency modes, final state vibrational populations, and the effects of excess energy and IVR. Section IV is a brief summary of conclusions. 11. Models
The calculation of ET rates for specific quantum numbers of medium- or high-frequency vibrations requires rate expressions that explicitly treat these vibrational modes, often referred to as inner-sphere modes. The primary consideration is that ET needs to be fast compared to intramolecular vibrational redistribution and vibrational relaxation processes, which can be as slow as 1-100 ps for the slower relaxing modes of complex systems. Fast ET rates are usually associated with strong coupling between electronic states and, therefore, can require adiabatic electron-transfer models. Models for the transition between adiabatic and nonadiabatic regimes of ET with solvent effects are under active i n v e ~ t i g a t i o n ? ~and - ~ ~a model has been p r o p o ~ e d where ~ ~ , ~ the ~ general rate constant, kET, including solvent-controlled effects, uses a multiplicative factor to correct nonadiabatic rate constants, kNA. Nonadiabatic rate constant models typically use Fermi Golden Rule expressions to incorporate inner-sphere vibrations, and these are convenient for our problem. The simple for kET is
where the adiabatic parameter, y, is given by
4nV2z, Y=-
Spears The theory of nonadiabatic electron transfer has had many contributors and historically was a natural extension of radiationless transition theory. These earliest results7-I2 provide useful expressions for calculations, and a number of treatments derive and compare quantum and semiclassical expressions for rates.11-12119-21 A key concept, often described as semiclassical, has been to separate the low-frequency modes of the solvent out of the fully quantum expressions and use the solvent reorganization energy, AS,as a parameter. The rate dependence on energy-matching conditions is included in a Gaussian weighting of the Franck-Condon sum. One study of such expression^^^ provided a comparison of rate predictions for equations that included one or two high-frequency modes in addition to the solvent mode. This work suggests that by adding additional vibrational modes to provide superior energy matching, the accidental energy-matchingresonances will be removed. We have used four vibrational modes in addition to the solvent mode (five-mode model) to minimize such artifacts of complex molecules. We find that two modes of small geometry change added to one or two modes of large geometry change remove large effects from energy matching. We have modified the usual rate expression^^^ for kNA in order to consider optically excited vibrational modes. The initial electronic state, a, has one specific vibration that is optically excited with quantum number v. The final electronic state, b, has v’ quanta in the optical mode and quantum numbers in a number of other acceptor modes,j . These other acceptor modes have a thermal distribution of populations, so m, is the change in acceptor quantum number. The acceptor modes are assumed to have no frequency change, just a geometry change given by S,, while the optically excited mode uses frequency and geometry changes to compute the Franck-Condon factor (FCF) (vlv’ ). All of these modes are explicit in the Gaussian energymatching expression, since both types of modes can be populated in the final state. The parameters x, and zj are used in eq 5 along with the dimensionless displacement parameter Sj and the modified Bessel function Zm(z). The temperatureweighted form of the Franck-Condon factors for acceptor modes is a useful feature of this representation, as will be seen later. In eqs 3-5,
u s
hw . x. =J zj = Sj csch(xj) (3) In this parameter, ZL is the solvent longitudinal dielectric I “BT relaxation time, As is the solvent reorganization energy, and V is the electronic coupling. This form has been m ~ d i f i e dto~ ~ . ~ ~ provide more complex expressions that are sums of rates into separate channels where the adiabatic parameter, y , is now weighted by a Franck-Condon factor for the specific channel. Recently, the validity of solvent control has been questioned for ultrafast ET,27 which suggests even more caution in considering the adiabatic limit. While more complex calculations may be advisable in specific cases, expressions for kNA +m are used here to investigate the vibrational quantum state dependence of ET. In our work, we assume rapid loss of vibrational coherence in comparison with electron-transfer rates. The issue of [-(-AI? As Cm,hm, - vhw, ~ ’ h m , , ) ~ predicting coherence effects has been previously raised for i# v electron-transfer model^.^^,^^ We assume that specific quantum state populations are present but without the interesting complexity of phase coherence between initial and final states. We effectively need to assume that phase coherence is lost by AS is the solvent reorganization energy, AE is the effective energy gap (positive sign), andfi is the force constant for a collisional processes with solvent or even by intramolecular normal mode j with frequency, vj = wj/2z, and a normal vibrational redistribution, leading to specific populations. We make a simple modification of nonadiabatic rate equations to coordinate change AQj. The above expression separates one treat IVR, but more extensive examination of IVR and coherence optical mode and many possible acceptor modes having possible effects is outside the scope of this paper. quantum number changes ranging over f m , where the effective
+ +
+
Models for Electron Transfer
J. Phys. Chem., Vol. 99, No. 9, 1995 2471
limits are defined by very small term contributions. The electronic coupling matrix element between states a and b is given by V. The computations for the above expression use both frequency and geometry changes for the initial quantum resolved modes, where the ratio of vibrational frequencies is given by g = w J wv'. In our specific case, we allow up to two optical modes, rather than the single mode shown in eq 5, and add another summation while including the second optical mode in the Gaussian. The FCF calculation uses specific formulas3I for the case of the frequency ratio g > 1 and g < 1. As suggested in this reference, a sum rule test of FCF expressions is desirable. Future work will consider the effects of vibrational anharmonicity.
TABLE 1: Parameters for Vibrational ModetFC
111. Results and Discussion
a Frequency ratio, g 2 =~0.944, g430 = 0.95. Two other acceptors of 525 and 555 cm-' had dimensionless displacements of s525 = 0.4315 and S555 = 0.1633 for all cases. For example, S2w = 0.75 is obtained with a displacement of 0.055 8, and a force constant of 19.7007 m d y d 8,. Experimental estimates for a single carbonyl mode have S2w
A. Quantum States for High-Frequency Modes. The parameters selected for use in our calculations provide systematic comparisons of the total vibrational reorganization energy and the fractional contribution of one or two high-frequency modes. We have selected a high frequency of 2000 cm-' since this corresponds to our experimental case of carbonyl frequencies in inorganic complexes; we also have briefly examined cases of 1200 cm-' and find similar trends. Equation 5 requires that we have a dimensionless geometry change, Sj, for each mode, where each mode contributes a vibrational reogranization energy, A,, defined by hw,S. In this case, the total vibrational reorganization energy, A,, is given by The contribution of n acceptor modes of identical frequency (and no frequency change) and the same value of Sj is obtained by using nS' in the formula. Therefore, it has become common to use values of 'S as parameters and to even use a single mode having a large (and unrealistic) Sj in such expressions. While we have used up to four modes in our calculation, we also have treated the value of S, as a parameter, although in our code we actually vary the force constant, 3, when we seek to define a particular percentage of a given mode in A,. As shown by recent Raman ~ o r k , ~it*may - ~ be ~ possible to define a more complete set of vibrations that contribute to electron transfer for specific tests of rate predictions. We have found that the general trends effected by quantum number changes in one or two modes having a large geometry change are not sensitive to the values of two to four other acceptor modes having modest or small geometry changes; however, the presence of these modes is essential to reduce the resonances in energy matching. The particular parameters used for the calculations are given in Table 1. Since our data entry is based on molecular parameters, the value of Sj is a computed quantity, and when constant percentages of a given mode are desired, the 3 is selected to fit. We have used AV values of 2000, 3500, and 5000 cm-' with contributions from a 2000-cm-I vibration that are 15, 30, 50, and 75%. A second mode of 430 cm-] is considered as an important middle-frequency contributor, which is representative of a metal-ligand type of vibration. We have assumed a solvent reorganization energy, As, of 5500 cm-' as representative of low dielectric solvents and inorganic ion pairs. This parameter makes little qualitative difference in the quantum effects, but it and A, together define the peak location in a plot of rate vs energy gap, AE, since the maximum rate is when the total reorganization energy equals the energy gap. The electronic coupling matrix element, V, was 100 cm-I, a value that is on the low side of a strongly coupled system. By using this number, additional values are easily scaled; however, from eq 1, one sees that for large coupling the rate equation would become more complex. The results are reported as rates, and
% in
,Iv, cm-I
2000 cm-l
430 cm-I
s 2 0 0 0
s430
2000 2000 2000 2000 3500 3500 3500 3500 5000 5000 5000 5000
15 30 50 75 15 30 50 75 15 30 50 75
69.1 54.1 34.1 9.1 15.9 60.9 40.9 15.9 78.7 63.7 43.1 18.7
0.15 0.3 0.5 0.75 0.2625 0.525 0.875 1.3125 0.375 0.75 1.25 1.875
3.2160 2.5 183 1.5880 0.4253 6.1811 4.9601 3.3322 1.2973 9.1462 7.402 5.0764 2.1694
between 0.25 and 0.43.
u,.
Energy Gap (1000 cm-')
Figure 1. Electron-transfer rate vs energy gap for zero quanta in the 2000-cm-I mode and a total vibrational reorganization energy, A,, of 3500 cm-I. Individual curves are for different percentages of the 2000cm-' mode in the constant 1".
the Franck-Condon sum is easily computed from eq 5 if you assume a multiplier of 2nV?h = 1.184 x loi6 and divide the rate by this quantity. When a range of compounds are available that have different energy gaps, a plot of ET rate vs energy gap has been one method of assigning model parameters such as the vibrational reorganization energy. In Figure 1, we show such a plot for no vibrational excitation in the initial state, where the total vibrational reorganization energy is constant at 3500 cm-I and the percentage of contribution from a 2000-cm-' mode is varied from 15 to 75%, with the 430-cm-' mode being the other major contributor. In these curves, the maximum rate is when the energy gap equals the sum of vibrational and solvent reorganization energies. One can see from this plot that quite a range of curves are found by having variable contributions of two strong modes, even though the total vibrational reorganization is constant. A number of experimental cases have used a single vibrational mode to fit data of this form,34,35while useful information is found, precise values of molecular parameters will not be obtained by fits with a single vibrational mode and even multiple modes will require more information on contributions in the Franck-Condon sum of eq 5. Another plot is shown in Figure 2, where we show the effect of quantum number on the ET rate for a particular case of 1,= 3500 cm-' and 50% contribution of the 2000-cm-' mode. The effect of quantum number is very dramatic at energy gaps differentffom the maximum. The effect is smallest for I , = 2000 cm-' with only 15% in the high-frequency mode, and this is shown in Figure 3.
2472 J. Phys. Chem., Vol. 99, No. 9, 1995 1
0
Spears 50% in 2000 cm-'
50% in 2000 cm-' mode 1 3 7
25.7 0 E=O
+
lnlo
'"
'
v=o -0-
+, 10
v(
h, =3500
f I
v=o +
2 5
v=2 t
a: Iw -
v=3 -3
5
v = l -Etv=2
+-
cm-l,
V = l -fr
1091f
10
15
20
'0
Energy Gap (1 000 cm-' )
mode
X, =ZOQO em-'
5000
10000
15000
2 ( 00
Energy ~ a (cm-') p
Figure 2. Electron-transfer rate vs energy gap for zero, one, two, and three quanta in the 2000-cm-' mode. All curves have 50% of the 1, = 3500 cm-I being contributed by the 2000-cm-I mode.
50% in 2000 cm-'
15
mode
>
15% in 2000 cm-'
-
2
mode
I
+,
10
v 0
z
f 5 4-
Q
'0
5000
10000 15000 20000
Energy Gap (cm-l)
+
Energy Gap (1000 cm-l)
Figure 3. Electron-transfer rate vs energy gap for zero, one, two, and three quanta in the 2000-cm-' mode. All curves have 15% of the = 2000 cm-' being contributed by the 2000-cm-' mode. 1013~
Initial Quantum No.=l
h v =3500
t
cm-'
0'2
0"
X
0Io
109
of 2000 cm-'
mode
30% -A-
75% -e-
5
10
15
20
Energy Gap (1 000 cm-' ) Initial Quantum 1013
h, =SSOO
X
of 2000 cm-'
Nod
cm-1
mode
5 10 15 Energy Gap (1000 cm-')
20
Figure 4. (a, top) Electron-transfer rate vs energy gap for one quanta in the 2000-cm-' mode and a total vibrational reorganization energy, A,, of 3500 cm-'. Individual curves are for different percentages of the 2000-cm-' mode in the constant a,. (b, bottom) Electron-transfer rate vs energy gap for three quanta in the 2000-cm-' mode and a total
vibrational reorganization energy, A,, of 3500 cm-l. In Figure 4, we show the importance of high-frequency-mode contributions for a constant Av = 3500 cm-I. Figure 4a shows the case for one quantum of population in the high-frequency mode. Figure 4b shows the effect of three quanta of population for the same range of percentages. These results show that quantum populations can make a dramatic effect in the ET rate,
Figure 5. (a, top) Ratio of electron-transfer rates as ( v l)/v, with v = 0, 1, and 2. All curves have 50% of the d) = 2000 cm-' being contributed by the 2000-cm-' mode. (b, bottom) Ratio of electrontransfer rates as ( v l)/v, with v = 0, 1, and 2. All curves have 50% of the d, = 3500 cm-' being contributed by the 2000-cm-' mode. Absolute rates are shown in Figure 2.
+
and the sensitivity to percentage contributions of highlfrequency modes suggests that vibrationally resolved data can be used to test specijic vibrational models of ET. The oscillatory structure as a function of energy gap is due to the optical FCF for large geometry changes. These effects can be considered as an increase in rate for large and small energy gaps due to increased Franck-Condon sum when initial quantum numbers are nonzero. In addition, such optical FCF show oscillatory behavior when the initial quantum numbers are non-zero. For example, the squares of the FCF for transitions from 3 quanta in the initial state into final state quanta ranging from 0 to 10 are 0.049, 0.227,0.066,0.137,0.0099,0.193,0.202,0.089,0.023,0.0038, and 0.0004, respectively. These results are not sensitive to the nature of the low-frequency modes and fine structure in energy matching, since changing to more low-frequency modes with values as low as 100 cm-' gives essentially the same behavior. From Figures 2-4, it is expected that ratios of ET rate as a function of vibrational quantum number would be useful experimental measures when the electronic coupling is factorable in the manner shown in eq 5 . In Figure 5 , we plot ratios of ET rate constants for the case of 50% contribution by highfrequency modes. Figure 5a shows the case for A" = 2000 cm-', and Figure 5b shows 1" = 3500 cm-l. The trends are similar for both cases, and the ratios obtained for several values of energy gap would provide a good test of the vibrational models of ET. However, the oscillatory trends with energy gap shown in Figures 2-4 are masked by these ratios of rates. B. Quantum States for Midfrequency Vibrations. The vibrational frequencies in the 400-500-cm-' range correspond to metal-ligand frequencies for inorganic systems and various deformation modes in organic molecules. For inorganic systems, such modes can have large geometry changes;32however, their close proximity to other modes may lead to efficient intramolecular redistribution processes. For direct optical excitation into charge-transfer bands, one may expect midfrequency modes to have large initial quantum numbers due to
J. Phys. Chem., Vol. 99, No. 9, 1995 2473
Models for Electron Transfer
E W
1-O9Y
9 # 430 cm-1 mode v=o -mv=l
\\ 1
*
c)
15
v
El0 c
Y
5
Energy Gap (cm-')
Figure 6. Electron-transferrate vs energy gap for zero, one, two, three, and five quanta in the 430-cm-' mode. All curves have 15% of the 1, = 2000 cm-l being contributed by the 2000-cm-' mode.
efficient FCF for optical excitation. The issue of intramolecular vibrational redistribution is very important for such modes and will be briefly discussed in a later section. In this section, we examine a series of calculations to see if the quantum effects found for high-frequency modes are also found for midfrequency modes. The high-frequency component of vibrational reorganization is kept at 15% of the total, and the 430-cm-' component is defined in Table 1 for different Av values and V = 100 cm-'. In Figure 6, we plot ET rates as a function of energy gap for quantum numbers of 0, 1, 2, 3, and 5 in the 430-cm-I mode when the total vibrational reorganization energy is 2000 cm-I. These results show similar behavior as the high-frequency mode, but for lower quantum numbers, the flattening of the curvature is much less. For larger geometry changes in the 430-cm-' mode, such as with 1"= 3500 cm-I, we find curves similar to Figure 6. However, when the initial quantum number reaches v = 5 and the energy gap is on the order of 12 000 cm-I, then significant contributions to the Franck-Condon sum occur for quantum numbers larger than our current limit of 20. Additional study of larger geometry changes in the midfrequency modes requires changes in our code. We conclude that midfrequency vibrations require larger quantum numbers to effect changes in the ET rate similar to high frequencies. The rate changes per unit quantum change, while small, could be easily measurable for large and small energy gaps. For cases where vibrational relaxation is on the same time scale as electron transfer, the quantum dependence in the midfrequency modes would show nonexponential behavior for ET rates. For cases of fast IVR, quantum resolution is not possible, although the fraction of excess energy in these modes will affect the rate. This problem is treated in section D.
-'4
-2
0
2
4
6
8
Coordinate Figure 7. Schematic diagram of neutral pair [AD] and ion pair [A+D-]
energy as a function of one vibrational coordinate; the high-frequency CO stretching mode is shown as a displaced oscillator, and optical excitation to near the zero level is shown by an arrow.
a rise time that is a convolution of the ET decay times from several levels in the initial state. For example, in Figure 7, we show a schematic diagram of one single vibrational mode for the case of optical excitation in a charge-transfer absorption band that changes an ion pair state to a neutral pair state. In this diagram, one considers optical excitation in the upper potential, where the absorption FCF give populations in several levels. We assume that all excess energy in the low-frequency modes rapidly decays by VR to leave specific quanta in the high-frequency mode of this diagram. Each of these quantum levels can decay by spontaneous ET to populate several quanta of the ion pair ground state. Of course, in a multimode vibrational case, we can absorb the final state energy with many vibrations so that large quantum numbers of 6-8 are unlikely in the final state. This type of experiment is similar to our recent reports,'* where we measured decay rates for ET from initial quantum numbers of 0, 1, and 2 and rise times for final vibrational states with quantum numbers of 2, 3, and 4. For experimental analysis purposes, it is of interest to compute the branching into final vibrational states, which is expected in such an experiment, and to compute the fractional contributions of each initial state to the rise time of any final state. We use an estimate of vibrational parameters appropriate to our prior experiments and present the results as an example; we require additional spectroscopic and theoretical work to make a precise comparison with our experiments. For these calculations, we use two cases, A,, = 2233 and 1467 cm-I. The models use four vibrations, 525- and 555-cm-I modes with properties shown in Table 1, and a 430-cm-' mode with S = 1.4634 and g = 0.95 for both cases. For Av = 2233 cm-I, the 2000-cm-' mode had S = 0.6434, g = 0.944, and 57.6% of the energy, while the 430-cm-' mode had 28.2% of C. Final State Vibrational Populations the energy. For Av = 1467 cm-', the 2000-cm-I mode had S = 0.2605 and 35.5% of the energy, while the 430-cm-' mode The final state quantum populations after the ET can be used had 42.9% of the energy. to test a vibrational model for electron transfer. Measurements of final state vibrational populations have been done r e c e n t l ~ , ' ~ * ~ ~ The results for the case of Av = 2233 cm-' are shown in Figure 8. Here we have plotted the final state quantum numbers but such measurements are intrinsically more convoluted than for the 2000 cm-' mode that are populated by initial state measurements of initial vibrational state decays. A common branching, where the initial state quantum numbers are labeled experimental difficulty arises because if the ET state is formed on the curves, These branching fractions are obtained by by ultrafast optical excitation, then several initial vibrational summing all contributions in eq 5 that start at a specified states are populated simultaneously. As shown by our work,'* quantum number and reach a specified final quantum number. the rates of ET from different vibrational states can be monitored This sum includes all combinations that connect the specified with transient infrared absorption spectroscopy. However, when quantum numbers, and for any initial state, the fractions sum the final states are being populated by ET that simultaneously to unity, The plots show how initial quantum numbers originates from several initial vibrational states, then each initial preferentially populate larger final quantum numbers due to the vibrational state has some probability of reaching each final significant geometry change in this mode. state. Therefore, in an experiment where the final vibrational Another analysis of the calculations is made in Figure 9, states are monitored by transient infrared absorption (when where we plot the fraction of optical absorption events in each sufficient anharmonicity allows probing each level), we expect
2474 J. Phys. Chem., Vol. 99, No. 9, 1995 0.8
Spears
Initial State Branching
0.7
Final Quantum Number(v')
Figure 8. Plot of fractional populations vs final state quantum numbers for the 2000-cm-' mode. Individual curves for initial state quantum numbers, v = 0, 1, 2, and 3. Final State Fraction
v'=2
.20
57.6% in 2000 cm-' v'=3
Intermediate Quantum Number of Neutral Pair Final Sfate Fraction
.-s 0.5 c
b 0.4
X, =1467 cm-' 3 5 . 5 ~in 2000 cm-' mode
n
3!eszL4 6 0.2
:
0.1
LL
0.00
v'=2
v'=3
1
2
Intermediate Quantum Number of Neutral Pair
Figure 9. (a, top) Fraction of optical absorption events vs intermediate (neutral pair) quantum states. Curves are for final state quantum numbers, v' = 0, 1, 2, 3, 4,and 5 . Each curve shows the fraction of absorption events that passed through different intermediate (neutral pair) quantum levels and ended in the given v' of the ion pair. A,,= 2233 cm-I. (b, bottom) Similar to a, with 2, = 1467 cm-I.
final state, v'. The fractions are a product of the absorption probability (FCF squared) and the branching ratio in ET. This is an oversimplified model of absorption followed by ET, but it is useful to demonstrate the composition of final states. The sum of all fractions on all the curves equals unity, and for any curve of v', the relative importance of different intermediate (neutral pair) quantum numbers is visually displayed. For example, in Figure 9a, we have the case for 1"= 2233 cm-', and we see that for v' = 1, there is little contribution from neutral pair states other than v = 0, while v' = 2 has contributions from both v = 0 and v = 1. However, for v' = 3, we see that the neutral pair state of v = 1 is dominant. From such a curve, we can expect the rise time of any final state such as v' = 3 to have a major contribution from only the v = 1 neutral pair state. In contrast to these results, we also show Figure 9b for ,Iv = 1467 cm-I. If this smaller geometry change was operative, we would expect that v' = 1 would be the largest population and that it would be almost purely derived from an initial v = 0. In addition, less population would be in states with larger v', and v' = 3 would have contributions from two states.
The results in Figures 8 and 9 show how specific models can be tested with experimental measurements of relative populations in final states and the rise times of the final vibrational states. D. Effects of Excess Energy and IVR. The question of interest is to estimate the change in fast ET rates when excess energy from optical excitation is contained in the molecule, but not with specific vibrational quantum numbers. We consider a more restricted case where the excess energy is in a subset of vibrational states that are interchanging energy rapidly through efficient IVR but where vibrational relaxation (VR) of the subset is slower than the possible ET. For slower VR (e.g., 1-10 ps), this effect could lead to a time-varying ET rate that is dependent on the total excess energy left in the molecule by VR. We note that when a subset of vibrational states is populated with a specific total energy, then the populations among modes in this vibrational pool can affect ET rates when there are one or two modes with large geometry change. For very broad charge-transfer absorptions, one could easily have excess energies in the 2000-5000 cm-I range with IVR over 3-7 modes. This type of problem has been briefly discussed for ET in the context of a kinetic model3' for vibrational relaxation of levels undergoing ET. Also, calculations for ET in gas-phase supermoleculesZ2included IVR onset and predicted an increase of ET rate with increasing excess energy. A number of different treatments of radiationless transition rates in the presence of VR have been done,38.39and one of these3*has used the concept of communicating states and an active promoting mode to explain radiationless rates for large molecules when no vibrational resolution is possible. We use a similar concept, IVR within a subset of modes characterized by a vibrational temperature, but we rely on eq 5 for our rate calculation. In particular, we compute an average temperature, (0, for the vibrational pool by statistical mechanics formulas for a Bose distribution of energy over the pool modes. In eq 6, we define the excess energy, E X S in , terms of (2"). We use ci = b i , where NP is the number of normal modes in the IVR pool. An iterative method is used to find (0 consistent with eq 6.
Our modification of eq 5 is to use (7) only for evaluating the temperature-dependentFranck-Condon factors, while using the temperature of the surroundings in the Gaussian and prefactor terms. Therefore, we replace T with (7) only for the parameter x, in eq 3. This approach is relatively simple, but to evaluate its range of accuracy, one needs to compare it with other methods. One case of interest relates to the quantum resolved results in prior sections, where we consider optical excitation of both specific quantum resolved high-frequency modes that do not have IVR or VR and a group of modes with fast IVR. If one midfrequency mode in the vibrational pool, such as the 430cm-' mode, has a large geometry change, under what conditions will initial excess energy speed up the ET rate? The case with no quanta in the high-frequency mode would be similar to the general case when no quantum resolution is possible. From the quantum resolved case of a 430-cm-' mode shown in Figure 6, it seems likely that excess energy should affect the rate differently depending on the energy gap, solvent reorganization energy, and vibrational reorganization energy. In Figure 10, we show a calculation for V = 100 cm-I, 1s = 5500 cm-I, and 1, = 2000 cm-I, where the mode parameters are similar to those of Table 1 with 15% of 1"in a 2000-cm-'
Models for Electron Transfer
lo"[
J. Phys. Chem., Vol. 99, No. 9, 1995 2475
5 Mode Model of IVR
I
4500
Y
u3000
s
c
el500 I-a
1, =5500 cm-'
4000
7500 -9'0
107
0
Figure 10. Electron-transfer rate vs excess energy, where IVR distributes the excess energy in a vibrational pool of five modes (555, 430, 525, 250, and 100 cm-I). A five-mode model is used for ET calculations, and five energy gaps are shown; I S = 5500 cm-I, dv = 2000 cm-l, and 69.1% of vibrational energy is in the 430-cm-I mode with a 2000-cm-' mode for the optical mode with I/ = 0.
4000
6000
8000
Excess Energy (cm-l)
2000 4000 6000 8000 Excess Energy (cm-')
2000
Figure 12. Temperature (K) of the vibrational pool required to give the specified excess energy. Displayed for the three-, five-; and sevenmode pools of Figure 11. 1013
75% in 1200 cm-' A,
E Gop=7000
2000
4000
mode =?SO0 cm-'
6000
8000
Excess Energy (cm-l)
Excess Energy (cm-')
Figure 11. Electron-transfer rate vs excess energy in IVR pools of either three, five, or seven modes. The solid symbols are for 12 000cm-I energy gap and the open symbols are for a 16 000-cm-' energy gap. The total reorganization energies are the same as for Figure 10.
vibration and $30 = 3.216. In eq 5, the 430-cm-' mode is now treated as the dominant acceptor without any frequency change, but the 2000-cm-' mode remains an optical excited mode with v = 0. From eq 5, only the optical mode is a pure FCF, without a temperature-dependent correction. The other two acceptor modes of 525 and 555 cm-' are the same as in Table 1. The vibrational pool that defines (qis given by five vibrations of 555, 430, 525, 250, and 100 cm-'. Five energy gaps are shown in Figure 10, where the maximum rate is expected at 7500 cm-' . The maximum rate is depressed slightly by excess energy, and the nearby energy gap of 4000 cm-' has only a slight increase followed by a decrease in rate. The largest effect of excess energy is for the rates smaller than the maximum. The competition with VR in this model requires that the ET rate be in the 10"-1O1*-s-' range. For V = 100 cm-I, it is not easy for excess energy to increase the ET rates to 10" for energy gaps of 16 000 cm-I. However, if V was 3-5 times larger, then excess energy could be very effective in increasing the ET rate to compete with VR. In general, energy gaps very different from the case of maximum rate are required to see a large effect of excess energy. The size of the vibrational pool is important, as is shown in Figure 11. Here we pick energy gaps of 12 000 and 16 000 cm-' and compare three-, five-, and seven-mode pools for the parameters of Figure 10. For the three-mode case, we use 555, 430, and 525 cm-I, while the seven-mode pool adds 250 and 100 cm-I to the five mode case. The three-mode case shows a drop in rate at large excess energies. Such a drop in rate also occurs with small energy gaps of 1000 and 4000 cm-I, which implies a real effect; however, we need to check for an artifact of our code, which uses m = - 10 to 20. Code limits were not
Figure 13. Electron-transfer rate vs excess energy, where IVR distributes the excess energy in a vibrational pool of seven modes (1200, 560, 520, 430, 300, 250, and 90 cm-I). A five-mode model is used for ET rate calculations and four energy gaps are shown; IS = 3500 cm-', I , = 3500 cm-', and 75% of the total vibrational energy is in the 1200-cm-' mode with a 430-cm-' mode as the optical mode with v = 0.
present for quantum resolved calculations of up to five quanta, but the three-mode case creates an unusually high temperature. In Figure 12, we plot the temperature equivalent of excess energy and note that temperatures near 2700 K have Bose factors in eq 6 with energy in the 430-cm-' mode of 3.88 quanta. Since a three-mode pool and excess energies of over 4000 cm-' would be unusual conditions, we will leave investigation of the rate reduction for future work, where this simple calculation method can be investigated more thoroughly and also compared with quantum distribution methods. Another issue is the predicted effect when the acceptor modes are dominated by a larger frequency, such as 1200 cm-' typical of organic molecules. We have defined a new acceptor frequency set with V = 100 cm-I, I S = 3500 cm-I, and A, = 3500 cm-' where the four acceptor mode parameters are similar to those in Table 1 but with 75% of the I , energy in a 1200cm-' vibration instead of a 2000-cm-' mode, with S12~= 2.1875. The 430-cm-I mode is the optical mode with v = 0 and 15.9% of the reorganization energy. We define a sevenmode pool with 1200, 560, 520, 430, 300, 250, and 90 cm-'. The results are shown in Figure 13, where we once again only see a modest increase of ET rate with excess energy for larger energy gaps. In summary, we have used a simple modification of an ET rate expression having temperature-weighted Franck-Condon factors to include vibrational temperature, (q, as characteristic of populations among a vibrational pool of IVR coupled modes. The ET rate predictions as a function of excess energy show a significant effect only when the energy gap is not equal to the sum of reorganization energies. For cases with one dominant acceptor mode of 430 or 1200 cm-I, the ET rate increases about an order of magnitude with excess energy of 2000-4000 cm-'
2476 J. Phys. Chem., Vol. 99, No. 9, 1995
Spears
only at large and small energy gaps, when the ET rate is slower. In this case, we expect that ET could compete with vibrational relaxation when the electronic coupling is larger than 100 cm-' and specific energy gaps are present to create a range of ET rates that bracket the VR rate. These parameters seem to be realistic for a number of molecules, and small excess energy effects may be measurable for very fast ET by varying the wavelength of excitation. However, for larger couplings, we also need to reconsider the effect of adiabatic behavior.
transfer, the excess energy slightly reduces the electron-transfer rate. In summary, rate effects from excess energy are likely to be small in typical situations. However, they could be more important for very fast electron transfer or environments with unusually slow vibrational relaxation. Additional work needs to be done with more sophisticated models; however, this simple modification of nonadiabatic ET models is useful for mapping excess energy effects on ET.
IV. Conclusion
Acknowledgment. We thank the U.S. Department of Energy, Office of Energy Research, Division of Chemical Sciences (Grant FG02-91ER14228) for support of this research.
In this work, we have studied how electron-transfer rates depend on populations in specific vibrational states. We have provided the first model calculations to show how experiments with vibrational resolution can provide new tests of electrontransfer models. In particular, structural contributions to electron transfer can be tested with vibrational models, and in specific cases, such tests might be used to reveal new electronic coupling mechanisms involving electronic-vibrational coupling. The electron transfer must be fast enough to compete with intramolecular and intermolecular vibrational relaxation processes, and only some systems will have appropriate rates. Our work excluded coherence effects of vibrational relaxation. Our specific models were for vibrational populations in a 2000-cm-' and a 430-cm-' vibrational mode, characteristic of metal complexes and similar to our experimental measurements where quantum resolved vibrational effects are important. We investigated electron-transfer rate variations with the energy gap, the total vibrational reorganization energy, and the percentage of vibrational energy in the quantum-selected mode. The rate was computed for nonadiabatic electron transfer by using semiclasical expressions based on the Fermi Golden Rule, where low-frequency solvent modes are incorporated into a solvent reorganization energy. For correlations of rate energy gap, we find a major increase in electron-transfer rate at small and large energy gaps when one or more quanta are in vibrations having large geometly change during the electron transfer. For energy gaps near the total reorganization energy, the ET rate is a maximum, and it decreases with additional quanta of population. The rate change from zero to one quantum of vibration has the largest unit quantum effect. The effect of a one quantum change is much larger for the 2000-cm-' mode than for the 430-cm-' mode. We also performed two model calculations demonstrating how rise times and vibrational populations in the final state can provide additional tests of vibrational models. We expect that measurements of electron-transfer rate for one to three quanta at one or more energy gaps will provide a good test of the vibrational part of ET theory, especially when coupled to measurements of the vibrational populations afer the electron transfer. We also addressed a related question of predicting the electron-transfer rates when excess vibrational energy is in the molecule but cannot be individually studied due to fast intramolecular redistribution in the molecule. We proposed a model that used the excess energy to define an average temperature for a subset of vibrations, where one of the vibrations had a large geometry change. We showed that excess energy of 2000-6000 cm-' in a vibrational pool of three, five, and seven modes can affect the electron-transfer rate an order of magnitude but only for rates that are significantly slower than the maximum rate. The energy gaps for order of magnitude effects can be small or large, but when the excess energy relaxes rapidly into the environment, only a narrow selection of energy gaps can allow the rate to increase enough to compete with vibrational relaxation times in the 1-IO-ps time scale. For energy gaps near the case with a maximum rate of electron
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