9674
J. Phys. Chem. 1992, 96, 9674-9677
Absorption Spectra of the Solvated Electron in Hydrocarbons H. Abramczyk,* B. Werner, and J. Krob Institute of Applied Radiation Chemistry, Technical University, 93-590 Wf, Wrbblewskiego 15, Poland (Received: February 25, 1992)
The absorption spectra of the excess electrons solvated in 3-meth~lpentane-h,~, 3-methylpentar~e-d~,, 2-methylbutane, 2,2-dimethylbutane,2,3-dimethylbutane,and 2,2,3-trimethylbutane at 71 K and n-propane-h,, n-propane-d, at 91 K are calculated in terms of the theory presented in our previous paper (Abramczyk, H. J. Phys. Chem. 1991,95,6149) and compared with the experimental data. We have shown that the band shapes and the spectral changes due to the isotope substitution or molecular structure in hydrocarbons can be explained on the basis of the coupling between an excess electron and intramolecular vibrational modes of the solvent in which the electron is stabilized. The effects of molecular structure and the isotope substitution on G(e;) (100 eV electron yield) is discussed.
Introduction The excess electrons stabilized in hydrocarbons have been the subject of many abs0rption,2-~ESR14,11-14 decay, bleaching, scavenger m e a ~ u r e m e n t s , 2 .and ~ J ~picosecond'ez3 ~~ and femtosecond laser studies.24 Among the questions raised by the trapping of electrons in hydrocarbons which still are not answered on a quantitative basis is the problem of the large differences in drift mobilitiesz5and the trapping yields (G(e;) values) in solvents of similar structure or after deuteration.6J1 Electrons stabilized in hydrocarbons display strong absorption bands in the near-IR, undergoing blue shift with time. This shift with time is smaller than in polar matrices. In 3-methylpentane, the absorption maximum at 76 K is observed at 0.62 and 0.73 eV 3 ps and 380 s after pulse, respectively.8 In 3-methylhexane, the absorption maximum is reporteds at 0.55 eV 3 ps after pulse and at 0.75 eV The mechanisms of at 76 K for the %o y-irradiated ~amp1es.l~ molecular reorientation and detrapping-retrapping are currently most favored in explaining the spectral shifts.8 Geminate recombination in hydrocarbons is faster than in polar matrices and occurs simultaneously with the spectral shift.6~~~ The band widths and are smaller than in water (0.82 eVz7) are about 0.5 eV6,7326 or in alcohols (1.4 eV27). The typical bandwidths in hydrocarbons are comparable with those in ethers and amines.z6*z8It has been found that the G(e,-) yields values for the production of trapped electrons in hydrocarbons range from 0.02 to 1.1 and even small changes in molecular structure of the matrix molecules such as the differences between 2-methylpentane, 3-methylpentane, or 3-ethylpentane are often accompanied by large changes in G(e;).I1 It has been reported6that the electron yields G(e;) are some 50% higher in the deuterated in respect to the protiated form of a given alkane. It has been suggested6that the changes of G(e;) caused by the structure of the matrix and the deuterium substitution are due to (a) differences in the probability of e; tunneling to the geminate cation or other acceptor or (b) different probability of e,- annihilation by migrating positive charge. It should be remembered, however, that all these results and conclusions are correct for hydrocarbons if we assume that the Gc values obtained from absorption measurements simply reflect changes in G and not in c. In the above-mentioned papers, the G(e,-) values were assigned on the assumption that the molar extinction coefficient c is 3 X lo4 L.mol-'.cm-l at the band maximum in each of the hydrocarbons and equals the value for 3MP at 77 K."7 This means that both in deuterated and protiated media e is assumed to be the same. However, we cannot eliminate the possibility that the changes in Gc are also due to the changes in c and not only changes in G. Recently, we have shown that for water and deuterated water the extinction coefficients are not the same because of different vibrational properties of protiated and deuterated matrices.z9 It seems that also for hydrocarbons vibrational properties of the matrices might be an important factor affecting the absorption and ESR spectra. If the Gc values reflect changes in c, and not in G, than it would be necessary to verify conclusions that the small differences in molecular structure (as in 3-methylpentane
and 3-ethylpentane, 3-methylhexane and 3-ethylhexane, or 2methylpentane and 3-methylpentane) and isotope effect are accompanied by large differences in the trapping yields G(e;).6 Recently, we have proposed' a model which assumes that the spectral changes are due predominantly to the coupling between an excess electron and the intramolecular vibrational modes of the matrix in which the electron is stabilized. If the coupling were an important factor, the changes in vibrational frequencies upon deuteration or substitution would cause the changa of the solvated electron spectrum. It would be easier to understand why there is no correlation between the electron trapping efficienciesof the hydrocarbon glasses and the density, viscosity, dielectric constant, mobilities, and the maximum of the absorption spectra." On the other hand, it is well-known that small changes in molecular structure or isotope effect can cause significant differences in the vibrational frequenciesof hydrocarbon The purpose of the present paper is to examine to what extent the experimental band shapes of e; and the spectral changes due to the isotope substitution or molecular structure of hydrocarbons can be understood in terms of our model.
Numerical Calculations In the framework of the linear response theory the extinction coefficient e is expressed as 2rIw c(w) -(1 3hcnV - exp(-Bhw))I+-(M+(t) -_ M(0))e-iufdt (1) The expression for the dipole moment correlation function (W(t) M ( 0 ) ) was presented in our previous paper.' The theoretical treatment considers the contribution to the absorption band profile from the vibrational coupling, tunneling, and inhomogeneous broadening due to the statistical variety of trapping sites. The vibrational coupling includes interaction between the solvated electron and the solvent molecules forming the trap (in terms of the 'cavity" or 'solvated anion" model) and is treated in the framework of the strong coupling limit theory. The displacement operator A diagonalizes the Hamiltonian for the q mode coupled with the electron when the electron is in its first excited state. The A operator creates the dynamic effects of the vibrational motion on electron between its Bom-Oppenheimer eigenstates. The time evolution of the operator A is governed by the Liouville equation for the vibrational density operator of the q mode coupled to the bath. The interaction between the q mode and the bath is described in terms of the resonance energy exchange and is characterized by the damping parameter r. Despite this indirect coupling of the solvated e l m o n to the bath (through the q mode), we can expect the direct coupling (inhomogeneous broadening). The direct coupling to the bath is responsible for the fluctuations of the potential well during the electronic transition (single or double potential well) and the electronic energy levels (dephasing). For equilibrium averaging over these fluctuations we have used the cumulant expansion procedure. The fluctuations of the energy levels (characterized by a and ra in eq 5) and the fluctuations of
0022-3654/92/2096-9674$03.00 f 0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 9615
Absorption Spectra of the Solvated Electron the energy potential well (b and 7 b in eq 5 ) are regarded as a random variable in the low-level system governed by the stochastic Liouville equation. The origin of the fluctuations of the potential well is due to the assumptions which were introduced in the strong coupling treatment of the electron-vibrational mode coupling: (i) the vibrational frequencies in the ground and excited states of the cavity (or anion) are the same, (ii) only one vibrational mode is modified in the absorption process, (iii) equilibrium geometry of the potential well in the excited electronic state is displaced linearly with respect to the normal vibrational coordinate. Generally, these assumptions may be unjustified. The effects (i)-(iii) were formally included in our model by regarding the electronic transition as a transition in the same potential well fluctuating with time. Using the model we have shown' that the dipole correlation function (M+( t ) M(0)) is given by (W(t) M(0)) = 4
T
u
u
m
/
e 2 wC CZ(Y) C { C C C C C A / p l c t j C k j , y u ( B ; l o l ) m=l u=Oj=O&=O/=Op=O
y=O
+
where
t
= exp(-Eo+/kT)/Tr pcl tI
= 1 -exp(-hwoo/kT)
€2
= exp(-+oo/kT)
The electron charge is given by e, pel is the electron density operator, Eo+has the meaning of the energy of the lower ground state (if there is the splitting of the energy levels due to tunneling) and the energy of the ground state if the electron motion occurs in a single potential well. The theoretical details can be found elsewhere.' Here we explain the meaning of the parameters which have been used in eqs 1-5 for calculations of the theoretical spectra. The correlation function (W(0)M ( t ) ) depends on the following molecular properties: (wb), woo, ao,r, T,, 7b, a, b, f,, ( w m f ) . We have that only a. and b are adjustable in our model. All of the others can be obtained from ab initio calculations ((&,f,, wmf) and IR and Raman measurements (om,fa), or they can be reasonably estimated (a, r, 7b). The phase angular speed characterizes the electronic transition of the solvated electron when there is no coupling to the intramolecular vibrational modes of the solvent. This parameter can be treated approximately as the depth of the trap. It was taken as equal to 0.84 eV for npropanahs and n-propand, and 1.06 eV for 3-methylpentane-h14, 3-methylpentane-d14,2-methylbutane, 2,2-dimethylbutane, 2,3dimethylbutane, and 2,2,3-trimethylbutane for the coupling with the CCC bending modes. The vibrational frequencies of the intramolecular modes of the solvent, which are characterized by the phase angular speeds woo were taken from IR and Raman measurements.*3s The coupling constant ao,which characterizes the strength of the coupling of the excess electron with the vibrational mode characterized by woo is treated as a fitting parameter and is chosen to reproduce the extinction coefficients t = 3 X lo4 L.mol-'.cm-' at maximum as was reported for 3-
(@A)
methylpentane.' The damping parameter r reflects the strength of the coupling between the vibrational mode woo and the thermal bath (translational, reorientational degrees of freedom, low-frequency intermolecular vibrational modes of the solvent). We have taken r as 130 cm-l for all of the hydrocarbons, which comsponds to the typical frequency of the torsional mode. The terms a = 2-'f2(Au2),and b = ( h 2 ) b l f 2 characterize the root mean square fluctuations of the electronic transition energy due to the dephasiig process and the fluctuations of the energy potential well during the transition. The correlation times 7, and 7 b characterize the time scale of these fluctuations. In terms of the stochastic Liouville equation used in our model, b is the change of the electronic angular speed, when the electron jumps from the ground to the first excited state with the probability per unit timefb = 1/76, The tunneling phase angular speeds (w")are defied by eqs 52-53 in ref 1 and depend on the splitting of the energy levels due to the tunneling in the double potential well. In the calculations we have assumed that the electron motions occurs in a single minimum potential well, the influence of the direct coupling to the bath (inhomogeneous broadening) is negligible, and the Born-Op penheimer potential well is static with time. It means that we put (w"u), a, and b as equal to zero. The transition matrix element f, was taken in harmonic approximation a s h = h/2m(ob). In all cases summation over y up to 20 was sufficient in order to achieve convergence in eq 2.
Results and Discussion Using eqs 1-5 we have calculated the theoretical spectra for the electron solvated in n-propane, 3-methylpentane, 2-methylbutane, 2,2-dimethylbutane, 2,3-dimethylbutane, and 2,2,3-trimethylbutane. The absorption bands were calculated at low temperatures. The reason for that is that the IR spectra in hydrocarbons are time dependent and only at low temperatures can they be regarded as quasi-stationary and treated in the framework of the linear response theory which applies to q u i librium processes. Indeed, a typical decay time is lo3 s at 4 K and 106-10-3 s at 77 K. Taking into account that time scale for the averaging over the vibrational states of the q mode coupled to the solvated electron is much shorter, the time evolution of the IR spectra at low temperatures can be treated as a quasi-stationary process. However, an interesting result obtained from our calculations performed so far is the fact that linear response theory applied even at higher temperatures, where the evolution of the band is much faster, is able to predict remarkably well the major portion of the nonequilibrium response. Our theory of course is not able to predict the time evolution of the peak position of the band, but only the band shape at a given time. We have calculated the spectra of the excess electron in hydrocarbons for the coupling with each of the normal modes for a given alkane. The normal modes of alkanes can be divided into the following group^:^^^^ (a) CH3 and CH2 symmetric and asymmetric stretching (2962-2853 cm-I (0.367-0.353 ev)); (b) HCH bending (1475-1274 cm-I (0.183-0.158 ev)); (c) C-C stretching, CH2 wagging, twisting, and rocking (1416-763 cm-'(0.175-0.946 ev)); (d) CH2 rocking (776-728 cm-' (0.096-0.090 ev)); (e) CCC bending (444-292 cm-'(0.0554.036 ev)); and (f) torsional modes (220-80 cm-' (0.027-0.009 ev)). We have compared the theoretical spectra with the experimental ones. We have found that both for n - p r ~ p a n eand ~ ~ 3-methylpentane only the coupling between the excess electron and the vibrational modes belonging to the groups d and e (CH2 rocking and CCC skeleton bending) give reasonable bandwidths and band shapes. The couplings with the modes having higher frequencies give unreasonably broad bands. In Figure 1 the theoretical spectra are presented for the coupling with the CCC bending mode (375 cm-'(0.046 eV) and 323 cm-' (0.040 eV) for n-propane-hs and -ds3s,respectively). The isotope effect calculated as the ratio of t for deuterated and protiated species is 1.24 for the coupling with the CCC bending and 1.7 for the coupling with the CH2 rocking mode, respe~tively,~~ indicating that the isotope effect on the extinction coefficient predicted from our model is similar to that reported for the isotope
Abramczyk et al.
9676 The Journal of Physical Chemistry, Vol. 96, No. 24, 1992
OS
VI
Figure 1. Absorption profiles of the electron solvated in n-propane-d8at theory. (ob) = 0.84 eV, I' = 0.016 eV, 91 K: (-) experiment;10 a0 = -2.415, n = 1.2898, a = 0.0, b = 0.0, 1 - woo = 375 cm-' (0.046 eV), 2 - woo = 323 cm-I (0.04 eV) ( d e ~ t e r a t e d ) . ~ ~ (-0)
effect on the electron yield G(e;) (1.3-1.6) in hydrocarbons.6 The absorption bands of 3-methylpentane at 76 K 3 ps and 380 s after a pulse are observed at 0.62 and 0.73 eV, respectively.* The spectra at 380 s are very similar to those determined after @To y-irradiation.* In Figure 2 we have shown the theoretical and experimental spectra6 for 3-methylpentane at 71 K. The theoretical spectra have been calculated for the coupling between the excess electron and the CCC bending (445 cm-I (0.055 eV)) (Figure 2A, curve 1) and the CH2rocking mode (776 cm-'(0.096 eV)) (Figure 2B, curve 1). The bandwidths calculated from the theoretical profiles in Figure 2, A and B, are 0.46 eV for the coupling with the CCC bending mode and 0.68 eV for the coupling with CH2rocking mode. It means that the coupling with the CCC bending gives better agreement with experiment (0.46 eV6) but none of them reproduce the characteristic long asymmetric tail on the high-frequency side. Assuming harmonic approximation, which gives the CCC bending mode at 314 cm-l (0.038 eV) and the CH2 rocking mode at 550 cm-' (0.068 eV) for deuterated 3-methylpentane, we have calculated the isotope effect on the absorption band (Figure 2A,B, curves 2). The deuterium substitution causes the blue shift by 0.09 eV, which is higher than the experimental one (0.01 eV6). The bandwidths for the coupling with the CCC bending modes at 445 and 314 cm-l (0.055 and 0.039 eV) are 0.46 and 0.4 eV for protiated and deuterated spectra, respectively, which are in excellent agreement with experiment.6 The isotope effect in 3-methylpentanecalculated from the theo-
retical profiles at 0.79 eV is given by 1.29 for the coupling with CCC bending and 1.53 for the coupling with the CH2rocking (see Figure 2, A and B). This value is in very good agreement with those reported for the isotope effect on the G(q-) by Wang and Willard! It means that all changes in Gc upon isotope substitution interpreted by Wang and Willard as the higher values of G(e,-) in the deuterated than the protiated matrices may reflect rather higher extinction coefficient c, not G. According to our model, these changes are due to the change8 in the zero pint of Vibrational energies of the matrix upon deuteration. Another important aspect of the electron stabilization in hydrocarbons is the effect of molecular structure of matrices on the optical spectra. There were reported6.9J1striking differences between compounds of very similar structure (e.g., 2-methylpentane, 3-methylpentane, and 3-ethylpentane). These dif'ferences were attributed by Willard et al.6*99"to differences in the probability of tunneling of the electrons to their geminate cations and other cations or radicals in protiated and deuterated matrices. However, in the same way as with deuterated and protiated ghsm, it is hard to understand why there is no correlation between the electron trapping yield and the density, viscosity, mobilities, and the maximum of the absorption spectra. Our model seems able to explain all of the spectroscopic characteristics on the basis of the vibrational properties of hydrocarbon matrices. We have shown that the absorption bands of the electron in n-propane and 3-methylpentane are reproduced best for the coupling with the CCC bending modes. It is w e l l - k n ~ w n ~that * ~ ~these modes are especially sensitive toward methyl and ethyl substitution. For example, the frequency of the CCC bending mode at 375 cm-I (0.046 eV) in n-propane is replaced by 367 cm-' (0.045 eV) and 433 cm-' (0.054 eV) and is assigned to the symmetric and asymmetric CCC bending upon methyl substitution in 2methylpr~pane.~~ In Figure 3 we have shown the theoretical spectra for 2-methylbutane, 2,2-dimethylbutane, 2,2,3-trimethylbutane, and 2,3-dimethylbutane assuming the coupling with each of the CCC bending and CH2rocking modes. They illustrate very well how strongly the absorption of the solvated electron depends on the vibrational frequencies upon methyl substitution.
Conclusions Our model seems to generate the band shape of the absorption spectra of the solvated electron in n-propane and 3-methylpentane with fair accuracy, indicating that the coupling between the excess electron and the intramolecular modes of the solvent molecules may play a dominant role in shaping of the absorption spectrum Ob
0.5
A) 0.5
04
\
,
0,lI
0.3 >
*
k w
c_
50.3
m z
x
0
i
,0.2
U
c v
2
a
L0
O O2 0.1 0.1 I
I
I~VI
Figure 2. Absorption profiles of the electron solvated in 3-methylpentane at 71 K: (-) experiment;6(--) theory. (A) (wb) = 1.06 eV, r = 0.016 eV, a. = -2.425, n = 1.3765, a = 0.0, b = 0.0, 1 - w o o = 444 cm-' (0.055 eV), 2 - ww = 314 cm-' (0.038 eV) (deuterated). (B) (a;)= 1.23 eV, r = 0.016 eV, a0 = -2.47, n = 1.3765, a = 0.0, b = 0.0, 1 - woo = 776 cm-' (0.096 eV), 2 - ww = 550 cm-' (0.068 eV) (deuterated).
The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 9611
Absorption Spectra of the Solvated Electron
of the deuteration on the extinction coefficient and similar to that reported for to the electron yield G(e;). If the Gt values reflect changes in t and not in G,it would be necessary to verify the conclusions for the electron solvated in hydrocarbons that have been made to explain the striking differences in the trapping yields G(e;) upon changes of molecular structure or deuteration. It has been suggested that the changes of G(e
