Absorption spectra of a solvated electron in ethers

Chemical Physics North-Holland

I57 ( 199 1) 373-379

Absorption spectra of a solvated electron in ethers Halina Abramczyk and Jerzy Kroh Technical University, Institute ofApplied Radiation Chemistty, 93-590 tbdi. Wrdblewskiego 15, Poland Received

8 May 199 1

The absorption spectra of a solvated electron in diethyl ether and tetrahydrofuran at 211 K were calculated in terms of the theory presented in our previous paper and compared with the experimental ones. We have considered the influence of the coupling of the electron with intramolecular vibrations of the solvent, the effect of fluctuation of the potential energy well and the influence of the bath.

1.

Introduction

The nature of a solvated electron in ethers has been the subject of many papers devoted both to pure liquids [l-9] and binary solvent systems [lo-131. Ethers are weakly polar liquids. In contrast to the most frequently studied matrices stabilizing electrons like water, alcohols, amines and ammonia they do not form H-bonds in pure liquids. They are however very good proton acceptors in H-bond complexes with water, alcohols, HCl and other compounds [ 14- 19 1. The optical spectra of the excess electron in ethers [ l-91 appear in the infrared (about 0.6 eV [ 9]), i.e. at much lower energies than in ammonia (0.89 eV [ 20]), amines (0.99 eV in methylamine [ 20]), water (1.73eV [21]),alcohols(1.95eVinmethanol [22]) at the same temperature (200 K). This peak position is close to that in trimethylamine (0.77 eV at 77 K [ 231) which is also a weakly polar matrix and does not form H-bonds. There is no correlation between the peak position and the bandwidth. The bandwidths in alcohols (about 1.3 eV [ 221) and in water (0.843 eV [ 221) are much larger than those in ethers (0.5 eV [ 91). The bandwidths in amines are a little broader (0.58 eV in methylamine [ 201) while the band in ammonia is narrower (0.398 eV [ 201). The band profiles of the electron solvated in ethers have a characteristic asymmetric shape with a long tail on the high-energy side. Recently, we have proposed a theoretical model [ 11, in which the excess electron motion occurs in a 0301-0104/91/%

03.50 0 1991 Elsevier Science Publishers

Born-Oppenheimer energy potential well and is coupled to the intramolecular vibrational modes of the solvent. The purpose of the present paper is to examine to what extent the experimental bandshapes of the solvated electron in ethers can be understood in terms of our theory. The experimental data for ethers compared with the theory should provide a broader test of the validity of our theoretical model and give some guidance with respect to the importance of the molecular structure of the matrix and the molecular structure of the cavity (or anion complex) in the selectivity of the vibron-electron coupling.

2. Numerical calculations We have calculated the absorption profile of the solvated electron from the linear response theory e(w)=

X

$$[I-

s

exp(--Bho)l

(M+(t)M(O))

e-‘“‘dt,

(1)

-co

where t is the extinction coefficient. The expression for the dipole moment correlation function (M+ ( 1) M( 0) ) has been developed in our previous paper [ 11. The theoretical development considers the contribution to the absorption band profile from the vibrational coupling, tunneling and inhomogenous

B.V. All rights reserved.

374

H. Abramczyk. J. Kroh /Absorption spectra of a solvated electron in ethers

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). The vibrational coupling is treated in the framework of the strong coupling limit theory, where the displacement operator A obtained with the aid of the canonical transformation creates the dynamic effects of the vibrational motion on the electron between its Born-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 dumping parameter ZY Despite this indirect coupling of the solvated electron to the bath (through the q mode) we can expect a direct coupling (inhomogenous broadening). The direct coupling to the bath is responsible for the fluctuations of the potential well (single or double potential well) and the electronic energy levels. For equilibrium averaging over these fluctuations we have used the cumulant expansion procedure. The fluctuations of the energy levels (electronic dephasing, characterized by a and r, in eq. (5) and the fluctuactions of the energy potential well (b and ?b in eq. (5) ) are regarded as a random variable in the two-level system governed by the stochastic Liouville equation. The origin of the fluctuations of the potential well is 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) the equilibrium geometry of the potential well in the excited electronic state is displaced linearly with respect to the normal vibrational coordinate. Generally, these assumptions are not justified. 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 this model we have shown [ 1] that the dipole correlation function (M+(t)M(O)) isgivenby

where A kc4 +(Y! ( _ 1

)k+j+’

’

2'(a-k)!(a-j)!(y-a+k)!(y-a+j)!

(3) (4)

X

exp [{

(y-cr)+2p+2j}

xexp[ -2nl(rz(exp( x exp[ --b’{r2(exp(

3

- t)-

- ;)-

1 l)+rJ}]

l)+rbl>].

(5)

The theoretical details can be found elsewhere [ 11. Here we explain the meaning of the parameters which have been used in eqs. ( 1 )-( 5) for the calculations of the theoretical spectra. The correlation function (M+ ( t ) M( 0) ) depends on the following molecular properties: , woo, ao, C 7@, 7b56, a, b,f,, (wmr>. We have shown [ 1,241 that only cxoand b are adjustable parameters in our model. All of the other parameters can be obtained from ab initio calculations ( (oh ), fm, ( comr) ), IR and Raman measurements (%o, 7,) or reasonably estimated (a, r, rb). The phase angular speed (WA) 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 0.93 eV and 1.091 eV for diethyl ether and tetrahydrofuran, respectively. The vibrational frequencies of the intramolecular modes of the solvent ( ooo) were taken from IR and Raman measurements [ 2528]. The coupling constant CY~,which characterizes

H. Abram&,

375

J. Kroh /Absorption spectra of a solvated electron in ethers

the strength of the coupling of the excess electron with the vibrational modes woo is treated as a fitting parameter and is chosen to reproduce the extinction coefftcients at maximum (3.2 x lo4 mol- ’ P cm-’ and 1.9x104mol-‘Qcm-‘at211 Kaswasreportedfor diethyl ether and tetrahydrofuran [ 91, respectively). The dumping parameter rreflects the strength of the coupling between the vibrational mode ooo and the thermal bath (translational, reorientational degrees of freedom, low-frequency intermolecular vibrational modes of the solvent). We have taken ras 70 or 120 cm- ’ which correspond to the typical torsional mode frequencies of ether matrices. We have assumed that the electron motion occurs in a single minimum potential well, the influence of the direct coupling to the bath (inhomogeneous broadening) is negligible and the Born-Oppenheimer potential well is static with time. So, this means that we take a, b and (oP) equal to zero. The transition matrix elementf, was taken in the harmonic approximation as J = h/2m ( 0: ) . In all cases summation over y up to 20 was sufficient in order to achieve convergence in eq. (2).

bands were obtained from a least-squares lit of a Lorentzian on the high-energy side and a Gaussian on the low one and the temperature dependence of the absorption maximum and the bandwidth [ 91. The bands are featureless, asymmetric with a long tail extending to high frequencies. The asymmetry factor W,/ W, characterizing the ratio of the bandwidth in the blue and red region is equal to 1.55 for both ethers. In table 1 we have compared the peak positions, W,,, bandwidths W, ,2, dipole moments p and dielectric constants t for ethers, ethanol, water, ammonia, methylamine and trimethylamine. Four spectroscopically distinguishable rotamers of diethyl ether are possible (fig. 1 ), but practically only TT and some TG are energetically favourable [ 261. The enthalpy difference between TT and TG was determined as AH= 1.1 kcal/mol and the assignment of IR and Ra-

H

HH (1,/

H

H

TT (Czv)

3. Results We have calculated the theoretical spectra at 211 K for diethyl ether and tetrahydrofuran and compared with experiment. The maxima of the absorption bands of the electron solvated in diethyl ether and tetrahydrofuran are found at 0.62 eV and 0.69 eV at 211 K, respectively [ 91. The experimental

H

‘i /

GG

TGfC,

(C2)

Gd

1

fCs)

Fig. I. The conformers of diethyl ether,

Table I The absorption band peak positions IV,_, bandwidths W,,2, dipole moments p and dielectric constants c

W, diethyl ether tetrahydrofuran ethanol water ammonia methylamine trimethylamine ” Ref. [ 91, at 2 1 I K. “Ref. [22],at300K.

(ev)

0.62 .’ 0.69 .’ 1.78 =’ 1.72 =’ 0.89 C” 0.99 Ii’ 0.73 e’

Wl12(ev)

P(D) b’

0.52 .’ 0.52 l’ 1.46 =’ 0.85 c’ 0.39 d’ 0.58 .” 0.59 =’

1.15 1.63 1.69 1.85 1.47 1.31 0.612

t b’

jllc

4.33

0.265

24.3 78.5 16.9 9.4 2.44

0.069 0.023 0.087 0.139 0.25 1

b’ At 300 K, CRC Handbook of Chemistry and Physics, 56th edition 1975-1976. d’Ref. [20],at203K. “Ref. [23],at77K.

H. Abramczyk, J. Kroh /Absorption spectra of a solvated electron in ethers

376

man spectra strongly supports only TT and TG rotamet-s [ 26 1. The vibrational modes of diethyl ether can be divided into the following groups: ( 1) The C-H stretching vibrations (3000-l 700 cm-‘) (symmetric and asymmetric C-H stretch in methyl CH3-C and methylene C-CH2=0 groups). (2 ) The C-H deformation vibrations ( 1700- 1200 cm-’ ) (symmetric and asymmetric HCH bend, methylene bend, methylene wag, methylene twist, methyl rock). ( 3 ) The skeletal vibrations ( 1200- 120 cm- ’ ) (CC stretch, C-O stretch, OCC bend, COC bend and C-O torsion ) . The best agreement with experiment both for the bandwidth and asymmetry ratio is obtained for the vibrational mode at 440 cm-’ (fig. 2, curve 1). The small contribution from inhomogeneous broadening due to various types of traps (b= 0. 124 eV) (fig. 2, curve 1) improves the agreement with experiment. This mode represents the OCC and COC bend and is active both in IR and Raman spectroscopy. The intensity of the Raman band is very strong. Taking into account that the physical nature of the electron-vibron coupling depends on the changes of the electrostatic field produced by the vibrations, i.e. the derivative of the dipole moment a@q (IR) or of the polarizability tensor aa/aq (Raman) with respect to

10

1.5

teV

20

I

Fig. 2. The absorption profiles of an electron solvated in diethyl etherat21lK.ptheory,---experiment.(l)a=O,b=0.124 eV, LY,,=-2.435, g= 1 x lo-l5 s; (2) a=O, b=O, CY,,=-2.43, (wb)=0.929eV,r=120cm-‘,woo=440cm-’,n=1.3526.

the normal coordinate we can expect that the mode at 440 cm- ’ produces the strong electron-vibron coupling. The vibration modes at 503,45 1,440, 376 and 3 18 cm - ’ describe the OCC bend and COC bend for TT (451 and 440 cm-‘) and TG (503,376 and 3 18 cm- ’ ) conformers. The bandwidths for the coupling with these modes are similar and change from 0.533 to 0.44 eV. It is very likely that the experimental band is the superposition of the bands coming from the coupling with these modes, but the bands at 503, 376 and 318 cm-’ are much weaker both in IR and Raman spectroscopy, what means that the contribution from these couplings is expected to be small. The best agreement with experiment for tetrahydrofuran was achieved for the coupling of the electron with the mode at 596 cm-’ (fig. 3). It is worth emphasizing that like in diethyl ether the mode 596 cm- ’ describing the in-plane ring bending also modifies the COC and OCC angles and as a consequence the dipole moment of the molecule. Inhomogeneous broadening due to the distribution of solvent environments seems to play a minor role for the electron solvated in tetrahydrofuran (a = b= 0 in our model ) . In figs. 4-7 we have shown the best fit of the spectrum of the electron solvated in diethyl ether calculated from eqs. ( 1 )-( 5) for the coupling with some of the other vibrational modes belonging to the groups

10

1.5

20

teV1

Fig. 3. The absorption profiles of an electron solvated in tetrahydrofuran at 2 11 K. theory, - - - experiment. a = 0, b = 0, (wb)=l.O91 eV, r=70 cm-‘, q,=-2.4, w,=596 cm-‘, n= 1.4050.

377

H. Abramczyk, J. Kroh /Absorption spectra of a solvated electron in ethers

\

I

\

,

‘. -.__

- -__

00 0.5

1.0

1.5

2.0

(eV) Fig. 4. The absorption profiles of an electron solvated in diethyl ether at 21 I K. theory, - - - experiment. a=O, b=O, (wA>=1.798eV, %=28OOcm-‘,cYo=-2.63, n= 1.3526; (1) r=70cm-‘, (2) r= 120cm-I.

Fig. 6. The absorption profiles of an electron solvated in diethyl ether at 21 I K. ~ theory, - - - experiment. a=O, b=O, (w&)=0.806 eV, r=120 cm-‘, CQ=-2.402; (I) mm=245 cm-‘, (2) 0~~231 cm-‘.

I

: I : :

1’ I

I \ \ \ \ \ \

\ \

:

1

I ,

I

/

0

0.5

z 1.0

1.S

2.0

leV)

>

1.0

15

20

(eV)

Fig. 5. The absorption profiles of an electron solvated in diethyl ether at 211 K. theory, - - - experiment. a=O, b=O, (I&) = 1.302 eV, r=70 cm-‘, aO= -2.51, ~=1200 cm-‘, n= 1.3526.

Fig. 7. The absorption profiles of an electron solvated in diethyl ether at 211 K. ~ theory, - - - experiment. a=O, b=O, (wb) ~0.744 eV, r=70 cm-‘, a,,= -2.373, ooo= 120cm-I.

1,2 and 3. We can see (fig. 4) that the experimental band profile is very poorly reproduced by the C-H stretch modes of the methyl and methylene groups giving a bandwidth much broader than the experimental one with substructure features, which are not observed experimentally [ 9 1. The band profile calculated for the coupling of the solvated electron with

the C-H deformation vibrations belonging to group 2 are represented in our calculations by the vibrational frequency w, = 1200 cm- ’ (fig. 5 ) . The same as for the C-H stretching vibrations the theoretical band profile does not reproduce the experimental spectrum. From these results we can conclude that

378

H. Abramczyk, J. Kroh /Absorption spectra ofa solvated electron in ethers

methylene or methyl groups in ethers are neither the sites of preferential attachment (in terms of the diffuse solvent anionic complex [ 20,28-3 1 ] ) nor the preferential groups to form the trap around the excess electron (in terms of the cavity models [ 32-341). The band profiles of the solvated electron coupled to the skeletal vibrations such as at 935 and 916 cm-’ describing the C-C and C-O stretch for both TT and TG conformers or the vibrational modes at 846 and 837 cm-’ describing also mainly the C-O stretch do not reproduce the experimental band shape. The same conclusions we have obtained for the coupling with the modes at 245 and 23 1 cm- ’ describing the methyl torsion (fig. 6 ) and with the mode 120 cm- ’ describing the C-O torsion (fig. 7 ). In both cases the bands are too narrow in comparison with the experiment. 4. Conclusions We have shown that our theoretical model [ 1 ] generates the absorption band profile of the solvated electron in diethyl ether and in tetrahydrofuran with fair accuracy. The bandwidths arise from the combination of the vibronic bands due to the coupling of the electron with the C-O-C and C-C-O bending modes laying at 440 cm- ’ for diethyl ether and at 596 cm- ’ for tetrahydrofuran. There is a similarity in this respect between ethers and alcohols where the excess electron is coupled with the bending mode of the COH group [24] lying at 1345 cm-’ and modifying the dipole moment of the molecule. However, due to the fact that the frequency of this mode is much higher than in ethers the vibrational coupling gives a bandwidth broader as much as three times in alcohols than in ethers. Alcohols seem to ignore the H-bond in creating the cavity [ 351 forming the dipole oriented trap with the dipole moments directed towards the electron localized at the center of the cavity. On the contrary, the other H-bond forming matrices like water, ammonia and amines seem to create H-bond oriented cavities with the OH or NH bonds oriented towards the center of the trap. Pure ethers do not form H-bonds and one can expect that the traps are dipole oriented. Our results support this expectation both for diethyl ether and for tetrahydrofuran because the bending modes at 440 and 596 cm-’ strongly modify the dipole moment of the molecules. The vibronic

bands are broadened by the coupling with the lowfrequency modes of the bath. The inhomogeneous broadening due to the distribution of solvent environments plays more important role in diethyl ether than in tetrahydrofuran.

Acknowledgement

This work was carried out within the project CPBP 01.19.

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H. Abramczyk, J. Kroh /Absorption spectra ofa solvated electron in ethers

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