A semi-continuum model for the solvated electron in methanol

cHEMICAL PHYSICS L~.'-~-q'TERS. ,

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A sEMI,CONTINUUM

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FOR

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SOLVATED

ELECTRON

IN METHANOL

Chemistryi F a c u l t y o:Eagineer¢ng, N a g o y a V n ¢ v e m t y , Nagoya, Japan

":"" ' " D.-FI'FENG and L.KEVAN * : Deper,'ment o f Chemistry, Wayne S t a t e University, Detroit, M i c h i g a n 4 8 2 0 2 , U S A

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MODEL

I S e P t e m b ~ 197]~

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Received 28May 1 9 7 i , . i

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' The semi-continuum'model for solvated elect~Ons has been a p p l i . - d t o m e t h a n o l at 300 K. The configura~onaI stability Of the ground state was established and various physical pr0perties of ~ e solvated electron have been calcuiated and compared w i t h - e x p e r i m e n t . . ,,

.,, .:, l a vre,

ous-papers i t , 2 1 we de

1opea a. semi-c0n-:

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t i n u u m m o d e l fo r t h e h y d r a t e d e l e c t r o n . In this w o r k ' " w ~ have a p p l i e d t h e m o d e l t o the s o l v a t e d e l e c t r o n i n > m e t h a n o l a n d have c a l c u l a t e d i t s p h Y s i c a l p r o p e r t i e s i a t 3 0 0 ~ K - ' T h e m o d e l a n d the m e t h o d o f e a l e u i a t i o n ' are e s s e n t i a l l y fl~e s a m e as t h o s e d e s c r i b e d in t h e p r e . v i o u s p a p e r [ 2 ] , T h e p h y s i c a l c o n s t a n t s used in t h e ' p r e s e n t c a i c u h t i o n s are as f o l l o w s : t h e p e r m a n e n t dip o l e m o m e n t o f t h e m e t h a n o l m o l e c u l e = 1:70 d e b y e ; .. t h e i s o t r o p i e p o l a r i z a b i l i t y = 3.23 A 3 , t h e o p t i c a l die l e c t r i c Constant o f m e t h a n 0 1 = 1.77. t h e static, dielee• t r i c c o n s t a n t = 33, a n d t h e s u r f a c e e n e r g y o f m e t h a n o l " .Imf u n i t a r e a = 2 2 . 6 erg e r a - 2 . T h e radius, r s , o f t h e m e t h a n o l m o l e c u l e i n t h e first s o l v a t i o n shell w a s .,

2= .N - ' 4 ~ \ ~ \ .

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' - : T h e r e s u l t s o f t i l e c a l c u l a t i o n s are g i v e n i n fig. 1 a n d : . ,. in t a b l e s . l a n d 2-1The C o n f i g u r a t i o n a l ~ a b i l i t y : OI7t h e . . . . . : " g r o u n d . s t a t e . o f the s o l v a t e d e l e c t r o n in m e t h a n o l w0s " i e s t a b l i ~ e d a t a f ' m i t e l c a v i t y r a d i u s / a s s h o w n in fig. I, . " . " b y . a p p l y i n g t h e Variatior, al p r o c e d u r e t o the t o t a l e n 2' - . . . . )- • " . . e r g y ( e l e c t r o n i c p l u s m e d i u m r e a r r a n g e m e n t energY) " ' ' - ' . , ". ' Of t h e s y S t e m ; A l l o f tt~e p h y s i c a l q u a n t i t i e z, e x c e p t . , , . , " " f o r h v, . ¢ . l i s t e d i r ~ L h e t a b l e s a r ¢ t h g s e c a l c u l a t e d f o r the: ~.,-" " " " ' "i "":: ' " ) "'? :" " i ................ : i - ~ : ....... c a t t y r a d i tuostaatl gwr ohui cnhd - the stateenergyis'. :" -, .o... .t . .. z . . . . 3 o 4 . .s.. ..' ,, s . ... • b e r o f t h e '.s o l v e n t m o l e . c u l e s ' ....rda.. ~aminimum~Nisthenum . . . . . .. .... ., -" -:, ,.. , . ;., , . : . . .. . . . : i n . t l a e f ~ s t s 0 I v a t t o n s h e l l , V 0 t h e e n e r g y o f t h e q u a s i - ~ ": " : , : , ",. :: . . , : . . " , ' : . "!,., i . : i ": . ", . : . . ~... : " ' • . . . :" ~ .: ' , : , . ~. " F i g . l:Cord~tgurationcoozdh'vatediag r a m f o t Lh¢ solvated", ' . ' ] - ) . . e : - o n G , ; . . o e n h e i m F e I I o w i970~-71 , . , .:... , . • . electronmmothanolat300 KforVoTl,O,¢ ... , , .... "

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,.Volume 10, nwnber 5

CHZMICAL PHYSICS LETTERS :

VO (eW

hv(eV)’

4

2.0 1.5 1.0 0.5 0.0 -0.5

2.51 2.28 2.03 1.77 1.51 1.25

0.85 0.83 0.83 0.85 0.89

20 1.5 1.0 0.5 0.0 -0.5

2.06 1.98 1.82 1.64 1.42 1.21

0.99 0.97 0.95 0.92 0.90 0.90

6

Charge distribution

-f

of the solvated~electron

3.44 3.07 2.72 2.37 2.05 1.76

1.12 1.12 1.11 1.08 0.99 0.84

1.67 1.20 1.34 1.49 1.65 1.84

I.1

0.956

1.1 1.1 1.L 1.1

0.955 0.953 0.951 0.947

0.29 0.25 0.22 0.19 0.15 0.12

3.64 3.26 2.87 2.50 2.14 1.82

1.20 1.18 1.14 1.07 0.99 0.84

I.10 1.20 1.31 1.43 l-56 1.71

1.7 1.7 1.7 I.7 1.7 1.7

0.938 0.938 0.937 0.935 0.932 0.930

in methanol

Table 2 at 300°K at radii corresponding of the ground state

to the co@urational

Cl,

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2.0 1.5 1.0 0.5 0.0 -0.5

0.255 0.241 0.226 0.208 0.187 0.161

0.752 0.735 0.713 0.686 0.650 0.601

0.943 0.936 0.926 0.912 0.892 0.861

0.027 0.019 0.014 0.010 0.007 0.005

2.0 1.5 1.0 0.5 0.0 -0.5

0.431 0.403 0.384 0.362 0.336 0.293

0.806 0.789 0.771 0.749 0.720 0.679

0.948 0.942 0.933 0.922 0.907 0.884

0.153 0.117 0.091 0.065 0.044 0.027

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Cl,

transition energy, f

the oscillator strength for the ls+2p transition, A the half-width of the absorption line arising from the

1~2~ transition,1 ,tie photoconductivity threshold energy, M’the heat of solution of the election, rt the void radius,
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czp

0.324 0.260

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0.165 0.133 0.108

0.694 0.6L.5 0.536 0.465 0.402 0.347

0.568 0.500 0.431 0.349 0.270 0.199

0.852 0.806 0.748 0.664 0.568 0.468

,0.207

(a) O@cal n-amitin ew. transition energy increases with best agreement

Cq,

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The c=&uIated I s*Zp increasing Vo. The

the calculated

value and the

observed transition energy, I.97 eV (3,4], at the absorption maximum is found forN = 4 an< V. = I .O eVorN=6and Vo= I.5eV. For hrger values of N, higher values of _Vo are required to fit the observed transition energy. Althou& quantitative estimates of VOhave not been.made, large v&es of V. are unlikely because ,there ia a~c&eMion effect between the kinetic energy and the electronic polarization energy of -. the quasi-free electron [S] . For sohated electrons in water [2] ax&in ainmonia [5] the’expk&ental op-

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minimum energy

Excited state

.C1,(r,o)

rd +‘rs.

0.957

0.21 0.17 0.14 0.11 0.09

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state, hv the Is++

I.1

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N

free electron

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Ground state

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I NW

A@V) 0.89

1971

Table 1. of the solvated electron in methanol at 300°K

Calculated properties N

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Volume 10, number 5

1 September 1971

CHEMICALPHYSICSLET.IE&

ticaltransition-energy is severaltenthsof an eVlower Thaithe calculatedvaluesfor Vi> 0. Tha energyhv,

(g)Charge&stributim Theamount of electron ctige enclosedwithin a specifiedradiusimeases with

is much less than hv and the 2p+ 1s transition energy is about 1.1 eV. However, since the dielectric relaxation time of liquid methanol is very short the medium will probably relax to yield the relaxed excited state before such a transition occurs {2]. (b) Oscillatorstrength. The calculated oscillator strength for N = 4 is in reasonable agreement with the experimental value, 0.78 [3]. The calculated values for N = 6 are somewhat higher. (c)Absorption line-width.The calculated half-width of the absorption line arising from the ls+2p transition is about 0.2 eV and much narrower than the observed band-width, 1.29 eV [4] . As can be seen from table 1, the difference in the ground-state energy (the negative of AH) between coordination number N = 4 and 6 is of the same order of magnitude as the thermal energy kT for V0 = 1.0 to 1.5 eV. Thus it is probable that both the electron cavities withN = 4 and 5 coexist. This situation would contribute to broadening of the absorption line. Taking this factor into account, the linewidth amountsto about 0.4 eV, which is still much less than the observed band-width. Further theoretical study is required to explain this situation. Transitions to higher excited states might contribute to broadening of the absorption band. (d) Photoconductihy threshold energy. The calc&ted threshold energy for photocpnductivity increases with increasing V, and is significantly higher than’ the 1s+2p transition energy. The difference between I and trv is quite simiiar to that for the solvated electron in water [2]. No experiment.aI data exist for I for electrons in liquid methanol. Recently, weak photoconductivity has been observed for trapped electrons in methanol at 77’K [6]. This observation implies that I- hvGO.1 eV;at least in the frozen mat:ix; (e) Heot of solution_The c&ulated heat of solution decreases with increasing Vo. Taking V. = 1 .OeV, we obtain the heat of solution of the electron in methanol to be about 1.3 eV. There are no experimental dataavaiiable for comparison with &thisprediction. (0 Void radius The void radius is insensitive to V,, anciitis 1.1.AforN=$and 1.7AforN=6.These values are greater than those calculated for the so&ted electron in water by about.0.5 A:

increasing Vo, and it is greater for N= 6 than for N=4. A major fraction of the eIectron population is found within radiusR, i.e., in the first salvation shell for the ground state; while in the excited state about half of the electron population is found in the first solvation shell (N= 4) and only a very small fraction of the electron population is found in the void. Thus, it can be said thar upon the optical excitation of the solvated electron, the electron orbital becomes much more diffuse than that in the ground state. These charge distributions are quite similar io those for the solvated electron in water except that in water the charge within r”V is much less. In conclusion, the semi-continuum model semiquantitatively accounts-for properties of the solvated electron in methanol except for the band-width of the absorption spectrum. In earlier model calculations [l] a freely adjustable parameter, cavity radius, was involved. However, in the present calculation, although V. is an adjustable parameter it is adjustable only over a limited range near v. = 0. In order to make a completely nonempirica! calculation of the solvated electron feasible, it is essential to develop a method to evaluate V. reliably for polar solvents. This research was supported by the Air Force Office of Scientific Research under Grant No. AFOSR-701852, by the U.S. Atomic Energy Commission under Contract No. AT (1 l-1)-2086 and by the Computing Centers at Wayne State University and Kansas University. L.K. thanks the University of Utah Chemistry Department for their hospitality and services.

References 111 K. Fueki, D.-F. Feng and L. Kevan, J. Phys. Chem. 74 (1970) 1976. 121 K. Fueki, D.-F. Feng. L. Kevan and R.E. Christoffersen, to be published. [3] M.C. Saver jr., S. Arai and L.M. Do&an, J. Chem. pfiys. 42 (1965) 708. 141, S. Arni and MC. Sauer Jr., J. Chem. Phys. 44 (1966) 2297, ; [S] D.A. CopelGd, N&. Kestner and J..Jortner; J. Chem. “ys. 53 (1970) 1189; [6] T. Huang, 1. Eisk and L. Kevan; to be published.

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