Solvated Electron

14 Solvent Dimer Model for Solvated Electrons LIONEL RAFF and HERBERT A. POHL*

Downloaded by UNIV OF NEW ENGLAND on February 11, 2017 | http://pubs.acs.org Publication Date: January 1, 1965 | doi: 10.1021/ba-1965-0050.ch014

Department of Chemistry, Oklahoma State University, Stillwater, Okla.

A dimer model consisting of two solvent molecules with an attached electron is presented as a representation of the solvated electron system.

Simple molecular orbital theory, using

scaled H + wave functions, is employed to 2

evaluate the geometries,

binding

energies,

and electronic transition energies for several series of homologous compounds.

The results

predict a general decrease of optical transition energies and

binding

energies with dipole

moment of the solvent.

The calculated transi-

tion energies when compared with existing experimental data give the trend expected from chemical experience, but tend generally to overestimate the magnitudes.

Jhe experimental study of the physical and chemical properties of the ionic species known as the solvated electron goes back many years (3, 5, 8, 9, 10). It includes the interesting aspects of solutions of alkali metals i n liquid ammonia and amines (5) and of irradiated water (8). Considerable experimental data is available, but the nature of the binding of the electron i n the solvents is still controversial. T h e most widely discussed model is one suggested by L a n d a u (18) and by Ogg (21) i n which the electron is pictured as being i n a cavity bound by the polarization of the dielectric medium. Ogg computed the cavity radius and binding energy using a simplified model of an electron i n a " s p h e r i c a l " box. T h i s calculation was much refined b y Lipscomb (19) to include électrostriction, electronic polarization, and surface energy. K a p l a n and K i t t e l (17) proposed that binding of the electron took place i n a cavity orbital associated with many (ca.50) protons of the N H solvent molecules. Jortner (12, 13, 14, 15) proposed a model for very dilute solutions of the solvated electron, e J ., i n which the electron is removed from the metal S

s

lv

* Department of Physics 173

Hart; Solvated Electron Advances in Chemistry; American Chemical Society: Washington, DC, 1965.

SOLVATED ELECTRON


174

cation and trapped i n a cavity by polarization of the medium. T h i s concept of " a n electron trapped by digging its own hole" is that intro duced by L a n d a u (4) i n which the medium is regarded as a continuous dielectric. A totally different view was proposed by Becker, Lindquist, and Alder (2) who considered the electron to be trapped at a solvated cation site i n which the solvated metal cation has its valence electron localized on the protons of the ammonia molecules of the innermost solvation layer. T h i s species, with its valence electron i n a sort of enlarged R y d berg orbital, may then dissociate or undergo dimerization. We wish here to inquire if a molecular view of the basic e„iv species might be taken. W e shall postulate, as i n the cavity model, that the solvated electron is removed from the metal cation, but we shall assume instead a molecular model—one i n which several solvent molecules are bound, forming the basic unit. The

Model

W e have i n mind the following model: the energetic electron as first produced —e.g., on irradiating a solvent—is preliminarily trapped or localized in the neighborhood of the positive regions of the dipoles of the polar solvent with the emission of energy by photons, etc., characteristic of the trap depth. Massey (20) notes that electron attachment for H 0 begins at an electron beam energy of 0.1 e.v. and continues to increase in cross-section with increasing beam energy up to 2.0 e.v. T h e existence of the H 0 ~ ion is thus implied. Following this quasi-localization of the electron, rapid reorientation of a local solvent molecule occurs to form a deeper trap. These deeper traps can be viewed as analogs of the hydro gen molecule ion (dimeric traps). F r o m dielectric relaxation measure ments (4), it is known that the relaxation times of solvent dipoles (such as H 0 ) is of the order 10 " to 10 " sec., so that it seems likely that the conversion of monomeric traps to the dimeric type takes place on such a time scale. Further clustering of the solvent molecules near the deeper trap then occurs almost simultaneously causing a modest further increase of the trap depth. T h e scheme may be written as: 2

2

9

2

(Monomer) (Dimer) (Multimer) (Destruction)

1 0

e~ + S S" S" + S S ~ S ~ + x.S S ~S S^S*" other 2

2

lifetime « 1 0 ~ sec. lifetime « 1 0 < sec. lifetime « 10"« sec. to « 11

5to

2

Z

where S represents a solvent molecule and χ is a small integer. T h e longer term properties of the solvated electron depend essen tially, i n this view, on the character of the dimer and multimer. F o r simplicity in calculating the transition energy between the ground and first excited state we shall examine the dimer states alone, although the spectroscopic properties of the solvated dimer can be expected to be some-


14.

RAFF AND POHL

175

Solvent Dimer Model

what modified from this by the accumulation of other solvent molecules owing to ion-dipole and ion-induced dipole forces. Specifically, for the calculation we choose a dimer model consisting of opposed molecular dipoles bound by a delocalized electronic charge, having a center of gravity lying between the positive ends of the original molecular dipoles. F o r example, the simplest case, ( H F ) ~ , could be written diagramatically as: 2


F*~ -

H* .. . β " . . . H +

,

+

-

F * " or [F -

H.. .H - F]~

T h i s proposed structure can be alternately viewable as resembling that of a perturbed H ion—e.g., 2

+

F-...H +...F2

or one with a scaled nuclear charge, F ~...H ,

2

Î , +

- ...F'~. 1

Analogous structures are proposed for other members of the homologous series of the solvated electron family: Χ

[X—H... .Η—X]-

L

\

/

Y—Η.

\

/

/ \Η_

Y = 0 , S, Se

-

Ζ = Ν , Ρ, A s

Ζ—Η. ...Η—Ζ

\

H—W—H. L

J

.

Η"

.Η

H

Ι Γ 1-

...Η—Υ

"Η

/

H

= F , Cl, Br, I

H

/ .

W

H—W—H

= C., S i

\ H

J

Calculation A full quantum mechanical calculation of the simplest dimer ion is a formidible task (—e.g., [ F — H . . . H — F ] with 21 electrons). W e are forced to examine the matter using various approximations. Using a molecular orbital approach we can view the problem as one of five elec trons associated with the several atomic cores. F o r [ F — H . . . H — F ] ~ , the simplest M O could be built from 2 P , orbitals on F and Is orbitals on H , as in Table I. B y symmetry, the orbital coefficients for the two F atomic orbitals will be identical i n magnitude as will those of the two H atomic orbitals. T h e orbital coefficients, c , can thus be chosen to be positive definite i f the sign convention i n Table I is used. T h e M O ' s should be -

ft


SOLVATED ELECTRON

176

chosen orthogonal to each other and to all other M O ' s of the molecule ion. Table I.


MO Wave No. Func Energy of tion Level Nodes 0 1 Ψι 2 1 3 2 Ψζ 4 3 ΦΑ

LCAO MO Scheme for [F—H . . . H—F]~ No. of Elec trons 2 2 1 (1*)

Orbital Coefficient for AO's F Η Η F (2) (4) (3) (ί) +c 1 2 +c 1 2 + C„ Cj2 +C21 +C22 -a, — C»2 — C»2 +c„ — C42 +C41 + C42 -c«

Sym metry g

u

—

g

u

In a first-order approximation one may view the problem i n a oneelectron approximation, fixing attention on the fifth electron, that in the lowest " u p p e r " molecular orbital, ψ ; and then argue that on physical grounds one expects C 3 1 < C 3 2 — i . E . , this electron orbital will have the electron charge distribution gathered more strongly on the electropositive hydrogen rather than on the electronegative fluorine cores. A s a crude approximation we can set Ο < C i C 3 2 so that the problem then further reduces to that of a perturbed H + ion (Orthogonality of the M O ' s is then lost i n this very rough approximation). 3

2

2

3

2

2

2

T h e effective charge, Z . , at the screened protons can be estimated eff

as Zeff. = (/Abond/^bond) · |β| where Mbond dbond

\e\

= dipole moment of the Η — Χ , Η — Y , or H — Ζ bond = bond length = electronic charge, 4.80 X 10 ~ e.s.u. 10

A n excellent summary of the arguments for and against using dipole moments to estimate the effective charge i n sigma-bonded systems is give by D e l R e (6). T h e Hamiltonian chosen is that for an electron in an L C A O - M O of two scaled hydrogenic functions seeing a potential at the four nuclei of F — H . . . H — F of ZFI, Z , Z 3 , Z 4 , respectively. PreHrninary calcula tion for ZFI Z F = —1, Z H — ZHS +1 showed the effect on the transition energy, ΔΕ, to be affected i n a minor way (25%) b y including Z terms; hence, these were dropped. T h e calculation then reduced to that for an electron seeing a potential at two shielded positive nuclei. T h e ground state energy, E , for an electron in the field of two positive charges, Z and Z using the L C A O - M O approximation can be shown to be H 2

=

F

H

2

=

2

F

s

H 2

m

*· " " Τ + Ï T Â

{à

[Z{1

+

δ)

- D - d +

z v -

+

(1 + l/ZR)e-» \ R


(1)

14.

RAFF AND POHL

Solvent Dimer Model

It has a minimum (in a.u.) at àE /àR s

177

= 0.

(a.u. = atomic units)

T h i s requires ^

Z/R

Λ

- e-*'* [2Z +

+ 1/ZR*}} + A * | J ^ + e - * ' ( l + ZR) - e~^{l e

+

(2)

Δ = e~* [l + Ζ Λ + Z R /S]

where

R

and Downloaded by UNIV OF NEW ENGLAND on February 11, 2017 | http://pubs.acs.org Publication Date: January 1, 1965 | doi: 10.1021/ba-1965-0050.ch014

|J - ( 1 + Δ ) ] + Ζ » Λ β -

= 0 = (1 + Δ) |1

Δ

1

2

= -e~ Z R(l BR

(3)

2

+ Z)/3.0

2

(4)

T h i s can be solved numerically for R = R at given Z . Using Λ , one evaluates E and A F = E — E , the transition energy, where e

s

22

A

β

s

2

ΔΕ = Î T ^ A i l e - ^ d

+ 2 Λ ) + A[-l/ZR

+ e~*

Solvated Electron

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