Conduction state energy of excess electrons in condensed media

Publication Date: December 1975. ACS Legacy Archive. Cite this:J. Phys. Chem. 79, 26, 2866-2874. Note: In lieu of an abstract, this is the article's f...
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2866

S.Noda, L. Kevan, and K. Fueki

(23) J. G. Kirkwood, J. Chem. Phys., 4, 592 (1936). (24) W. Hauser, "Introduction to the Principles of Electromagnetism", Addison-Wesley, Reading, Mass., 1971, p 195. (25) Reference 24, p 145. (26) Reference 14, pp 225-226. (27) One of the reviewers pointed out that the local electric fleld produced by a point charge immersed in a continuous dielectric (without cavities) is simply 9/?. In this instance, the dipole screening function would be unity. The argument proceeds as follows. The local electric field is given by the field due to the point charge, namely, q/r?, plus the sum of the fields due to microscopic electric dipoles induced in the dielectric. Inasmuch as all the polarization in the dielectric is directed radially toward

the charge, its contribdion to the local field cancels out, making the local electric field simply 9/?. This argument can in fact be verified by replacing the fluid radial distribution function in eq 11 of ref 4 by unity and performing the indicated integration. A fluid characterized by a radial distribution function, which is unity everywhere, is a fluid in which the molecules may overlap. The physical impossibility of molecules overlapping is in some sense taken into account in the cavity model used in this paper. That is to say, the cavlty boundary surrounding the induced multipoles prevents the uniform dielectric representing the remaining molecules from approaching the multipoles too closely. To this extent, the repulsive aspect at the intermolecular force has been taken into account.

Conduction State Energy of Excess Electrons in Condensed Media. Liquid Methane, Ethane, and Argon and Glassy Matrices Shojl Noda, Larry Kevan,* Department of Chemistry, Wayne State University, Detroit, Michigan 48202

and Kenji Fueki Department of Synthetic Chemistry, Faculty of Engineering, Nagoya University, Nagoya, Japan (Received July 3, 1975)

The temperature dependence of the conduction electron energy V Oin liquid methane and ethane has been measured, and the theoretical model of Springett, Jortner, and Cohen (SJC) for the temperature dependence of Vo is shown to apply to liquid alkanes as well as rare gases. Criteria for electron localization in liquid rare gases, molecular gases, and alkanes are critically discussed. The SJC criterion for electron localization in liquid rare gases is shown not be extendable to liquid alkanes or to molecular gases. In alkanes we assume that transient localization arises from rotational and translational fluctuations leading to cavities of molecular size. Then consideration of the kinetic and electronic polarization energies with a cavity radius given by the Wigner-Seitz radius leads to a new localization criterion that seems generally applicable to alkane systems. Electrons in molecular gases probably move as molecular anions so neither of the above criteria apply to them. An empirical relation between the electron mobility and V Ois found that seems valid for all alkanes a t room temperature for both localized and quasi-free electron states. Values of VOfor polar and nonpolar glassy matrices at 77 K are derived from experimental data by two independent methods. It is found that V Obecomes more positive as the glassy matrix polarity decreases.

Introduction Excess electrons are excellent probes of the complex electron-molecule interactions found in condensed media. Electron mobilities and conduction state energy levels relative to vacuum are the major experimental quantities available to probe the weak interactions of excess electrons, while optical and magnetic resonance spectroscopy of excess electrons can probe stronger interactions leading to electron localization.' Rare gases constitute the simplest condensed media. There the electron mobilities and conduction state energies are fairly well understood theoretically2 although discrepancies do remain, notably in liquid heli~m.~ Condensed alkanes are much more complex and electron interactions in them are the focus of much current interest. Electron mobilities p have been measured in both and phases for a wide variety of alkanes, and conThe Journal ef Physical Chemistry, Vol. 79, No. 26, 1975

duction electron energies VOhave been reported for several higher molecular weight a l k a n e ~ . ~ The - I ~ basic question is to what degree the theories of V0I3and of p,14 which seem rather successful for rare gases, apply to condensed molecular media. Attempts have been made to extend these theories to liquid alkanes.15-17 It has also been suggested both experimentallygJs and t h e o r e t i ~ a l l ythat ~ ~ there ~ ~ ~ is a general correlation between p and Vo for rare gas and alkane liquids. In this work we report measurements of VOfor both liquid ethane and methane over a limited range of temperature to compare with the measured mobilities in these liquids and to test the generality of a correlation between ru and VO.The correlation appears to fail for methane.21 We present a simple successful model of the temperature dependence of V Oand discuss a new approach to the relation between p and VO.Finally, we consider photoionization

Conduction State Energy of Excess Electrons in Condensed Media data in a variety of glassy matrices and derive estimated values of V Ofor organic and aqueous glasses.

2887

TABLE I: Temperature Dependence of V , in Liquids T , Ka

Liquid

Experimental Section Methane (99.99% Matheson research grade), ethane (99.94% Phillips research grade), and argon (99.995% Matheson research grade) were purified by passing through activated silica gel (activated at 300°C in vacuo for 48 hr) and activated charcoal (activated at 2OOOC in vacuo for 48 hr) columns at 195 K. Methane and ethane were stored over activated silica gel at 77 K, and argon was stored over activated charcoal a t 77 K. Conduction state energies were measured by work function changes of zinc electrodes in vacuo compared to the liquid. Our apparatus was similar to that described by Holroyd and Alleng with modifications for low-temperature work. Light from an air-cooled Philips SP-9OOW mercury lamp passed through a Bausch and Lamb high-intensity uv monochromator and appropriate filters to eliminate scattered light and higher order diffractions and was reflected 90° by a mirror up through the unsilvered flat bottom of a quartz dewar onto the bottom quartz window of the photoelectric cell. The light passed through an electron collector electrode made of brass mesh to strike the zinc-coated emitting electrode. The electrodes were separated by a 0.5 mm Mylar spacer. The applied voltage was obtained from a Keithley 245B power supply, and the photocurrent was measured with a Keithley 610B electrometer. Temperatures were obtained with liquid argon or by cold nitrogen gas flow and were monitored with a calibrated digital thermocouple thermometer. The cell and vacuum line were flushed with purified gas Torr). Zinc was freshly and evacuated to 6. This pseudopotential excludes the excess electron from the hard core radius d of the atoms. Up is the polarization potential from the atom inside and the atoms outside rgwhich is approximated by

where a is the isotropic polarizability. Then ( 2 ) may be solved to yield

V O= Up

+ h2ko/2m = Up + To

(4)

where the electron wave vector ko is obtained from the Wigner-Seitz boundary condition (d@o/dr),,,, = 0 which gives the equation tan ko(r, - a ) = kor,

(5)

To is the zero point energy which arises because the excess electron is excluded from the hard core region of radius d in each Wigner-Seitz sphere. Assuming the polarizability and density are known, VO may be calculated from d and vice versa. VOmay be directly determined experimentally but d cannot. Thus, derived The Journal of Physical Chemistry, Vol. 79, No. 26, 1975

experimental values of must come from V Omeasurements. However, d can be independently obtained from the calculated effective electron scattering cross section by only the Hartree-Fock potential of an atom or molecule via UHF = 4 d 2 . Note that UHF is not the experimental gasphase scattering cross section because the real total potential includes both Hartree-Fock and polarization components. There have been few measurements of VOin liquid rare gases. The ones extant including ours in argon are summarized in Table 11. Direct photoelectric injection methods are felt to be the most accurate. We have used the Fowler function22 to find the thresholds and believe that this should give more accurate results than thresholds from simple linear extrapolations. Our value (-0.17 eV) is less than a previous direct experiment (-0.33 eV)24in which a linear extrapolation was used. If we try to fit Halpern et al.’s dataz4to a Fowler plot, we find that it is all far above the Fowler threshold. This perhaps explains some of the discrepancy between the two sets of results; if we use a linear extrapolation on our data we find VO -0.26 eV. Raz and JortnerZbdetermined V Oin argon by an interesting indirect method. They interpret the vacuum uv absorption spectrum of 3 ppm Xe impurity in liquid argon as a hydrogenic series and obtain the series ionization limit I1 from a Rydberg constant modified by the dielectric constant of the medium. Using the expression 11 = I , P+ (6) known values of the gas-phase ionization potential I,, and calculated values of the polarization energy of the positive ion, P+, they obtain VO.Their experimental uncertainty comes from the spectral analysis, but it seems that the calculated values of P c in the solid (-1.10 eV), which were assumed to be the same in the liquid, could well be uncertain by several tenths of an electron volt. Table I1 also shows values of d derived from the experimental VOvalues via eq 1-5. Note that d is relatively insensitive to V Obut that the converse is not true. Thus theoretical VOvalues obtained by using a theoretical calculation or estimate of d13 seem subject to large errors which can exceed a factor of 2. E. Temperature Dependence of VO.Vo is only a function of density if d is assumed independent of donsity. Within the framework of the VOtheory presented, d is density independent. Thus a test of the validity of the VOtheory may be provided by the density dependence of VOas obtained by changing temperature or pressure. Recall that V O= Up To. Up and To change in opposite directions with density, so in principle V Omay increase, decrease, or be independent of density. We will find that in alkanes V Obecomes more negative with increasing temperature (decreasing density), but that V Ois approximately independent of temperature in argon. Changes in Up are always dominated by P , - ~ or n4I3.The term involving a n inside the bracket in eq 3 changes much more slowly for all reasonable values of a and n in atomic or molecular media; and, in fact, this term changes Up in the opposite direction from the dominant n4/3dependence. So Up becomes less negative as temperature increases. To becomes less positive as temperature increases, and i t depends on both n and d . Note that if d is changed To changes but Up does not, so the temperature dependence of V Ois rather sensitive to ii. This suggests that the temperature dependence of V Omay serve as a rather sensitive indication of the validity of a particular d.

-

+ + vo

+

Conduction State Energy of Excess Electrons in Condensed Media p e 2 10 cni2 V-l sec-l 31

:t

p,

5 1 cm2 V-l sec-l

quasidree state stable

(8)

localized state stable

(9)

excess electrons are localized in the liquid phase of helium (3He and 4He), neon, hydrogen, and most alkanes, and are quasi-free in argon, krypton, xenon, methane, neopentttne, and probably 2,2-dimethylbutane and 2,2,4,4-tetramethylpentane.26 Let us now examine how to obtain Et for various media. We will use the general formulation for Et given by the semicontinuum r n 0 d e l . ~ 3Et~ ~is given by

/To

Et = Ek

Figure 3. Calculated density dependence (or temperature dependence) of Vo in liquid argon at 55 atm pressure. The density dependence of the To and Up components of VOis also shown. The open points correspond to a = 0.928 A as determined from VO = -0.17 eV at 87 K at 1 atm. The solid points correspond to a = 0.904 A as determined from Vo = -0.33 e V at 85 K at 0.8 atm.

These points are illustrated in Figure 3 for argon over an extremely large density range. If (z = 0.928 A, as determined from V O= -0.17 eV a t 87 K under 1 atm pressure, Vois constant with temperature. But if a = 0.904 8, as determined from Vo = -0.33 eV at 85 K under 0.8 atm, Vo becomes less negative by -0.1 eV (25%) with increasing temperature. The only temperature-dependent measurements of VOin liquid rare gases are in helium. V Ois temperature independent from 1.1 to 2.2 K. However, this is a trivial test since there is less than 0.7% change in density over this temperature range. I t seems that a critical test of the V Otheory can be carried out for liquid alkanes. Table I shows that the calculated temperature dependence of Vo in methane and particularly in ethane agrees well with theory. This is further substantiated by similar good agreemeht for propane through n-hexane and several other branched alkanes.Il Thus, all alkanes, including methane, seem to obey the V O theory of Springett et al.,13 at least over the normal range of liquid densities. Furthermore, the temperature-indepandent values of the hard core radii a can be regarded as having some general validity, and trends between these radii are probably meaningful. It would be interesting to measure V Ovs. temperature in argon and other rare gases to further test the Vo theory, although we expect the theory to be even more valid for the rare gases. The alkane measurements do not extend into the critical region. There we might expect density fluctuations to be more important and to find a failure of the Vo theory which ignores these. This could be most-easily tested in the liquid rare gases. C. Electron Localization Criteria. Given a value for Vo in a condensed medium, one can then ask whether electron localization is possible. The general criterion for localization is that the ground-state energy level be less than the conduction state energy level or

Et

< Vo

(7)

where Et is the total ground-state energy relative to vacuum of the electron in the medium. If Et > Va, we then describe the electron as quasi-free and able to move in the conduction band of the medium. An experimental criterion of localization is the electron mobility and we may specify approximate limits as follows. From this experimental point of view

+ EeS-t E,' + Vo(1 - C,) + Ems+ E,' + E , + E,,

(10)

where Ek is the kinetic energy, EeS and E: are the shortrange and long-range attractive electronic energies, Vo(1 C,) is the short-range repulsive energy between the medium and the electron whose charge density within a given radius r is C,, Ems and Em1are the short-range and longrange medium rearrangement energies due to polarization interactions, E , is the void or surface tension energy, and E,, is the pressure-volume energy associated with void formation. In media consisting of polar molecules like water, alcohols, etc., the E," term due to charge-dipole interaction is large and negative (-2 to -4 eV) so that the localized state is always energetically favored. This situation has been treated in some detailz7 and will not be considered further: In media consisting of nonpolar molecules (rare gases, homonuclear diatomics, and alkanes) EeSand Ems = 0 because there are no permanent dipoles (or negligibly small ones in the case of alkanes) and E,' = 0 because the static and optical dielectric constants are equal. We will also consider media a t low external pressures (E,'

+ 1?h~/8rnr,~

(16)

This simply states that the total energy is nonpositive for electron binding to occur. In the case of interest here the bound state must be less than the conduction electron level so our criterion for electron localization becomes

We then have the following localization criteria where approximate mobility ranges are ( VO- VO*)> 0

-

localization

(pe 5

1cm2 V-1 sec-1) (18)

The Journal of Physical Chemistry, Vol. 79, No. 26, 1975

-

S.Noda, L. Kevan, and K. Fueki quasi-free (we Z 10 cm2 V-l sec-l) (19)

given in accordance with the range above which electron band motion is considered valid. We have therefore found that the general localization criterion, Et < VO, can be replaced by the localization criterion VO* < VO. The advantage is that we have a simple expression for VO*that can be simply evaluated, whereas evaluation of Et requires the nontrivial solution of an eigenvalue equation. Table I11 summarizes the data on mobility, VO,VO*,and the two components of VO*from eq 17 for electrons in the alkanes plus tetramethylsilane for which VOmeasurements have been reported. Methane, neopentane, and tetramethylsilane are clearly predicted to exhibit the quasi-free electron state, and this is consistent with the high dectron mobility in these liquids. The quasi-free electron state is also predicted for 2,2-dimethylbutane and 2,2,4,4-tetramethylpentane as is consistent with their reasonably high electron mobility, but this prediction is borderline. The other alkanes are correctly predicted to exhibit electron localization and a consequent low electron mobility. As discussed above, VOis temperature dependent; for alkanes it decreases with increasing temperature. Thus we must also consider the temperature dependence of VO*and how it affects the above correlation. Vo* depends on the temperature dependence of both r, and Dop, and for alkanes VO* increases very slowly with increasing temperadecreases ture (Table IV). The net result is that ( VO- VO*) with increasing temperature. This suggests that the quasifree state becomes more favorable a t higher temperature. In ethane the onset of the quasi-free state is predicted to occur near 180 K; p does approach unity near this temperat ~ r e In . ~general, Table IV shows that the criteria (18) and (19) still hold over the temperature ranges for which VOhas been measured in alkane systems. We point out that our localization criteria, (18) and (19), are not expected to and do not apply to rare gas liquids since our assumption that R equals the Wigner-Seitz radius is then not valid. If one tries to apply (18) and (19) to rare gases, the quasi-free state is always predicted, even for Ne and He. The liquified molecular gases H2, D2, 0 2 , and N2 all have low electron mobilities (

272

-1.11

228 245 286 275

-1.34 -2.44 -2.23

-0.01 -0.17 -0.33 -0.78 -0.38

states a t the bottom of the conduction band to the density of the localized states is near unity. Equation 21 predicts a linear plot for log p vs. E , a t constant temperature. We may interpret E , as IEd(transient) referenced to VO.We thus expect that (VO VO*)will qualitatively correlate with the mobility activation energy, E,. This correlation is only approximate since (VO- VO*)is weakly temperature dependent. Figure 4 appears to support this picture by a roughly linear correlation for a wide variety of alkanes near room temperature, although other interpretations may be possible. It is striking that this correlation seems to include electrons in both localized and quasi-free states. Methane a t 109 K is also included in this figure; its mobility has a weak temperature dependence, and if p ct T-3/2is assumed, the extrapolated mobility a t 295 K is 100 cm2 V-l sec-l. (Vo - Vo*) will also become somewhat more negative so the extrapolated point to room temperature will also roughly fit the correlation. The idea of transient trapping of electrons in alkanes has

-

The Journal of Physical Chemistry, Vol. 79, No. 26, 1975

0 3:2

!(n-CgH141neo-CgHI2

'-ZSHl2

-I-

-0.4 -0.3

-0.2

!

0 +O.l vo-vo lev)

-0.1

*

+0.2

+0.3

Figure 4. Correlation plot of electron mobility at room temperature (log scale) and the proposed stability criterion ( VO VO")for excess electrons in liquid hydrocarbons. The electron mobilities and ( VO Vo*)values were taken from Table 111. The triangles and circles, for the same compound represent two different reported mobilities. The points connected by a line for a single compound correspond to different reported Vo values. The abbreviations for the compounds may be deduced from the names given in Table 111. The values in parentheses indicate results not at room temperature (-295 K). The n-C4H10results are for 271 K and would not change much when extrapolated to room temperature. The CH4 results are for 11 1 K and would extrapolate to higher temperature in the direction of the arrow. The squares represent data from ref 6 and 18 for n-hexaneneopentane mixtures with the mole ratio shown.

-

-

been used before to suggest quantitative correlations between p and V0.12Jg,20The model of Kestner and Jortnerlg and that of Schiller20assume a two-state picture involving high and low mobility regions. Kestner and Jortner emphasize that the medium is microscopically inhomogeneous while Schiller emphasizes energy fluctuations of the electron states in a homogenous medium. Both approaches lead to a nonlinear correlation between log and Vo with two20 or fourlg parameters. Although a two-state picture of mobility would be expected to extrapolate to the high mobility limit (quasi-free state), both of these correlations predict V O5 -0.8 eV for the quasi-free state in methane which does not agree with experiment. Schiller has suggested that these two-state mobility models may be brought into agreement with experiment by making other assumptions about the parameters involved.35I t also seems that the effect of temperature on these correlations should be investigated. Holroyd and Tauchert? have noted that the following empirical relation holds approximately for alkanes, L./ = 0.35 exp(-15.2Vo), where VOis in electron volts and p is in cm2 V-l sec-l. This relation gives V O= -0.47 eV for liquid methane and V O= -0.06 eV for liquid ethane and does not agree with the present experiments. Again, the effect of temperature on the above relation should also be explored. F. VOin Glassy Matrices. Attempts were made to measure Vo in various polar and nonpolar glassy matrices, but the photoemission currents were too small to measure in our apparatus. The addition of a chopped light beam with lock-in detection still did not improve the sensitivity enough. In %methyltetrahydrofuran (MTHF) and 3-meth-

Conduction State Energy of Excess Electrons in Condensed Media TABLE VI: Derived Experimental Values of Conduction Electron Energies (V,) in Glassy Matrices at 77 K (Method 1;See Text) Matrix

AI

Do, -Pt,eV

V,,eV

Meth ylcyclo hexane 0.85a 2.28 2.09 1.24 3-Methylpentane 0.85a 2.22 2.05 1.20 2-Methyltetrahydrofuran 1.08a 2.23 2.06 0.98 1-Propanol 1.65b 2.17 2.01 0.36 Ethanol 1.70b 2.08 1.94 0.24 Methanol 1.85b 1.97 1.84 -0.01 Ice (2.0)b 1.78 1.64 -0.36 1 0 M KOH-H,O 2.2C 1.96 1.83 -0.37 a A. Bernas, M. Gauthier, and D. Grand, J. Phys. Chem., 76, 2236 (1972). b A. Bernas, M. Gauthier, D. Grand, and G. Parlant, Chem. Phys. Lett., 17, 439 (1972)khe value for ice is extrapolated. C R. Santus, A. Hblene, C. Helene, and M. Ptak,J. Phys. Chem., 7 4 , 550 (1970).

2873

TABLE VII: Derived Experimental Values of Conduction Electron Energies (V,) in Nonpolar Glassy Matrices at 77 K (Method 2; See Text) Matrix

p,

g/cm3

r,,a

E , Aa

V,, eV

Methylcyclohexane 1.02 3.37 2.22 1.21 0.88 3.39 2.16 0.82 3-Methylpentane a R. A. Holroyd and R. L. Russell, J. Phys. Chem., 7 8 , 2128 (1974).

es removes the last remaining parameter from the semicontinuum model with the not unexpected consequence of slightly less good agreement with experiment. In the photoionization of TMPD in liquid alkanes the value of A I is much larger than in the corresponding glassy matrices. For example, AI = 1.8 eV in liquid methylcyclohexane (MCH) and in liquid 3-methylpentane (3MP).11 Since the P+ energy only decreases by -0.2-0.3 eV in the ylpentane (3MP) glasses at 77 K, the photoemission curglass compared to the liquids, the major factor in changing rent yield was >lo5 lower in the glass compared to the vacA I is VO,We have shown that the temperature dependence uum. of V Oin alkanes seems satisfactorily accounted for by the Since direct attempts to measure V Oin glassy matrices VOtheory of Springett et al.,13 so we may use this theory failed, we have used the indirect method of eq 6 to obtain with directly measured values of VOin liquids to derive exVO.Bernas et a1.36-3s have measured the photoionization perimental values of Vo in glassy matrices. The calculated threshold I , of tryptophan and N,N,N’,N’-tetramethyl-pvalues of VOat 77 K are given in Table VI1 together with phenylenediamine (TMPD) solutes in a range of glassy mavalues of the matrix density, p, the Wigner-Seitz radius, rs, trices. Since the gas-phase ionization potential I, is known and the hard core radius, &. This independent method of -~~ for these solutes, values of A I = I , - I, are r e p ~ r t e d . ~ ~ deriving VOvalues for glassy matrices gives results in reaThen by calculating the cation polarization energy P+ we sonable agreement with the method summarized in Table can obtain VO.We have calculated P+ from40 VI. The large positive value of VOin 3MP and MCH (-1 eV) p+ = -e2 (1 (22) is consistent with electron localization and the observed 2ro Do, low electron mobilities in such g l a s ~ e s . ~ I t, ~also implies where ro is the effective radius of the positive ion that the potential a mobile electron “sees” is very “rough” (TMPD+) in the matrix. Instead of estimating a value of r; because of the large repulsive forces implied by the large from geometrical considerations, as is typical, we obtain ro positive VO.Macroscopic voids in alkane glasses will act as 1 1.93 A as the only unknown in eq 6 from experimental low spots in the potential or as so-called preexisting traps. values of 11,I,, and V Oin liquid n-pentane.Q41 We implicHowever, the potential will be considerably modified by itly assume ro to be temperature independent. Values of bond dipole orientation and molecular configurational reDo, a t 77 K were obtained from nd2 a t 77 K, where nd is arrangement as indicated by e x ~ e r i m e n t a and l ~ ~theoret~~~ the refractive index. Room-temperature values of nd were ica130 studies. extrapolated to 77 K by analogy to the observed temperature dependence of nd in glycerol.42The temperature coefAcknowledgment. This research was supported by the d ~ ~extropc!,? ficients (dnd/dT) mec.ared I n the l i ~ u i UWY U S . Energy Research and Development Agency under ed to the glass transition temperature, and below that the Contract No. E(11-1)-2086. temperature coefficient was assumed to be zero. The values References and Notes of Do, used are given in Table VI. Table VI summarizes the indirect experimental values of (1)L. Kevan, Adv. Radiat. Chem., 4, 181-306 (1974). (2)J. Jortner, Ber. Bunsenges. Phys. Chem., 75,696 (1971). VOobtained by this analysis together with AI and P+. For (3) M. Silver, private communication. ice and 10 A4 KOH tryptophan was used to measure AI and (4) W. F. Schmidt, G. Bakale, and U. Sowada, J. Chem. Phys., 81, 5275 ro for tryptophan+ was assumed to be the same as for (1975). (5) M. G. Robinson and G. R . Freeman, Can. J. Chem., 52,440 (1974). TMPD+. The matrices are listed in order of increasing po(6)R. M. Minday, L. D. Schmidt, and H. T. Davis, J. Phys. Chem., 76, 442 larity, and there appears to be a clear trend of decreasing (1972),and earlier references cited therein. (7)Y. Maruyama and K. Funabashi, J. Chem. Phys., 56,2342 (1972). VOwith increasing polarity. (8)T. Huang and L. Kevan, J. Chem. Phys., 61,4660(1974). It is interesting to compare these V Ovalues with the op(9)R. A. Holroyd and M. Alien, J. Chem. Phys., 54, 5014 (1971). timized V Ovalues used in the semicontinuum model of (IO)R. A. Holroyd, J. Chem. Phys., 57,3007 (1972). (1 1) R. A. Holroyd and R. L. Russell, J. Phys. Chem., 78,2128 (1974). trapped electron energy levels.27In this model V Ois a lim(12)R. Schiller, Sz.Vass, and J. Mandics, Int. J. Radiat. Phys. Chem., 5,491 ited adjustable parameter. Although the agreement is not (1973). (13)E. E. Springett, J. Jortner, and M. H. Cohen, J. Chem. Phys., 48, 2720 quantitative, the trends are similar in polar matrices, and (1968). the more positive optimized VOvalues in the glass com(14)M. Cohen and J. Lekner, Phys. Rev., 158,305(1967). pared to the liquid are in agreement with what we know of (15) K. Fuekl, D. F. Feng, and L. Kevan, Chem. Phys. Lett., 13,413 (1972). (16)H. T. Davis, L. D. Schmidt, and R. M. Minday, Chem. Phys. Letf., 13, 413 the temperature dependence of VO.In MTHF, however, (1972). there is significant disagreement between the optimized Vo (17)H.T. Davis and L. D. Schmidt, Can. J. Chem., 51,3443 (1973). R. A. Hoiroyd and W. Tauchert, J. Chem. Phys., 60,715 (1974). (18) value from the semicontinuum model and the derived ex(19)N. R. Kestner and J. Jortner, J. Chem. Phys., 59,26 (1973). perimental value which deserves further investigation. (20)R. Schiller, J. Chem. Phys., 57,2222 (1972). Thus, in general, the derivation of Vo values for polar glass(21)S.Noda and L. Kevan, J. Chem. Phys., 61,2467 (1974).

2)

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S. Noda, L, Kevan, and K. Fueki

(22)R. H. Fowler, Phys. Rev., 38, 45 (1931). (23)A. M. Brodskii and Y. Y. Gurevich, Sov. Phys. JETP (fngl. Trans.), 27, 114 (1968). (24)B. Halpern, J. Lekner, S. A. Rice, and R. Gomer, Phys. Rev., 156, 351 41967). (25)8 . Raz and J. Jortner, Chem. Phys. Lett., 4, 155 (1969). (26)W. F. Schmidt in “Electron-Solvent and Anion-Solvent Interactions”,L. Kevan and B. Webster, Ed., Elsevier, New York, N.Y., 1975,Chapter 7. (27)K. Fueki, D. F. Feng, and L. Kevan, J. Am. Chem. SOC., 05, 1398 (1973). (28)L. I. Schiff, “Quantum Mechanics”,McGraw-Hill, New York N.Y., 1955, pp 76-77. (29)A. Gedanken, B. Raz, and J. Jortner, Chem. Phys. Lett., 14, 326 (1972). (30)D. F. Feng, H. Yoshida, and L. Kevan, J. Chem. Phys., 61, 4440 (1974). (31) B. Halpern and R. Gomer, J. Chem. Phys., 51, 1031 (1969). (32)J. Lekner, Phys. Rev., 158, 130 (1967). (33)J. Bardeen and W. Shockley, Phys. Rev., 60, 72 (1950). (34)H. T. Davis, L. D. Schmidt, and R. M. Minday, Phys. Rev. A, 3, 1027 (1971). (35)R. Schiller, private communication. (36)A. Bernas, J. Blais, M. Gauthier, and D. Grand, Chem. Phys. Lett., 30, 383 (1975). (37)A. Bernas, M. Gauthier, and D. Grand, J. Phys. Chem., 76, 2236 (1972). (38)A. Bernas, M. Gauthier, D. Grand, and G. Parlant, Chem. Phys. Lett., 17, 439 (1972). (39)R. Santus, A. Helene, C. Helene, and M. Ptak, J. Phys. Chem., 74, 550 (1970). (40)J. Jortner, J. Chem. Phys., 30, 839 (1959). (41)Here lg = 8.6 eV was used to be consistent with the Alvalues reported by Bernas et a1.37*38 If I, = 6.2eV is used, @ as well as A/ are both decreased by -0.4 eV so the derived value of VOis unchanged. (42)K. Schultz, KolioidZ., 138, 75 (1954). (43)Landolt-Bornstein, “Zahlenwerte und Funktionen,” Band II, 8 Tell, Springer-Verlag,Berlin, 1962,p 572;when dndldT was not given for a

particular compound, it was estimated from data of similar compounds.

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Discussion M. H. COHEN.The transition in the mobility from quasi-free electron behavior to that associated with electrons bound within well-defined configurations of the liquid occurs continuously. In the transition region, Anderson localization occurs. Electrons are indeed transiently trapped in fluctuations in the liquid, but the latter are not well defined and cover a broad range of configurations. Moreover, the mobility contains contributions from both extended, i.e., quasi-free, states and from localized states. Thus, arbitrarily to select one as a mobility marking a transition from quasifree electron behavior to localized behavior when Vo = VO*is an oversimplification. Further, it is only well after the mobility is dominated by localized states that the liquid configuration or trapping configuration becomes sharply enough defined to be obtained from an energy minimization criterion. Any such criterion has to include the energy required to form the fluctuation, which goes over continuously to the surface energy contribution.

L. KEVAN.I agree that the transition from delocalized to localized states ol electrons in different alkanes is not a sharp one. And as our correlation between log ge vs. V O Vo* shows, the trend of the points between clearly localized electron states, as in n-pentane, to clearly delocalized electron states, as in methane, is roughly continuous. What we think we have provided is a simple criterion that does validly predict both extremes of electron behavior and provides a connection over the entire mobility region. In the mobility range of -1 to -10 cm2 V-l sec-l, we clearly have intermediate behavior as far as localization goes, and likewise when our V O- VO*criterion is say --0.05 to -+0.05 eV, we clearly have an

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The Journal of Physical Chemistry, Vol. 79, No. 26, 1975

intermediate range where the predictability of the criterion is not sharp. We have indeed ignored the energy of the fluctuations which produce the transient cavities of molecular size. This assumption seems valid for the extreme cases of electron localization and delocalization which we seem to have accounted for, and this assumption becomes less good in the transition region.

J. JORTNER. (1) Concerning the Vo for dense rare gases I’m somewhat worried by your statement that this quantity calculated by the SJC model is density independent in liquid Ar due to mutual cancellation between changes in the To and Up energy terms. We have recently determined experimentally for spectroscopic and photoemission data the value VO= +0.3 f 0.1 eV for solid Ar, while the corresponding value in the liquid is - 0.33to -0.17 eV. (2) Your electron localization picture for hydrocarbons differs from the SJC model originally applied to simple dense fluids. Also in heavier rare gases polarization effects have to be incorporated in the calculation of the energy of the localized state. The main difference between your approach and the SJC model lies in a different treatment of short-range repulsive interactions. I wonder whether a physical argument can be advanced concerning this point. If your model does not work for electron localization in the simple case of liquid rare gases, its detailed justification for hydrocarbons is crucial. (3) In segregating excess electron states in liquid hydrocarbons into two categories, localized ( p < 1 cm2 V-l sec-l) and extended ( p > 10 cm2 V-l sec-l), you disregard the most interesting aspect of the continuous variation of the mobility with the change of molecular structure in these liquids. On the other hand, two such extreme situations prevail for liquid rare gases where g = 10-2-10-3 cm2 V-l sec-l for liquid He and Ne (i.e,, localized state) while g e 400-2000 cm2 V-l sec-’ for liquid Ar, Kr, and Xe (Le., extended quasi-free state), such a sharp distinction does not prevail in liquid hydrocarbon. In the latter case we face the interesting physical situation where local fluctuation (probably orientational fluctuation) determines the transport properties. This is a most interesting and challenging problem.

L. KEVAN.(1)Figure 3 is a plot of the cancellation of the To and Up terms for liquid argon a t 55 atm pressure. As for your VOmeasurement in solid Ar, it would be interesting to measure Vo vs. temperature in the solid phase and see how this would compare with the SJC model. (2) I probably overemphasized that our approach does not apply to liquid rare gases. In fact, it does apply to the heavier rare gases (argon, krypton, and xenon) where it correctly predicts electron delocalization. This is what you would expect since our approach emphasizes the polarization contribution, and this contribution is undoubtedly important in the heavier rare gases. It strikes me as perhaps fortuitous that the SJC electron localization criterion “works” for these heavier rare gases when the polarization contribution is ignored. (3) I think the intermediate range of mobility is indeed an interesting question. What we have done in our simple approach is to explain the two extremes of electron behavior in liquid alkanes which does not appear to have been done before and which is not explained by the model applied to rare gases. In the intermediate mobility range, our model is certainly overly simplified and it is an interesting question as to how far it may be extended. We have of course considered the role of local fluctuations by making them the driving force to form our transient traps of molecular size. By using the Wigner-Seitz radius to “size” these transient traps we have emphasized fluctuations in the intermediate mobility range by considering a distribution of “important” fluctuations.