J. Phys. Chem. 1987,91, 3425-3430
+ 2a2] - [a2+ (6 - A)2][6A - 2aW] 4r(a2 + A2) [a2 + (6 + A)2][6A - 2 a w - W[6A - 2a2]
where w ( z ) = exp(-z2) erfc (-iz); as in eq 25
W[6A
d, = d2 =
(115) zj =
-Qj+ iyj
(1 16)
+ A2) - A2 f 2iAWj’/2 a f i6 = [ E 2 + r = (a2 + b2)1/2 tp + = -+ + + a, tb2 = -+ + - a 2 2 4r(a2
2P’/2
(1 17) (118)
ES
6bl
3425
(119)
Y,= 16 - AI, Y2 = 16 + AI
(120)
Employing the integral representation
Y:
E,, D, and P are given by expressions 88, 29, and 22, respectively. Thus, presently we have derived the current expression in terms of various system parameters such as coverage factor 6, induced dipole moment of the adsorbate p , reorganization energy E,, various orbital energy levels and coupling parameters, etc. From (123), it is clear that the functional dependence of the current on these parameter is rather involved, though the form of the current expression is similar to the one obtained for the loneadsorbate, single orbital model (cf. eq 19). Therefore, the procedures discussed in section 4 can be applied to eq 124 for arriving at the various limiting cases. 7. Summary and Conclusions
in (1 13), interchanging the order of integration from (t, n,q,, e, 7) to (e, T, t, r Y q Y )and , thereafter evaluating the ayqUr t, and T integrations, we obtain the following analytical expression for the anodic current.
d,
+ c,a Re w ( z l ) + d2 - c2a Re w ( z 2 )- c, Im w ( z I )YZ
1
(1) Ignoring direct coupling between reactant and substrate, we consider the electron transfer only through the reactant coupling to the chemisorbate. We solve for the current as a function of various energy levels involved in the model, as well as the several coupling coefficients. This brings to the fore the appropriate scaling of energies/temperature in this problem. (2) The results of Marcus, both the homogeneous and heterogeneous versions, besides Schmickler’s are deduced as limiting cases, thereby defining the range of validity of the earlier results. (3) Our model Hamiltonian is different from that used by Schmickler for the same problem and we provide a critical discussion of the latter’s analysis; (4) The two-adsorbate results in chemisorption are transposed in this context and the dependence of the current on the neighboring adsorbates is studied. ( 5 ) Even though most of the results given here employ the HF expressions for adsorbate GFs for the sake of simplicity, the formalism is general enough to accommodate the correlation effects.
Electron Transfer through Film-Covered Surfaces. Case of Monolayer/Submonolayer-A Coherent Potential Approximation Formalism A. K. Mishra and S. K. Rangarajan* Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560 012, India (Received: July 17, 1986) The problem of electron transfer mediated by the chemisorbed layer with random occupancy on a two-dimensional lattice is taken up. The analysis employs linear response formalism and coherent potential approximation for evaluating the current. Two cases referring to (i) the delocalized land (ii) the localized electronic states in the chemisorbed layer are treated.
1. Introduction In an earlier report,l the problem of electron transfer via the adsorbed intermediates has been considered. Therein, the model system contained only a few (one or two) adspecies. In the present paper, we extend the formalism to a case wherein an aggregate of adspecies forms a submonolayer or monolayer film on the electrode. In general, the problem here is twofold, viz., (i) characterization of adsorbate layer through appropriate parameters (1) Mishra, A.
K.; Rangarajan, S. K., preceding article in this issue.
and (ii) investigation of the functional dependence of the electrochemical rate on these parameters. The first step concerns the equilibrium state of the adlayer whereas the electron dynamics involving the depolarizer and the adlayer states need to be considered for obtaining the required functional dependence. Herein, we develop a formalism at the microscopic level for describing the effect of a chemisorbed submonolayer or monolayer on the electrochemical reaction rate. The case when the electrons in the adlaver are delocalized and the case of the localized electronic states in the adsorbate’s film are considered separately. As considered earlier,, the direct coupling between the depolarizer
0022-3654/~7/2091-3425$01.50/0 0 1987 American Chemical Society
3426
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987
and the electrode is ignored and only the indirect electronic coupling via the adlayer has been taken into account. The aim here is to find out the dependence of the current not only on the macroscopic parameter 6, but also on the fundamental parameters, viz., adsorbate's energy level, adsorbate-substrate coupling strength, interadsorbate interaction in the adlayer, etc. In the next section, the modelling aspects concerning the system are described and an appropriate system Hamiltonian has been provided. The chemisorbed species are considered to occupy randomly the adsorption sites on the electrode. The solvent has been described in terms of various classical polarization modes. The anodic current expression for the case of the delocalized electronic states in the mediating adlayer has been considered in section 3. Two alternate formalisms, one based on the concept of the configuration-averaged GF, and the other based on the restricted configuration-averaged G F approach has been considered therein. In section 4, the formalism has been extended to a case wherein the solvent polarization mode, adsorbate interactions, become important. Section 5 provides a summary and conclusions.
2. Modelling the System Components-The Hamiltonian We specify below the physical models used for the characterization of the system components: 1. The interfacial solvent layer is modelled as a structureless dielectric medium of dielectric constant e l . 2. Chemisorbed species are distributed randomly on the various adsorption sites on the substrate.2 These sites are considered to form a two-dimensional lattice in a medium of (effective) dielectric constant t , and are commensurate with the underlying substrate. As a result, the adspecies occupy the on top position^.^ At coverages less than unity, the unoccupied sites on the two-dimensional lattice are considered as the second component of the a d l a ~ e r . The ~ energy levels associated with the vacant sites are taken to be infinity. This ensures that no electron can transfer to or through a vacant site. 3. The random distribution of the adsorbates leads to the diagonal randomness in the system, in the sense that site energies (which are the diagonal component of the Hamiltonian) acquire random characteristics depending on the "occupancy" or the "vacancy" state of site. 4. We consider that only the diagonal randomness is present in the s y ~ t e m . ~This . ~ implies that various off-diagonal terms associated with the lattice sites are deterministic quantities and they do not depend on the occupancy state of the ~ i t e . ~ ~ ~ 5. No randomness is associated with the underlying substrate and it is modelled as a semiinfinite medium. 6. The depolarizer is situated beyond the OHP and is coupled weakly to a site T on the adlayer lattice, with no direct coupling to electrode. The occupancy state of the site is unspecified; Le., the site i may be occupied or vacant. In the latter case, as the energy associated with the site would be. infinite, no charge transfer via this site will take place. 7. Distribution of the adsorbates in the two-dimensional lattice and distribution of the depolarizer (or reactant) are taken to be uncorrela ted. 8. The bulk solvent is modelled classically in terms of various polarization modes, to which both the adspecies and the reactant are coupled. After describing the various steps involved in the system we now specify the Hamiltonian for the system as
Mishra and Rangarajan =
+
ckflka
1,u
[semiinfinite electrode term]
c
+
+
rci(lqul)nic
ko
(P:
[random, diagonal site terms for 2-D lattice]
+ 4:) +
+
ernr
[free reactant term]
Z:hw,[&pr + E0"lq"
-+
U
[free solvent term]
C
Vijcw+cja
[reactant-solvent coupling term] + [ y k C i a + C k o + HC] +
i#j [hopping term between the sites on 2-D lattice]
k,i,a
Z[vricr+qm+ HCI
(1)
u
[reactants' coupling term to site on the 2-D lattice] In the above Hamiltonian, site representation is used for the adsorbate lattice and (ckr,ciu]is assumed to form a complete orthonormal set. The core charge of the adspecies is taken to be zero,' though its inclusion in the formalism is a straightforward task. The only random quantity appearing in the above Hamiltonian is the term ti((qY1). When a site i is occupied by an adsorbate, it takes the value ti(lqv1) = ea
+ hEuvgavqp
(2)
where g,, is the coupling coefficient corresponding to the adsorbate's electron and the solvent polarization mode interaction. In case i is vacant, ti((qu1)
---*
(3)
3. The Electrochemical Current When the Electronic States in the Chemisorbed Layer Are Delocalized 3.1. Anodic Current via Configuration-Averaged GF Formalism. When the electrons are delocalized in the chemisorbed layer, their coupling strengths with the solvent polarization modes are negligible and are ignored. Also, the electrons now effectively screen the adsorbate's core charge. Therefore, the interaction between the adlayer and the solvent modes can be neglected.* Next, the only mechanism responsible for the transitions involving the reactant state is provided by the "reactant orbital-site ? coupling { V,,, Vra)which is considered to be weak. Thus, the linear response formalism considered earlier' can be once again used for obtaining the electrochemical current in the present case. We write the anodic current expression corresponding to the transition via site 1 as (cf. eq 3-7 of ref I )
( ...)F in (4) implies that averages are taken considering the solvent coordinates as fixed parameters (Franck-Condon approximation) and (...)ec implies that the averages over the solvent modes are finally evaluated by treating them as classical variables. Using the relation (cf. eq 11 of ref 1)
The expression for the current can be written as
(2) Schultze, J. W.; Dickertmann, D. Faraday Symp. Chem. Soc. 1977, N-12. 36. ..
Einstein, T. L.; Schrieffer, J. R. Phys. Reu. B 1973, 7 , 3629. Rangarajan, S . K. J . Electroanal. Chem. 1977, 82, 93. Ehrenreich, H.; Schwartz, L. M. Solid State Phys. 1976, 31, 149. Elliott, R. J.; Krumhansl, J. A,; Leath, P. L. Rec. Mod. Phys. 1974,
The G F associated with the site 1 is defined as (7)
46, 465.
(7) Schmickler, W. J . Electroanal. Chem. 1980, 113, 159. (8) Here, the situation is similar to "electrode's delocalized electronssolvent mode" interactions.
where 10) is the fermion vacuum state and Ho is the Hamiltonian describing the metal-adsorbate system.
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987 3427
Electron Transfer through Film-Covered Surfaces
Ho = ztknk,, k,e
+ z€i’ni,, + i,s
K’(E)=
+
[VikCi,,+Ckr HC] i,k.o
+ i#j,r E VjijCi,,+Cj,
with ti’
m
if site i is vacant
k’(c) = (ilK’(e)li)
(19)
The self-consistency expression for k‘ (which is site-independent) in the more general case when the energy t, associated with the vacant site m is5S6 % 1-% =o (20) ( e , - k9-l - Gii (ea - k7-I - Gii
+
= e, if site i is occupied =
Ek’(t)cis+cisr i,r
(8)
(9)
From (7-9), it is clear that 6 in (6) contains the random parameter t[, and therefore, for obtaining the deterministic current expression, configuration average need to be performed in the expression 6. For this purpose, we employ the coherent potential approximation (CPA) formalism. From a physical point of view, this formalism implies that the random adsorbate layer has been replaced by an effective medium to which the depolarizer is coupled. A point to note here is that whereas the Hamiltonian Ho for the metal-adlayer system does not contain the energy parameter t, the effective medium Hamiltonian depends on e. Remembering that only q in (6) is random quantity, the configuration-averaged current expression is written as
+
The above equation in the limit 1-%
k‘r: e , - -
becomes I-% , or Gii = E, - k’
Gii
E”
-+
m
Equation 21 represents the self-consistency equation for k’. Prior to considering the general case, we first consider the values of k’ in the limit of % 0 and % 1. From (21), it is obvious that k’ = t, when % 1 (22)
-
-
+
The physical meaning of the above identity becomes transparent when we realize that, at the monolayer coverage, the problem becomes completely deterministic and the energy associated with each site in the absence of interaction is tal. Next, for obtaining k’ in the 0 0 limit, we write W a s
-
w = w, + w, where
(23)
where
and (...), denotes the configuration averaging. Thus, the next step in the current evaluation is the determination of 4. 3.2. Evaluation of Configuration-Averaged G Ff o r the Electrode-Adsorbate System. It can easily be seen that the G F matrix elements corresponding to sites on two-dimensional lattice obey the following relationship Gij = GijOGij
+ CGiiOWilGI, I
w, =
c
~jCi,+Cj,
i#j,s
Using (23), eq 18 can be rewritten as
G
=
GO + GOw,G
where
Go = ( t
(12)
- Wl - K3-I
(27)
Since @ varies as 0 when % 0, the second term on the right-hand side of (27) is at least proportional to %2 and hence can be dropped in the present limit. Thus, we have
where
+
G
i,s
N
(28)
GO
or
Clearly Gijo = 0 if i # j
-e
- ea’
if the site i is occupied
= 0 if the site i is vacant
(14)
Wi, in (12) is given as
From (21) and (29), we have ( E - t, - Wii) k‘= e - Wii when % %
-
0
(30)
Finally, we consider the problem of determining K’ and G for a general average 0. Since the effective medium Hamiltonian for a two-dimensional lattice is periodic, the GF matrix elements become diagonal in the two-dimensional Bloch representationg
I/ikVkl
wi, = v,, + E-
k c-tk
and is energy dependent but deterministic. Equation 12 can be written as an operator equation G = GO
+ GOWG
(16)
which holds good only for the two-dimensional lattice. From (16) 1 G = (17) e-
C
ei’nio - W
i,r
The use of coherent potential approximation (CPA) for configuration averaging leads to 1 (G), G = E - W - K’(t) where the self-energy (or coherent potential) operator K is a deterministic quantity and depends on the energy parameter 6 . It also has a diagonal representation in the site basis (i), i.e.,
where the summation is restricted inside the Brillouin zone of the two-dimensional lattice and Nilis the total number of sites on this lattice. W(t,u) is the Fourier transform of Wlj,Le.,
~ ( t , u=)
&Rji
J
wij(c)
(32)
Rii is the position of jth site from ith site. Substitution of (31) in (21) leads to the following self-consistent equation for k’(e):Io (9) As the effective medium has two-dimensional periodicity, for it does not contain any random component, the matrix elements of the Greenian for every \i,i‘]. operator G is same for all the sites, is., Gii = (10) In the case of monolayer coverage, k’(e) in (33) is replaced by e, and the substitution of the resulting expression in (10) will lead to the configuration-averaged current.
342% The Journal of Physical Chemistry, Vol. 91, No. 12, 1987
k-=1-e ea -
1 Nl1
?
k’(c)
1 c - k’(t) - W(c,u)
(33)
The above self-consistent expression for the evaluation of &’(e) is exact, but requires tedious summations over u and the metal states k. Tsukada” has simplified the right-hand side of (33) using the following approximations: (i) Adsorbate is coupled to only the nearest substrate atom. Hence
W(c,u) =
v2 -z(c N, k,
- €u,kZ)-’;
(34)
cu,k, I ck
where v is the adsorbate-nearest surface site coupling strength and is given as vi, = ve‘t‘R>[N N Ill -1 .L where N , is the number of atomic layers in the substrate in the direction perpendicular to the surface. k, is the component of the vector k perpendicular to the surface. (ii) direct hopping term between the lattice sites, viz., are neglected. (iii) cu,k, is assumed to be
v,
cu,k,
1 -zS(c
(iv)
=
cUli
- cull) = Y2A,,for
+ ekzL 161
(35)
< A,,, 0 for 161 > A,l (36)
Nll u
(37)
The condition (iii) implies the separability of metal states energies in the directions parallel and perpendicular to the surface. The densities of states in the direction parallel and perpendicular to the surface are approximated by expression 36 and 37, respectively. From the above approximations, the self-consistent expression 33 for k’( t) becomes* 1-e --- l x c - k’(e) ea - k’(e)
(I
V2
-
2All(~- k’(e))
1
ir) - v2 ( t - k ’ ( t ) ) ( t + A,, - ir) - v2 (t
In
- k’(c))(c - All -
(38)
Substitution of the self-consistent value of k’(t), as obtained from the above expression in (31) finally leads to the configurationaveraged G F matrix element & in the CPA limit. 3.3. Configuration-Averaged Anodic Current.Expression. The final expression for the anodic current is obtained by substituting
and thereafter evaluating the average corresponding to the solvent modes (cf. section 3, ref 1) and time integration in (10). The resulting expression for the current is
Mishra and Rangarajan eo and cR are the energy of the reactant in the oxidized and reduced forms, gRyand go,are the coupling coefficients of these two forms of the reacting species to the solvent polarization mode, and E, is the usual reorganization energy for the direct heterogeneous electron-transfer reaction. From (36-39), it is clear that the anodic current expression in the present case and in the case of direct heterogeneous electron-transfer reaction (cf. section 3, ref 1) are similar in form. The only difference is that the density of states for the electrode has been replaced by -1m G(c),the density of states at the site 1in the effective medium. The dependence of the current on the average as well as on the other parameters characterizing the metaladlayer system are contained in G“, which ought to be determined via the self-consistent expression 38 and the expression 31. The limiting values of 4 corresponding to the complete monolayer coverage is obtained via the expressions 22 and 31. Similarly, the and hence the anodic current in the low-coverage regime are obtained from the expressions 29, 30, and 40, provided the electrons are delocalized in the entire adlayer-a condition probably difficult to be satisfied in the low-coverage regime. For such cases, one needs to consider the “adsorbate-solvent polarization mode” interactions because of the localized nature of the adsorbate electrons. Finally, it can be stated that the activation energy in the present case, and in the case of direct heterogeneous electron-transfer reaction, remains unchanged. This is because the activation energy in the present types of model basically depends on the charge-solvent polarization mode interactions, and presently, the interaction (strengths) of the additional charge centers on the adlayer sites with the solvent modes are negligible. The case when the adsorbates are interacting strongly with the solvent modes will be considered in the next section. But prior to that, in the following subsection, we provide an alternate formalism based on the concept of restricted configuration-averaged G F for the evaluation of the anodic current. Therein, we also estahlish the equivalence of the newer approach and the method prescribed in section 3.1. 3.4. An Alternate Approach for Current Evaluation Using Restricted Configuration-Averaged GF. The whole process of the charge transfer to electrode, mediated by a chemisorbed layer, can be viewed from a slightly different angle, though finally it leads to the same result for the electron-transfer rate. We start with the condition that site 7, which is coupled to the reactant, is occupied by an adsorbate. The occupancy states of the remaining sites are unspecified. We replace the random medium encompassing the remaining sites by an effective medium using a restricted sites configuration average.12 The picture which emerges now is the one in which the reactant is coupled to an adsorbate occupying the site 1 and embedded in an effective medium. If we calculate the current using this model and take the further average at site 1 (or multiply the final expression by 8, the probability that site 1 is occupied by an adsorbate) we get back the earlier result, wherein no such a priori constraint has been imposed on the system. Mathematically, the link between the two approaches can easily be seen in terms of the following relationship13 = a = adsorbate (44) where ( . . . ) i = a implies that while obtaining the configuration average, the site i, which is occupied by an adsorbate, has been excluded from the averaging. Using the restricted averaging and the linear response formalism with respect to { V,,, V,,),we get the following expression for the partially averaged current
4,
f(c))
(41) (42) (43) ___ (1 1) Tsukada, M. J . Phys. SOC.Jpn. 1976, 41, 899.
(c,+(O) c , ( t ) ) , Im ( G i ) r = a ) B d (45)
where
(C,)i=a = G;;[1 - ( e ; - &9&]-l
(46)
(12) Restricted in the sense that, since the occupancy state of the site i is predetermined, we are deleting it from the configuration average. (13) We should remember that the above result is a limiting case of the more general result. The other term pertaining to vacancy vanishes because of the condition ci m for a vacant site.
-
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987 3429
Electron Transfer through Film-Covered Surfaces Next, as has already been stated, multiplica_tion of (45) by 6 corresponds to the further averaging for site i. Thus, the total averaged current becomes
where (%)!=a
= l/{[Gi(c, { q u h I - ' + E'
(€1- (ca + Chuugavqv)) (53) Y
As we have already seen, the coherent potential E'and 4depends on the variables (4")(cf. eq 49 and 51). Now we make the approximation that (qu)dependence of k'and 6 is treated in an average sense; i.e., Now, using relation 44, the identity between the current expressions 10 and 47 can be established.
4. Current Corresponding to Localized Electronic States in the Adlayer As we have already discussed in section 1 the incorporation of the adsorbatesolvent mode interactions in the formalism becomes essential in the present case. The system is now described by the full Hamiltonian ( 1 ) and the use of linear response formalism leads to the following expression for the averaged anodic current
E'(€,
(4"))
@4(qu)l)
(54)
4(44ul)
4(4(qu)l)
(55)
The procedure for obtaining k'(e,((q,)])and Cii(c,(( 4 " ) ) )has already been delineated in approach A. Now the only term conwhich appears in taining the variables (qv)in (53) is ~,,hwYga,,q,, the adsorbate's energy at the site i. Physically the present approximation means that, at the site i, where the adspecies is considered to be present, the (qY)dependence has been treated explicitly, whereas the further (q,) dependence associated with the inherent randomness at the system has been considered in an averaged sense. S ~ b s t i t u t i n g ' ~ [%]-I
where
4is now given as 4= ( O h u (
@=
tknkr k,a
E-@
+ Cti((qu))niu + l,U
c
)c
+,.OI
[ v k c ~ u + cku
C
JI'
+ iX2(t,6)
(56)
1 ' 2 1'.
-x2(d')
(49)
Im
(50)
Replacing the right-hand side of (57) by its integral representation, substituting the resulting expression in (52), and writing down the explicitly leads to the following expression for the anodic current
+ HC1 +
IZJP
xl(t,e)
in (53), we obtain
)
t.k,u
+ E'=
=
[xl(c,e) - ea
- Chuvgavqv12 + [x2(c,8)12
(57)
and the random quantity cl((qv))is given by expressions 2 and 3. The form of expression 40 is similar to (10) corresponding to a noninteracting adlayersolvent system, and therefore, the analysis presented in section 3 c p be applied to the present case. For evaluating 4,we require k'(e), to be obtained via the self-consistent expression (cf. eq 33)
Y
Self-consistent evaluation of k'via the above expression implies a complicated (q,,)dependence of 4. As a result, the evaluation of ( ...)BC in (48), which requires integrations over the variables {qJ, becomes quite difficult. Hence, we are forced to resort to Below some approximation schemes for evaluating k' and we report two such approximations for this purpose. (A) When the coupling between the adspecies and the bulk solvent is weak, we can replace (qv)in (51) by its average value
eii.
where g u = gov
+ grv
((4Y)l'
As a first approximation (4") can be evaluated by using the classical density function
This approximation amounts to neglecting the dynamical effects arising due to the adsorbate-solvent coupling. In the present approximation scheme, expression 5 1 reduces to expression 33, The provided the ea therein is reinterpreted as t, + ~,hwvga,(qv). formalism developed therein can now be directly applied in the present context in a straightforward manner and therefore we will not repeat the various steps involved in the evaluation of the current. (B) A better approach, wherein the dynamical effects of the adspeciessolvent interaction is partially realized, can be obtained if we use the restricted configuration-averaged formalism (see section 3.3). The configuration-averaged anodic current expression is now written as (cf. eq 47) 26elv l2 7,IA =-L $ _ I d t I m d t ((1 ah2
f(t))e"'lh(
and T is the integration variable appearing in the integral representation of (57). Expression 39 has also been used to arrive at (58). Comparing (58) with eq 16, we observe that both of them are of identical forms. Therefore, proceeding in a manner similar to one described in section 3, we obtain the following expression for the anodic current.
-.
L
1
I"
h2Er "'X,dc
-P[t
sgn [X2(e,6)] X
+ F,,'- FR' + ErI2 4Er
1
(P)-1/2
-m
( cr+(O) Cr(t))F Im (G(c,(qJ) );=a)BC)
(52)
(14) &dependence in ( 5 6 ) appears because Gii and k'depend on 8.
J . Phys. Chem. 1987, 91, 3430-3433
3430
leads to eq 19 of ref 1, the current corresponding to the loneadsorbate system.I5 5. Summary and Conclusions The coherent potential approximation is employed to study the electron-transfer rates through random adlayers. The self-consistent analysis for generating the effective medium potentials (and, hence, the current) contains implicitly the complex dependence on the coverage. The model employed in the present analysis differs from the other treatment@ of the electron transfer via the adsorbed layer in the sense that we specifically handle the strong interaction between the adlayer and the substrate and consider the coverage dependance of the current, though in an implicit way.
w(z) = exp(-z2) erfc (-iz) Q(tv0) = [ ( X I ( G ~-) ea
+ Chuvgvgau)(Cmvgr,Z) Y
-(e -
€1
Y
-I-Ch~ugr,g,)(C~,ga,g,,)I(C~,g,,~)-’ (63) Y
Y
Y
Thus, we observe that the current expression given by (60) depends on the various parameters characterizing the metal adlayer system via the term P1I2 Re w(z). The coverage dependence of the current is determined by the parameter z and a multiplicative factor 0. Also, the anodic currents in the present case and in the case when mediation is realized via a “a lone adsorbate” system are essentially of the same form. It may be interesting to note 0 limit (cf. eq 29 and 30) that in the 0
(15) We should remember that since the adsorbate core charge z, is presently taken to be zero, the coresolvent mode coupling strength gc. vanishes. Also, the current corresponding to the 0 0 limit in the present case contains an additional multiplicative factor 6, which is absent in eq 14 of ref 1. This discrepancy can be traced to the fact that whereas the probability of finding one adsorbate at the mediating site i is 6 in the present case, the presence of an adsorbate at the mediating site has been presumed while obtaining eq 19 in ref I . (16) Bockris, J. O M . ; Khan, S . U. M. Quantum Electrochemistry; Plenum‘ New York, 1979; Chapter 8, p 235.
-
-
[G$’ + E’(€) = e
- w,j
(64)
Now, the evaluation of anodic current via (52), (53), and (64)
Reaction Fields and Solvent Dependence of Nuclear Quadrupole Coupling Constants L. Huis,* J. Bulthuis,* G. van der Zwan,* and C . MacLean* Chemical Laboratory of the Free University, De Boelelaan 1083, 1081 H V Amsterdam, The Netherlands (Received: October 20, 1986)
Quadruple coupling constants of nuclei in liquid-phase molecules often show large deviations from the gas-phase values. Dipolar and quadrupolar reaction fields are invoked t o explain these phenomena. Estimates concur with the experimental findings.
It is well-known that reaction field theory has been very successful in interpreting dielectric phenomena in liquids.8 In this article we investigate how far reaction field theory can explain the above-mentioned solvent dependence of qcc’s. Consider a molecule in a liquid. The molecule polarizes its surroundings and generates its own “reaction” field.8 The reaction field rotates with the molecule, and if this field is inhomogeneous, an extra contribution to the field gradient is generated, which may add to or subtract from the molecular values, thus modifying the qcc. Furthermore, the reaction field can polarize the bonds, an effect also capable of changing qcc’s. (The influence of a crystalline electric field on qcc’s has been discussed, e.g., by B e r ~ o h n ; ~ see also ref 1.) Reaction fields are often huge, say 107-10s V/cm; order of magnitude estimates indicate that local electronic densities may be modified substantially by such fields. Two types of reaction field will be considered: of a nonpolarizable dipole and of an ideal quadrupole. More often than not, the role of quadrupoles (and of higher multipoles) has been underestimated, if considered at all. For instance, in discussing solvent effects on indirect couplings, Smithlo only invokes dipolar reaction fields. In such cases, the reaction field at a particular site in the cavity of the solute molecule is required, rather than its average value over the cavity, as in dielectric theory, which leads to zero quadrupolar (and higher) fields.]’ Many years ago, Buckingham pointed out that reaction fields due to molecular quadrupoles may
Introduction Nuclear quadrupole coupling constants (qcc), measured in the liquid phase, are often different from the gas-phase values, and they may show a substantial solvent dependence.’ For example, the *H qcc of deuteriated benzene as a vapor is 225 kHz; the liquid-state value is 186 k H z 2 2H qcc’s of molecules dissolved in mixtures of liquid crystals may show variations3 of 20-30%. I4N qcc’s are notably solvent dependent. The gas-phase values of methyl isocyanide (CH3NC) and methyl cyanide (CH3CN) are +489.4 f 0.4 kHz and -4.2244 f 0.0015 MHz, re~pectively.~ For methyl isocyanide in the nematic phase of Merck‘s ZLI 1 167 the I4N qcc varies between 150 kHz (50 “C) and 180 kHz (10 0C).5 Yannoni reported a value of 272 f 2 kHz in a different liquid crystaL6 For acetonitrile analogous changes have been reported; nematic-phase experiments5 gave qcc (I4N)= 3.6 f 0.1 MHZ.~ (1) Lucken, E. A. Nuclear Quadrupole Coupling Constants; Academic: London, 1969. Loewenstein, A. Advances in Nuclear Quadrupole Resonance; Heyden: London, 1983; Vol. 5. (2) Zijl, P. C. M. van; MacLean, C . ;Skoglund, C. M.; Bothner-By, A. A. J . Magn. Reson. 1985, 65, 316. Oldani, M.; Ha, T.-K.; Bauder, A. Chem. Phys.-ktt. 1985, 115, 317. (3) Jokisaari, J.; Hiltunen, Y.; J . Magn. Reson. 1986.67, 319. Bjorholm, T.: Jacobsen, J. P.Mol. Phvs. 1984, 51, 65. Jokisaari. J.; Diehl, P.; Amrein, J.; Ijtis, E.J . Magn. Reson. 1983, 52, 193. (4) Kukolich, S . G . J . Chem. Phys. 1972.57.869. Kukolich, S . G.;Ruben, D. J.; Wang, J. H . S.; Williams, R. J . Chem. Phys. 1973, 58, 3155. Emsley, J. W.; Lindon, J. C. N M R Spectroscopy Using Liquid Crystal Solvents; Pergamon: Oxford, 1975. (5) Barbara, T. M. Mol. Phys. 1985, 54, 651. (6) Yannoni, C. S. J . Chem. Phys. 1970, 52, 2005. (7) Gerace, M. J.; Fung, B. M. J . Chem. Phys. 1970, 53, 2984.
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(8) Bottcher, C. J. F.; Bordewijk, P. Theory of Electric Polarization; Elsevier: Amsterdam, 1978; Vols. I, I1 and literature cited therein. (9) Bersohn, R. J . Appl. Phys. 1962, 33, 286. (10) Smith, S. L. Fortschr. Chem. Forsch. 1971, 27, 117. (1 1) Linder, B. In Advances in Chemical Physics; Hirschfelder, J. O., Ed.; Interscience: New York, 1967; Vol. XII, p 225.
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