5588
J . Phys. Chem. 1986, 90, 5588-5597
the radical as compared to that of the parent molecule. The small values of the isotropic hyperfine coupling constants observed experimentally imply that the @-protonsare located close to the radical plane. Small couplings have been observed for the &protons in C H R C H 2 0 H and are probably due to a preferred orientation of the C,-OH bond close to the orbital axis.24 In that case, A , = 49.0 MHz corresponding to pz = 0.70 and A, = 21.8 MHz. A coupling of A , = (0.53/0.70)27.8= 21.0 MHz would thus be expected in the present case. The experimental values are somewhat smaller (Table I). The reasons for the (slight) discrepancy are hard to assess. This geometry is probably preferred also for structures 111 and IV discussed here. At this stage, we cannot distinguish between the two models 111 and IV. Such an analysis must depend on knowledge of the mechanisms of radical formation, as discussed briefly below. Structure I1 can probably be ruled out unless one of the small couplings is assigned to a y-proton. The coupling seems somewhat too large, however.)O Radical 2. According to the ENDOR data, there exists another radical characterized by two small couplings, probably due to P-protons. In Figure 2 is shown a plot of the line positions at Q band when the crystal is rotated about the ( b ) axis. A resonance assigned to radical 2 was observed on the high field side of radical 1 for several orientations (Figure 2 ) . It consists of four lines (dashed curves). The g anisotropy of the resonance assigned to radical 2 is comparable to that of radical 1. There are, however, some indications that more hyperfine couplings could be involved. The Q-band data (Figure 2 ) contain lines on each side of the central resonance of radical 1 at certain orientations. As already mentioned, weak ENDOR lines due to an a-H were observed from radical 2 at a few directions. According to a recent analysis by Brustolon and Cassol, the ENDOR intensities and line widths of an a-proton could be very anisotropic, depending on mixing of nuclear spin states.)I This causes "forbidden" ESR lines so that relaxation rates for these transitions affect the ENDOR intensity. It is tentatively concluded that radical 2 is characterized by one a- and two P-H hyperfine
couplings. The g-factor anisotropy is larger than for a typical > C H fragment. This leads us to assign radical 2 to the similar structure -CH2-CH-C0- proposed for radical 1. A more detailed interpretation is not possible before measurements on the sucrose components (glucose and fructose) have been carried out. Mechanistic Aspects. The radicals observed at room temperature are secondary products. The sequence of reactions starting from the primary (oxidation and reduction) products involves several steps. The details of these steps in sucrose are presently under study.25 It is clear, however, that elimination of water is a characteristic reaction of carbohydrate radicals,23a fact which was clearly understood by Graslund and Lofroth.* They were led to propose a radical structure (11) based partly on mechanistic considerations. Their proposed precursor radical (radical A in ref 8) is, indeed, a dominating product at low temperatures (10 K), but it does not convert directly to the radicals observed at room t e m p e r a t ~ r e . ~ ~ Considerable knowledge about carbohydrates radical reactions in the solid state has accumulated over the past 5 years. This is in part due to some very detailed ESR/ENDOR studies of glucose derivatives11*'2*26 and in part to the studies of the characteristic^^*^' and fate'3,14*28129 of trapped electrons in carbohydrates. These investigations very clearly illustrate the generally complex scheme of radical reactions in these compounds. A similarly complex sequence of reactions takes place in sucrose. It is clear that radical A in sucrose*at low temperatures is formed upon the release of a primary trapped electron9 by visible light bleaching or thermal annealingz5 It must thus be designated a reduction product. On the oxidation side, two alkoxy radicals are present, of which one previously has been analyzed in detail.'O In the temperature region 10-300 K, at least five radical reactions occur. At present, it is not possible unambiguously to state whether radical 1 originates from a primary reduction or oxidation. As noted above, a full discussion will appear e l ~ e w h e r e . ~ ~ Registry No. Sucrose, 57-50-1.
Vibronlc Coupling Model for Mixed-Valence Compounds. Extension to Two-Site Two-Electron Systems Kosmas Prassides,*t Paul N. Schatz,**Kin Y. Wong,ts and Peter Dayt The Inorganic Chemistry Laboratory, Oxford University, Oxford OX1 3QR, England, and the Chemistry Department, University of Virginia, Charlottesville, Virginia 22901 (Received: March 10, 1986)
The two-site one-electron vibronic coupling model of Piepho, Krausz, and Schatz (PKS) for mixed-valencesystems is extended to two-site two-electron dimeric systems. The full dynamic problem is solved and characterization of the system reduces to a simple matrix diagonalization. For strong vibronic coupling, a perturbation treatment is also presented which leads to simple analytical expressions for the eigenvalues and eigenfunctions. The model is used to fit the absorption, resonance Raman, and luminescence spectra of the onedimensional chlorine-bridged mixed-valence compound [PtnL4][PtNL4Cl2]C1,-4H2O (L = ethylamine), Wolffram's Red Salt.
I. Introduction Mixed valency is one of the various names used to describe compounds that contain ions of the same element in two different formal states of oxidation. The first models of mixed-valence systems were proposed by Allen and Hush' and by Robin and Day.Z The latter also proposed a classification scheme which is University of Oxford. f University of Virginia. Present address: General Electric Company, Seminole Trail, Charlottesville, VA 22906.
0022-3654/86/2090-5588$01.50/0
now widely used. The full vibronic problem for a mixed-valence dimer was later solved by Piepho, Krausz, and Schatz) (PKS). Since then the PKS model has been used to calculate a variety of mixed-valence properties, including intervalence band contours,)~~ electron t r a n ~ f e r ,magnetic ~.~ su~ceptibilities,~.~ resonance ( 1 ) Allen, G. C.; Hush, N. S . Prog. Inorg. Chem. 1967, 8, 357-444. (2) Robin, M. B.; Day, P. Adu. Inorg. Chem. Radiochem. 1967, 10, 247-422. (3) Piepho, S.B.; Krausz, E. R.; Schatz, P. N. J . Am. Chem. Soc. 1978, 100. 2996-3005.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 22, 1986 5589
Vibronic Coupling Model for Two-Electron Systems Raman spectra,8-1° and solvent effects.Il Its formulation and use, however, has been largely restricted to dimeric systems which contain a single electron (or hole). Recently, Prassides and DayI2 solved the PKS model in the static limit (zero nuclear kinetic energy) for dimeric mixed valence systems which contain two electrons. The interest in these systems in which there is a two-electron difference between ions of differing oxidation states stems from the fact that they are simple examples of the so-called "negative-U" effect,I3 Le., valency disproportionation of the following form has taken place: (e)Al
+ (elel
-P
(e)A2 + (e)Bo
In this paper we solve the full vibronic problem for this case and show that the model can be used to calculate a variety of mixed-valence properties. We explicitly consider the absorption, resonance Raman, and luminescence spectra of Wolffram's Red Salt as an illustration of the applicability of the model. For such cases of strong vibronic coupling, a perturbation treatment is also presented which leads to simple analytical results.
chloroantimonate(II1,V) compounds and Wolffram's Red Salt (and analogues) are examples of such systems with N = 5 and 4,respectively. We define the zero of energy by WNoA+ WN-,OB = WN-20A wNoB = 0 and define 2 w ( WN-,OA WN-,OB) ( WNoA WN-,OB). As before: we introduce the sum and difference coordinates
+
+
+
Q*
-
*
= (1/2'j2)(Q~
(5)
QB)
We now make the crucial assumptions (which also apply for A
B) kNA= kN-lA= kN-2A= k = y2(l"4
l,,A
+ IN-2'4) = I
(6)
With eq 6, W,, wb, and W, give identical parabolas in Q+ space. Thus the Q+ coordinate decouples from the problem and may be dropped.5 We choose origins by setting QA and QBto zero at the respectively. Introducing the diminima of WN-2Aand WN-2B, mensionless variables
11. Naive Formulation of the Problem
q = 2~(v-/h)'/~Q-
We proceed in close analogy with an earlier treatmentS and start with the following simple picture. Suppose we have two moieties A and B associated with formal oxidation states N and N - 2 , respectively. We write the electronic Hamiltonian operators associated with the two moieties as HeIAand He?, respectively. Then
X
(8~~hv_3)-'/~I
2W = 2W/hv-
+ X2/2
(7)
where V- = (2?r)-Ik1i2 is the vibrational frequency associated with the crucial coordinate Q-, we obtain (after changing the energy zero by X 2 / 2 )
+ Wb/hV- = 2w + y2q2
HelAJ/NA= WNAJ/~A
Wa/hv- = Y2(q
If we assume that there is no interaction between the two subunits, the electronic Schrodinger equation for the system is given by
(HeIA + He?)J/NAJ/N-2B (HelA + HeIB)J/a= (WNA+ WN-zB)J/a
WaILa
(2)
If one or two electrons are transferred from B to A, the zerothorder electronic wave functions of the new states are respectively
Wc/hv- = 1/2(4-
(8)
Now let us remove the assumption that there is no interaction between centers A and B. We hold the nuclei at q = 0 and define
v:
(J/ilVABl#,)o; i, j = a, b, c
(9)
In the symmetrical case, Va> = Vc:. We define the parameters and e' by the relations
E
J/b
--
-
J/c
= J/N-IAJ/N-IB =
ILN-ZAJ/NB
-
- -
Then equations exactly analogous to ( 2 ) apply if the interchanges a b, ( N - 2 ) ( N - l), N ( N - l ) , and a c, A B, B A are made, respectively. We make the harmonic approximation and assume that the subunits have the same point group symmetry in all three oxidation states. Then it is only necessary to consider totally symmetric normal coordinates. We restrict ourselves to a single totally symmetric normal coordinate on each moiety (QA and Q B ) .The vibrational potential energy of subunit j in oxidation state i can then be written
W / = W:j
+ l/Qj + '/zk/Q/z; j
E
Vabo/hv-
E'=
Va:/hv-
(3)
= A, B; i = N , N - 1, N - 2 (4)
where W:j W / ( Q j = 0). Let us initially assume that centers A and B are equivalent (the symmetrical case). The hexa(4) Wong, K. Y.; Schatz, P. N.; Piepho, S. B. J. A m . Chem. SOC.1979, 101, 2793-2803. ( 5 ) Wong, K. Y.;Schatz, P. N. Prog. Inorg. Chem. 1981, 28, 369-449. (6) Wong, K. Y.; Schatz, P. N. Chem. Phys. Lett. 1980, 71, 152-157. (7) Schatz, P. N . In Mixed Valence Compounds, Brown, D. B., Ed.; Reidel: New York, 1980 pp 115-150. (8) Wong, K. Y.; Schatz, P. N. Chem. Phys. Lett. 1980, 73, 456-460. (9) Wong, K. Y.; Schatz, P. N. Chem. Phys. Let?. 1981, 80, 172-177. (IO) Wong, K. Y.; Schatz, P. N . Chem. Phys. Lett. 1984, 108,484-489. (1 1) Wong, K. Y.; Schatz, P. N. In Mechanistic Aspects of Inorganic Reactions, Rorabacher, D. B., Endicott, J. F., Eds.; American Chemical Society: Washington, DC, 1982; ACS Symp. Ser. No. 198, pp 281-299. (12) Prassides, K.; Day, P. J. Chem. SOC.,Faraday Trans. 2 1984, 80, 85-95. (13) Anderson, P. W. Phys. Rev. Lett. 1975, 34, 953-955.
(10)
Changing the zero of energy by Vmo/hv-and substituting the above definitions and eq 8 into eq 2 , we obtain the vibronic matrix in the J / b , J/, electronic basis: 2 e
*'
The constant term (Vbt - Va2)/hv-has been absorbed into 2W. Matrix (1 1) becomes identical with the one proposed by Prassides and Day12 if E' is set to zero. In the following section, we demonstrate that this vibronic matrix has a more general physical meaning. 111. Exact Equivalence with the Two-Site Two-Electron Model14 Consider a general system consisting of two closed-shell moieties, two additional electrons, and the surrounding deformable lattice medium (a mixed-valency dimer in which there is a twoelectron difference between oxidation states is a special case of such a system). Electronic basis functions with associated spin for moieties A and B are simply I,&, P G A p ,J/%and where a and GAp represent the spin functions. The zeroth-order electronic wave functions of the system are
5590 The Journal of Physical Chemistry, Vol. 90, No. 22, 1986 singlets:
+:
do not consider the triplet state further despite the possibility of spin-orbit interaction with the singlets. This interaction is in fact a higher order effect because implicit in eq 12 is the assumption is nondegenerate and thus has no that the orbital $ (qAor orbital angular momentum. Thus for example in our later applications to Wolffram’s Red Salt, $ is the nondegenerate (al,) Pt d,z orbital.
= I$Ba$BB)
$2 = (1/21/2)(I$A. J.BB) + WBaItAS), $,“ =
I$A.
triplets:
IPa $Ba) = (1/21’2)(1py4a $BB) - I$’BaPs)) =
$aT
$bT
$d = IJPB $BB)
(12)
where the two-electron kets are normalized Slater determinants. The electronic Hamiltonian for the dimeric system is
+
+
He! = HeIA HelB
e2 VAB r12
+
(13)
where HelA,He?, and PB have the same meaning as before and $ / r I 2is the electron repulsion operator. We construct the singlet vibronic matrix elements by evaluating ($:lHcll$i); i, j = a, b, c. To connect this procedure to the previous treatment, make the ,$ : $N-29NB $,“, $N-lA$N-lB identifications, $N9N-ZB tbS. We use (HeiA+ He?) in the sense of eq 2 and make the usual approximation of zero overlap between centers, ($+bB) = 0. We again consider one totally symmetric mode for each site (QA,QB) and make the harmonic approximation. As before, Q+ decouples from the system when we assume equal force constants and = (IN 1N-2)/2. Changing to dimensionless variables as before with energy in units of hv- we obtain
-
+
-
+
+ A)’
= ‘/2(4
($;IHeil$bS)
E
$0 = $b (19) Tn(q)commutes with $*, $o (Le. ($ilTnl$j)= “Tn for i , j = +, -, 0) with ($il$j) = 6,. We write the vibronic wave functions as @u
= $+X+.”
(20)
=
C Cj,vnxn(q) n-0
(21)
We separate the vibronic wave functions into those that change (W,,) and those that do not change (PU) sign under interchange: @+v
= $+
C
rvnxn + $-
C
r‘vnxn + $0
r’vn~n+ $-
E
r‘unxn + $0
n=evcn
@-v
(14)
= $+
C
n=dd
n=dd
n=even
n=even
C
n=odd
SvnXn
s’vnxn (22)
Proceeding in exact analogy to the simple PKS t r e a t ~ n e n t we ,~ find that the coefficients runand sVnare determined by the following secular equations:
v,o ($b”lvABl$b”)
($;IVABl$;)
+ +OXO,” + $ - x - , v
The xi,, (j = +, -, 0) are expanded in the complete orthonormal set of harmonic oscillator functions Xn(q),n = 0, 1, 2, ..., m:
= ($b”lHcll$,”)
= 6’
(18)
Tn(q)is the kinetic energy operator for the crucial coordinate q. A . The Symmetrical Case. To take advantage of the interchange symmetry (A = B), we use the electronic basis:
xj.u
in which
Vbbo E
W c i + Tn(q))@u= E“*U
- A2/2
(Jl:IHell$cS) Va,O =
IV. “Exact” Solution of the Dynamic Problem The two-site two-electron system has been studied by RiceI5 within the small X region (weak electron-phonon coupling) with particular application to organic charge-transfer salts. ToyozawaI* has studied the same system over the whole region of A, t, and Win the adiabatic approximation. We proceed in the spirit of the PKS model to solve the full dynamic problem, i.e.
m
+ Va,O/hv- + U ($bSIHcil$bs) = y2q2 + Vbbo/hv- + u’ ($,“lHcll$2) = ‘/2(q- A)’ - X2/2 + V,O/hv- + U
($:iHell$:)
Prassides et al.
=
m
=o =O m = 0 , 1 , 2 ...; v = 0 , 1 , 2 ,...
We assume that centers A and B are sufficiently separated that the intersite electron repulsion is negligible. Thus U’= 0. Note that if we now add (A2/2 - Vmo/hv_- v) to the diagonal elements in eq 14, we obtain (1 l ) , with the identification 2w = x=/2 - u
(16)
The quantity (1/2)X2 represents the lattice stabilization energy due to the electron-phonon interaction and hence the parameter W is a direct measure of the competition between the on-site component U of the interelectronic Coulomb repulsion and the electron-phonon interaction. If we set e’ = 0, matrix (1 1) is identical with the result of T ~ y o z a w aand ’ ~ the relation between our parameters and his is easily established:
= 2 8 = 2 s ; e = -21/2T (17) The energy of the triplet state in eq 12 is also readily evaluated and is equal to ($> (I/JX2 + W, W, has a single minimum at q = 0 (Figure Id). The first excited-state surface W, shows two minima at f q 2 for small values of le1 (Figure Ib) that eventually coalesce as le1 increases (Figure 1, c and d). W,, the second excited state surface, has a single minimum at q = 0 in all cases. A similar pattern is followed in the case W < 0 (Figure 1, e, f, and g). If 0 C (’/&’ + W, Wl has a single minimum at q = 0 for large ltl; otherwise it has three minima, the one at q = 0 being deepest. If 0 > (1/4)X2 + W, Wl always has a single minimum at q = 0. A nonzero value for t’ will lead to a smaller potential barrier in the ground surface (Figure lb). An illuminating discussion of the “phase diagram” of these systems has been given by ToyozawaI4 (for the case t’ = 0). B. The Absorption Band. A characteristic feature of mixedvalence systems (classes I1 and 111)2 is the existence of an ab-
The Journal of Physical Chemistry, Vol. 90, No. 22, 1986 5593
Vibronic Coupling Model for Two-Electron Systems
t
I 45t
(b)
A
/
i
1
tc r I ,\
-8
-6
-4
-2
2
0
4
6
8
9Potential energy surfaces (units of hv) in q space for e = -0.5, = -0.5, X = 5.0, W = 4.1. The vertical arrows indicate the approximate energies of the first (IVBl) and second (IVB2) intervalence bands described, respectively, by eq 42 and 43. The horizontal lines show the location of three of the vibronic levels -onn each surface. The tunneling splittings of the vibronic lines are too small to be seen to the scale of the figure for this strongly localized case. Figure 2. c’
sorption band (the so-called intervalence band) which is not present in the absorption spectrum of either of the single-valence moieties. If there is a one-electron difference between the two centers, the intervalence band is associated with the intermolecular chargetransfer process MN+(A)
h + M(N-~)+(B)-Y, M(N-~)+(A) + MN+(B)
(41)
whence an electron is transferred from the donor site B to the acceptor site A. In the two-site, two-electron case, two such processes and hence two intervalence bands are possible, namely the one-electron charge transfer governed by t MN+(A)
h + M(N-~)+(B)A M(N-I)+(A)+ M(N-~)+(B)
and the two-electron charge transfer governed by MN+(A)
(42)
t’
h
+ M(N-~)+(B)2M(N-~)+(A)+ MN+(B) (43)
The absorption spectrum corresponding to these processes is readily calculated in our treatment because we can calculate the eigenfunctions and eigenvalues associated with the interacting potential surfaces. The band intensity as a function of frequency can be written Z(V) = CvCDJ;(v)
(44)
I
in which D, is the dipole strength of the ith transition whose line shape is given byf;:(v), and the summation is over all transitions. C is a constant which depends upon the units used and simply serves as a scaling factor in our case since we only calculate relative intensities. In our calculations we choosef;(v) = (1/A7r1l2) exp[-(v - V ~ ) ~ / A ?a]normalized , Gaussian which is supposed to represent the various interactions which give each line its breadth. A, is chosen to produce a smooth band and is usually assigned a fixed of eq value for all lines in a band. In the symmetrical case, 25 is calculated from eq 23 if the diagonalization (“exact”) procedure is used or from eq 34-36 if the perturbation treatment is applicable. Likewise, eq 29 or eq 37 is used in the unsymmetrical case. [A very important check of both our exact and perturbational approaches was explicit numerical confirmations that both
0
18
54
36
72
90
E/hu
Calculated absorption contours at 4.2 K showing the two sets of intervalence transitions in the case of nonzero electronic coupling constants c and e’. Each vibronic transition (vertical bar) is assumed to be a Gaussian with half-width 1 . 2 ~ :(a) u = u I = u2 (see eq 47) with parameters X = 5.0, W = 4.1, c = -0.5, c’ = -0.5; (b) u = u , # u2 (see eq 47 and 48) with parameters X = 5.8, W = 23.4, u = 307.4 cm-I, 7 = 0.84, and c = c‘ anywhere in the range 0 to -1.0. The very low energy infrared (tunneling) transitions are of negligible intensity to the scale of the figure. Figure 3.
gave (essentially) the same eigenvalues, eigenvectors, and dipole strengths for strongly localized cases (6, c’ G-FI~
(52)
PO
-
where vs is the frequency of the scattered light and (ap,JGdF is F, with p and u the polarizability tensor for the transition G the polarizations of the incident and scattered radiation: ( ‘ Y ~ , ~ ) G - F=
[
(hc)-’ E’ E
(FIm,lE) (Elm,lG ) vGE
- vL + irE
(Elm,IG) + ( FlmulE) vFE + vL + irE
]
(53)
ma is the component of the electric dipole operator, vL is the frequency of the incident light, and rE is a damping factor associated with intermediate state IE). In principle, the summation runs over the complete vibronic manifold of all electronic states IE) but is restricted in the resonance case to the electronic state in resonance with the exciting light. Since the PKS model yields a “comp1ete”set of vibronic eigenfunctions and eigenvalues if the parameters e, X, W, and v are specified, one can routinely evaluate eq 53 for a specified value of rEand thus explore the predicted characteristics of mixed-valence resonance Raman spectra. These calculations show that in strongly localized systems, in addition to the expected long harmonic vibrational progression (band R, Figure 5), another series of Raman lines can occur whose Stokes shifts correspond to transitions between the two vibronically coupled surfaces. The intensity of these “electronic” Raman transitions relative to the vibrational progression depends critically on the magnitude of t. It was thus argued1° that band L originates
from these “electronic” Raman transitions. Tanino and Kobayashi had, on the other hand, proposed that band L be identified as an emission from a self-trapped state to which the system has relaxed following excitation into the intervalence band.26 A difficulty with the “electronic” Raman assignment’Oarises when the location and width of band L is calculated. While parameters can be found that correctly locate the intervalence band and correctly describe band R and the relative intensities of bands R and L, the calculations consistently overestimate the band R-band L separation, and the width of the latter. It was suggested’O that these discrepancies might be accounted for in a treatment which properly included the Pt”’-Pt”’ surface (Wb), i.e. the treatment we have done in earlier sections of this paper. We have therefore repeated the resonance Raman calculations using eq 52 and 53. J G ) , IF), and IE) all correspond to the eigenfunctions we obtain by coupling the surfaces in eq 49 in accord with the treatment in sections 1V.A and/or V.A (taking note of eq 50 and 51 when T # 1). We use a sufficient basis that truncation of the sum in eq 53 does not significantly affect our results. Despite strenuous efforts, the difficulties encountered in the simpler model persist. Figure 6 shows a typical case where the parameters (e = 4 . 1 0 , X = 10.0, W = 4.1, T = 1.0,v 1 = 307.4 cm-l) have been chosen. These account correctly for the location of the intervalence band and the relative intensities within band R and between bands R and L. However, the calculated separation between the first line of band R and the maximum of band L remains substantially higher than observed, viz. =50vl = 1.906 eV vs. =33vl = 1.258 eV for excitation frequency vL = 6 3 . 2 4 ~ ~ = 2.410 eV. The calculated width is also about twice that observed. All attempts to move band L closer to band R fail if the correct location of the intervalence band is maintained. (If e’ # 0, a second electronic Raman band is predicted with a still larger Stokes shift.) We thus conclude that the present model also cannot account for the location of band L on the basis of an electronic Raman effect. We therefore now use our model to examine the proposal of Tanino and Kobayashi.26 Following excitation into the intervalence band, let us suppose the system relaxes rapidly to the ground vibrational level of the (111,111) potential energy surface (see inset, Figure 5b). We then assume that emission from this state is responsible for band L. (Band B could then be assigned to hot luminescence during the above relaxation process.) We can easily synthesize band L since, by choosing values of e, e’, A, W, T , and v l , we determine the “complete” vibronic manifold. For a strongly localized system like WRS, the perturbation treatment (section VA) can be used.
5596
The Journal of Physical Chemistry, Vol. 90, No. 22, 1986
The intensity of each vibronic emission line Eo+ from
I(&+ -,a;)
0:
-
EOu4)(Xo+)m,la;)12
is obtained (54)
where EO"is the difference in energy between the emitting and the final vibronic state, and the transition moment matrix element is given by eq 50. Each emission line is convoluted with a Gaussian broadening function of the same width arbitrarily chosen to obtain a smooth, unstructured profile. In Figure 5b we show a typical attempt to account for the observed spectrum (but not band B). The agreement is quite good. The relative intensities within band R and the separation of bands R and L are accounted for well although the calculated width of band L is about 50% greater than observed. It is not possible to calculate the relative intensities of bands R and L, and these have been arbitrarily scaled. The parameters used also correctly locate the intervalence band. [In fact the separation between bands R and L can be correctly predicted for a range of values of our parameters (A = 5.4-6.0, W = 22.3-23.4, 7 = 0.77-1.0) which also correctly locate the intervalence band.] We thus conclude that the Tanino-Kobayashi hypothesis provides a basis for explaining the observed W R S spectrum (Figure 5a). There are, however, difficulties in the details of our treatment. First, the value of X used, 4.4-6.0, predicts a value of lxoAlwhich is only about half the observed value (see eq 167 et seq., ref 5). Second, using the parameters giving rise to Figure 5b, we obtain U = - 3 O q = -1.1 eV from eq 16. A simple view of the model requires U > 0, but we have neglected intersite electron repulsion and considered only one coordinate q. It could well be that a positive U would emerge when all degrees of freedom are considered, as in calculation^^^ on the mixed-valence salt Cs2SbC16where all vibrational degrees of freedom and the lattice coulomb energy were included. Clearly, in the present approximate calculations we must regard parameters such as U and X as effective quantities. We note in a recent treatment of W R S and related systems that Nasu)' suggested the parameters 2T S with 2T, U, and S all of the order of 1 eV. Translating to our units via eq 17, this is equivalent to X2 = 2(21/2)1tl, i.e. an intermediate case where vibronic and electronic coupling are comparable. It seems to us that such a possibility can be dismissed because it is incompatible with the observed vibrational resonance Raman spectrum of WRS. Specifically, the latter spectrum shows a very harmonic progression in v, going up to 16 quanta. This means that the lower potential surface is very harmonic up to at least 16q = 4900 cm-' = 0.61 eV, which would not be compatible with the parameters suggested by Nasu (see, for example, Figure 5b, ref 31). It would also seem difficult to understand the low electrical conductivity28 of W R S with such a large electronic coupling ( e ) . Our treatment will not account for the observed26slight shift of band L with excitation frequency. Tanino and Kobayashi suggest that this is due to motion of the excited 111-111 state along the chain, a phenomenon outside the scope of our model. We do note that, in a number of similar 11-IV systems, no shift of band L with excitation frequency is ~ b s e r v e d . ' ~ The predictiong of a possible electronic Raman effect in WRS and similar systems remains unconfirmed. Our calculations (Figure 6) indicate that such a band will be Stokes shifted an additional -0.65 eV with respect to band L. In addition, the predicted intensity of this band decreases rapidly in intensity relative to band R as It1 increases? Thus if le1 is appreciably greater than 0.1, the intensity of this band may be very small. C. The Composite Nature of the Intervalence Band in WRS. The intervalence band in W R S has a rather strange shape. It 1 and then stays almost flat to 2 3 rises very sharply at ~ 2 . eV eV where it is obscured by a strongly rising background absorption (seeFigure 2, ref 26 and Figure 9, ref 33). (The latter also reports
-
-
Prassides et al.
an unexplained smaller band at 1.63 eV which appears at low temperature.) The shape of the band suggests overlapping transitions. We can model this possibility by allowing e' # 0. Such a calculation is illustrated in Figure 3b for the case t'/t = 1.0. (The calculation is sensitive to the ratio but not to the numerical values of t and e' over the range 0 to -1 .O.) It is not possible to compare this contour quantitatively with the experimental band because of the background absorption. But there is a quite reasonable qualitative resemblance. (The parameters used to obtain Figure 3b are the same as those used to obtain Figure 5b except that e' = t instead of t' = 0. Because ItJ,181
