J. Phys. Chem. 1989, 93, 83-89 the photoisomerization in hexane, a nonpolar solvent (dielectric constant e = 1.890), is much less efficient than in chloroform, a polar solvent (t = 4.806).57 The ground state of all-trans-@carotene is n-bonded with conjugation extending along the length of the polyene chain. In the lowest excited singlet state (n-n*), the conjugation is broken, and the molecule may undergo a rotation around the bond at the center of the chain. This motion resuls in a charge separation between the two groups, and the twisted excited state generated by rotation thereby shows zwitterionic character. The interaction between the twisted, charge-separated, excited state and polar solvent (e.g., chloroform) may lower the excited energy surface,51and the transition rate from the excited twisted state to the ground-state surface increases, resulting in a increase in the photoisomerization yield. In nonpolar solvents, the energy gap between the excited twisted state and ground-state surface is not affected, the transition rate between them is small, and the other decay paths are dominant the relaxation of the excited state. In order to confirm this explanation, further studies of photoisomerization of all-trans-@-carotenein a series of solvents of different polarity are planned. While the anharmonic thermal grating technique allows accurate determination of photoisomerization yield, subpicosecond absorption or Raman spectroscopies could provide more direct details of the photoisomerization dynamics. (57) CRC Handbook of Chemistry and Physics; CRC: Boca Raton, FL, 1983-1984.
83
Conclusions
We have studied nonlinear absorption of all-trans-@-carotene solutions in hexane and chloroform by the laser-induced anharmonk dynamic thermal grating technique. Observation of diffraction signals at the second Bragg angle provides a method to detect the nonlinear absorption with zero background. The measured angular sensitivity and decay rates of diffraction transients are in good agreement with the Bragg theory. A two-step absorption model is developed to explain our experimental results. The nonlinear absorption of @-carotene in hexane is understood to be due to the excited-state absorption and while the saturation observed in chloroform is attributed to formation of a long-lived photoisomer. We have estimated an upper limit of the ground-state recovery time is 35 ps in hexane, in agreement with recently reported results. Photoisomerization of trans-@carotene is a significant process in chloroform but was not detected in hexane; the quantum yields of photoisomerization are estimated to be 0.026 and 1 7 X IO4, respectively, in the two solvents. Acknowledgment. We express our appreciation to Dan McGraw and Jack Simons for helpful discussions. This work has been supported in part by the National Science Foundation under Grant CHE-8506667, and by a fellowship grant (to J.M.H.) from the Alfred P. Sloan Foundation. Registry No. all-trans-@-Carotene, 7235-40-7; hexane, 1 10-54-3;
chloroform, 67-66-3.
Vibronic Coupling Model for Mixed-Valence Compounds. Extension to the Multimode Case Kosmas Prassides**+ and Paul N. Schatz**t The Chemistry Department, University of Crete, 711 10 Heraklion, Greece, and the Chemistry Department, University of Virginia, Charlottesuille, Virginia 22903 (Received: February 4, 1988; In Final Form: June 15, 1988)
The two-site two-electron vibronic coupling model for mixed-valencesystems is formulated for an arbitrary number of modes. In both the delocalized and localized limits simple analytical expressions are presented for the positions and intensities of the charge-transfer-induced infrared and intervalence transitions. The two-site one-electron model is also easily extended to the multimode case. The model is used to fit some preliminary infrared data on the (TCNQ-), and (TCNQ),- dimeric moieties isolated in inert gas matrices at low temperatures.
I. Introduction The full vibronic problem for a mixed-valence dimer was solved several years ago by Piepho, Krausz, and Schatz (PKS).’ The original PKS formulation was a two-site one-electron model using a single effective vibrational coordinate to discuss the complete ground vibronic manifold of the system. This manifold was completely characterized by the specification of three (or four) parameters related to properties of the individual monomers A and B: (i) the electron-phonon (vibronic) coupling parameter A; (ii) the electronic coupling parameter t; (iii) the frequency of the single effective mode v-, the antisymmetric combination of a single totally symmetric breathing mode on each monomer; (and in the unsymmetric case A # B; (iv) the difference in zero-point energy between the two-coupled states of the mixed-valence dimer)., Several extensions of the model have now been made: (i) inclusion of spin-orbit and low symmetry effect^;^ (ii) extension to two-electron systems with the inclusion of an on-site electron +University of Crete. *University of Virginia.
0022-3654/89/2093-0083$01.50/0
repulsion parameter U-the two-site two-electron (iii) introduction of a second effective mode to take account of solvent effects6 In addition, Ondrechen and co-workers have made important advance^^,^ by explicitly including the bridging ligand as a third site (with two effective modes in the more general case). (1) Piepho, S. B.; Krausz, E. R.; Schatz, P. N. J . Am. Chem. SOC.1978, 100, 2996. ( 2 ) Wong, K. Y.;Schatz, P. N. Prog. Inorg. Chem. 1981, 28, 369. (3) Neuenschwander, K.; Piepho, S. B.; Schatz, P. N. J. Am. Chem. SOC. 1985, 107, 7862. (4) Prassides, K.; Day, P. J . Chem. SOC.,Faraday Trans. 2 1984,80, 85. (5) Prassides, K.; Schatz, P. N.; Wong, K. Y.; Day, P. J . Phys. Chem. 1986, 90, 5588. (6) Wong, K. Y.; Schatz, P. N. In Mechanistic Aspects of Inorganic Reactions; Rorabacher, D. C., Endicott, T. F., Eds.; ACS Symposium Series 198; American Chemical Society: Washington, DC, 1982; p 281. (7) (a) Root, L. J.; Ondrechen, M. J. Chem. Phys. Lett. 1982, 93, 421. (b) Ondrechen, M. J.; KO,J.; Root, L. J. J. Phys. Chem. 1984.88, 5919. (c) KO, J.; Ondrechen, M. J. Chem. Phys. Lett. 1984, 112, 507. (8) (a) KO,J.; Ondrechen, M. J. J . Am. Chem. SOC.1985,107,6161. (b) KO, J.; Zhang, L.-T.; Ondrechen, M. J. J . Am. Chem. SOC.1986, 108, 1712. (c) Zhang, L.-T.; KO,J.; Ondrechen, M. J. J . Am. Chem. SOC.1987, 109, 1666. (d) Ondrechen, M. J.; KO, J.; Zhang, L.-T. J . Am. Chem. SOC.1987, 109, 1672.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989
Prassides and Schatz
However, it is a feature of all these treatments that only a small number of modes (generally one or two) can be considered in practice since a numerical procedure (usually a matrix diagonalization) is required to extract the eigenvalues and eigenfunctions of the system. The size of the vibrational basis in such procedures grows exponentially with the number of modes, and the treatment very rapidly becomes intractable. This places a potentially severe limitation on all of these treatments because one expects a priori that many modes may participate in the vibronic coupling of a mixed-valence system. Indeed it has been demonstrated in the case of (TCNQ),- and (TCNQ-)2 that all 10 ag modes of the ring play an important TCNQ (7,7,8,8-tetracyanoquinodimethane) and explicit r0le.~9’~ It is the purpose of this paper to show that in both the localized and delocalized limits it is possible to give explicit analytical solutions for the two-site two-electron problem for any number of modes. Fortunately, most real systems seem to fall into one of these two regimes, and thus an important limitation of these models is easily removed. (The two-site one-electron case (simple PKS) is also easily treated.) It also seems reasonable to expect that a similar approach can be applied in more complicated
We now permit interaction between subunits A and B and write
84
11. Two-Center Two-Electron Systems
Following our earlier treatment,5 we consider a mixed valence dimer AB and associate formal oxidation states N a n d N - 2 with subunits A and B, respectively. In the limit of no electronic interaction, we write the zeroth-order electronic wavefunctions as
with the nuclei fixed at all q, = 0. Defining a new energy zero by Va/aaO = Vm0= 0, we obtain (9), the vibronic matrix in the +a, +b, +c electronic basis for an arbitrary number of vibrational modes (cf. eq 11 of ref 5 for the one-mode case). The constant term
(9) vb? has been absorbed into -U. The parameter U can be identified5 as the difference betwee! the intrasite U and the intersite U’electron repulsion, i.e. U = U - U’. If wtassume that centeEs A and B are sufficiently separated, then U >> U’and U = U. A . The Nearly Delocalized Case. To take advantage of the interchange symmetry (A = B) we use the electronic basis +i
= ( l /2i’2)(k
*
+c); $0
=
+b
and rewrite the vibronic matrix in the ++, t,b0,
(10)
+- basis as (1 1).
= h”+N-2B $b = +N-lAIC/N-iB #c = +N-2A$~B (1) restricting ourselves to singlet states. The vibrational potential energy of subunit j in oxidation state i can be written as +a
W: = W:J
+ Cla/Qaj + C’/zka:Qat; a
j = A, B; i=N,N-l,N-2
(2)
where W,“j = Cawa:( Q , = 0) and Qaj is the a t h totally symmetric normal coordinate of thejth subunit. We have made the harmonic approximation, have assumed that the two subunits have the same point group symmetry in all three oxidation states, and for simplicity consider the symmetric case (A = B). We define the sum and difference coordinates for each mode
In the limit of a fully delocalized system, all A, are equal to zero. However, we see in this case that only uncouples; ++ and q0remain coupled through the term 2’I2VO.We therefore choose a new electronic basis
+-
\ko = ci-++
\k, = \k2
a
= (1/21/2)(QaA f QaB)
Qia
(3)
and the parameter U as
+ c2-+0
*-
= ci+++
(12)
+ c2++0
and require that the coefficients cl*, c2* diagonalize ( 1 1) when all A, = 0. The results are W2,o = C’/2q2hva- 1/2(U - V,l) f ‘/2R
u = ( W j p + W?+,OB) - (WN-,OA + WN-,OB)
(4) We further set WPA= WYB,laiA = la?, ka? = k,?; i = N, N 1, N - 2. We choose origins by setting WNoA + WNW2OB = 0 and la(N-2)A= 1a(N-2)B = 0. If we make the assumptions that kaiA e k, and (5) = y2(laNA+ la(N-2)A)= I , the potential surfaces W,, W,, and Wc give identical parabolas in Q+OLspace, and the problem becomes separable with respect to Q+, and Q-* modes. As b e f ~ r ewe , ~ introduce the dimensionless variables q, = 2 n ( ~ , / h ) ~ / ~ Q - ”A;, = (8n2hv,3)-’/21a (6)
a
+ t2)‘/2]/(1 + [t F (1 + t2)i/2]2)’/2(13) with t s ( U + Vd)/2(2)i/2V0,R = [ ( U + Vi)’ + 8V02]1/2, (cli)2 + (c~’)~= 1, and (\kllHeil\ki),i = 0, 1, 2. W2and Woapply cli = [t
7
(1
respectively for all upper and all lower signs in eq 13. Hence the vibronic matrix (1 1 ) may be written in the \ko, \kl, \k2 electronic basis as (14). We note that cl* commute with the nuclear kinetic
where Y, = ( 2 ~ ) - ’ k , ~is/ ~the vibrational frequency associated with Q-“. Then we obtain for the potential surfaces in q, space: Wa = C(’/2qa2+ Aaqa)hva a
wb
= Cf/2q2hva a
u
(7)
Wc = C(’/24,2 - Aaqa)hua a
(9) Rice, M. J.; Yartsev, V. M.; Jacobsen, C. S. Phys. Rev. B Condens. Matter 1980, 21, 3437. (10) Rice, M. J. Solid State Commun. 1979, 31, 93.
energy operator since they are not explicit functions of the q,. The potential surfaces that arise when (14) is diagonalized are shown in Figure 1 for three cases in the one-mode limit. We proceed now to solve the full dynamic problem: (He1 + TAqi . . . S i ) ) @ ” = Ev@v
(1 5 )
where T, is the kinetic energy operator in coordinates qa, a = 1, 2, . . ., i. The vibronic wave functions are written 9”= *OXO.” +
\klXl,”
+ *2X2,”
(16)
Vibronic Coupling Model for Mixed-Valence Systems
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 85
The x (j= 0, 1, 2) are expanded in the complete orthonormal sets o h a r m o n i c oscillator functions: xn, (q,), n, = 0,1,2, . . ., so that
Examining (14),we note that if all A, = 0, Wo,W,, and W2are independent potential surfaces and the exact vibronic wave functions are
aj,?= \Ejxj,v, j
= 0, 1 , 2
(18)
The corresponding eigenenergies are =
E(n, + Y2)hva + K
E2,nl...n,O =
E(%+ Y 2 W , + rJ
Eo,n
U
where K -1/2(U-Vd)- 1 / 2 Rand u = -‘l2(V- Vd) + I/2R. We may use &>, c$],: &,” as the unperturbed basis and treat the vibronic coupling constants A, as the perturbation. Our assumption in considering the solution “on” the lower surface is that the energy gap between the middle and lower surfaces is much larger than the energy spacings “on” the lower surface, i.e. using eq 14 IW, - Wol >> hv,, for all a
(20)
Similarly for the solution “on” the middle surface, the conditions IWl - Wol
-
IW2
- Wll >> hv,
(21)
have to be satisfied for all a. Using eq 19 and setting Vd = 0 for simplicity, we obtain W, - Wo = ’ / 2 U + ’ / 2 ( f l 8V02)’/’ and W2- W , = - I / 2 U + ‘/2(@ + 8V02)1/2.So, we can see that condition (20) is always satisfied in the nearly delocalized limit, whereas condition (21) is applicable only when 2(2)1/21Vol>> U. Under these circumstances we obtain for the energies to second order by using standard perturbation theory
+
Eo,nl...ni =
We can now calculate transition moments between vibronic levels “on” the same or different surfaces. Transitions between levels “on” the lower surface give rise to the so-called charge-transfer-induced infrared (CTIIR) absorptions. Their dipole strengths are given by (using eq 23)
DIR,E
I(ao,nt...nIImlIa0, n l . , . n ~ + l . . . n i ) 1 2
= 2 ~ ( ~ ~ - ) 4+~ 1 ):( K(, ~+, t,’)2/[(Ka
+ td)2 - 1 1 2
(24)
where M ($+lm,l$-). Hence if there are i totally symmetric vibrational modes for each monomer we expect the appearance in the infrared of i CTIIR absorptions at energies lower than the monomer ag modes, given by ( K , and e,’ are > 2(2)1/21V01(Figure IC), we note from eq 19 that the energy gap between the upper and middle surfaces is approximately (2V;/U) (Vd = O for simplicity). Consequently our perturbation treatment cannot be used to describe the energies and wave functions “on” the middle and upper surfaces (Figure 1); that is, the electronic basis of eq 12 is no longer appropriate. Instead we set Vo= 0, the two upper surfaces become degenerate, and our electronic basis is
q0 = +o, \kl = +-, and \k2 = ++ (29) Using perturbation theory, we obtain for the energies (cf. eq 22) “on” the three surfaces (in the case of U >> IVol >> A,): Eo,nl,,,n,=
t..-IO
CU ( n u + ‘/z)hva- u - (2Voz/V)
E1,n,,,,nt=
Ca ( n a + ‘/z)hva- C’/2X,Zhva U
~
II.; 1 0 0
I
(30)
E24 l...n, = C(na + Y2)hva - EY’/2X,ZhVa+ (2VO2/U) a
a
and for the eigenfunctions (cf. eq 23) @o,nI,,.n# =
*oyxnjqj)
- (21’2~o/U)*2v~n,(~j) !5 c
1
-4
3
0 0
-2 0
2 0
-
,
I
4 0
q.
Figure 1. Potential energy surfaces (in units of hv,) in q, space for a range of values of A,, V,, and t, (with €011 = 0): (a) A, = 0.5, V , = 1.O, tu = -10.0; (b) A, = 0.5, Va = 5.0, t u = -1.5; (c) A, = 0.5, U, = 10.0, tu = -1.0.
where the vibrational wave functions x,,,~, xnb=, and xk,. are harmonic oscillator functions centered about q, = -Aa, 0, and A,, respectively. Their corresponding eigenenergies are
Using the above expressions, we can easily derive the positions and intensities of all allowed transitions. A noteworthy result is that to first order, the transition moment of the CTIIR transitions (i.e. @ ~ , n l . . , n w , , n , @o,nl ...n,,+l,.,n,) is zero. E. The Nearly Localized Case. In this case we start from the other extreme, setting Vo= V,l = 0. The appropriate electronic basis5 is then +a, +b, q0 as defined in eq 1 and 2. Using the vibronic matrix of (9), wa, wb, and Wc (eq 7) are three independent harmonic potential surfaces. The corresponding exact vibronic eigenfunctions are -+
@ n l . . . n ~ o= +aqxkJ, @nl...n;O
=
+bVXn&
@nl.,.np = +c?Xd (32)
Enl...ny = C(na + Y2)hva a
u
(33)
Vois then treated as the perturbation using the zeroth-order basis functions
@nl..,ny = (1 /21’2)(@nl,,,n~0 *~-~)v~n~,.,n~o~ I
~ n l , . , = n ~@
( v = even,
+; v = odd, -)
(34)
with v = Canuand coupling (+) vibronic levels with (+) and (-) with (-). The energies to first order are Eanl,,?* =
+ 1/21 C(na a
Ezn,.”,+ =
* (-l)’Vd~(xn.~Ixn~)
C(na + YZ + Y2A,2)hva- U a
(35)
Vibronic Coupling Model for Mixed-Valence Systems
The Journal of Physical Chemistry, Vol. 93, No. 1. 1989 87
where ( xrlx,.) is the overlap integral between displaced harmonic osci lators of equal force constant in the ath mode.” Using the first-order eigenfunctions, we can evaluate the transition moments. Following our one-mode treatment in the localized limit,5 we again distinguish three sets of transitions: (i) the so-called tunneling transitions between the @nl,,,n: split pairs for which
tional quantum number “on“ the ground surface. These comprise the CTIIR absorptions for which the energies and dipole strengths are
D I R ~I( @ n l...nl+ImrI@n,...niT) I’ = M2
(36)
(ii) a set of intervalence transitions governed by the interaction parameter, Vo, formally involving the transfer of one electron between states DIVEE =
EIRa/hVa = 1
DIRu= 8(na + 1)t,2X,2M2/(4ca2 - 1)’
U
(37) and (iii) a set of intervalence transitions governed by Vd, formally involving a two-electron transfer D I V BE ~I( @ n l ~ , . . n ; * I ~ ~ I @ n l . . . n ~ ) I ’
= ~ V , ~ ’ M ~ ~ I ( X ~ ~ ~ I X~na)hva12 ,)I~/[C(~ (38) ,‘ J
U
DIVE= [I -
Ua n a + Y2)X,2/(4t,2 - 1)I2M2
The approach we have employed in section I1 can easily be applied to the simple PKS model.’ In this case, there is only one electron that can be transferred between the subunits. Perturbation results in the one-mode case have been quoted for localized systems by Wong and Schatz,’ and delocalized systems have been treated by Wong.I2 The extension to many modes is immediate since each mode contributes independently to first order. We here present a brief treatment for bothdimits. A . The Nearly Delocalized Case. In this limit, the vibronic matrix (9) can be written in the $+, $- basis as (39),where qa,
(44)
and the energy is
ElvB= -2V0 - C2X2(hva)ta(2na+ 1)/(4c2 - 1) (45) a
The rest of the allowed transitions involve second-order terms and consequently make small contributions to the intervalence band. For completeness they are summarized in Appendix A2. B. The Nearly Localized Case. The appropriate electronic basis in this case is $a and $b (eq 40). The exact vibronic eigenfunctions are
@nl,,,ny = ( Y ~ ” ’ ) ( $ aJ ~ x n ~* (-I)V~?X&)
111. Two-Center One-Electron Systems
(43)
where M = ($alm,l$b). Similarly, the major contribution to the intervalence band arises from transitions “between” the two surfaces with no change in vibrational quantum numbers. The dipole strength is
I ( ~ n~, . . . ~ ; * I ~ ~ I ~ n ~ . . . n ~ ) I ’
~~O’MZIII(X,~X~~)~~/[C(~,’ - n, - &b%va - u]’ I
+ 2~,X2/(4t2 - 1)
(46)
when Vo(=tahva)is set equal to zero and v = Cana. When Vois turned on, it couples vibronic levels (+) with (+) and (-) with (-). The energies are
Eo,,
“t*
=
m a
a
+ Y21hV.3 f (-1)”Vo?(x,lIx&)
(47)
In analogy with eq 36 and 37, we obtain for the dipole strengths of the CTIIR absorptions DIRa = I($almzl$b)l’
E
hf
(48)
and for the intervalence band DIVB
Xa are defined as in eq 6, but Vo now is given by Vo = = ($aIVABl$b)’ (cf. eq 8). Furthermore, we use the definitions $i
=
(Y2”’)($a
$b)
(40) = $NA$N-IB; $b = J/N-IA+NB Following section HA, we set all X, equal to zero whence $+ and $- uncouple and can be used as the unperturbed wave functions. Then we obtain for the energy to second order in the perturbations X, $a
with
t,
Vo/hva,and for the first-order wave functions
Using eqs 41 and 42, we calculate transition energies and moments. One set of transitions involves a change of + 1 in a single vibra(11) Fulton, R. L.; Gouterman, M. J . Chem. Phys. 1961, 35, 1059. (12) Wong, K. Y . Inorg. Chem. 1984, 23, 1285. (13) Wong, K. Y.; Schatz, P. N.; Piepho, S.B. J . Am. Chem. SOC.1979, 101, 2793.
=
~ ~ , ~ M ~ ~ I ( xa ~ ~-~n,)Y I x ~ )(49) I’/[C(~
IV. Discussion We have presented a perturbation treatment valid for any number of vibrational modes coupled to the electronic transition between the two centers of a mixed-valence dimer. The treatment applies in both the (strongly) localized and (strongly) delocalized regimes and for both the one- and two-electron cases. Since most physical systems fall into one of these regimes, a major limitation of the onemode (or two-mode) model is removed. (We have made numerical checks of our analytic expressions by comparing them in the one-mode limit with appropriate exact In the simple PKS case (two-center one-electron systems) the situation is straightforward. One can generally divide the absorption spectrum into two regions. The first involves transitions “between” the ground- and excited-state potential surfaces and comprises the intervalence band. The transition energy is >> hva, where v, is the vibrational frequency of the a t h antisymmetric combination of the modes of the two monomers. The second involves transitions in the infrared, the CTIIR absorptions occurring between split levels “on” the lower potential surface with transition energies Shv,. Both kinds of transitions are correctly handled in both limits by our treatment. The situation is again straightforward for strongly localized mixed-valence systems in the two-center two-electron case (eq 36-38). In nearly delocalized systems, however, we must distinguish two cases: (i) IVol >> U and (ii) U >> lPol, as illustrated in Figure la,c. Our perturbation treatment is applicable in both these limits. A noteworthy result in the first case is the existence of a set of transitions “between” the ground- and second excitation-state surface giving rise to a second intervalence band even if Vd = 0 (eq 28). This occurs on the high-energy side of the main intervalence band, and its intensity is predicted to be of the same order of magnitude as the CTIIR absorptions.
88 The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 TABLE I: Vibronic Coupling Model Parameters for (TCNQ), Dimers mode unlcrn-la Eraulcm-lb Anc And Amc 0.16 0.14 0.34 0.23 0.35 0.37 0.25 0.25 0.38 0.26 0.28 electronic coupling constant, Vo/crn-': -1780, -1580, -1864 2
3 4 5 7
2221 1604 1422 1206 720
2196 1602 1379 1193 698
0.11 0.05 0.25 0.17
Average of corresponding ag frequencies of TCNQ and TCNQ- in an Ar rnatrix.17 bobserved for (TCNQ)T prepared by codepositing Na and TCNQ with a large excess of Ar.I7 CUsingeq 45 and the first equation of (43). d(MEM)(TCNQ)2;MEM is N-methyl-N-ethylmorpholinium (ref 9). (TeEA)(TCNQ),; TeEA is tetraethylammonium (ref 16).
-
A further possibility in the case of nearly delocalized systems is U 2(2)1/21Vo(>> A, (Figure lb). In this case we cannot use perturbation theory to treat the two excited surfaces. A complete treatment would involve use of the exact PKS model to treat the two excited states while the ground state could be dealt with perturbationally. Individually, these problems become intractable in the many-mode case. However, we can still use eq 24-27 to obtain the energies and intensities of the CTIIR and IVB transitions at zero K as long as the second-order terms in eq 26 and 27 are ignored. V. An Application to TCNQ Dimers To illustrate the use of our formalism, we apply it to TCNQ dimers. In general, the planar TCNQ molecules stack to form linear conducting c h a i n ~ . ~ J ~Typically, J ~ ' ~ these chains contain repeating TCNQ dimer units with weak interdimer interactions. Most commonly we can formally associate either one-half or one unpaired electron with each T C N Q moiety to obtain either (TCNQ),- or (TCNQ-),. These are respectively two-center one-electron and two-center two-electron systems. Both TCNQO and TCNQ- have DZhsymmetry with 10 totally symmetric (a,) infrared active vibrational modes. In the mixed-valence dimers, an intervalence band and 10 CTIIR absorptions are ob~erved,9.10,14-16 Rice and c o - w o r k e r ~ have ~ ~ ' ~used linear response theory to extract the electron-phonon coupling parameter for each CTIIR mode observed and the electronic coupling constant and, where appropriate, the effective electron repulsion parameter U. We can perform the same kind of analysis. However, our treatment has the important advantage of providing explicit analytical expressions (eq 24, 25, 43). Furthermore, their range of validity may be assessed by perturbation theory. (A numerical comparison between linear response theory, the perturbation expressions of eq 43, and an exact diagonalization also suggests that the perturbation treatment is a bit more accurate.I2) We here fit some recent experimental results of Rosei7 on (TCNQ),- and (TCNQ-), isolated in Ar, Kr, and Xe matrices. These anions were preparedI7 by codepositing TCNQ and an alkali metal (Cs, Na, K) with a large excess of noble gas on an alkali halide window. Thus, interdimer interactions should be smaller than in the crystalline salts. By varying the concentration of TCNQ and the metal, it is possible to control which dimeric species is formed predominantly. In general the bands are sharper because of the low temperature (1 1 K) and the isolation of the dimers.I7 T C N P s T C N Q Dimers. The intervalence band is observed at -3800 cm-', and five CTIIR absorptions can be unambiguously assigned to the one-electron dimer." Using the analytical expression, eq 43, for the position of the CTIIR absorptions and eq 45 for the IVB position, we can extract the five electron-phonon (14) Rice, M. J.; Lipari, N. 0.;Strassler, S.Phys. Reu. Lett. 1977, 39, 1359
(15) Tanner, D. B.; Miller, J. S.;Rice, M. J.; Ritsko, J. J. Phys. Rev. B: Condens. Matter 1980, 21, 5835. (16) Zelezny, V.; Petzelt, J.; Swietlik, R. Phys. Status Solidi B 1987, 140, 595. (17) Rose, J. L. Ph.D. Thesis, University of Virginia, 1987
Prassides and Schatz TABLE 11: Vibronic Coupling Model Parameters for (TCNQ-), Dimers mode u,/cm-l" ElRU/crn-lb Amc And Ane 2 3 4 5 7
2192 1613 1389 1195 124
2185 1582 1345 1186 721
0.29 0.85 1.18 0.62 0.60
0.57 0.82 0.82 0.55 0.47 electronic coupling constant, V0/crn-': -3080 on-site electron repulsion, U/crn-I: 8000
0.74 0.67 0.99 0.58
llag frequency of TCNQ- in an Ar rnatrix.17 bobserved for (TCNQ-), prepared by codepositing M and TCNQ with a large excess of Ar;17 M = Na, K, Cs. CUsingeq 25 and 27 and fixing U = 8000 cm-'. d K (TCNQ) (ref 10). '(DMeFc)(TCNQ); DMeFc is 1:l decamethylferroceniurn (ref 15). coupling constants A, and the electronic coupling constant Vo. The results are summarized in Table I together with the values extracted from data on crystalline salts. Except for mode 3, there is reasonable agreement between the different sets of parameters. Each vo has been chosen as the average of the corresponding TCNQO and TCNQ- aBfrequencies." If the relative intensity of the intervalence and CTIIR bands could also be determined, we could use eq 44 and the second equation of (43) to check the internal consistency of the model. T C N Q o T C N Q Dimers. In this case, there are at present no experimental data in the energy region where the intervalence band of the matrix-isolated dimer is expected to occur. For this reason, we assume that the position of the intervalence band is -1.2 eV.10,15Again, five out of the 10 expected CTIIR absorptions can be unambiguously assigned to the two-electron matrix-isolated dimer.I7 Using eq 25 and 27, we extract the electron-phonon coupling constants A,. These are presented in Table I1 together with values from crystalline salts. Using values of 8000 cm-' for the on-site electron repulsion, U, and -3000 cm-I for the electronic coupling constant, V,, we obtain for the position of the intervalence band, E = 9350 cm-l (=1.16 eV). Our parameters (A,, V,) are related to the ones used by Ricelo by 2g, = A,hv,
and
Vo = -2'i2f
(50)
VI. Conclusions In this paper, we have investigated the vibronics of simple dimeric systems with one or two electrons coupled to an arbitrary number of vibrational modes. In the limits of strong or weak delocalization, each mode contributes independently to first order, and the simple analytic expressions allow the treatment of multimode systems. We have illustrated our model with an application to the organic a acceptor, TCNQ, which we view as a prototype many-mode weakly vibronically coupled mixed-valence system. Our model is clearly applicable to the whole family of charge-transfer organic semiconductors?JO which almost invariably show both IR-enhanced and charge-transfer absorption. Indeed in such cases a single-mode model is a priori inadequate. Further, the model removes the assumptions involved in relating effective vibronic coupling parameters and effectiue vibrational modes to observed bond lengths and frequencies. O n the other hand, no CTIIR absorptions have been of sufficient intensity in localized mixed-valence systems to have been observed experimentally. In such cases, it is more difficult to unravel the effects of more than one important vibrational coordinate, except when there are obvious broadening or asymmetry effects6 Potentially informative methods under such circumstances are the study of the temperature4 and/or pressureI8 dependence of the half-width, energy maximum, and integrated intensity of the intervalence band. Indeed, we have unambiguously demonstrated4 the inadequacy of a one-dimensional model and the necessity of coupling low-frequency lattice motions to the intervalence transition in the case of SbC1,3--SbC16- moieties through a careful (18) Sinha, U.; Lowery, M. D.; Ley, W. W.; Drickamer, H. G.; Hendrickson, D. N. J . A m . Chem. SOC.1988, 110, 2471.
J . Phys. Chem. 1989, 93, 89-94 analysis of the temperature dependence of the intervalence band moments. The model will also clearly be applicable to other systems as new data accumulate. For example, Collman et aLl9have recently reported the presence of both an intervalence band and a CTIIR absorption in a partially oxidized sample of [Os(OEP)(pyz)ln (OEP is octaethylporphyrin), and Boekelheide and co-workers20*21 have reported the synthesis of mixed-valence dimers with a two-electron difference in oxidation state of the metal atom. It would also be possible to extend our approach to more complicated cases, for example a many-mode treatment of the three-center model7**or a many-mode system in which one of the modes must be treated exactly.
Acknowledgment. This work was supported in part by the National Science Foundation (P.N.S.) under Grants CHE8400423 and CHE8700754. We also acknowledge receipt of a NATO travel grant (RGO 146/87).
89
(ii) An, = f 2 , AnBza = 0: (TM)*2 =
-y'(
Kn,
+ 1) f 11P2
f
c1-)2X,*M
K,
+
7
C',
(
1
(cI-)2
K,
+
C',
F
+
1
(CI+)'
u,
+
6,'
F 1
(iii) An, = fl, An, = f l : (TM)*l.*l =
+ 1 ) f 1)'/2((2ng+ 1) 1)112+ 7 1)(Kp + €81 7 1) ~ ~ ( C , - ) ( C ~ + ) ~ X , X , M (+ ( ~ 1) ~ , f 1)'/2((2n, + 1) f 1)1/2 x -y2(C1-)3XaXBM
(K,
6',
Appendix A1 Second-Order Terms in Eq 26. Second-order contributions to the dipole strength arise for three cases as follows (D N (TM12): (i) An, = 0: Appendix A2 Secondorder Terms in Eq 44. Second-order contributions arise as follows (D N ITMI'):
(i) An, = f 2 , AnBZa= 0: (n, (K,
+
' ,€
+ 1)/2
- l)(U,
+ t,'
(TM),, = -'/zXa2M((n, f l)(n,
+ 1 f l))lI2/(2~,7 1)'
+I)
(A41 (ii) An, = fl,An, = fl:
(19) Collman, J. P.; McDevitt, J. T.; Leidner, C. R.; Y e , C. T.; Torrance, J.; Little, W. A. J . Am. Chem. SOC.1987, 109, 4606. (20) Voegeli, R. H.; Kang, H. C.; Finke, R. G.; Boekelheide, V. J . Am. Chem. Soc. 1986, 108, 7010. (21) Plitzko, K. D.; Boekelheide, V. Angew. Chem., In!. Ed. Engl. 1987, 26, 700.
(TM)*l,*l = -'/zV@4K2na + 1 f 1) x (2n, 1 f 1))'/'/[(2t,
+
T
1)(2t8 7 l ) ] (A5)
Registry No. (TCNQ)y, 523 15-68-1; (TCNQ-),, 34464-96-5.
Electronic States and Potential Energy Surfaces of ASH,:
Comparison with AuH,
K. Balasubramanian*s+and M. Z. Liao Department of Chemistry, Arizona State University, Tempe, Arizona 85287- 1604 (Received: February 8, 1988)
Complete active space MCSCF (CASSCF) followed by second-order configuration interaction (SOCI) and multireference single plus double configuration interaction (MRSDCI) which include excitations from the d shells are carried out on the two low-lying states of AgH, (,B2 and 'A,). The bending potential energy curves of the two states with bond lengths optimized for all angles are presented. The 'Bz surface contains a double minimum with acute and obtuse H-Ag-H angles. The 2Al surface contains a large barrier for the insertion of an Ag(,S) atom into Hz to form the linear 2Zg+state of AgH,. The excited Ag(,P) atom spontaneously inserts into H, to form an acute-angled weak complex of Ag with H2 and another more stable ,B2state with obtuse bending angle. The d shell correlation lowers the 'B2 obtuse-angled structure significantly. The potential energy surfaces and electronic states of AgH, are compared with those of AuH2. Relativistic mass-velccity effects are significant for AuH, in comparison to AgH,, while d correlation effects are more significant for AgH, in comparison to AuH2. The Mulliken population analyses of the electronic states of AgH, reveal considerable 5p participation. The bending potential energy surfaces of the 'B, and 'Al states of both the molecules cross, which would lead to avoided crossing of the 2E,/2components if the spin-orbit term is included. The effect of f-type polarization functions is also investigated by carrying out MRSDCI calculations which included the ten-component f functions in the basis sets.
1. Introduction The electronic properties, geometries and reactivities of transition-metal clusters are topics of intense activity in recent years. The investigation of metal atom insertion into hydrogen bonds Alfred P. Sloan Fellow; Camille and Henry Dreyfus Teacher-Scholar.
0022-3654/89/2093-0089$01.50/0
could provide considerable insight into the reactivities of small clusters with H2.I4 Further, such calculations could also provide (1) Daudey, J. P.; Jeung, G.; Ruiz, M. E.; Novara, 0. Mol. Phys. 1982, 46, 67. (2) Muller, E. W.; Tsong, T. T. Prog. Surf. Sci. 1973, 4, 1.
0 1989 American Chemical Society
