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Electron mobility in weak donor-acceptor complexes
\‘otumr 67. number I
ELECTRON
Ctil:\tICAL
MOBILITY
IN WEAK
DONOR-ACCEPTOR
Received 79 Ma_\ 1979: m tinzl form 1 hugzst
A Jmple qruntum-mechsniui along rhe donor-acceptor srxhs cept of ‘fuperrransfsr” inroh-ins
! So%ember 1979
PHYSICS LEITCRS
COMPLEXES
1979
model is propowd
to account for rhr e~perimenrally
observed Irtrge elecrron mobilities
in rtnthracene-PUD.4 rhe %irru;ll
2nd phen~nthrene-PMDA a_vstals_ The model is based on the concharge-rranxfer stana creared by polari~rion -
I _ Introduction in ZI recent
that the coupling of Frenkel excitons with the charge-transfer (CT) states carrier can result in increased charge carrier mobilities in the crystals of some weak chargetransfer compkes. This effect was expected to be operative for the compleses crystallizing in alternating _..ADADAD___donor-acceptor stacks, and to affect the charge carrier mobilities along the stack axis. As long JS cr>sral polarization is disregarded, the transport of, e_=_ = an electron along the stack axis. shouid be limited to jumps between the acceptor molecules. xkith the donor molecules not participating in the process_ Crystal polarization into the CT states (via the Frenkel states) enables the donor molecules to contribute to electron transfer aIong the stack. This is due to the processes, M here a mobile electron annihilates a “hoIe” on the donor molecuIes involved in d virtwl CT state created by polarization [ 1J _ As the distance from the acceptor molecule to the donor molecule is smaller (by more or less the factor of twoj than the distance to the next acceptor moIecule, the corresponding matris eiement should be larger. Hence. eiectron mobility shouid be increased due to the contribution from some CT states to crystal polarization *_ A possible consequence would be an anomalously high (with respect to the other crystal direction Is) charge carrier mobility along the stack asis (11x)_ Such anomalous electron mobility was observed indeed in the crystals of the complexes of pyromellitic dhn(,u,,~ = 0.15, hydride (P&IDA) with phenanthrene (~_t,t~= 0.018, pli, = 0.0075 cm2 V-1 s-t [3,rf]) and anthracene pLr =O.O? cm? V-* s-* [S]). It ws conjectured from these results [3] that the neutral donor molecules could contribute to the electron mobiIity, i.e. that the electron wavefunction was partially delocalized over the donor sites. which was intuitilelk rather surprising. The objectite of this letter is to demonstrate that electron mobility of the order of those observed in PMDA complexes can reasonably be accounted for in terms of a simple model postulating the involvement of the virtual CT states created by polarization_ xijacent
paper
[I ] we suggested
to the charge
2_ The hamiltonian Let us consider ’ This conddention
a rigid linear crystal is
rehted to the concept
consisting
of a
of alternating
“supertransfer”
acceptor
introduced
and donor
by Harer
molecules_
The spacing
be-
et al_ 131 in their discussion of rhe migm-
tion of CT states_
149
rihere .-I& Bit are crcxtion operators oft charge carrier (i-e. an electron) Ioated rtt the acceptor molecule of the nrth unit cell_ and of ;I Frenkei es&on Ioated at the donor moIecuIe of the txtb unit cell. respectively. with the axresponding energies Cr and E_ Dtt,t and D$,t are creation operators for the ionic pair (CT) smtes consisting of ZIhole located at the donor molecule of the tttlth unit cell and ;Lnelectron located tit the acceptor moIecuIe of the (ttr+l) st and tnth unit &I. respectively. Both of than are assumed to ha\e the energy kfl_ The hermitmn conju@es
stand for the corresponding annihihtion operarors. R(ttz-tz) is rhe charge carrier-esciton rxx~pIing constant whic!l describes the polarization of surrounding donor molecules by the electron located af the tlth acceptor molecute- G1 (m--It) an3 C1(ttz-tz) describe the Coulomb interaction between the eiectron located at the tzth accep-
for motecuk and the surrounding CT states.
The matrix elements A? and Xz describe the dissocktion
of Frenkel
excitons
into charge-transfer
states. We
will reject from the furrbcr treatment IIre CT states which are destsbilired by rhe interaction with the mobiie efectron_ In this aim we put: XL = X
=0
if ttt > At.
otherwise,
X2 = X
if ttt Gi;‘.
=0
otherwise,
and G1 (m-~2) = Gl (m-n)
=o
if ttr > IV, otherwise,
Gz(ttz-tz)
= G~(t~r-tt)
if m
= 0
otherwise,
3 being the actual position OF the mobiIe electron_ It is assumed that only one charge carrier is present in the crystal_ Yis the charge-transfer integral which describes the propagation of the electron from one acceptor molecule to anorher. We will assume Y= 0, since it involves the transfer of the charge to the second-nearest-neighbour molecule_ The 1st term of the hamiltonian, proportional to W, describes the above-mentioned processes where the charge is ailowed to jump to the nearest-neighbour molecule involved in the CT state_ Tbe form (l)-(3) of the bamiIton%m is entirely analogous to that used previously for a linear crystal consisting of identical moIecuIes [I ] _
150
V~~lume67. number 1
CHIXICAL
1 November 1979
PHYSICS LETI-ERS
3. The trarlsformation
x=x’
+.I?,
K(m)
K’(ttt)
= K’ (HZ) •t K’(m),
= c G’(tz-tn)A;A,,, t, f tt1
and next E = eiSne-iS
S=-i
(3 RZo(ttz,k) EB k
R1o(ttz.k)
c Alla, nr Sk
O,w+
E:_
o,,
1
(6)
+ Il_C__
where
R’“(tit.tt) = - v~,,~R.(ttt-n),
RB(ttt,tz) = u,,,R(ttz-tt), q;,
=f~E+ECr,K(tn)t[(CCT+K(ttt)-E)2t4X~]I’)).
-
c
k%
R@(ttt.k)Rfl(tt,k) + R”(ttt.k)R
lo (tt.k) + R’*(ttt,k) R’*(tt,k) I
Ep
Ep
Ef
A+ A A+A _ tt1 IW It I1
(7)
The propagation of the eicctron aIong the stack is governed by the renormalized matrix element whose value folIONS from the transformation of Htnmfer: f?(;,itl)
= W-~e%l,fAi+,(Dfi
. + D14 -)e-slitl),
(8)
Ltihere Ii> denotes the state kector corresponding to the state with an electron located at the ith acceptor molecule. The ma&is element is readily ewztluated according to the method used in the former papers [I ,6,7]. The result is G(i. i+l)
= It’[2
CO.S(~D,,~/~)~~
(i,i)
+
sin ((?t,i/2) ~r(i,i)] expB.
(9)
where
y[(i,k)=
-2R
(i-k)sin
(4 +yk)/{E+fl
tG’(k-i)
-
[(p
+ G’(k-i)
-E)’
+4(X’)‘]“‘),
151
In order to ekxhrrtte the efttctiw matrix clement of eq. (‘1) \\r’ need therefore the numerial wiues oC1-“r. E. G. R and _Y_ Al1 the% pzmmeters 131 be tytinltlted from independent Jata. f+ T and E are taken from the energia of the cwresponding electronic transitions. For the p~lenantIlrcrle-PRIDr\ crysttll fl= 2-9 eV [SI_ There xc three low-energetic tr.msitions_in phen.mthrene molecule [‘,I _ The lowest Frenhcl state corresponds to E = _;_a7 eV_ and the most intense to E = ?_SJ hr. The s.impIe caicul~tions are performed for both ~ducs. G follows fr.ml simpic electrostatic considerations. The DA distance is ;~ssunIed to be 3.5 r\ which is the !-sum n interphnar separation along the stack ais_ There is some doubt regrrrdmg the b”~Iue of microscopic diekctric constrtnt in the ~ystal_ IL seems r+ysonabIe LO assume E = 1 end E = 3 xs possible limitins wIues_ The corresponding G bikes caIcuIated in the point-chargs niodei are G = -2 eV rind C = -0.7 eV. respecthely. R can ah be estimated from simple ctectrostatic considerations. according to the method proposed b) Cilaihin et a!. [IO]. is_ tiom rhc chuic~l bindins energy of the escitonic polaron_ For E = 3. this )ieIds R = I.06 eV and R = ! 25 eV for E = 3_-? e\f and E = -t_SieV. respcctircly. The most difficult is the estimation of X_ Acwrding co Trt~r~ka [I 11. we use the appro_tim.iLe expression X = !5S(eV). whereS is the intermoleculrtr okerhp iniegmI_ For the anthracene-P&IDA complex a rough estimate is S- 0-I [Ij_This mfue is rather an upper bound. sirxz lhe interphnar distance was assumed to be 32 cl (kecsus ;Ictual3.5 _%).and the SIater esponent I .625_ The cstimrttrs for other weak CT compIexes [2.11] gile values smiler by almost an order of magnitudc_Therefore a reasonable estimate (perhaps on the small side) for anthrxenePMDA and phcmmthrene-PhlDA complexes is X = 0.2-0.5 eV_ For the sake of simplicity. we set it’= X_ Typiwl vahtes of the effective mstris element for electron transfer CJL c&ted from eq_ (9). xe coliecred in table I _As these are only order-of-magnitude estim.aes the same pxm~eter
Table
t
Effecti\= matrix eknients -_--
for etectron twnsfer along the st.tck. ka -_------_--c
x 0.2 0.7 0.1 0.3 03 0-3 0.4 O-4
IS2
3 1 3
-03 7 27
t
-2
3
-0.7
I 3 !
-2 --0-1 -1
= 2.9 eV_ All ener$x are in eV --E
R
IV
3.47 3.47 4.64 4.64 3-47 3.47 3.47 3-47
i-06 3.16 I.15 3.75 1.06 3.16 1.06 3.18
0_0183 o.os9-I 0_0173 0.0731 oa402 O_IMI 0.0546 0.1669
Vohmtr 67. number 1
ClIE\lICAL
PH‘tSICS
I Novenlber 1979
Lk-l-l-ERS
sets can be apphed both for the anthracenc .md phendnthrene complexes [I 1]_ The results indicate that the effecriw electron transfer mtegr& are of the order of the largest transfer integrals in anthracene cryst.d (0.03-0.05 eV), or eben larger [IS-- IS] _ The transfer int egrais perprndicuiar fo the stack axis ..we expected to be smalier than in mthracene crystal due to less favourable irltermoIecu1.x overlap and hence sm4ler than l?_ The large electron mobility along the stack is therefore not surprisin g, and need not involve any contribution from the neutral donor molecules. The effect is due to the involvement of the virtual CT states created by polarization_ The experimental %aIues of the mobiiities in P&IDA complexes (0.01 S cm2 V-I s-1 in phenanthrene-PblDA .md 0.15 cm2 VW* s-l in anthracrrle--PblDA_ measured Along the stack a_sis) are smaller than in the anthracene cqstal (O_-l--I -5 cm2 V-l s-I_ measured parJle1 to the b and c’ crystal axis, respectively [161)_ This is most probabIy due pxtly to the band narrowing resultq from escitonic polaron efftxts involving the CT states more distant from the charge carrier r6.71, and partly to rhe coupling with phonons_ This effect would also contribute to the retnarkble difference between electron mobilittes in the anthracene-PhlDA and phenanthrene-PMDA complexes [Z--5 I_
Acknolrledgement Fmancial
support
from
the Polish
Academy
of Sciences
is gratefully
acknowledged_
References [ 11 P. Petelmz. Ph\s_ Star. Sot. 90b (197s) 63.5. 121 D. Hurer. M.R. Phtlpott .tnd H. ~Iora~r~tz, J. Chem. Phls. 63 (1975) 5238. [3 1 11. X5ltxdd. D. llurer and G. Catro, Chem_ Ph>s. Letters 32 (1975) -$33_ [4] D. Haxer and H. MBh~~ald, Ph)s. Re%_ Letters 34 (1975) 1447. [S] N. Karl and J_ Ziepler, Chem. Ph>s. Letters 31 (1975) 43S_ [6] P. Petelrnr. Ph)s. Stat. Sol. S3b (1977) 169. 171 P. f’eteknz, Ph>s. Srat. Sol. SSb (197s) 669. [Sl VK. Eondrsto\, G.I\I. Ktrpin, AX_ Lipstova and N-D. Rusyxto~n. Zh. Fiz. Khim. 50 (1976) 749. [9J IL Otoboza\;l.S_ 1nom.n~. X_ .\likami and X Ito. Bull. Chem. Sot. Japan 50 (1977) X99. 1101 P.1\1-Cl~tl&t.A.1‘_C~rito and AJ. Heeger, Ph>s. Re\_ B5 (1971) 4966. il I] M.Tatsh. BulLChem.Soc_ Japan51 (197s) 1001. [Ill D.IIx1rer,J_Chem_Phps_67(1977)1076_ [ 131 R. Silbey. 1. Jortncr. S.A. Rice and hI_T_ Vale Jr.. J. Chem. Phys_ 41 (1965) 733~43 (1965) 1925. [I?] D-C. S&h and S C. Jlsthttr, Mol. Crlst. Liquid Crlst. 27 (1974) 55_ 1151 A.Ttber&ienand G. Dehcote. J. Phls. (Paris) 31 (1971) 637. [ 16 j W_ Me>. TJ_ Sonnoscine. D-L_ Mosel and _&.X1_ Harmann, J. Chem. Phys. 58 (1973) 7541.
153
ELECTRON
Ctil:\tICAL
MOBILITY
IN WEAK
DONOR-ACCEPTOR
Received 79 Ma_\ 1979: m tinzl form 1 hugzst
A Jmple qruntum-mechsniui along rhe donor-acceptor srxhs cept of ‘fuperrransfsr” inroh-ins
! So%ember 1979
PHYSICS LEITCRS
COMPLEXES
1979
model is propowd
to account for rhr e~perimenrally
observed Irtrge elecrron mobilities
in rtnthracene-PUD.4 rhe %irru;ll
2nd phen~nthrene-PMDA a_vstals_ The model is based on the concharge-rranxfer stana creared by polari~rion -
I _ Introduction in ZI recent
that the coupling of Frenkel excitons with the charge-transfer (CT) states carrier can result in increased charge carrier mobilities in the crystals of some weak chargetransfer compkes. This effect was expected to be operative for the compleses crystallizing in alternating _..ADADAD___donor-acceptor stacks, and to affect the charge carrier mobilities along the stack axis. As long JS cr>sral polarization is disregarded, the transport of, e_=_ = an electron along the stack axis. shouid be limited to jumps between the acceptor molecules. xkith the donor molecules not participating in the process_ Crystal polarization into the CT states (via the Frenkel states) enables the donor molecules to contribute to electron transfer aIong the stack. This is due to the processes, M here a mobile electron annihilates a “hoIe” on the donor molecuIes involved in d virtwl CT state created by polarization [ 1J _ As the distance from the acceptor molecule to the donor molecule is smaller (by more or less the factor of twoj than the distance to the next acceptor moIecule, the corresponding matris eiement should be larger. Hence. eiectron mobility shouid be increased due to the contribution from some CT states to crystal polarization *_ A possible consequence would be an anomalously high (with respect to the other crystal direction Is) charge carrier mobility along the stack asis (11x)_ Such anomalous electron mobility was observed indeed in the crystals of the complexes of pyromellitic dhn(,u,,~ = 0.15, hydride (P&IDA) with phenanthrene (~_t,t~= 0.018, pli, = 0.0075 cm2 V-1 s-t [3,rf]) and anthracene pLr =O.O? cm? V-* s-* [S]). It ws conjectured from these results [3] that the neutral donor molecules could contribute to the electron mobiIity, i.e. that the electron wavefunction was partially delocalized over the donor sites. which was intuitilelk rather surprising. The objectite of this letter is to demonstrate that electron mobility of the order of those observed in PMDA complexes can reasonably be accounted for in terms of a simple model postulating the involvement of the virtual CT states created by polarization_ xijacent
paper
[I ] we suggested
to the charge
2_ The hamiltonian Let us consider ’ This conddention
a rigid linear crystal is
rehted to the concept
consisting
of a
of alternating
“supertransfer”
acceptor
introduced
and donor
by Harer
molecules_
The spacing
be-
et al_ 131 in their discussion of rhe migm-
tion of CT states_
149
rihere .-I& Bit are crcxtion operators oft charge carrier (i-e. an electron) Ioated rtt the acceptor molecule of the nrth unit cell_ and of ;I Frenkei es&on Ioated at the donor moIecuIe of the txtb unit cell. respectively. with the axresponding energies Cr and E_ Dtt,t and D$,t are creation operators for the ionic pair (CT) smtes consisting of ZIhole located at the donor molecule of the tttlth unit cell and ;Lnelectron located tit the acceptor moIecuIe of the (ttr+l) st and tnth unit &I. respectively. Both of than are assumed to ha\e the energy kfl_ The hermitmn conju@es
stand for the corresponding annihihtion operarors. R(ttz-tz) is rhe charge carrier-esciton rxx~pIing constant whic!l describes the polarization of surrounding donor molecules by the electron located af the tlth acceptor molecute- G1 (m--It) an3 C1(ttz-tz) describe the Coulomb interaction between the eiectron located at the tzth accep-
for motecuk and the surrounding CT states.
The matrix elements A? and Xz describe the dissocktion
of Frenkel
excitons
into charge-transfer
states. We
will reject from the furrbcr treatment IIre CT states which are destsbilired by rhe interaction with the mobiie efectron_ In this aim we put: XL = X
=0
if ttt > At.
otherwise,
X2 = X
if ttt Gi;‘.
=0
otherwise,
and G1 (m-~2) = Gl (m-n)
=o
if ttr > IV, otherwise,
Gz(ttz-tz)
= G~(t~r-tt)
if m
= 0
otherwise,
3 being the actual position OF the mobiIe electron_ It is assumed that only one charge carrier is present in the crystal_ Yis the charge-transfer integral which describes the propagation of the electron from one acceptor molecule to anorher. We will assume Y= 0, since it involves the transfer of the charge to the second-nearest-neighbour molecule_ The 1st term of the hamiltonian, proportional to W, describes the above-mentioned processes where the charge is ailowed to jump to the nearest-neighbour molecule involved in the CT state_ Tbe form (l)-(3) of the bamiIton%m is entirely analogous to that used previously for a linear crystal consisting of identical moIecuIes [I ] _
150
V~~lume67. number 1
CHIXICAL
1 November 1979
PHYSICS LETI-ERS
3. The trarlsformation
x=x’
+.I?,
K(m)
K’(ttt)
= K’ (HZ) •t K’(m),
= c G’(tz-tn)A;A,,, t, f tt1
and next E = eiSne-iS
S=-i
(3 RZo(ttz,k) EB k
R1o(ttz.k)
c Alla, nr Sk
O,w+
E:_
o,,
1
(6)
+ Il_C__
where
R’“(tit.tt) = - v~,,~R.(ttt-n),
RB(ttt,tz) = u,,,R(ttz-tt), q;,
=f~E+ECr,K(tn)t[(CCT+K(ttt)-E)2t4X~]I’)).
-
c
k%
R@(ttt.k)Rfl(tt,k) + R”(ttt.k)R
lo (tt.k) + R’*(ttt,k) R’*(tt,k) I
Ep
Ep
Ef
A+ A A+A _ tt1 IW It I1
(7)
The propagation of the eicctron aIong the stack is governed by the renormalized matrix element whose value folIONS from the transformation of Htnmfer: f?(;,itl)
= W-~e%l,fAi+,(Dfi
. + D14 -)e-slitl),
(8)
Ltihere Ii> denotes the state kector corresponding to the state with an electron located at the ith acceptor molecule. The ma&is element is readily ewztluated according to the method used in the former papers [I ,6,7]. The result is G(i. i+l)
= It’[2
CO.S(~D,,~/~)~~
(i,i)
+
sin ((?t,i/2) ~r(i,i)] expB.
(9)
where
y[(i,k)=
-2R
(i-k)sin
(4 +yk)/{E+fl
tG’(k-i)
-
[(p
+ G’(k-i)
-E)’
+4(X’)‘]“‘),
151
In order to ekxhrrtte the efttctiw matrix clement of eq. (‘1) \\r’ need therefore the numerial wiues oC1-“r. E. G. R and _Y_ Al1 the% pzmmeters 131 be tytinltlted from independent Jata. f+ T and E are taken from the energia of the cwresponding electronic transitions. For the p~lenantIlrcrle-PRIDr\ crysttll fl= 2-9 eV [SI_ There xc three low-energetic tr.msitions_in phen.mthrene molecule [‘,I _ The lowest Frenhcl state corresponds to E = _;_a7 eV_ and the most intense to E = ?_SJ hr. The s.impIe caicul~tions are performed for both ~ducs. G follows fr.ml simpic electrostatic considerations. The DA distance is ;~ssunIed to be 3.5 r\ which is the !-sum n interphnar separation along the stack ais_ There is some doubt regrrrdmg the b”~Iue of microscopic diekctric constrtnt in the ~ystal_ IL seems r+ysonabIe LO assume E = 1 end E = 3 xs possible limitins wIues_ The corresponding G bikes caIcuIated in the point-chargs niodei are G = -2 eV rind C = -0.7 eV. respecthely. R can ah be estimated from simple ctectrostatic considerations. according to the method proposed b) Cilaihin et a!. [IO]. is_ tiom rhc chuic~l bindins energy of the escitonic polaron_ For E = 3. this )ieIds R = I.06 eV and R = ! 25 eV for E = 3_-? e\f and E = -t_SieV. respcctircly. The most difficult is the estimation of X_ Acwrding co Trt~r~ka [I 11. we use the appro_tim.iLe expression X = !5S(eV). whereS is the intermoleculrtr okerhp iniegmI_ For the anthracene-P&IDA complex a rough estimate is S- 0-I [Ij_This mfue is rather an upper bound. sirxz lhe interphnar distance was assumed to be 32 cl (kecsus ;Ictual3.5 _%).and the SIater esponent I .625_ The cstimrttrs for other weak CT compIexes [2.11] gile values smiler by almost an order of magnitudc_Therefore a reasonable estimate (perhaps on the small side) for anthrxenePMDA and phcmmthrene-PhlDA complexes is X = 0.2-0.5 eV_ For the sake of simplicity. we set it’= X_ Typiwl vahtes of the effective mstris element for electron transfer CJL c&ted from eq_ (9). xe coliecred in table I _As these are only order-of-magnitude estim.aes the same pxm~eter
Table
t
Effecti\= matrix eknients -_--
for etectron twnsfer along the st.tck. ka -_------_--c
x 0.2 0.7 0.1 0.3 03 0-3 0.4 O-4
IS2
3 1 3
-03 7 27
t
-2
3
-0.7
I 3 !
-2 --0-1 -1
= 2.9 eV_ All ener$x are in eV --E
R
IV
3.47 3.47 4.64 4.64 3-47 3.47 3.47 3-47
i-06 3.16 I.15 3.75 1.06 3.16 1.06 3.18
0_0183 o.os9-I 0_0173 0.0731 oa402 O_IMI 0.0546 0.1669
Vohmtr 67. number 1
ClIE\lICAL
PH‘tSICS
I Novenlber 1979
Lk-l-l-ERS
sets can be apphed both for the anthracenc .md phendnthrene complexes [I 1]_ The results indicate that the effecriw electron transfer mtegr& are of the order of the largest transfer integrals in anthracene cryst.d (0.03-0.05 eV), or eben larger [IS-- IS] _ The transfer int egrais perprndicuiar fo the stack axis ..we expected to be smalier than in mthracene crystal due to less favourable irltermoIecu1.x overlap and hence sm4ler than l?_ The large electron mobility along the stack is therefore not surprisin g, and need not involve any contribution from the neutral donor molecules. The effect is due to the involvement of the virtual CT states created by polarization_ The experimental %aIues of the mobiiities in P&IDA complexes (0.01 S cm2 V-I s-1 in phenanthrene-PblDA .md 0.15 cm2 VW* s-l in anthracrrle--PblDA_ measured Along the stack a_sis) are smaller than in the anthracene cqstal (O_-l--I -5 cm2 V-l s-I_ measured parJle1 to the b and c’ crystal axis, respectively [161)_ This is most probabIy due pxtly to the band narrowing resultq from escitonic polaron efftxts involving the CT states more distant from the charge carrier r6.71, and partly to rhe coupling with phonons_ This effect would also contribute to the retnarkble difference between electron mobilittes in the anthracene-PhlDA and phenanthrene-PMDA complexes [Z--5 I_
Acknolrledgement Fmancial
support
from
the Polish
Academy
of Sciences
is gratefully
acknowledged_
References [ 11 P. Petelmz. Ph\s_ Star. Sot. 90b (197s) 63.5. 121 D. Hurer. M.R. Phtlpott .tnd H. ~Iora~r~tz, J. Chem. Phls. 63 (1975) 5238. [3 1 11. X5ltxdd. D. llurer and G. Catro, Chem_ Ph>s. Letters 32 (1975) -$33_ [4] D. Haxer and H. MBh~~ald, Ph)s. Re%_ Letters 34 (1975) 1447. [S] N. Karl and J_ Ziepler, Chem. Ph>s. Letters 31 (1975) 43S_ [6] P. Petelrnr. Ph)s. Stat. Sol. S3b (1977) 169. 171 P. f’eteknz, Ph>s. Srat. Sol. SSb (197s) 669. [Sl VK. Eondrsto\, G.I\I. Ktrpin, AX_ Lipstova and N-D. Rusyxto~n. Zh. Fiz. Khim. 50 (1976) 749. [9J IL Otoboza\;l.S_ 1nom.n~. X_ .\likami and X Ito. Bull. Chem. Sot. Japan 50 (1977) X99. 1101 P.1\1-Cl~tl&t.A.1‘_C~rito and AJ. Heeger, Ph>s. Re\_ B5 (1971) 4966. il I] M.Tatsh. BulLChem.Soc_ Japan51 (197s) 1001. [Ill D.IIx1rer,J_Chem_Phps_67(1977)1076_ [ 131 R. Silbey. 1. Jortncr. S.A. Rice and hI_T_ Vale Jr.. J. Chem. Phys_ 41 (1965) 733~43 (1965) 1925. [I?] D-C. S&h and S C. Jlsthttr, Mol. Crlst. Liquid Crlst. 27 (1974) 55_ 1151 A.Ttber&ienand G. Dehcote. J. Phls. (Paris) 31 (1971) 637. [ 16 j W_ Me>. TJ_ Sonnoscine. D-L_ Mosel and _&.X1_ Harmann, J. Chem. Phys. 58 (1973) 7541.
153