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Electron transfer optical bands of charge-ordered mixed valence compounds. Quasidynamical model
Chemical Physics North-Holland
166 (1992)
97-106
Electron transfer optical bands of charge-ordered mixed valence compounds. Quasidynamical model B.S. Tsukerblat a, S.I. Klokishner b and B.L. Kushkuley b a Institute of Chemistry, Academy of Sciences of Moldova, Grosul street 3. 277028 Kishinev, Moldova b Moldavran State Unrversrty, Sadovaya street 60, 277014 Kishinev, Moldova Received
14 October
199 1; in final form 9 June 1992
An approximate quasidynamical model is proposed to describe electron transfer bands of mixed-valence charge-ordered crystals. Within the scope of this model the adtabatic approximation IS employed for order parameter calculation. The dynamical (quantum-mechanical) single-cluster problem is solved in the molecular field approach. The electron-transfer (intervalence) opttcal band shape is investigated in detail for different phase states of a mixed-valence crystal. Intervalence absorption band shape and positron provide important data on key parameters of mixed-valence crystals, i.e. double exchange, vibronic and intercluster interaction parameters. Comparison of results obtained for the semiclassical and quasidynamical models is presented for different values of vibronic coupling, double exchange and intercluster interaction. The semiclassical approximation is shown to be not applicable for the description of some important spectral regions of intervalence bands.
1. Introduction Mixed-valence (MV) clusters contain metal ions with different oxidation degrees. In the fundamental papers [ l-3 ] the mechanism leading to “extra” electron migration (tunneling) was referred to as double exchange. MV clusters usually exhibit characteristic absorption bands within infrared or visible spectral regions. These bands are related to light induced transfer of the “extra” electron from one metal ion to another. This absorption is called intervalence, while the corresponding bands are denoted as electron transfer bands. Since a mononuclear complex cluster moiety does not exhibit intervalence absorption, the latter is one of the most important features of the phenomenon of mixed valency. A classical example is the so-called Creutz-Taube ions [ (NH3)5R~NON-Ru ( NH3 ) 5] 5+ [ 4 1. Recently more complex compounds have been investigated of the type
Correspondence to: B.S. Tsukerblat, Institute of Chemistry, Academy of Sciences of Moldova, Grosul street 3, 277028 Kishinev, Moldova. 0301-0104/92/%
05.00 0 1992 Elsevier Scrence Publishers
(X=NH3, py; Y=NH,, py; m=4=6) as dimeric complexes [ 6 ] :
[5], as well
NC-RU”(~~~),-CN-RU”‘(~~~)~-CN~+
,
NCRu”(bpy)2-CN-Ru’n(phen)2-CN2+
.
Since the “extra” electron strongly deforms the crystal surroundings , the charge transfer bands are electron-vibrational ones and have an appreciable width and a characteristic shape. The MV cluster vibronic model (PKS model) was proposed in the pioneer works of Piepho, Krausz and Schatz (see refs. [ 710 1. It comprises two tunnel electronic levels of opposite parity mixed by a single antisymmetric (“outof-phase”) mode. The problem of the pseudo-JahnTeller effect was solved by numerical diagonalization of the vibronic interaction matrix. This procedure made it possible to derive the shape of the electronvibrational light absorption band in molecular MV clusters [ 8-101. At present a wide class of crystals is known for which the concept of isolated clusters is not adequate. In refs. [ 1 l- 19 ] a strong intercluster interaction was revealed and investigated leading to phase
B.V. All rights reserved.
98
B.S. Tsukerblat et al / Chemcal Physm 166 (1992) 97-106
transition to the so-called charge-ordered state. The charge-ordered state has been identified in dialkylbiferrocenium triiodide crystals with substituent ions X=H, CH2CH3, (CHl)&H3, (CH2)3CH3, dihalobiferrocenium triiodide and dibromiodide crystals with substituent ions X=Br, I. The charge-ordering phenomenon has also been detected in crystals containing trinuclear MV complexes Fe30where (py) is pyridine (OZCCH,),(PY),(PY)> [ 20,2 1 ] _ The charge-ordered state was investigated by Mossbauer spectroscopy, electron paramagnetic resonance, and several other methods [ 11-l&20,2 11. In the theoretical papers [22-281 the dipole-dipole mechanism of intercluster interaction was investigated in detail. Heisenberg-type exchange interaction was taken into consideration as well as the double exchange. This made it possible to elucidate conditions of existence of charge ordering. Partial or total double exchange suppression and ferroelectric or antiferroelectric ordering are shown to be inherent for the MV crystal charge-ordered state. An attempt to calculate intervalence band shapes of charge-ordered crystals was made in refs. [ 29,30 1. The quantum-mechanical approach to band shape calculation based on the numerical solution of the pseudo-Jahn-Teller problem cannot be applied to the description of a many-particle system of interacting clusters. Therefore in refs. [ 29,301 both the conventional molecular field method and the semiclassical adiabatic approximation were employed. Within the framework of this approach only influence of the molecular field on the adiabatic potential of each cluster was taken into account (an analogous approach was used for calculation of the inelastic neutron scattering cross sections [ 3 1 ] ). However, as is well known, the validity of adiabatic approximation for the optical band problem of electron-vibrational Jahn-Teller systems is restricted to the case of strong vibronic coupling and/or high temperatures. Detailed studies also show that some light absorption spectrum regions, for whose calculation the non-adiabatic effects are important, cannot be described even in the favourable case of high temperature and strong vibronic coupling. A characteristic example is singletdoublet transition in cubic and trigonal Jahn-Teller impurity centers. In ref. [ 321 (see also references cited therein) the semiclassical approximation is shown to be not applicable for the description of the
wings of the optical line, nor for the central part of the double-humped band related to transitions within the region of electronic non-adiabatic mixing terms. Physically similar problems appear in the description of the electron transfer bands in MV systems with double-well adiabatic potentials. To overcome these difficulties an approximate quasidynamical approach is proposed here to describe electron transfer bands in MV charge-ordered systems. The idea of the new approach lies in the use of the adiabatic approximation, when nuclei motion is treated classically, to the calculation of the order parameter of a chargeordered crystal. The second stage within the framework of the molecular field approximation method consists in solving the dynamical (quantum-mechanical) vibronic problem for a single cluster affected by the molecular field produced by the chargeordered crystal. Then the vibronic wavefunctions are utilized for electron transfer band shape calculation. Since electron transfer bands are one of the basic indications of mixed valency, the absence of experimental data on infrared spectra of charge-ordered crystals seems to be surprising. Meanwhile the position and shape of these bands for isolated clusters can provide immediately valuable information about MV key characteristics, such as electron activation energy and the double exchange parameter. At the same time these parameters also determine the phase transition characteristics in charge-ordered crystals. We hope that the peculiarities of electron transfer bands revealed below may stimulate experimental work within this region,
2. Hamiltonian of crystal Let us consider a crystal, whose structural units are MV dimeric clusters. For simplicity we shall confine ourselves to the case of a cluster containing a single d electron over the tilled ion shells (the d’d’ system). The energy spectrum of such a system consists of two tunnel electronic levels with energies fp (energy gap 2p) and wavefunctions w+ = ( 1I$ ) ( qa + h ) , where p is the transfer (double exchange) parameter, and pa and o)bcorrespond to the states of electrons localized on a and b ions. In the vu+, v/_ basis the electronic Hamiltonian of the Ith crystal cell (Ith cluster) can be written in the following form:
B S Tsukerblat et al. /Chemcal
H; =pfiwa;
,
(1)
where a, is the Pauli matrix, p is the dimensionless double exchange parameter, Aw is the energy of the vibrational quantum (see below). Following refs. [ 7- 10,33-35 1, we shall introduce the intracluster vibronic interaction with a single localized vibration. The tunnel states v/+ and I,Y_are mixed by the antisymmetric (out-of-phase) mode q= ( l/$)(Q,-Qb), where Q, and Q,, are the totally symmetric (“breathing”) modes of the two cluster’s moieties. In the adopted basis the electronvibrational (vibronic) Hamiltonian of the Ith crystal cell has the form Ht
= hwq,at
,
(2)
where v is the dimensionless (in units of fiw) vibronic coupling constant. Finally, the free vibrations of the Ith cluster with the frequency w are assumed harmonic, (3) As was shown in refs. [22-241 the intercluster dipole-dipole interaction may be appreciable, its Hamiltonian being v,,=-td;x
K(l-k)o:a:.
(4)
Here do=eR,,/2 is the dipole moment of the cluster with the completely localized electron, K( I- k) = R,g3(3 COS~I~~~1); R,k is the distance between the fth and kth cluster, 19,~is the lateral angle of the vector Rkl. The full Hamiltonian of a crystal consisting of interacting clusters may be thus represented in the form H= C H;+ /
1 HdL+ C H:+V,,. 1 I
(5)
The essential difference of the model under consideration from that of refs. [ 22-241, where ordering in a rigid lattice was investigated, consists in the detailed consideration of vibronic interaction.
3. Molecular field approximation in semiclassical vibronic model The conventional
molecular
field approximation
99
Physm 166 (1992) 97-106
[ 361 consists in substituting in the intercluster interaction Hamiltonian the quantity aLo”; with (see for instance eq. (2.12) in ref. [ 19 ] ) a’ak=,=j,++a’~-$ x .r zi r
(6)
The quantity d= doa= do4, is the mean cluster dipole moment. The mean value (Tis a dimensionless order parameter and is defined by the relation ‘=
Tr[exp( -fl/kT)aj,] Tr[exp(-Ej/kZ)]
.
(7)
Here E? is the Hamiltonian of the crystal in the molecular field approximation, i.e. the Hamiltonian H from eq. ( 5 ) in which the interaction V,, is replaced is a by pddd= -Ld@C, a;, where L=x:,K(/-k) structural parameter depending on the mutual disposition of clusters in the crystal lattice. Due to the crystal invariance under translation the structural parameter L does not depend on the number of the crystal site 1. Within the framework of this approximation the total Hamiltonian fi can be expanded as a sum of single-site (single-cluster) Hamiltonians fl,: if,=H;+H;+H;,-Ld;aaj,.
(8)
In order to calculate the trace in eq. (7), it is necessary to know the eigenvalues of the Hamiltonian fi. For a certain d value the Hamiltonian (8) describes a two-level electronic subsystem with off-diagonal vibronic interaction (see eq. (2 ) ). To obtain the quantum solution one faces the problem of diagonalizing the vibronic interaction matrix within the framework of the PKS model. The last term in eq. (8), which represents the interaction of the “extra” electron with the molecular field, plays the role of “asymmetry parameter” in terms of the PKS model [ 7 1. In accordance with the molecular field approximation it is necessary to find the eigenvalues of the Hamiltonian (8) for an arbitrary value of the order parameter 6, i.e. at any temperature. Then these eigenvalues should be substituted in eq. (7 ). The selfconsistent equation thus obtained provides the temperature dependence of the order parameter. The selfconsistent equation for the order parameter cannot be derived explicitly due to the non-adiabatic (pseudo-Jahn-Teller) vibronic interaction in a single cluster affected by the molecular field. To overcome this difficulty when the order parameter is obtained from eq. (7), we employ the adi-
B.S. Tsukerblatet al. /ChemicalPhysrcs 166 (1992) 97-106
100
abatic approximation within the scope of the molecular field method. The adiabatic approximation consists in neglecting the nuclear kinetic energy in the Hamiltonian ( 8 ) . Hence, the single-site (single-cluster) adiabatic Hamiltonian takes the form ~C=pAoo,+~oqa,+1Aoq2-Ld~a,d,
(9)
where the vibrational coordinate q plays the role of parameter and the symbol 1 is omitted. To calculate the trace in eq. (7) it is convenient to use the adiabatic wavefunctions diagonalizing the Hamiltonian ( 9 ). These functions are: Y ,(2)={f[l+PlW(4)l}1’2V/+
*{t[lTPlw9)l)“2y/-
2
W(q)=Aw[p2+(uq-~c?)2]“2,
(10)
where [= Ldi /Am is the dimensionless parameter of intercluster interaction. The total single-site energies in the adopted approximation represent the adiabatic potentials u ,(2) = &ilq2*
W(q)
.
i=z-l
7 dqexp(-$$)[&sinh($) -CC v2q*
cash
+ d2(q)kT
HI ‘(‘)
kT
(13)
’
where d(q)=fiw(p*+v~q~)“2.
4. Temperature dependence of order parameter The temperature behavior of the mean dipole moment is determined by three dimensionless parameters: v, p/c, p. The numerical solution of eq. ( 12) shows that there is a range of these parameters where charge ordering occurs. The temperature dependence of the order parameter is presented in fig. 1. The family of curves 6( T) =6( T) /do shown in fig. 1a illustrates the influence of vibronic interaction on the temperature de-
(11)
In the semiclassical approach, summation over electron-vibrational states in eq. ( 7 ) is replaced by integration over the vibrational coordinate q. We hence obtain a self-consistent equation for the order parameter: CC gfic~
5 dqexp(
- %)((a-vq)
-m
x
.,.
sinh1W(q)lkT1 WC?) ’
(12)
,.
,
-
qzz
0.5
where ~~2
7 dqexp( --CC
- $$)cosh
(F)
is the partition function in the semiclassical approximation. At the phase transition temperature the order parameter vanishes. Expanding the right side of eq. ( 12 ) in a power series of 8 one obtains the equation for the phase transition temperature Tc:
Fig. 1. Temperature
dependence
of the order parameter.
101
B.S. Tsukerblat et al. /Chemical Physrcs 166 (I 992) 97-106
pendence of the order parameter. Fig. 1a shows that vibronic coupling growth leads to an increase in phase transition temperature and mean dipole moment. If the intercluster interaction (2p/c= 1.6) is relatively weak, tunneling in the charge-ordered phase is partially suppressed, and for v= 0.1 we obtain a( 0) =0.4. Vibronic interaction leads to additional localization of the extra electron and when u increases from 0.5 to 1.O the maximum dipole moment increases from 0.73 to 0.87. Finally, for v= 3 at low temperature the dipole moment attains a value close to the maximum (d= 1). It should be noted that for v= 0.1 the phase transition temperature kT, = 0.92 x 2pAw = 0.74Ldi and for v=3 it is kT,= 1.25pAw=Ldi. (These values of phase transition temperature have been obtained while calculating the temperature dependence of the dipole moment (eq. (12), see also eq. (13).) This shows the physical role of the vibronic interaction. When this interaction is weak the phase transition temperature is determined by the competition between the stabilizing effect of intercluster interaction and tunnel intracluster interaction. If the vibronic interaction is strong the double exchange is suppressed and the phase transition temperature is close to the value of intercluster interaction energy. For the parameter values under consideration the phase transition is possible even in static (rigid) lattices (v= 0)) since the condition p < c holds. For weak intercluster interaction (p > c) there is no charge-ordered phase in the rigid lattice (v= 0 ). Fig. lb shows that inclusion of vibronic interaction leads to charge ordering even in that region of parameters p and [ for which ordering cannot take place in the rigid lattice. The vibronic interaction increase is always accompanied by an increase in the phase transition temperature as well as an increase of the maximum dipole moment value. For example, for v= 0.1 the phase transition temperature is equal to 0.32Ldij, and for v= 3 kT,= Ld& Thus, from the physical point of view, the order parameter determined above in the semiclassical approximation reveals a regular temperature behaviour. The approach developed above makes it possible to investigate the influence of the vibronic effects on charge ordering characteristics.
5. Dynamical vibronic problem Our next step is the approximation procedure. The essence of the suggested approximation consists in the quantum-mechanical approach of the solution of the single-site vibronic problem with the order parameter defined semiclassically. Mathematically this is the same as solving the vibronic problem with the Hamiltonian (9) including the kinetic nuclei energy X = HL +pfzoa= + &oqaX - @Ma,
.
(14)
Here d is determined from the semiclassical equation ( 12) for the order parameter. To solve the dynamical problem we shall use the mathematical procedure proposed within the framework of the PKS model [ 7 1. The presence of a molecular field removes the inversion symmetry, so that we are dealing with the so-called asymmetric case of the PKS model here. The temperature-dependent value Lad0 plays the role of perturbation leading to the disappearance of the inversion centre. Thus for every given temperature we face the PKS problem for an asymmetric dimeric MV cluster. Nevertheless, in contrast to the isolated cluster problem, in the case under consideration the asymmetry parameter essentially depends on temperature. Vibronic wavefunctions are represented as an expansion in terms of unperturbed electronic and vibrational states:
@Y= f
II=0
tC”nW+(r)Xn(q) +Lw-
(r),%(q) 1 ,
(15)
where x,,(q) is the harmonic oscillator wavefunction, the index v numerates the hybrid cluster states in the molecular field. As in the PKS model the eigenvalues E, and the eigenvectors 0, are determined from the system of equations c,,(m+f+p-E,/fio)+
f
b,,A,,=O,
fl=O
b,,( m+ f -p-EJAo)
+ f
c,,,&,
=O ,
(16)
?I=0
where &7n=V(~&l,n+l m=O, 1,2, .... In computer
>
+JR=&%l,,-,)+l&Z, v=O, 1,2, ... .
calculations
200 unperturbed
(17) vibra-
102
B.S. Tsukerblat
et al. /Chemrc*al Physics 166 (1992) 97-106
8 \I/ 6
‘4
\
2
LI
-6
I I LJib -4
I
,Q
I -2
, A.
, 2
4
69
t
-L-4:
-4 0.2
0.2
0.;
ula
a
T/T=
T=O
Fig. 2. Temperature dependence of vibronic energy levels within the framework of the molecular field approximation: i= 1, p= 0.5, v=2.5.
tional states have been taken as the basis set. The temperature dependence of the lowest cluster vibronic levels in the molecular field is shown in fig. 2 for v=2.5, p=O.5, c= 1. Qualitatively, the hybrid spectrum peculiarities are quite clear from fig. 3, where the adiabatic potentials at T= 0 and T= T, are also represented for the same parameters. At low temperatures the first two vibronic levels correspond to located states close to the harmonic oscillator levels. The energies of the lower adiabatic potential sheet minima flatten out as the temperature increases, leading to tunnel level splitting. It should be noted that within the scope of the adopted approach the quantum properties of the vibronic states in a self-consistent field are taken into account. It is reasonable therefore to call the proposed approximation quasidynamical. The traditional semiclassical approximation deals with electronic states in the quantum way, while the motion of the nuclei is treated classically. The vibronic states obtained within the scope of the quasidynamical approximation are hybrid, i.e. retaining the quantum properties of both electronic and vibrational states.
6. Intervalence transfer bands
The band shape function of the inter-valence cal absorption is described by the expression
Fig. 3. Temperature dependence of the adiabatic potentials in the molecular field approximation: i= 1, p=O.5, u=2.5. (a) T=O, (b) T= T,. Vibronic levels of the lower sheet are indicated schematically.
F(Q)=
c cF ”
d&7,,, (a)
“=-“I
x (c,,,h, +LJ”n)
9
(18)
where NV is the equilibrium population of the vth vibronic level, Z= E,exp( -E,/kT) is the partition function, pyy. is the line shape connected with the transition between vibronic states Y and v’. This line shape is assumed to be Gaussian:
pyy,(Q;2) = opti-
To smooth the quantum discrete structure of the entire band the second moment of the individual line il
B.S. Tsukerblat et al. /Chemcal
should be comparable with the fro value. As well as in ref. [ 351 we put here 1= Aw. The temperature dependence of the absorption band envelope is represented in fig. 4. Fig. 4a shows the band shape for relatively strong coupling with deep adiabatic potential minima both for ordered and disordered states. At high temperatures TX T, the band has two maxima. The high-frequency maximum is related to the Franck-Condon transition. In terms of the semiclassical adiabatic approximation this transition can be associated with the vertical transition from the minimum of the adiabatic potential lower sheet to the upper sheet. For relatively small p and strong vibronic coupling (p < v2) the frequency of this maximum can be roughly estimated by the semiclassical expression [ 29,301
103
Physrcs 166 (1992) 97-106
(v2+[dy-
x
.
(20)
Eq. (20) shows that the temperature dependence of the band maximum is related to two physical factors: ( 1) explicit temperature dependence in the ordered as well as in the disordered phase and (2) temperature dependence of the order parameter. As is clear from fig. 4a, the frequency of the right maximum approximately satisfies relation (20) (w,,, x 13.5fiw). The low frequency maximum is the envelope of the lines arising from the transitions between the tunnel split levels. These tunnel states appear only at temperatures close to the phase transition temperature, when the potential surface minima are energetically equivalent. When the temperature decreases the band
d Fig, 4.
Temperature dependence
~=1,p=l,v=2;(d)~=l,p=l,u=lS.
of the absorption
coeffkient
(in arbitrary
units).
(a) c= 1, P=O.~,
u=2.%
(b) c= 1, P=2,
V= 1; (c)
B.S. Tsukerblat et al. / Chemcal Physics 166 (1992) 97-106
104
narrows and shifts to the high-frequency range. In the case of weak vibronic interaction the absorption coefficient is presented in fig. 4b. The lower sheet of the adiabatic potential has one minimum, at any temperature the optical band is bell-shaped, but not Gaussian. The molecular field contributes to the level splitting, and when the temperature decreases the band shifts to the high-frequency range. In figs. 4c and 4d the band shape for moderate vibronic coupling is shown. When the vibronic coupling parameter decreases, the low-frequency maximum of the optical curves (in the vicinity of r,) disappears and turns into a shoulder (see fig. 4). If coupling is further reduced, this maximum is absent at any temperature. For all parameters and temperatures the charge transfer bands remain essentially asymmetric and possess a long tail in the high-fre-
quency range. At the phase transition point the order parameter vanishes, and the results obtained coincide exactly with those of the PKS model. As is shown in fig. 4a, at temperatures T-c T, the position of the high-frequency maximum changes considerably when the temperature increases. Therefore the temperature shift of the high-frequency maximum of the band as well as the intervalence band-shape transformation may be considered as a manifestation of intercluster interaction.
7. Comparison with semiclassical
results
We shall now compare the results obtained with the semiclassical results obtained within the framework of classical nuclei motion assumption. The semiclas-
K(Q)
Fig. 5. Comparison of the results of the quasidynamical model with those of semiclassical ~~2.5; (b) C=l,p=l, v=2; (c) C=l,p=l,u=lS; (d) l=l.p=2,v=l.
adiabatic
approximation.
(a) C= 1, ~=0.5,
B.S. Tsukerblat et al. / Chemrcal Physrcs 166 (I 992) 97-106
sical shape band is given by the relation
approximation can hardly be applied to the case of moderate vibronic coupling. The intervalence band features elucidated above corroborate the existence of the charge-ordered state and provide unique information about the key parameters of interacting clusters in crystal charge-ordered state and phase transitions.
[ 32 ]
s
‘30
X
dqexp( - G/W
-cc
x~(U,(9)-U2(9)-fi-Q)
3
105
(21)
where 2 is the semiclassical partition function. The charge-ordered crystal band shape calculated from eq. (2 1) is discussed in detail in refs. [ 29,301. The optical curves obtained in these papers possess nonphysical peculiarities resulting from the inapplicability of the semiclassical approximation for the description of certain spectral ranges. In fig. 5 the results of the band shape calculations for the quasidynamical model are compared with those obtained from eq. (2 1). For strong coupling (fig. 5a) and low temperatures both the semiclassical and quasidynamical model yield bell-shaped curves which are similar in general. These curves however differ markedly in shape and maximum position. At T= T, the overall view of contours of the classical and quantum bands are also similar. A significant difference takes place in the region O/wz2p, where the semiclassical curve is divergent. When the vibronic coupling decreases (fig. 5b) the non-physical region @/wz2p expands, and the discrepancy between semiclassical and quasidynamical results becomes more significant. This discrepancy takes also place at low temperatures. In the semiclassical approximation within the intermediate and weak coupling range (figs. SC and 5d) the high temperature ( Tz T,) absorption band decreases monotonically, beginning from the divergence region (Q/o= 2~). The optical curves in the quasidynamical model are quite different from the semiclassical ones within this range. The quasidynamical curves are bell-shaped, and significant absorption takes place in the classically forbidden region B/w < 2p. Thus, for moderate and weak vibronic coupling the semiclassical approximation cannot describe the charge transfer band contour in charge-ordered crystals, both at low temperatures and within the range of phase transition. It should be stressed that at high temperatures the semiclassical
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B.S. Tsukerblat et al. /Chemical Physm 166 (1992) 97-106
[22] S.I. Klokishner and B.S. Tsukerblat, Pis’ma Zh. Eksp. Teor. Fiz. 45 (1987) 25. [23] S.I. Kloktshner and B.S. Tsukerblat, Fiz. Tverd. Tela 29 (1987) 2679. [24] S.I. Klokishner and B.S. Tsukerblat, Chem. Phys. 125 (1988) 11. [ 251 S.I. Klokishner, A.V. Koryachenko and B.S. Tsukerblat, Fiz. Tverd. Tela 32 ( 1990) 3 11. [26] B.S. Tsukerblat. M.I. Belinskii and S.I. Klokishner, Zh. Strukt. Khim. 31 (1990) 137. A.V. Koryachenko and B.S. Tsukerblat, ~27 ] S.I. Klokishner, Phys. Stat. Sol. (b) 160 (1990) 641. S.I. Kloktshner and B.S. Tsukerblat, ]28 ] A.V. Koryachenko, Chem. Phys. 150 ( 1991) 295. Fiz. Tverd. Tela 31 ]29 ‘1 S.I. Klokishner and B.S. Tsukerblat, (1989) 85.
[30] S.I. Klokishner and B.S. Tsukerblat. Spectry. Letters 23 (1990) 637. [ 3 1 ] S.I. Klokishner. A.V. Palii and B.S. Tsukerblat. Solid State Commun. 71 (1989) 879. 132 1Yu.E. Perlin and B.S. Tsukerblat, in: The dynamical JahnTeller effect m locahzed systems, eds. Yu.E. Perlin and M. Wagner (North-Holland, Amsterdam, 1984) pp. 25 l-346. R.L. Fulton and M.J. Gouterman, Chem. Phys. 41 ( 1964) 2280. R.L. Fulton and M.J. Gouterman, Chem. Phys. 35 ( 196 1) 1059. P.N. Schatz, in: Mixed-valence compounds. Theory and applications in chemistry, physics, geology, and biology. ed. D.B. Brown (NATO Advanced Study Institute, Oxford, 1979). G.S. Smart, Effective field theories of magnetism (Saunders. Philadelphia, 1966).
166 (1992)
97-106
Electron transfer optical bands of charge-ordered mixed valence compounds. Quasidynamical model B.S. Tsukerblat a, S.I. Klokishner b and B.L. Kushkuley b a Institute of Chemistry, Academy of Sciences of Moldova, Grosul street 3. 277028 Kishinev, Moldova b Moldavran State Unrversrty, Sadovaya street 60, 277014 Kishinev, Moldova Received
14 October
199 1; in final form 9 June 1992
An approximate quasidynamical model is proposed to describe electron transfer bands of mixed-valence charge-ordered crystals. Within the scope of this model the adtabatic approximation IS employed for order parameter calculation. The dynamical (quantum-mechanical) single-cluster problem is solved in the molecular field approach. The electron-transfer (intervalence) opttcal band shape is investigated in detail for different phase states of a mixed-valence crystal. Intervalence absorption band shape and positron provide important data on key parameters of mixed-valence crystals, i.e. double exchange, vibronic and intercluster interaction parameters. Comparison of results obtained for the semiclassical and quasidynamical models is presented for different values of vibronic coupling, double exchange and intercluster interaction. The semiclassical approximation is shown to be not applicable for the description of some important spectral regions of intervalence bands.
1. Introduction Mixed-valence (MV) clusters contain metal ions with different oxidation degrees. In the fundamental papers [ l-3 ] the mechanism leading to “extra” electron migration (tunneling) was referred to as double exchange. MV clusters usually exhibit characteristic absorption bands within infrared or visible spectral regions. These bands are related to light induced transfer of the “extra” electron from one metal ion to another. This absorption is called intervalence, while the corresponding bands are denoted as electron transfer bands. Since a mononuclear complex cluster moiety does not exhibit intervalence absorption, the latter is one of the most important features of the phenomenon of mixed valency. A classical example is the so-called Creutz-Taube ions [ (NH3)5R~NON-Ru ( NH3 ) 5] 5+ [ 4 1. Recently more complex compounds have been investigated of the type
Correspondence to: B.S. Tsukerblat, Institute of Chemistry, Academy of Sciences of Moldova, Grosul street 3, 277028 Kishinev, Moldova. 0301-0104/92/%
05.00 0 1992 Elsevier Scrence Publishers
(X=NH3, py; Y=NH,, py; m=4=6) as dimeric complexes [ 6 ] :
[5], as well
NC-RU”(~~~),-CN-RU”‘(~~~)~-CN~+
,
NCRu”(bpy)2-CN-Ru’n(phen)2-CN2+
.
Since the “extra” electron strongly deforms the crystal surroundings , the charge transfer bands are electron-vibrational ones and have an appreciable width and a characteristic shape. The MV cluster vibronic model (PKS model) was proposed in the pioneer works of Piepho, Krausz and Schatz (see refs. [ 710 1. It comprises two tunnel electronic levels of opposite parity mixed by a single antisymmetric (“outof-phase”) mode. The problem of the pseudo-JahnTeller effect was solved by numerical diagonalization of the vibronic interaction matrix. This procedure made it possible to derive the shape of the electronvibrational light absorption band in molecular MV clusters [ 8-101. At present a wide class of crystals is known for which the concept of isolated clusters is not adequate. In refs. [ 1 l- 19 ] a strong intercluster interaction was revealed and investigated leading to phase
B.V. All rights reserved.
98
B.S. Tsukerblat et al / Chemcal Physm 166 (1992) 97-106
transition to the so-called charge-ordered state. The charge-ordered state has been identified in dialkylbiferrocenium triiodide crystals with substituent ions X=H, CH2CH3, (CHl)&H3, (CH2)3CH3, dihalobiferrocenium triiodide and dibromiodide crystals with substituent ions X=Br, I. The charge-ordering phenomenon has also been detected in crystals containing trinuclear MV complexes Fe30where (py) is pyridine (OZCCH,),(PY),(PY)> [ 20,2 1 ] _ The charge-ordered state was investigated by Mossbauer spectroscopy, electron paramagnetic resonance, and several other methods [ 11-l&20,2 11. In the theoretical papers [22-281 the dipole-dipole mechanism of intercluster interaction was investigated in detail. Heisenberg-type exchange interaction was taken into consideration as well as the double exchange. This made it possible to elucidate conditions of existence of charge ordering. Partial or total double exchange suppression and ferroelectric or antiferroelectric ordering are shown to be inherent for the MV crystal charge-ordered state. An attempt to calculate intervalence band shapes of charge-ordered crystals was made in refs. [ 29,30 1. The quantum-mechanical approach to band shape calculation based on the numerical solution of the pseudo-Jahn-Teller problem cannot be applied to the description of a many-particle system of interacting clusters. Therefore in refs. [ 29,301 both the conventional molecular field method and the semiclassical adiabatic approximation were employed. Within the framework of this approach only influence of the molecular field on the adiabatic potential of each cluster was taken into account (an analogous approach was used for calculation of the inelastic neutron scattering cross sections [ 3 1 ] ). However, as is well known, the validity of adiabatic approximation for the optical band problem of electron-vibrational Jahn-Teller systems is restricted to the case of strong vibronic coupling and/or high temperatures. Detailed studies also show that some light absorption spectrum regions, for whose calculation the non-adiabatic effects are important, cannot be described even in the favourable case of high temperature and strong vibronic coupling. A characteristic example is singletdoublet transition in cubic and trigonal Jahn-Teller impurity centers. In ref. [ 321 (see also references cited therein) the semiclassical approximation is shown to be not applicable for the description of the
wings of the optical line, nor for the central part of the double-humped band related to transitions within the region of electronic non-adiabatic mixing terms. Physically similar problems appear in the description of the electron transfer bands in MV systems with double-well adiabatic potentials. To overcome these difficulties an approximate quasidynamical approach is proposed here to describe electron transfer bands in MV charge-ordered systems. The idea of the new approach lies in the use of the adiabatic approximation, when nuclei motion is treated classically, to the calculation of the order parameter of a chargeordered crystal. The second stage within the framework of the molecular field approximation method consists in solving the dynamical (quantum-mechanical) vibronic problem for a single cluster affected by the molecular field produced by the chargeordered crystal. Then the vibronic wavefunctions are utilized for electron transfer band shape calculation. Since electron transfer bands are one of the basic indications of mixed valency, the absence of experimental data on infrared spectra of charge-ordered crystals seems to be surprising. Meanwhile the position and shape of these bands for isolated clusters can provide immediately valuable information about MV key characteristics, such as electron activation energy and the double exchange parameter. At the same time these parameters also determine the phase transition characteristics in charge-ordered crystals. We hope that the peculiarities of electron transfer bands revealed below may stimulate experimental work within this region,
2. Hamiltonian of crystal Let us consider a crystal, whose structural units are MV dimeric clusters. For simplicity we shall confine ourselves to the case of a cluster containing a single d electron over the tilled ion shells (the d’d’ system). The energy spectrum of such a system consists of two tunnel electronic levels with energies fp (energy gap 2p) and wavefunctions w+ = ( 1I$ ) ( qa + h ) , where p is the transfer (double exchange) parameter, and pa and o)bcorrespond to the states of electrons localized on a and b ions. In the vu+, v/_ basis the electronic Hamiltonian of the Ith crystal cell (Ith cluster) can be written in the following form:
B S Tsukerblat et al. /Chemcal
H; =pfiwa;
,
(1)
where a, is the Pauli matrix, p is the dimensionless double exchange parameter, Aw is the energy of the vibrational quantum (see below). Following refs. [ 7- 10,33-35 1, we shall introduce the intracluster vibronic interaction with a single localized vibration. The tunnel states v/+ and I,Y_are mixed by the antisymmetric (out-of-phase) mode q= ( l/$)(Q,-Qb), where Q, and Q,, are the totally symmetric (“breathing”) modes of the two cluster’s moieties. In the adopted basis the electronvibrational (vibronic) Hamiltonian of the Ith crystal cell has the form Ht
= hwq,at
,
(2)
where v is the dimensionless (in units of fiw) vibronic coupling constant. Finally, the free vibrations of the Ith cluster with the frequency w are assumed harmonic, (3) As was shown in refs. [22-241 the intercluster dipole-dipole interaction may be appreciable, its Hamiltonian being v,,=-td;x
K(l-k)o:a:.
(4)
Here do=eR,,/2 is the dipole moment of the cluster with the completely localized electron, K( I- k) = R,g3(3 COS~I~~~1); R,k is the distance between the fth and kth cluster, 19,~is the lateral angle of the vector Rkl. The full Hamiltonian of a crystal consisting of interacting clusters may be thus represented in the form H= C H;+ /
1 HdL+ C H:+V,,. 1 I
(5)
The essential difference of the model under consideration from that of refs. [ 22-241, where ordering in a rigid lattice was investigated, consists in the detailed consideration of vibronic interaction.
3. Molecular field approximation in semiclassical vibronic model The conventional
molecular
field approximation
99
Physm 166 (1992) 97-106
[ 361 consists in substituting in the intercluster interaction Hamiltonian the quantity aLo”; with (see for instance eq. (2.12) in ref. [ 19 ] ) a’ak=,=j,++a’~-$ x .r zi r
(6)
The quantity d= doa= do4, is the mean cluster dipole moment. The mean value (Tis a dimensionless order parameter and is defined by the relation ‘=
Tr[exp( -fl/kT)aj,] Tr[exp(-Ej/kZ)]
.
(7)
Here E? is the Hamiltonian of the crystal in the molecular field approximation, i.e. the Hamiltonian H from eq. ( 5 ) in which the interaction V,, is replaced is a by pddd= -Ld@C, a;, where L=x:,K(/-k) structural parameter depending on the mutual disposition of clusters in the crystal lattice. Due to the crystal invariance under translation the structural parameter L does not depend on the number of the crystal site 1. Within the framework of this approximation the total Hamiltonian fi can be expanded as a sum of single-site (single-cluster) Hamiltonians fl,: if,=H;+H;+H;,-Ld;aaj,.
(8)
In order to calculate the trace in eq. (7), it is necessary to know the eigenvalues of the Hamiltonian fi. For a certain d value the Hamiltonian (8) describes a two-level electronic subsystem with off-diagonal vibronic interaction (see eq. (2 ) ). To obtain the quantum solution one faces the problem of diagonalizing the vibronic interaction matrix within the framework of the PKS model. The last term in eq. (8), which represents the interaction of the “extra” electron with the molecular field, plays the role of “asymmetry parameter” in terms of the PKS model [ 7 1. In accordance with the molecular field approximation it is necessary to find the eigenvalues of the Hamiltonian (8) for an arbitrary value of the order parameter 6, i.e. at any temperature. Then these eigenvalues should be substituted in eq. (7 ). The selfconsistent equation thus obtained provides the temperature dependence of the order parameter. The selfconsistent equation for the order parameter cannot be derived explicitly due to the non-adiabatic (pseudo-Jahn-Teller) vibronic interaction in a single cluster affected by the molecular field. To overcome this difficulty when the order parameter is obtained from eq. (7), we employ the adi-
B.S. Tsukerblatet al. /ChemicalPhysrcs 166 (1992) 97-106
100
abatic approximation within the scope of the molecular field method. The adiabatic approximation consists in neglecting the nuclear kinetic energy in the Hamiltonian ( 8 ) . Hence, the single-site (single-cluster) adiabatic Hamiltonian takes the form ~C=pAoo,+~oqa,+1Aoq2-Ld~a,d,
(9)
where the vibrational coordinate q plays the role of parameter and the symbol 1 is omitted. To calculate the trace in eq. (7) it is convenient to use the adiabatic wavefunctions diagonalizing the Hamiltonian ( 9 ). These functions are: Y ,(2)={f[l+PlW(4)l}1’2V/+
*{t[lTPlw9)l)“2y/-
2
W(q)=Aw[p2+(uq-~c?)2]“2,
(10)
where [= Ldi /Am is the dimensionless parameter of intercluster interaction. The total single-site energies in the adopted approximation represent the adiabatic potentials u ,(2) = &ilq2*
W(q)
.
i=z-l
7 dqexp(-$$)[&sinh($) -CC v2q*
cash
+ d2(q)kT
HI ‘(‘)
kT
(13)
’
where d(q)=fiw(p*+v~q~)“2.
4. Temperature dependence of order parameter The temperature behavior of the mean dipole moment is determined by three dimensionless parameters: v, p/c, p. The numerical solution of eq. ( 12) shows that there is a range of these parameters where charge ordering occurs. The temperature dependence of the order parameter is presented in fig. 1. The family of curves 6( T) =6( T) /do shown in fig. 1a illustrates the influence of vibronic interaction on the temperature de-
(11)
In the semiclassical approach, summation over electron-vibrational states in eq. ( 7 ) is replaced by integration over the vibrational coordinate q. We hence obtain a self-consistent equation for the order parameter: CC gfic~
5 dqexp(
- %)((a-vq)
-m
x
.,.
sinh1W(q)lkT1 WC?) ’
(12)
,.
,
-
qzz
0.5
where ~~2
7 dqexp( --CC
- $$)cosh
(F)
is the partition function in the semiclassical approximation. At the phase transition temperature the order parameter vanishes. Expanding the right side of eq. ( 12 ) in a power series of 8 one obtains the equation for the phase transition temperature Tc:
Fig. 1. Temperature
dependence
of the order parameter.
101
B.S. Tsukerblat et al. /Chemical Physrcs 166 (I 992) 97-106
pendence of the order parameter. Fig. 1a shows that vibronic coupling growth leads to an increase in phase transition temperature and mean dipole moment. If the intercluster interaction (2p/c= 1.6) is relatively weak, tunneling in the charge-ordered phase is partially suppressed, and for v= 0.1 we obtain a( 0) =0.4. Vibronic interaction leads to additional localization of the extra electron and when u increases from 0.5 to 1.O the maximum dipole moment increases from 0.73 to 0.87. Finally, for v= 3 at low temperature the dipole moment attains a value close to the maximum (d= 1). It should be noted that for v= 0.1 the phase transition temperature kT, = 0.92 x 2pAw = 0.74Ldi and for v=3 it is kT,= 1.25pAw=Ldi. (These values of phase transition temperature have been obtained while calculating the temperature dependence of the dipole moment (eq. (12), see also eq. (13).) This shows the physical role of the vibronic interaction. When this interaction is weak the phase transition temperature is determined by the competition between the stabilizing effect of intercluster interaction and tunnel intracluster interaction. If the vibronic interaction is strong the double exchange is suppressed and the phase transition temperature is close to the value of intercluster interaction energy. For the parameter values under consideration the phase transition is possible even in static (rigid) lattices (v= 0)) since the condition p < c holds. For weak intercluster interaction (p > c) there is no charge-ordered phase in the rigid lattice (v= 0 ). Fig. lb shows that inclusion of vibronic interaction leads to charge ordering even in that region of parameters p and [ for which ordering cannot take place in the rigid lattice. The vibronic interaction increase is always accompanied by an increase in the phase transition temperature as well as an increase of the maximum dipole moment value. For example, for v= 0.1 the phase transition temperature is equal to 0.32Ldij, and for v= 3 kT,= Ld& Thus, from the physical point of view, the order parameter determined above in the semiclassical approximation reveals a regular temperature behaviour. The approach developed above makes it possible to investigate the influence of the vibronic effects on charge ordering characteristics.
5. Dynamical vibronic problem Our next step is the approximation procedure. The essence of the suggested approximation consists in the quantum-mechanical approach of the solution of the single-site vibronic problem with the order parameter defined semiclassically. Mathematically this is the same as solving the vibronic problem with the Hamiltonian (9) including the kinetic nuclei energy X = HL +pfzoa= + &oqaX - @Ma,
.
(14)
Here d is determined from the semiclassical equation ( 12) for the order parameter. To solve the dynamical problem we shall use the mathematical procedure proposed within the framework of the PKS model [ 7 1. The presence of a molecular field removes the inversion symmetry, so that we are dealing with the so-called asymmetric case of the PKS model here. The temperature-dependent value Lad0 plays the role of perturbation leading to the disappearance of the inversion centre. Thus for every given temperature we face the PKS problem for an asymmetric dimeric MV cluster. Nevertheless, in contrast to the isolated cluster problem, in the case under consideration the asymmetry parameter essentially depends on temperature. Vibronic wavefunctions are represented as an expansion in terms of unperturbed electronic and vibrational states:
@Y= f
II=0
tC”nW+(r)Xn(q) +Lw-
(r),%(q) 1 ,
(15)
where x,,(q) is the harmonic oscillator wavefunction, the index v numerates the hybrid cluster states in the molecular field. As in the PKS model the eigenvalues E, and the eigenvectors 0, are determined from the system of equations c,,(m+f+p-E,/fio)+
f
b,,A,,=O,
fl=O
b,,( m+ f -p-EJAo)
+ f
c,,,&,
=O ,
(16)
?I=0
where &7n=V(~&l,n+l m=O, 1,2, .... In computer
>
+JR=&%l,,-,)+l&Z, v=O, 1,2, ... .
calculations
200 unperturbed
(17) vibra-
102
B.S. Tsukerblat
et al. /Chemrc*al Physics 166 (1992) 97-106
8 \I/ 6
‘4
\
2
LI
-6
I I LJib -4
I
,Q
I -2
, A.
, 2
4
69
t
-L-4:
-4 0.2
0.2
0.;
ula
a
T/T=
T=O
Fig. 2. Temperature dependence of vibronic energy levels within the framework of the molecular field approximation: i= 1, p= 0.5, v=2.5.
tional states have been taken as the basis set. The temperature dependence of the lowest cluster vibronic levels in the molecular field is shown in fig. 2 for v=2.5, p=O.5, c= 1. Qualitatively, the hybrid spectrum peculiarities are quite clear from fig. 3, where the adiabatic potentials at T= 0 and T= T, are also represented for the same parameters. At low temperatures the first two vibronic levels correspond to located states close to the harmonic oscillator levels. The energies of the lower adiabatic potential sheet minima flatten out as the temperature increases, leading to tunnel level splitting. It should be noted that within the scope of the adopted approach the quantum properties of the vibronic states in a self-consistent field are taken into account. It is reasonable therefore to call the proposed approximation quasidynamical. The traditional semiclassical approximation deals with electronic states in the quantum way, while the motion of the nuclei is treated classically. The vibronic states obtained within the scope of the quasidynamical approximation are hybrid, i.e. retaining the quantum properties of both electronic and vibrational states.
6. Intervalence transfer bands
The band shape function of the inter-valence cal absorption is described by the expression
Fig. 3. Temperature dependence of the adiabatic potentials in the molecular field approximation: i= 1, p=O.5, u=2.5. (a) T=O, (b) T= T,. Vibronic levels of the lower sheet are indicated schematically.
F(Q)=
c cF ”
d&7,,, (a)
“=-“I
x (c,,,h, +LJ”n)
9
(18)
where NV is the equilibrium population of the vth vibronic level, Z= E,exp( -E,/kT) is the partition function, pyy. is the line shape connected with the transition between vibronic states Y and v’. This line shape is assumed to be Gaussian:
pyy,(Q;2) = opti-
To smooth the quantum discrete structure of the entire band the second moment of the individual line il
B.S. Tsukerblat et al. /Chemcal
should be comparable with the fro value. As well as in ref. [ 351 we put here 1= Aw. The temperature dependence of the absorption band envelope is represented in fig. 4. Fig. 4a shows the band shape for relatively strong coupling with deep adiabatic potential minima both for ordered and disordered states. At high temperatures TX T, the band has two maxima. The high-frequency maximum is related to the Franck-Condon transition. In terms of the semiclassical adiabatic approximation this transition can be associated with the vertical transition from the minimum of the adiabatic potential lower sheet to the upper sheet. For relatively small p and strong vibronic coupling (p < v2) the frequency of this maximum can be roughly estimated by the semiclassical expression [ 29,301
103
Physrcs 166 (1992) 97-106
(v2+[dy-
x
.
(20)
Eq. (20) shows that the temperature dependence of the band maximum is related to two physical factors: ( 1) explicit temperature dependence in the ordered as well as in the disordered phase and (2) temperature dependence of the order parameter. As is clear from fig. 4a, the frequency of the right maximum approximately satisfies relation (20) (w,,, x 13.5fiw). The low frequency maximum is the envelope of the lines arising from the transitions between the tunnel split levels. These tunnel states appear only at temperatures close to the phase transition temperature, when the potential surface minima are energetically equivalent. When the temperature decreases the band
d Fig, 4.
Temperature dependence
~=1,p=l,v=2;(d)~=l,p=l,u=lS.
of the absorption
coeffkient
(in arbitrary
units).
(a) c= 1, P=O.~,
u=2.%
(b) c= 1, P=2,
V= 1; (c)
B.S. Tsukerblat et al. / Chemcal Physics 166 (1992) 97-106
104
narrows and shifts to the high-frequency range. In the case of weak vibronic interaction the absorption coefficient is presented in fig. 4b. The lower sheet of the adiabatic potential has one minimum, at any temperature the optical band is bell-shaped, but not Gaussian. The molecular field contributes to the level splitting, and when the temperature decreases the band shifts to the high-frequency range. In figs. 4c and 4d the band shape for moderate vibronic coupling is shown. When the vibronic coupling parameter decreases, the low-frequency maximum of the optical curves (in the vicinity of r,) disappears and turns into a shoulder (see fig. 4). If coupling is further reduced, this maximum is absent at any temperature. For all parameters and temperatures the charge transfer bands remain essentially asymmetric and possess a long tail in the high-fre-
quency range. At the phase transition point the order parameter vanishes, and the results obtained coincide exactly with those of the PKS model. As is shown in fig. 4a, at temperatures T-c T, the position of the high-frequency maximum changes considerably when the temperature increases. Therefore the temperature shift of the high-frequency maximum of the band as well as the intervalence band-shape transformation may be considered as a manifestation of intercluster interaction.
7. Comparison with semiclassical
results
We shall now compare the results obtained with the semiclassical results obtained within the framework of classical nuclei motion assumption. The semiclas-
K(Q)
Fig. 5. Comparison of the results of the quasidynamical model with those of semiclassical ~~2.5; (b) C=l,p=l, v=2; (c) C=l,p=l,u=lS; (d) l=l.p=2,v=l.
adiabatic
approximation.
(a) C= 1, ~=0.5,
B.S. Tsukerblat et al. / Chemrcal Physrcs 166 (I 992) 97-106
sical shape band is given by the relation
approximation can hardly be applied to the case of moderate vibronic coupling. The intervalence band features elucidated above corroborate the existence of the charge-ordered state and provide unique information about the key parameters of interacting clusters in crystal charge-ordered state and phase transitions.
[ 32 ]
s
‘30
X
dqexp( - G/W
-cc
x~(U,(9)-U2(9)-fi-Q)
3
105
(21)
where 2 is the semiclassical partition function. The charge-ordered crystal band shape calculated from eq. (2 1) is discussed in detail in refs. [ 29,301. The optical curves obtained in these papers possess nonphysical peculiarities resulting from the inapplicability of the semiclassical approximation for the description of certain spectral ranges. In fig. 5 the results of the band shape calculations for the quasidynamical model are compared with those obtained from eq. (2 1). For strong coupling (fig. 5a) and low temperatures both the semiclassical and quasidynamical model yield bell-shaped curves which are similar in general. These curves however differ markedly in shape and maximum position. At T= T, the overall view of contours of the classical and quantum bands are also similar. A significant difference takes place in the region O/wz2p, where the semiclassical curve is divergent. When the vibronic coupling decreases (fig. 5b) the non-physical region @/wz2p expands, and the discrepancy between semiclassical and quasidynamical results becomes more significant. This discrepancy takes also place at low temperatures. In the semiclassical approximation within the intermediate and weak coupling range (figs. SC and 5d) the high temperature ( Tz T,) absorption band decreases monotonically, beginning from the divergence region (Q/o= 2~). The optical curves in the quasidynamical model are quite different from the semiclassical ones within this range. The quasidynamical curves are bell-shaped, and significant absorption takes place in the classically forbidden region B/w < 2p. Thus, for moderate and weak vibronic coupling the semiclassical approximation cannot describe the charge transfer band contour in charge-ordered crystals, both at low temperatures and within the range of phase transition. It should be stressed that at high temperatures the semiclassical
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