Dielectric theory of weak charge-transfer crystals. II. Polarization energies

Chemical Physics 135 ( 1989) 15-26 North-Holland, Amsterdam

DIELECTRIC THEORY OF WEAK CHARGE-TRANSFER II. POLARIZATION ENERGIES

CRYSTALS.

R.W. MUNN and R.J. PHILLIPS Department of Chemistry and Centrefor Electronic Materials, UMIST, Manchester h460 IQD, UK Received 27 December 1988

Polarization energies and charge-quadrupole energies are calculated for a single excess charge in the complexes of anthracene with PMDA (pyromellitic dianhydride) and TCNB ( 1,2,4$tetracyanobenzene) using different sets of molecular polarizabilities from part I. Comparison with experimental crystal ionization energies and photoconductivity band gaps allows preferred polarizabilities to be identified. Relaxation energies of about 0.8 eV per charge are deduced in anthracene-PMDA and 0.4 eV in anthracene-TCNB, consistent with strong electron-phonon coupling. Polarization, Coulomb and charge-quadrupole energies are calculated for charge-transfer configurations with a hole on anthracene plus an electron on the acceptor (heteromolecular CT), on another anthracene (homomolecular CT), or symmetricalIy split between two acceptors (split CT). In anthracene-PMDA split CT configurations have energies comparable with heteromolecular CT while in anthracene-TCNB the split ones lie somewhat higher. In both crystals the lowest homomolecular CT configurations lie about 1 eV above the lowest heteromolecular ones and should interact strongly with them. This interaction could reconcile the calculated CT transition energies with observed values some 0.5 eV lower.

1. Introduction

The polarization energy is a molecular energy change associated with the induced electric dipoles on the molecules, which plays an important role in many phenomena. Neutral molecules produce polarization through their permanent electric multipole moments, as in the electronic Stark effect [ l-31 and in lattice energies and lattice dynamics [ 45 1. Molecular ions produce polarization through their charges, as in the localized ionic states from which tight-binding carrier bands are constructed and in the chargetransfer states implicated in carrier photogeneration and recombination [ 6 1. The first successful calculations of polarization energy used the self-consistent polarization field method [ 6-9 1. Subsequently a Fourier-transform method was developed to give results directly, in effect by calculating the inverse dielectric function of the crystal. By using effective molecular polarizabilities derived from the measured crystal dielectric tensor, the method was able to provide an internally consistent account of both the microscopic and the macroscopic polarization. Algebraic results were obtained for single car-

tiers in perfect [ lo] and imperfect [ 111 crystals and for charge-transfer pairs [ 12,13 ] in perfect crystals, and evaluated for the aromatic hydrocarbons. The Fourier-transform method has already been applied to tetracene and pentacene [ lo- 13 1, which contain two crystallographically independent molecules in the unit cell, and to the charge-transfer complex of anthracene with tetracyanobenzene [ 141. In the latter case only algebraic results were obtained since effective polarizabilities for the separate molecules were not available. However, techniques to derive effective molecular polarizabilities have now been devised [ 15 ] and applied to the weak charge-transfer crystals of anthracene with the acceptors pyromellitic dianhydride 7,7,&S-tetracyanoquinodimethane (PMDA), (TCNQ), and 1,2,4,5_tetracyanobenzene (TCNB ). Given this essential input information, we have calculated polarization energies for various localized ionic states of these chargc-transfer crystals. Results for anthracene-PMDA and anthracene-TCNB are reported here; those for anthracene-TCNQ are not considered reliable because they depend on lattice sums for which we have not been able to obtain sat-

0301-0104/89/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

16

R. W. Mum, R.J. Phillips /Dielectric

isfactory convergence. Preliminary results for anthracene-PMDA were given elsewhere [ 16 1. In practice, these calculations provide insight not only into the localized ionic states but also into the effective polarizabilities. Analysis of the dielectric tensor of these weak charge-transfer crystals yields several pairs of effective molecular polarizabilities each compatible with the macroscopic dielectric behaviour [ 15 1. We have used the various sets of polarizabilities to calculate single-carrier polarization energies in anthracene-PMDA and anthraceneTCNB. Comparison with experiment shows that only one polarizability solution for each crystal is acceptable. This solution is then used for subsequent calculations. Charge-transfer crystals offer a richer variety of localized ionic states than do homomolecular crystals. A single carrier of either sign may be located on either component, and the resulting energies may be compared both with one another and with those in the homomolecular crystals. This allows the effectiveness of dielectric screening by the two molecules to be compared, and provides insight into photoemission and conduction properties [ 17 1. Charge-transfer excitations yielding anion and cation states are dominant spectroscopic features, and are also important in charge-carrier photogeneration. In addition to the obvious excitations to donor cation (D+ ) and acceptor anion (A- ) states, one can consider those to states with cation and anion both on donor or acceptor molecules. The exciton-phonon interaction is generally strong for CT excitons, and in competition with strong carrier transfer interactions may lead either to asymmetric polar states most simply described as dimers D+ A- or to symmetric non-polar trimer states describable as D”2+A-D”2+ or A”*-D+A”*[ 18,191. In the present paper we explore all these aspects of localized ionic states. We do not consider uniformly ionized crystal states in which all donors carry a charge ps and all acceptors a charge -pe (0
theory of weak CT crystals. II

imental information are made in each section, and general conclusions are drawn in section 5. We conclude that electron transfer to anthracene may play a significant role in CT states.

2. Method 2.1. Polarization energy Let charges qm be located at submolecule sites m= (l,k,,&), where 1, denotes the unit cell, k, the molecule in that cell and j, the submolecule in that molecule. Then the total polarization energy is [ lo] P= - (1/2e0v)N-i

c 1 4&&Y,P,,~(Y) y mm’

x exp(2rtiy*r,,,).

(1)

Here the sum runs over all N wavevectors y in the first Brillouin zone, and P,,.(y) is the (k,,,j,,,, k,,,,j,,,. ) element of the supermatrix

P(y)=u(-y)T.[a-‘-t(y)]-‘.u(y).

(2)

For Z formula units of the complex each with a total Of s = sd -t s, submolecules with sd on the donor and s, on the acceptor, P is of dimension SZXSZ. The 3sZxsZ supermatrix u(v) is formed from lattice monopole sums ~(y; kj, k’j’ ) calculated between submolecules with intramolecular contributions excluded. The 3sZx 3sZ supermatrix a is formed from submatrices ak&d,. involving the submolecule reduced polarizabilities ak,=akj/c,v, with u the unit cell volume. The expressions were evaluated much as previously [ 10 1. Convergence of the Brillouin zone sum was tested by increasing n in the sequence 2,4,8, 16; calculating lattice sums beyond la= 16 demanded excessive computer time. Individual sums had not always converged satisfactorily for n= 16, but adequate convergence was achieved by using each consecutive pair of sums to estimate the sum as n-+co by the Romberg method [ 211. For anthraceneTCNQ even this procedure gave unsatisfactory results and further work on this complex was abandoned.

11

R. W. Munn, R. J. Phillips / Dielectric theory of weak CT crystals. II

2.2. Submolecule polarizabilities

action between the charges. This is given by [ 12,13 ] Wc= (1/4x%) 1 qokjqikf1rokj.Ik.j B’

(4)

Polarizabilities for the separate donor and acceptor molecules in CT crystals have been derived [ 15 ] using a submolecule treatment to calculate lattice dipole sums. For present purposes, polarizabilities for individual submolecules are essential to take account of the marked variation in electric field between different submolecules near a molecular ion. The method is described in appendix A.

for the interaction between ion k in the origin unit cell and ion k’ in the unit cell l’, where rokj,rk.f is the distance between submolecules Okjand I’k'j' .For the centrosymmetric fractionally charged CT states there is a term like (4) for each pair of ions.

2.3. Charge-quadrupole energy

3. Singly charged states

The energy of an ionic state depends not only on the polarization energy but also on the interaction energy between the charge and the permanent multipole moments of the surrounding molecules. For centrosymmetric molecules, the leading term is the charge-quadrupole energy given by [ 22 ]

Polarization energies have been calculated for an excess charge on each site in each crystal for each of the nonzero effective polarizability solutions found previously [ 15 1. The results are given in table 1 for anthracene-PMDA and in table 2 for anthraceneTCNB. Consecutive pairs of polarizability solutions have the molecular polarizabilities almost exactly interchanged, and this feature is carried through into the polarization energies, which vary considerably. The polarization energy is larger in magnitude when the excess charge is on the less polarizable molecule and hence immediately surrounded by the more polarizable molecules. For comparison, the polarization energy of a single carrier in the anthracene homomolecular crystal was calculated to be - 1.19 eV

WQ=(~/~~OV) g,

9kj:Lkj.k’j’qk.j.

.

(3)

Here @., is the quadnrpole moment of submolecule kj and qki’ is the charge of submolecule k’j’. It is assumed that only one molecule carries a charge, uniformly distributed between submolecules. For chargetransfer states, W, consists of a term like ( 3 ) for each charged molecule, plus a correction factor to allow for the interaction of the charge on each ion with the quadrupole moment(s) of the other ion(s), which will in general differ from those of the neutral molecules. Quadrupole moments for anthracene, PMDA, and TCNB were obtained using ab initio MO calculations at STQ/4-31G level using the POLYATOM suite of programs [ 23 1. The quadrupole moments in the molecular LMN axis system were 29.9, 56.3 and - 86.2~ 10m40C m2 for anthracene, - 133.4, 12.9 and 120.4x10-40Cm2forPMDA,and -2.9, -31.2 and 34.0 x 1O-4o C m2 for TCNB. Quadrupole moment changes in the ions were calculated by previous methods [ 12,13 ] ; see appendix B.

[ 101. Hole-quadrupole energies are also given in tables 1 and 2. They may be compared with -0.22 eV in the anthracene homomolecular crystal (this is slightly larger in magnitude than that reported previously [ 111 which used a quadrupole moment calculated at STG/3G level). The difference in sign at the anthracene site in the complexes is explained by the differTable 1 Polarization energies P”=F and hole-quadrupole energies W% = - Wb at the anthracene (A) and PMDA (P) sites in the anthracene-PMDA crystal. The polarization energies are calculated using different effective polarizability solutions [ 15 1. AU energies are in eV Site

2.4. Coulomb energy The leading contribution to the energy of the charge-transfer states is the Coulomb energy of inter-

A P

Ph

W:,

(i)

(ii)

- 1.56 -1.10

-1.18 - 1.62

0.67 -0.76

R. W. Mum, R.J. Phillips /Dielectric theor?, of weak CT crystals. II

18

Table 2 Polarization energies Ph=Pe and hole-quadrupole energies Wb = - W; at the anthracene (A) and TCNB (T) sites in the anthraceneTCNB crystal. The polarization energies are calculated using different effective polarizability solutions [ 151. All energies are in eV Site

A T

Ph

W:,

(i)

(ii)

(iii)

(iv)

(v)

(vi)

-1.08 - 1.95

-2.37 - 1.07

- 2.02 -1.11

-1.15 -1.84

- 1.76 -1.18

- 1.25 - 1.63

ent molecular packing, which modifies the field gradients. The generally larger magnitude of W& stems from the close approach of the molecules in the parallel mixed stacks to distances less than the sum of the van der Waals radii. The similar values of W 6 at the acceptor sites in the two crystals reflect similar molecular environments sandwiched between anthracene molecules. On the other hand, W!j at anthracene is noticeably larger in the PMDA complex than in the TCNB complex because PMDA has a much larger quadrupole moment than TCNB. However, the PMDA quadrupole moment is roughly equal and opposite to that of anthracene, and because the two molecules occupy similar sites which therefore experience very similar field gradients from the excess charge, the values of Wb at the two sites in anthracene-PMDA are roughly equal and opposite. Polarization energies cannot be measured directly. The apparent or effective polarization energy is the energy of the ion in the crystal compared with that in free space, which can be written for quadrupolar molecules as [ 10 ] w=p+

w, +

w, .

(5)

Here W, is the relaxation energy, a lowering of energy due to changes in the molecular separations and orientations under the influence of the polarization and charge-quadrupole forces; it can also be regarded as the binding energy of polaron formation [ 24 1. For holes we have Wh=I, -I, )

(6)

where Zgand Z, are the ionization energies in the gaseous and crystalline states, while for electrons we have We=A,

-A,,

(7)

where A, and A, are the gas and crystal electron affinities. In practice, I,, Z, and A, can be determined ex-

0.28 -0.77

perimentally but not A,, and it is then useful to consider also the conductivity band gap [ 6,9, lo], EG =I, -A, =I,--A,+

(8) Wh+ W’.

(9)

Note that as defined here EG is the adiabatic band gap determined from photoconductivity studies, and is to be distinguished from the optical band gap determined from electromodulation studies, which by the Franck-Condon principle refers to processes occurring on a time scale too short to allow lattice relaxation and so contains no contribution form W, [24,25]. Electron photoemission measurements from anthracene_pMDA give I, = 6.0 eV [ 26 1, compared with an average I,= 7.42 eV for anthracene [24]. Hence Wk = - 1.42 eV requiring Wk = -0.53 eV for solution (i) in table 1 and W”, = -0.91 eV for solution (ii ). Photoemission studies on a series of complexes with PMDA as acceptor lead to an estimate w;= -1.58eV [17],requiring W&=-1.24eVfor solution (i) and - 0.72 eV for solution (ii). The activation energy for photoconduction in anthracenePMDA has been determined as 0.14 eV [ 271 which with the optical CT transition energy of 2.27 eV [ 28) implies a lower limit of 2.4 eV for EG., while an upper limit of 2.7 eV has been estimated. With A,=2.04 eV for PMDA [ 291 these imply net relaxation energies Wk + we, of - 1.80 to - 1.50 eV with solution (i ) and - 1.66 to - 1.36 eV with solution (ii), comparing well with - 1.77 and - 1.63 eV respectively obtained from the relaxation energies estimated separately above. Deciding between solutions (i) and (ii) for anthracene-PMDA amounts to deciding between the relaxation energies they imply. All are negative, as required physically, and all are significantly larger in

19

R. W. Mum, R.J. Phillips /Dielectric theory of weak CTcrystals. II

magnitude than in anthracene and naphthalene [ 10 1. Large relaxation energies are implied by the strong electron-phonon coupling shown by the broad chargetransfer band in the optical spectrum. Overall, solution (ii) implies somewhat smaller magnitudes of the relaxation energies, gives the larger hole relaxation energies, consistent with offsetting the destabilising effect of the hole-quadiupole energy as much as possible, and gives a value of P at anthracene less in magnitude than that at PMDA, as calculated previously [ 301, and close to that in homomolecular anthracene. Experimental information on ionic states in anthracene-TCNB is rather sparse. Photoconductivity studies lead to a value of 2.8 eV for the band gap EG and 2.4 eV for the optical CT transition energy [ 3 11. A highly simplified treatment using these values yields an estimate of - 1.6 eV for the (apparent) polarization energy [ 321. With A,= 2.00 eV for TCNB [ 291, the various solutions in table 2 yield net relaxation energies of -0.23 to -0.79 eV. These are much smaller than in anthracene-PMDA because the net polarization energies are significantly larger in magnitude and Wb at anthracene is significantly smaller. From the solutions (i) to (vi) we discard the three with Pat anthracene greater than at TCNB, in accord with indications from previous calculations [ 301, and select solution (vi) as having the largest net relaxation energy of -0.79 eV (though by only 0.11 eV). The breadth of the CT absorption again indicates a relaxation energy larger than found [ lo] in anthracene and naphthalene. A smaller relaxation energy than in anthracene-PMDA is consistent with the smaller hole-quadrupole energy here if this serves as an effective driving force for relaxation. Relaxation in anthracene-TCNB may be affected by the disorder in the structure [33,34]. Solution (vi) and the chosen solution (ii) for anthracene-PMDA give essentially the same polarization energy at the acceptor, where the immediate environments of anthracene donor molecules are very similar. The results for the two complexes are summarized in the form of energy level diagrams in figs. 1 and 2. These show the electronic polaron levels (without relaxation) and the vibronic polaron levels (with relaxation); the separations between corresponding electron and hole levels give the optical and adiabatic band gaps respectively. In the absence of suitable photoemission data for anthracenc-TCNB, the indi-

I

I

Aq=2.04

Fig. 1. Energy level diagram for heteromolecular ionic states in anthracene-PMDA. Superscripts e and h denote electrons and holes; I, and I, are the gas and crystal-phase ionization energies of anthracene and A, is the gas-phase electron affinity of PMDA, P is the polarization energy, W, the charge-quadrupole energy, and W, the relaxation energy. The levels labelled S and R correspond to electronic and vibronic polarons respectively, with EF’ and Eg the optical and adiabatic band gaps.

T

I Fig. 2. Energy level diagram for heteromolecular ionic states in anthracene-TCNB (notation as in fig. 1). Note that the relaxation energies are not determined separately but have been fixed by taking I, to be the same as in anthracene-PMDA.

vidual relaxation energies are not determined. They have been futed here by assuming that the vibronic polaron levels for the hole on anthracene (and hence the crystal ionization energies) are the same in both complexes. This yields a plausible division of the total relaxation energy between the electron and the hole

20

with each contribution cene-PMDA.

R. W. Mum, R.J. Phillips /Dielectric theory of‘weak CTcrystals. II

roughly half that in anthra-

4. Charge-transfer states All the charge-transfer states investigated have the hole on the donor anthracene at the origin. Three types of states were investigated: with the electron on the acceptor (heteromolecular CT, D+A- ), on another “donor” anthracene (homomolecular CT, D’D-), or split between two acceptors symmetrically located either side of the donor (split CT, A”2-D+A’/2). Over 100 states of each type were studied for anthracene-PMDA, and over 30 for anthracene-TCNB. Various energies have been calculated as a function of the vector r between the hole and electron sites. The principal quantities calculated are the electronhole polarization energy Peh (r) and the Coulomb energy W,(r), the sum of which we call the net energy W,(r). The screened Coulomb energy Ws( r) is obtained by subtracting from the net energy the separate electron and hole polarization energies:

Ws(r)= Wc(r)+Peh(r)-Pe-Ph.

(10)

By adding to the screened Coulomb energy the correction A W, (r) to the separate electron and holequadrupole energies for the change in quadrupole moment in the ions, one obtains the energy of the CT state relative to that of the separate charges, which is the potential V(r) for the CT state [ 12,13,35]:

V(r)= Ws(r)+AWQ(r)

.

(11)

The single-charge polarization energies are obtained from tables 1 and 2, except for the electron polarization energy for split CT. Polarization energies are not pairwise additive: if they were, eq. ( 10) shows that the Coulomb energy would be unscreened. Hence the total polarization energy for the two half-electrons in split CT differs from that for one electron by a contribution from their mutual polarization which depends on r and tends to zero as r tends to infinity. Results presented here for split CT all include the total polarization energy calculated exactly. This gives the energy of the split CT state relative to the separated hole and two half-electrons. Results for the first few CT states of each type are

given in table 3 for anthracene-PMDA and in table 4 for anthracene-TCNB; full details are given elsewhere [ 361. For sufficiently large r, the screened Coulomb energy in a given direction defined by direction cosines (2, p, Y) relative to the principal dielectric axes takes the form

W,(r) x -q2/4xfA,,r. Here tlllu is the apparent direction, given by [ 13 ]

(12) dielectric

t~,,=(E2t3~2+t3E,~2+~,E*Y*)“2.

constant

of this

(13)

We have analysed all the results and obtain satisfactory agreement with ( 12 ) . As previously we find deviations at small r, where the molecular separations are comparable with molecular dimensions so that a continuum treatment is inapplicable, and at large r where the precision of the results is inadequate [ 12,131. Examination of tables 3 and 4 shows that each type of CT state is much the same in the two crystals. For heteromolecular CT, the tint state has much the smallest r, corresponding to the nearest-neighbor separation. It has the largest (negative) Coulomb energy but the smallest (negative) Pehand ends up with the largest (negative) potential energy. Were it not for its large positive A W,, the first state would be even more separated in energy from the others; indeed, in anthracene-TCNB a sizeable negative A W, for the electron at ( - 1, 0, l/2) brings this state, with r= 10.35 A, within 0.1 eV of the lowest, with r= 3.71 A. For homomolecular CT, the values of Vvary rather erratically as r increases. They are generally smaller in magnitude than for heteromolecular CT, though the largest in magnitude is exceeded only by one heteromolecular CT state for anthracene-PMDA and by two for anthracene-TCNB. For split CT, the values of V tend to vary erratically with r again and to be smaller still in magnitude. However, in anthracenePMDA the value of V for the half-electrons at f. (0, 0, 5) is actually larger in magnitude than for the nearest-neighbour heteromolecular CT state, - 1.25 eV as against - 1.13 eV. The nearest-neighbour split CT state also has the next largest magnitude of V for any state, -0.79 eV. These results indicate that invocation of split CT states is by no means energetically implausible in anthracene-PMDA. Charge-transfer excitations can be studied spectro-

21

R. W. Mum, R. J. Phillips /Dielectric theory of weak CT crystals. II

Table 3 Mean separations r, polarization energies Ixh, unscreened and screened Coulomb energies WC and Ws, charge-quadrupole energy changes A W, and potential energy V for different types of charge transfer in anthracene-PMDA. All energies are expressed in eV Lattice vector

peh

WC

WS

AWQ

V

10.2973 10.9500 10.9704 11.4985

-1.0919 -1.4138 -1.7132 - 1.8774 - 1.8212 - 1.8241 - 1.9936 - 1.7769 -2.0021

-3.2316 - 1.9831 - 1.5269 - 1.4082 - 1.4921 - 1.3833 - 1.2813 - 1.4353 - 1.2262

- 1.5253 -0.5987 -0.4410 -0.4874 -0.5151 -0.4092 -0.4767 -0.4140 -0.4301

0.3932 -0.0190 - 0.0474 - 0.0066 -0.1075 -0.0579 - 0.0500 -0.0153 -0.0107

-1.1321 -0.6177 -0.4884 0.4808 -0.6226 -0.3513 -0.4267 - 0.4293 - 0.4408

7.3000 7.6000 7.9560 9.8238 10.0000 10.7472 11.3471

- 1.0885 -1.1938 -1.1075 -1.3179 - 1.4524 - 1.4466 - 1.6188

-

1.8908 1.7809 1.I636 1.6013 1.4256 1.3801 1.2615

-0.6135 -0.608 1 -0.5053 -0.5534 -0.5122 - 0.4609 -0.5137

-0.1578 -0.0144 -0.8751 -0.2180 -0.0433 -0.0032 -0.0579

-0.4557 -0.6225 - 0.4296 -0.7714 -0.5555 -0.4577 -0.5716

3.6500 6.8707 9.7174 9.1441 9.9654 10.2973 10.9500 10.9704

-0.4519 -0.7851 - 1.0361 - 1.1905 - 1.1333 - 1.1411 - 1.2982 - 1.0892

-2.7738 - 1.4372 -1.1892 - 1.5137 - 1.0932 - 1.5098 - 2.0934 -1.2961

- 1.0679 -0.0527 -0.1042 -0.5930 -0.1062 -0.5347 - 1.2887 -0.2748

0.2781 -0.0105 - 0.0343 0.0056 - 0.0866 0.0445 0.0379 -0.0129

-0.7898 -0.0632 -0.1385 -0.5874 -0.1928 -0.4902 - I .2508 -0.2877

r(A)

heteromolccular CT (O,O, (O,O, (091, (-l,O, (l,l, (l,O,

l/2) l/2) l/2) l/2) l/2) 3/2)

(0, 0,3/2) (1, 1,3/2) (0, - 1, l/2)

3.6500 6.8707 9.7174 9.7441 9.9654

homomolecular CT

(0, 0, - 1) (-l,O,O) (-l,O, -1) (-1, -1, -1) (0, -1,O) (0, -1, -1) (-1, -1,O) split CT *(O,O, l/2) +(I, 0, l/2) f (0, 1, l/2) k(-l,O, l/2) * (1, 1, l/2) *(1,0,3/2) * (0,0,3/2) + (1, 1,312)

scopically. In the simplest picture, the excitation energies of the CT states are obtained by adding the potential V(r) to the optical band gap Ez’. For anthracene-PMDA the band gap is 4.0 1 eV (fig. 1) and the lowest heteromolecular CT state has V= - 1.13 eV , so that the excitation energy is estimated as 2.88 eV. This compares with experimental values of 2.27 eV [37,38] and2.31 eV [39].Foranthracene-TCNB the band gap is 3.59 eV and the lowest heteromolecular CT state has V= -0.63 eV, so that the CT excitation energy is estimated as 2.96 eV. This compares with experimental values of 2.4 eV [ 3 1 ] and 2.44 eV [ 401. Consideration of the next level of sophistication in deducing the energies of the CT states shows that this discrepancy may be regarded as satisfactory. What have so far been called simply “CT states”

are better regarded as CT conjgurations for which the

electrostatic potential V has been calculated. The CT states proper are the eigenstates of the crystal Hamiltonian with this potential [ 12,131. These eigenstates involve mixing between CT states and Frenkel exciton states of appropriate symmetry, and splitting by electron and hole transfer integrals. Even for homomolecular anthracene these interactions could be treated only within a simplified model [ 12,13 1. In the CT crystals there are the further complications of separate donor and acceptor states and the different types of CT configuration, one simplification being the unambiguous polarization of the lowest CT transition along the stack axis. Hence only brief consideration is given here to the possible contribution of homomolecular CT confguration and transfer integrals.

R. U’. Mum. R. J. Phillips /Dielectric theory of break CT crystals. II

22

Table 4 Mean separations, r, polarization energies P”, unscreened and screened Coulomb energies I+‘,:and U’s, charge-quadrupole energy changes A Wo and potential energy V for different types of charge transfer in anthracene-TCNB. All energies are expressed in eV Lattice vector heteromolecular (090, l/2) (l/2, l/2,1/2) (l/2, l/2, l/2) (l,O, l/2) (-190, l/2) (O,O, 3/2)

CT

homomolecular (0, 0, 1) (l/2, l/2,0) (l,O, 0) (l/2, l/2, 1) (l/2,1/2, -1) (l,O, 1)

CT

split CT * (020, l/2) *(l/2, l/2, l/2) &(-l/2,1/2,1/2) +(l,O, l/2) k(-l,O, l/2) * (o,o, 3/2)

r(A)

Peh

WC

WS

Awn

V

3.7085 8.6868 8.8586 10.0541 10.3495 11.1255

-0.8872 - 1.6507 - 1.5643 - 1.7919 - 1.6868 - 2.0476

-3.0318 - 1.6235 - 1.6830 - 1.4548 - 1.5363 - 1.2485

- 1.0363 -0.3996 -0.3727 -0.3640 -0.3404 -0.3134

0.4095 0.0439 -0.0115 0.0107 -0.1864 0.0406

-0.6268 -0.3557 -0.3842 -0.3522 -0.3728 - 1.3728

7.4170 7.9507 9.5058 10.7337 11.0109 11.8038

-1.1893 - 1.1627 - 1.2338 - 1.5194 - 1.5014 - 1.5990

- 1.8622 - 1.7874 - 1.6363 - 1.3027 -1.3159 -1.2110

-0.5528 -0.4514 -0.3714 - 0.3234 -0.3186 -0.3113

0.1924 -0.0331 -0.1368 0.0502 -0.0056 -0.0403

- 0.3404 -0.4845 -0.5082 - 0.2732 -0.3242 -0.2710

3.7085 8.6868 8.8586 10.0541 10.3495 11.1255

-0.3148 - 1.0039 -0.9197 -1.1059 - 1.0031 - 1.3416

-

0.6015 -0.1759 -0.1542 -0.1839 -0.167 -0.2538

0.3351 0.0322 -0.0147 0.0139 -0.1428 0.0336

- 0.2664 -0.1437 -0.1689 -0.1700 -0.3045 -0.2202

For homomolecular CT, the optical band gap in figs. 1 and 2 is not appropriate, as it refers to an anthracene cation and a PMDA or TCNB anion. For an anthracene anion, the anthracene electron affinity of 0.55 eV [ 4 1] should be used with values for IP”i- Wb 1of 1.85eVinanthracene-PMDAand 1.53 eV in anthracene-TCNB, from tables 1 and 2. The resulting values of Ez’ are 4.51 eV in anthracenePMDA and 4.37 eV in anthracene-TCNB. The lowest homomolecular CT excitation energies in the simple picture are then obtained from tables 3 and 4 as 3.74 eV in anthracene-PMDA and 3.86 eV in anthracene-TCNB. These lie within 1 eV of the lowest heteromolecular CT excitation energies calculated in the same way. Although the acceptors have a much greater electron affinity than anthracene (by about 1.5 eV) and Y is lower in magnitude (less negative ) for homomolecular CT, these factors are significantly offset by the change in sign of W& from destabilizing on the acceptors to stabilizing on anthracene. Given that the lowest homomolecular and heter-

2.5970 1.3951 1.4591 1.2747 1.3576 1.0889

omolecular CT configurations are not too distant in energy, they could interact significantly. For example, the configurations with the electron at (0, 0, f ) and (0,0, 1) are related by electron transfer between adjacent acceptor and anthracene molecules in the alternating stack, which is probably rather strong. Some information relevant to transfer interactions can be deduced from carrier mobility measurements in anthracene-PMDA [ 27 1. Above 80 ’ C,where trapping becomes ineffective, the electron mobility is 0.15 cm* V-’ s-’ along the stack axis and 0.02 cm* V- ’ s- ’ perpendicular to the axis. These values are comparable with or a little smaller than those in other organic crystals. However, it has already been noted that the electronphonon coupling is strong in CT crystals [ 37 1, consistent with the sizeable relaxation energies W, deduced earlier. Carrier mobility is the result of competition between carrier transfer interactions and electron-phonon coupling. If the electron mobility in anthracene-PMDA is much the same as in other crystals while the electron-phonon coupling is much

R. W. Mum, R.J. Phillips /Dielectric theory of weak CT crystals. II

stronger, this implies that the transfer interactions must also be much stronger. Those in aromatic hydrocarbon crystals reach 0.1 eV, and hence by this argument those in anthracene-PMDA must be nearer 1 eV. This is sufficient to produce substantial interaction between the heteromolecular and homomolecular CT states, yielding one level lower than either, as required to explain the observed CT transition energy of 2.3 eV: in particular configurations at 2.88 and 3.74 eV with an interaction energy of 1 eV give levels at 4.40 and 2.22 eV. Previous estimates directed towards anthracene-TCNB indicated significant coupling (0.2-0.4 eV) between heteromolecular CT configurations and Frenkel excitons [ 42 ] but did not treat homomolecular CT, where stronger coupling might be expected. Photoconductivity measurements for anthraceneTCNB [ 3 1 ] reveal some unusual features which provoked a simple model analysis of the CT states [ 32 1. This suggested that the nearest-neighbour heteromolecular CT state did not tit the screened Coulombic dependence on distance of other CT states, and that the nearest-neighbour CT state was not the state of lowest energy. The present results show in contradiction that the nearest-neighbour configuration does have the lowest potential energy V. However, since the configuration with the electron at ( - 1, 0, 4 ) lies only 0.1 eV higher, it could possibly contribute to an eigenstate lower than any based on the nearest-neighbour configuration.

5. Discussion We have used single-carrier polarization energy calculations in comparison with experimental data to deduce preferred effective polarizability solutions from the sets obtained in part I for anthracenePMDA and anthracene-TCNB. Because the donor and acceptor molecules occupy similar sites, the solutions come in pairs with essentially the same polarizabilities but exchanged between the molecules. A single carrier can be used to “probe” an individual molecule, and this allows one of each pair of solutions to be discarded, in practice the one which implies a smaller rather than a larger polarizability of the donor anthracene. Uncertainty is introduced into the process by the unknown relaxation energies, but

23

application of fairly general criteria allows the selection to be completed. Deducing effective polarizabilities for the separate molecules in weak charge-transfer crystals will normally require information in addition to the dielectric tensor, which defines a number of acceptable polarizabilities. Provided the molecules are sufficiently different in polarizability (as one might expect for a donor and an acceptor), a reasonably clear selection is likely to ensue. If the molecules are little different in polarizability, no clear selection is likely -but then no significant difference is likely in any derived properties. The present approach is therefore likely to be of general applicability and utility. In this regard, more electron photoemission data on charge-transfer crystals would be helpful. From the effective polarizabilities, we have constructed energy level diagrams for the charged states. These give a rather complete picture of the various levels accessible by experiment, including the relaxed and unrelaxed separated charge-pair states which determine the adiabatic and optical band gaps. Because of the large relaxation energies, these gaps differ much more than in anthracene [ 24,25 1. They may therefore offer more scope for optical and electrical measurements to explore the roles of the various levels in the mechanism of photoconductivity. Charge-transfer excitations are important in photoconductivity as well as in the optical spectra of these crystals. We have calculated the energies of CT configurations not only for the obvious heteromolecular CT but also for homomolecular CT and the symmetry-preserving split CT. Broadly speaking the different types of configuration increase in energy in the order given. However, the lowest configuration in anthracene-PMDA is a split one. One way of treating the states of symmetrically stacked structures is to include the two heteromolecular CT configurations with the electron on one side or the other of the hole [ 28,421. These configurations interact strongly and the present results suggest that in some cases the split CT configuration may offer a better starting point for calculations. Homomolecular CT configurations might seem of little significance: an acceptor molecule is surely a much more favourable site for an electron than a donor molecule. In fact, anthracene can be regarded as “amphoteric”, with a respectable positive electron

24

R. W

Mum,

R.J. Phillips /Dielectric theoty of weak CT qvstals. II

affinity. All things being equal, this would still leave homomolecular CT configurations at least 1.5 eV higher in energy than heteromolecular ones, even apart from their generally lower stabilization energies. However, the two types of configuration approach within 1 eV because of charge-quadrupole interaction effects. The acceptors, with their electronegative groups around the molecular periphery, have quadrupole moments opposite to those of anthracene but occupy similar sites. Thus the electron-quadrupole energies I%‘&at the two sites are opposite in sign. As a result, WC reinforces the stabilizing effect of the polarization energy for homomolecular CT but opposes it for heteromolecular CT. From these arguments, a similar mechanism seems likely in other weak CT crystals. Passing from the potential energy of CT configurations to observable CT eigenstates is not straightforward. Given the uncertainties, differences of 0.5 eV between the energy of the lowest heteromolecular CT configurations in the two crystals and those observed experimentally for the CT transition are satisfactory. In particular, the lowest homomolecular CT configurations seem to have energies and strong interactions which would make them strong contributions to a CT level at about the right energy. If so, the lowest CT state would involve significant electron transfer from the donor to its nearest and next-nearest neighbours in the stack. Detailed calculations of the potential energy of CT configurations in the aromatic hydrocarbons have provided the input necessary for detailed quantitative theoretical study of CT levels in these crystals [ 12,131. We hope that the present calculations for these two complex crystals will provide a similar stimulus to both theory and experiment. In conclusion, we have obtained the following results. Single-carrier energies in these weak CT crystals can be calculated using polarizabilities from part I. Comparison with experiment indicates sizeable relaxation energies. These imply large differences between the optical and photoconductivity band gaps. CT excitation energies can also be calculated, for heteromolecular, homomolecular and split configurations. All these configurations may play a role in the actual CT states and hence in the observed properties of the states.

Acknowledgement This work was supported by the UK Science and Engineering Research Council through the award of a Studentship to RIP.

Appendix A. Submolecule polarizabiities The crystal susceptibility is written as x= z

akj’dkj=

1

ak-dk

>

(A.1)

k

where d is the local field tensor. We assume that all submolecules on chemically equivalent molecules contribute equally to x, and then ak,‘dkj=8/,1ak’dk.

(A.2)

We define the molecular local field as the average of the submolecule local fields dk=Sk’ 1 dkj

(A.3)

J

and then the submolecule and molecule polarizabilities are related by 1 ak,’ .

a,‘=s,*

(A.4)

i

Finally we have the submolecule local fields as dkj = 1 + 1

LkJ,k’j’

‘akfr

‘dk.7

,

(A.5)

k’j

where is a Lorentz-factor tensor. In a CT crystal we set &j&j.

aa.d, =R-a,*d,

,

(‘4.6)

where the matrix R relating the donor and acceptor contributions to x is obtainable from the molecular polar&abilities derived numerically [ 15 1. This allows the left-hand side of (A.2) to be expressed in terms of the known quantities x and R. Substitution in (A.5) gives dki in terms of x, R and known Lorentz-factor tensors, which then allows ay to be obtained by substituting back in (A.2 ) . The result yields for Z= 1 as in anthracene-PMDA aG’=Sd[X-‘.(l

+R)+&,,+b,,-R]

a;‘=s,[x-“(1

+R-l)+&,,+LJ,d-R-l],

,

(A.7) (A.8)

25

R. W. Munn, R. J. Phillips /Dielectric theory of weak CT crystals. II

Table 5 Components of the quadrupole moment 8,, from ab initio STO/4-3 1G calculations, and calculated changes A@,,, referred to the molecular centre and Se,, referred to the submolecule centres for the anion (in e AZ).Changes for the cation are equal and opposite. All components are expressed in the molecular L MN axis system Molecule

AB

Quantity

MM

LL

1.864 -9.061 -4.605

3.514 2.590 0.362

-5.378 6.471 4.234

PMDA

-8.325 - 1.227 -2.423

0.808 1.287 -0.337

7.517 5.940 2.760

TCNB

-0.178 - 7.658 - 3.503

- 1.945 2.458 1.592

1.123 5.200 1.911

anthracene

eArI A@,, 68.4,

where Lkj.

k’

=sG’

NN

QM= (q/2s) C Y,’. 1

(B.5)

I Lkj,k.j.

.

The results are shown in table 5.

j’

For Z= 2 as in anthracene-TCNB, similar results are obtained in terms of fx and sums of Lorentz-factor tensors over both donors or acceptors in the unit cell.

Appendix B. Submolecule quadmpole moments

The total quadrupole moment change A8 in the ion is proportional to the fractional change in the number of valence electrons assuming no change in the anisotropy of the electronic charge distribution from the neutral molecule to the ion [ 12,13 1. The submolecule quadrupole moment changes 68 have to be transformed from the molecule origin to the submolecular origins. For an ion of charge q distributed over a submolecule j at positions (Xj, yj, 0) in the molecular LMN axis system (all molecules being planar), one obtains

(B.1) U3.2) (B.3)

(B-4)

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R. W. Mum, R. J. Phillips /Dielectric theory of weak C’Tcrystals. II

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