Optimal parametrisation of the Pariser–Parr–Pople Model for benzene and biphenyl

18 September 1998

Chemical Physics Letters 294 Ž1998. 305–313

Optimal parametrisation of the Pariser–Parr–Pople Model for benzene and biphenyl Robert J. Bursill a

a,1

, Christopher Castleton

b,2

, William Barford

b,3

School of Physics, UniÕersity of New South Wales, Sydney, NSW 2052, Australia b Department of Physics, The UniÕersity of Sheffield, Sheffield, S3 7RH, UK Received 22 July 1998; in final form 4 August 1998

Abstract We obtain a parametrisation of the Pariser–Parr–Pople model of the p-conjugated systems which is optimal for benzene, biphenyl and polyŽ para-phenylene.. We first optimise agreement with experiment for a number of low-lying excitations of benzene, leading to a phenyl transfer integral of 2.539 eV, an on-site Coulomb energy of 10.06 eV and a relative error of 2.8%, compared with 7.4% using the standard values. We next optimise agreement for the long axis polarised optical transitions of biphenyl with absorption data, leading to a bridging bond transfer integral of 2.22 eV. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction The semi-empirical approach to p-conjugated electronic systems dates back to the 1950s and to the work of Pariser, Parr and Pople w1,2x. The Pariser– Parr–Pople ŽP–P–P. model is a one band model of the p-electron system with only nearest neighbour hybridisation t i j and long range, direct Coulomb interactions Vi j . In the language of a solid state physicist this model is known as the extended Hubbard model. The Coulomb interaction is an interpolation between long range 1rr behaviour and short range behaviour which models the shape of the

1

E-mail: [email protected] Current address: European Synchrotron Radiation Facility, B.P. 220, F-38043 Grenoble Cedex, France. E-mail: ´ [email protected] 3 E-mail: [email protected] 2

atomic orbitals. Various interpolations have been proposed including Ur Ž 1 q a ri2j . w3x and UrŽ1 q a < ri j <. w4x. The hopping integral, t i j , for neighbouring sites i and j, is usually taken to be a linear function t i j s t p Ž1 q d Ž rp y ri j .. where t p and rp are the phenyl transfer integral and bond length respectively. U, t p and d are the adjustable parameters in the model, chosen to best fit experimental features, while a is determined by the correct 1rr behaviour. The values most used in the literature are t s 2.40 ˚ y1 w8x Žthis leads eV, U s 11.26 eV and d s 1.22 A y2 ˚ ., although somewhat lower valto a s 0.6117 A ues of U Ž10.84 eV. and higher values of t Ž2.50 eV. have also been used w5x. An extension of the P–P–P model, which includes some differential overlap, has been parameterised by Warshel and Kaplan w6x and Martin and Freed w7x. Although the P–P–P model has generally been superseded by ab initio methods for small molecules, it is in fact remarkably succesful in predicting excita-

0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 9 0 3 - 8

(

R.J. Bursill et al.r Chemical Physics Letters 294 (1998) 305–313

306

tions to within a few tenths of an eV w8,9x. The only ab initio method which comes close to, or betters, this accuracy is the CASPT2 approach. The recent theoretical interest in linear, organic semi-conductors such as polyacetylene, polydiacetylene and the phenyl based systems, coupled to recent advances in computational methods for solving quantum lattice Hamiltonians which are most easily applied to semi-empirical models, has highlighted the utility of semi-empirical approaches. Thus, while a powerful and robust ab initio approach is highly desirable, at present it is limited to small molecules, and for the understanding of larger molecules a semi-empirical model is the only feasible approach. However, although the standard P–P–P parametrisation is good, it is not excellent, and so the purpose of this letter is to optimise its parametrisation by fitting its predictions to the known experimental excitations of benzene. This will lead to a parametrisation for U and t p . Since our ultimate goal is to derive a robust model of the phenyl-based semiconductors, we will then turn our attention to biphenyl and parameterise ts , the single bond hybridisation Žthis is equivalent to fixing d .. We will compare our optimised results to previous P–P–P calculations, CASPT2 calculations and experiment. In addition, we will show how the low lying excitations of biphenyl are derived from the primitive excitations of benzene. We will also note that a closer fit to experiment can be obtained by including inter-phenyl next-nearest-neighbour hopping. In the next section the P–P–P model will be introduced and solved for benzene, using both the standard and optimised parameters. In Section 3 the optimised parametrisation will be applied to biphenyl, and the nature of the excitations discussed. Finally, we conclude. 2. Benzene The P–P–P Hamiltonian is written as Hsy

Ý

t i j c†i s c js q h.c. q U Ý Ž n i ≠ y 12 .

-ij) s

= Ž n i x y 12 . q

i

1 2

Ý Vi j Ž n i y 1. Ž n j y 1. , i/j

Ž 1.

where c†i s creates a p-electron with spin s on carbon site i, n i s s c†i s c i s , n i s n i ≠ q n i x and -) represents nearest neighbours. Notice that this model is particle-hole symmetric. The electron-ion potential exactly cancels the electron-electron interaction in the mean field limit, and thus the average particle density on each site is unity. We use the Ohno parametrisation for the Coulomb interaction w3x, Vi j s

U

(1 q a r

2 ij

,

Ž 2.

where 2

a s Ž Ur14.397 . ,

Ž 3. 2

thus ensuring that Vi j ™ e r4pe 0 ri j as ri j ™ `. ri j ˚ the C–C bond is the inter-atomic distance in A, ˚ and U has units of eV. length is taken as 1.40 A The irreducible symmetry of benzene is D6 h , whence the excitations are characterised by the angular quantum number j. Optically allowed transitions occur between the ground state Ž j s 0. and the j s "1 states. Weakly Žphonon. allowed transitions also occur to the j s "2 and the j s "3 states. In addition to being eigenstates of the rotation and total spin operators, the states are also eigenstates of the ˆ with eigenvalues particle-hole symmetry operator, J, J s q1 Žcovalent. or J s y1 Žionic.. Optical transitions occur between states of opposite particle-hole symmetry. The Hamiltonian Ž1. is exactly diagonalised using the conjugate gradient method. Table 1 lists the low lying vertical excitations of benzene obtained from the P–P–P model with both standard and optimised parameters, the CASPT2 method and from experiment. The optimised parameters are found by performing a least squares fit comparison of the P–P–P predictions to experiment. ŽNote, however, that the E2 g states are not included in the fit owing to their experimental uncertainty.. This results in values of ˚ y2 . t p s 2.539 eV, U s 10.06 eV and a s 0.4881 A Although these values are relatively close to those used in the standard parametrisation, they result in an average relative error of 2.75% compared to an error of 7.35% for the standard parametrisation. Moreover, the improvement is systematic: Quite generally, the standard parametrisation yields systematically low energies for the triplets and the weakly allowed

R.J. Bursill et al.r Chemical Physics Letters 294 (1998) 305–313

307

Table 1 Low lying excitation vertical energies Žin eV. of benzene, as calculated with standard and optimised P–P–P theory, CASPT2 theory w10x and from experiment State

< j<

s Ž xy .

s Ž xz .

Optimised

Standard

CASPT2

Experiment

11 Bq 2u 11 B1yu 11 E1yu Ž z . 11 E1yu Ž y . 11 Eq 2g 11 Eq 2g

3 3 1 1 2 2

q y y q q y

y q q y q y

4.75 5.47 6.99 6.99 7.60 7.60

4.23 5.52 7.00 7.00 6.81 6.81

4.84 6.30 7.03 7.03 7.90 7.90

4.90 a 6.20 b 6.94 c 6.94 c 7.8 " 0.2 7.8 " 0.2

13 B1qu 13 E1qu Ž z . 13 E1qu Ž y . 13 By 2u 13 Eq 2g 13 Eq 2g

3 1 1 3 2 2

y y q q q y

q q y y q y

4.13 4.76 4.76 5.60 6.71 6.71

3.52 4.32 4.32 5.58 5.94 5.94

3.89 4.49 4.49 5.49 7.12 7.12

3.94 d 4.76 d 4.76 d 5.60 d 7.49 " 0.25 7.49 " 0.25

The experimental assignments are those of Ref. w10x.

a

Ref. w11x.

singlets. In particular, the optimised parameters give rise to a notable improvement in the prediction of the location of the 2-photon states Žthe E2 g .. For instance, the optimised parameters predict that the 11 E2 g states lie above the 11 E1 u states, in agreement with experiment. The standard parameters, in contrast, predict the reverse ordering. The CASPT2 method has a relative error of 2.54%, although the results are more consistent than those of the opti-

b

Refs. w11,12x.

c

Refs. w11,13x.

d

Ref. w14x.

mised P–P–P approach, the errors being more narrowly distributed. In particular, there is a fairly large error in the P–P–P model prediction of the 11 B1 u energy. The ab initio effective valence shell Hamiltonian, introduced by Martin and Freed w7x, has a relative error of 2.75%, identical to ours. However, this model includes three and four centre integrals, so goes beyond the simplicity of the P–P–P model. Another test with experiment is the comparison of the calculated electron affinity minus ionisation Ž I y A.. This is 11.38 eV and 11.44 eV, respectively for the optimised and standard parameters. Although it is natural to use the D6 h symmetry group to label the states, we are also interested in the excitations of biphenyl, and the way they are derived from their parent benzene excitations. It is thus useful to label the benzene states as eigenstates of the x–y and x–z plane reflection operators, as illustrated in Fig. 1. The eigenvalues of the reflection operators are also indicated in Table 1.

3. Biphenyl

Fig. 1. The geometry used for biphenyl, showing the convention adopted for the definition of symmetry operators and the transfer integrals.

We calculate the vertical transition energies of the biphenyl molecule by exactly diagonalising the P– P–P hamiltonian Ž1. assuming the geometry depicted in Fig. 1. Biphenyl belongs to the D 2 h symmetry group. We adopt the convention that the z-axis is the

R.J. Bursill et al.r Chemical Physics Letters 294 (1998) 305–313

308

Table 2 Vertical, dipole allowed, long axis polarised singlet excitation energies Žin eV. of planar biphenyl calculated using the benzene optimised P–P–P theory and various values of the single bond hopping ts ts

11 B1yu

21 B1yu

2.00 2.10 2.15 2.18 2.20 2.22 2.25 2.30 2.50 3.00

4.87 4.85 4.83 4.81 4.81 4.80 4.79 4.76 4.68 4.48

6.23 6.23 6.22 6.22 6.22 6.22 6.21 6.21 6.19 6.16

Standard P–P–P CASPT2 Experiment

4.88 4.63 4.80

6.22 5.76 6.16

Also tabulated are the CASPT2 results w10x, the results of standard P–P–P theory and experimental results derived from reflection spectra on crystalline biphenyl w17x.

long axis, while the y-axis is the short axis. The symmetries used in diagonalising H are the ˆ and the x–y and x–z particle-hole symmetry J, planes of reflection. The states of biphenyl arise from the delocalisation and coupling of the benzene states within each spatial, particle-hole and spin symmetry sector. The reader is referred to Ref. w16x for both a review of the experimental evidence and a

discussion of the exciton model of biphenyl excitations. In order to derive an optimised P–P–P theory we take the optimal values of t p and U from the preceding benzene calculation and regard the inter-phenyl hopping Žacross the bridging single bond. as an adjustable parameter, ts . In Table 2 we list the energies of the two long axis polarised singlet transitions Ž11 B1yu and 21 B1yu . for a number of values of ts . Also listed are the results of the P–P–P model with standard parameters, CASPT2 results w15x and the experimental, vertical transition results for crystalline biphenyl where the geometry is closest to planar. Our calculations of these states with the standard parameters agree with those in w8x. We find that a choice of ts s 2.22 eV Ži.e. d s ˚ y1 . gives the best agreement with experi1.142 A ment. We note that a substantially larger value of ts would be required in order to fit the CASPT2 results, which are systematically lower than the experimental results. In w15x this discrepancy is explained by the fact that a twisting about the long axis is observed in biphenyl in the crystalline form at sufficiently low temperatures, which would lead to a blue shifting in the energies compared to the planar form. However, for the vapour phase, CASPT2 theory with the experimentally observed twist incorporated still underestimates the experimentally observed energy of the 11 B1yu state w15x. Moreover, Ramasesha and coworkers w8x attribute the overestimation of the 11 B1yu

Table 3 Vertical singlet excitation energies Žin eV. of biphenyl calculated within P–P–P theory using optimised and standard parametrisations State

s Ž xy .

s Ž xz .

Optimised

Standard

CASPT2

Experiment

11Aq g 11 Bq 2u 11 Bq 3g 11 B1yu 21Ay g 21 B1yu 21 By 3g 31Aq g 21 By 2u 1 q 3 B3 g 31 Bq 2u 31 B1qu 41Ay g

q q y y q y y q q y q y q

q y y q q q y q y y y q q

0.00 4.55 4.58 4.80 5.57 6.22 6.28 6.30 6.66 7.07 7.08 7.17 7.30

0.00 4.08 4.11 4.88 5.60 6.22 6.32 5.76 6.69 6.50 6.50 6.47 7.30

0.00 4.35 4.04 4.63 5.85 5.76 5.07 5.85 5.69 5.07 – – –

0.00 4.20 Ž0–0. a –4.59 4.11 Ž0–0. a 4.80 b 4.71–5.02 c 6.14 d , 6.16 b – ca. 6.0 Žmax. c 5.85 b , 5.96 d – – – –

b

Also given are the CASPT2 results of w15x, as well as experimental data. a Crystal absorption spectrum at 4.2 K w18x. b Reflection spectrum of crystalline biphenyl w17x. c Two-photon excitation spectrum in ethanol w16x. d U.V. spectrum in stretched film w20x.

R.J. Bursill et al.r Chemical Physics Letters 294 (1998) 305–313

and 21 B1yu energies by the standard P–P–P theory to the solid state red shift occuring in crystalline biphenyl w8x. Given that biphenyl is a substantially more complicated system to solve by ab initio means than benzene and that convergence with basis size is not explicitly demonstrated in w15x, we cannot regard the results of w15x as an unquestionable benchmark for the evaluation of semi-empirical approaches to biphenyl. We therefore choose the value ts s 2.22 eV for our optimised parametrisation of biphenyl, and polyŽp-phenylene. oligomers and polymers. We now turn to the remainder of the low energy spectrum of biphenyl which is given in Tables 3 and 4 for our optimised as well as standard P–P–P theory, CASPT2 theory and experiment. We first point out that our triplet results obtained with the standard parametrisation conflict with those calculated in w8x Že.g. these authors found the lowest triplet state to be of B2 u symmetry at 2.84 eV, whereas we found it to be of B1 u symmetry at 3.16 eV.. There is also some disagreement in the nature and positions of the high energy singlet states Ži.e. our results for the first 7 singlets agree precisely with those of w8x, but for a different assignment of the 1 q 21Ay g state, which they assign as 2 B3 g. Beyond the 1 y 2 B1 u state there is disagreement in both the nature and the energy of the singlets.. We believe that the results of w8x in conflict with ours are incorrect. That is, we have run our program for a reduced U–V model of biphenyl and we reproduce all the results Žsinglets and triplets. obtained by another group w21x,

309

we have calculated the triplet state in different ways Žusing different values of the total z-spin, the spin-flip operator etc.. and we have run our program at high precision Žreal)16 in FORTRAN. to check for spurious convergence in the highly excited states. Both the optimised and standard parametrisations of the P–P–P model fail to predict the correct ordering of the two lowest singlet states, with the bulk of experimental evidence w16x indicating that the 11 B3 g state lies below the 11 B2 u state. The standard parameters appear to somewhat underestimate these energies, while the optimised parameters somewhat overestimate them. Likewise, there is experimental evidence to support a short axis polarised Ži.e. B2 u . excitation on the low energy side of the second lowest long axis polarised excitation Ži.e. the 2 B1 u state. w16x. However, both of the P–P–P model parametrisations predict the 21 B2 u state to lie ca. 0.45 eV higher than the 21 B1 u state. In contrast, the CASPT2 calculation correctly predicts this ordering. The P–P–P model does, however, predict a low-lying A g state at ca. 5.6 eV which predominately originates from the bonding combinations of the benzene 11 B1 u and 11 E1 u Ž z . states. ŽThe biphenyl 11 B1 u state arises from the anti-bonding combination.. There is experimental evidence for two weak two-photon allowed states at 4.71 and 5.02 eV w16x. In addition, there is another A g state originating from the benzene 11 E2 g states at 6.30 eV and 5.76 eV for the optimised and standard parameters, re-

Table 4 Vertical triplet excitation energies Žin eV. of biphenyl calculated within P–P–P theory using optimised and standard parametrisations State

s Ž xy .

s Ž xz .

Optimised

Standard

CASPT2

Experiment

13 B1qu 13Aq g 23 B1qu 13 Bq 2u 13 Bq 3g 23Aq g 3 y 2 B2 u 23 By 3g 33 B1qu 33 Bq 3g 33 Bq 2u 33Aq g

y q y q y q q y y y q q

q q q y y q y y q y y q

3.63 4.25 4.56 4.56 4.56 4.80 5.32 5.37 6.24 6.38 6.39 6.83

3.16 3.65 4.12 4.17 4.17 4.35 5.32 5.36 5.57 5.71 6.05 6.82

3.10 3.95 3.94 4.14 3.89 – – – – – – –

ca. 3.5 Žmax. – – 3.93 Ž0–0. a – – – – – – – –

Also given are the CASPT2 results of w15x, as well as experimental data. w19x.

a

a

EELS for biphenyl deposited on a thin film of argon at 20 K

310

R.J. Bursill et al.r Chemical Physics Letters 294 (1998) 305–313

spectively. Notice that the standard parameter result is much lower than the optimised parameter result because, as we have seen in Section 2, the former underestimates the benzene 11 E2 g states by ca. 1 eV. There is a strong, broad, 2-photon absorption starting at 5.33 eV and extending to the spectral range of the spectrometer Ži.e. at 6.2 eV., but with a dominant peak at 6.01 eV w16x. The CASPT2 calculation, however, predicts a doubly degenerate A g state at 5.85 eV. The low-lying triplet spectrum has recently been determined by EEL spectroscopy w19x. There is a structureless band starting at 3.0 eV with a rough maximum at ca. 3.5 eV which corresponds to the long axis polarised 13 B1 u exciton. The standard parameters place this state at ca. 0.5 eV lower than the optimised parameters and in closer agreement with the CASPT2 calculation. There is a better resolved 0–0 short axis polarised transition, 13 B2 u , at 3.93

eV. As in the case of the 31A g state, the standard parameters predict lower triplet energies than the optimised parameters owing to the former’s underestimation of the parent benzene triplet states. As pointed out in w16x, the six low lying singlet excitations of benzene result in twelve low-lying biphenyl singlet states, and likewise in the triplet sector. Figs. 2 and 3 show how the low lying excitations of biphenyl are derived from the parent benzene excitations in the singlet and triplet sector, respectively, using the results of the optimised P–P– P model. Before concluding this section, we return to the problem of the incorrect ordering of the 21 B2 u and 21 B1 u states with the optimised P–P–P model parameters. It might be expected that the inclusion of next-nearest-neighbour ŽNNN. hopping would lead to a reversal of the predicted order of the 21 B1 u and 21 B2 u states, since it gives an explicit hybridisation

Fig. 2. Showing how the low energy singlet spectrum of biphenyl is derived from the low energy singlet spectrum of benzene in the optimised P–P–P model.

R.J. Bursill et al.r Chemical Physics Letters 294 (1998) 305–313

311

Fig. 3. Showing how the low energy triplet spectrum of biphenyl is derived from the low energy triplet spectrum of benzene in the optimised P–P–P model.

of short axis polarised benzene states. We repeat our calculation for biphenyl with all NNN terms included. With t p s 2.539 eV and U s 10.06 eV, we attempt to re-optimise ts and t 2 s t 2p s t 2s Žas defined in Fig. 1. simultaneously, using the least squares deviation as before. We fit to the values 11 B1 u s 4.80 eV, 21 B2 u s 5.905 eV, 21 B1 u s 6.15 eV. Using values of around t 2 s 1.35 eV and ts s 2.20 eV we are indeed able to obtain a reversal of the states, but at the expense of introducing a host of spurious states not seen by experiment in the 3 ™ 6 eV region. Closer investigation shows that introduction of a t 2p term into the benzene calculation results in a significant lowering of the 11 E2 g and 13 E2 g states, and it is these states that lead to the spurious new states in biphenyl. Optimising t p , t 2p and U simultaneously for benzene gives t 2p s 0.00 " 0.01

eV, either with or without the inclusion of the E2 g states in the optimisation. Much better results for biphenyl can be obtained by setting the intra-phenyl t 2p s 0.00 and simply using the inter-phenyl t 2s terms. Here, the optimisation leads to the choice of t 2s s 1.45 eV with t s s 1.73 eV. In Figs. 4 and 5 the singlet and triplet energy level diagrams are shown as a function of t 2s . When t 2 s s 1.45 eV an improvement is clear for most states, with the relative error for the states 11 B1 u , 21 B2 u and 21 B1 u changing from 4.40% to 0.45%. In particular, the ordering of the states 21 B1 u , 31A g and 21 B1 u has been reversed in accordance with experiment, as has the ordering of the 11 B2 u and 11 B3 g states. For the triplets there is also a marked improvement in the value of the 13 B2 u level. The main problem with this new set of parameters is the

312

R.J. Bursill et al.r Chemical Physics Letters 294 (1998) 305–313

21A g level, which was previously somewhat too high, and is now somewhat too low, lying below the 11 B3 g and 11 B2 u levels instead of above. In practice, a spread of physically reasonable values of t 2s is from ca. 1.20 to 1.35 eV where the ordering of the singlet states is the most consistent with experiment. Finally, we note that we have also tried fitting the biphenyl data with the inclusion of different site energies on the phenylene bonding sites, Žwhich are in principle not equivalent to the other sites. or with different values for t p on the intra-phenyl bonds to these sites. Neither approach produces the desired shift in the spectrum at physically reasonable values of the parameters. We conclude this section by summarising the results on biphenyl. The standard P–P–P theory gives lower energies for the lowest two singlets Ž11 B2 u and 11 B3 g ., the 31Aq g , and most of the triplet states than the optimised theory. This is a consequence of the fact that the standard theory underestimates the energies of the related parent states in benzene. For the other singlet states the two parametrisations give very similar results. Overall, the CASPT2 calculation appears to underestimate the excitation energies, although the relative ordering of states is correct. The P–P–P model, on the other hand, does well for the dominant long axis polarised

Fig. 4. The lowest 8 singlet energy levels as a function of the inter-phenyl next nearest neighbour hopping integral Ž t 2 s .. Us 10.06 eV, t p s 2.539 eV and t s s1.73 eV. Symbols: q for A g states, ( for B1 u states, I for B2 u and = for B3 g states. Filled symbols indicate states for which the normalised oscillator strength from the ground state is G 0.1.

Fig. 5. The same as Fig. 4, but for the triplet energy levels.

singlet states, but gets the ordering of some states wrong: the most significant failure being the prediction that the 21 B2 u state lies above the 21 B1 u state. The P–P–P model can be improved by the inclusion of inter-phenyl NNN hopping. This results in a reversal of the predicted order of the 21 B1 u and 21 B2 u states and the 11 B2 u and 11 B3 g states, and moves several other states nearer their experimental values.

4. Conclusion In this letter we have introduced a parametrisation of the P–P–P Hamiltonian which is systematically optimised for benzene, biphenyl and the oligophenylenes. A choice of t p s 2.539 eV for the phenyl transfer integral and U s 10.06 eV for the on-site Coulomb repulsion leads to a systematically improved description of the low energy excitations of benzene, particularly in the case of the forbidden or weakly allowed states, the relative error being decreased by a factor of 3. The single bond transfer integral is optimally determined to be ts s 2.22 eV by fitting exact diagonalisation results to the two lowest long axis polarised singlet transitions of biphenyl. It is shown that an inter-phenyl nextnearest-neighbour hopping term of approximately 1.25 eV is required in order to achieve better agreement with experiment, where the P–P–P model fails to predict the correct ordering of states for biphenyl. This is at the expense of decreased transferability of the modelling.

R.J. Bursill et al.r Chemical Physics Letters 294 (1998) 305–313

Acknowledgements WB and CC acknowledge financial support from the EPSRC ŽU.K.. ŽGRrK86343.. RJB acknowledges the support of the Australian Research Council. Computations were performed on the SGI Power Challenge facilities at the New South Wales Centre for Parallel Computing and at the University of Sheffield.

References w1x w2x w3x w4x w5x

R. Pariser, R.G. Parr, J. Chem. Phys. 21 Ž1953. 446. J.A. Pople, Trans. Faraday Soc. 42 Ž1953. 1375. K. Ohno, Theor. Chim. Acta 2 Ž1964. 219. K. Nishimoto, N. Mataga, Z. Phys. Chem. 12 Ž1957. 335. G.L. Bendazzoli, S. Evangelsti, L. Gagliardi, Int. J. Quant. Chem. 51 Ž1994. 13. w6x A. Warshel, M. Karplus, J. Am. Chem. Soc. 94 Ž1972. 5612. w7x C.H. Martin, K.F. Freed, J. Chem. Phys. 101 Ž1994. 5929.

313

w8x S. Ramasesha, I.D.L. Albert, B. Sinha, Mol. Phys. 72 Ž1991. 537. w9x Z.G. Soos, S. Ramasesha, D.S. Galvao, ˜ S. Etemad, Phys. Rev. B 47 Ž1993. 1742. w10x J. Lorentzon, P. Malmqvist, M. Fulscher, B.O. Roos, Theor. ¨ Chim. Acta 91 Ž1995. 91. w11x A. Hiraya, K. Shobatake, J. Chem. Phys. 94 Ž1991. 7700. w12x E.N. Lassettre, A. Skerbele, M.A. Dillon, K.J. Roos, J. Chem. Phys. 48 Ž1967. 5066. w13x P.G. Wilkinson, Can. J. Phys. 34 Ž1956. 596. w14x J.P. Doering, J. Chem. Phys. 51 Ž1967. 2866. w15x M. Rubio, M. Merchan, ´ E. Orti,´ B.O. Roos, Chem. Phys. Lett. 234 Ž1995. 375. w16x B. Dick, G. Hohlneichler, Chem. Phys. 94 Ž1985. 131. w17x T.G. McLaughlin, L.B. Clark, Chem. Phys. 31 Ž1978. 11. w18x R.M. Hochstrasser, R.D. McAlpine, J.D. Whiteman, J. Chem. Phys. 58 Ž1973. 5078. w19x P. Swiderek, M. Michaud, G. Hohlneicher, L. Sanche, Chem. Phys. Lett. 187 Ž1991. 583. w20x J. Sagiv, A. Yogev, Y. Mazur, J. Am. Chem. Soc. 99 Ž1977. 6861. w21x Y. Anusooya, S.K. Pati, S. Ramasesha, Symmetrized DMRG studies of the properties of Low-lying states of poly-paraphenylene ŽPPP. system, Preprint.