An ab initio study of the vibrational spectra of Li-doped thiophene, bithiophene, benzene and biphenyl as model systems for (bi)polaronic defects

THEO CHEM Journal of Molecular Structure (Theochem) 364 (1996) 15-31

An ab initio study of the vibrational spectra of Li-doped thiophene, bithiophene, benzene and biphenyl as model systems for (bi)polaronic defects Stephan

Irle, Hans Lischka*

Institut fir Theoretische Chemie und Strahlenchemie der Universitiit Wien, Wiihringerstrasse 17, A-1090 Wien, Austria

Received 9 October 1995; accepted 7 December 1995

Abstract Ab initio SCF investigations of the IR and Raman spectra of the charge-transfer complexes of Li atoms with thiophene, bithiophene, benzene and biphenyl are reported. Computed geometries show an aromatic -+ quinoid transition during the formation of the complexes. The force constants reflect this behavior. As a consequence, the vibrational frequencies are considerably rearranged. IR and Raman intensities are strongly enhanced in the complexes as compared to the undoped compounds. The inter-ring vibrational modes occur at frequencies close to 1600 cm-’ both for di-Li-BT and di-Li-BP. This suggests a reconsideration of the assignment of that mode in the experimental Raman spectrum of the p-oligophenyl dianions. Keywords: Ab initio calculation; Bipolaron; Charge transfer complex; Polaron; Scaled force field; Vibrational spectra

1. Introduction Polythiophene, poly(p-phenylene), polypyrrole and many other related n-conjugated organic polymers display metallic conductivity upon doping with oxidizing, e.g. AsFS or Br2, or reducing, e.g. electropositive metals, agents [l-5]. Charged defect structures like polarons (radical ions) and bipolarons (dications or dianions) are believed to play a major role in the conduction mechanism of these compounds [6-81. The structural changes are governed by quinoidic distortions of the originally aromatic rings. However, due to the amorphous nature of these doped polymers the experimental characterization of their structures and of * Corresponding author.

their spectroscopic properties is still not satisfactorily solved. This situation has led to studies of smaller oligomers showing metallic conduction properties that are similar to those of the polymers (see, for example, Refs. [9-131). Of particular relevance to our work is the recent Raman study on sodiumdoped poly-p-phenylene and the radical anions and dianions ofp-oligophenyls [14]. The advantage of the study of oligomers is that one can investigate a series of compounds having well-defined degrees of polymerization under more controllable conditions allowing a better understanding of the polymer properties. Several ab initio SCF [ 15- 191 and semiempirical calculations [:!O-231 on the charged oligomers with or without inclusion of counterions show evidence for the importance of such defect states.

0166-1280/96/%15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI Ol66-1280(95)04465-S

S. Irle, H. LischkalJournal of Molecular Structure (Theochem)

16

Counterions significantly polarize the electron density of the doped oligomers and influence the localization of charge on the chain, especially in the case of n-doping because of the large polarizability of negative charge created in the conjugated T-system. In a recent study on the structures, the electron densities and energetic stabilities of monoand di-alkali-doped oligophenyls and oligothiophenes [18], we were able to show that the charge transfer is localized mainly within the monomer unit above which the alkali atom is located. The localization of the defect is much more pronounced in systems with explicit inclusion of counterions than in comparable doubly-charged systems without screening by counterions [17]. Vibrational spectroscopy is, of course, one of the most important ways to study doped and undoped oligomers and polymers [24,25]. Various ab initio [26-301 and semiempirical [31,32] calculations on the vibrational spectra of the undoped oligomers have been reported. Recently, the vibrational spectra of positively charged closed-shell bipolaronic defects in a series of oligothiophenes have been studied by Ehrendorfer and Karpfen [33-351 by means of ab initio SCF and semiempirical PM3 calculations. Cuff et al. [19] presented calculations on the vibrational spectra of doped poly( p-phenylene) by means of their scaled quantum mechanical oligomer force field method. Our present investigation on the vibrational spectra of polarons and bipolarons is a continuation of our previous study on the electronic structure and energetics of such systems [18], explicitly taking counterions into account. In these investigations we not only treated closed-shell bipolaronic but also open-shell polaronic defects. Because of what has been said in the preceding paragraph about the localized structure of the alkali-doped compounds, we believe we are justified in investigating the vibrational modes of the smallest possible doped oligomers (Li-benzene, Li-thiophene, di-Li-biphenyl and di-Li-bithiophene) as basic model cases for negative polarons and bipolarons.

364 (1996) 15-31

with the TURBOMOLE program [36]. For the remaining CdCUhtiOUS, the GAUSSIAN 92 Suite of programs [37] was used. Closed-shell molecules were treated on the spin-restricted Hartree-Fock (RHF) level of theory, whereas for open-shell systems the spinunrestricted Hartree-Fock formalism of Pople and Nesbet [38] was employed. Equilibrium structures were obtained by full geometry optimizacriterion of tions, applying a convergency 1 x 10d3 atomic units for the root mean square gradient value. The double-zeta basis set of Ahlrichs and coworkers [39] polarized with one dshell for S and C (o(S) = 0.55, a(C) = 0.80) was chosen for all calculations. The Li DZ basis set was augmented by two p-functions with exponents of o, = 0.55 and o2 = 0.06 which where obtained by exponent optimization for di-Li-bithiophene [18]. These polarization functions play an important role in the charge transfer process from Li to the acceptor molecule [40]. An extended version of the program SCALES developed by Pongor and coworkers [41] was used for the fitting procedure of scaling factors for force constants according to Pulay et al. [42].

3. Equilibrium structures

The molecular structures of undoped and Lidoped thiophene, bithiophene, benzene and biphenyl are depicted in Figs. 1 and 2. The equilibrium geometries of these compounds have been discussed in detail in Ref. [ 181.The main structural change due to the charge transfer is the introduction of a quinoid character which can be described in general terms by the following valence-bond-type formulae:

2. Methodology The geometry

optimizations

were performed

Formula

1.

S. Irle, H. LischkajJournal of Molecular Structure (Theochem)

LC*ClC1'C*'= 456 Lc2c1cI'ce = -134.4

Fig. 1. Atomic numbering schemes, bond lengths in A and torsional angles in degrees for thiophene, bithiophene, benzene and biphenyl.

For Li-thiophene (see Figs. 1 and 2) the C-S bonds become stretched by 0.080 to 1.805 A. the C2-C3 bond is stretched by 0.070 to 1.417 A and the C3-C4 bond is shortened by 0.051 to 1.383 A. However, 7r-conjugation is not completely destroyed since these bonds do not reach the standard lengths for C-S (1.82 A), C-C (1.54 A) and C=C (1.35 A). Compared to these large differences, the remaining geometrical parameters experience only minor changes. Upon doping bithiophene with two Li atoms, a double bond of length 1.332 A is formed between Sl

Fig. 2. Atomic numbering schemes, bond torsional angles in degrees for Li-thiophene, Likbenzene and di-Li-biphenyl.

lengths in A,, and di-Li-bithiophene,

364 (1996) 15-31

17

the rings. Also, the SlC2C2’Sl’ dihedral angle is almost planar at 178”. The quinoid bond length pattern is not fully established within the thiophene rings for the syn-facial arrangement of the two lithium atoms above the two thiophene rings. A change to the anti-facial arrangement also fully introduces these quinoid bond length alternations [18]. Li-benzene exhibits the same electronic and geometrical features as the benzene radical anion where ab initio SCF and CI calculations have already been reported by Hinde et al. [43]. Two structures of C6H6-, close in energy, are found corresponding to the occupation of either of the two degenerate benzene LUMOs of e2u type. The molecular point group reduces to D2h. One of the two states (2Biu) shows a quinoid bond length alternation pattern and the other (2Au) an anti-quinoid pattern. This situation can be analyzed in terms of the Jahn-Teller theorem [44,45] and on the basis of the nodal properties of the singly occupied lezu orbitals (see Ref. [43] and references cited therein). Analogous to Hinde et al. [43], we restrict ourselves to calculations within the framework of the Born-Oppenheimer approximation, neglecting coupling between different electronic states. This procedure can certainly be regarded as only a first approximation, especially when calculating the vibrational frequencies reported later on. For the Li-benzene complex, one obtains two geometries with the same characteristics as for C6H6- (see Ref. [ 181).The point group is further reduced to Ch due to the asymmetric polarizing effect of the Li+ counterion. The two structures correspond to the 2A1 and 2A2 states, respectively. Vibrational analysis shows that the 2Ai state is a local minimum, whereas the other structure is a saddle point (one imaginary frequency of 2451’cm-‘) for the interconversion of one local minimum structure to an equivalent one obtained by cyclic permutation of carbon-carbon single- and double-bonds. The 2A, state is the one with the quinoid geometry. Its bond length alternation pattern is more pronounced by M 0.02A than in the Li-thiophene case, and the molecule is found to be boat-shaped by z 15” with the atoms Cl and C4 oriented towards Li. In the following, we only consider the ‘Ai minimum structure further. For di-Li-biphenyl, the bond length alternation pattern within the rings is comparable to the

18

S. Irle. H. LischkalJournal of Molecular Structure (Theochem)

Li-benzene case, the intra-ring double bonds even being slightly shorter. The rings in di-Li-BP possess chair-conformation with a torsional angle of 14”, in contrast to Li-benzene. The molecule possesses CzV symmetry, and the torsional angle for C2ClCl’C2’ is 0”. Shortening the ortho-hydrogen bond lengths by 0.02 A relaxes the strain only a little,’ the distance between them still being smaller than their sum of van der Waals radii. This steric hindrance leads to stretching of the interring double bond by 0.037 A relative to di-Libithiophene. Our previous SCF calculations on the open-shell systems have been performed with the ROHF method [18]. Since for an analytical calculation of force constants only the UHF method was available, all geometries were reoptimized using UHF. Only a relatively small spin contamination was found ((S2) = 0.82 and 0.79 for Li-thiophene respectively). Therefore, the and Li-benzene, ROHF and UHF geometries are very similar and are not explicitly tabulated here. Electron correlation is very important for the energetic stability of the charge transfer systems [18]. At the SCF level (even with very large basis sets), these complexes are not bound with respect to Li atoms and the neutral molecule. Electron correlation calculations at the MP2 and MP3 levels show the charge-transfer systems to be stable [18] (Li-benzene is probably metastable in the gas phase). However, all structures investigated here are local minima even at the SCF level. A comparison of SCF and MP2 geometries (see Ref. [18]) for the monomer and dimer subsystems shows only relatively small effects due to electron correlation. Since we are mainly interested in the vibrational frequency shifts of the intramolecular modes within the monomers and dimers and not in the intermolecular vibrations of lithium, we consider it justified to use the much more economical SCF method for the calculation of force constants.

4. Vibrational spectra 4.1. Scaling procedures

Scaling of the force constants according to Pulay

364 (1996) 15-31

Table I SCF/DZP force constant scaling factors for thiophene and bithiophene (BT), benzene and biphenyl (BP), all-trans-hexaand all-trans-octatetraene Type of coordinate

Thiophene/BT

Benzene/BP

Oligoenes

CS c-c c=c CC aromatic CH Ring bends CCH bends CH out-of-plane Ring torsions

0.927 0.902 0.729

0.841

0.867 0.707

CC/CC CC/CC CC/CC c=c/c=c CC/C-C c=c/c-c

ortho meta para

0.832 0.806 0.826 0.736 0.815

0.836 0.834 0.829 0.832 0.756 0.777 0.791 0.455 0.643 1.754 1.178 1.158

et al. [42] is a very useful way to account for systematic errors such as the neglect of anharmonicity effects, basis set defects and electron correlation. In many cases, “conventional scaling” is sufficient, and typical values for the scaling factors range between 0.7 and 0.9. This kind of scaling was used for the thiophene systems. However, for special cases like acyclic conjugated compounds (1,3-butadiene, 1,3,5_hexatriene and higher homologues of this series), it has been demonstrated that separate scaling factors larger than unity have to be chosen for the coupling constants between CC single and double bonds [46,47]. Moreover, the so-called “Kekule constraint” of Scherer and Overend [48], i.e. the equality of ortho-, metaand para-coupling constants, has been proven to be unjustified [29]. Therefore, individual scaling factors for the CC/CC coupling force constants have been employed for benzene, biphenyl and the oligoenes. The scaling factors for the undoped compounds were obtained from fits to the experimental frequencies of thiophene [49], bithiophene (BT) [50], benzene [29] and biphenyl (BP) [51]. They are given in Table 1 together with the carboncarbon scaling factors derived from all-trans1,3,5hexatriene and all-trans-1,3,5,7_octatetraene

S. Irle, H. LischkalJournal of Molecular Structure (Theochem)

364 (1996) 15-31

Table 2 Vibrational frequencies, IR intensities and Raman activities of thiophene and Li-thiophene

Thiophene Al

B2

A2

BI

RMSD errorC Li-thiophene A’

AN

a Frequencies in cm’,

and approximate assignments’

Raman act.

Exptb

Approx. assignment

3.2 4.2 19.4 0.1 4.6 1.0 33.7 1.5 0.7 5.1 0.2 8.5 3.4 1.6 0.0 0.0 0.0 0.0 1.4 132.8 1.9

233.9 92.3 57.0 4.2 12.1 6.0 9.7 8.3 1.9 90.0 0.8

3126 3098 1409 1360 1083 1036 839 608 3125 3086 1504 1256 1085 872 763 898 683 565 867 712 452

C-H str. C-H str. c=c C=C str., CCH bend CCH bend C-C str., CCH bend C-S str. Ring def. C-H str. C-H str. C=C str. CCH bend CCH bend C-S str. Ring def. C-H oop CH oop Ring def. C-H oop C-H oop Ring def.

3.5 4.5 4.6 1.4 1.4 8.6 27.0 70.4 157.7 27.5 50.4 1.1 17.8 1.7 0.3 4.7 17.9 43.3 0.5 0.4 0.3 80.6 84.3 39.7

246.7 96.9 97.8 11.8 45.7 66.5 10.6 24.9 33.4 7.7 32.0 20.3 19.8 42.7 82.8 6.9 5.6 1.2 1.5 4.1 7.8 0.6 5.8 38.2

Calc.

IR int.

3121 3092 1424 1358 1075 1025 835 603 3118 3080 1502 1270 1075 867 755 897 693 564 878 696 442

0.1 8.7 0.4 4.9 2.4 1.7 0.4 0.4 0.3 1.9

19

7.9 3095 3077 1415 1291 1063 1009 762 721 638 534 502 353 209 3091 3062 1369 1206 1000 852 829 686 546 509 249

C-H str. C-H str. C=C str. CCH bend, C-C str. CCH bend C-C str. C-S str. C-H oop C-H oop Ring bend Li-Ring str. Li tors. Ring def. C-H str. C-H str. CCH bend CCH bend C-C str. Ring def. C-H oop C-S str. Ring tors. C-H oop Li tors.

IR intensities in km mol-’ , Raman activities in A4 amu-‘.

b Ref. [49]. ’ Defined as [Ciwi(v~’ - r~~“)~/C~w~]“~where veXpand zlcB”are the experimental and calculated frequencies, and Wi= l/vf”” x 1000.

S. Irle, H. Lischka/Journal of Molecular Structure (Theochem) 364 (1996) 15-31

20

inverted bond length alternation into account. Scaling factors for the C3-C4 and C4-C5 bonds which are intermediate between single and double bonds were computed as the geometric mean of the single and double bond scaling factors. The scaling factors chosen for benzene/biphenyl are very specific for the aromatic case. Since in the doped compounds there is a strong quinoid character, the “aromatic” scaling factors are certainly not well suited in the latter case. Therefore, we decided to apply the scaling factors determined for polyenes (Table 1) for the carbon backbone and to take those involving hydrogen atoms and ring bending/torsional modes from the aromatic systems. Scaling of coupling force constants was done in two ways: in scheme I the standard scaling procedures were used without any special modifications of CC coupling force

[52,53]. The accuracy of the fits is quite satisfactory for our purposes. In all cases the 1/v-weighted root mean squares deviation (RMSD) error is about 8 cm-’ The transfer of scaling constants from the undoped, aromatic compounds to the doped, quinoid species is not straightforward. The doped thiophene and bithiophene cases are probably not so problematic because the aromaticity of thiophene is much lower in comparison to benzene (see, for example, Refs. [54] and [55] and references cited therein). No extra scaling of coupling force constants was considered necessary here, in line with the calculations by Ehrendorfer and Karpfen [33-351 on the vibrational spectra of positive bipolaron defects in oligothiophenes. Thus, the scaling factors from thiophene/bithiophene were used for Li-thiophene and di-Li-BT, taking the

Raman activity [A4amd]

IR intensity [km mol-‘1

60-

50

Thiophene

40 30 20.

120

160 140 120

100

Li-thiophene

Li-thiophene

80 100 80 60 40 20

2000

1800

,600

1400

,*0*

1000

800

600

400

200

0

2000

infrared

and Raman

,800

1600

1400

!

1200

I

I

1000

BOO

600

Wavenumber [cm-l]

Wavenumber [cm11 Fig. 3. Theoretical

I I

II

0

line spectra

of thiophene

and Li-thiophene

excluding

CH stretches.

400

200

0

S. Me, H. Lischka/Journal of Molecular Structure (Theochem)

constants; in scheme II the coupling force constants determined from the polyenes were applied. Because of lack of experimental spectra, it is difficult to decide which of these schemes is to be preferred. For most modes, results are very similar. Scheme I involves less assumptions and we therefore will always refer to these frequencies in the subsequent discussion and indicate differences to scheme II as necessary. All force constants belonging to internal coordinates of Li were left unscaled. 4.2. Thiophene and Li-thiophene For thiophene, the strongest IR band is by far the one corresponding to the C-H out-of-plane mode at 696 cm-’ (see Table 2 and Fig. 3). Next in intensity comes the symmetric C-S stretching mode at 835 cm-‘. The CC stretching vibrations in the region of 1025 to 1502 cm-’ show only intensities up to 20 km mol-‘. The highest of these bands correspond to a stretching of the C=C double bonds. In the Raman spectrum, the symmetric C=C stretching mode gives rise to the strongest band (excluding CH stretches) with an activity of 57 A4/amu. All other Raman-active modes not corresponding to C-H stretching modes show activities of only up to 12 A4/amu. Due to the nonplanar structure of Li-thiophene, new bands appear both in the IR and Raman spectra. IR and Raman intensities are found to be generally enhanced due to uptake of charge in the ring system (see Table 2 and Fig. 3). The C-H out-ofplane mode (originally at 696 cm-‘) is still by far the most IR-intensive vibration, but it is red-shifted to 638 cm-‘. The red shift of this band might be caused by the high negative extra charge on the a-carbon atoms (-0.20 e-) weakening the C-H bond. Probably for the same reason, the C-H out-of-plane band at 721 cm-’ has moved from 878 cm-’ and rises sharply in intensity from 1 to 70 km mol-‘. C-H stretching modes are shifted uniformly by z 20 cm-’ towards lower frequencies. The C-S band is down-shifted to 762 cm-‘, thereby indicating the weakening of the C-S bonds. CC stretching bands are found to be generally lower than those of neutral thiophene, located within a region of 1000 to 1415 cm-‘. The lowest of

364 (1996) 15-31

21

these bands shows a remarkably high IR intensity of 43 km mol-’ . The highest one corresponds to a stretching of the C3-C4 bond for which the largest force constant is obtained (see below). In the Raman spectra, the most noticeable effects can be found in modes corresponding to symmetric C-C single bond stretching that increase in activity by a factor of 10 (in the region of 1000 to 1080 cm-‘). Also, vibrations appearing at smaller wavenumbers, e.g. the strong IR-active C-H out-ofplane band, and some ring-bending modes, show large variations in activity. Since their absolute scattering activity is small, these changes are not spectacular. Vibrations involving lithium appear at rather low frequencies (249, 353 and 502 cm-‘), typical for ionic interactions. These fre.quencies might be even lower since lithium modes were left unscaled. The most intensive of these modes is the Li-ring stretch motion. Force constants of stretching coordinates undergo drastic changes when thiophene becomes charged. Those of the C-S bonds decrease by 1.0 mdyn A-’ with a value of 3.2 mdyn A-’ for Lithiophene, and the former double bonds C2=C3 and C4=C5, with force constants of 7.0 mdyn A-‘, show values of 5.3 mdyn A-’ which is even below that of the former single bond C3-C4 (5.7 mdyn A-‘). Instead, the force constant of this bond js the only one to rise slightly by 0.2 mdyn A-‘. Therefore, the force constants of Li-thiophene are in line with the small bond length alternations which form, nevertheless, a quinoidic pattern. 4.3. Bithiophene and di-Li-bithiophene As for thiophene, the most intensive IR-active band corresponds to a symmetric C-H out-ofplane motion at 691 cm-’ (see Table 3 and Fig. 4). Again, CC stretching modes are located in the range of 1041 to 1562 cm-’ and are comparatively low in IR intensity (at most 36 km mol-‘). The most IR-intensive symmetric C-S stretching mode appears at 833 cm-‘, similar to thiophene. The Raman spectrum is dominated by two totally symmetric vibrations (excluding CH stretches) of the double bond skeleton. The most intensive vibration of B symmetry appearing in the Raman

22

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364 (1996) IS-31

Table 3 Vibrational frequencies, IR intensities and Raman activities of bithiophene and approximate assignmentsa

A

RMSD errorC

Calc.

IR int.

Raman act.

Exptb

Approx. assignment

3114 3089 3075 1562 1454 1383 1258 1219 1074 1041 894 854 824 754 691 674 567 471 359 284 118 37

0.2 0.5 0.0 0.2 2.1 0.0 0.2 5.7 0.0 0.1 0.8 14.3 5.7 7.0 132.4 1.1 0.1 3.2 0.1 0.0 2.3 0.3

364.2 179.9 144.6 362.0 829.0 21.2 12.0 3.4 25.5 28.3 1.2 1.9 11.0 8.8 0.5 27.1 1.5 2.3 2.6 2.5 1.4 5.6

3107 3080 3074 1557 1446

C-H str. C-H str. C-H str. C=C str. C=C str. C-C str., CCH bend CCH bend Interring str. CCH bend CC str., CCH bend C-H oop C-S str. C-H oop Ring def. CH oop Ring def. Ring tars. Ring tors. Ring def. Ring def. Ring tom. Torsion

3114 3088 3076 1498 1432 1326 1205 1075 1043 898 893 843 833 752 691 608 587 523 280 108

2.7 4.9 19.8 9.1 36.1 0.4 17.0 4.0 4.8 0.2 0.0 2.1 58.6 10.0 0.1 2.3 3.1 2.2 0.1 0.3

5.4 26.6 13.4 1.6 2.9 1.4 2.0 2.8 1.0 9.8 2.0 2.9 0.6 1.2 1.1 1.3 5.6 1.9 4.5 2.1

1250 1227 1083 1056 894 863 815 740 703 675 _ 458 _ _ _ 3104 3076 3064 1498 1413 1325 1205 1078 1047 894 _ 826 _ _ _ _

8.1

a Frequencies in cm-‘, IR intensities in km mol-‘, Raman activities in A4 amu-’ b Ref. [50]. ’ As defined in Table 2.

CH str. C-H str. C-H str. C=C str. C-C str. C=C str., CCH bend C=C str., CCH bend CCH bend C-C str. C-S str., Ring def. C-H oop C-H oop C-S str. Ring def. C-H oop Ring def. Ring tors. Ring tors. Ring tors. Ring def.

S. Me, H. LischkalJournal of Molecular Structure (Theochem)

the dominant feature of the Raman spectrum, being one order of magnitude more Raman-active than all other vibrations. Modes corresponding to lithium motions are found in the region of 145 to 571 cm-t, where the peak at 470 cm-’ possesses the highest IR intensity with 231 km mol-‘. C-H stretching frequencies are red-shifted as in the case of thiophene, the lowest two being even more strongly affected, with shifts of nearly 30 cm--‘. We note that the theoretical spectra of doubly positive charged BT without counterions reported by Ehrendorfer and Karpfen [33] are rather similar in shape to those of di-Li-BT, with differences of about +30 cm-’ on average. In particular, we also note that in their calculations, the highest Ramanactive mode (excluding CH stretches) at around 1600 cm-’ is related to inter-ring stretching. These authors also report a remarkably fast

spectrum below 1600 cm-’ corresponds to C-S stretching and is located at 898 cm-‘. Turning to the spectra of di-Li-BT (Table 4 and Fig. 4) one immediately recognizes new intensive bands covering the whole range of wavenumbers and reaching up to 795 km mall’ in IR intensity and 4385 A4/amu in Raman activities. The prominent C-H out-of-plane mode at 668 cm-’ no longer gives rise to the strongest band in the IR spectrum. The most intensive peak is located at 1449 cm-‘, corresponding to an asymmetric stretching of C3-C4, while the symmetric mode located 13 cm-’ above shows an intensity of only 16 km mol-’ . Apart from hydrogen stretching vibrations, the highest band at 1590 cm-’ can clearly be assigned to the inter-ring stretching which is extremely blue-shifted by 371 cm-’ due to the formation of the central double bond. It is

Raman activity [A4amu-l]

IR intensity [km moV]

140

1

120

BT

BT

di-Li-BT

3500

di-Li-BT

1.,! !, 2000

1800

23

364 (1996) 15-31

1600

1400

,200

1000

BOO

600

400

200

0

2000

1s00

1600

infrared

1200

1000

800

600

Wavenumber [cm-l]

Wavenumber [cm-l] Fig. 4. Theoretical

1400

! :, \ , ?1..,1.1!.. .

and Raman

line spectra

of BT and di-Li-BT

excluding

CH stretches.

400

! 200

, 0

24

S. Irle. H. LischkajJoumal

of Mdecular

Table 4 Vibrational frequencies, IR intensities and Raman activities di-Li-bithiophene and their approximate assignmentsa

A

B

Calc.

IR int.

Raman act.

Approx.

3106 3080 3046 1590 1462 1346 1247 1145 1059 1036 908 812 693 668 628 571 516 491 409 335 293 273 145 97 63

0.2 3.1 4.9 8.5 16.1 18.3 2.5 11.4 1.6 1.3 3.7 43.1 169.9 44.1 7.7 19.0 14.1 50.8 0.4 1.2 0.4 0.4 2.5 0.6 5.6

482.6 127.7 224.2 4385.4 201.8 210.7 15.2 321.9 21.5 59.1 70.6 9.3 94.2 124.2 46.9 139.9 140.1 307.1 12.8 5.0 53.3 21.5 143.1 2.4 6.0

C-H str. C-H str. C-H str. Interring str. C=C, C-C str. C-C str., CCH C-C str., CCH C-C str., CCH CCH C-C str. C-H oop Ring def. C-S str. C-H oop Ring def, C-C str. Ring tom., Li tars. Li-Ring str. Ring tars. Ring def. Li-Ring str. Ring tars. Ring tow Li tars. Torsion Ring def.

3106 3079 3048 1449 1386 1267 1100 1060 1018 908 865 768 694 661 629 550 519 506 470 318 150 131 117

10.1 3.2 18.0 794.7 277.8 81.2 37.5 8.2 22.3 13.5 10.0 16.8 15.3 21.2 11.1 170.2 194.6 17.3 231.0 59.5 144.2 5.0 14.9

114.6 32.6 49.4 149.4 20.7 7.4 1.8 0.3 0.4 9.6 0.7 7.6 1.1 0.1 5.1 20.4 24.5 12.9 70.8 25.9 13.7 13.7 5.6

C-H str. C-H str. C-H str. C=C str. C-C str., CCH CCH C=C str., CCH C-C str., CCH c-c SW. C-H oop Ring def. C-S str. C-S str. C-H oop Ring tars. Ring def. Ring def. Ring tors. Li-Ring str., C-H Li-Ring str. Li tom. Ring tars. Ring def.

a Frequencies in cm-‘, activities in A4 amu-’

IR intensities

of

assignment

in km mol-‘,

Structure

(Theochem]

364 (1996)

15-31

convergence of the Raman spectra with chain length. The similarity between the computed vibrational spectra for di-Li-BT and those of the BT dication is certainly related to the fact that both systems show quinoidic distortions. Force constants of neutral BT are very close to the corresponding values of thiophene. The interring force constant is typical for a single bond with 5.4 mdyn A-‘. The largest one in di-Li-BT belongs to the inter-ring double bond (7.5 mdyn A-‘), followed by those of C4-C5 (6.5 mdyn A-‘) and C3-C4 (5.8 mdyn A-‘). Thus, the incompleteness of the quinoid distortion on the edges of doped BT is, therefore, also reflected by the force constants. Unexpectedly, the C2-C3 force constant is higher by 0.2 mdyn A-’ than in the thiophene case, although its bond length is larger by 0.051 A.

Table 5 Vibrational frequencies, IR intensities and Raman benzene and their approximate assignmentsa

AIs A,, hu B2u

Q3

E I”

A 2u %

% E2u

oop

Raman

RMSD errorC

Calc.

IR int.

Raman act.

3080 999 1366 3043 1010 1314 1135 3054 1603 1162 606 3069 1490 1027 663 988 691 838 964 402

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 46.1 11.4 3.3 110.4 0.0 0.0 0.0 0.0 0.0

370.6 56.4 0.0 0.0 0.0 0.0 0.0 132.4 11.9 5.7 4.8 0.0 0.0 0.0 0.0 0.0 0.0 1.4 0.0 0.0

Expt.b

Approx.

3056 1599 1178 606 3064 1482 1037 673 990 707 846 967 398

C-H Ring CCH C-H CC C-C C-C CH C-C CCH Ring CH C-C CCH C-H C-H Ring CH CH Ring

activities

of

assignment

str. breath bend str. str., CCH bend str. (Kekule) str. str. str. bend def. str. str. bend oop oop tars. oop oop tors.

8.2

a Frequencies in cn~‘, IR intensities activities in A4 amu-‘. b Ref. [29]. ’ As defined in Table 2.

in km mol-‘,

Raman

S. Irle. H. LischkalJournal of Molecular Structure (Theochem)

25

364 (1996) 15-31

The IR and Raman spectra of Li-benzene (Table 6 and Fig. 5) are rich in visible bands due to the reduced molecular symmetry. IR intensities are somewhat enhanced, whereas Raman activities increase by one order of magnitude. Below 1600 cm-‘, the highest two frequencies at 1551 and 1434 cm-i can be clearly assigned to stretching of C2=C3 and C5=C6. Both are IR and Raman allowed. The ring breath and the Kekule modes are red-shifted with respect to benzene by 100 and 80 cm-‘. C-C single bond stretching modes occur at low frequencies of 775 and 999 cm-‘. The IRintensive CH out-of-plane mode of benzene is red-shifted by z 30 cm-‘, which is only half of the corresponding shift in thiophene. It still gives rise to the highest peak of the IR spectrum. C-H stretching modes are red-shifted in general, but some of them are affected only to a small extent (Ar,,Bi,). The Raman spectrum is dominated by

4.4. Benzene and Li-benzene The IR and Raman spectra of benzene (see, for example, Refs. [29,56,57] are well known. The results of our calculations are given in Table 5 and Fig. 5. Only four fundamentals appear in the IR spectrum belonging to Et, and AZu symmetries, the most intensive one being the AZu C-H out-ofplane mode at 663 cm-‘. The Raman spectrum shows 7 peaks; the totally symmetric ring breath mode at 999 cm-’ is clearly apparent beside the hydrogen stretching modes, with an activity of 56 A4/amu. Although the diagonal CC force constant of benzene is only 6.6 mdyn A-’ (vs. 7.0 mdyn A-’ for the C=C double bond in thiophene), the largest stretching frequency for benzene at 1603 cm-’ exceeds the highest one of thiophene by 103 cm-‘, thus indicating the strong coupling effects in benzene.

Raman activity [A4amu-I]

IR intensity [km mol-7

60-

“’ i

1cJo

50

Benzene

Benzene

40

80

30.

60 20

40

2000

1800

1600

1400

1200

1000

800

600

‘loo

200

0

2000

1800

1600

1400

1200

1000

1000

800

600

400

200

0

2000

1800

1600

1400

1200

1000

BOO

600

400

200

0

Li-benzene

2000

1800

1600

1400

1200

infrared

and Raman

600

Wavenumber [cm-l]

Wavenumber [cm’] Fig. 5. Theoretical

800

line spectra

of benzene and Li-benzene

excluding

CH stretches.

400

200

0

26 Table 6 Vibrational

S. Irle. H. Lischka/Journal of Molecular Structure (Theochem)

frequencies,

IR intensities

Caleb

Al

B2

A2

Bl

3075 3053 1551 1141 900 738 636 532 479 286 3070 3038 1419 999 934 918 621 542 278 3038 1327 1286 944 775 597 433 3058 1434 1234 1115 989 759 213

(1567) (953)

(918) (895)

(719)

(1208) (1105) (775)

a Frequencies in cm-‘, IR intensities b Scheme I (scheme II in parentheses

and Raman

activities

of LiPbenzene

364 (1996) 15-31

and their approximate

assignments”

IR int.

Raman act.

Approx.

3.1 0.8 1.8 0.1 1.8 4.4 147.4 18.6 68.7 7.9 16.2 4.2 0.6 1.5 3.4 65.2 45.2 133.6 40.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 52.1 3.7 59.1 3.5 0.9 0.1 22.8

283.6 180.4 502.0 160.5 437.7 367.0 172.9 10.5 166.9 200.2 45.7 0.6 0.2 0.6 15.0 1.4 0.2 15.7 117.9 155.3 0.2 1.6 0.0 23.2 8.1 0.9 15.5 56.2 58.7 3.0 2.4 6.1 0.2

C-H str. C-H str. C=C str. C-C str., CCH bend Ring breath C-H oop C-H oop Ring def. Ring tars, LiPring str. LiPring str. C-H str. C-H str. C-C str., CCH bend C-C str. C-H oop Ring def. C-H oop Ring tars. Li tars. C-H str. CCH bend, C-C str. CCH bend C-H oop C-C str., Ring def. Ring def. Ring tors. C-H str. C=C str. CCH bend, C-C str. (Kekulk) CCH bend CCH bend C-H oop Li-Ring str.

assignment

in km mol-‘, Raman activities in A4 /amu-’ where differences exceed 10.0 cm-‘).

the C=C A, symmetric stretching mode at 1551 cm-’ and by the ring breath mode, both showing scattering activities of more than 400 A4/amu. For the Li-benzene complex, the larger of the two CC stretching force constants is that for the C2=C3 and C5=C6 double bonds with 6.2 mdyn A-‘. However, this value is even smaller compared to the value of 6.6 mdyn A-’ in benzene. The lower one is the absolute lowest of all the CC bonds discussed here, with only 4.4 mdyn A-‘, although the corresponding bond length is not the largest. These relatively small force constants are certainly an

indication of the loss of aromaticity by charge transfer. Li modes are found in the same region as for thiophene systems. The Li-ring stretching mode is located a little lower (479 cm-‘) indicating a weaker interaction, in agreement with our previous calculations [ 181. The two different sets of coupling force constants (scheme I and scheme II) mostly affect the CC stretch and ring deformation modes of Li-benzene. The ring breath and Kekule modes are calculated at 900 and 1234 cm-’ with scheme I and at 953 and

S. Me, H. LischkalJournal of Molecular Structure (Theochem)

1208 cm-’ with scheme II. Even more affected is the A2 ring deformation located at 775 cm-’ (scheme I) as compared to 719 cm-’ with scheme II. In total, there are eight modes that differ by more than 10 cm-’ in both scaling schemes. The RMSD between the frequencies calculated with the two scaling schemes is only 9.7 cm-‘. 4.5. Biphenyl and di-Li-biphenyl Similar to the spectrum of benzene, the most Table 7 Vibrational

A

RI

R?

frequencies,

IR intensities

Calc.

IR int.

Raman act.

3080 3067 3049 1619 I523 1287 1173 1030

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

1000

0.0

967 837 736 408 303 56

0.0 0.0 0.0 0.0 0.0 0.0

547.1 122.8 51.5 279.7 10.3 103.3 4.9 20.5 0.1 80.4 1.3 11.9 2.1 4.8 11.2

3071 3064 3057 1603 1498 1169 1041 1012 998 967 837 609 403

30.0 40.7 6.4 14.4 41.6 0.4 0.8 0.1 5.7 0.5 0.1 1.6 0.3

63.6 28.4 7.0 2.3 0.0 1.1 0.0 0.0 1.4 7.2 1.0 0.9 0.1

3078 3074 1573 1443

45.2 9.4 5.4 11.5

0.2 40.2 0.9 0.6

and Raman

Expt.’

_ 1613 1505 1282 1190 1029 1003 964 838 740 403 315 _ _ _ 1595 1481 1174 1042 1007 990 964 838 609 403 _ _ 1567 1439

a Frequencies in cm-‘, IR intensities b Ref. [51]. ’ As defined in Table 2.

activities

Approx.

str. str. str. str. str. bend str., ring def. str., ring def. def. oop oop tors. tors.

C-H C-H C-C C-C

str. str. str. str.

in km mol-‘,

and their approximate

assignment

C-H C-H CH C-C C-C CCH CC CC Ring C-H C-H Ring Ring

B3

activities

assignments”

Calc.

IR int.

1344 1270 1146 1076 980 921 774 687 624 541 266 93

2.5 0.0 0.0 4.5 0.1 1.2 8.1 12.4 0.0 3.9 0.0 0.2

0.7 3.1 0.0 0.7 1.1 0.1 1.6 1.9 1.5 4.9 4.5

3054 3048 1587 1469 1340 1311 1146 1072 980 904 733 689 612 496 352 119

11.4 0.3 0.3 2.3 0.0 0.0 0.1 0.3 0.1 2.6 81.7 45.3 0.0 8.6 0.3 1.0

14.5 195.5 5.7 3.1 1.2 0.0 14.0 0.6 0.1 0.0 0.3 0.1 9.5 0.0 0.0 2.1

RMSD errorC

Raman

21

15-31

IR-intensive bands (excluding CH stretches) of biphenyl belong to two C-H out-of-plane motions (see Table 7 and Fig. 6). CC stretch modes appear in the region between 1450 and 1620 cm-‘. The most important Raman active modes are located at 1619 cm-’ (intra-ring CC stretching), 1287 cm-’ (inter-ring stretching), and 967 cm-’ (symmetric intra-ring breath). The IR and Raman spectra of di-Li-BP (Table 8 and Fig. 6) show extraordinarily enhanced intensities and scattering activities compared to BP. IR

of biphenyl

C-H str. C-H str. C-H str. C-C str. C-C str. 4-4 str. CCH bend C-C str., CCH bend Ring def., C-C str. CH oop CH oop Ring def. Ring tors. Ring def. Torsion

364 (1996)

in A4 /amu-’

8.1

Raman act. 1.0

Expt.h

1337 I266 1155 1072 979 917 778 670 628 543 269 112 _ 1595 1455 1376 1317 1158 1090 964 902 735 698 615 486 367

Approx.

CCH CCH CCH CCH C-H C-H C-H C-H Ring Ring Ring Ring

assignment

bend, C-C bend bend bend oop oop oop oop def. tors. tors. tors.

str.

CH str. C-H str. C-C str. C-C str. CCH bend, C-C str. CCH bend CCH bend CCH bend, ring def. C-H oop Ring tars., C-H oop Ring tars., C-H oop C-H oop Ring def. Ring tors. Inter-ring bend Inter-ring bend

S. Irle, H. Lischka/Journal of Molecular Structure (Theochem)

28

Raman activity [A4amw1]

IR intensity [km mol-l]

I, 2000

Ill00

,600

I_

1400

364 (1996) lS-31

I

II.,

I

1200

1000

I, 600

800

400

200

0

700 12000 600

di=Li-BP

10000

500 8000 400 6000

300

4000

200

2000

100

,.

0 2000

la00

1800

1400

I 1200

I;,1 ;..,.. 11 1000

500

500

400

200

0

infrared

and Raman

,500

,500

1400

1200

1000

500

800

400

200

( 0

Wavenumbor [cml]

Wavonumbor [cm’] Fig. 6. Theoretical

I

u.:. .,I . ...! ..,. :?. !.. ’..,...

0 2000

di-LbBP

line spectra

intensities increase by one, and Raman activities by two orders of magnitude. Inter-ring stretching occurs as the highest (scaling scheme I) and most prominent peak in the Raman spectrum at 1578 cm-‘. However, this wavenumber is somewhat lower than that of di-Li-BT inter-ring stretching due to steric hindrance caused by the hydrogens, which is absent in di-Li-BT. In addition to its high Raman activity, the corresponding IR intensity is also high, with 160 km mol-’ . This band has been blue-shifted by about 300 cm-’ relative to BP and is fortuitously located close to the position of the most prominent Raman peak of the undoped molecule. This drastic blue-shift stems from the change of the interring single bond in BP to a double bond in doped BP. An intra-ring C=C stretching mode of B2 symmetry is also located rather close to the

of BP and di-Li-BP

excluding

CH stretches.

inter-ring frequency. It has, however, a much lower Raman activity. The originally dominant Raman peak at 1619 cm-’ is significantly red-shifted in the doped complex. C-H stretches do not show the clear trend towards red-shifting as for the complexes discussed above. The highest symmetric mode involves mainly stretching of ortho-hydrogens and is blue-shifted by 16 cm-’ due to the planarization of the two phenyl rings upon doping. Diagonal force constants of BP are 4.5 mdyn A-’ for the inter-ring and x 6.6 mdyn A-’ for the intra-ring C-C bonds. For the lithium-doped complex, yalues of 6.9 mdyn ApI0 (inter-ring), 4.7 mdyn A-’ (CJ-C2), 7.9 mdyn A-’ (C2-C3) and 5.4 mdyn A-’ (C3-C4) are found. The geometry parameters and force constants are therefore in nearly perfect agreement, indicating the

S. Irle. H. LischkalJournal of Molecular Structure (Theochem) Table 8 Vibrational

Al

IQ

frequencies,

IR intensities

and Raman

activities

of di-Li-biphenyl

and their approximate

Caleb

IR int.

Raman act.

Approx.

assignment

Caleb

3096 3081 3030 1578 1482 1365 1174 979 956 955 741 694 659 638 504 346 318 245 54

5.1 0.0 8.5 159.6 100.4 9.3 39.8 1.2 3.6 20.7 361.7 7.7 7.9 52.5 21.5 0.0 0.0 4.0 1.4

292.5 472.9 176.6 12620.3 683.4 116.2 598.3 3.4 207.4 795.5 20.0 158.8 327.3 566.6 912.3 10.1 254.3 58.3 9.5

C-H str. C-H str. C-H str. f#-4 str. C=C str. C-C str., CCH CCH C-C str., ring def. C-H oop Ring def. C-H oop C-H oop C-H oop Ring tors. Li-ring str. Ring tors. Ring def. Li tars. 4-4 bend

472 292 89

7.2 0.0 31.1 677.1 6.8 126.5 3.9 149.9 0.1 72.3 14.0 10.3 183.3 9.1 6.9

43.8 3.0 13.9 801.2 15.7 49.3 0.0 60.1 35.4 35.2 107.5 9.1 144.4 1.6 85.5

(1563) (1455) (1354) (1184) (965) (935) (712)

3082 3071 3029 1523 (1579) 1437 (1427) 1141 (1156) 1002 (992) 975 938 907 756 692 648 562 522

a Frequencies in cm-‘, IR intensities b Scheme I (scheme II in parentheses

C-H C-H C-H C=C C=C CCH Ring. C-C Ring Ring C-H C-H C-H Ring Ring

str. str. str. str. str. def. str., ring def. def. def. oop, ring tors. oop oop def. tars.

3071 3031 1464 1384 1249 1215 1084 1047 936 748 606 470 399 265 84

A2

BI

IR int. (438)

(1444) (1294) (1239) (1106) (1001) (956)

3094 3032 1469 (1430) 1371 1290 (1277) 1200 1064 1012 991 763 615 483 256 162

29

364 (1996) 15-31

assignmentsa Raman act.

Approx.

assignment

78.9 0.3 6.4

39.3 25.4 62.8

Li-ring str Li tors. 4-4 tors.

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

2.2 341.8 7.1 1.3 2.4 2.7 1.6 8.3 3.3 0.6 3.0 1.9 0.1 4.8 0.9

C-H str. C-H Str. C=C str. C-C str. CCH CCH CCH C-C str. C-H oop Ring tars. Ring. def. Ring tors. Ring def. Li tars. 4-p tors.

29.2 59.6 10.4 178.5 210.4 8.8 63.2 1.6 7.9 18.6 0.8 27.6 0.5 2.1

5.6 24.1 3.8 0.3 0.1 3.2 13.1 0.1 3.0 0.2 1.6 19.9 54.4 0.3

C-H str. C-H str. C=C str. C-C str., CCH CCH CCH CCH C-C str. C-Hoop C-H oop Ring def. Ring tors., LI tors Li tors. +r$ bend

in km mol-‘, raman activities in A4 /amu-’ where differences exceed 10 cm-‘).

quinoid distortion. The diagonal intra-ring CC stretch force constants in di-Li-BP are roughly 1 mdyn A-’ larger than the respective ones in Li-benzene. Twenty frequencies of di-Li-BP differ by more than 10 cm-’ between scaling schemes I and II. Most of them are located in the CC stretching region between 955 and 1578 cm-‘. The biggest differences amount to about 56 cm-’ and concern C=C stretching modes. Interestingly, the B2

Li-ring stretch frequency is also affected by the different scaling schemes. The RMSD between both schemes is only 5.1 cm-’ compared to an RMSD value of 9.7 cm-’ for Li-benzene.

5. Conclusions Information on the structures spectra of lithium-doped aromatic

and vibrational systems serving

30

S. Irle, H. LischkalJournal

ofMolecular Structure (Theochem) 364 (1996) 15-31

as models for polaronic and bipolaronic defects has been presented. As a consequence of the formation of quinoidic structures from the conjugated aromatic rings, a significant rearrangement of the vibrational modes takes place. Moreover, because of the charge-transfer from Li to the conjugated r-system, the intensities of the IR and Raman lines are increased substantially (one order of magnitude and more). This effect is much more pronounced in the case of the di-Li complexes. One of the main features for di-Li-BT and di-Li-BP is the formation of an inter-ring CC double bond due to the charge transfer. Concomitantly, the inter-ring vibrational mode is blue-shifted by M 300 cm-‘. It is located a little below 1600 cm-’ and constitutes the most prominent peak in the Raman spectrum for both the doped BT and BP cases. This finding agrees well with the theoretical results by Ehrendorfer and Karpfen [35] who also report a remarkably fast convergence of the Raman spectra with respect to chain length. Assuming a similar behavior for the oligophenyl series, we expect the inter-ring vibration for the bipolaron defects in p-oligophenyls to be in the region between 1550 and 1600 cm-‘. Raman spectra of p-oligophenyl dianions in solution have been reported by Furukawa et al. [14]. In this work the inter-ring stretching mode was assigned to bands in the region between 1320 and 1357 cm-‘. This assignment is certainly not in agreement with our calculations. From our results obtained from the two scaling schemes (1578 and 1563 cm-‘), we cannot imagine that our calculated values, which are already corrected empirically by scaling, would be shifted in more accurate calculations by more than 200 cm-’ towards 1320 or 1357 cm-’ as required by the experimental assignment. Also, the aforementioned fast convergence of the Raman spectra with chain length on oligothiophene dications [35] does not support any significant reductions of the inter-ring frequency. Of course, calculations on larger p-oligophenyls would be required in order to investigate the actual chain length dependences in more detail. Moreover, great care has to be exercised when comparing our quantum chemical calculations on isolated complexes with measurements in solution.

Acknowledgment

This work was carried out with the support of the Austrian “Fonds zur Fiirderung der wissenschaftlichen Forschung”, project no. P9569-CHE. The calculations were carried out in part on the DEC Alpha 2100 4/275 cluster of the Vienna University Computer Center.

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S. Irle, H. LischkalJournal of Molecular Structure (Theochem) [23] C. Fredriksson and S. Stafstrtim, J. Chem. Phys., 101 (1994) 9137. [24] G. Zerbi, M. Gussoni and C. Castiglioni, in Ref. [4]. [25] C. Castiglioni, M.D. Zoppo and G. Zerbi, J. Raman Spectrosc., 24 (1993) 485. 1261 C. Kui, M. Kertesz and H. Eckhart, Synth. Met., 41 (1991) 3491. [27] M. Kofranek, T. Kovar, H. Lischka and A. Karpfen, J. Mol. Struct. (Theochem), 259 (1992) 181. [28] F. Negri and M.Z. Zgierski, J. Chem. Phys., 100 (1994) 2571. [29] P. Pulay, G. Fogarasi and J.E. Boggs, J. Chem. Phys., 74 (1981) 3999. [30] L. Cuff and M. Kertesz, Macromolecules, 27 (1994) 762. [31] J.T.L. Navarette and G. Zerbi, Synth. Met., 28 (1989) C18. [32] J.T.L. Navarette, B. Tian and G. Zerbi, Synth. Met. 38 (1990) 299. [33] C. Ehrendorfer and A. Karpfen, Vib. Spectrosc. 8 (1995) 293. [34] C. Ehrendorfer and A. Karpfen. J. Mol. Struct., 349 (1995) 417. [35] C. Ehrendorfer and A. Karpfen, J. Phys. Chem., 99 (1995) 5341. [36] R. Ahlrichs, M. Bar, M. Haser, H. Horn and C. Kolmel, Chem. Phys. Lett., 162 (1989) 165. [37] M.J. Frisch, G.W. Trucks, M. Head-Gordon, P.M.W. Gill, M.W. Wong, J.B. Foresman, B.G. Johnson, H.B. Schlegel, M.A. Robb, E.S. Replogle, R. Gomperts, J.L. Andres, K. Raghavachari, J.S. Binkley, C. Gonzales, R.L. Martin, D.J. Fox, D.J. Defrees, J. Baker, J.J.P. Stewart and J.A. Pople, Gaussian 92, Rev. B, Gaussian, Inc., Pittsburgh, PA, 1992. [38] J.A. Pople and R.K. Nesbet, J. Chem. Phys., 22 (1954) 571. [39] A. Schafer, H. Horn and R. Ahlrichs, J. Chem. Phys., 97 (1992) 2571.

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