Symmetry distortion of extended 1-D π-electron systems

Chemical Physics 310 (2005) 297–301 www.elsevier.com/locate/chemphys

Symmetry distortion of extended 1-D p-electron systems N. Tyutyulkov a

a,b

, N. Drebov b, F. Dietz

a,*

Wilhelm-Ostwald-Institut fu¨r Physikalische und Theoretische Chemie, Universita¨t Leipzig, Johannisallee 29, D-04103 Leipzig, Deutschland, Germany b Faculty of Chemistry, Chair of Physical Chemistry, University of Sofia, 1, J. Bourchier Blvd, BG-1126, Sofia, Bulgaria Received 4 August 2004; accepted 27 October 2004 Available online 30 November 2004

Abstract It has been shown that a symmetry lowering from a state with higher symmetry to a state with lower symmetry (D(2m)d ! C(2m)) occurs in 1-D polymers with polymethine fragments as elementary units when electron–vibration interaction is taken into account. The investigations are carried out using an extended Su–Schrieffer–Heeger method, where the electron–electron interaction is taken into account.
2004 Elsevier B.V. All rights reserved. Keywords: 1-D polymers; Electron–vibration interaction; Symmetry lowering (Peierls distortion)

1. Introduction The first-order Jahn–Teller distortion effect [1] is a well investigated phenomenon of different molecular systems with degenerate quantum states [2]. The symmetry lowering has been interpreted as determined by the electron–vibration interaction [1,2]. The symmetry distortion of extended p-electron systems (1-D polymers) with non-degenerate MOs and closed-shell ground state has been associated traditionally with the Peierls distortion [3]. The symmetry lowering of the classical example, of polyacetylene (PA), is a second order effect, i.e., the symmetry distortion is determined by the electron–vibration coupling, as follows from the investigations of Su, Schrieffer and Heeger [4]. References of the Peierls distortion of 1-D polymers with a large number of p-centers within the elementary unit (EU) are given in the paper of Klein et al. [5] (see also [6–10]). In the papers [5,6] the dimerization (spin-

*

Corresponding author. Tel.: +49 341 97 36403; fax: +49 341 97 36399. E-mail address: [email protected] (F. Dietz). 0301-0104/$ - see front matter
2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.10.044

Peierls distortion) has been explained with a Heisenberg-like Hamiltonian model. The Peierls distortion phenomenon is a special case of a symmetry distortion in extended systems with a symmetrical ground state and has bearing on solitonic excitation and conductivity of organic polymers. The problem of the relation between the symmetry of a 1-D system, in particular Peierls distortion, and the structure, mobility and excitation of solitons requires investigations with non-trivial methods. A general formulation of the problem is given in [11]. Other (different) types of symmetry lowering may occur in molecular systems. Arguments in favour of such speculations follow from the Herzberg–Teller theorem [12], and the considerations of Bader [13] and Pearson [14] (see also the recent paper of Hoffmann et al. [15]). Tolbert [16] has considered the symmetry distortion (symmetry collapse) in polymethine cyanine molecules with a large number of methine groups. In this paper which has a preliminary character we consider the symmetry lowering (distortions) of 1-D p-electron polymers with odd-numbered and also even-numbered polymethine fragments as elementary units.

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2. Methods of investigation

clm ¼ e2 =ða þ DRlm Þ,

2.1. Extended Su–Schrieffer–Heeger method

clm ¼ e2 =ða2 þ DR2lm Þ

An extended Su–Schriefer–Heeger (ESSH) method [17] was used with linear electron–lattice coupling and a harmonic bond-strain potential as described in the original SSH model [4] and taking into account the p–p electron interaction corresponding to the Pariser–Parr– Pople (PPP) approximation [18]. The formalism is discussed in more detail in [17]. The adiabatic energy of the considered 1-D systems is given by: E ¼ Ep þ Ep
r þ Er þ Ep
p ¼ EðSSHÞ þ Ep–p :

ð1Þ

The one electron part of Eq. (1), E(SSH), is given in terms of the original SSH model [4]: X X EðSSHÞ ¼ 2ðb þ aDrij Þpij þ 1=2 K ij Dr2ij : ð2Þ hi,ji hi,ji In Eq. (2) b is the standard resonance integral, a is the electron–lattice coupling constant, K is the spring constant, the pij are the p-bond orders, and Drij is the bond distortion parameter, i.e., the deviation of the corresponding bond length from the standard value ˚. r0 = 1.40 A X X Ep–p ¼ 1=2 cii ðqi
1Þqi þ 1=2 cij ðqi
1Þ i i,j X  ðqj
1Þ
1=2 cij p2ij , ð3Þ i,j

ð4Þ 1=2

ð5Þ

,

with a = 2e2/(cll + cmm). If the screening parameter is D = 1, then Eq. (4) is identical with the Mataga–Nishimoto approximation [22], and Eq. (5) is identical with the Ohno approximation [23]. The following standard values of the one-center Coulomb integrals have been used: cCC = 10.84 eV, cNN = 12.28 eV [20].

3. Objects of investigations For the 1-D polymers investigated in this paper the cyclic Born–Karmann conditions are fulfilled. The polymers are rota polymers, i.e., the polymers are viewed to be embedded in a cylindrical surface exhibiting at least CN symmetry, where xj = 2pj/N denotes the argument of the character of rotation in the irreducible representation Cj of this group. We consider two types of 1-D polymers: • polymers PM with an odd number of methine groups between the N-atoms (Fig. 1), and • polymers TM-1 and TM-2 with an even number of methine groups between the N-atoms (Fig. 2).

where qi = pii is the p-electron charge on the site i. The relative computational simplicity of the ESSH method allowed us to describe 1-D systems with up to N  102 p-centers.

d

+

+ N

2.2. Parametrization

N n

Sets of different values of the electron–lattice coupling constant, a, and the spring constant, K, for the carbon–carbon bond have been published:

H

n

D(2m)d

H

˚ , K = 24.6 eV/A ˚ 2 [17]. a = 3.21 eV/A ˚ , K = 49.7 eV/A ˚ 2 [4]. a = 6.31 eV/A ˚ , K = 60.0 eV/A ˚ 2 [19]. a = 7.0 eV/A ˚ ) = bC–N(r0 = 1.40 A ˚)= Values b0 = bC–C(r0 = 1.40 A
2.4 eV [20] have been used for the resonance integrals between the 2pp–2pp AOs. The parameter of the coulomb integral of the nitrogen atom is aN = aC + b0 [20,21]. For the two-center atomic Coulomb repulsion integrals, clm, for the calculation the energy of the electron–electron interaction the following approximations have been used:

+ N

H

+ N

C(2m)

H

Fig. 1. Investigated 1-D polymers PM with odd numbers of methine groups between the N atoms. n = 0, 1, 2, 3,. . . The vertical dashed lines define the elementary unit.

N. Tyutyulkov et al. / Chemical Physics 310 (2005) 297–301

4.1. Polymers with odd methine groups between the N-atoms

d

+ N

+ N

In Table 1 are given the calculated values of the bond distortion parameters Drij, (numbering of the atoms, see Scheme 1) and DE, the calculated values of the energy gap in the equilibrium state, and the calculated values of the energy lowering per p-site upon bond length distortions, with different approximations of the Coulomb integrals (see Eqs. (4) and (5)). The results are given for an 1-D polymer (PM) with five methine p-centers in the EU. The electron–vibration interaction is connected with a symmetry lowering of the undistorted 1-D lattice – a transition to (or towards) states belonging to a lower symmetry group, i.e., the D(2m)d ! C(2m) transition (see Fig. 1). There exist two degenerate non-mixing Kekule phases, A and B (Scheme 2) with opposite signs of the bond length distortion parameter Dr (see Eq. (2)). As is shown in Table 2, the charge density distribution is also unsymmetric. Qualitatively, the above results for the PM-structure of the EU are valid also for all 1-D polymers with n = 0, 1, 2, . . . . The results are qualitatively the same also for calculations carried out with different values of the bond parameter given in Section 2 ([4,17,19]). However, the Dr values are different.

N + n

n

+ N

TM-1 + N

N + n

n d

+ N

N + n

N + n

TM-2

+ N

N + n

299

N +

1

n

Fig. 2. Investigated 1-D polymers TM-1 and TM-2 with even numbers of methine groups between the N-atoms: n = 0, 1, 2, 3, . . . .

4. Results and discussion The numerical results given below are obtained using the set of parameter a and K (see Section 2.2) tested for the ESSH approach ([17]) and with different potentials for the calculation of the energy of the electron–electron interaction, Ep–p, given in Eq. (1). Su and Epstein [24] have investigated the Peierls distorted form of polyaniline (pernigraniline base polymer). As in this paper of Su and Epstein [24] we treat the C–N bond parameters K (spring constant) and a (electron–lattice coupling constant) like those of a C–C bond.

2

3

4

5

6

1

+

.

.+

N

N

H

H

1

PM 10

.

+ N

3

9

+ N

.

N +

.

2 4

5

6

7

8

TM-1

Scheme 1. Numbering of the p-centers of the EU of polymers PM and TM-1.

Table 1 ˚ (Scheme 1) of polymer PM with different approximations of the Coulomb integrals c Calculated values of the bond distortion parameter Drij · 102 A (M Eq. (4), O Eq. (5)) and different values of the screening parameter D c

D

DE

Dr1–2

Dr2–3

Dr3–4

Dr4–5

Dr5–6

Dr6–1

M M M O

1 2 3 3

4.39 2.74 1.98 2.50


5.8
5.1
4.4
5.1

5.8 5.1 4.4 5.1


6.1
5.3
4.6
5.3

6.1 5.3 4.6 5.3


6.2
5.4
4.7
5.4

6.2 5.4 4.7 5.4

DE (in eV) are the calculated values of the energy gap in the equilibrium state.

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N. Tyutyulkov et al. / Chemical Physics 310 (2005) 297–301

+ N

N

+ N

N

A

q2 = 0.938

q1 = 0.974

B N

Scheme 2. Degenerate non-mixing Kekule phases of the EU of polymer PM.

38 0.0

0.003

-0. 0

33

0.043

6 03 -0.

Table 2 Calculated values of the p-electron charge densities of the 2nd and 6th p-centers of the polymer PM (see Table 1) c

D

q2

q6

M M M

1 2 3

0.873 0.902 0.915

0.949 0.958 0.961

N

0.0

04

3 03 -0.

-0.

06

4

43 0.0

-0.

03

3

N

Scheme 4. Distortion parameter of the bonds and selected charge densities of the EU of emeraldine.

5. Conclusions Table 3 Calculated values of the charge densities at the p-centers 5 and 7 and ˚ of the 1-D system TM-1 the bond distortion parameter Drij · 102 A (Scheme 1) q5

q7

Dr1–2

Dr10–1

Dr2–3

Dr9–10

Dr3–4

Dr8–9

0.986

0.925

5.1


4.9


5.0

5.1


4.9

5.1

The results have been obtained with the screened Mataga–Nishimoto approximation for the coulomb integrals (Eq. (4), D = 3). The energy gap in the equilibrium state is DE = 2.05 eV.

4.2. Polymers with even methine groups between the N-atoms As in the case of polymers with an odd number of methine groups between the N-atoms in the EU, the electron–vibronic coupling is also connected with a symmetry lowering of polymers with an even number of methine groups in the undistorted EU for all values of n (see Fig. 2). The results for an example, namely the TM polymer are given in Table 3 (numbering of the atoms see Scheme 1). The Kekule formulas which can be associated to TM1 and TM-2 polymers are shown in Fig. 2 A polymer of this type is emeraldine (EM, Scheme 3), a special form of polyaniline [25]. In Scheme 4 are shown the calculated values of the ˚ obtained with bond distortion parameter Drij · 102 A screened Mataga–Nishimoto potentials (Eq. (4), screening parameter D = 3).

NH

NH n EM Scheme 3. EU of emeraldine.

The structures (bond lengths) and charge distributions of 1-D polymers with polymethine chains of different (odd and even) number of methine groups within the elementary unit (EU) have been investigated theoretically taking into account the electron–vibration interaction. The following conclusions can be drawn: A symmetry lowering from a state of higher symmetry (D(2m)d) to a state of lower symmetry (C(2m)) has been found when the electron–vibration interaction is considered. Using an extended Su–Schrieffer–Heeger formalism a bond distortion parameter has been calculated which shows a moderate bond length alternation within the EU. Alternating charge densities within the polymethine chain with odd or even numbers of methine groups of the EU show that also the electronic structure is distorted. The symmetry lowering has been demonstrated for a well-known polymer with an even-numbered polymethine EU, namely a special form of polyaniline (emeraldine) which shows alternation of the bond lengths and charge densities, respectively.

Acknowledgement This work was supported Forschungsgemeinschaft.

by

the

Deutsche

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