Electronic charge density distributions in tetrathiafulvalene derivatives

PII:

Eur. Polym. J. Vol. 34, No. 3-4, pp. 455±462, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0014-3057/98 $19.00 + 0.00 S0014-3057(97)00124-9

ELECTRONIC CHARGE DENSITY DISTRIBUTIONS IN TETRATHIAFULVALENE DERIVATIVES I. V. KITYK,1 B. SAHRAOUI,2 G. RIVOIRE,2 J. KASPERCZYK,1* M. CZERWINÂSKI3 and M. SALLE2 Physics Institute, Pedagogical University, Al. Armii Krajowej 13/15, PL-42201 Cze° stochowa, Poland, Laboratoire POMA, Universite d'Angers, 2 Boulevard Lavoisier, F-49045 Angers Cedex, France and Chemistry Institute, Pedagogical University, Al. Armii Krajowej 13/15, PL-42201 Cze° stochowa, Poland 1

2

3

(Received 1 November 1996; accepted in ®nal form 29 January 1997) AbstractÐWe have calculated electronic charge density distributions for typical representatives of tetrathiafulvalenes. We have applied a combined semi-empirical and ab initio method to perform these calculations because of the complexity of these molecules. Calculations have unambiguously shown an essential in¯uence of modi®cations of the back-bone aromatic group on the distribution of electronic charge density. Moreover, we have observed a substantial change of the charge density gradients under the in¯uence of aromatic ring modi®cations. This leads to a strong nonuniformity of the electronic charge density distribution. It is surprising that the main charge density change is observed on the other molecule end than the structural modi®cation (e.g. a substitution) has been made. # 1998 Elsevier Science Ltd. All rights reserved

linear optical behavior of tetrathiafulvalene derivative compounds.

INTRODUCTION

An increasing interest in organic materials showing nonlinear optical properties has recently been observed [1±5]. The search for new materials with desired physical properties could be simpli®ed if our knowledge of the electronic charge density distribution in such materials could be increased. The major properties needed for good nonlinear optical materials are as follows: high values of nonlinear optical susceptibilities, a wide spectral range of optical transparency and a short response time. These requirements are closely related with sucient noncentrosymmetry of the electronic charge density distribution in these molecules. Tetrathiafulvalenes are very promising systems [1] since they possess highly nonlinear optical properties. Moreover, their charge-transfer salts demonstrate remarkable conducting and nonlinear optical properties [2]. The chemical formulae of the investigated molecules are presented in Fig. 1. For example, a large increase (by four orders of magnitude as compared to CS2) of the second-order susceptibility was observed in derivative compounds of the acetylenic analogues of TTF for l = 532 nm [3]. We have performed an optimization of the molecular structure using a molecular dynamics procedure. We have also calculated electronic charge density distributions of tetrathiafulvalene derivative compounds by means of an approach which was a combination of ab initio and semiempirical quantum chemistry calculations. The fundamental physical properties (e.g. molecular structures, electronic charge distributions, etc.) should be investigated in order to understand the origin of the unusual non-

GEOMETRIC OPTIMIZATION

Structural optimizations are usually carried out using standard molecular dynamics methods, such as MOPAC [6] or GAUSSIAN [7]. However, our previous investigations of the hyper®ne structure together with other results of optical investigations have unambiguously shown that it is necessary to consider a superposition of di€erent structural clusters which give their contributions to the observed values of the nonlinear optical susceptibilities. Therefore, we have carried out a complex optimization of the molecular structural arrangement based on a molecular dynamics approach. The main di€erence of the proposed approach compared with traditional ones consists of taking into account many structural con®gurations and their possible interactions. So, we have built structural clusters possessing di€erent (optimized) potential energies Ui. In the next step, we have applied a self-consistent procedure with continuously increasing values of the potential energy of the considered molecular system. Partial contributions have been renormalized using the proper Boltzmann weighting factors pi, which are proportional to the appearance probability of a given phase at temperature T: .X exp…ÿUi =kB T†; …1† pi ˆ exp…ÿUi =kB T† i

where Sipi=1 and kB is the Boltzmann constant. The summation was stopped if the value of pi became less than 0.02.

*To whom all correspondence should be addressed. 455

456

I. V. Kityk et al.

Fig. 1. Chemical formulae of a1, e1, e2 and e3 tetrathiafulvalene derivatives.

Another diculty which appeared in these calculations was the need for taking into account intercluster coupling. We have performed the minimization of the corresponding energies based on linear couplings between the clusters. It was, however, necessary to increase the number of coordination spheres to 6. At this level, the calculations were carried out using an ab initio approach and afterwards we have included the values of external potentials. A large number of di€erent structure con®gurations has been obtained for the materials under consideration. This number ranged between 50 and 78 depending on the chemical composition. A proper inclusion of the in¯uence of the cluster boundary conditions is very important in this case

since the boundary conditions are very sensitive to the cluster size. To overcome this diculty, we have proposed a special procedure in order to avoid potential jumps on the boundaries. We have calculated interface potential gradients, taking into account the condition of a continuous variation of the ®rst and second derivative of the potential. The molecular dynamics optimization has been carried out in two steps. The ®rst one consisted of the use of a standard method of molecular dynamics with a force ®eld MM+ [8] in a version represented by the HYPERCHEM 4.0 computer package [9]. The second step consisted in the application of an unrestricted Hartree±Fock method (UHF) in a version of MOPAC 6.0 package [10] with PM3 parametrization [11]. The geometry was

Charge density distributions in tetrathiafulvalene derivatives

457

Fig. 2. Typical a1 geometry of example tetrathiafulvalene derivative molecule. Oxygen, carbon and hydrogen atoms are denoted by large grey, large black and small empty circles, respectively. Sulphur atoms are additionally denoted by letter S.

successfully optimized using a derivative procedure presented in MOPAC program and a Broyden± Fletcher±Goldfarb±Shanno method (BFGS) [12± 15]. The iteration procedure was continued until the energy and the charge density convergence became smaller than 10ÿ8. We have used the BFGS method in order to minimize the system energy with the relative changes of energy gradients being ®nally smaller than 10ÿ8. Correlation e€ects were also included by a proper choice of adjustable parameters since we have applied semiempirical methods for the geometry optimization. Based on these methods, approximate molecular structures were obtained and one example structure of the a1 type is presented in Fig. 2. Oxygen, carbon and hydrogen atoms are denoted by large grey, large black and small empty circles, respectively. Sulfur atoms are additionally denoted by letter S. It is clearly seen that the stoichiometric relations are very complicated for these molecules. Their structures are, on the other hand, very similar to the structures of many biological systems. The central chain is not linear and has a very complicated shape. The aromatic rings (consisted of three carbon and two sulfur atoms) are situated at di€erent planes and the dihedral angle between these planes is of about 308. COMPUTATIONAL DETAILS

The main part of our calculations has been carried out using the DMol quantum chemistry software package [6], which allows theoretical calculations for a wide range of compounds, including metal clusters, biological compounds, organometallic and organic compounds. The DMol package enables one to obtain a variational selfconsistent solution within a density functional theory (DFT) and using a numerical basis for atomic orbitals. The solutions of these equations provide molecular wavefunctions and electron charge densities which can be used to evaluate energetics, electronic and magnetic properties of the system. The procedure of self-consistency was ensured by using an unrestricted Hartree±Fock method. The density functional theory is based on a theorem by Hohenberg and Kohn [16], which states that

all ground-state properties are functionals of the charge density r. In particular, the total energy Et can be written as: Et ‰rŠ ˆ T‰rŠ ‡ U‰rŠ ‡ Exc ‰rŠ;

…2†

where T[r] is the kinetic energy of an electronic system of charge density r, U[r] is a classical electrostatic energy due to the Coulomb interaction and Exc[r] includes all many-body contributions to the total energy, in particular, exchange and correlation energies which play a key role in ab initio iteration procedures. The charge density is given by the following simple sum: X r…r† ˆ jfi …r†j2 : …3† i

Here we use the same fi for electrons with di€erent spin rates. The third term of equation (2), the exchange-correlation energy, requires, however, an approximation within the DFT method in order to be computationally tractable. A local density approximation is simple but e€ective. It is based on the well-known exchange-correlation energy for the uniform electron gas. Analytical expressions can be obtained by integrating the result of the uniform electron gas. This yields: exc ‰rŠ ˆ inf r…rexc ‰r…r†Šd3 r;

…4†

where exc[r] is the exchange-correlation energy per particle for the uniform electron gas and r is the electronic density. This implementation uses a form derived by von Barth and Hedin [17]. The total energy can be written as: X Et ‰rŠ ˆ fhfi j ÿ D=2jfi i ‡ hr…ri † i

‡ Ve …ri †=2 ÿ VN †ig ‡ VNN ;

…5†

where Ve(ri) is the electronic potential at point ri, which originates from the Coulomb interaction between all electrons (the division by 2 was introduced to avoid a double taking into account this energy). VN is the potential acting on the electrons from the nuclei side and VNN denotes the interaction energy between the nuclei. The ®rst term of

458

I. V. Kityk et al.

Fig. 3. Total electronic charge density distributions of e1, e2 and e3 tetrathiafulvalene derivatives. Each density is presented as a cut by a given plane and atomic positions are projections since a part of the atoms is situated outside the cut plane. The lowest charge density line is always shown for ÿ0.4 e/AÊ3 (e = elementary charge, AÊ=0.1 nm) and the charge density di€erence between neighboring lines corresponds to 0.04 e/AÊ3. The ®gures were obtained using the Hyperchem 4.5 computer package ensuring a standard accuracy of the results.

equation (5) is the usual electronic kinetic energy, where D denotes a Laplacean with respect to the position vector ri components and, for simplicity, the electron mass and the Dirac's constant are trea-

ted as units. The kinetic energy of nuclei was neglected (the so-called adiabatic approximation). To determine actual energies, variations of Et must be optimized with respect to variations of r

Charge density distributions in tetrathiafulvalene derivatives

459

Fig. 4. (a) HOMO and (b) LUMO electronic charge density distributions of e1 tetrathiafulvalene derivative.

(subject to an orthonormality condition): XX dEt =dr ÿ eij hfi jfj i ˆ 0; i

X Et ˆ fei ‡ hr…ri †…exc ‰rŠÿmxc ‰rŠÿVe …ri †=2†ig ‡ VNN …6†

i

…9†

j

where eij are Lagrangian multipliers. This procedure leads to a set of coupled equations: fÿD=2 ÿ VN ‡ Ve ‡ mxc ‰rŠgfi ˆ ei fi :

…7†

The term mxc is the exchange-correlation potential which results from di€erentiating exc: mxc ˆ @…rexc †=@r:

…8†

A use of the eigenvalues of equation (7) leads to a reformulation of the energy expression:

which is close to the form of Kohn and Sham [18]. In practice, it is convenient to expand molecular orbitals (MOs) in terms of atomic orbitals (AOs): X Cim wm : …10† fi ˆ m

The atomic orbitals wm are called atomic basis functions. Several choices are possible for the basis set, including Gaussian, Slater and numerical orbitals. In these calculations, numerical orbitals of a DND basis set (double-numerical basis functions together with polarization functions) were used. This basis is

460

I. V. Kityk et al.

Fig. 5. (a) HOMO and (b) LUMO electronic charge density distributions of e2 tetrathiafulvalene derivative.

comparable in its quality to Gaussian 6-31G** sets. The local correlation functional of Hedin± Lundqvist/Janak±Morruzi±Williams [19] or Vosko± Wilk±Nusair [14] were used in the DMol calculations. RESULTS AND DISCUSSION

It is important to point out that the presence of aromatic rings leads to a very complicated structure of the molecules under consideration (see Fig. 2). The aromatic rings are situated near the ends of the molecules and their presence strongly in¯uences molecular as well as solid-state properties. One can

easily notice that the main di€erences between the considered molecules are observed on their ends. The e1 molecule possesses two additional CO2CO3 fragments, while the e2 one possesses an additional benzene ring compared with the e3 molecule. Other parts of the molecules remain unchanged. We have calculated the one-electron energies, in particular, of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) for all considered tetrathiafulvalene derivatives. In order to check the quality of our calculations, we have compared the di€erence between the HOMO and the LUMO energies [the so-called HOMO±LUMO (energy) gap] with the

Charge density distributions in tetrathiafulvalene derivatives

461

Fig. 6. (a) HOMO and (b) LUMO electronic charge density distributions of e3 tetrathiafulvalene derivatives.

available experimental data of the fundamental absorption edge. We observe agreement between calculated and measured values within a tolerance lower than 0.3 eV. Electronic charge density distributions are presented in Fig. 3 for three molecules. Each density is presented as cut by a given plane and atomic positions are projections since a part of the atoms is situated outside the cut plane. The lowest charge density line is always shown for ÿ0.4 e/AÊ3 (e = elementary charge, AÊ=0.1 nm) and the charge density di€erence between neighboring lines corresponds to 0.04 e/AÊ3. The ®gures were obtained using the Hyperchem 4.5 computer package ensuring the standard accuracy of the results. First of all, it is necessary to underline that the additional benzene ring (in the case of e2) leads to a decrease of the electronic charge density gradients between the 18th and 11th atom. Therefore, we can state that

the right-side additional benzene ring makes the electronic charge density distribution more homogenous. The largest gradients of the electronic charge density are observed on the molecule backbones. The presence of CO2CO3 groups leads to the appearance of large electronic charge density gradients near the above-mentioned group and near the middle of molecule, i.e. the 26th and the 17th atom. Moreover, central double carbon bounds are the main detectors of the charge density gradients and they are very sensitive to the weakest changes in surrounding atoms arrangement. A sketch of investigated molecules was chosen in such a manner as to ensure the best observation of the electronic charge density gradients. It is surprising that the electronic charge density does not directly correlate with the structural changes in the case of the e3 molecule. Besides the

462

I. V. Kityk et al.

double C1C chemical bonds, such a large electronic density gradient can also be observed near the left-side atomic fragments. So, an essential in¯uence of the right-side group on the total charge density distribution has been found. However, states corresponding to the HOMO and the LUMO states are of importance from an optical point of view. Corresponding HOMO and LUMO charge density distributions are presented in Fig. 4, 5 and 6. Contrary to the case of the total density for the e2 molecule, one can observe a substantial electronic density gradient (in the case of neighboring C1C groups) near the left-side fragments. In our opinion, this re¯ects a correspondence of these electronic charge density distributions. The important role of atomic fragments lying between the 20th and 15th atom should be underlined. This molecular fragment strongly in¯uences linear as well as nonlinear optical properties. The right-side groups play an important role in all these molecules due to their high polarizabilities which strongly in¯uence the total charge density distributions, especially in the case of the e1 molecule. The essential nonuniformity of the electronic charge density distribution is clearly seen in Fig. 4. Larger charge density gradients are observed in vicinity of the 20th and 21st double carbon groups in the case of the LUMO. On the other hand, the HOMO does not show such large electronic density gradients at the same places. The electronic charge density gradients located in the vicinity of the left-side aromatic rings are essentially di€erent. This di€erence can lead to the occurrence of the larger electronic dipole moments which mainly determine the values of the nonlinear optical susceptibilities. The e3 molecule demonstrates a quite di€erent behavior with regard to the total electronic charge density distribution. As in previous cases, the largest gradients are observed near double carbon bonds (between the 23rd and the 24th atom). So, we can state a key role of the left-side carbon bonds which, in all three cases, demonstrate a very strong impact on the electronic charge density distributions.

CONCLUSIONS

We have applied the combined ab initio and semiempirical method for the calculation of the electronic charge density distributions in molecules based on tetrathiafulvalenes. The charge density distributions of TTF derivatives favor the appearance of the nonlinear optical susceptibilities because of a strong nonuniformity of the electronic charge density near the aromatic rings. The above statement is strongly supported by appropriate experimental data [3] obtained recently for the TTF derivative samples. We have also estimated the in¯uence of structural modi®cations on the HOMO and LUMO

charge density distributions for the TTF derivatives of di€erent contents. The large nonuniformity of the electronic charge density distributions in tetrathiafulvalene derivatives is the main source of their high hyperpolarizabilities and nonlinear optical properties. The present study allows us to perform a qualitative insight as concerns the latter. However, a detailed numerical study of the TTF derivative hyperpolarizabilities should be carried out in order to have a quantitative approach to the optical nonlinearity. Our present semiempirical calculations have unambiguously shown good agreement between measured electronic energy gas and calculated HOMO±LUMO energy di€erences. The theoretical results for these molecules agree well with spectroscopic absorption data, although a possible dispersion of the optical absorption fundamental edge has to be taken into consideration in the solid state samples that can explain, at least in part, the existing theory±experiment discrepancies. This dispersion is, of course, caused by intermolecular interactions and quasi-phonon vibrations in solids, but this is outside a scope of the molecular description presented here. However, the problem is currently under investigation and the expected results will be published soon. REFERENCES

1. Marder, S. R., Sohn, J. E. and Stucky, C., Am. Chem. Symp. Scr., 1992, 455, . 2. Zyss, J. and Chemla, D. S., Quantum Electronics (Principles and Applications), Vols 1 and 2, Academic Press, New York, 1987. 3. Sahraoui, B., Sylla, M., Bourdin, J. P., Rivoire, G., Zaremba, J., Nguen, T. T. and Salle, M., Mod. Optics, 1995, 42, 2095. 4. Ledoux, I., Onde Electr., 1994, 74, 5. 5. Dubois, J. C. and Faure, J., Onde Electr., 1994, 74, 4. 6. Stewart, J. J. P., Program MOPACÐversion 6.0no. 455, 1990, QCPE, 455. 7. Stewart, J. J. P., J. Comput. Chem., 1989, 10, 209. 8. Allinger, N. L., J. Am. Chem. Soc., 1977, 99, 8127. 9. Hyperchem2, Computational Chemistry, Publ. HC 4000-30-00, Hypercube, Inc., 1994. 10. Broyden, C. G., J. Inst. Math. Appl., 1970, 6, 222. 11. Fletcher, R., Comput. J., 1970, 13, 317. 12. Versluis, L. and Ziegler, T., J. Chem. Phys., 1988, 88, 3322. 13. Pople, J. A. and Nesbet, R. K., J. Chem. Phys., 1954, 22, 571. 14. Vosko, S. J., Wilk, L. and Nusair, M., J. Phys., 1980, 58, 1200. 15. Hirshfeld, F. L., Theor. Chim. Acta B, 1977, 44, 129. 16. Hohenberg, P. and Kohn, W., Phys. Rev. B, 1964, 136, 864. 17. von Barth, U. and Hedin, L., J. Phys. C, 1972, 5, 1629. 18. Kohn, W. and Sham, L. J., Phys. Rev. A, 1965, 140, 1133. 19. Hedin, L. and Lundqvist, B., J. Phys. C, 1971, 4, 2064.