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Large effect of dopant level on second hyperpolarizability of alkali-doped polyacetylene chains
Chemical Physics Letters 412 (2005) 217–222 www.elsevier.com/locate/cplett
Large effect of dopant level on second hyperpolarizability of alkali-doped polyacetylene chains Milena Spassova
a,b,*
, Benoıˆt Champagne b, Bernard Kirtman
c
a
b
Institute of Organic Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Laboratoire de Chimie The´orique Applique´e, Faculte´s Universitaires Notre-Dame de la Paix, rue de Bruxelles, 61, B-5000 Namur, Belgium c Department of Chemistry and Biochemistry, University of California, Santa Barbara, CA 93106, USA Received 11 May 2005; in final form 21 June 2005 Available online 20 July 2005
Abstract Hartree–Fock calculations are reported for the effect of donor dopant concentration on the static longitudinal second hyperpolarizability of individual polyacetylene chains using explicit alkali (Li, Na, K) atoms and varying chain lengths. Enhancement factors approaching two orders of magnitude, with respect to the undoped chain, are found for potassium at the largest concentrations that can be sustained without dissociation. Sample MP2 calculations indicate that the Hartree–Fock structure–property relations are preserved. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction Thirty years have now passed since the discovery that the conductivity of p-conjugated polymers such as polyacetylene (PA) can be increased to metallic levels [1] by doping with electron donors or acceptors. Much has been determined since that time regarding the conductivity mechanism and other properties of these remarkable systems [2]. Yet a good deal more remains to be learned. One area of special interest is the nonlinear optical (NLO) response of conducting polymers, particularly the cubic response which is governed by the second hyperpolarizability c at the microscopic level. There is reason to believe that a substantial enhancement of c with respect to the undoped polymer might be achievable for a suitable system and doping strategy. This is of potential importance in the current widespread search for materials with large NLO properties that could be used in optoelectronic and pho*
Corresponding author. E-mail addresses: [email protected], orgchm.bas.bg (M. Spassova).
mspasova@
0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.06.106
tonic devices. Recent analyses suggest that molecular c values measured thus far fall well below fundamental physical limits [3]. The earliest theoretical simulations for the hyperpolarizability of prototypical doped PA implicitly assumed that each dopant molecule transferred a single electron to or from the polymer chain, to form a charged soliton, and that the presence of the resulting charged dopant counterion [4–6] could be ignored. Under such circumstances a potentially large enhancement of c was, then, demonstrated on the basis of semiempirical calculations. Since that time several groups [7–14] have carried out ab initio treatments which address the situation in more detail. Yet, until quite recently, most studies considered only the bare charged PA chain. In order to remedy that deficiency we previously undertook a set of ab initio Hartree–Fock calculations using an explicit alkali atom (Li, Na, K) as the dopant [14]. This work included vibrational as well as electronic hyperpolarizabilities and examined how the degree of electron charge transfer, along with the size of the alkali atom, determines the effect of electrostatic pinning by the counterion for PA chains of varying length. Since each chain was doped
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M. Spassova et al. / Chemical Physics Letters 412 (2005) 217–222
with a single alkali atom one key aspect not investigated is the role of dopant concentration, which is the subject of this Letter. For exploratory purposes we focused on just the electronic contribution using the same dopant species that we employed previously. As it happens the results obtained turn out to be quite interesting.
2. Computational and theoretical aspects The static electronic longitudinal second hyperpolarizability of alkali-doped (Li, Na, K) PA chains of increasing length was determined for geometries that were fully optimized at the Hartree–Fock/6-31G level of approximation with a tight convergence threshold. In constructing the chains we wish to take into account the fact that each metal atom bonds to three carbon atoms, thereby forming a local C3H3M moiety [14,15]. In addition, it is desirable to have a uniform distribution of counterions, which is consistent with the experimental work of Winokur et al. [16] and Murthy et al. [17] on high concentration doping. Both criteria are satisfied by structures containing 1, 2, 3, . . ., C–C formal double bonds between successive C3H3M units leading to dopant concentrations y = 1/5, 1/7, 1/9, . . . Although there are other possibilities we chose to use the chains H–[C3H3M–C2H2]N–C3H3M–H, H–[C3H3M–C4H4]N– C3H3M–H, and H–[C3H3M–C6H6]N–C3H3M–H, where N is a positive integer, for the three concentrations explicitly listed. Two different configurations were considered. In the first, the alkali atoms alternate above and below the plane of the PA backbone (Fig. 1) whereas, in the second, they are all on the same side. When they are all on the same side the plane of the chain bends due to soliton formation just as it does for a single defect [14]. On the other hand, for doping on opposite sides, the bend becomes a gentle undulation and the plane containing the entire set of M atoms makes a dihedral angle of about 60° with respect to the plane of the PA backbone. This is similar to what Baughman et al. [18] found for the X-ray structure of Na-doped PA and all numerical results reported here pertain to the latter configuration. As a basis for interpretation, charge distributions were obtained by means of a Mulliken Population analysis. Although different charge analysis methods will lead to different results it has been shown that, with the exception of atoms in molecules (AIM) and generalized atomic polar tensor (GAPT), the Hartree–Fock derived charges are similar for all commonly used definitions [19]. Static electronic second hyperpolarizabilities were evaluated at the Coupled-Perturbed Hartree–Fock (CPHF) level using the same 6-31G basis set employed in the equilibrium geometry calculations. As in [14] we focus on the longitudinal (see below) second hyperpolar-
Fig. 1. Sketch of doped PA chains. (a) Dopant atoms alternate above and below the plane of the PA backbone; (b) all alkali atoms are located on the same side of that plane.
izability, which becomes the dominant tensor component when the chain is sufficiently lengthened. From now on that component is denoted simply by c. The longitudinal direction is defined for all chains by the line passing through the midpoint of the first and last C@C bond. We calculate c by applying the finite field (FF) procedure [20] to the field-dependent CPHF static electronic longitudinal polarizability and/or first hyperpolarizability [21]. Our procedure employs a Romberg fit, which removes higher-order contaminants and enables us to reach an accuracy of 0.1% or better for c (see [14] and references cited therein). All calculations were performed using the GAUSSIAN-98 [22] program.
3. Results and discussion It is important to note that equilibrium structures could not be obtained for the larger alkali atoms at
M. Spassova et al. / Chemical Physics Letters 412 (2005) 217–222
the highest dopant concentrations due to the formation of M2 molecules that dissociate from the chain. The missing cases are (NC is the number of carbon atoms): (i) y = 1/5, K (all NC); (ii) y = 1/5, Na (NC P 38); and (iii) y = 1/7, K (NC P 45). We find that uniform doping leads to small calculated changes in geometry with respect to chains bearing a single dopant atom. For example, the converged C–Li distance for the bond at the ˚ at y = 0 center of the soliton defect varies from 2.13 A ˚ at (single dopant atom per chain) [14], to 2.120 A ˚ y = 1/9, and 2.101 A at y = 1/5. Slightly larger changes ˚ , y = 0; 2.477 A ˚ , y = 1/9) and K are found for Na (2.49 A ˚ ˚ (2.95 A, y = 0; 2.907 A, y = 1/9). The important bond length alternation (BLA) parameter is shown for representative chains in Fig. 2. Since the solitons are very well localized (at points where BLA = 0.0) the maximum BLA is reflective of a bare neutral PA chain. Hence, ˚ ) regardless the value is essentially the same (0.11–0.12 A of the dopant or its concentration or position along the chain. In fact, these variables make very little difference in the entire plot of BLA vs. position except at the chain ends. The charge transfer associated with uniform doping behaves quite differently from the geometry. First of
219
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Fig. 3. Average HF/6-31G Mulliken charge on alkali atoms for different doping levels and chain lengths.
all, Fig. 3 shows that the average amount of charge transferred per alkali atom increases markedly with the size of the alkali atom. This is in contrast to the usual models which assume complete charge transfer in all cases [23]. On the other hand, there is little dependence on the dopant level (except for concentrations
˚ ) along a doped PA chain as determined at the HF/6-31G level. (a) 93 C atom chains at dopant level y = 1/9; (b) 87 C atom chains at Fig. 2. BLA (A Li, Na dopant level y = 1/7; (c) 48 C atom chain at Li dopant level y = 1/5.
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close to the stability limit). We determine the average charge per CH unit as the product of the value in Fig. 3 and the number of metal atoms, divided by the number of CH units. The result is essentially independent of chain length. In the case of Li-doped chains the CH charges are 0.105, 0.079, and 0.063 |e| for y = 1/5, 1/7, and 1/9, respectively. For Na-doping, the values are 0.120 |e| (y = 1/5), 0.100 |e| (y = 1/7), and 0.080 |e| (y = 1/9) whereas for K-doping they are 0.108 |e| (y = 1/7) and 0.095 |e| (y = 1/9). Thus, the average charge per CH unit increases significantly with dopant level. There appears to be a limiting value, equal to a little bit more than 0.12 |e| per CH unit, above which the doped chains become unstable at all lengths with respect to dissociation of alkali diatomic molecules. The charge added to the chain is not uniformly distributed, although our Hartree–Fock values, shown in Fig. 4 for y = 1/9, may somewhat overemphasize the oscillations [19,24]. Qualitatively similar results were obtained at the other dopant levels. We see that the maximum negative charge is on the CH units adjacent to the defect center. On the other hand, once the defect is created the increase in negative charge due to replacing Li with Na or K is distributed almost uniformly along the chain except at the ends. Now we turn to the second hyperpolarizability which, like charge transfer, is quite sensitive to the choice of dopant. For Li-doping there is a small increase in c/NC with respect to undoped PA (10%) regardless of doping level and chain length. In the long chain limit the largest enhancement is attained for the lowest doping level considered here. Fig. 5a shows a comparison between Li, Na, and K for y = 1/9. There is evidently a substantial increase in c due to replacing Li by Na and, particularly, by K. For the longest chains (93 carbon atoms) the values of c are in the ratio 1.00:1.99:4.83. Furthermore, at concentrations in the range 1/5 > y > 1/9 the value of c increases for Na and
K (as opposed to Li) as the doping level increases. This is seen in Fig. 5b where we have plotted the overall enhancement factor with respect to undoped PA [25] as a function of the number of carbon atoms (to obtain the proper number of carbon atoms cubic spline fits were used for undoped PA where necessary). A rather large maximum theoretical enhancement factor of 60 occurs for K-doped chains at a length of 24 carbon atoms and doping level y = 1/7. In general, this factor diminishes with chain length until reaching its asymptotic value. In the case of K-doping at a level y = 1/9, for example, it is approximately halved in going from 24 carbons to an (extrapolated) infinite chain value of 5.3. Since the charge per CH unit increases with dopant level it appears that the best enhancement strategy, besides optimizing the chain length, is to maximize the charge transferred to the chain and minimize the electrostatic interaction between dopant and chain. The latter might be accomplished, for instance, by using a dopant which has a very delocalized excess charge.
.
.
.
.
.
.
Fig. 4. HF/6-31G charge distribution along PA chains for y = 1/9 and different alkali atoms.
Fig. 5. (a) CPHF/6-31G longitudinal c per C atom for different dopants at y = 1/9, (b) enhancement factor for CPHF/6-31G c of Naand K-doped vs. undoped PA chains as a function of chain length.
M. Spassova et al. / Chemical Physics Letters 412 (2005) 217–222
Since electron correlation effects have been found to be substantial for the c of undoped PA chains [26] finite field MP2/6-31G calculations were performed on the smallest systems in order confirm the structure–property relationships deduced from our CPHF results. In either case the same geometry was used. For y = 1/7, the MP2/ 6-31G second hyperpolarizabilities of 24 C atom chains doped with Li, Na, and K are in the ratio 1.00:6.64:84.5, respectively, whereas the corresponding CPHF ratios are 1.00:4.24:43.6. Similarly, for y = 1/9 and chains of 30 C atoms, the MP2 values are in the ratio 1.00:3.16:14.2 compared to 1.00:2.38:7.31 at the CPHF level. These results show that the general CPHF structure–property relationships for varying dopant and dopant level are preserved when electron correlation is taken into account while the difference between dopants is magnified. Although basis set effects on the second hyperpolarizabilities of PA chains – as well as other conjugated oligomers – have several times been shown to become negligible when the chains grow [21,26,27], the presence of counterions could alter this situation. We, therefore, perform calculations using the larger basis sets 6-31G*, 6-311G*, 6-31+G, and 6-31+G*. Due to the computational requirements, these calculations were limited to small systems. Before summarizing our results it is important to note that larger basis sets do not guarantee more accurate hyperpolarizabilities [28]. Bearing that in mind, we find that increasing the size of the basis set beyond 6-31G leads to an increase in c for Li-doped chains, but a decrease for Naand K-doping. The effect is always less than about 20% in the case of Li and Na. For the former it diminishes as the chain is lengthened and the dopant concentration is reduced; for the latter it is almost independent of chain length and dopant concentration. The situation is different for K-doping because using larger basis sets leads to a reduction of c by factors as large as 3 at the higher concentrations, a result which depends little on chain length. Fortunately, the effects of including electron correlation and employing more extended basis sets tend to cancel one another as far as the structure–property relationships are concerned. In order to substantiate this tendency a final set of calculations was carried out at the MP2/631G* level of approximation. For y = 1/7, the MP2/ 6-31G* second hyperpolarizabilities of 17 C atom chains doped with Li, Na, and K are in the ratio 1.00:6.33:48.4, respectively, whereas the corresponding CPHF/6-31G ratios are 1.00:4.86:36.7. Similarly, for y = 1/9 and chains of 21 C atoms, the MP2/6-31G* values are in the ratio 1.00:3.40:9.58 compared to 1.00:2.89:8.97 at the CPHF/6-31G level. Although the CPHF/6-31G ratios may not be quantitative, we conclude that they are sufficient for predicting structure– property relationships.
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We have not yet considered the role of the vibrational contribution to c or of interchain interactions. However, the issue of altering the spatial distribution of dopant atoms so that they are all on the same side of the PA plane, rather than opposite sides, has been addressed. It turns out that this makes little difference. Our results indicate a potentially large effect for one particular system that already has a large second hyperpolarizability. The effect of doping on other systems remains to be explored.
Acknowledgments This study results from a scientific cooperation established and supported by the Bulgarian Academy of Sciences, the Belgian National Fund for Scientific Research (FNRS) and the Commissariat Ge´ne´ral aux Relations Internationales (CGRI) de la Communaute´ franc¸aise Wallonie-Bruxelles. This work has been financially supported by the Bulgarian Fund for Scientific Research under the Project X-827. M.S. thanks the Federal Science Policy for the financial support through IUAP No. 5-03. B.C. thanks the FNRS for his Senior Research Associate position. The calculations have been performed on PCs installed in the Institute of Organic Chemistry of the Bulgarian Academy of Sciences as well as on the Interuniversity Scientific Computing Facility (ISCF), installed at the Faculte´s Universitaires NotreDame de la Paix (Namur, Belgium), for which the authors gratefully acknowledge the financial support of the FNRS-FRFC and the ÔLoterie NationaleÕ for the convention No. 2.4578.02, and of the FUNDP.
References [1] H. Shirikawa, W.J. Louis, A.G. MacDiarmid, C.K. Chiang, A.J. Heeger, J. Chem. Commun. 16 (1977) 578. [2] A.J. Heeger, J. Phys. Chem. B 105 (2001) 8475. [3] M.G. Kuzyk, Phys. Rev. Lett. 85 (2000) 1218; Phys. Rev. Lett. E 90 (2003) 039902. [4] C.P. de Melo, R. Silbey, Chem. Phys. Lett. 140 (1987) 537. [5] C.P. de Melo, R. Silbey, J. Chem. Phys. 88 (1988) 2567. [6] A. Villesuzanne, J. Hoarau, L. Ducasse, L. Olmedo, P. Hourquebie, J. Chem. Phys. 96 (1992) 495. [7] M. Nakano, I. Shigemoto, S. Yamada, K. Yamaguchi, J. Chem. Phys. 103 (1995) 4175. [8] C.P. de Melo, T.L. Fonseca, Chem. Phys. Lett. 261 (1996) 28. ¨ hrn, J. Chem. Phys. 107 (1997) [9] B. Champagne, E. Deumens, Y. O 5433. [10] Z. An, K.Y. Wong, J. Chem. Phys. 114 (2001) 1010. [11] L.N. Oliveira, O.A.V. Amaral, M.A. Castro, T.L. Fonseca, Chem. Phys. 289 (2003) 221. [12] Z. An, K.Y. Wong, J. Chem. Phys. 119 (2003) 1019. [13] M. Nakano, S. Yamada, M. Takahata, K. Yamaguchi, J. Phys. Chem. A 107 (2003) 4157. [14] B. Champagne, M. Spassova, J.B. Jadin, B. Kirtman, J. Chem. Phys. 116 (2002) 3935.
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[15] J.L. Bre´das, R.R. Chance, R. Silbey, J. Phys. Chem. 85 (1981) 756. [16] M. Winokur, Y.B. Moon, A.J. Heeger, J. Barker, D.C. Bott, H. Shirakawa, Phys. Rev. Lett. 58 (1987) 2329. [17] N.S. Murthy, L.W. Shacklette, R.H. Baughman, Phys. Rev. B 40 (1989) 12550. [18] R.H. Baughman, N.S. Murthy, G.G. Miller, J. Chem. Phys. 79 (1983) 515. [19] V. Monev, M. Spassova, B. Champagne, Int. J. Quantum Chem. 104 (2005) 354. [20] H.D. Cohen, C.C.J. Roothaan, J. Chem. Phys. 43 (1965) S34. [21] G.J.B. Hurst, M. Dupuis, E. Clementi, J. Chem. Phys. 89 (1988) 385. [22] M.J. Frisch et al., GAUSSIAN-98, Revision A.11, Gaussian, Inc., Pittsburgh, PA, 1998.
[23] G. Sun, J. Ku¨rti, M. Kertesz, R. Baughman, J. Chem. Phys. 117 (2002) 7691. [24] B. Champagne, M. Spassova, Phys. Chem. Chem. Phys. 6 (2004) 3167. [25] B. Kirtman, J.L. Toto, K.A. Robins, M. Hasan, J. Chem. Phys. 102 (1995) 5350. [26] J.L. Toto, T.T. Toto, C.P. de Melo, M. Hasan, B. Kirtman, Chem. Phys. Lett. 244 (1995) 59. [27] B. Champagne, B. Kirtman, in: H.S. Nalwa (Ed.), Handbook of Advanced Electronic and Photonic Materials and Devices, Nonlinear Optical Materials, vol. 9, Academic Press, San Diego, 2001, p. 63 (Chapter 2). [28] See, for example M. Torrent-Sucarrat, M. Sola, M. Duran, J.M. Luis, B. Kirtman, J. Chem. Phys. 120 (2004) 6346.
Large effect of dopant level on second hyperpolarizability of alkali-doped polyacetylene chains Milena Spassova
a,b,*
, Benoıˆt Champagne b, Bernard Kirtman
c
a
b
Institute of Organic Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Laboratoire de Chimie The´orique Applique´e, Faculte´s Universitaires Notre-Dame de la Paix, rue de Bruxelles, 61, B-5000 Namur, Belgium c Department of Chemistry and Biochemistry, University of California, Santa Barbara, CA 93106, USA Received 11 May 2005; in final form 21 June 2005 Available online 20 July 2005
Abstract Hartree–Fock calculations are reported for the effect of donor dopant concentration on the static longitudinal second hyperpolarizability of individual polyacetylene chains using explicit alkali (Li, Na, K) atoms and varying chain lengths. Enhancement factors approaching two orders of magnitude, with respect to the undoped chain, are found for potassium at the largest concentrations that can be sustained without dissociation. Sample MP2 calculations indicate that the Hartree–Fock structure–property relations are preserved. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction Thirty years have now passed since the discovery that the conductivity of p-conjugated polymers such as polyacetylene (PA) can be increased to metallic levels [1] by doping with electron donors or acceptors. Much has been determined since that time regarding the conductivity mechanism and other properties of these remarkable systems [2]. Yet a good deal more remains to be learned. One area of special interest is the nonlinear optical (NLO) response of conducting polymers, particularly the cubic response which is governed by the second hyperpolarizability c at the microscopic level. There is reason to believe that a substantial enhancement of c with respect to the undoped polymer might be achievable for a suitable system and doping strategy. This is of potential importance in the current widespread search for materials with large NLO properties that could be used in optoelectronic and pho*
Corresponding author. E-mail addresses: [email protected], orgchm.bas.bg (M. Spassova).
mspasova@
0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.06.106
tonic devices. Recent analyses suggest that molecular c values measured thus far fall well below fundamental physical limits [3]. The earliest theoretical simulations for the hyperpolarizability of prototypical doped PA implicitly assumed that each dopant molecule transferred a single electron to or from the polymer chain, to form a charged soliton, and that the presence of the resulting charged dopant counterion [4–6] could be ignored. Under such circumstances a potentially large enhancement of c was, then, demonstrated on the basis of semiempirical calculations. Since that time several groups [7–14] have carried out ab initio treatments which address the situation in more detail. Yet, until quite recently, most studies considered only the bare charged PA chain. In order to remedy that deficiency we previously undertook a set of ab initio Hartree–Fock calculations using an explicit alkali atom (Li, Na, K) as the dopant [14]. This work included vibrational as well as electronic hyperpolarizabilities and examined how the degree of electron charge transfer, along with the size of the alkali atom, determines the effect of electrostatic pinning by the counterion for PA chains of varying length. Since each chain was doped
218
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with a single alkali atom one key aspect not investigated is the role of dopant concentration, which is the subject of this Letter. For exploratory purposes we focused on just the electronic contribution using the same dopant species that we employed previously. As it happens the results obtained turn out to be quite interesting.
2. Computational and theoretical aspects The static electronic longitudinal second hyperpolarizability of alkali-doped (Li, Na, K) PA chains of increasing length was determined for geometries that were fully optimized at the Hartree–Fock/6-31G level of approximation with a tight convergence threshold. In constructing the chains we wish to take into account the fact that each metal atom bonds to three carbon atoms, thereby forming a local C3H3M moiety [14,15]. In addition, it is desirable to have a uniform distribution of counterions, which is consistent with the experimental work of Winokur et al. [16] and Murthy et al. [17] on high concentration doping. Both criteria are satisfied by structures containing 1, 2, 3, . . ., C–C formal double bonds between successive C3H3M units leading to dopant concentrations y = 1/5, 1/7, 1/9, . . . Although there are other possibilities we chose to use the chains H–[C3H3M–C2H2]N–C3H3M–H, H–[C3H3M–C4H4]N– C3H3M–H, and H–[C3H3M–C6H6]N–C3H3M–H, where N is a positive integer, for the three concentrations explicitly listed. Two different configurations were considered. In the first, the alkali atoms alternate above and below the plane of the PA backbone (Fig. 1) whereas, in the second, they are all on the same side. When they are all on the same side the plane of the chain bends due to soliton formation just as it does for a single defect [14]. On the other hand, for doping on opposite sides, the bend becomes a gentle undulation and the plane containing the entire set of M atoms makes a dihedral angle of about 60° with respect to the plane of the PA backbone. This is similar to what Baughman et al. [18] found for the X-ray structure of Na-doped PA and all numerical results reported here pertain to the latter configuration. As a basis for interpretation, charge distributions were obtained by means of a Mulliken Population analysis. Although different charge analysis methods will lead to different results it has been shown that, with the exception of atoms in molecules (AIM) and generalized atomic polar tensor (GAPT), the Hartree–Fock derived charges are similar for all commonly used definitions [19]. Static electronic second hyperpolarizabilities were evaluated at the Coupled-Perturbed Hartree–Fock (CPHF) level using the same 6-31G basis set employed in the equilibrium geometry calculations. As in [14] we focus on the longitudinal (see below) second hyperpolar-
Fig. 1. Sketch of doped PA chains. (a) Dopant atoms alternate above and below the plane of the PA backbone; (b) all alkali atoms are located on the same side of that plane.
izability, which becomes the dominant tensor component when the chain is sufficiently lengthened. From now on that component is denoted simply by c. The longitudinal direction is defined for all chains by the line passing through the midpoint of the first and last C@C bond. We calculate c by applying the finite field (FF) procedure [20] to the field-dependent CPHF static electronic longitudinal polarizability and/or first hyperpolarizability [21]. Our procedure employs a Romberg fit, which removes higher-order contaminants and enables us to reach an accuracy of 0.1% or better for c (see [14] and references cited therein). All calculations were performed using the GAUSSIAN-98 [22] program.
3. Results and discussion It is important to note that equilibrium structures could not be obtained for the larger alkali atoms at
M. Spassova et al. / Chemical Physics Letters 412 (2005) 217–222
the highest dopant concentrations due to the formation of M2 molecules that dissociate from the chain. The missing cases are (NC is the number of carbon atoms): (i) y = 1/5, K (all NC); (ii) y = 1/5, Na (NC P 38); and (iii) y = 1/7, K (NC P 45). We find that uniform doping leads to small calculated changes in geometry with respect to chains bearing a single dopant atom. For example, the converged C–Li distance for the bond at the ˚ at y = 0 center of the soliton defect varies from 2.13 A ˚ at (single dopant atom per chain) [14], to 2.120 A ˚ y = 1/9, and 2.101 A at y = 1/5. Slightly larger changes ˚ , y = 0; 2.477 A ˚ , y = 1/9) and K are found for Na (2.49 A ˚ ˚ (2.95 A, y = 0; 2.907 A, y = 1/9). The important bond length alternation (BLA) parameter is shown for representative chains in Fig. 2. Since the solitons are very well localized (at points where BLA = 0.0) the maximum BLA is reflective of a bare neutral PA chain. Hence, ˚ ) regardless the value is essentially the same (0.11–0.12 A of the dopant or its concentration or position along the chain. In fact, these variables make very little difference in the entire plot of BLA vs. position except at the chain ends. The charge transfer associated with uniform doping behaves quite differently from the geometry. First of
219
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.
.
.
.
.
.
.
Fig. 3. Average HF/6-31G Mulliken charge on alkali atoms for different doping levels and chain lengths.
all, Fig. 3 shows that the average amount of charge transferred per alkali atom increases markedly with the size of the alkali atom. This is in contrast to the usual models which assume complete charge transfer in all cases [23]. On the other hand, there is little dependence on the dopant level (except for concentrations
˚ ) along a doped PA chain as determined at the HF/6-31G level. (a) 93 C atom chains at dopant level y = 1/9; (b) 87 C atom chains at Fig. 2. BLA (A Li, Na dopant level y = 1/7; (c) 48 C atom chain at Li dopant level y = 1/5.
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close to the stability limit). We determine the average charge per CH unit as the product of the value in Fig. 3 and the number of metal atoms, divided by the number of CH units. The result is essentially independent of chain length. In the case of Li-doped chains the CH charges are 0.105, 0.079, and 0.063 |e| for y = 1/5, 1/7, and 1/9, respectively. For Na-doping, the values are 0.120 |e| (y = 1/5), 0.100 |e| (y = 1/7), and 0.080 |e| (y = 1/9) whereas for K-doping they are 0.108 |e| (y = 1/7) and 0.095 |e| (y = 1/9). Thus, the average charge per CH unit increases significantly with dopant level. There appears to be a limiting value, equal to a little bit more than 0.12 |e| per CH unit, above which the doped chains become unstable at all lengths with respect to dissociation of alkali diatomic molecules. The charge added to the chain is not uniformly distributed, although our Hartree–Fock values, shown in Fig. 4 for y = 1/9, may somewhat overemphasize the oscillations [19,24]. Qualitatively similar results were obtained at the other dopant levels. We see that the maximum negative charge is on the CH units adjacent to the defect center. On the other hand, once the defect is created the increase in negative charge due to replacing Li with Na or K is distributed almost uniformly along the chain except at the ends. Now we turn to the second hyperpolarizability which, like charge transfer, is quite sensitive to the choice of dopant. For Li-doping there is a small increase in c/NC with respect to undoped PA (10%) regardless of doping level and chain length. In the long chain limit the largest enhancement is attained for the lowest doping level considered here. Fig. 5a shows a comparison between Li, Na, and K for y = 1/9. There is evidently a substantial increase in c due to replacing Li by Na and, particularly, by K. For the longest chains (93 carbon atoms) the values of c are in the ratio 1.00:1.99:4.83. Furthermore, at concentrations in the range 1/5 > y > 1/9 the value of c increases for Na and
K (as opposed to Li) as the doping level increases. This is seen in Fig. 5b where we have plotted the overall enhancement factor with respect to undoped PA [25] as a function of the number of carbon atoms (to obtain the proper number of carbon atoms cubic spline fits were used for undoped PA where necessary). A rather large maximum theoretical enhancement factor of 60 occurs for K-doped chains at a length of 24 carbon atoms and doping level y = 1/7. In general, this factor diminishes with chain length until reaching its asymptotic value. In the case of K-doping at a level y = 1/9, for example, it is approximately halved in going from 24 carbons to an (extrapolated) infinite chain value of 5.3. Since the charge per CH unit increases with dopant level it appears that the best enhancement strategy, besides optimizing the chain length, is to maximize the charge transferred to the chain and minimize the electrostatic interaction between dopant and chain. The latter might be accomplished, for instance, by using a dopant which has a very delocalized excess charge.
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Fig. 4. HF/6-31G charge distribution along PA chains for y = 1/9 and different alkali atoms.
Fig. 5. (a) CPHF/6-31G longitudinal c per C atom for different dopants at y = 1/9, (b) enhancement factor for CPHF/6-31G c of Naand K-doped vs. undoped PA chains as a function of chain length.
M. Spassova et al. / Chemical Physics Letters 412 (2005) 217–222
Since electron correlation effects have been found to be substantial for the c of undoped PA chains [26] finite field MP2/6-31G calculations were performed on the smallest systems in order confirm the structure–property relationships deduced from our CPHF results. In either case the same geometry was used. For y = 1/7, the MP2/ 6-31G second hyperpolarizabilities of 24 C atom chains doped with Li, Na, and K are in the ratio 1.00:6.64:84.5, respectively, whereas the corresponding CPHF ratios are 1.00:4.24:43.6. Similarly, for y = 1/9 and chains of 30 C atoms, the MP2 values are in the ratio 1.00:3.16:14.2 compared to 1.00:2.38:7.31 at the CPHF level. These results show that the general CPHF structure–property relationships for varying dopant and dopant level are preserved when electron correlation is taken into account while the difference between dopants is magnified. Although basis set effects on the second hyperpolarizabilities of PA chains – as well as other conjugated oligomers – have several times been shown to become negligible when the chains grow [21,26,27], the presence of counterions could alter this situation. We, therefore, perform calculations using the larger basis sets 6-31G*, 6-311G*, 6-31+G, and 6-31+G*. Due to the computational requirements, these calculations were limited to small systems. Before summarizing our results it is important to note that larger basis sets do not guarantee more accurate hyperpolarizabilities [28]. Bearing that in mind, we find that increasing the size of the basis set beyond 6-31G leads to an increase in c for Li-doped chains, but a decrease for Naand K-doping. The effect is always less than about 20% in the case of Li and Na. For the former it diminishes as the chain is lengthened and the dopant concentration is reduced; for the latter it is almost independent of chain length and dopant concentration. The situation is different for K-doping because using larger basis sets leads to a reduction of c by factors as large as 3 at the higher concentrations, a result which depends little on chain length. Fortunately, the effects of including electron correlation and employing more extended basis sets tend to cancel one another as far as the structure–property relationships are concerned. In order to substantiate this tendency a final set of calculations was carried out at the MP2/631G* level of approximation. For y = 1/7, the MP2/ 6-31G* second hyperpolarizabilities of 17 C atom chains doped with Li, Na, and K are in the ratio 1.00:6.33:48.4, respectively, whereas the corresponding CPHF/6-31G ratios are 1.00:4.86:36.7. Similarly, for y = 1/9 and chains of 21 C atoms, the MP2/6-31G* values are in the ratio 1.00:3.40:9.58 compared to 1.00:2.89:8.97 at the CPHF/6-31G level. Although the CPHF/6-31G ratios may not be quantitative, we conclude that they are sufficient for predicting structure– property relationships.
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We have not yet considered the role of the vibrational contribution to c or of interchain interactions. However, the issue of altering the spatial distribution of dopant atoms so that they are all on the same side of the PA plane, rather than opposite sides, has been addressed. It turns out that this makes little difference. Our results indicate a potentially large effect for one particular system that already has a large second hyperpolarizability. The effect of doping on other systems remains to be explored.
Acknowledgments This study results from a scientific cooperation established and supported by the Bulgarian Academy of Sciences, the Belgian National Fund for Scientific Research (FNRS) and the Commissariat Ge´ne´ral aux Relations Internationales (CGRI) de la Communaute´ franc¸aise Wallonie-Bruxelles. This work has been financially supported by the Bulgarian Fund for Scientific Research under the Project X-827. M.S. thanks the Federal Science Policy for the financial support through IUAP No. 5-03. B.C. thanks the FNRS for his Senior Research Associate position. The calculations have been performed on PCs installed in the Institute of Organic Chemistry of the Bulgarian Academy of Sciences as well as on the Interuniversity Scientific Computing Facility (ISCF), installed at the Faculte´s Universitaires NotreDame de la Paix (Namur, Belgium), for which the authors gratefully acknowledge the financial support of the FNRS-FRFC and the ÔLoterie NationaleÕ for the convention No. 2.4578.02, and of the FUNDP.
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