Vibronic Interactions in Negatively Charged Polyacetylene - American

Vibronic Interactions in Negatively Charged Polyacetylene. Takashi Kato* and Tokio Yamabe. Institute for InnoVatiVe Science and Technology, Graduate S...
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J. Phys. Chem. B 2005, 109, 10620-10630

Vibronic Interactions in Negatively Charged Polyacetylene Takashi Kato* and Tokio Yamabe Institute for InnoVatiVe Science and Technology, Graduate School of Engineering, Nagasaki Institute of Applied Science, 3-1, Shuku-machi, Nagasaki 851-0121, Japan ReceiVed: October 25, 2004

Electron-phonon interactions in the monoanions of polyacetylenes such as C2H4 (2tpa), C4H6 (4tpa), C6H8 (6tpa), and C8H10 (8tpa) are studied and compared with those in the monoanions of polyacenes. The C-C stretching Ag modes around 1500 cm-1 the most strongly couple to the lowest unoccupied molecular orbitals (LUMO) in polyacetylenes. The estimated total electron-phonon coupling constants for the monoanions (lLUMO) are 0.579, 0.555, 0.463, and 0.401 eV for 2tpa, 4tpa, 6tpa, and 8tpa, respectively. The lLUMO values for polyacetylenes are much larger than those for polyacenes. Furthermore, the lLUMO value for polyacetylene with C2h geometry is estimated to be 0.254 eV, and is larger than that (0.024 eV) for polyacene with D2h geometry. The phase patterns difference between the LUMO of polyacenes localized on the edge part of carbon atoms, and the delocalized LUMO of polyacetylenes is the main reason for the calculated results. The single charge transfer through the molecule in polyacetylenes are also discussed. The reorganization energies between the neutral molecule and the corresponding monoanion are estimated to be 0.164, 0.144, 0.125, and 0.113 eV for 2tpa, 4tpa, 6tpa, and 8tpa, respectively. Such reorganization energy decreases with an increase in molecular size. The conditions under which the attractive electron-electron interactions are realized in the monoanions of polyacetylenes and polyacenes are discussed. In terms of the electron-phonon interactions and the reorganization energies, the relationships between the normal and possible superconducting states are briefly discussed. We find that the monoanions with smaller molecular size cannot easily become good conductors, however, the conditions under which the interactions between two electrons are attractive are more easily realized in the monoanions with smaller molecular size than in the monoanions with larger molecular size.

Introduction interaction1-3

Analysis of vibronic is important for the prediction of electronic control of nuclear motions in degenerate electronic systems. Application of vibronic interaction theory covers a large variety of research fields such as spectroscopy,4 instability of molecular structure, electrical conductivity,5 and superconductivity.5,6 Vibronic interactions in discrete molecules can be viewed as the coupling between frontier orbitals and molecular vibrations, while those in solids are the coupling between free electrons near the Fermi level and acoustic phonons. There is a close analogy between them. In modern physics and chemistry, the effect of vibronic interaction1 in molecules and crystals has been an important topic. Electron-phonon coupling1-3 is the consensus mechanism for attractive electron-electron interactions in the BardeenCooper-Schrieffer (BCS) theory of superconductivity.5,6 Since Little’s proposal for a possible molecular superconductor based on exciton mechanism,7 the superconductivity of molecular systems has been extensively investigated. Although such a unique mechanism has not yet been established, advances in design and synthesis of molecular systems have yielded a lot of BEDT-TTF-type organic superconductors,8,9 where BEDTTTF is bis(ethylenedithio)tetrathiafulvalene. An inverse isotope effect due to substituting hydrogen by deuterium in organic superconductivity was observed by Saito et al.10 Goddard et al. proposed that the mechanism for superconductivity of BEDT* To whom correspondence should be addressed: E-mail: [email protected]. Telephone: +81-95-838-4363. Fax: +81-95-838-5105.

TTF-type organic molecules involves the coupling of charge transfer to the boat deformation mode.11 It was found that the alkali-doped A3C60 complexes12 exhibit superconducting transition temperatures (Tcs) of more than 3013 and 40 K under pressure.14 In superconductivity in alkali-doped fullerenes,15 pure intramolecular Raman-active modes have been suggested to be important in a BCS-type6 strong coupling scenario. For a long time, the possibility of superconductivity in polymers has been an open problem.7 It is of renewed urgency due to the recognition of important similarities between conducting polymers and high-Tc and organic superconductors. Even ceramic or powder samples of the latter materials are good superconductors while it is demonstrated by observation of superconductivity in the inorganic polymer (SN)x that the typical fibrillar morphology of polymers is not detrimental to superconductivity.16 The crossover between density waves and superconductivity in the one-dimensional (1D) electron gas has been studied.17-19 Starting from standard 1D models of conducting polymers, Voit discussed20 under what conditions the competition or cooperation of electron-electron and electronphonon interactions can bring about singlet superconductivity. They suggested that the dimerized charge-density-wave (CDW) ground state of the Su-Schrieffer-Heeger (SSH) model21 is stable against electron-electron interactions and phonon quantum fluctuations for any band filling. On the other hand, they also suggested that dominant superconducting state fluctuations are found for a Holstein on-site electron-phonon coupling22 describing the coupling to the vibrational degrees of freedom of a monomer. It is believed that electron-electron interactions

10.1021/jp0406823 CCC: $30.25 © 2005 American Chemical Society Published on Web 05/05/2005

Negatively Charged Polyacetylene SCHEME 1

J. Phys. Chem. B, Vol. 109, No. 21, 2005 10621 TABLE 1: C-C Distances in the Neutral Polyacetylenes 2tpa 4tpa 6tpa 8tpa

in polymers like (CH)x, are weak (though not negligible) and well screened.23 The search for new organic metals and superconductors has attracted a great deal of attention in synthetic chemistry and material science since the discovery of high electrical conductivity in conjugated polymers such as polyacetylene.24 Many theoretical studies9,24-26 have been carried out in order to understand the mechanism of conductivity and superconductivity in the conjugated polymers and related materials and to reveal the interconnection between chemical and electronic structures. The role of the vibronic interactions in the normal and superconducting states of conjugated polymers was qualitatively indicated by one of the authors.26 However, the continuity of properties of vibronic interactions and the electron transfer, from molecule to crystal, depending on the size of the system, has not necessarily been clear. In previous work, we have analyzed the vibronic interactions and estimated possible Tcs in the monocations of acenes based on the hypothesis that the vibronic interactions between the intramolecular vibrations and the highest occupied molecular orbitals (HOMO) play an essential role in the occurrence of possible superconductivity in positively charged nanosized molecular systems.27 Recently, based on an experimental study of ionization spectra using high-resolution gas-phase photoelectron spectroscopy, the electron-phonon interactions in the positively charged acenes were well studied.28 The experimental results showed that our predicted frequencies for the vibrational modes which play an essential role in the electron-phonon interactions27 as well as the predicted total electron-phonon coupling constants27 are in excellent agreement with those obtained from the experimental research.28 The purpose of this paper is to discuss the electron-phonon interactions in the monoanions of trans-polyacetylenes such as C2H4 (2tpa), C4H6 (4tpa), C6H8 (6tpa), and C8H10 (8tpa) (Scheme 1). We compare these calculated results with those for the monoanions of acenes27 in order to investigate how the properties of the electron-phonon interactions are closely related to the molecular sizes and structures from the point of view of the molecular orbitals. We will also discuss the single charge transfer through molecule, and estimate the reorganization energy for elementary charge transfer, and will discuss the vibration effect onto the charge-transfer problem. In terms of the electron-phonon interactions and the reorganization energies, we briefly discuss the relationships between the normal

d1

d2

d3

d4

1.330 1.341 1.343 1.344

1.458 1.450 1.448

1.352 1.355

1.440

and possible superconductivity states. We will discuss the conditions under which the attractive electron-electron interactions are realized in the monoanions of polyacetylenes and polyacenes. Furthermore, we will discuss how the conditions under which the monoanions crystals become good conductor are related to the molecular sizes and structures. There is an interesting paradox in conventional superconductivity; the higher resistivity at room temperature, the more likely it is that a metal will be a superconductor when cooled.5a However, this paradox has not been explained in detail from the point of view of physical chemistry, as far as we know. We will explain this paradox in molecular systems in detail in terms of the vibronic interactions and the reorganization energies. Finally, we will suggest the possible electron pairing and Bose-Einstein condensation in the negatively charged polyacetylenes. Theoretical Background We discuss a theoretical background for the orbital vibronic coupling constants1(a) in 2tpa with D2h geometry and 4tpa, 6tpa, and 8tpa with C2h geometry. The dimensionless orbital vibronic coupling constant of the mth Ag mode in polyacetylenes is defined by

gLUMO(ωm) )

〈

1 LUMO pωm

||(

) || 〉

∂hAgm

∂qAgm

LUMO

(1)

0

where qAgm is the dimensionless normal coordinate29 of the mth vibrational mode and hAgm is the vibronic coupling matrix of the mth Ag mode of polyacetylenes. Assuming that the conduction band of the monoanion crystals of polyacetylenes consists of the LUMOs because polyacetylenes would consist of strongly bonded molecules arranged on a lattice with weak van der Waals intermolecular bonds, electron-phonon coupling constants of the mth Ag modes in polyacetylenes are defined as follows:

lLUMO(ωm) ) g2LUMO(ωm)pωm

(2)

Optimized Structures of Polyacetylenes The structure of neutral 2tpa is optimized under D2h symmetry, and those of neutral 4tpa, 6tpa, and 8tpa are optimized under C2h symmetry, using the hybrid Hartree-Fock/densityfunctional-theory method of Becke30 and Lee, Yang, and Parr31 (B3LYP) and the 6-31G* basis set.32 The GAUSSIAN 98 program package33 was used for our theoretical analyses. Optimized structures of these molecules are listed in Table 1. Each structure was confirmed to be a minimum on each energy surface. According to our calculations, the energy differences between the HOMO and the LUMO of 2tpa, 4tpa, 6tpa, and 8tpa are 7.77, 5.62, 4.49, and 3.79 eV, respectively, and those of benzene C6H6 (1a), naphthalene C10H8 (2a), anthracene C14H10 (3a), tetracene C18H12 (4a), and pentacene C22H14 (5a) are 6.80, 4.83, 3.59, 2.78, and 2.21 eV, respectively. Therefore, the HOMO-LUMO gaps in polyacetylenes are smaller than those in the same size of polyacenes. Let us look into optimized structures of polyacetylenes. The C-C distance observed from experimental work in 2tpa are reported to be 1.34 Å.34 Therefore, our estimated C-C bond

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Figure 1. Phase patterns of the frontier orbitals in polyacetylenes.

length is in agreement with those observed from experimental research. We can see from Table 1 that there is a distinct variation (bond alternation) in the C-C distances in polyacetylenes. This result is reasonable in view of the orbital patterns of the HOMOs. The orbital patterns of the HOMO and the LUMO in polyacetylenes are shown in Figure 1. The C-C distances between two neighboring carbon atoms whose atomic orbitals are combined in phase (out of phase) in the HOMOs of polyacetylenes are short (long). But it should be noted that the long (short) C-C bond lengths become slightly shorter (longer) with an increase in molecular size in polyacetylenes. This can be understood as follows. The orbital coefficients of each carbon atom in the HOMO become smaller with an increase in molecular size, and thus the orbital interactions between two neighboring carbon atoms become weaker with an increase in molecular size in polyacetylenes. This is the reason the long (short) C-C bond lengths become slightly shorter (longer) with an increase in molecular size in polyacetylenes. Vibronic Interactions in the Monoanions of Polyacetylenes We carried out vibrational analyses of polyacetylene at the B3LYP/6-31G* level of theory. We next calculated first-order derivatives at this equilibrium structure on each energy surface by distorting the molecule along the Ag modes of 2tpa, 4tpa, 6tpa, and 8tpa in order to obtain orbital vibronic coupling constants defined by eq 1.1a We can estimate the electronphonon coupling constants from the dimensionless orbital vibronic coupling constants by using eq 2. The calculated electron-phonon coupling constants in the monoanions of 2tpa, 4tpa, 6tpa, and 8tpa are shown in Figure 2. Furthermore, the calculated reduced masses and the electron-phonon coupling

constants for the monoanions of 2tpa and 4tpa are listed in Tables 2 and 3, respectively. The estimated frequencies of the vibronic active modes for the C-H bending, the C-C stretching, and the C-H stretching modes are 1396, 1720, and 3168 cm-1, respectively in 2tpa, while those for the C-C stretching and the C-H stretching modes, observed from experimental research in the gas phase of 2tpa are 1623 and 3026 cm-1, respectively.34 Furthermore, the experimentally reported frequencies of the vibronic active modes for the C-C bending, the C-H bending, the C-C stretching, and the C-H stretching modes are 7001250, 1340-1465, 1620-1680, and 2850-2960 cm-1, respectively, in polyacetylenes.34 Therefore, the estimated frequencies in this work are in agreement with the experimental results. But it should be noted that very low-frequency modes, which are analogous to acoustic modes of phonon in solids, appear with an increase in molecular size in polyacetylenes. Let us first look into the electron-phonon interactions between the Ag modes and the b2g LUMO in 2tpa. We can see from Table 2 that the C-C stretching Ag modes of 1396 and 1720 cm-1 very strongly couple to the b2g LUMO in 2tpa. This can be understood in view of the orbital patterns of the LUMO in 2tpa. The selected vibronic active Ag modes in polyacetylenes are shown in Figure 3. When 2tpa is distorted along the Ag modes of 1396 and 1720 cm-1 toward the same direction as shown in Figure 3, the antibonding interactions between two neighboring carbon atoms become weaker, and thus the b2g LUMO is significantly stabilized in energy by such distortions. This is the reason the C-C stretching Ag modes of 1396 and 1720 cm-1 afford large electron-phonon coupling constants in the monoanion of 2tpa. But it should be noted that the Ag mode of 1720 cm-1 more strongly couples to the b2g LUMO than the Ag mode of 1396 cm-1 in 2tpa. This is because the displacements of carbon atoms in the Ag mode of 1720 cm-1 are larger

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Figure 2. Electron-phonon coupling constants in the monoanions of polyacetylenes.

TABLE 2: Calculated Reduced Masses, Vibronic Coupling Constants, and Electron-Phonon Coupling Constants in the Monoanion of 2tpa reduced masses gLUMO(ωm) lLUMO(ωm) (eV)

Ag (1396)

Ag (1720)

Ag (3168)

total

1.23 1.228 0.261

3.11 1.204 0.309

1.07 0.153 0.009

0.579

than those in the Ag mode of 1396 cm-1 in 2tpa. The C-H stretching Ag mode of 3168 cm-1 hardly couples to the b2g LUMO in 2tpa. This can be understood as follows. The displacements of hydrogen and carbon atoms are very large and small, respectively, in the C-H stretching Ag mode of 3168 cm-1 in 2tpa. The LUMO is completely localized on carbon atoms in 2tpa. It is rational that the frequency mode in which the displacements of carbon atoms are very small, cannot strongly couple to the LUMO completely localized on carbon atoms. This is the reason the C-H stretching Ag mode of 3168 cm-1 hardly couples to the b2g LUMO in 2tpa. Let us next look into the electron-phonon interactions between the Ag modes and the au LUMO in 4tpa. We can see from Table 3 that the C-C stretching Ag mode of 1730 cm-1 the most strongly couples to the au LUMO in 4tpa. The Ag modes of 517 and 1241 cm-1 also strongly couple to the au LUMO in 4tpa. The reduced masses for the Ag modes of 517, 1241, and 1730 cm-1 are 2.69, 1.95, and 4.45, respectively,

and are larger than those for other Ag modes in 4tpa. That is, the displacements of carbon atoms are larger in these modes than in the other modes. Also, the au LUMO is completely localized on carbon atoms. This is the reason the Ag modes of 517, 1241, and 1730 cm-1 afford large electron-phonon coupling constants in the monoanion of 4tpa. In a similar way, the Ag modes of 351, 1232, and 1712 cm-1 strongly couple to the bg LUMO in 6tpa, and the Ag modes of 223, 1226, and 1687 cm-1 strongly couple to the au LUMO in 8tpa. Total Electron-Phonon Coupling Constants in Negatively Charged Polyacetylenes Let us next discuss the total electron-phonon coupling constants in the monoanions (lLUMO) of polyacetylenes, and compare the calculated results for polyacetylenes with those for polyacenes. The lLUMO values for polyacetylenes are defined as

lLUMO )

lLUMO(ωm) ) ∑g2LUMO(ωm)pωm ∑ m m

(3)

The lLUMO values as a function of the number of carbon atoms in polyacetylenes and polyacenes are shown in Figure 4. In this figure, circles and triangles represent the lLUMO values for

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Figure 3. Selected vibronic active Ag modes of polyacetylenes.

TABLE 3: Calculated Reduced Masses, Vibronic Coupling Constants, and Electron-Phonon Coupling Constants in the Monoanion of 4tpa reduced masses gLUMO(ωm) lLUMO(ωm) (eV)

Ag (517)

Ag (907)

Ag (1241)

Ag (1329)

Ag (1497)

Ag (1730)

Ag (3147)

Ag (3166)

Ag (3247)

total

2.69 1.401 0.126

1.82 0.040 0.000

1.95 0.842 0.109

1.35 0.451 0.034

1.22 0.341 0.022

4.45 1.106 0.262

1.08 0.065 0.001

1.07 0.051 0.001

1.12 0.023 0.000

0.555

polyacenes and polyacetylenes, respectively. The lLUMO values are estimated to be 0.579, 0.555, 0.463, and 0.401 eV for 2tpa, 4tpa, 6tpa, and 8tpa, respectively, and those were estimated to be 0.322, 0.254, 0.186, 0.154, and 0.127 eV for 1a, 2a, 3a, 4a, and 5a, respectively.27 Therefore, the lLUMO values decrease with an increase in molecular size in both polyacenes and polyacetylenes. The lLUMO values for polyacetylenes are much larger than those for polyacenes. In particular, the C-C stretching vibronic active modes in polyacetylenes couple much more strongly to the LUMO than the C-C stretching vibronic active modes in polyacenes. This can be understood as follows. The LUMO of polyacenes is localized on the edge part of carbon atoms, and the LUMO of polyacenes has nonbonding characteristics. On the other hand, the LUMO of polyacetylenes is delocalized, and electron is distributed evenly over carbon structures. Therefore, the orbital interactions between two neighboring carbon atoms

are much stronger in the LUMO of polyacetylenes than in the LUMO of polyacenes, and the energy levels of the LUMO more significantly change in case that polyacetylenes are distorted along the C-C stretching vibronic active modes than in case that polyacenes are distorted along the C-C stretching vibronic active modes. This is the reason the lLUMO values for polyacetylenes are much larger than those for polyacenes. Let us next estimate the lLUMO value for polyacetylene with C2h geometry. The lLUMO values for polyacetylenes with C2h geometry as a function of 1/N values are shown in Figure 5, where N is the number of carbon atoms in polyacetylenes with C2h geometry. We can see from this figure that the lLUMO values for polyacetylenes with C2h geometry are approximately inversely proportional to the number of carbon atoms in polyacetylenes, as suggested in previous research.28 From this figure, the lLUMO value for polyacetylene (N f ∞) with C2h geometry is estimated to be 0.254 eV, while that for polyacene with D2h

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Figure 4. Total electron-phonon coupling constants as a function of number of carbon atoms in the monoanions of polyacenes and polyacetylenes. Circles and triangles represent the values for polyacenes and polyacetylenes, respectively.

Figure 5. Total electron-phonon coupling constants as a function of 1/N in the monoanions of polyacetylenes.

geometry is estimated to be 0.024 eV. Therefore, the lLUMO value for polyacetylene is estimated to be much larger than that for polyacene. As described above, the orbital patterns difference between the LUMO of polyacene localized on the edge part of carbon atoms, and the delocalized LUMO of polyacetylene is the main reason for the calculated results. The Logarithmically Averaged Phonon Frequencies Let us next estimate the logarithmically averaged phonon frequencies ωln, which measures the frequency of the vibrational modes which play an essential role in the electron-phonon interactions, and corresponds to the Debye frequency appearing in the prefactor in the McMillan’s formula in the theory of superconductivity.35,36 The logarithmically averaged phonon frequency, ωln,37 for the monoanions of polyacetylenes is defined as

ωln,LUMO ) exp

{

∑ m

}

lLUMO(ωm) ln ωm lLUMO

(4)

The ωln,LUMO values for the monoanions of polyacenes and polyacetylenes as a function of molecular weight Mw are shown in Figure 6. We can see from this figure that the ωln,LUMO values

Figure 6. Logarithmically averaged phonon frequencies ωln,LUMO as a function of molecular weights Mw. Circles and triangles represent the values for polyacenes and polyacetylenes, respectively.

are estimated to be 1576, 1210, 1147, and 1076 cm-1 for 2tpa, 4tpa, 6tpa, and 8tpa, respectively, and those were estimated to be 1390, 1212, 1023, 926, and 869 cm-1 for 1a, 2a, 3a, 4a, and 5a, respectively.27 Therefore, the ωln,LUMO values decrease with an increase in the Mw values in both polyacenes and polyacetylenes. This can be understood as follows. The C-C stretching modes around 1500 cm-1 less strongly couple to the LUMO with an increase in molecular size in polyacetylenes because the electron density on each carbon atom in the LUMO decreases with an increase in molecular size. On the other hand, the low-frequency modes, which are analogous to the acoustic modes of phonon in solids, would couple to the LUMO, regardless of such weakened orbital interactions between two neighboring carbon atoms with an increase in molecular size. This is the reason the ωln,LUMO values decrease with an increase in the Mw values in both polyacenes and polyacetylenes. Let us next compare the calculated results for the monoanions of polyacetylenes with those for the monoanions of polyacenes. The ωln,LUMO values for polyacenes are larger than those for polyacetylenes. The ωln,LUMO values for polyacenes with D2h geometry and polyacetylenes with C2h geometry would converge and be estimated by extrapolation in Figure 6 to be 124 and 753 cm-1, respectively, assuming that the ωln,LUMO values are approximately inversely proportional to the square root of the Mw values in each series. The estimated ωln,LUMO value for polyacetylene would be much larger than that for polyacene. Therefore, the ωln,LUMO values for polyacetylenes decrease much less significantly with an increase in the molecular weights than those for polyacenes. Vibronic Interactions and Electron Transfer in the Negatively Charged Polyacetylenes Electron Transfer in the Negatively Charged Polyacetylenes. Let us next discuss the single charge transfer through the molecule under consideration, which is of interest for possible nanoelectronics applications. Here, we will estimate the reorganization energy for elementary charge transfer, and will discuss the vibration effect onto the charge-transfer problem. We optimized the structures of the monoanions of polyacetylenes. The optimized structures of the monoanions of polyacetylenes are listed in Table 4. The estimated ionization energy, electron affinity, hopping barrier, and reorganization energy between neutral molecules and the corresponding monoanions in polyacetylenes are listed in Table 5.

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TABLE 4: C-C Distances in the Monoanions of Polyacetylenes

which λ ) µ* is satisfied are listed in Table 6. For example, considering large lLUMO values estimated above and the usual µ* values (∼0.20), eq 5 is satisfied if n(0) > 0.499 for 8tpa-. Therefore, we can expect that the attractive interaction between two electronic states can dominate over the repulsive Coulomb interaction between two electronic states on separate molecules in the monoanions of polyacetylenes under consideration. On the other hand, considering the lLUMO values for 5a and the usual µ* values (∼0.20), eq 5 is satisfied if n(0) > 1.575 for 5a-. We have considered that the n(0) value for the 2-fold degenerate electronic state in the monocation of benzene is approximately 4, and thus those for the nondegenerate electronic states in the monocations of acenes are approximately 2.27 On the other hand, the n(0) values are obviously sensitive to the overlap (the transfer integral) between the LUMOs on neighboring molecules, and consequently to the distance and the orientation between the molecules and to the extent and the position of the nodes of the LUMO. Therefore, the n(0) values are changeable compared with the physical values which are related to the intramolecular characteristics such as the lLUMO and ωln,LUMO values. Actually, there is the very interesting dependence38 of Tc upon cell parameter in a variety of systems containing mixtures of alkali metal atoms as fullerene dopants. The overlap integral between adjacent molecules decreases with an increase in the distance between the two adjacent monoanions. The density of states at the Fermi level n(0) values much increase with a decrease in overlap integral because the total integrated density of states is constant. Assuming that ωln,LUMO and lLUMO remain invariant to molecular crystal structure differences, then Tc varies simply with n(0). Since the bandwidth decreases with an increase in the distance between polyacetylene units, there will be some limiting separation where the delocalized model becomes inappropriate and the electrons localize at large distance between two neighboring molecules. Such an electronic instability will destroy normal metallic behavior as well as superconductivity. In more general terms, superconductivity, usually associated with high n(0) values, often competes with other electronically driven processes in solids. The possibility of electron localization in polyacetylenes as the interunit separation becomes large would be noted, but such high n(0) values often also signal geometrical distortions of the CDW type and ferromagnetic behavior. Let us next look into the relationships between the electron transfer and the electron-phonon interactions in the negatively charged polyacetylenes. In Figure 7, the reorganization energies as a function of the total electron-phonon coupling constants are shown. In view of Figure 7, a plot of the reorganization energies against the total electron-phonon coupling constants is found to be nearly linear. The reorganization energies decrease with an increase in molecular size in polyacetylenes. For a good conductor with rapid electron transfer, the overlap of the LUMOs of these molecules should be sufficiently large. This requires interaction energies greater than the reorganization energies between the neutral and the corresponding monoanions. This means that the larger the molecular size of polyacetylenes is, the better conductor with rapid electron transfer this negatively charged molecule is, if we assume that the LUMOs between two neighboring polyacetylenes do not significantly depend on the molecular size. Therefore, in order that a monoanion crystal becomes good conductor, the larger overlap integral (i.e., smaller distance) between two adjacent molecules is needed with a decrease in molecular size. Therefore, we can expect that the negatively charged polyacetylenes with larger molecular size can become better conductor than those with

d1 2tpa4tpa6tpa8tpa-

d2

d3

d4

1.444 (+0.114) 1.405 (+0.064) 1.408 (-0.050) 1.383 (+0.040) 1.411 (-0.039) 1.407 (+0.055) 1.373 (+0.029) 1.417 (-0.031) 1.401 (+0.046) 1.402 (-0.038)

a The values in parentheses indicate the change of the C-C distances by electron doping in polyacetylenes.

Considering the Marcus-type electron transfer diagram, reorganization energies between the neutral molecule and the corresponding monoanion are estimated to be 0.164, 0.144, 0.125, and 0.113 eV for 2tpa, 4tpa, 6tpa, and 8tpa, respectively. It should be noted that electronic interactions (orbital overlap) and steric interactions are ignored when we estimate the reorganization energies between the neutral molecules and the monoanions. Vibration Effects and the Optimized Structures of the Negatively Charged Polyacetylenes. Let us next look into the optimized structures of the monoanions of 2tpa, 4tpa, 6tpa, and 8tpa. We optimized the structure of the monoanion of 2tpa under D2h geometry, and those of the monoanions of 4tpa, 6tpa, and 8tpa under C2h geometry. We can see from Table 4 that the long (short) C-C bond lengths in the neutral polyacetylenes become shorter (longer) by electron doping, and the bond alternation is suppressed in the monoanions of polyacetylenes. This can be understood in view of the orbital patterns of the HOMO and LUMO in polyacetylenes. The bonding and antibonding interactions between two neighboring carbon atoms in neutral polyacetylenes are weakened by electron doping. This is the reason the long (short) C-C bond lengths in neutral polyacetylenes become shorter (longer) by electron doping, and bond alternation is suppressed in the monoanions of polyacetylenes. However, it should be noted that the changes of the C-C bond distances by electron doping become smaller with an increase in molecular size in polyacetylenes. This is because the orbital coefficients of each carbon atom in the LUMO become smaller, and the orbital interactions between two neighboring carbon atoms become weaker with an increase in molecular size, and thus the effects of suppression of bond alternation by electron doping become less effective with an increase in molecular size in polyacetylenes. Let us discuss the vibration effect onto the charge-transfer problem. We can see from Figure 3 and Table 4 that the Ag modes of 1720, 1730, 1712, and 1687 cm-1 are the main modes converting the neutral structures to the monoanions in 2tpa, 4tpa, 6tpa and 8tpa, respectively. This can be also confirmed from our calculated results that these modes the most strongly couple to the LUMO in these molecules. The Condition of Attractive Electron-Electron Interactions Once the attractive interaction between two electrons dominates over the repulsive screened Coulomb interaction as expressed by eq 5, the system would produce as many Cooper pairs as possible to lower its energy.

-λ + µ* ) -n(0)lLUMO + µ* < 0

(5)

where µ* is the Coulomb pseudopotential describing the electron-electron repulsion, usually used as a fitting parameter, and ranges between about 0.10 and 0.20 in conventional superconductivity. The n(0) values as a function of µ* under

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J. Phys. Chem. B, Vol. 109, No. 21, 2005 10627

TABLE 5: Estimated Ionization Energy, Electron Affinity, Hopping Barrier, and Reorganization Energy between Neutral Molecules and the Monoanions in Polyacetylenes

(2tpa)(2tpa-) f (2tpa-)(2tpa) (4tpa)(4tpa-) f (4tpa-)(4tpa) (6tpa)(6tpa-) f (6tpa-)(6tpa) (8tpa)(8tpa-) f (8tpa-)(8tpa)

ionization energy (eV)

electron affinity (eV)

hopping barrier (eV)

reorganization energy (eV)

0.337 0.286 0.246 0.223

-0.317 -0.288 -0.253 -0.228

0.654 0.574 0.499 0.451

0.164 0.144 0.125 0.113

TABLE 6: Necessary Minimum n(0) Values as a Function of µ* Which Satisfy Eq 5 (-λ + µ* e0) 2tpa4tpa6tpa8tpa1a2a3a4a5a-

µ* ) 0.1

µ* ) 0.2

µ* ) 0.3

µ* ) 0.4

µ* ) 0.5

µ* ) 0.6

µ* ) 0.7

µ* ) 0.8

0.173 0.180 0.216 0.249 0.311 0.394 0.538 0.649 0.787

0.345 0.360 0.432 0.499 0.621 0.787 1.075 1.299 1.575

0.518 0.541 0.648 0.748 0.932 1.181 1.613 1.948 2.362

0.691 0.721 0.864 0.998 1.242 1.575 2.151 2.597 3.150

0.864 0.901 1.080 1.247 1.553 1.969 2.688 3.247 3.937

1.036 1.081 1.296 1.496 1.863 2.362 3.226 3.896 4.724

1.209 1.261 1.512 1.746 2.174 2.756 3.763 4.545 5.512

1.382 1.441 1.728 1.995 2.484 3.150 4.301 5.195 6.299

Figure 7. Reorganization energies as a function of the total electronphonon coupling constants in the monoanions of polyacenes and polyacetylenes. Circles and triangles represent the values for polyacenes and polyacetylenes, respectively.

smaller molecular size of polyacetylenes in terms of the intramolecular characteristics. On the other hand, in polyacetylenes, we can see from eq 5 and from the fact that the lLUMO value decreases with an increase in molecular size, the condition of attractive electron-electron interactions becomes more difficult to be realized with an increase in molecular size, assuming that the overlap of the LUMOs of molecules under consideration do not significantly depend on the molecular size. In addition, the Tc values would become lower with an increase in molecular size in the monoanions of molecular systems such as polyacetylenes because the lLUMO values decrease with an increase in molecular size in polyacetylenes. Furthermore, the reorganization energies between the neutral molecule and the corresponding monoanion for 1a, 2a, 3a, 4a, and 5a were estimated to be 0.099, 0.064, 0.049, 0.039, and 0.032 eV, respectively.27b Therefore, the reorganization energies between the neutral molecule and the corresponding monoanion for polyacetylenes are larger than those for polyacenes. This means that the negatively charged polyacenes would be better conductor with rapid electron transfer than the negatively charged polyacetylenes in terms of only intramolecular parameters. On the other hand, the conditions under which eq 5 is satisfied are more difficult to be realized in the monoanions of polyacenes

than in the monoanions of polyacetylenes because the lLUMO values for polyacenes are smaller than those for polyacetylenes. In addition, the Tc values for the monoanions of polyacetylenes would be larger than those for the monoanions of polyacenes if these anions could exhibit superconductivity caused by the intramolecular vibronic interactions between the vibronic active modes and the LUMO. In summary, the monoanions with smaller molecular size cannot easily become good conductors, but the conditions under which the electron-electron interactions become attractive are realized more easily in the monoanions with smaller molecular size than in those with larger molecular size. This means that the monoanions with smaller molecular size cannot easily become good conductors, however, once the condition under which they can become metallic and superconducting states is realized, they would become higher temperature superconductor than the monoanions with larger molecular size. There is an interesting paradox in conventional superconductivity; the higher resistivity at room temperature, the more likely it is that a metal will be a superconductor when cooled.5a Therefore, the calculated results for the reorganization energies and the lLUMO (Tc) values in the negatively charged polyacetylenes and polyacenes are in qualitative agreement with such an interesting paradox, if these anions could exhibit superconductivity caused by the intramolecular vibronic interactions. Possible Electron Pairing in the Monoanions of Polyacetylenes Conduction electrons are subject to the Pauli principle. Condensation into a zero momentum state may be realized if the two electrons form a bound state via the attractive electronelectron interaction and the resultant pair of electrons behaves as a single particle obeying the Bose statistics in the monoanions. Let us next look into the possible electron pairing in the negatively charged molecular systems. As can be seen in Figure 8, it has been generally considered that a possibility of electron pairing is that between two electrons with opposite discrete wave vectors (kLUMO,j and opposite spins in negatively charged molecular systems. Thus, we have thought that intermolecular electron pairing can occur between two electronic states with opposite molecular wave vectors and opposite spins on separate molecules, as shown in Figure 8. In analogy with the BCS theory, the wave function in the monoanion is constructed by taking a linear combination of many normal-state configurations

10628 J. Phys. Chem. B, Vol. 109, No. 21, 2005

Kato and Yamabe

Figure 8. Electron pairing between electrons with discrete wave vectors on separate molecules.

in which the Bloch states are occupied by a pair of opposite momenta and spins;

Φ(rLUMO,i,rLUMO,j) )

∑ ∑ aijΦ(+ kLUMO,i,s, - kLUMO,j,-s) i,j s)(1/2

(6)

where |aij|2 represents the probability of finding a pair of electrons with the states +kLUMO,i,s and -kLUMO,j,-s. A Cooper pair is expressed by the wave function in eq 6. Therefore, it is important to realize that each Cooper pair is composed of the Bloch states with all possible wave vectors + kLUMO,i. Once the attractive interaction between two electrons dominates over the repulsive screened Coulomb interaction, the system would produce as many Cooper pairs as possible to lower its energy. The ground-state wave function in the superconducting state by using a Hartree-like approximation can be constructed as a product of the individual Cooper pair wave functions given by eq 7:

ψ0(r1,r2,...,rn0) )

∑

Φ(r1,r2) Φ(r3,r4)...Φ(rn-1,rn) )

n

∑ ∏ Φ(ri,rj)

(7)

i,j(i*j)

where n is the total number of electrons participating in the superconducting, ri the position of the ith electron and Φs on the right-hand side are the same for all pairs, and summation in eq 7 means that all possible two electrons pairings in n electrons such as Φ(r1,r2)Φ(r3,r4)...Φ(rn-1,rn) + Φ(r1,r3)Φ(r2,r4)...Φ(rn-1,rn) + ... are considered. The square of manyelectron wave function in eq 7 gives the probability of finding superconducting electrons at r1,r2,...,rn0 regardless of their momenta. The Cooper pair Φ(rn-1,rn) involved in eq 7 can be regarded as a single particle obeying the Bose-Einstein statistics. Because of their resultant zero momentum, the system is in an ordered state. The superconducting state is in a constrained condition such that we cannot alter the momentum of the paired electrons at will. As a consequence, for the paired electrons, the scattering which changes the direction of the wave vector is prohibited.

Once a current is induced, the same velocity wave vector W in parallel to the applied field is acquired by each Cooper pair. Thus, the drift velocity of all Cooper pair becomes W. Thus, all the Cooper pairs acquire the same momentum. Such a current flowing without disturbing the ordered state results in a resistanceless conduction. Concluding Remarks We studied electron-phonon interactions in the monoanions of polyacetylenes. Our calculated results show that the C-C stretching Ag modes around 1500 cm-1 can the most strongly couple to the LUMO in polyacetylenes. The estimated lLUMO values are 0.579, 0.555, 0.463, and 0.401 eV for 2tpa, 4tpa, 6tpa, and 8tpa, respectively, while those were estimated to be 0.244, 0.173, 0.130, 0.107, and 0.094 eV for 1a, 2a, 3a, 4a, and 5a, respectively. Therefore, the lLUMO values for polyacetylenes are much larger than those for polyacenes. This is mainly because the C-C stretching vibronic active modes in polyacetylenes couple much more strongly to the LUMO than those in polyacenes. The lLUMO value for polyacetylene with C2h geometry is estimated to be 0.254 eV, while that for polyacene with D2h geometry is estimated to be 0.024 eV. Therefore, the lLUMO value for polyacetylene is estimated to be larger than that for polyacene. The phase patterns difference between the LUMO of polyacenes localized on edge part of carbon atoms and the delocalized LUMO of polyacetylenes is the main reason for the calculated results. We also discussed the single charge transfer through the molecule under consideration, which is of interest for possible nanoelectronics applications. The reorganization energies between the neutral molecule and the corresponding monoanion are estimated to be 0.164, 0.144, 0.125, and 0.113 eV, for 2tpa, 4tpa, 6tpa, and 8tpa, respectively. The C-C stretching Ag modes of 1720, 1730, 1712, and 1687 cm-1 are the main modes converting the neutral structures to the monoanions in 2tpa, 4tpa, 6tpa, and 8tpa, respectively. This can be also confirmed from our calculated results that these C-C stretching Ag modes strongly couple to the LUMO in polyacetylenes. We discussed the conditions under which the attractive electron-electron interactions are realized in the monoanions

Negatively Charged Polyacetylene of polyacetylenes and polyacenes. Furthermore, we discussed how the conditions under which the monoanions crystals become good conductors are related to the molecular sizes and structures. We also discussed the relationships between the electron transfer and the electron-phonon interactions in the negatively charged polyacetylenes and polyacenes. We found that a plot of the reorganization energies against the total electron-phonon coupling constants is found to be nearly linear. The reorganization energies decrease with an increase in molecular size in polyacetylenes. This means that the larger the molecular size of polyacetylenes is, the better conductor with rapid electron transfer this negatively charged molecule is, if we assume that the overlaps of the LUMOs between two neighboring polyacetylenes do not significantly depend on the molecular size. Therefore, in order that a monoanion crystal becomes good conductor, the larger overlap integral (i.e., smaller distance) between two adjacent molecules is needed with a decrease in molecular size. On the other hand, since the lLUMO value decreases with an increase in molecular size in polyacetylenes, the condition of attractive electron-electron interactions becomes more difficult to be realized with an increase in molecular size, assuming that the overlaps of the LUMOs of molecules under consideration do not significantly depend on the molecular size. In addition, the estimated Tc values would become lower with an increase in molecular size in the monoanions of molecular systems such as polyacetylenes because the lLUMO values decrease with an increase in molecular size in polyacetylenes. The reorganization energies between the neutral molecule and the corresponding monoanion for polyacetylenes are larger than those for polyacenes. This means that the negatively charged polyacenes would be better conductor with rapid electron transfer than the negatively charged polyacetylenes, assuming that the overlap of the LUMOs between two neighboring polyacetylenes is not significantly different from that between two neighboring polyacene molecules. In summary, the monoanions with smaller molecular size cannot easily become good conductors, but the conditions under which the electron-electron interactions become attractive are realized more easily in the monoanions with smaller molecular size than in those with larger molecular size. This means that the monoanions with smaller molecular size cannot easily become good conductors, however, once the condition under which they can become metallic and superconducting states is realized, they would become higher temperature superconductor than the monoanions with larger molecular size. There is an interesting paradox in conventional superconductivity; the higher resistivity at room temperature, the more likely it is that a metal will be a superconductor when cooled. Therefore, the calculated results for the reorganization energies and the lLUMO (Tc) values in the negatively charged polyacetylenes and polyacenes are in qualitative agreement with such an interesting paradox, if these anions could exhibit superconductivity caused by the intramolecular vibronic interactions. Acknowledgment. This study was performed under the Project of Academic Frontier Center at Nagasaki Institute of Applied Science. This work is partly supported by a Grant-inAid for Scientific Research from the Japan Society for the Promotion of Science (JSPS-15350114, JSPS-16560618). References and Notes (1) (a) Bersuker, I. B. The Jahn-Teller Effect and Vibronic Interactions in Modern Chemistry; Plenum: New York, 1984. (b) Bersuker, I. B.;

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