18166
J. Phys. Chem. B 2006, 110, 18166-18179
Electron-Phonon Interactions and Intra- and Intermolecular Charge Mobility in the Monocations of Annulenes 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: January 6, 2006; In Final Form: May 16, 2006
Possible electron pairing in π-conjugated positively charged annulenes such as (CH)18 (18an) and (CH)30 (30an) is studied and compared with that in the positively charged acenes. The total electron-phonon coupling constants in the monocations (lHOMO) for 18an and 30an are estimated. The E2g modes of 1611 and 1201 cm-1 most strongly couple to the highest occupied molecular orbitals (HOMO) in 18an and 30an, respectively. The lHOMO values for annulenes are larger than those for acenes. The phase pattern difference between the HOMO of acenes localized on the edge part of carbon atoms and the delocalized HOMO of annulenes is the main reason for the calculated results. In view of the calculated results of the lHOMO values, intramolecular electron mobility (σintra,HOMO), and the reorganization energies (REHOMO) in the positively charged molecules, the monocations of annulenes cannot easily become good conductors compared with the monocations of acenes, but the condition of the attractive electron-electron interactions is realized more easily in the monocations of annulenes than in the monocations of acenes. The hypothetical intramolecular supercurrent originating from both intramolecular and intermolecular vibrations in the monocations of annulenes and acenes in a case where the distance between two adjacent molecules is too large for the molecular crystal to become normal metallic state, is also discussed.
Introduction The search for new organic metals and superconductors has attracted a great deal of attention since the discovery of high electrical conductivity in conjugated polymers such as polyacetylene in synthetic chemistry and material science.1 The effect of vibronic interactions2 in molecules and crystals has been an important topic in modern physics and chemistry. A large variety of research fields such as spectroscopy, instability of molecular structure, electrical conductivity,3 and superconductivity are covered by the analysis of vibronic interaction theory. Electronphonon coupling2-4 is the consensus mechanism for attractive electron-electron interactions in the Bardeen-CooperSchrieffer (BCS) theory of superconductivity.5,6 One of the authors qualitatively discussed the role of the vibronic interactions in the normal and superconducting states of conjugated polymers.7 Since a hypothetical molecular superconductor based on an exciton mechanism has been proposed by Little,8 the superconductivity of molecular systems has been extensively investigated and many bis(ethylenedithio)tetrathiafulvalene (BEDT-TTF)-type organic superconductors9,10 have been yielded by advances in the design and synthesis of molecular systems. It was found that the alkali-doped A3C60 complexes11 exhibit superconducting transition temperatures (Tc) of more than 30 K (ref 12) and 40 K under pressure.13 Pure intramolecular Raman-active modes have been suggested to be important in a Bardeen-Cooper-Schrieffer (BCS)-type5,6 strong coupling scenario in superconductivity in alkali-doped fullerenes.14 The rehybridization of the π orbitals and the 2s atomic orbitals which may be useful in order to understand the high Tc superconductivity of the A3C60 complexes was discussed by Haddon.15 The * To whom correspondence should be addressed. E-mail: kato@ cc.nias.ac.jp.
electron-phonon interactions were proposed to dominate the charge transport in the crystals of naphthalene (C10H8) (10ac), anthracene (C14H10) (14ac), tetracene (C18H12) (18ac), and pentacene (C22H14) (22ac).16 The possible superconductivity of polyacene has been proposed from a theoretical point of view.17 It is very intriguing to investigate the possible superconductivity in various molecular crystals. Since the end of 1950s, cyclic polyenes, i.e., annulenes, have been extensively investigated via molecular orbital (MO) theory by many authors.18-23 It is well-known that the diamagnetic anisotropy of aromatic hydrocarbons and annulenes can be attributed to the induced ring currents in their π-electronic systems.24-26 The relationship between the ring current and the virtual superconducting state in these molecular systems has been discussed. For example, Haddon27 demonstrated an analytical relationship between the resonance energies and ring currents of the (4n + 2)π-electronic systems and suggested the analogy between the MO wave function of 6an and the BCStype5,6 paired configuration. The condensation of electrons into boson-like Cooper pairs has been further discussed to explain the aromatic stabilization energy.28 We have analyzed the vibronic interactions and estimated possible Tc values 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 superconductivity in positively charged nanosized molecular systems.29 Electron-phonon interactions in positively charged acenes were well studied recently, on the basis of an experimental study of ionization spectra using high-resolution gas-phase photoelectron spectroscopy.30 Our theoretically predicted frequencies for the vibrational modes, playing an essential role in the electronphonon interactions,29 as well as the predicted total electron-
10.1021/jp068000u CCC: $33.50 © 2006 American Chemical Society Published on Web 08/30/2006
Electron Pairing in Annulenes
J. Phys. Chem. B, Vol. 110, No. 37, 2006 18167
phonon coupling constants29 are in excellent agreement with those obtained from experimental research.30 The purpose of this paper is to discuss the Jahn-Teller effects, electron-phonon interactions, intramolecular electron mobility, and the single charge transfer through molecule by estimating the reorganization energy for elementary charge transfer, and to discuss the vibration effect onto the charge transfer problem, which is of interest for possible nanoelectronics applications, in the positively charged annulenes such as 6an, [18]annulene (18an), and [30]annulene (30an). We also discuss the conditions of attractive electron-electron interactions in the monocations of annulenes. We compare the calculated results for the positively charged annulenes with those for the positively charged acenes such as 10ac, 14ac, 18ac, and 22ac to investigate how the properties of the electron-phonon interactions, intramolecular electron mobility, the electron transfer, and attractive electron-electron interactions are changed by geometrical differences. Finally, we briefly discuss the possibility of intramolecular supercurrent in the monocations of annulenes.
where qE2gγm (dimensionless) is the dimensionless normal coordinate31 of the mth vibrational mode defined as
qE2gγm ) (ωm/p)1/2QE2gγm
and hE2gm (eV) is the vibronic coupling matrix of the mth E2g mode in annulenes defined as
(
-QE θm QE m hE2gm ) Am Q 2g Q 2g E2gm E2gθm
∑i
gi
(1)
Here, we consider the frozen orbital approximation. Considering the one-electron approximation and that the first-order derivatives of the total energy vanish in the ground state at the equilibrium D6h structure in neutral 6an, 18an, and 30an (gneutral ) ∑HOMO gi ) 0) and that one electron must be removed from i the HOMO to generate the monocations, the vibronic coupling constant gmonocation(ωm) (dimensionless) of the vibronically active modes to the electronic states of the monocations of annulenes can be defined by
gmonocation(ωm) ) gHOMO(ωm)
(2)
where ωm is the frequency of the mth vibronically active mode. The numbers of the vibronically active E2g modes (NE2g) for 6an, 18an, and 30an are 4, 12, and 20, respectively, and those of the A1g modes (NA1g) for 6an, 18an, and 30an are 2, 6, and 10, respectively. In such a case, we must consider multimode problems, but in the limit of linear vibronic coupling, one can treat each set of modes (i.e., each mode index m) independently.2 Let us look into the vibronic coupling of the E2g vibrational modes to the HOMO in annulenes. The dimensionless orbital vibronic coupling constant gHOMO(ωm) (dimensionless) of the mth mode in annulenes is defined by using the reduced matrix element
gHOMO(ωm) )
〈
( )
∂hE2gm 1 HOMO| |HOMO pωm ∂qE2gγm 0
m ) 1, 2, ..., NE2g, γ ) θ,
〉
(5)
Here, Am (eV/(xg/mol Å)) is the reduced matrix element and the slope in the original point (i.e., QE2gm ) QE2gθm ) 0) on the energy sheet of the HOMO at the ground state of the neutral annulenes. This is defined as
〈
( ) ∂hE2gm
Am ) HOMO|
∂QE2gγm
|HOMO
0
〉
(6)
m ) 1, 2, ..., NE2g, γ ) θ, In a similar way, the dimensionless orbital vibronic coupling constant gHOMO(ωm) (dimensionless) of the mth A1g mode in annulenes is defined by using qA1gm (dimensionless), hA1gm (eV), and Bm (eV/(xg/mol Å))
gHOMO(ωm) )
occupied
gelectronic state )
)
m ) 1, 2, ..., NE2g
Theoretical Background Vibronic Interactions. We discuss a theoretical background for the orbital vibronic coupling constants2a in 6an, 18an, and 30an with D6h geometry. Here, we take a one-electron approximation into account; the vibronic coupling constant gelectronic state (dimensionless) of the vibronically active modes to the electronic states in the monocations of annulenes is defined as a sum of orbital vibronic coupling constants from all the occupied orbitals i2a
(4)
〈
( )
∂hA1gm 1 HOMO| |HOMO pωm ∂qA1gm 0
〉
(7)
(m ) NE2g + 1, NE2g + 2, ..., NE2g + NA1g) qA1gm ) (ωm/p)1/2QA1gm hA1gm ) Bm
(
QA1gm 0 QA1gm 0
(8)
)
(9)
m ) NE2g + 1, NE2g + 2, ..., NE2g + NA1g
〈
( )
Bm ) HOMO|
∂hA1gm
∂QA1gm
|HOMO
0
〉
(10)
m ) NE2g + 1, NE2g + 2, ..., NE2g + NA1g Electron-Phonon Interactions. In the previous section, the vibronic interactions in free annulenes were discussed. We set up an assumption to apply the calculated vibronic coupling constants to the solid state properties of annulenes. We assume that the conduction band of the monocation crystals of annulenes consists of the HOMOs because annulenes would consist of strongly bonded molecules arranged on a lattice with weak van der Waals intermolecular bonds. In addition to these assumptions, we consider only intraband scattering and focus on q ) 0 processes.14 On the basis of these assumptions, we can derive the dimensionless electron-phonon coupling constant λ appearing in the theory of superconductivity as discussed in previous research.14 We use a standard expression for λ14
(3) λ)
2
∑∑ N(0) m,q k,k′
1
|hk,k′(m,q)|2δ(k)δ(k′)
2ωm,q
2
(11)
18168 J. Phys. Chem. B, Vol. 110, No. 37, 2006
Kato and Yamabe
where ωm,q is the vibrational frequency for the mth phonon mode of wave vector q, hk,k′ is the corresponding electron-phonon matrix element between electronic states of wave vectors k and k′, k and k′ are the corresponding energies measured from the Fermi level, and N(0) is the total density of states (DOS) per spin. The Fermi level is essentially composed of two E2u (E1g) states of 18an (30an) as assumed and we can write in the form of Bloch sum
ψ(k) )
nνν′(0) ) (1/2)n(0)δνν′, and thus λ takes the form of eq 20
1
λ ) n(0) 4
xN
cν(k) eikRφν,R ∑ ∑ ν)1 R
(12)
h00γm ) hE2gγm (m ) 1, 2, ..., NE2g)
hk,k′ ) 〈ψ(k)|h|ψ(k′)〉 ) 1 ν,ν′ cν*(k)cν′(k′) ei(k′-k)R hR,R (m,q) (13) ν,ν′ N R
[
]
∑ ∑
h00γm ) hA1gγm, (m ) NE2g + 1, NE2g + 2, ..., NE2g + NA1g) vibronic coupling matrix, hE2gm, derived by eq 5 with respect to the mode amplitude, QE2gγm, as
hE2gγm )
∂ ∂QE2gγm
ν,ν′ (m,q) ) 〈φν,R|h|φν′,R〉 hR,R
1 iqR ν,ν′ e h00m xN
(15)
ν,ν′ h00m cν*(k)cν′(k - q) ∑ ν,ν′
λ)
1
2
∑ ∑ ∑∑ N(0) k,k-q ν ,ν ′ ν ,ν ′ m
× N 2ωm2 cν1*(k)cν2(k)cν2′*(k - q)cν1′(k - q)δ(k)δ(k-q) (17) 1
1
2
2
The partial DOS per molecule at the Fermi level, nνν′(0), can be rewritten as
nνν′(0) )
1
∑ cν*(k)cν′(k)δ(k) N k
(18)
We can derive eq 19 from eqs 17 and 18 using n(0) ) N(0)/N
λ)
2
∑ n(0) m
1
νν′ νν′ h00m h00m *nν ν (0)nν ′ν ′(0) ∑ ∑ ν ,ν ′ ν ,ν ′ 1 1
2ωm
2
2 2
1 2
1
1
2
Am2
m)1
ωm2
λ)
2
1
(19)
2
where n(0) is now the DOS per spin and per annulene molecule. Since we consider 2-fold degenerate states, we can assume that
)
∂ h (23) ∂QE2gγm E2gm
∑
+
n(0) 2
NE2g+NA1g
Bm2
m)NE2g+1
ωm2
∑
λm ) ∑ n(0)lHOMO(ωm) ) n(0)lHOMO ∑ m m lHOMO(ωm) ) gHOMO2(ωm)pωm
(24)
(25) (26)
m ) 1, 2, ..., NE2g
(16)
ν2ν2′ h00m *
)]
Using eqs 3 and 7, we finally get the relation between the nondimensional electron-phonon coupling constant λ, the electron-phonon coupling constants lHOMO(ωm) (eV) closely related to vibronic stabilization energy, and the intramolecular vibronic coupling constant as
1 lHOMO(ωm) ) gHOMO2(ωm)pωm 2
We now proceed to calculate λ by inserting eq 16 in eq 11 considering that ωm,q is independent of q. We then obtain ν1ν1′ h00m
NE2g
(14)
We insert eq 15 in eq 13 taking the condition k′ ) k - q into account to get
xN
-QE θm QE m Am Q 2g Q 2g E2gm E2gθm
λ ) n(0)
For the one-phonon mode with wave vector q, this term takes the following form ν,ν′ (m,q) ) hR,R
[(
Since Tr ∑γhE2gγm2 ) 4Am2 and Tr ∑γhA1gγm2 ) 2Bm2, one can rewrite eq 20 as
ν,ν′ where hR,R is the intramolecular coupling matrix and
hk,k-q )
(21)
(22)
where R defines the cell position, ν runs over the two degenerate states per annulene molecule, N is the number of molecules in the crystal, and φν,R is the molecular orbital at the cell position R. If we neglect the intermolecular electron-lattice coupling, hk,k′ can be reduced to
1
(20)
ωm2
where Here hE2gγm is the derivative matrix element of the
2
1
∑ m,γ
Trh00γm2
(27)
m ) NE2g + 1, NE2g + 2, ..., NE2g + NA1g Total electron-phonon coupling constants lHOMO (eV) in the monocations of annulenes is defined as NE2g
lHOMO )
1
NE2g+NA1g
gHOMO (ωm)pωm + gHOMO2(ωm)pωm ∑ ∑ 2 m)1 m)N +1 2
E2g
(28)
Orbital Interactions in Frontier Orbitals in Annulenes and Acenes Conjugated π systems in carbon compounds represent one of the few areas where simple algebraical expressions may be easily derived for the orbital energies and wave functions. The basis of Hu¨ckel’s approach is very simple indeed. The energy of each carbon π orbital before interaction (Coulomb integral) is put equal to R and the interaction energy between two adjacent pπ orbitals is put equal to β, the resonance integral. Annulenes. Here we consider [N]annulenes, i.e., finite (CH)N rings. We assume here that the [N]annulenes have planar D6h geometries with equidistant C-C bonds. The π-MO by the
Electron Pairing in Annulenes
J. Phys. Chem. B, Vol. 110, No. 37, 2006 18169 CHART 2
CHART 1
CHART 3
simple Hu¨ckel theory18,32 can be written as
ψ(kj )
N
1
∑ exp{(ikja(r - 1)}χr
xN r)1
(29)
where a is the one-dimensional unit vector of translational symmetry around the ring as shown in Chart 1; the azimuthal number is j ) 0, (1, (2, ..., ((N/2 - 1), N/2; χr the π atomic orbital of site r; and kj the discrete wave vector (kj ) 2πj/Na) defined in the range [-π/a, π/a]. For j * 0 and j * N/2, the degenerate pair MOs φ(kj correspond to two independent traveling waves going around the molecular ring, one clockwise and the other counterclockwise. The orbital energies are expressed by
(kj ) R + 2β cos kja ψAHOMO )
x∑ 2
N r)1
x∑ 2
x
2
N
{( ) } π
1
-
2
1
N
(r - 1) χr (31)
x
N
{( ) } π
1
2
-
1
N
(r - 1) χr (32)
{ ( ) } { ( ) } (33) 2 1 1 1 1 ) sin{π( - )(r - 1)} sin{π( - )r}β (34) N 2 N 2 N
Polyacenes. Here, we will build up in general the π orbitals of a conjugated chain of N carbon atoms. There is a simple expression32 for the energy levels and orbital coefficients of a one-dimensional chain containing Nchainπ orbitals. The π-MO by the simple Hu¨ckel theory32 can be written as Nchain
φj )
4
φj )
sin{kN/4-1/2a(r - 1)}χr ) 2
cjrχr ) ∑ r)1
x
2
Nchain
∑ {sin(kjar)}χr
Nchain + 1 r)1
(37)
In the case of butadiene (Nchain ) 4), π-MO by the simple Hu¨ckel theory18,32 can be written as
1 1 1 1 2 cos π - (r - 1) cos π - r β N 2 N 2 N
B Sr,r+1
(36)
The orbital energies are expressed by
N
N r)1
jπ (Nchain + 1)a
j ) R + 2β cos kja cos{kN/4-1/2a(r - 1)}χr )
∑ sin N r)1
A Sr,r+1 )
kj )
N
∑ cos N r)1
ψBHOMO )
(30)
orbital of site r; and kj the discrete wave vector defined in the range [0, π/a].
(35)
where a is the one-dimensional unit vectors as shown in Chart 2; the azimuthal number is j ) 1, 2, ..., Nchain; χr the π atomic
∑ r)1
cjrχr )
x { ( )} 2
4
∑ 5 r)1
sin
jrπ 5
χr
(38)
φ1 ) c1χ1 + c2χ2 + c2χ3 + c1χ4
(39)
φ2 ) c2χ1 + c1χ2 - c1χ3 - c2χ4
(40)
φ3 ) c2χ1 - c1χ2 - c1χ3 + c2χ4
(41)
φ4 ) -c1χ1 + c2χ2 - c2χ3 + c1χ4
(42)
where
x25 sin(π5) ) 0.3717 c ) x25 sin(2π5 ) ) 0.6015 c1 )
(43)
2
(44)
The orbital φ2 is illustrated in Chart 3. Here we consider finite polyacene systems. We assume here that polyacenes have planar D2h geometries with equidistant C-C bonds. Let us next assemble the band structure of polyacenes considering the orbitals of “butadiene-like” units as shown in Chart 3. Using Bloch’s theorem, the unperturbed tightbonding wave function ψj may be written as
18170 J. Phys. Chem. B, Vol. 110, No. 37, 2006
ψj ) C
∑s eik bsφj,s l
Kato and Yamabe
(45)
where kl is the discrete wave vector defined in the range [0,π/ b]
kl )
lπ (Nunit + 1)b
(46)
The HOMO of polyacenes (Chart 3) can be defined as
ψHOMO ) ψ2 ) C C
∑s eik bsφ2,s ) l
∑s eik bs(c2χ1,s + c1χ2,s - c1χ3,s - c2χ4,s) l
(47)
The phase patterns of the HOMO of polyacenes and annulenes are shown in Figure 1. We apply the useful fragment molecular orbital (FMO) method to the characterization of the electronic structures of anthracene. Anthracene can be theoretically partitioned into naphthalene and butadiene fragments in different ways, although other partitions are possible.33 We now consider the reason why the HOMOs of anthracene are rather localized on the edge part of carbon atoms. Anthracene is derived by connecting the two terminal carbons (1 and 4 sites) of butadiene to the 2 and 3 sites of naphthalene, as indicated in Figure 1a. That is, the HOMO of naphthalene in which the electron density on the edge part of carbon atoms is rather high, and the HOMO of butadiene in which the electron density on the edge part of carbon atoms is rather high, interact nicely at the connecting sites, so that the orbital interactions between connected two neighboring carbon atoms are weak. Similar discussions can be made in larger molecular size of polyacenes. Furthermore, it should be noted that apart from prefactor eiklbs of eq 47, there are orbital overlaps between two neighboring carbon atoms with orbital coefficients c1 and c1, or c1 and c2 (i.e., c12 or c1c2), but are no orbital overlap between two carbon atoms with large orbital coefficients of c2 and c2 (i.e., c22) in the HOMO of polyacenes. In summary, we can see from Figure 1 and eq 47 that the HOMO is rather localized on the edge part of carbon atoms and thus have characteristics of the nonbonding orbital interactions. Therefore, the orbital interactions between two neighboring carbon atoms in the HOMO in polyacenes are very weak. On the other hand, the HOMO of annulenes is delocalized and we can see from eqs 33 and 34 that the orbital overlap between some two neighboring carbon atoms (for example, S4,5 and S5,6 in 18an and S7,8 and S8,9 in 30an) in the HOMO in annulenes is strong. Electron-Phonon Interactions in the Monocations of Annulenes We calculated first-order derivatives for the equilibrium structure on each orbital energy surface by distorting the molecule along the vibronically active modes using the hybrid Hartree-Fock (HF)/density-functional theory (DFT) method of Becke,34 and Lee, Yang, and Parr (B3LYP),35 and the 6-31G* basis set.36 The Gaussian 98 program package37 was used for our theoretical analyses. The calculated electron-phonon coupling constants in the monocations of annulenes are shown in Figure 2. And the estimated electron-phonon coupling constants in the monocations of annulenes are listed in Tables 1 and 2. Let us look into the electron-phonon interactions in the monocation of 18an. We can see from this table that the C-C stretching E2g mode of 1611 cm-1 can most strongly couple to
the HOMO in 18an. This can be understood as follows. When 18an is distorted along the C-C stretching E2g mode of 1611 cm-1, the bonding interactions between two neighboring carbon atoms in the HOMO (A) (HOMO (B)) become stronger (weaker), and the antibonding interactions between two neighboring carbon atoms in the HOMO (A) (HOMO (B)) become weaker (stronger), and thus the HOMO (A) and HOMO (B) are significantly stabilized and destabilized in energy, respectively, by such a distortion. This is the reason the C-C stretching E2g mode of 1611 cm-1 most strongly couples to the HOMO in 18an. The E2g modes of 1306 and 1377 cm-1 also strongly couple to the HOMO in 18an. It should be noted that the 2-fold degenerate E2g modes much more strongly couple to the HOMO than the nondegenerate A1g modes in 18an as can be seen from Table 1. Let us next look into the electron-phonon interactions in the monocation of 30an. We can see from Figure 2 that the E2g mode of 1201 cm-1 most strongly couples to the HOMO in 30an. In addition to this mode, the E2g modes of 1246, 1513, and 1556 cm-1 also strongly couple to the HOMO in 30an. It should be noted that the 2-fold degenerate E2g modes much more strongly couple to the HOMO than the nondegenerate A1g modes in 30an, as can be seen from Table 2. Total Electron-Phonon Coupling Constants in the Monocations of Annulenes We next estimate the total electron-phonon coupling constants lHOMO in the monocations of annulenes. The estimated lHOMO values in annulenes and acenes are shown in Figure 3. The lHOMO values are estimated to be 0.244, 0.137, and 0.121 eV for 6an, 18an, and 30an, respectively. Therefore, the lHOMO value decreases with an increase in molecular size in annulenes. Let us next compare the calculated results for the monocations of annulenes with those for the monocations of acenes. The lHOMO values were estimated to be 0.173, 0.130, 0.107, and 0.094 eV for 10ac, 14ac, 18ac, and 22ac, respectively.29 Therefore, the lHOMO values for annulenes decrease with an increase in molecular size less rapidly than those for acenes, and thus the lHOMO values for annulenes are larger than those for acenes. This can be understood in view of the phase patterns of the HOMO of annulenes and acenes, as described in the previous section. The HOMO is localized on an edge part of the carbon framework in acenes. The nonbonding characteristics are important in the HOMO in acenes, and the orbital interactions between two neighboring carbon atoms in the HOMO are weak in acenes. On the other hand, the HOMO is delocalized and the electron is distributed evenly over whole carbon framework in the HOMO of annulenes. Therefore, orbital interactions between two neighboring carbon atoms in the HOMO in annulenes are stronger than those in acenes. Therefore, the energy levels of the HOMO in annulenes are changed more significantly than those in acenes in the case where annulenes are distorted along the C-C stretching modes around 1500 cm-1. This is the reason the electron-phonon coupling constants originating from the C-C stretching modes around 1500 cm-1 in the monocations of annulenes are larger than those in the monocations of acenes and the reason the lHOMO values for annulenes are larger than those for acenes. Let us next estimate the lHOMO value for annulene (N f ∞) with D6h geometry. The lHOMO values as a function of 1/N values in annulenes and acenes are shown in Figure 4, where N denotes the number of carbon atoms in annulenes and acenes. We can see from Figure 4 that the lHOMO values for annulenes with D6h geometry are approximately inversely proportional to the number of carbon atoms, as suggested in the previous research.29 The
Electron Pairing in Annulenes
J. Phys. Chem. B, Vol. 110, No. 37, 2006 18171
Figure 1. The HOMO in acenes and annulenes and selected vibronically active modes in 18an.
lHOMO value for annulene (N f ∞) with D6h geometry is estimated to be 0.088 eV. The lHOMO value for polyacene with D2h geometry was estimated to be 0.033 eV.29 Therefore, the lHOMO values for annulenes (N f ∞) with D6h geometry are estimated to be larger than those for polyacenes (N f ∞) with D2h geometry. The orbital pattern difference between the HOMO localized on the edge part of carbon atoms in carbon framework in acenes and the delocalized HOMO in annulenes is the main reason for the calculated results. Attractive Electron-Electron Interactions Once the attractive interaction between two electrons dominates over the repulsive screened Coulomb interaction as expressed by eq 48, the system would produce as many Cooper
pairs as possible to lower its energy
-λ + µ* ) -n(0)lHOMO + µ* < 0
(48)
where µ* is the Coulomb pseudopotential usually used as a fitting parameter, which ranges about 0.10-0.20 in conventional superconductivity. The density of states at the Fermi level n(0) values as a function of µ* under which λ ) µ* is satisfied are listed in Table 3. For example, considering large lHOMO values estimated above and the usual µ* values (∼0.20), eq 48 is satisfied if n(0) > 1.653 for 30an. 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 monocations of annulenes under consideration. On the other hand, considering
18172 J. Phys. Chem. B, Vol. 110, No. 37, 2006
Kato and Yamabe
Figure 2. Electron-phonon coupling constants in the monocations of annulenes.
TABLE 1: The Reduced Masses, Vibronic Coupling Constants, and Electron-Phonon Coupling Constants in the Monocation of [18]Annulene E2g (116)
E2g (405)
E2g (647)
E2g (1127)
E2g (1268)
E2g (1306)
E2g (1377)
E2g (1500
E2g (1611)
red. masses gHOMO (ωm) lHOMO (ωm)
6.18 0.571 0.005
4.11 0.058 0.000
5.26 0.122 0.001
1.81 0.152 0.003
1.94 0.150 0.004
2.00 0.438 0.031
1.58 0.402 0.028
2.17 0.169 0.005
2.29 0.490 0.048
E2g (3157)
E2g (3174)
E2g (3202)
A1g (297)
A1g (1037)
A1g (1337)
A1g (1570)
A1g (3176)
A1g (3235)
total
1.09 0.008 0.000
1.09 0.003 0.000
1.08 0.001 0.000
5.98 0.529 0.005
2.34 0.226 0.003
1.68 0.185 0.003
5.11 0.068 0.000
1.09 0.066 0.001
1.08 0.007 0.000
0.137
TABLE 2: The Reduced Masses, Vibronic Coupling Constants, and Electron-Phonon Coupling Constants in the Monocation of [30]Annulene E2g (38)
E2g (179)
E2g (376)
E2g (526)
E2g (668)
E2g (1069)
E2g (1201)
E2g (1246)
E2g (1296)
6.34 0.531 0.001
5.14 0.114 0.000
5.18 0.134 0.001
3.62 0.107 0.001
4.08 0.037 0.000
2.70 0.079 0.001
1.79 0.591 0.052
1.48 0.293 0.013
1.69 0.025 0.000
E2g (1313)
E2g (1325)
E2g (1381)
E2g (1513)
E2g (1556)
E2g (1649)
E2g (3149)
E2g (3153)
E2g (3161)
E2g (3167)
E2g (3175)
1.44 0.151 0.004
2.62 0.045 0.000
1.40 0.092 0.001
2.21 0.331 0.020
2.53 0.276 0.015
3.87 0.018 0.000
1.09 0.004 0.000
1.09 0.003 0.000
1.09 0.007 0.000
1.09 0.005 0.000
1.09 0.002 0.000
red. masses gHOMO (ωm) lHOMO (ωm)
A1g (171)
A1g (448)
A1g (1114)
A1g (1306)
A1g (1342)
A1g (1401)
A1g (1645)
A1g (3154)
A1g (3169)
A1g (3178)
total
6.27 0.689 0.005
4.60 0.105 0.000
1.87 0.149 0.002
5.58 0.140 0.002
1.59 0.082 0.001
1.54 0.102 0.001
4.01 0.024 0.000
1.09 0.031 0.000
1.09 0.040 0.000
1.09 0.019 0.000
0.121
the lHOMO values for 18ac and the usual values (∼0.20), eq 48 is satisfied if n(0) > 1.869 for 18ac+. 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 2.29 And the n(0) values for the 2-fold degenerate electronic states in the monocations of 18an and
30an would be approximately 4. On the other hand, the n(0) values are obviously sensitive to the overlap (the transfer integral) between the HOMOs 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 HOMO. Therefore, the n(0) values are changeable compared with the physical values which are related to the intramolecular
Electron Pairing in Annulenes
J. Phys. Chem. B, Vol. 110, No. 37, 2006 18173
Figure 3. Vibronic energy gain as a function of the number of carbon atoms in acenes and annulenes. The open circles and triangles represent the lHOMO values for acenes and annulenes, respectively, and the closed circles and triangles represent the ∆G0HOMO values for acenes and annulenes, respectively.
Figure 4. Electron-phonon coupling constants as a function of 1/N in annulenes and acenes. The circles and triangles represent the values for the monocations of acenes and annulenes, respectively.
characteristics such as the lHOMO and ωln,HOMO 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. Intramolecular Electron Mobility in Annulenes Here, let us consider one-electron transfer assisted by molecular vibrations as a molecular model of the interaction of conduction electrons with vibration. From analogy with the definition of the intrinsic intramolecular conductivity σintra,HOMO as discussed in previous research,29 according to the definition suggested by Kivelson and Heeger,3,7 we define that for the monocations of annulenes as
σintra,HOMO ∝
ωHOMOMCN lHOMO2
(49)
where MC denotes the mass of carbon atom and ωHOMO denotes the frequencies of the vibrational modes playing an important role in the electron-phonon interactions. The calculated σintra,HOMO values as a function of N are shown in Figure 5. Let us first consider the case where ωHOMO corresponds to the logarithmically averaged phonon frequencies (ωln,HOMO),14,29 which measure the frequencies of the vibronically active modes playing an essential role in the electron-phonon interactions in the monocations of annulenes, and have been used widely in
order to measure such frequencies. We can see from this figure that the σintra,HOMO values slightly increase with an increase in molecular size from 6an (σintra,HOMO ) 14.1 × 105) to 30an (σintra,HOMO ) 280.1 × 105). Therefore, the σintra,HOMO values for annulenes are smaller than those for acenes. This is because the lHOMO and ωln,HOMO values for annulenes are larger and smaller, respectively, than those for acenes. The σintra,HOMO values for annulenes increase with an increase in molecular size less rapidly than those for acenes. This is because the lHOMO values for annulenes decrease with an increase in molecular size less significantly than those for acenes and the ωln,HOMO values do not significantly change both in annulenes and acenes, as described in the previous section. In this paper, we introduced the ωln,HOMO values, which are usually used in the McMillan’s formula39 in order to measure the frequency of the vibrational mode that plays an essential role in the electron-phonon interactions. Of course, other definitions of averaged phonon frequencies as well as the ωln,HOMO values are also available if the values approximately indicate the frequencies of the vibrational modes that play an essential role in the electronphonon interactions. In this sense, it should be noted that the ωln,HOMO values may not have strict physical meaning but may be the approximate values by which we qualitatively discuss which frequency modes play an essential role in the electronphonon interactions. Let us next consider the σintra,HOMO values estimated by considering the case where the ωHOMO values correspond to the frequencies of the vibrational modes that play the most important role in the electron-phonon interactions (i.e., 1656, 1611, and 1201 cm-1 for 6an, 18an, and 30an, respectively, and 1630, 1610, 1594, and 1572 cm-1 for 10ac, 14ac, 18ac, and 22ac, respectively). Even though the σintra,HOMO values estimated by considering the frequency of only the vibrational modes playing the most important role in the electron-phonon interactions are somewhat larger than those estimated by considering the ωln,HOMO values, essential qualitative results obtained by considering the frequency of only vibrational modes playing the most important role in the electron-phonon interactions are similar to those obtained by considering the ωln,HOMO values. Therefore, the σintra,HOMO values are much more sensitive to the strengths of the electron-phonon interaction than to the frequencies of vibrational modes in the molecules under consideration. Vibration Effects and Electron Transfer in Positively Charged Annulenes Electron Transfer in the Positively Charged Annulenes. 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 monocations of annulenes. The optimized parameters of these molecules are listed in Table 4. The estimated ionization energy, electron affinity, hopping barrier, and reorganization energy between neutral molecules and the monocations in annulenes are listed in Table 5. Considering the Marcus-type electron transfer diagram, the reorganization energies between the neutral molecules and the corresponding monocations (REHOMO) are estimated to be 0.037 and 0.033 eV for 18an and 30an, respectively. For a good conductor with rapid electron transfer, the overlap of the HOMOs of these molecules should be sufficiently large. This requires interaction energies greater than the REHOMO values. The REHOMO values were estimated to be 0.046, 0.034, 0.027,
18174 J. Phys. Chem. B, Vol. 110, No. 37, 2006
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TABLE 3: Necessary Minimum n(0) Values in the Monocations of Acenes and Annulenes as a Function of µ* That Satisfy Eq 33 6an+ 18an+ 30an+ 10ac+ 14ac+ 18ac+ 22ac+
µ* ) 0.05
µ* ) 0.10
µ* ) 0.15
µ* ) 0.20
µ* ) 0.25
µ* ) 0.30
µ* ) 0.35
µ* ) 0.40
µ* ) 0.45
µ* ) 0.50
0.205 0.365 0.413 0.289 0.385 0.467 0.532
0.410 0.730 0.826 0.578 0.769 0.935 1.064
0.615 1.095 1.240 0.867 1.154 1.402 1.596
0.820 1.460 1.653 1.156 1.538 1.869 2.128
1.025 1.825 2.066 1.445 1.923 2.336 2.660
1.230 2.190 2.479 1.734 2.308 2.804 3.191
1.434 2.555 2.893 2.023 2.692 3.271 3.723
1.639 2.920 3.306 2.312 3.077 3.738 4.255
1.844 3.285 3.719 2.601 3.462 4.206 4.787
2.049 3.650 4.132 2.890 3.846 4.673 5.319
TABLE 4: The Optimized Structures (in Å) of the Neutral and Monocations of Annulenes 6an 6an+A 6an+B 18an 18an+A 18an+B 30an 30an+A 30an+B
dr,r+1 dr,r+1 ∆dr,r+1 Sr,r+1 dr,r+1 ∆dr,r+1 Sr,r+1 dr,r+1 dr,r+1 ∆dr,r+1 Sr,r+1 dr,r+1 ∆dr,r+1 Sr,r+1 dr,r+1 dr,r+1 ∆dr,r+1 Sr,r+1 dr,r+1 ∆dr,r+1 Sr,r+1
r)1
r)2
r)3
r)4
r)5
r)6
r)7
r)8
1.397 1.432 +0.035 +0.167 1.391 -0.006 (0.000 1.396 1.401 +0.005 +0.019 1.394 -0.002 (0.000 1.396 1.398 +0.002 +0.007 1.395 -0.001 (0.000
1.397 1.372 -0.025 -0.084 1.454 +0.057 +0.250 1.410 1.401 -0.009 -0.018 1.424 +0.014 +0.037 1.396 1.391 -0.005 -0.007 1.401 +0.005 +0.014
1.396 1.413 +0.017 +0.052 1.382 -0.014 -0.033 1.406 1.416 +0.010 +0.020 1.398 -0.008 -0.013
1.396 1.378 -0.018 -0.043 1.417 +0.021 +0.062 1.396 1.385 -0.011 -0.019 1.408 +0.012 +0.026
1.410 1.436 +0.026 +0.065 1.391 -0.019 -0.046 1.396 1.411 +0.015 +0.030 1.383 -0.013 -0.024
1.396 1.382 -0.014 -0.027 1.412 +0.016 +0.034
1.396 1.413 +0.017 +0.036 1.380 -0.016 -0.029
1.406 1.389 -0.017 -0.030 1.425 +0.019 +0.037
TABLE 5: The Estimated Ionization Energy, Electron Affinity, Hopping Barrier, and Reorganization Energy between Neutral Molecules and the Monocations in Annulenes
(18an)(18an+) f (18an+)(18an) (30an)(30an+) f (30an+)(30an)
ionization energy
electron affinity
hopping barrier
reorganization energy
0.074 0.067
-0.075 -0.065
0.149 0.132
0.037 0.033
and 0.022 eV for 10ac, 14ac, 18ac, and 22ac, respectively. The REHOMO values decrease with an increase in molecular size in annulenes and acenes. This means that the smaller the molecular
Figure 5. Intramolecular electron mobility as a function of the number of carbon atoms in acenes and annulenes. The open circles and triangles represent these values estimated by considering the ωln,HOMO values for acenes and annulenes, respectively, and the closed circles and triangles represent these values estimated by considering the frequencies of the vibrational modes which play the most essential role in the electron-phonon interactions for acenes and annulenes, respectively.
sizes of annulenes and acenes are, then the larger orbital overlap is needed for the monocation molecule to become good conductor with rapid electron transfer. The REHOMO values for annulenes are larger than those for acenes. This means that the larger orbital overlap is needed for the positively charged annulenes to become a good conductor with rapid electron transfer than those for the positively charged acenes. The phase pattern differences between the HOMO localized on edge part of carbon atoms in acenes and the delocalized HOMO in annulenes are the main reason for the calculated results. The REHOMO value for annulene (N f ∞) is estimated to be 0.020 eV, and that for acenes (N f ∞) was estimated to be 0.001 eV,29 assuming that the REHOMO value is approximately inversely proportional to the number of carbon atoms. Vibration Effects and the Optimized Structures of the Positively Charged Annulenes. Let us next look into the optimized structures of the monocations of annulenes. We can see from Table 4 that the d1,2 and d2,3 values become larger and smaller, respectively by hole doping in 6an+A. This can be understood in view of the phase patterns of the HOMO in 6an. In the HOMO (A), the atomic orbitals between two neighboring carbon atoms whose bond lengths are d1,2 and d2,3 are combined in phase and out of phase, respectively, and thus the bonding and antibonding interactions, respectively, are significant. When an electron is removed from the HOMO (A), such bonding and antibonding interactions become weaker, and thus the d1,2 and d2,3 values become larger and smaller by hole doping in 6an+A. Similar discussions can be made in 6an+B.
Electron Pairing in Annulenes
Figure 6. The reorganization energies as a function of the total electron-phonon coupling constants for the monocations of acenes and annulenes. The circles and triangles represent the values for the monocations of acenes and annulenes, respectively.
Let us next look into the optimized structures of 18an. We can see from Table 4 that the C-C bond lengths which become larger (smaller) by hole doping in 18an+B become smaller (larger) by hole doping in 18an+A. This is because the phase patterns of the HOMO (B) are completely different from those of the HOMO (A) in 18an; the atomic orbitals between two neighboring carbon atoms combined in phase (out of phase) in the HOMO (B) are combined out of phase (in phase) in the HOMO (A) in 18an. It should be noted that the larger the |Sr,r+1| values are, the larger the changes of each C-C bond length (|∆dr,r+1|) are, as expected. Similar discussions can be made in the monocation of 30an. But it should be noted the |∆dr,r+1| value becomes smaller with an increase in molecular size, as expected. The C-C stretching E2g modes around 1200-1500 cm-1 are the main modes converting the neutral structures to the monocations of annulenes. Relationships between Normal Metallic States and Attractive Electron-Electron Interactions Let us next look into the relationships between the electron transfer and the electron-phonon interactions in the positively charged annulenes and acenes. In Figure 6, the REHOMO values as a function of the lHOMO values are shown. Intramolecular electron mobility, intermolecular charge transfer, and attractive electron-electron interactions are schematically shown in Figure 7. In view of Figure 6, a plot of the REHOMO values against the lHOMO values is found to be nearly linear. The REHOMO values for annulenes are larger than those for acenes. The Intra- and Intermolecular Electron Mobility. As discussed in the previous section, the σintra,HOMO values for annulenes are smaller than those for acenes. This is because the lHOMO and ωln,HOMO values for annulenes are larger and smaller, respectively, than those for acenes. The REHOMO values for annulenes are larger than those for acenes. This means that in order for the positively charged annulenes to become a good conductor, the larger overlap integral is needed (the n(0) values should not be too large) compared with those for the positively charged acenes (Figure 7). That is, the positively charged acenes would be a better conductor with rapid electron transfer than the positively charged annulenes in terms of only intramolecular parameters. These calculated results can be rationalized from the fact that the orbital interactions between two neighboring carbon atoms in the HOMO of annulenes are stronger than those for acenes. In other words, the strong bonding interactions in the HOMO
J. Phys. Chem. B, Vol. 110, No. 37, 2006 18175 tend to make an electron be localized on the regions where the positive orbital overlap is large and thus such an electron cannot easily move. Attractive Electron-Electron Interactions. The conditions under which eq 48 is satisfied are more difficult to be realized in the monocations of acenes than in the monocations of annulenes because the lHOMO values for acenes are smaller than those for annulenes. We can see from eq 48 that large n(0) and lHOMO values are favorable for the realization of the condition under which the electron-electron interactions become attractive (Figure 7b,c). On the other hand, since the orbital overlap decreases with an increase in the distance between annulenes units, there will be some limiting separation where the delocalized model becomes inappropriate and the electrons localize at large distance between two neighboring molecules (Figure 7c). That is, when the n(0) values are large (the orbital overlap is too small), the molecular crystal does not become metallic. Furthermore, considering that the lHOMO values are proportional to the REHOMO values, the overlap integral between two neighboring molecules must be large and thus the n(0) values should not be too large in order that a molecular crystal becomes metallic. And the σintra,HOMO values are inversely proportional to lHOMO2. That is, a large lHOMO value would not be favorable for the normal metallic behavior. Therefore, large n(0) and lHOMO values are favorable for the realization of the condition under which the electron-electron interactions become attractive, at the same time, are not favorable for the realization of the conditions under which a molecular crystal becomes normal metallic (Figure 7c). In summary, the monocations of annulenes cannot easily become good conductors, but the conditions under which the electron-electron interactions become attractive are realized more easily in the monocations of annulenes than in the monocations of acenes. This means that the monocations of annulenes cannot easily become a good conductor; however, once the condition under which they can become metallic is realized, they would more easily become a superconductor than the monocations of acenes. There is an interesting paradox in conventional superconductivity; the higher the resistivity at room temperature, the more likely it is that a metal will be a superconductor when cooled.6a Therefore, the calculated results for the σintra,HOMO values, REHOMO values, and the lHOMO values in annulenes and acenes are in qualitative agreement with the interesting paradox, if these cations could exhibit superconductivity caused by the intramolecular vibronic interactions. Hypothetical Intramolecular Supercurrent in the Monocations of Annulenes Let us next look into the possibility of intramolecular supercurrent in molecular systems as shown in Figure 7c. As discussed in the previous section, the σintra,HOMO values for annulenes are smaller than those for acenes. Therefore, intramolecular electron mobility in annulenes would be smaller than that in acenes at high temperature. Such conduction electrons are subject to the Pauli exclusion principle. On the other hand, below the temperature at which electron-electron interactions become attractive, two electrons form electron pairing between electrons with discrete wave vectors on separate molecules as shown in Figure 7c. However, it should be noted that since the electrons localize at one molecule, only intermolecular vibrations which would afford much smaller electron-phonon coupling constants than intramolecular vibrations play an essential role in forming of possible electron pairs as shown in Figure 7c. Since the total electron-phonon coupling constants originating
18176 J. Phys. Chem. B, Vol. 110, No. 37, 2006
Kato and Yamabe
Figure 7. Intramolecular electron mobility, intermolecular charge transfer, and attractive electron-electron interactions in the monocations of annulenes. (a) The condition under which normal metallic states are realized, but the condition under which the attractive electron-electron interactions are not realized (n(0) value is too small for the attractive electron-electron interactions to be realized). (b) The condition under which normal metallic state and attractive electron-electron interactions are realized (n(0) value is appropriate for both metallic state and the attractive electronelectron interactions to be realized). (c) The condition under which normal metallic states are not realized, but the condition under which the attractive electron-electron interactions are realized (n(0) value is too large for the normal metallic state to be realized). The larger the lHOMO values are, the smaller the appropriate n(0) values satisfying the condition (b) are.
from only intermolecular vibrations would be very small, such electron pairing would be realized only at sufficiently low temperatures. Let us next investigate the possible condition under which such intramolecular supercurrent occurs at higher temperatures. If the condition of normal metallic state and the attractive electron-electron interactions are realized as shown in Figure 7b at the initial stage, intramolecular vibrations would play an essential role in such attractive electron-electron interactions, as shown in Figure 7c. If we increase the distance between two neighboring molecules from the initial stage as shown in Figure 7b, electrons localize at one molecule and the electron-electron pairing originating from the intramolecular vibrations would occur, as shown in Figure 7c. Since the total electron-phonon coupling constants originating from the intramolecular vibrations would be much larger than those originating from intermolecular vibrations, hypothetical electron pairing originating from the intramolecular vibrations would occur at higher temperatures. Condensation into a zeromomentum state may be realized if two electrons form a bound state via the attractive electron-electron interaction and the
resultant pair of electrons behave as a single particle obeying Bose statistics in the monocations. Such a pair can be regarded as a single particle obeying Bose-Einstein statistics. Because of their resultant zero spin, such a pair can behave as Bose particles, and because of resultant zero momentum, the system is in an ordered state. The hypothetical intramolecular supercurrent 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 that changes the direction of the wave vector v is prohibited. Once a current is induced, the same velocity vector that is parallel to the applied field is acquired by each pair. Thus, the drift velocity of all pairs becomes v. Thus, all pairs acquire the same momentum. Such a current flowing without disturbing the ordered state in a molecule results in a resistanceless intramolecular conduction. All pairs are condensed into a single energy level in the ground state at 0 K. At 0 K, all electrons partially occupying the HOMO would form electron pairs as discussed above, and thus we can expect to observe the intramolecular supercurrent in all molecules. If the pair receives enough energy to destroy the electron
Electron Pairing in Annulenes
J. Phys. Chem. B, Vol. 110, No. 37, 2006 18177
TABLE 6: New Vibrational Normal-Mode Frequencies νm Arising after Vibronic Coupling of the Monocation of [18]Annulene in the Case of e X (nE) E2g (116)
E2g (405)
E2g (647)
E2g (1127)
E2g (1268)
E2g (1306)
E2g (1377)
E2g (1500)
E2g (1611)
E2g (3157)
E2g (3174)
E2g (3202)
total
gHOMO (ωm) 0.571 0.058 0.122 0.152 0.150 0.438 0.402 0.169 0.490 0.008 0.003 0.001 2 1.36 9.63 26.04 28.53 250.55 222.53 42.84 386.80 0.20 0.03 0.00 1006.33 ωmgHOMO (ωm) (cm-1) 37.82 νm (cm-1) 0 290 411 661 1136 1271 1342 1473 1528 3157 3174 3202 ωm - νm (cm-1) 116 115 236 466 132 35 35 27 83 0 0 0 1245
pair, then some pairs are broken into two independent electrons. The resulting electrons, which are called quasi-particles, are subject to Fermi statistics. At finite temperatures, there are several molecules in which we cannot observe intramolecular supercurrent, and the number of such molecules increases with an increase in temperature. Electron pairs are gradually destroyed with increasing temperature and, accordingly, the energy gap ∆g decreases and vanishes at Tc. At Tc, no electrons are condensed into zero-momentum states and electron pairs are not formed, and intramolecular supercurrent would not be observed in any molecule. Such hypothetical intramolecular supercurrent originating from intramolecular vibrations in the monocations would be realized at higher temperature in annulenes than in acenes because the lHOMO values for annulenes are larger than those for acenes, even though the σintra,HOMO values for acenes are larger than those for annulenes. Multimode Problem To investigate how the consideration of the multimode problem is closely related to the characteristics of the electronphonon interactions, let us next handle the difficult multimode problem, which we ignore in the previous sections in this paper. It is important for us to treat the multimode problem because the vibronic coupling determines the vibrational frequencies. According to ref 40, the relevant ground-state energetics for the multimode problems in the case of e X (nE) are defined as
∆EHOMO ) lHOMO +
∆G0HOMO
(50)
0 The ∆GHOMO represents the lowest order quantum corrections that originate from the change of the frequencies of the vibronic modes as a consequence of the vibronic coupling. For example, the relevant ground-state energetics for the multimode problems in the case of e X (nE) are defined as
∆EHOMO )
1
pωmg2HOMO(ωm) + ∑ p(ωm - νm) ∑ 2 m m
(51)
where νm is the new frequency that is redetermined as a consequence of the vibronic coupling and is the solution (ν) of the equation
(
ωmg2HOMO(ωm))-1 ∑ ∑ m m
g2HOMO(ωm)ωm3 ωm2 - ν2
)1
(52)
This method is based on the approximate calculation of the new vibrational normal-mode frequencies, νm, arising after vibronic coupling. Of these modes, n - 1 falls in the interval between the successive original bare mode frequencies; the remaining one is a zero mode ν1 ) 0. The νm values for E2g vibrational modes in 18an are listed in Table 6. The relevant ground-state energetics, ∆EHOMO, for the multimode problems by using eq 51 in the monocations under consideration are listed in Table 0 values as well as the lHOMO 7. Furthermore, the ∆GHOMO 0 values are shown in Figure 3. The ∆GHOMO values are
TABLE 7: The Relevant Ground-State Energetics (in meV) for Multimode Problems in the Monocations ∆EHOMO (eq 51) ∆EHOMO (eq 53)
18an
30an
238.1 243.1
207.0 206.6
estimated to be 0.101 and 0.086 eV for 18an and 30an, 0 values were estimated respectively. Furthermore, the ∆GHOMO to be 0.110, 0.094, 0.093, 0.089, and 0.083 eV for 6an, 10ac, 14ac, 18ac, and 22ac, respectively. Therefore, compared to the 0 lHOMO values, the ∆GHOMO values do not change significantly with an increase in molecular size. The ∆EHOMO values are estimated to be 0.238 and 0.207 eV for 18an and 30an, respectively. Furthermore, the ∆EHOMO values were estimated to be 0.354, 0.268, 0.223, 0.197, and 0.177 eV for 6an, 10ac, 14ac, 18ac, and 22ac, respectively. Therefore, fundamental aspects of the relationships between the ∆EHOMO values and the molecular sizes and structures are similar to those between the lHOMO values and the molecular sizes and structures. This 0 is because the ∆GHOMO values do not change significantly with an increase in molecular size. Let us next look into O’Brien’s effective-mode approxima0 values as tion.41 According to ref 41, we estimate the ∆GHOMO well as the lHOMO values, defined in eq 53, for example, in the case of e X (nE)
∆EHOMO )
pωmg2HOMO(ωm) + ∑ m
{
p 4
}
ωm2g2HOMO(ωm) ∑ ωmg2HOMO(ωm) ∑ m m +
∑ m
ωmg2HOMO(ωm)
∑ m
g2HOMO(ωm)
(53)
The ∆EHOMO values estimated by this equation are listed in Table 7. We can see from this table that the ∆EHOMO values estimated by this equation are in excellent agreement with those estimated by eq 51. Concluding Remarks We studied the possible electron pairing in π-conjugated positively charged hydrocarbons, annulenes such as 18an and 30an. We estimated the electron-phonon coupling constants in the monocations of annulenes. The C-C stretching E2g mode of 1611 cm-1 that most strongly couples to the HOMO in 18an, and the E2g mode of 1201 cm-1 that most strongly couples to the HOMO in 30an. The 2-fold degenerate E2g modes much more strongly couple to the HOMO than the nondegenerate A1g modes in both 18an and 30an. The lHOMO values are estimated to be 0.137 and 0.121 eV for 18an and 30an, respectively. The lHOMO values for annulenes are larger than those for the same size of acenes. This can be understood in view of the phase patterns of the HOMO of annulenes and acenes. The HOMO is localized on an edge part of the carbon framework in acenes. The nonbonding characteristics are important in the HOMO in
18178 J. Phys. Chem. B, Vol. 110, No. 37, 2006 acenes, and the orbital interactions between two neighboring carbon atoms in the HOMO are weak in acenes. On the other hand, the HOMO is delocalized and the electron is distributed evenly over whole carbon framework in the HOMO of annulenes. Therefore, orbital interactions between two neighboring carbon atoms in the HOMO in annulenes are stronger than those in acenes. Therefore, the energy levels of the HOMO in annulenes are changed more significantly than those in acenes in the case where annulenes are distorted along the C-C stretching modes around 1500 cm-1. This is the reason the electron-phonon coupling constants originating from the C-C stretching modes around 1500 cm-1 in the monocations of annulenes are larger than those in the monocations of acenes and the reason the lHOMO values for annulenes are larger than those for acenes. The ωln,HOMO values are estimated to be 1231 and 1139 cm-1 for 18an and 30an, respectively. The ωln,HOMO values do not significantly decrease with an increase in molecular size in annulenes and acenes. We estimated the intrinsic intramolecular conductivity σintra,HOMO in the monocations of annulenes. The σintra,HOMO values for annulenes are smaller than those for acenes. This is because the lHOMO and ωln,HOMO values for annulenes are larger and smaller, respectively, than those for acenes. The σintra,HOMO values for annulenes increase with an increase in molecular size less rapidly than those for acenes. This is because the lHOMO values for annulenes decrease with an increase in molecular size less significantly than those for acenes and the ωln,HOMO values do not significantly change with an increase in molecular size in both annulenes and acenes. We also discussed the single charge transfer through the molecules under consideration, which is of interest for possible nanoelectronics applications. The REHOMO values are estimated to be 0.037 and 0.033 eV for 18an and 30an, respectively. The REHOMO values decrease with an increase in molecular size in annulenes. This means that the smaller the molecular sizes of annulenes are, the larger orbital overlap is needed for the monocation molecule to become a good conductor with rapid electron transfer. The REHOMO values for annulenes are larger than those for the positively charged acenes. This means that a larger orbital overlap is needed for positively charged annulenes to become a good conductor with rapid electron transfer than those for the positively charged acenes. The phase pattern differences between the HOMO localized on edge part of carbon atoms in acenes and the delocalized HOMO in annulenes are the main reason for the calculated results. The C-C stretching E2g modes around 1200-1500 cm-1 are the main modes converting the neutral structures to the monocations of annulenes. We discussed the conditions under which the attractive electron-electron interactions are realized in the monocations of annulenes. The σintra,HOMO values are inversely proportional 2 to the lHOMO values, and the REHOMO values are proportional to the lHOMO values. Therefore, large lHOMO values are favorable for the realization of the condition that the electron-electron interactions become attractive in the monocation crystals, however, not for the realization of the condition that the monocation crystals become normal metallic. The lHOMO values for annulenes are larger than those for acenes. Therefore, the monocations of annulenes cannot easily become good conductors, but the conditions under which the electron-electron interactions become attractive are realized more easily in the monocations of annulenes than in the monocations of acenes. There is an interesting paradox in conventional superconductivity; the higher the resistivity at room temperature, the more likely
Kato and Yamabe it is that a metal will be a superconductor when cooled.6a Therefore, the calculated results for the σintra,HOMO, REHOMO, and lHOMO values in the positively charged annulenes and acenes are in qualitative agreement with the interesting paradox; the resistivity of annulene monocation crystals would be higher at room temperature than those of acene monocation crystals; on the other hand, the condition of attractive electron-electron interactions would be more easily realized in annulene monocation crystals than in the monocation crystals of acene monocation crystals when cooled. Finally, we discussed the hypothetical intramolecular supercurrent originating from both intramolecular and intermolecular vibrations in the monocations of annulenes and acenes in the case where the distance between two neighboring molecules is too large for the molecular crystal to become a normal metallic state. 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-16560618, JSPS-17350094). References and Notes (1) (a) Shirakawa, H.; Louis, E. J.; MacDiarmid, A. G.; Chiang, C. K.; Heeger, A. J. J. Chem. Soc., Chem. Commun. 1977, 1977, 578. (b) Chiang, C. K.; Park, Y. W.; Heeger, A. J.; Shirakawa, H.; Louis, E. J.; MacDiarmid, A. G. Phys. ReV. Lett. 1977, 39, 1098. (c) Chiang, C. K.; Druy, M. A.; Gau, S. C.; Heeger, A. J.; Louis, E. J.; MacDiarmid, A. G.; Park, Y. W.; Shirakawa, H. J. Am. Chem. Soc. 1978, 100, 1013. (2) (a) Bersuker, I. B. The Jahn-Teller Effect and Vibronic Interactions in Modern Chemistry; Plenum: New York, 1984. (b) Bersuker, I. B.; Polinger, V. Z. Vibronic Interactions in Molecules and Crystals; Springer: Berlin, 1989. (c) Bersuker, I. B. Chem. ReV. 2001, 101, 1067. (d) Bersuker, I. B. The Jahn-Teller Effect; Cambridge University Press: Cambridge, 2006. (3) Kivelson, S.; Heeger, A. J. Synth. Met. 1988, 22, 371. (4) Grimvall, G. The Electron-Phonon Interaction in Metals; NorthHolland: Amsterdam, 1981. (5) (a) Schrieffer, J. R. Theory of SuperconductiVity; Addison-Wesley: Reading, MA, 1964. (b) de Gennes, P. G. SuperconductiVity of Metals and Alloys; Benjamin: New York, 1966. (c) Mizutani, U. Introduction to the Electron Theory of Metals; Cambridge University: Cambridge, 1995. (6) (a) Kittel, C. Quantum Theory of Solids; Wiley: New York, 1963. (b) Ziman, J. M. Principles of the Theory of Solids; Cambridge University Press: Cambridge, 1972. (c) Ibach, H.; Lu¨th, H. Solid-State Physics; Springer: Berlin, 1995. (7) Yamabe, T. Conjugated Polymers and Related Materials; Oxford University Press: Oxford, 1993; p. 443. (8) Little, W. A. Phys. ReV 1964,. 134, A1416. (9) (a) Je´rome, D.; Mazaud, A.; Ribault, M.; Bechgaad, K. J. Phys. Lett. (Paris) 1980, 41, L95. (b) Ribault, M.; Benedek, G.; Je´rome, D.; Bechgaad, K. J. Phys. Lett. (Paris) 1980, 41, L397. (c) Je´rome, D.; Schulz, H. J. AdV. Phys. 1982, 31, 299. (d) Ishiguro, T.; Yamaji, K. Organic Superconductors; Springer: Berlin, 1990. (e) Williams, J. M.; Ferraro, J. R.; Thorn, R. J.; Karlson, K. D.; Geiser, U.; Wang, H. H.; Kini, A. M.; Whangbo, M.-H. Organic Superconductors; Prentice Hall: Englewood Cliffs, NJ, 1992. (10) Oshima, K.; Urayama, H.; Yamochi, H.; Saito, G. J. Phys. Soc. Jpn. 57, 1988, 730. (11) (a) Hebard, A. F.; Rosseinsky, M. J.; Haddon, R. C.; Murphy, D. W.; Glarum, S. H.; Palstra, T. T. M.; Ramirez, A. P.; Kortan, A. R. Nature 1991, 350, 600. (b) Rosseinsky, M. J.; Ramirez, A. P.; Glarum, S. H.; Murphy, D. W.; Haddon, R. C.; Hebard, A. F.; Palstra, T. T. M.; Kortan, A. R.; Zahurak, S. M.; Makhija, A. V. Phys. ReV. Lett. 1991, 66, 2830. (12) Tanigaki, K.; Ebbesen, T. W.; Saito, S.; Mizuki, J.; Tsai, J. S.; Kubo, Y.; Kuroshima, S. Nature 1991, 352, 222. (13) Palstra, T. T. M.; Zhou, O.; Iwasa, Y.; Sulewski, P. E.; Fleming, R. M.; Zegarski, B. R. Solid State Commun. 1995, 93, 327. (14) (a) Varma, C. M.; Zaanen, J.; Raghavachari, K. Science 1991, 254, 989. (b) Lannoo, M.; Baraff, G. A.; Schlu¨ter, M.; Tomanek, D. Phys. ReV. B 1991, 44, 12106. (c) Asai, Y.; Kawaguchi, Y. Phys. ReV. B 1992, 46, 1265. (d) Faulhaber, J. C. R.; Ko, D. Y. K.; Briddon, P. R. Phys. ReV. B 1993, 48, 661. (e) Antropov, V. P.; Gunnarsson, O.; Lichtenstein, A. I. Phys. ReV. B 1993, 48, 7651. (f) Gunnarsson, O. Phys. ReV. B 1995, 51, 3493. (g) Gunnarsson, O.; Handschuh, H.; Bechthold, P. S.; Kessler, B.;
Electron Pairing in Annulenes Gantefo¨r, G.; Eberhardt, W. Phys. ReV. Lett. 1995, 74, 1875. (h) Dunn, J. L.; Bates, C. A. Phys. ReV. B 1995, 52, 5996. (i) Gunnarsson, O. ReV. Mod. Phys. 1997, 69, 575. (j) Devos, A.; Lannoo, M. Phys. ReV. B 1998, 58, 8236. (k) Gunnarsson, O. Nature 2000, 408, 528. (15) Haddon, R. C. Acc. Chem. Res. 1992, 25, 127. (16) (a) Silinch, E. A.; Capek, V. Organic Molecular Crystals; AIP: New York, 1994. (b) Warta, W.; Karl, N. Phys. ReV. B 1985, 32, 1172. (c) Warta, W.; Stehle, R.; Karl, N. Appl. Phys. A 1985, 36, 163. (17) (a) Kivelson, S.; Chapman, O. L. Phys. ReV. B 1983, 28, 7236. (b) Mishima, A.; Kimura, M. Synth. Met. 1985, 11, 75. (18) Salem, L. Molecular Orbital Theory of Conjugated Systems; Benjamin: New York, 1966. (19) Ooshika, Y. J. Phys. Soc. Jpn. 1957, 12, 1238, 1246. (20) Longuet-Higgins, H. C.; Salem, L. Proc. R. Soc. London 1959, A251, 172. (21) Cizek, J.; Paldus, J. J. Chem. Phys. 1967, 47, 3976. (22) (a) Binsch, G.; Tamir, I.; Hill, R. D. J. Am. Chem. Soc. 1969, 91, 2446. (b) Binsch, G.; Tamir, I. J. Am. Chem. Soc. 1969, 91, 2450. (23) (a) Nakajima, T. Topics in Current Chemistry; Springer: Heidelberg, Germany, 1972; Vol. 32, p 1. (b) Nakajima, T.; Toyota, A.; Fujii, S. Bull. Chem. Soc. Jpn. 1972, 45, 1022. (24) Pauling, L. J. Chem. Phys. 1936, 4, 673. (25) London, F. J. Phys. Radium 1937, 8, 397. (26) McWeeny, R. Mol. Phys. 1958, 1, 311. (27) Haddon, R. C. J. Am. Chem. Soc. 1979, 101, 1722. (28) (a) Squire, R. H. J. Phys. Chem. 1987, 91, 5149. (b) Yamabe, T.; Yoshizawa, K.; Matsuura, Y.; Tanaka, K. Synth. Met. 1995, 75, 55. (29) (a) Kato, T.; Yamabe, T. J. Chem. Phys. 2001, 115, 8592. (b) Kato, T.; Yamabe, T. Recent Research DeVelopments in Quantum Chemistry; Transworld Research Network: Kerala, 2004. (c) Kato, T.; Yamabe, T. Recent Research DeVelopments in Physical Chemistry; Transworld Research Network: Kerala, 2004. (d) Kato, T.; Yamabe, T. J. Chem. Phys. 2004, 120, 7659. (e) Kato, T.; Yamabe, T. J. Chem. Phys. 2003, 118, 3804.
J. Phys. Chem. B, Vol. 110, No. 37, 2006 18179 (30) (a) Coropceanu, V.; Filho, D. A. da Silva; Gruhn, N. E.; Bill, T. G.; Bre´das, J. L. Phys. ReV. Lett. 2002, 89, 275503. (b) Bre´das, J. L.; Beljonne, D.; Coropceanu, V.; Je´rome, C. Chem. ReV. 2004, 104, 4971. (31) Conwell, E. M. Phys. ReV. B 1980, 22, 1761. (32) (a) Albright, T. A.; Burdett, J. K.; Whangbo, M.-H. Orbital Interactions in Chemistry; Wiley: New York, 1985. (b) Burdett, J. K. Chemical Bonding in Solids; Oxford University: New York, 1995. (33) Yoshizawa, K.; Yahara, K.; Tanaka, K.; Yamabe, T. J. Phys. Chem. B 1998, 102, 498. (34) (a) Becke, A. D. Phys. ReV. A 1988, 38, 3098. (b) J. Chem. Phys. 1993, 98, 5648. (35) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785. (36) (a) Ditchfield, R.; Hehre, W. J.; Pople, J. A. J. Chem. Phys. 1971, 54, 724. (b) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 213. (37) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.; Paterson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P. M. W.; Johnson, B. G.; Chen, W.; Wong, M. W.; Andres, J. L.; Head-Gordon, M.; Replogle, E. S.; Pople, J. A.; Gaussian 98; Gaussian Inc.: Pittsburgh, PA, 1998. (38) Fischer, J. E.; Heiney, P. A.; Smith, A. B. Acc. Chem. Res. 1992, 25, 112. (39) McMillan, W. L. Phys. ReV. 1968, 167, 331. (40) Manini N.; Tosatti, E. Phys. ReV. B 1998, 58, 782. (41) O’Brien, M. C. M. Phys. ReV. B 1996, 53, 3775.
