Extended Hückel Method Calculation of Polarization Energies: The

Article pubs.acs.org/JPCA

Extended Hü ckel Method Calculation of Polarization Energies: The Case of a Benzene Dimer Raphael M. Tromer and José A. Freire* Departamento de Física, 81531-990, Curitiba-Pr, Brazil

ABSTRACT: We adapted Hoffmann’s extended Hückel method to an interacting molecular system and use this approach to compute the electron affinity and ionization potential of benzene dimers. We restrict the added charge to one of the molecules and argue that the dimer energy computed in this manner is the relevant energy in any meaningful thermally activated hopping rate expression. The dimer electron affinity and ionization potential differs from the isolated molecule corresponding quantity by what is called polarization energy. The polarization energy normally stabilizes the anion and this is particularly relevant for benzene, given that its isolated anion is unstable with respect to charge detachment. We found that the anionic benzene dimer is only stabilized in certain conformations, suggesting that the stabilization of a benzene anion in an amorphous environment is very unlikely. The modest computational cost of the method makes it a viable alternative to compute the energy of charged molecules in amorphous molecular films, a central issue in the problem of charge transport in organic electronics.

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final quantum states and λ is the combined reorganization energy of the two molecules. What concerns us in this paper are the energies Ei and Ef. In the initial state molecule i is charged and molecule f is neutral and in the final state the reverse is true. The energy difference in the hopping rate can therefore be viewed as the difference in adiabatic electron affinities (in the case of electron hops) or adiabatic ionization potentials (in the case of hole hops) of the two molecules in the amorphous environment. The energies Ei and Ef contain the following: (i) the internal energies of the charged molecule and its neutral neighbors and (ii) the electrostatic interaction among the molecules. The electron affinity (ionization potential) of a molecule in the condensed phase is usually larger (smaller) than the corresponding quantities in the gas phase, see Figure 1. The energy difference is called polarization energy and is usually estimated to be of the order of 1.5 eV in typical films of organic molecules; see for instance ref 6. The polarization energy of a given molecule is a direct function of its particular environment, the dispersion in the polarization energies is a measure of the

INTRODUCTION Amorphous films of organic molecules are used in organic field effect transistors,1 light-emitting diodes,2 memory devices3 and various other applications. The calculation presented in this paper is relevant to the problem of charge transport in this type of system. The electronic coupling between the molecules in the amorphous film is usually very weak, which leads to localized electronic states confined to the molecules. The charge transport then occurs via thermally activated hops, the electron or hole absorbs or emits vibrational energy to move from one molecule to another. The theoretical description of the charge transport in such systems usually resort to model hopping rates. The Marcus rate, or small-polaron hopping rate,4,5 is a favorite when dealing with hopping between organic molecules, where the molecular reorganization energy is expected to play an important role. The Marcus rate for the hopping from molecule i to molecule f is Wi → f = |Jifep |2

⎡ (E − E + λ)2 ⎤ π f ⎢− i ⎥ exp 2 ⎢⎣ ⎥⎦ 4λkT ℏ λkT

(1)

Received: October 17, 2013 Revised: December 4, 2013 Published: December 4, 2013

In this expression, Jep if is the matrix element of the electronic part of the electron−phonon coupling between the initial and © 2013 American Chemical Society

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electrostatic environment, the general aspects of which are discussed in ref 23. A central assumption of this work is that the charge is entirely confined to one of the molecules. The justification behind this assumption is that there are in fact two time-scales for the charge transfer between two molecules in an amorphous environment: W−1, due to the thermally activated process, and ℏ/J, due to the residual quantum coupling between the two molecules (this J is related to the purely electronic part of the Hamiltonian, it is not equal to the Jep in eq 1). The energies to be obtained here are the relevant energies for a pair of molecules where the thermal process is faster than the purely quantum process (W−1 < ℏ/J). When this does not hold one must consider the dimer as a single molecular entity and use its ground state as initial or final state in eq 1.

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THE EXTENDED HÜ CKEL METHOD CALCULATION OF POLARIZATION ENERGIES The adiabatic electron affinity and ionization potential of the benzene dimer (Ad and Id) will be displayed below in reference to the corresponding quantities of the isolated molecule (Ag and Ig). The polarization energy is the difference between these energies. More precisely

Figure 1. Illustration of the polarization energy concept. The gas phase electron affinity (Ag) and ionization potential (Ig) are normally enhanced and depressed respectively in the condensed phase (Ac and Ic). In an ordered crystal the polarization energy shows little or no dispersion (the molecular level just gives rise to a band of width ∼4J), whereas in an amorphous molecular environment the dispersion can be quite significant. The anion and the cation polarization energies are EP− and EP+ respectively. Adapted from ref 8.

morphological disorder of the film and is typically of the order of 100 meV.7,8 Recently one of us suggested9 that the polarization energy provides a physical mechanism for the correlation in the molecular energy distribution. This correlation was shown to be the cause behind the charge mobility dependence on the exponential of the square-root of the applied electric field observed in a variety of disordered organic materials in time-offlight measurements.10−12 To predict the charge carrier mobility in an amorphous molecular environment one must then be able to compute the adiabatic electron affinity and ionization potential of each molecule in the system. An atomistic description of the charged molecule and its nearby environment is necessary, whereas the more distant molecules may be properly described by a polarizable continuum model (PCM).13,14 For recent atomistic computations of the charge mobility in disordered molecular systems see refs 15−19. This paper proposes an atomistic method to compute the electron affinity and the ionization potential of a molecule surrounded by neutral molecules using Hoffmann’s extended Hückel model.20 This method differs from recently published methods15,19 in that it describes the electrons quantummechanically. The method is here applied to compute the adiabatic electron affinity and ionization potential of benzene dimers. We chose benzene as a prototype of the organic molecules actually used in organic devices and also to investigate the issue of the stability of the benzene anion in the condensed phase. The benzene anion is unstable against charge detachment in the gas phase since its electron affinity is negative, Ag = −1.12 eV.21 Recently it was argued that this anion may be stabilizedi.e., its affinity may become positivein the vicinity of charged electrodes.22 In this work, we will evaluate the possibility of the benzene anion being stabilized in an amorphous environment by polarization effects of neighboring neutral molecules. Both situations are similar and deal with a molecule in an

EP− = Ag − Ad = Ed( −) − Ed(0) + Ag

(2)

EP+ = Ig − Id = Ed(0) − Ed( +) + Ig

(3)

In this expression, Ed is the dimer total energy and the argument is the dimer charge. When EP is negative the ion is stabilized in the dimer. Figure 1 displays the expected behavior in the condensed phase, E−P < 0 and E+P < 0. This expectation is based solely on the sign of the charge-induced dipole interaction, ignoring other electrostatic interactions and internal energies. We shall see that these other energies play a decisive role in certain conformations of the benzene dimer. In the results presented below the geometry of the benzene molecules, in all three charge states, were obtained with the GAMESS package24 using DFT(B3LYP/TZV). In order (i) to describe the electrons quantum mechanically, (ii) to have the charge confined to a given molecule in the dimer, and (iii) to do the calculation in the most economical way, we opted for the extended Hückel method (EHM).20 Abinitio methods have difficulties with point ii because, with the charge entirely confined to one of the molecules, the quantum state we are considering is not an eigenstate of the electronic Hamiltonian. Besides, ab initio methods would be too costly to handle situations of a charged molecule surrounded by several other molecules, which is where we envisage applying the method presented here. Other semiempirical methods could be tried but the EHM is the most economical alternative. The EHM uses a minimum set of Slater-type atomic orbitals as a basis and computes the molecular orbitals from the generalized eigenvalue problem H ·cn = EnS ·cn

(4)

The vector cn contains the coefficients of the nth molecular orbital. The atomic orbitals overlap matrix S and ionization potentials Iμ completely determine the Hamiltonian matrix (H0) in the absence of external fields: 0 Hμμ = −Iμ

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Figure 2. Mulliken atomic charges in a negatively charged benzene dimer (the left benzene has total charge −e) with a molecular separation of 10 Å. The numbers are the charges of the isolated molecules, the change caused by the interaction is shown in parentheses. Charge is in units of e. The polarization effect is much more pronounced in the carbons of the neutral benzene. 0 Hμν = −KSμν

with

(Iμ + Iν)

(μ ≠ ν )

2

where E(k) levels is the sum of the occupied levels energies of molecule k, Unn is the Coulomb interaction energy between nuclei in different molecules and Uee is the Coulomb interaction energy between the Mulliken electronic charges in different molecules. The last two terms are needed because the sum of the two Elevels misses Unn and includes Uee twice. In fact, each Elevels contains an internal energy part and an electrostatic energy part. The internal energy part can be viewed as the sum of the isolated molecule ground-state energy and the (positive) energy cost of polarizing the isolated molecule. The electrostatic part corresponds to the interaction energy between the molecule’s polarized electrons and all nuclei and polarized electrons of the other molecule. Therefore, Ed contains the internal energies of all molecules and all intermolecular electrostatic interactions. Since the energy calculation to be performed demanded a proper description of the electron affinity and ionization potential of benzene, as well as a proper description of polarization effects, we optimized the constant k in eq 7 for the neutral, cationic and anionic benzene so as to fit the following: (i) the neutral benzene experimental average polarizability; (ii) the cationic and anionic benzene average polarizabilities obtained from ab initio DFT(B3LYP/TZV); (iii) the experimental (vertical) electron affinity; (iv) the experimental (vertical) ionization potential. The target values and the fit obtained are displayed in Table 1. The method is easily extended to a large molecular ensemble. Its computational cost is dominated by the solution of the

(6)

25

K = k + Δ2 + (1 − k)Δ4

(7)

where k is a constant and Δ = (Iμ − Iν)/(Iμ + Iν). We used the ionization potentials and atomic orbitals zetas of the YAeHMOP package.26 The parameter k was adjusted as explained below. When an external electric field is present, E(r) = −∇φ(r), one adds matrix elements of the operator −eφ(R̂ ) to the matrix H0. The new Hamiltonian matrix (H) is defined as follows: 0 Hμμ = Hμμ − eφi

(μ in atom i)

0 Hμν = Hμν + e Ei ·⟨μ|δ R̂ i|ν⟩ 0 Hμν = Hμν − eφ ij Sμν

(8)

(μ , ν in atom i)

(9)

(μ in atom i , ν in atom j)

(10)

φi and Ei are the electrostatic potential and the electric field in atom i and φ⟨ij⟩ is the potential at the midpoint between atoms i and j. δR̂ i is the position operator from the center of atom i, its matrix elements between Slater-type orbitals can be found in ref 27. The second term in eq 9 is a hybridization induced by the external field that was missed in ref 28, but that is crucial to make a planar molecule such as benzene polarizable perpendicular to its plane. For the dimer calculation we zeroed the overlap matrix elements between atomic orbitals belonging to different molecules, so as to confine the charge to a single molecule, and considered the electrostatic potential at each atom as being due to the Mulliken atomic charges of the other molecule. The eigenvalue problem of eq 4 then separated into a set of eigenvalue problems, one per molecule, coupled only through the Mulliken charges. The system was solved iteratively until convergence of the Mulliken charges. Figure 2 shows an example of the converged Mulliken charges of a charged dimer showing the polarization effect. The total energy of the dimer is (1) (2) Ed = E levels + E levels + Unn − Uee

Table 1. Molecular Parameters Used To Fit the Constants k of the Extended Hückel Method for Neutral and Ionic Benzenea α0 [Å3] target EHM

10.00 9.03

29

α+ [Å3] 8.62 11.10

α− [Å3] 11.71 10.38

Avert [eV] g −1.12 −1.10

21

Ivert [eV] g 9.2430 8.93

a

The ionic benzene target polarizabilities were obtained from ab-initio DFT(B3LYP/TZV). The k-values that produced the fit were k0 = 1.99, k+ = 2.02, and k− = 1.93. With these values the EHM adiabatic electron affinity is −0.67 eV (an anion relaxation energy of 430 meV) and the adiabatic ionization potential is 8.76 eV (a cation relaxation energy of 170 meV).

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Figure 3. Polarization energy of an anionic benzene dimer as a function of the molecular separation, the left benzene has a charge −e. This can be viewed as the energy difference between the electron affinity of an isolated benzene and the electron affinity of the dimer. A negative (positive) polarization energy means that the charged dimer is stabilized (destabilized) against charge detachment. The red curve corresponds to the extended Hückel calculation. The black curves correspond to a calculation using just the electrostatic interaction between the fixed (FC, dashed line) and polarized (PC, solid line) Mulliken atomic charges of the two molecules.

Figure 4. Polarization energy of a cationic benzene dimer as a function of the molecular separation. The left benzene has a charge +e. This can be viewed as the energy difference between the ionization potential of an isolated benzene and the ionization potential of the dimer. A negative (positive) polarization energy means that the charged dimer is stabilized (destabilized) against charge detachment. The red curve corresponds to the extended Hückel calculation. The black curves correspond to a calculation using just the electrostatic interaction between the fixed (FC, dashed line) and polarized (PC, solid line) Mulliken atomic charges of the two molecules.

the cost of computing the field at each atom, which scales as NiterNmol2Nat2. Fixed-Charge Energy vs Polarized-Charge Energy. As mentioned above, part of the dimer energy can be viewed as being due to the intermolecular electrostatic interaction among polarized electrons and nuclei. This energy is always lower than

eigenvalue problem (4). If Nmol molecules of Nat atoms each are present, one has Nmol eigenvalue problems of size Nat to solve. Taking the number of iterations necessary for convergence as Niter we estimate this computational cost as NiterNmolNat3. In addition to the cost of solving the eigenvalue problem there is 14279

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the electrostatic energy among the fixed (unpolarized) electrons and nuclei, this later energy being the electrostatic energy that would be present if the charges of the isolated molecules were kept fixed while the molecules were brought together. In ref 15, a method was proposed to compute the energy of a charged molecule surrounded by neutral molecules using just this former energy. The lowering of the electrostatic interaction when molecules are brought together, brought about by the negative chargeinduced dipole interaction present in EP(±), is what leads one to naively expect the destabilization of a cation in a molecular cluster, EP+ = −Ed(+) + Ed(0) + Ig > 0, and the stabilization of an anion, EP− = −Ed(−) − Ed(0) + Ag < 0, as shown schematically in Figure 1.31 However this analyses ignores the fixed-charges energy and the internal energies also present in Ed. The former, in the case of a charged dimer of apolar molecules like benzene, is a charge-quadrupole interaction that can be positive or negative depending on the molecules relative orientation. In fact, at large molecular separations R, the charge-quadrupole energy in Ed(±) goes as ∼R−3 and dominates over the electrostatic relaxation energy that goes as ∼R−4. The fixed-charge energy in Ed(0) is a much weaker quadrupole−quadrupole interaction.

The cation dimer results are more interesting since this dimer actually exists, it was first observed experimentally in the gas phase by Ohashi and Nishi.32 Recently the cation dimer morphology was elucidated by the ab initio calculations of Tachikawa.33 In this work the whole ionization process of a neutral dimer was investigated. The neutral benzene dimer most favorable configuration is a T-shape conformation similar to conformations b and c in both figures, but with the hydrogen of one benzene directly above the center of mass of the other benzene. The center-of-mass separation was determined experimentally34 to be 4.96 Å. In ref 33, it was shown that (i) in the cationic dimer, 93% of the positive charge remains in the benzene whose hydrogen points to the center of the other benzene, similar (but not exactly equal) to Figure 4c, and (ii) the ionization energy of benzene is decreased with dimer formation by 100 meV, in our notation EP+ = Ig − Id = 100 meV. The first result further justifies our interest for studying charge entirely confined to a single molecule and is reproduced in our results since, of the two T-shaped dimers of Figures 4b and 4c, the one in Figure 4c has the lowest energy. The second result is only qualitatively reproduced since, at 5 Å separation in Figure 4c, we obtain EP+ = 163 meV. But it must be remembered that the dimer geometries are not exactly the same and that in our calculation the charge is fully confined to one of the benzenes. The preference of the hole for one of the benzenes in the Tshaped dimer is mirrored by the preference of the electron for precisely the other benzene in the T-shaped dimer; compare parts b and c of Figure 3. These preferences are entirely due to polarization effects and raise the question about the possibility of spontaneous exciton dissociation in a T-shaped benzene dimer. In a hopping rate expression this would imply that the hole and electron hopping rates would not be symmetrical between two benzenes in a T-shaped geometry, the electron and the hole tending to go in opposite directions.

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RESULTS In parts a−d of Figure 3, we show the polarization energy (EP−) of four different conformations of an anionic benzene dimer as a function of the distance between the molecule centers. In parts a−d of Figure 4, we show the polarization energy of the same dimers in their cationic version (EP+). In all figures, we compare the full EHM result (red curves) with the part of the polarization energy that can be attributed to the electrostatic interaction between fixed (unpolarized) charges (dashed black curves) and the part that can be attributed to the electrostatic interaction between polarized charges (solid black curves). In trying to read the charged dimer energy from these figures one must remember that EP− = −Ed(−) − Ed(0) + Ag and that EP+ = −Ed(+) + Ed(0) + Ig. Hence, whereas the dashed black lines in Figure 3a−d basically express the charge−quadrupole interaction between the charged and neutral benzenes, the dashed black lines in Figure 4a−d express minus this same interaction. The lowering of the electrostatic interaction due to the polarization effect appears as a negative energy difference between solid and dashed black lines in Figure 3a−d and as a positive energy difference in Figure 4a−d. In conformations c and d of both figures, the neutral benzene is polarized perpendicular to its plane and the energy difference is negligible. In both figures the energy difference between the EHM result (red curve) and the polarized charge result (solid black curve) expresses the molecules internal energy. We see that only in conformations 3a and 3b the anion is stabilized by the presence of the neutral benzene, but the stabilization is not strong enough to compensate for the (−670 meV) and render Ad = Ed(0) − Ed(−) negative Aadia g positive. The anion dimer is unstable against charge detachment. The question raised in the introduction about the possible stabilization of the benzene anion in an amorphous environment must therefore be answered negatively, since the tendency to stabilization of certain conformations, such as 3a and 3b, will be counteracted by other conformations, such as 3c and 3d, in the amorphous state.

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CONCLUSIONS We have presented a new way to compute the polarization energy of a charged molecule in the vicinity of neutral molecules using a quantum-mechanical description of the molecules via Hoffmann’s extended Hückel method. We have confined the charge to one of the molecules and argued that the polarization energy computed in this manner is the relevant energy in any hopping rate expression. The choice of the extended Hückel approach was based on a compromise between describing the molecules quantum-mechanically, confining the charge to a single molecule and lowering as much as possible the computational cost. The results obtained for four different conformations of a benzene dimer revealed that: (i) the anionic dimer is unstable against charge detachment, (ii) the benzene anion is unstable in an amorphous environment, (iii) the cationic dimer is stable and, in the T-shape geometry, the charge gets spontaneously confined to the benzene that cuts through the face of the other, and (iv) the method produces a reduction of the ionization potential in reasonable agreement with ab initio calculations. The method is certainly viable to be applied to large molecular ensembles (Nmol ∼ 103) and our next step will be to investigate the influence of polarization effects on the distribution of site-energies in an amorphous molecular system, similar to what was done in ref 19. 14280

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AUTHOR INFORMATION

Corresponding Author

*(J.A.F.) E-mail: jfreire@fisica.ufpr.br. Telephone: 55-41-33613002. Fax: 55-41-3361-3418. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS R.M.T. thanks CNPq for financial support and Thiago C. Freitas and Cristiano F. Woellner for fruitful discussions. J.A.F. thanks INEO for financial support.

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