We analyze the effect of an external electric field on the electronic structure of molecules which have been recently studied as molecular wires or diodes. We use a self-consistent tight binding technique which provides results in good agreement with ab initio calculations and which may be applied to a large number of molecules. The voltage dependence of the molecular levels is mainly linear with slopes intimately related to the electronic structure of the molecules. We emphasize that the response to the applied voltage is an important feature which governs the behavior of a molecular device.
Recent measurements of single molecule transport properties [1][2][3] represent important contributions to the eventual realization of molecular electronics. This is accompanied by an increasing theoretical effort to understand the relationship between the electronic structure of the molecules and the current-voltage I(V ) characteristics. Until recently, most of the calculations based on tight binding (TB) or ab initio methods 1 did not explicitly include the effect of the applied voltage on the electronic structure of the molecules. This is a severe drawback since several volts can be applied to the electrodes in contact with a single molecule. In the some cases 4 , the energy of the molecular electronic levels is approximated by a linear function of the voltage but its slope is not calculated or is approximated by 1/2.
An approach to overcome these limitations has been proposed 5,6 recently on the basis of a Hubbard Hamiltonian 7 and has been applied to the azulene molecule 5 and to polyacene wires 6 . But the Hubbard Hamiltonian is a model Hamiltonian which only gives qualitative predictions. On the other hand, ab initio methods cannot be easily applied to complex systems. Thus there is a need for simpler methods which correctly predict the response of the molecules and the dependence of the electronic levels as function of the applied bias. This is a prerequisite to calculate the I(V ) characteristics of a molecular device. In this paper, we present self-consistent TB calculations of various molecules. We study the evolution of the dipole and of the electronic levels with the voltage. We obtain results in good agreement with ab initio calculations in the local density approximation (LDA). This is a good test of our method and allows to extend it to large systems. We conclude that our self-consistent TB technique is a promising approach to study the transport properties in a molecular device and that it may be applied to a wide range of systems.
The molecules considered in this work are shown in Figure 1. In order to compare with previous results 5,6 , we study tetracene and azulene molecules as prototypes of molecular wires. The calculations are also applied to quinolinium tricyanoquinodimethanide (Q-3CNQ) and to 3,5-dinitrobenzyl 7-(1-oxohexylamino)-pyren-2-ylcarbamate (OHAPy-C-DNB) which are respectively D-π-A and D-σ-A molecules. D and A are respectively elec-tron donor and acceptor. π and σ are respectively “pi” and “sigma” bridges. These two molecules are intensively studied to make molecular diodes 8,9 . The electronic structure of all the molecules is obtained in LDA and in TB. In LDA, we use the DMOL code 11 with a double numerical basis set (two atomic orbitals for each occupied orbital in the free atom) together with polarization functions (2p for H, 3d for N, O and C). The exchange-correlation energy is approximated by the density functional of ref 12 . The self-consistent TB technique is presented in another publication 13 where we calculate the electronic structure of thienylenevinylene oligomers, showing a good agreement with LDA calculations and experiments. C, N, O atoms are represented by one s and three p atomic orbitals, and H atoms by one s orbital. The non-diagonal terms of the Hamiltonian matrix H are restricted to first nearest-neighbor interactions and to two-center integrals. They depend on the interatomic distance following Harrison’s rules 14 . The self-consistency is incorporated in the diagonal terms H iα,iα (α is the orbital index, i is the atomic index)
where H 0 iα,iα define the atomic levels, U is the intra-atomic Coulomb energy, R ij is the distance between atoms i and j, and Q j is the net charge on the atom j. The TB parameters are determined by a fit of the LDA electronic structure of simple molecules 13 . The parameters for C-C and C-H interactions are given in Ref 13 , those for C-N, N-H, C-O, N-O interactions are in Table I. The selfconsitency is obtained with an usual iterative method which is converged when the atomic charges Q j in eq. ( 1) correspond to the charges calculated from the eigenstates of the Hamiltonian. V ext in eq. ( 1) is the electrostatic potential resulting from the applied electrostatic field which, for simplicity, is assumed homogeneous, corresponding to a situation where the electrodes would be far from the molecule. Obviously, to calculate the I(V ) curve, we should consider the chemical and electrostatic interactions between the molecule and the electrodes. This can be done in TB but this is beyond the scope of the present paper.
The applied electric field E is parallel to the long axis (z) of the molecules (Figure 1).
The origin of the electrostatic potential (V ext = 0 ) is defined at 2 Å from the left side of the molecules. Thus V ext (i) = Ez i where z i is the coordinate of the atom i along the axis z. All the results presented in this paper are plotted as a function of the electrostatic potential at 2 Å from the right side of
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