Role of the virtual orbitals and HOMO-LUMO gap in mean-field approximations to the conductance of molecular junctions

A. Cehovin, H. Mera, J. H. Jensen, K. Stokbro, and T. B. Pedersen

Phys. Rev. B 77, 195432 – Published 22 May 2008

Abstract

We investigate the role of virtual orbitals on the molecular conductance at the static mean-field level of approximation in the nonequilibrium Green’s functions (NEGFs) formalism, within a model system with wide-band leads. We use a projector operator formalism to define corrections to the molecular Hamiltonian that solely act on its virtual-orbital subspace, leaving ground-state properties invariant. In its most trivial limit, these corrections reduce to the scissor operator. More generally, we also propose a parameterless correction aimed toward improved fundamental energy gap estimates in the context of local self-energy approximations. We demonstrate how within any mean-field description dramatical differences in conductance values can be generated with the application of the scissor operators. For our model system, we show how the dramatical difference in conductance values obtained from the restricted Hartree–Fock and Hartree–Fermi–Amaldi approximations can be related to the different HOMO-LUMO gaps of each method and to a lesser extent differences in the molecular orbitals. Finally, we comment on the impact of different coupling-strength regimes and illustrate how the $E_{g}$ -related differences in the two mean-field descriptions gradually lose their impact on the conductance with increasing molecular coupling to the leads. Our analysis points out some implications of the pragmatic use of the Kohn–Sham (KS) potentials in the NEGF-KS theory, and how to modify the scheme toward better physical consistency so that the resulting Hamiltonian is more physical and the position of the transmission resonances is improved.

Received 21 November 2007

DOI:https://doi.org/10.1103/PhysRevB.77.195432

Authors & Affiliations

A. Cehovin¹, H. Mera¹, J. H. Jensen², K. Stokbro³, and T. B. Pedersen⁴

¹Niels Bohr Institute and Nano-Science Center, University of Copenhagen,Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
²Department of Chemistry, H. C. Ørsted Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
³Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen Ø, Denmark
⁴Atomistix A/S, c/o Niels Bohr Institute, Juliane Maries Vej 30, DK-2100 Copenhagen Ø, Denmark

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Vol. 77, Iss. 19 — 15 May 2008

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Figure 1
Model system considered in this work. A diaminobenzene molecule is attached to metallic leads, which are accounted for by means of model self-energies $Σ_{L / R}$ [see Eq. (5)]. Carbon and nitrogen atoms are labeled by their chemical symbols in the plot, while hydrogen atoms are represented by small black dots. The leads are assumed to couple only to the nitrogen atoms. The isolated DAB molecule is closed shell, simplifying both the calculation and the analysis of our results.Reuse & Permissions
Figure 2
Transmission spectra of the system in Fig. 1, with three different choices of Hamilton operators in Eq. (2). The Hartree approximation is the Hartree-type field of Eq. (7), while the Hartree–Fock is the standard RHF. The dotted line is the Hartree–Hamiltonian with shifted virtual orbitals, as defined in Eqs. (15, 16, 17) so as to fit $E_{g}^{RHF}$ . The Fermi energy in each approximation is determined by requiring molecular charge neutrality.Reuse & Permissions
Figure 3
(a) Linear-response conductances of the system in Fig. 1 as a function of coupling strength parameter $Γ$ . The Hamilton operator is approximated by using the parametrized Hartree–Fermi–Amaldi approximation (HA) of Eq. (7) (solid), HA with virtual orbitals shifted so that for the isolated molecule $E_{g} = E_{g}^{HF}$ (dashed), and RHF (dotted). The vertical line shows the position of the Hartree–Fock charging energy, which is defined as $U_{HF} = E_{g}^{RHF} / 2$ . This corresponds to the crossover regime, $Γ / U = 1$ , for approximations with $E_{g} = E_{g}^{RHF}$ . (b) In this figure, the parameter-dependent scissor operator has been replaced with the extended HA, which is defined by Eq. (20). The result is shown to be similar to the scissor-corrected HA in (a), with larger conductances in the weak and crossover regimes.Reuse & Permissions
Figure 4
Linear-response conductances of the system in Fig. 1 as a function of $E_{g}$ for the self-consistent Hamilton operator in Eq. (2). The gap is varied with the scissor operators in Eqs. (15, 16, 17). The Hamilton operator is represented in the RHF and the HA. The vertical dashed lines show the gap values obtained from the HA ( $E_{g}^{HA}$ , left) and RHF ( $E_{g}^{HF}$ , right) when $\hat{O} \hat{A} \hat{O} = 0$ . The transmission values in these points are ${T_{RHF} (ϵ_{F}; E_{g}^{HF}) = 7.5 \times 10^{- 4} G_{0}, T_{HA} (ϵ_{F}; E_{g}^{HA}) = 5.5 \times 10^{- 2} G_{0}}$ . The difference between the RHF and the Hartree curves is approximately accounted for by a factor of 4. The star represents the conductance-gap value obtained by means of the parameter-free extended HA with the correction, which is defined by Eqs. (20, 16). For this approximation, $G = 8.6 \times 10^{- 3} G_{0}$ . The extended HA is roughly equivalent to a scissor operator shift to the same $E_{g}$ .Reuse & Permissions

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