^{1,a)}, Gemma C. Solomon

^{1}, Joseph E. Subotnik

^{1}, Vladimiro Mujica

^{1,2}and Mark A. Ratner

^{1}

### Abstract

The Landauer approach has proven to be an invaluable tool for calculating the electron transport properties of single molecules, especially when combined with a nonequilibrium Green’s function approach and Kohn–Sham density functional theory. However, when using large nonorthogonal atom-centered basis sets, such as those common in quantum chemistry, one can find erroneous results if the Landauer approach is applied blindly. In fact, basis sets of triple-zeta quality or higher sometimes result in an artificially high transmission and possibly even qualitatively wrong conclusions regarding chemical trends. In these cases, transport persists when molecular atoms are replaced by basis functions alone (“ghost atoms”). The occurrence of such ghost transmission is correlated with low-energy virtual molecular orbitals of the central subsystem and may be interpreted as a biased and thus inaccurate description of vacuum transmission. An approximate practical correction scheme is to calculate the ghost transmission and subtract it from the full transmission. As a further consequence of this study, it is recommended that sensitive molecules be used for parameter studies, in particular those whose transmission functions show antiresonance features such as benzene-based systems connected to the electrodes in *meta* positions and other low-conducting systems such as alkanes and silanes.

The authors would like to thank Thorsten Hansen, Jeffrey R. Reimers, and Matthew G. Reuter for helpful comments and discussions. C.H. gratefully acknowledges funding by a Forschungsstipendium by the Deutsche Forschungsgemeinschaft (DFG). M.A.R. thanks the Chemistry and Materials Research Divisions (MRSEC program) of the NSF and the DOE for support.

I. INTRODUCTION

II. THEORY AND ALGORITHMIC DETAILS

A. The Landauer approach in a nonorthogonal basis

B. Details of the implementation

C. Ghost transmission

III. GHOST TRANSMISSION IN ALKANE AND SILANE JUNCTIONS

IV. EXPLAINING GHOST TRANSMISSION IN A SMALL MODEL SYSTEM

V. IMPLICATIONS FOR MOLECULAR JUNCTIONS

A. Basic considerations

B. Influence of the basis function type and the size of the extended molecule

VI. DISCUSSION: GHOST TRANSMISSION AS A BIASED DESCRIPTION OF VACUUM TUNNELING

VII. CONCLUSION AND OUTLOOK

### Key Topics

- Electrodes
- 37.0
- Gold
- 25.0
- Basis sets
- 19.0
- Transport properties
- 12.0
- Electronic structure calculations
- 10.0

## Figures

(a) Partitioning of the full electrode-molecule-electrode system as employed in a finite cluster approach. Two alternative definitions of the central system are shown, one that only comprises the molecule , and one that comprises the molecule and several electrode atoms . (b) Definition of the central subsystem Hamiltonian and the coupling matrices as submatrices of the full one-particle Hamiltonian.

(a) Partitioning of the full electrode-molecule-electrode system as employed in a finite cluster approach. Two alternative definitions of the central system are shown, one that only comprises the molecule , and one that comprises the molecule and several electrode atoms . (b) Definition of the central subsystem Hamiltonian and the coupling matrices as submatrices of the full one-particle Hamiltonian.

Schematic illustration of the ghost basis setup for a model junction consisting of a Au–H–H–H–H–Au chain, where the central region is defined as the four H atoms. While a full calculation (left) contains both the basis functions (for simplicity, only one function per atom is shown, denoted by the blue lobes) and the atomic nuclei with all electrons associated with the neutral atoms (denoted by golden and red circles), the ghost calculation (right) has the atomic nuclei and the electrons removed in the central region.

Schematic illustration of the ghost basis setup for a model junction consisting of a Au–H–H–H–H–Au chain, where the central region is defined as the four H atoms. While a full calculation (left) contains both the basis functions (for simplicity, only one function per atom is shown, denoted by the blue lobes) and the atomic nuclei with all electrons associated with the neutral atoms (denoted by golden and red circles), the ghost calculation (right) has the atomic nuclei and the electrons removed in the central region.

Transmission for (a) octanedithiolate chains in two different conformations and (b) their silane analogs using clusters to mimic the coupling to gold electrodes. The LANL2DZ ghost transmission is in all cases too low to be displayed. KS-DFT(BP86); various Gaussian-type atom-centered basis sets. Electronic structure program: QCHEM.

Transmission for (a) octanedithiolate chains in two different conformations and (b) their silane analogs using clusters to mimic the coupling to gold electrodes. The LANL2DZ ghost transmission is in all cases too low to be displayed. KS-DFT(BP86); various Gaussian-type atom-centered basis sets. Electronic structure program: QCHEM.

Ball-and-stick model of the model junction.

Ball-and-stick model of the model junction.

Transmission for the model junction using (a) different exponents for the four -type Gaussian basis functions and (b) different combinations of matrix blocks. and refer to exponents of (diffuse) and (local), respectively. See Fig. 1 for definitions of the matrix blocks. KS-DFT—BP86/TZVP(Au), -type ghost basis function (H).

Transmission for the model junction using (a) different exponents for the four -type Gaussian basis functions and (b) different combinations of matrix blocks. and refer to exponents of (diffuse) and (local), respectively. See Fig. 1 for definitions of the matrix blocks. KS-DFT—BP86/TZVP(Au), -type ghost basis function (H).

Isosurface plots of the central subsystem MOs of the model junction (see Appendix B for a definition of subsystem MOs).

Isosurface plots of the central subsystem MOs of the model junction (see Appendix B for a definition of subsystem MOs).

Transmission for the *meta*-connected benzene derivatives using clusters to mimic the coupling to gold electrodes. KS-DFT(BP86); various Gaussian-type atom-centered basis sets. Electronic structure program: QCHEM.

Transmission for the *meta*-connected benzene derivatives using clusters to mimic the coupling to gold electrodes. KS-DFT(BP86); various Gaussian-type atom-centered basis sets. Electronic structure program: QCHEM.

Influence of the EM size on transmission functions for (a) *para*- and (b) *meta*-connected benzene derivatives using clusters to mimic the coupling to gold electrodes. KS-DFT(BP86), TZVP. Electronic structure program: QCHEM.

Influence of the EM size on transmission functions for (a) *para*- and (b) *meta*-connected benzene derivatives using clusters to mimic the coupling to gold electrodes. KS-DFT(BP86), TZVP. Electronic structure program: QCHEM.

Influence of the EM size on the transmission functions for (a) *para*- and (b) *meta*-connected benzene derivatives using clusters to mimic the coupling to gold electrode. KS-DFT(BP86) in combination with the scalar-relativistic ZORA approximation; small frozen cores; TZ2P basis set (DZ for comparison). Electronic structure program: ADF.

Influence of the EM size on the transmission functions for (a) *para*- and (b) *meta*-connected benzene derivatives using clusters to mimic the coupling to gold electrode. KS-DFT(BP86) in combination with the scalar-relativistic ZORA approximation; small frozen cores; TZ2P basis set (DZ for comparison). Electronic structure program: ADF.

Influence of the EM size and the LDOS value on transmission functions for (a) *para*- and (b) *meta*-connected benzene derivatives using clusters to mimic the coupling to gold electrodes. The values in the legend refer to the constant LDOS in Eq. (6). KS-DFT(BP86), TZVP. Electronic structure program: QCHEM.

Influence of the EM size and the LDOS value on transmission functions for (a) *para*- and (b) *meta*-connected benzene derivatives using clusters to mimic the coupling to gold electrodes. The values in the legend refer to the constant LDOS in Eq. (6). KS-DFT(BP86), TZVP. Electronic structure program: QCHEM.

## Tables

Central subsystem Fock matrix for the small-exponent (“D”) and the large-exponent (“L”) calculations. Entries are given in eV.

Central subsystem Fock matrix for the small-exponent (“D”) and the large-exponent (“L”) calculations. Entries are given in eV.

and at for the individual eigenvectors (MOs) of the central subsystem for the small-exponent (diffuse) and the large-exponent (local) calculations. Entries are given in eV.

and at for the individual eigenvectors (MOs) of the central subsystem for the small-exponent (diffuse) and the large-exponent (local) calculations. Entries are given in eV.

Energies and couplings and at the Fermi energies for the individual eigenvectors (MOs) of the central subsystem. Energies are given in eV. The assignments HOMO and LUMO are given by similarities in shape with the corresponding MOs calculated for the isolated molecule.

Energies and couplings and at the Fermi energies for the individual eigenvectors (MOs) of the central subsystem. Energies are given in eV. The assignments HOMO and LUMO are given by similarities in shape with the corresponding MOs calculated for the isolated molecule.

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