^{1,a)}

### Abstract

The source and sink potential method of Goyer *et al.* [J. Chem. Phys.126, 144104 (2007)] is extended to the case of multichannel coupling to leads. The formulation leads to a nonlinear equation for just one (the elastic)reflection coefficient. Solution of this equation, in general, requires repeated computation of an *n × n* determinant, where *n* is the number of supermolecule basis functions directly coupled to the source lead, as opposed to a determinant with order equal to the full size of supermolecule basis. The method is applied to a Hückel model of two-channel polyacene conduction. A simple model of resonance lineshapes is developed in case of weak coupling to leads. The model accurately relates peak characteristics to orbital probabilities associated with the eigenvectors of the isolated molecule Hamiltonian. The model shows how orbital probabilities that give rise to transmission resonances (i.e., 100% transmission), in the case of single-channel conduction, give rise to equal probabilities (of 1/4) for the two reflections and two transmissions, in the case of two-channel conduction. The model also shows how splitting of degenerate eigenvalues of the isolated molecule Hamiltonian results in overlapping resonances characterized by a single complex lineshape.

The author thanks Adam Lewis and Alex Klenov for assistance relating to this work, and Matthias Ernzerhof and Reza Khorasani for helpful discussions.

I. INTRODUCTION

II. THE SOURCE–SINK POTENTIAL METHOD FOR MULTICHANNEL CONDUCTION

A. The Hamiltonian

B. The source and sink potentials

C. The determinantal equation—A nonlinear eigenvalue problem

III. APPLICATIONS TO TWO-CHANNEL HÜCKEL SYSTEMS

A. The Hückel models

B. Case of weak coupling to leads

C. Symmetric transmission through benzene

D. Two-channel transmission through ethenylbenzene and 1-ethenylnaphthalene

E. Benzene with four uncoupled single-channel leads

V. SUMMARY

### Key Topics

- Lead
- 45.0
- Eigenvalues
- 38.0
- Wave functions
- 22.0
- Reflection coefficient
- 18.0
- Transmission coefficient
- 13.0

## Figures

A two-channel molecular wire constructed from tetracene. Other polyacenes or derivatized polyacenes could also be used. Portions of the leads and molecule are labeled according to the partitioning of the lead–molecule introduced in Sec. II. The *j,k* indices are shown for the atoms in *L*´ and *L*´´.

A two-channel molecular wire constructed from tetracene. Other polyacenes or derivatized polyacenes could also be used. Portions of the leads and molecule are labeled according to the partitioning of the lead–molecule introduced in Sec. II. The *j,k* indices are shown for the atoms in *L*´ and *L*´´.

Hückel parameters on the left for a two-channel molecular wire. Analogous parameters apply to the right side of the wire.

Hückel parameters on the left for a two-channel molecular wire. Analogous parameters apply to the right side of the wire.

Probability of symmetric reflection (red solid lines) and transmission (blue dotted lines) through benzene. Antisymmetric transmission and reflection probabilities are identically zero because of the top–bottom mirror symmetry of benzene. The molecule Hückel parameters are *α* = −6.5 eV and *β* = −2.7 eV. The lead parameters are , , , while lead–molecule couplings are (top panel) and −3 eV (bottom panel).

Probability of symmetric reflection (red solid lines) and transmission (blue dotted lines) through benzene. Antisymmetric transmission and reflection probabilities are identically zero because of the top–bottom mirror symmetry of benzene. The molecule Hückel parameters are *α* = −6.5 eV and *β* = −2.7 eV. The lead parameters are , , , while lead–molecule couplings are (top panel) and −3 eV (bottom panel).

Reflection and transmission through ethenylbenzene. Symmetric channels are depicted as in Fig. 3, while antisymmetric reflection and transmission probabilities are denoted by magenta-dashed and cyan-dotted-dashed lines, respectively. The latter two probabilities are equal because of the left-right mirror symmetry of ethenylbenzene. Hückel parameters are as in Fig. 3.

Reflection and transmission through ethenylbenzene. Symmetric channels are depicted as in Fig. 3, while antisymmetric reflection and transmission probabilities are denoted by magenta-dashed and cyan-dotted-dashed lines, respectively. The latter two probabilities are equal because of the left-right mirror symmetry of ethenylbenzene. Hückel parameters are as in Fig. 3.

Molecular portion of MED wavefunctions (on the left) at resonance peak energies for the lowest (top panel) and third lowest energy resonances (bottom panel). Corresponding (i.e., associated eigenvalues close to the resonance energies) isolated molecule eigenvectors are shown on the right. The parameters are those of the top panel of Fig. 4 (i.e., the weak coupling case of ethenylbenzene). Each circle radius indicates the relative size of the modulus of the coefficient for that atomic orbital. The phase of each coefficient is indicated (only for the MED molecular wavefunctions) by the direction of the straight line from the circle center—e.g., a line to the right indicates a real positive coefficient. Green solid circles indicate phase closest to real positive, while red dotted circles indicate phase closest to real negative.

Molecular portion of MED wavefunctions (on the left) at resonance peak energies for the lowest (top panel) and third lowest energy resonances (bottom panel). Corresponding (i.e., associated eigenvalues close to the resonance energies) isolated molecule eigenvectors are shown on the right. The parameters are those of the top panel of Fig. 4 (i.e., the weak coupling case of ethenylbenzene). Each circle radius indicates the relative size of the modulus of the coefficient for that atomic orbital. The phase of each coefficient is indicated (only for the MED molecular wavefunctions) by the direction of the straight line from the circle center—e.g., a line to the right indicates a real positive coefficient. Green solid circles indicate phase closest to real positive, while red dotted circles indicate phase closest to real negative.

As in Fig. 5, except these molecular wavefunctions correspond to the second lowest (top panel) and fourth lowest energy (bottom panel) resonances. Also, note that cyan-dotted-dashed and magenta-dashed circles indicate orbital coefficients closest to the positive and negative imaginary axes, respectively.

As in Fig. 5, except these molecular wavefunctions correspond to the second lowest (top panel) and fourth lowest energy (bottom panel) resonances. Also, note that cyan-dotted-dashed and magenta-dashed circles indicate orbital coefficients closest to the positive and negative imaginary axes, respectively.

Three resonances seen in the reflection and transmission probabilities of the top panel of Fig. 4, compared with weak-coupling model lineshapes—indicated by truncation to neighborhoods of the three resonances.

Three resonances seen in the reflection and transmission probabilities of the top panel of Fig. 4, compared with weak-coupling model lineshapes—indicated by truncation to neighborhoods of the three resonances.

As in the bottom panel of Fig. 4, except these data are for 1-ethenylnaphthalene.

As in the bottom panel of Fig. 4, except these data are for 1-ethenylnaphthalene.

Four independent leads attached to benzene. As in the top panel of Fig. 3, except . Symmetric and antisymmetric channels are replaced by bottom and top channels, respectively. The incoming wave is in the bottom channel. Here, we use. Otherwise, parameters are as in other figures.

Four independent leads attached to benzene. As in the top panel of Fig. 3, except . Symmetric and antisymmetric channels are replaced by bottom and top channels, respectively. The incoming wave is in the bottom channel. Here, we use. Otherwise, parameters are as in other figures.

The resonance, seen in Fig. 9, associated with the lower degenerate eigenvalue of benzene, compared with weak-coupling model lineshapes (see Appendix A). The model lineshapes are very close to the complete lineshapes here—they are not discernable as separate lines.

The resonance, seen in Fig. 9, associated with the lower degenerate eigenvalue of benzene, compared with weak-coupling model lineshapes (see Appendix A). The model lineshapes are very close to the complete lineshapes here—they are not discernable as separate lines.

The molecular portions of the MED wavefunction (on the left) at three energies near the resonance depicted in Fig. 10, *E* = −9.4, −9.2, and −9.0 eV. Orbital coefficients are depicted as in Fig. 5. Also shown (on the right) are the projections of these molecular wavefunctions onto the two-dimensional eigenspace of isolated benzene with eigenvalue = −9.2 eV.

The molecular portions of the MED wavefunction (on the left) at three energies near the resonance depicted in Fig. 10, *E* = −9.4, −9.2, and −9.0 eV. Orbital coefficients are depicted as in Fig. 5. Also shown (on the right) are the projections of these molecular wavefunctions onto the two-dimensional eigenspace of isolated benzene with eigenvalue = −9.2 eV.

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