Carmen Herrmann et al- Ghost transmission: How large basis sets can make electron transport...

8/3/2019 Carmen Herrmann et al- Ghost transmission: How large basis sets can make electron transport calculations worse

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Ghost transmission: How large basis sets can make electron transportcalculations worse

Carmen Herrmann,1,a Gemma C. Solomon,1 Joseph E. Subotnik,1 Vladimiro Mujica,1,2

and Mark A. Ratner11Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113,

USA2

Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, USA andDepartment of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287, USA

Received 26 October 2009; accepted 11 December 2009; published online 12 January 2010

The Landauer approach has proven to be an invaluable tool for calculating the electron transportproperties of single molecules, especially when combined with a nonequilibrium Greens functionapproach and KohnSham density functional theory. However, when using large nonorthogonalatom-centered basis sets, such as those common in quantum chemistry, one can find erroneousresults if the Landauer approach is applied blindly. In fact, basis sets of triple-zeta quality or highersometimes result in an artificially high transmission and possibly even qualitatively wrongconclusions regarding chemical trends. In these cases, transport persists when molecular atoms arereplaced by basis functions alone ghost atoms. The occurrence of such ghost transmission iscorrelated with low-energy virtual molecular orbitals of the central subsystem and may beinterpreted as a biased and thus inaccurate description of vacuum transmission. An approximatepractical correction scheme is to calculate the ghost transmission and subtract it from the fulltransmission. As a further consequence of this study, it is recommended that sensitive molecules beused for parameter studies, in particular those whose transmission functions show antiresonancefeatures such as benzene-based systems connected to the electrodes in meta positions and otherlow-conducting systems such as alkanes and silanes. 2010 American Institute of Physics.doi:10.1063/1.3283062

I. INTRODUCTION

From early work in the late 1940s1 and the first proposalof a single-molecule device in 1974,2 the field of molecularelectronics has grown into a very active area of research.Molecular electronics may be defined as the study of elec-tronic circuits containing individual molecules as buildingblocks.319 The most prominent example of such a circuit is amolecular transport junction, where one molecule is sand-wiched between two macroscopic electrodes. In a typical ex-periment, the electron current I through this device is mea-sured as a function of bias voltage V I-V curve and oftenalso as a function of gate voltage. The molecular charac-teristics of the junction are revealed particularly in deviationsfrom Ohmic behavior, i.e., in nonlinear I-V curves.

A strong motivation for theoretical calculations of mo-lecular conduction is that they may provide further insight

into the mechanisms of electron transport, often interpretedin terms of auxiliary, nonmeasurable quantities such as mo-lecular orbitals MOs, and therefore may help to design newtypes of molecular devices.2023 Apart from this practical im-portance for interpreting and designing experiments, the the-oretical description of current through molecules is equallyappealing from a purely scientific rather than technological

point of view: a satisfying description of the process requiresknowledge of electronic structure theory for stationarystates,24 electron dynamics,25 and the physics of open sys-tems away from equilibrium.26,27 There is also a strong con-nection to various other fields of research, in particular, aformal analogy may be made2830 between electron transportthrough molecular junctions and intramolecular electrontransfer ET in donor-bridge-acceptor systems.3134

Ideally, theory-based predictions should not rely on anysystem-specific input data and be sufficiently accurate for thedesired purpose. In principle, this combination is achievedby methods based on first-principles quantum mechanics, us-ing natural constants as the only input quantities obtainedfrom experiment. For devices of practical interest, however,the resulting equations of motion are very far from beingsolvableone very obvious reason is that a straightforwardapplication of quantum mechanics would require describing

macroscopic electrodes on the same footing as the singlemolecule placed between them. Therefore, several approxi-mations have to be made in practice, and the errors thusintroduced are often poorly understood.

Altogether, the theory of electron transport is in a ratherearly stage of development compared with more establishedareas of research such as molecular electronic structuretheory,24 vibrational spectroscopy,35,36 or moleculardynamics.37,38 This may be ascribed to the complex nature ofan open system under nonequilibrium conditions. Nonethe-less, in recent years, one set of approximations has evolved

aAuthor to whom correspondence should be addressed. Electronic mail:[email protected]. FAX: 1-847-467-4996. Tel.: 1-847-467-4984.

THE JOURNAL OF CHEMICAL PHYSICS 132, 024103 2010

0021-9606/2010/1322 /024103/17/$30.00 2010 American Institute of Physics132, 024103-1

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into what may be regarded as the standard approach tofirst-principles calculations in molecular electronics, and hasbeen shown to give a reasonable agreement with experimen-tal results.39 This is the combination of the LandauerImryelastic scattering treatment4042 with a nonequilibriumGreens function NEGF approach27,43 and KohnShamdensity functional theory KS-DFT,44,45 or other electronic-structure descriptions which give a simple MO picture, such

as HartreeFock HF theory, tight-binding approaches,46

orextended Hckel theory. Recently, the LandauerImry ap-proach has also been combined with MllerPlesset pertur-bation theory.47

The limitations of the Landauer approximation, such asits effective single-particle nature, and its restriction to co-herent electron transport processes are well-known. Althoughconsiderable progress has been made in the recent years, itstill remains a challenge, even for very sophisticated imple-mentations of the Landauer approximation such as thosedescribed, e.g., in Refs. 4851, to predict the order of mag-nitude of the measured zero-voltage conductance correctly.Undoubtedly, this is in part due to the unknown contactstructure. It also appears that more elaborate approaches donot necessarily perform better than less sophisticated ones.39

It is therefore hard to establish the extent to which the ap-proximations made in these transport approaches limit theaccuracy of the method. Our goal in this paper is to ensurethat at a minimum, no additional errors are introduced by thechoice of technical settings in the electronic structure calcu-lation, such as the atom-centered basis set in a KS-DFT cal-culation. This benchmarking will be particularly importantfor molecules with a complicated electronic structure, suchas transition metal complexes and other open-shell systems.

Several parameter studies within the Landauer approach

have been published, focusing on the influence of the densityfunctional,5255 the basis set,53,54,56 and the number of elec-trode atoms included in the electronic-structurecalculation.54,5759 Reference 56 is the first example of a ba-sis set study including basis sets of triple-zeta quality andpolarization functions. In contrast to Ref. 60, which consid-ers a double-zeta basis set with polarization functions as suf-ficient, Ref. 56 points out that to achieve convergence withrespect to the basis set size, a basis set of at least triple-zetaquality is necessary. This coincides with experience fromisolated-molecule studies, which suggest that a double-zetabasis set may be far from adequate.61

In Refs. 56 and 60, molecules based on para-connectedphenyl rings are chosen as test systems, In this work, weshow that when studying molecules with lower conductanceand more structure-sensitive transport properties, such asmeta-connected phenyl-based molecules, and when employ-ing large atom-centered, nonorthogonal basis sets of Slater orGaussian type as commonly used in quantum chemistry, alarge basis set may lead to artificially high transmission. Thishigh transmission is also observed when the molecules at-oms are replaced by ghost atoms i.e., basis functions only,a result that we here call ghost transmission. This may berelated to other accounts of situations where employing alarger basis set does not necessarily lead to more accurate

results, in particular when calculating nonvariational proper-ties, but also for dimerization energies.62

This manuscript is organized as follows. First, the imple-mentation of the Landauer approximation employed here isbriefly outlined, and the concept of ghost transmission isintroduced. In Sec. III ghost transmission is shown to be ofpractical concern for predicting the qualitative difference intransmission between alkanes in two different conforma-

tions. The causes of ghost transmission are investigated indetail for a small model junction Sec. IV. The insightgained so far is applied in Sec. V to a meta-connected ben-zene junction. Finally, in Sec. VI, the nature of the virtualMOs causing ghost transmission is discussed. Additional in-formation on the implications of large basis sets on the ap-proximate description of the bulk electrodes in the wide-band-limit WBL approximation, as well as details on thecomputational methodology, can be found in the appendices.The supplementary material63 provides data on the impor-tance of different molecular regions for ghost transmission,ghost transmission at the basis set limit, the effect of exactexchange admixture in the exchange-correlation functional

and the distance between the electrodes, and the change inpartial charges with the basis sets employed here as well asthe importance of interelectrode coupling.

II. THEORY AND ALGORITHMIC DETAILS

The combination of LandauerImry approach and NEGFformalism employed here for calculating currents through aquantum system is based on the following assumptions andapproximations.

The system is described within an effective one-particleapproach such as HF theory or KS-DFT.

The coupling between the quantum system and the elec-trodes is large enough to prevent substantial accumula-tion of charge which would be considered the Coulombblockade regime.

The electrodes are reflectionless, i.e., there is no back-scattering once a charge has passed the central region-electrode interface.

The transport is coherent, i.e., no phase breaking eventsoccur. This is usually a good assumption when dealingwith short molecular bridges, low temperatures, and alarge separation between the one-particle energy levelsand the Fermi energies of the electrodes.

The system is in a steady state, i.e., the number of elec-trons in the central region is constant over time.

The entire electrode-central region-electrode system isdescribed by a basis whose elements can be clearly at-tributed to either electrode or the central region.

The original Landauer equation relates the zero-temperature, zero-voltage conductance of a quantum systemto the value of a transmission function at the Fermienergy,4042 without giving a detailed prescription of how todeal with particular bridges. This was achieved by the intro-duction of techniques to calculate the transmission function

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based on a set of local orthogonal basis functions, employingKeldyshs perturbation theory for nonequilibrium systemsi.e., a NEGF approach.6466 It was obviously desirable to

combine this transport formalism and the experience gainedwith molecular electronic structure calculations using nonor-thogonal atom-centered Gaussian- or Slater-type basis sets,67

and such implementations have become available in the pastdecade.49,50,6878 For an approach to calculating ET matrixelements in a nonorthogonal basis, see Ref. 79. Since thederivations of the equations to be implemented have beendiscussed extensively in the literature references cited above,they shall only be briefly summarized here.

A. The Landauer approach in a nonorthogonal basis

The current Is of electrons of spin s s , throughthe molecular junction at a given bias voltage V is calculatedas

IsV =e

h

dETsE,VfE EF,sL fE EF,s

R , 1

where the transmission function Ts for electrons of energy Eis given as

TsE,V = trR,sGC,sL,sGC,s , 2

and f denotes the Fermi function. The appropriate definitionof the Fermi energy EF,s

X for electrons of spin s in electrode X

is an issue under discussion in the literature band lineupproblem80. One possible answer might be that if a moleculeis attached to sufficiently large finite metal clusters such asthe about 500 atoms discussed in Ref. 50; compare also Ref.81, the Fermi energy of the electrodes can be estimated asthe energy halfway between the highest-energy occupied mo-lecular orbital HOMO and the lowest-energy unoccupiedmolecular orbital LUMO of the metal cluster.50 Anotherapproach is to solve the Poisson equation for the entire sys-tem; this gives EF,s

X directly.82

The coupling matrices for the left and the right elec-trodes, X,sXR ,L, are calculated from the imaginaryparts of the corresponding self-energies X,s,

X,s = iX,s X,s = 2 ImX,s , 3

which in turn may be obtained as

X,s = ESXC VX,sgX,sESXC VX,s, 4where gX,s is the retarded Greens function of the isolatedelectrode X see Eq. 6 below for details. SX and VX are theoverlap matrix and the one-particle Hamiltonian or Fockmatrix involving electrode X and central system basis func-tions see Fig. 1.

The central subblock of the retarded Greens function iscalculated as

Gs,C= ESC HC R L1 , 5

where SC and HC are the central subblocks of the overlapand one-particle Hamiltonian matrices.

B. Details of the implementation

After neglecting effects such as molecular vibrations andexternal fields, electron transport through a molecular bridgeis determined by three factors: the geometric and electronicstructure of the molecule, the geometric and electronic struc-ture of the electrodes, and the coupling between electrodesand molecule. The goal of our approach is to describe thecontribution the molecule makes to the transport properties,irrespective of the measurement environment. That is, toidentify and compare the transport properties of differentmolecules regardless of the electrode material, the details of

their binding to the electrodes, and the question of whetherthey are investigated in a transport junction, in a scanningtunneling microscope STM setup, or an ET experiment.Among the many ways to carry out such comparative stud-ies, we choose one that gives a reasonably good descriptionof a very popular way of measuring transport properties, amolecular junction employing gold electrodes.

The Greens functions of the electrodes are calculated inthe WBL approximation,83

gXij = i LDOSconst ij , 6

assuming a local density of states LDOS which is indepen-dent of the energy, and which in our case is chosen equal to

C

C

WBL approximation

or

bulk calculation

g

V

g

RV

R

L

L

C

L C R

L

C

R

H =

V

V V

V

H

L

L

R

R

(a) (b)

FIG. 1. a Partitioning of the full electrode-molecule-electrode system as employed in a finite cluster approach. Two alternative definitions of the centralsystem are shown, one that only comprises the molecule C, and one that comprises the molecule and several electrode atoms C. b Definition of thecentral subsystem Hamiltonian and the coupling matrices as submatrices of the full one-particle Hamiltonian.

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the LDOS of the 6s band as calculated for gold.84 Our imple-mentation is based on a finite-cluster approach, i.e., the mol-ecule is attached to two finite clusters of metal atoms on eachside see Fig. 1, and no periodic boundary conditions areenforced.

For all calculations reported here, we use approach C inFig. 1 with Au9 clusters mimicking the coupling to a Au111surface according to Eq. 4, and with the Greens functions

of the bulk electrodes described by Eq. 6 see Appendix Cfor details.Testing the basis set limit using more sophisticated de-

scriptions of the transportfor example, the gold clustersizes of up to 500 atoms as used in Ref. 50would makesuch a study almost prohibitive within the computational re-sources available today. It may be anticipated, however, thatthe main basis set effects can be observed regardless ofwhich model is used to describe the open-boundary condi-tions of a molecular junction, which is why we focus on asimple implementation of the WBL approximation in thiswork.

The only quantities needed from an electronic structure

code are therefore the overlap matrix S and the one-particleHamiltonian matrix i.e., the Fock matrix H for a moleculecoupled to a finite number of electrode atoms on two sides.Since our focus is on the molecule, the central region of oursetup contains either the molecule only or the molecule and afew electrode atoms. This causes problems with the bandlineup,50 but may be considered an acceptable trade-off forhaving the transmission properties of the molecule filteredout. The coupling matrices VR,L in Eq. 4 are the elements ofthe Fock matrix in the central region-electrode blocks. Fur-thermore, the open nature of the system and the effect of anybias voltage are not taken into account in the electronicstructure calculations, i.e., the density matrix in the self-

consistent field SCF algorithm is calculated, as usual inelectronic structure theory of closed systems, from the MOcoefficients, and not from the central subsystem block of thelesser Greens function as often done for open systems.48,49,75

This allows a transport code to be constructed as a postpro-cessing tool for electronic structure calculations. That is, ourtransport calculations consist of two steps: 1 electronicstructure calculation and 2 calculation of transmissionfunction and, if desired, current and conductance.

C. Ghost transmission

In this work, two types of transport calculations are car-ried out, denoted as full and ghost. A full transport cal-culation corresponds to the regular approach describedabove, where a molecule is put between two metal clustersand the transmission and current are subsequently calcu-lated. In a ghost transport calculation, the same metal-molecule-metal junction is considered, but all atomic nucleiand electrons associated with atoms in the central region areremoved in the electronic structure calculation, so that allthat remains are the basis functions centered on these atomssee Fig. 2.

This corresponds to the ghost basis employed in thecounterpoise correction scheme for the basis set superposi-

tion error.85 It should be noted that depending on the numberof atoms in the electrode, these ghost transmission calcula-tions may have to be carried out in a different spin state fromthe full onesfor example, when employing Au9 clusters asdone here, the full calculations were carried out with no un-paired electrons, while the ghost transmission was calculatedhaving two unpaired electrons one on each gold cluster.For the remainder of this paper, the ghost transmissioncurves will always be reported for spin up electrons, because

only minor differences were obtained for the spin down elec-trons. Convergence of the SCF algorithm has been found tobe difficult in some cases, but the resulting ghost transmis-sion curves were not affected to any significant extent byconvergence issues.

As detailed below, we will show that the transmissioncurves of transport calculations using large atom-centeredbasis sets may be interpreted approximately as the sum of amolecular transmission and the ghost transmission. We there-fore consider the ghost transmission as significant when it islarge enough that when subtracted from the full transmission,the shape of the transmission curve changes substantively.

III. GHOST TRANSMISSION IN ALKANEAND SILANE JUNCTIONS

A striking example of the ghost transmission problem isthe comparison of transport calculations for octanedithiolatein two different conformations, one with all carbon centers inan anti conformation, and one with two of them in a synconformation to form a local U-shaped structure see leftpanel of Fig. 3.

Whereas a calculation using the double-zeta quality ba-sis set LANL2DZ gives the qualitative prediction that thetransmission function of the all-anti conformation is higher

than the syn one at all energies considered, the transmissionis predicted to be about the same for both conformationswhen employing the triple-zeta basis set with polarizationfunctions TZVP for all energies above 5.5 eV, a rangewhich encompasses all common choices for the Fermi en-ergy within a KS-DFT approach.

When calculating the transmission using the ghost basissetup described in Sec. II C, significant values can be ob-tained, despite the fact that no molecule is present in the

junction. The ghost transmission is nearly constant over theenergy range considered, and it is about the same for bothconformations. The similarity in the transmission curves ob-tained from the full and the ghost basis TZVP calculations

full ghost

FIG. 2. Schematic illustration of the ghost basis setup for a model junctionconsisting of a AuHHHHAu chain, where the central region is definedas the four H atoms. While a full calculation left contains both the basisfunctions for simplicity, only one p function per atom is shown, denoted bythe blue lobes and the atomic nuclei with all electrons associated with theneutral atoms denoted by golden and red circles, the ghost calculationright has the atomic nuclei and the electrons removed in the central region.




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from the full transmission, it is mandatory to understand whyghost transmission is generated to be able to prevent it fromleading to erroneous predictions. The possible causes of

ghost transmission could include self-interaction errors in theapproximate exchange-correlation functional, problemsstemming from an improper use of pseudopotentials, quali-tative changes in the electron density distribution, near lineardependencies in the basis set, or overlap between basis func-tions centered on the two electrodes. All of the aforemen-tioned possibilities have been excluded, however, by identi-fying cases of molecular junctions with a significant ghosttransmission when using density functionals with differentexact exchange admixture and when using frozen cores in-stead of pseudopotentials see Fig. 9, as well as by checkingthe partial charges on the atoms close to the interface, thecondition number of the overlap matrix for the central sub-systems, and the largest elements of the electrode-electrodeblocks of Fock and overlap matrix see supplementarymaterial.63

In order to elucidate the cause of ghost transmission, wehave chosen a model system similar to the one schematicallydepicted in Fig. 2: it consists of a AuuH4

ghostuAu junction

with a zigzag structure see Fig. 4, where Hghost refers to aghost atom with only one s-type Gaussian basis functionwith varying exponents centered on it. For the gold atoms,the same TZVP basis set employed in the calculations in Sec.III is used. The interatomic distances have been set todHuH =1.01 and dHuAu=2.09 , and all bond

angles were chosen to be 109.4 the linear analog sufferedfrom SCF convergence issues.

As shown in Fig. 5, when using a comparatively largeexponent of 1.0 a.u.2 and thus local functions, denotedby L in Fig. 5 for the basis functions on the ghost atoms, theghost transmission is low in the energy range from 8 to 2eV. When choosing a smaller exponent of 0.1 a.u. 2, andthus more diffuse D ghost basis functions, the ghosttransmission values are considerably higher around 0.3 in-stead of around 104 to 103.

The differences observed must stem from either the Fockor the overlap matrix in Eqs. 4 and 5. As can be seen fromthe right-hand side of Fig. 5, after replacing the couplingblocks of both the overlap and the Fock matrix SXC and VX;see definitions in Fig. 1 in the D-D-D-D calculation by thosefrom the L-L-L-L one, the high transmission remains essen-tially unchanged, so the coupling parts do not seem to be

responsible at least not alone. When replacing the centralpart of both matrices, however, the transmission goes downconsiderably. To determine whether this is due to the Fock orthe overlap matrix, both have been replaced individually.While the transmission curve with the central part of theFock matrix replaced nearly coincides with the one obtainedwhen both matrices were replaced, the one with the centralpart of the overlap matrix replaced is still very high, so itmay be concluded that it is mainly the central part of theFock matrix which is responsible.

From the Fock matrices of the central region, which aregiven in Table I, it can be seen that the main difference

between the two are the values on the diagonals: while theorbital energies are slightly negative or slightly positive for

-8 -7 -6 -5 -4 -3 -2E / eV

10-3

10-2

10-1

100

T

D- D-D -D HC

and SC

from L-L-L-L

D- D-D -D VX

and SXC

from L-L-L-L

D- D-D -D SC

from L-L-L-L

D- D-D -D HC

from L-L-L-L

D- D-D -D HC

(diag) from L-L-L-L

-8 -7 -6 -5 -4 -3 -2E / eV

10-4

10-3

10-2

10-1

100

T

L-L-L-LD-D-D-DD-L-L-D

(a) (b)

FIG. 5. Transmission for the AuuH4ghostuAu model junction using a different exponents for the four s-type Gaussian basis functions and b different

combinations of matrix blocks. D and L refer to exponents of 0.1 a.u.2 diffuse and 1.0 a.u.2 local, respectively. See Fig. 1 for definitions of the matrixblocks. KS-DFTBP86/TZVPAu, s-type ghost basis function H.

TABLE I. Central subsystem Fock matrix for the small-exponent D and the large-exponent L calcula-tions. Entries are given in eV.

D-D-D-D L-L-L-L

H1 H2 H3 H4 H1 H2 H3 H4

H1 1.566 0.730 0.042 0.542 36.122 1.983 0.735 0.003H2 0.730 0.754 0.769 0.042 1.983 38.387 1.582 0.735H3 0.042 0.769 0.754 0.730 0.735 1.5832 38.387 0.983H4 0.542 0.042 0.730 1.566 0.003 0.735 1.983 36.122




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the calculation with the high ghost transmission D-D-D-D,they are strongly positive for the one with the low ghosttransmission L-L-L-L.

By replacing only the values of the diagonal elements ofthe central subsystems Fock matrix in the D-D-D-D calcu-lation by the approximate values of the corresponding ele-ments in the L-L-L-L calculation 37.0 eV, the ghost trans-mission is brought down significantly to a curve thatcoincides with the one obtained by replacing the whole cen-tral region Fock matrix see right-hand side of Fig. 5. Also,if the central Fock matrix of the D-D-D-D calculation issubstituted into the L-L-L-L one, the transmission goes up byorders of magnitude. This suggests that the diagonal ele-ments of the central regions Fock matrix are mainly respon-sible for the high ghost transmission. To check again thecontribution of the more diffuse basis functions to a poten-tially larger coupling, the diffuse basis functions were usedon the two ghost atoms closest to the gold atoms only, whilethe two middle ones were described using the local basisfunction. This results in a fairly low ghost transmission seeleft-hand side of Fig. 5, again pointing to a minor role of thecoupling. However, given that the combination of the L-L-

L-L central Fock and overlap matrices with the electrode-central region coupling blocks from the D-D-D-D calculationdoes not bring the transmission to the same low values as theoriginal L-L-L-L calculation, it is clear that the coupling,although not solely responsible, does play a role in ghosttransmission.

To determine which MOs are responsible for the hightransmission in the small-exponent calculation D-D-D-D,the transmission through the individual central subsystemMOs has been calculated see Appendix B for details. Sinceformally there are no electrons on the central subsystem, allMOs are virtuals. It turns out that for the small-exponentcalculation D-D-D-D, MOs 1, 2, and 4 have a transmission

of 1 over the whole energy range under study 8.0 to2.0 eV, whereas MO 3 has a slightly lower one 0.62 atE=5.0 eV. The transmission close to 1 obtained for theindividual MOs can be understood from the coupling ofthese MOs to the electrodes, i.e., from the matrices L andR which in the case of transport through individual MOs aresimply numbers L and R see Table II: these are consid-erably smaller for MO 3 than for the others.

Note that the total transmission is not the sum of trans-missions through individual MOs. Thus, when the subsystemMOs with individual transmissions of 1 are considered to-gether, the transmission may be considerably lower than 1.For a AuuH4

ghostuAu model junction with the diffuse

function on the ghost atoms adjacent to the gold atoms onlyD-L-L-D there are two strongly coupled and two weaklycoupled MOs. The strongly coupled ones have a transmis-sion of nearly 1 each over the energy range under studywhen considered individually, but the transmission through asystem consisting of both MOs is considerably lower thanthat T5.0 eV =1.4103. Combining a high-couplingMO with a low-coupling one leads to a constant transmissionof 1, however. The two high-coupling MOs differ from eachother in symmetryone is symmetric and one is antisym-metric with respect to the mirror plane normal to the trans-port direction, thus resembling a bonding-antibonding MOpair see rightmost panel in Fig. 6. This is why their contri-butions to the transmission can cancel although the indi-vidual MOs have a transmission of 1 for a discussion oforbital contributions to transmission and cancellation ofthese contributions through interference, see, for example,Refs. 2023 and 86. On a side note, the MOs displayed inFig. 6 also explain why in the D-D-D-D calculation, thecoupling, and thus the transmission of MO 3, is smaller thanfor the other MOs: this MO is more strongly localized on thecentral subsystem.

If the coupling terms to the left and to the right elec-trodes are equal L =R =, as in the examples above, sothat

L =12 R + L = , 7

the transmission through one central subsystem MO j can bewritten as

TE = Gjjr EGjj

a E

= 1

E j i

1

E j + i

=

2

E j2

+ 2 , 8

TABLE II. L and R at E= 5.0 eV for the individual eigenvectors MOs of the central subsystem for the small-exponent diffuse and the large-exponentlocal calculations. Entries are given in eV.

MO

D-D-D-D L-L-L-L D-L-L-D

i R L i R L i R L

1 2.94 742.80 742.80 26.76 0.74 0.74 1.74 323.54 323.562 1.88 585.17 585.16 32.89 1.28 1.28 1.61 584.5 584.503 3.68 11.07 11.07 43.57 0.78 0.78 36.52 46.95 46.95

4 4.16 256.30 256.30 53.94 0.19 0.19 48.90 13.29 13.29

4

1 2

3 4

1 2

3 4

1 2

3

DDDD LLLL DLLD

FIG. 6. Isosurface plots of the central subsystem MOs of theAuuH4

ghostuAu model junction see Appendix B for a definition of sub-

system MOs.




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which, for absolute values of considerably larger than thedifference between the MO eigenvalue and the energy E,leads to a transmission close to 1 over a certain energy range.

The calculations on the AuuH4ghostuAu model junc-

tion suggest that the major contribution to a high ghost basistransmission arises from small diagonal elements of the cen-tral subsystem Hamiltonian and a subsequent downshift ofthe energies of the central subsystem MOs. However, it can-not be excluded that a large coupling of the central sub-system to the electrodes is a necessary prerequisite for ghosttransmission.

V. IMPLICATIONS FOR MOLECULAR JUNCTIONS

To check whether the conclusions drawn for the smallmodel systems can be applied to realistic molecular junc-tions, ghost transmission has been investigated for a benzenemolecule coupled to the electrodes via ethynyl thiolatespacer groups in the meta position. This system has beenchosen for three reasons. 1 It is known to have a muchlower transmission than its para-connected analog,21,87,88 andis thus more sensitive to parameter effects. 2 In contrast tothe alkanes and silanes investigated in Sec. III, where the two

systems of interest differ in their conformation, which is hardto control experimentally,89,90 phenyl rings with differentsubstitution are much more accessible to experimental stud-ies. 3 It has an interesting combination of transport throughMOs of and symmetry.22,91 Since the data discussed inSec. III involve systems where all bonds are bonds, it is anopen question whether ghost transmission affects-conducting molecular bridges to the same extent.

A. Basic considerations

As can be seen in Fig. 7, ghost transmission is significantin the meta-connected benzene system when using the TZVP

basis set, and leads to full transmission values up to almostan order of magnitude above the ones obtained from theLANL2DZ calculation.

However, when only the most diffuse functions in theTZVP basis set on the carbon atoms, which have p symme-try, are left out in the electronic structure calculations, thetransmission goes down to values even below the LANL2DZones. Therefore, it may be concluded that these diffuse basisfunctions are responsible for the ghost transmission, and, if

the results from Sec. IV apply for realistic systems, thenshifting the diagonal parts of the original TZVP Fock matrixfor these basis functions from their original values of about13 eV to higher values 0 eV in our example should bringthe transmission down. Indeed it does, and it also does sowhen only shifting the elements for the two diffuse p basisfunctions oriented in the molecular plane, i.e., the ones thatcontribute to the MOs the latter curve is shown in Fig. 7.Thus, it may be concluded that ghost transmission tends toaffect the system more than the system. Note howeverthat for the meta-connected benzene, the system is themajor contributor at low voltages due to interference in the system.20,21

Again, the question arises whether the ghost transmis-sion can be traced back to individual subsystem MOs. Incontrast with Sec. IV, we concentrate on the full transmissioninstead of the ghost one, keeping in mind that the unalteredTZVP calculation is the only one of those summarized inTable III which shows significant ghost transmission. Thisallows the role of occupied versus virtual subsystem MOs tobe assessed. The most striking difference in the MOs of thecentral subsystem between the unaltered TZVP calculationand the one with the diffuse p functions on C removed arethe energies: leaving out the diffuse basis function leads toupward shifts in virtually all orbital energies see Table IIIfor representative examples; the LANL2DZ results are also

-8 -7 -6 -5 -4 -3 -2E / eV

10-4

10-3

10-2

10-1

100

T

LANL2DZTZVPTZVP without diffuse p(C)

TZVP with shifted diffuse in-plane p(C)

ghost TZVP

TZVP minus ghost

FIG. 7. Transmission for the meta-connected benzene derivatives using Au9 clusters to mimic the coupling to gold electrodes. KS-DFTBP86; variousGaussian-type atom-centered basis sets. Electronic structure program: QCHEM.




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TABLE III. Energies and couplings L and R at the Fermi energies for the individual eigenvectors MOs ofthe central subsystem. Energies are given in eV. The assignments HOMO and LUMO are given by similaritiesin shape with the corresponding MOs calculated for the isolated molecule.

i MO i R L TEF

TZVP50 HOMO1 6.29 106.54 106.54 1.0051 HOMO 6.05 140.57 140.57 1.0052 LUMO+2 2.93 956.92 957.43 1.00

53 LUMO+3 2.76 963.62 963.33 1.0054 LUMO 2.69 268.03 268.07 1.0055 LUMO+1 2.31 324.42 324.42 1.00

LANL2DZ40 HOMO1 6.08 3.54 3.54 0.7041 HOMO 5.77 3.68 3.68 0.7742 LUMO 2.09 1.95 1.95 0.5843 LUMO+1 1.67 1.59 1.59 0.37

TZVP without diffuse pC50 HOMO1 5.47 33.58 33.58 1.0051 HOMO 5.18 34.01 34.01 1.0052 LUMO 1.52 32.89 32.89 1.00

53 LUMO+1 1.10 33.15 33.15 1.0054 LUMO+2 0.38 531.94 531.97 1.0055 LUMO+3 0.21 625.72 625.68 1.00

TZVP with in-plane pC shifted50 HOMO1 6.29 106.55 106.55 1.0051 HOMO 6.05 140.57 140.58 1.0052 LUMO 2.69 268.04 268.08 1.0053 LUMO+1 2.31 324.43 324.42 1.0054 LUMO+7 0.89 432.33 432.37 1.0055 LUMO+2 0.56 641.43 641.70 1.00

AuAu

Au

Au

Au

Au

S

Au

Au

Au

Au

Au

AuS

Au

Au

Au

Au

Au

Au

AuAu

Au

Au

Au

S

Au

Au

Au

Au

Au

Au

AuS

Au

Au

Au

Au

Au

Au

-8 -7 -6 -5 -4 -3 -2E / eV

10-4

10-3

10-2

10-1

100

full - 0 Au - 0.036full - 3 Au - 0.036ghost - 0 Au - 0.036

ghost - 3 Au - 0.036

-8 -7 -6 -5 -4 -3 -2E / eV

10-4

10-3

10-2

10-1

100

T

full - 0 Au - 0.036full - 3 Au - 0.036

ghost - 0 Au - 0.036ghost - 3 Au - 0.036

(b)(a)

FIG. 8. Influence of the EM size on transmission functions for a para- and b meta-connected benzene derivatives using Au9 clusters to mimic the couplingto gold electrodes. KS-DFTBP86, TZVP. Electronic structure program: QCHEM.




10/17

given for comparison. Note that the assignments HOMOand LUMO are given by similarities in shape with thecorresponding MOs calculated for the isolated molecule.Also, the order of the orbitals changes: while in the TZVPcalculation without diffuse functions, the energetic order-ing of the subsystem MOs corresponds to the one in theisolated system, the LUMO+2 and the LUMO+3 replace theLUMO and LUMO+1 when including these functions. Thesame qualitative picture of orbital shifts is also seen whenperforming analogous calculations with the two basis sets on

isolated benzene, so the shifts are not caused by the molecu-lar junction setup.Interestingly, the energies as well as the couplings of the

HOMO1 to LUMO+1 in the calculation with the Fockmatrix elements of the in-plane diffuse p functions on Cshifted are nearly the same as in the unaltered TZVP calcu-lation. The difference is rather the energies and the order ofthe virtual subsystem MOs above the LUMO+1, whichare at higher energies than the ones from the nonshifted cal-culation. In other words, shifting the diffuse p functionscauses the LUMO+2 and LUMO+3 to move out of theHOMO-LUMO gap.

The implication of these results is that the energetically

low-lying virtual subsystem MOs are causing the ghosttransmission. The couplings are large enough in both cases toresult in transmission of 1 over the energy range considered,so large couplings do not appear to be a sufficient but pre-sumably a necessary condition for ghost transmission. Fur-thermore, the assignments of the HOMO, LUMO, etc. aremade based on a TZVP calculation on the isolated moleculein its dithiol form. Thus, the reordering of the virtual sub-system MOs in the TZVP calculation on the junction must bedue to the effect of the gold clusters on the central sub-system, and thereby the electrode-molecule couplings alsohave an indirect influence on the ghost transmission.

In analogy to the D-L-L-D calculation carried out in Sec.

IV, we performed calculations using the TZVP basis set oncertain parts of the metal-molecule-metal system and theLANL2DZ basis set on others see supplementary materialfor details.63 The conclusions from this study were that it isessential to have the TZVP basis set on all atoms in thecentral region to obtain significant ghost transmission,whereas it only makes a minor difference which basis set ischosen for the gold atoms. Also, having the large basis set onthe gold atoms and all molecular atoms close to the interfacedoes not lead to ghost transmission, so that a larger

electrode-molecule coupling due to the larger basis set can-not be the sole factor responsible for ghost transmission.

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

Another aspect of ghost transmission is its sensitivity tothe type of basis set Slater versus Gaussian functions andthe size of the extended molecule EM molecule only ver-sus three gold atoms included on each side.

In all calculations so far, the central region, i.e., the EM,included the dithiolate molecule only. As shown on the right-hand side of Fig. 8, including the three gold atoms closest to

the thiolate adsorption site in the EM causes the ghost trans-mission to be insignificant when using the Gaussian-type ba-sis set.

However, it increases from an unproblematic to a signifi-cant contribution when performing the analogous ADF calcu-lations employing the Slater-type TZ2P basis set see right-hand side of Fig. 9. This illustrates that it is very difficult topredict whether or not ghost transmission will play a role fora given calculation, and in particular there is no systematicinfluence of either the size of the EM or the type of basisfunctions employed.

On the left-hand sides of Figs. 8 and 9, transmission dataare also given for the para-connected benzene system. The

AuAu

Au

Au

Au

S

Au

Au

Au

Au

Au

Au

AuS

Au

Au

Au

Au

Au

Au

-8 -7 -6 -5 -4 -3 -2E / eV

10-4

10-3

10-2

10-1

100

DZTZ2Pghost TZ2P

TZ2P - 3 Aughost TZ2P - 3 Au

-8 -7 -6 -5 -4 -3 -2E / eV

10-4

10-3

10-2

10-1

100

T

DZTZ2Pghost TZ2P

TZ2P - 3 Aughost TZ2P - 3 AuTZ2P minus ghost - 3 Au

(a) (b)

AuAu

Au

Au

Au

Au

S

Au

Au

Au

Au

Au

AuS

Au

Au

Au

Au

Au

Au

FIG. 9. Influence of the EM size on the transmission functions for a para- and b meta-connected benzene derivatives using Au9 clusters to mimic thecoupling to gold electrode. KS-DFTBP86 in combination with the scalar-relativistic ZORA approximation; small frozen cores; TZ2P basis set DZ for

comparison. Electronic structure program: ADF.




11/17

ghost transmission is always sufficiently well below the fulltransmission in the para system to be considered as insignifi-cant. The comparison to the meta-connected system, as wellas to the ghost transmission curves for the alkane in twodifferent conformations on the left-hand panel of Fig. 3, sug-gests that for similar molecules ghost transmission is often,but not always, of the same order of magnitude. It also rein-forces the conclusion drawn in Sec. V A that the system

has fewer ghost transmission problems than the system.For all calculations on meta benzene where ghost trans-mission plays a role, subtracting it from the full transmissionbrings the resulting curve close to the one obtained with theLANL2DZ basis set. This suggests that the simple correctionscheme introduced in Sec. III is indeed reasonable for caseswhere the ghost transmission is not significantly larger thanthe full transmission obtained from a small basis set.

VI. DISCUSSION: GHOST TRANSMISSION AS ABIASED DESCRIPTION OF VACUUM TUNNELING

Ghost transmission is one cause for the counterintuitiveresult that transport calculations can lose quality as the atom-centered basis set is enlarged. Of course, the static electronicstructure calculations on the metal-molecule-metal system dobecome more accurate, but this is not necessarily true for thecalculation of transport properties based on them.

As discussed in Secs. IV and V, the main reason for thisis the appearance of low-energy virtual MOs of the centralsubsystem when introducing diffuse basis functions. Twotypes of virtual MOs may be distinguished:92 1 chemicallyrelevant virtual MOs such as antibonding MOs in a diatomicmolecule and 2 virtual MOs arising from the basis func-tions that go beyond a minimal basis set. These may be

called valence virtual and hard virtual MOs, respectively seeRef. 92 and compare also Refs. 93 and 94 and referencestherein. In the limit of a complete basis set, the latter woulddescribe the continuum of unbound solutions to the HF orKS equations, and for a system with negative electron affin-ity, the LUMO should be of that type, having an energy ofzero.95 Altogether, the physical significance of virtual MOsmust be considered more questionable than occupied MOs,since they are not variationally optimized.

As pointed out in Ref. 95, there are two types of atom-centered basis functions which can lead to continuumlikevirtual MOs: 1 very diffuse basis functions and 2 basisfunctions centered on dummy atoms at some distance from

the molecule i.e., ghost basis functions. The ghost basiscalculations are an extreme example of the second case, andall the subsystem MOs are by definition continuum MOssince there are no bonds and therefore no antibondingMOs. Thus, it must clearly be continuum MOs which areresponsible for the high transmission in the ghost basis cal-culations.

As mentioned in Sec. I, there are other areas in compu-tational physics and chemistry where considering larger basissets does not necessarily lead to better results.62 In principlethis is true of the calculation of any nonvariational propertysuch as, e.g., response properties or transmission functions,for which virtual MOs play a significant role. Apart from

sum rules, there is often no general criterion as to how toestimate the error introduced by changing the size of thebasis set.

Since there is no molecule in the junction in a ghosttransmission calculation, it corresponds to pure through-space or vacuum transmission. In principle, the presence ofghost transmission could thus have some physical basis. Thisis also supported by the fact that when elongating the junc-

tion, an exponential decay of ghost transmission is predictedsee Section III and supplementary material.63 However, thedecay constants of 0.28 and 0.35 1 obtained for ghosttransmission in Sec. III and in the supplementary material,63

respectively, are much lower than the 3 5 1 typically re-ported for vacuum tunneling,96 and also less than half thevalue of typical decay constants for alkanes.97

Furthermore, it is unexpected that pure through-spacetransmission should dominate the transport properties ofmolecules such as meta-connected benzene systems al-though it cannot be strictly excluded, since controlling theinterelectrode distance when measuring pure vacuum trans-port is difficult, so that no reliable experimental reference

data are available. The results discussed in Secs. IIIV thussuggest that although of relevance in principle, the magni-tude of the through-space transmission may be overestimatedconsiderably by the transport approach employed here incombination with an extended atom-centered basis set.

Whether the transmission will converge to a meaningfulresult in the limit of a complete basis set is a difficult ques-tion for several reasons. First, for a basis of atom-centeredGaussian or Slater-type functions, it is not clear how a com-plete basis that can describe unbound states in an appropriateway should be constructed in practice. For example, is itmore important to add diffuse functions centered on the at-oms, or should dummy atoms be introduced? How small dothe exponents need to be, and which l quantum numbersneed to be considered? There is no obvious way of answer-ing these questions analytically, and it may be anticipatedthat attempts to address them numerically will be hinderedby linear dependencies in the basis set. Furthermore, the par-titioning problem see below may become more importantwhen adding functions with larger exponents, and it is un-clear to what extent this may counteract a possible beneficialeffect of increasing the basis set. One way to avoid the prob-lems associated with atom-centered basis sets would be toemploy a basis of appropriately spaced grid points. Thiswould appear to be a more straightforward way to deal with

through-space transmission, and might avoid the ghost trans-mission problem altogether. Such numerical basis sets havebeen employed in connection with DFT calculations,98 and itwill be very interesting to see how they perform in combi-nation with transport calculations. A practical drawback hereis the fact that these calculations may require a large compu-tational effort, so that in practice often smart grids are usedwith an increased density of points in regions of high elec-tron density, which again introduces a certain bias.

Another issue which may contribute to an unbalanceddescription of vacuum transmission is the fact that the inter-face through which electron transport is calculated is basisset dependent and becomes less and less well defined when




12/17

enlarging an atom-centered basis set. When performingtransport calculations, the goal is to calculate the rate ofcharge flow through an interface between an electrode andthe central subsystem, defined in three-dimensional Cartesiancoordinate space. However, the charge flow actually calcu-lated using Eqs. 16 is the one between subspaces of theHilbert space of basis functions. The larger the atom-centered basis set, the more Slater- or Gaussian-type basis

functions with small exponents, i.e., with a large spatial ex-tent, will usually be included. In particular, basis functionscentered on atoms close to the electrode-central subsysteminterface will have significant values on the other side of theinterface. This partitioning problem may affect the couplingof central subsystem MOs to the electrodes, which in turnmay affect the way the virtual MOs contribute to ghost trans-mission. Thus, it seems a worthwhile goal for future work tosolve this problem in a way which avoids any mixing ofcentral subsystem and electrode basis functions for one suchapproach, which, however, does not avoid the mixing en-tirely, see, e.g., Ref. 68.

For the sake of completeness, it should also be men-

tioned here that problems may occur because when combin-ing the Landauer approximation with KS-DFT calculations,effective transport of noninteracting fermions instead ofelectrons is described.44 Furthermore, it is not clear howwell MOs obtained from approximate exchange correlationfunctionals designed to yield accurate total energies aresuited for describing transport properties. Recent work byVan Voorhis et al . suggests that common exchange-correlation functionals are not suitable for transport calcula-tions because of their local nature.99 However, when employ-ing the exact functional, the KS MO energies may beinterpreted as ionization energies.100102

VII. CONCLUSION AND OUTLOOK

When performing transport calculations within the Lan-dauer approximation, in combination with a NEGF approachusing KS-DFT, we have made the counterintuitive observa-tion that transmission obtained by employing a larger basisset is often to be trusted less than that obtained using asmaller basis. The reason for this unexpected result has beenidentified as ghost transmission. Ghost transmission was de-fined as the non-negligible transmission up to 0.3 in themodel systems investigated here obtained from transport

calculations in which the central subsystem part consists ofghost basis functions only, i.e., without atomic nuclei or mo-lecular electrons in the central region. The similarity betweenthe nearly energy-independent ghost transmission and flat re-gions of transmission curves obtained in regular transportcalculations with extended basis sets suggested that ghosttransmission was also present in the latter. Our resultspointed to low-energy virtual MOs of the central subsystemas the cause for ghost transmission. Since these virtual MOsdo not give a good description of the continuum of unboundsingle-electron states, ghost transmission may be interpretedas an inaccurate description of through-space or vacuumtunneling. Large couplings between the central region and

the electrodes could not be excluded as equally meaningfulculprits, but are surely not the only source of ghost transmis-sion.

At the same time, relatively small changes, such as omit-ting only one set of p basis functions, using a smaller basisset on a few atoms only, employing Slater-type instead ofGaussian-type basis functions, or enlarging the central regionfrom including zero to including three gold atoms on each

side, have been shown to yield considerable changes in theimportance of the ghost transmission. Thus, for a given cal-culation, we cannot in general predict whether ghost trans-mission will change qualitative conclusions or not. However,the following may be regarded as rough rules of thumb: sofar, we have never obtained a significant ghost transmissionfor a calculation employing a basis set of double-zeta qualitywithout polarization functions. On the other hand, ghosttransmission is particularly likely to occur when using basisfunctions with comparatively small exponents such as0.1 a.u.2 in p-type Gaussian functions. Furthermore, ghosttransmission affects transport through orbitals of symmetryconsiderably more than those of symmetry. Finally, the

larger the distance between the electrodes, the smaller thevalues of ghost transmission tend to be which, of course,also holds for through-MO transmission.

In general, flat parts in the transmission as a function ofenergy may suggest significant ghost transmission althoughsuch features have also been found when the dominant con-tribution to transmission changes from the to the systemin cases of interference20. As a simple heuristic remedy, itmay then be recommended to subtract the ghost transmission

from the full transmission. Fortunately, although the conver-gence of the SCF algorithm may be difficult in some cases,loose convergence criteria do not seem to alter ghost trans-mission curves to a large extent. While this correction mayprove to be of little value in cases where the ghost transmis-sion is more than one order of magnitude larger than the fulltransmission calculated from a small basis set, it may pro-vide a good approximation for the true transmission in otheraffected cases.

Another conclusion from this work is that moleculeswith interference features in the transmission, such as meta-connected benzene derivatives, as well as other low-conducting systems such as alkanes, are considerably moresensitive to basis set effects and presumably to other param-eters of the calculation than those which have a consistentlyhigh transmission over the energy range of interest. There-

fore, basis set studies should be carried out on these sensitivemolecules rather than on their more robust counterparts.The numerical agreement between calculated ghost

transmission and full transmission in all problematic casesdiscussed in Secs. III and V suggests that ghost basis calcu-lations can lead to a good estimate of the way in which alarger basis set modifies the contribution of the virtual MOs.MO interference20,22 makes it hard, however, to settle thisissue satisfactorily by considering transmission through sub-sets of MOs, and thus deciding to what degree modificationsof the chemically relevant virtual MOs are responsible forthe ghost transmission, and to what degree the increasingnumber of hard virtuals leads to problems.




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Altogether it becomes clear that when used blindly, theimplementation of the Landauer approach adopted here maynot be suitable for large nonorthogonal atom-centered basissets as commonly used in quantum chemistry. This adds tothe other computational and formal uncertainties which mayprevent this implementation from giving quantitatively reli-able results. These include the band alignment problem, con-ceptual issues with KS-DFT, and self-interaction errors in

exchange correlation functionals also called delocalizationerrors103. To circumvent difficulties when using large atom-centered basis sets, a better description of the vacuum maybe needed to achieve a more balanced description ofthrough-space versus through-bond transmission, for ex-ample, by placing an array of ghost basis functions betweenthe electrodes, or by employing a numerical grid or planewaves as a basis. It may also be necessary to employ a dif-ferent partitioning strategy to describe the surface throughwhich the transmission and thus the current are to be cal-culated in a more realistic way.

A formal analogy may be made2830 between ET indonor-bridge-acceptor systems104,105 and molecular conduc-

tion, and consequently the interpretation of ghost transmis-sion in terms of a biased description of vacuum tunnelingmay be related to through-space coupling in ET reactions.The effects of basis set size,106,107 virtual orbitalcontributions,108 and bridge length109 in pathway formula-tions of bridge-mediated donor-acceptor coupling calcula-tions have been studied, but the connection between largebasis sets and an overestimation of through-space couplinghas not been discussed in this context. Therefore, the consid-erations discussed here apply, in principle, to ET reactionsand more generally to all cases where electronic coupling viamolecular bridges is of importance.

ACKNOWLEDGMENTS

The authors would like to thank Thorsten Hansen, Jef-frey R. Reimers, and Matthew G. Reuter for helpful com-ments and discussions. C.H. gratefully acknowledges fund-ing by a Forschungsstipendium by the DeutscheForschungsgemeinschaft DFG. M.A.R. thanks the Chemis-try and Materials Research Divisions MRSEC program ofthe NSF and the DOE for support.

APPENDIX A: A FORMAL PROBLEM WITH THESIMPLE IMPLEMENTATION OF THE WIDE-BAND-LIMIT APPROXIMATION

In the WBL approximation, the assumption is made thatthe density of states DOS of the electrodes is energy inde-pendent, which is a good approximation for gold with its flatDOS around the Fermi energy. In combination with elec-tronic structure calculations, a LDOS is assigned to eachbasis function centered on a gold atom. Formally, theseLDOS values add up to the total DOS see below, but inactual transport calculations it often turns out that the LDOSfor the gold 6s band may be assigned to each basis functionregardless of its l quantum number without significantly al-tering the results.83,110 The values chosen for the LDOS intransport calculations employing the WBL approximation are

commonly those obtained from DFT calculations under pe-riodic boundary conditions using the local-densityapproximation.84 These values sum up to the total DOS whenassigned to the functions in a minimal basis set. It is clear,however, that the larger the basis set, the more this sum willexceed the total DOS. Since a larger DOS leads to a largertransmission, this effect may contribute to an overestimationof the transmission function when using large basis sets in

combination with the implementation of the WBL approxi-mation presented here.

1. Formal considerations

Equation 6, which defines how the electrodes Greensfunctions are calculated within the WBL approximation, canbe derived as follows. The retarded Greens function opera-tor gE of a one-electron system described by the Hamil-

tonian h is defined as

gE = lim0+

E h + i1 , A1

which can be expressed in a basis of eigenfunctions j of has

gE = lim0+

j

jE j + i1j , A2

where j is the energy of eigenfunction j.

The operator DE associated with the DOS DE of thesystem is defined in the same basis as

DE = j

jE jj . A3

Of course, the total DOS is obtained as its expectation value,which for a single-Slater determinant wave function reads

DE = DE = j,k

kjE jjk

= j,k

jkE jjk = j

E j . A4

From the definition of the Dirac delta function as

x = lim0+

1

x2 + 2, A5

the relationship

x = 1

lim0+

Imx + i1 A6

can be derived, from which it follows that

DE = 1

ImgE . A7

The LDOS DE, i.e., the DOS projected onto the basisfunction which may be chosen to be local can be de-fined as

DE = 1

ImgE , A8

which may be reformulated as




14/17

ImgE = DE. A9

By choosing a constant value for DE and setting all off-diagonal imaginary and all real terms to zero, Eq. 6 is ob-tained.

By adding up the LDOS values for all basis functions,the total DOS is recovered,

DE = DE. A10

2. Effect of the LDOS on the transmission

The constant LDOS value of 0.036 eV1 employedthroughout this paper is taken from Ref. 84 and correspondsto the DOS of either spin up or spin down electrons, pro-

jected onto the 6s band of gold. It should be noted that theLDOS value of 0.072 eV1, which can sometimes be foundin the literature, refers to the LDOS for all electrons, not tothe spin-resolved one, so it seems reasonable to multiply itby 1/2 when using an effective one-electron picture comparealso Ref. 111.

Since there is no clear prescription on how to distribute

the 6s LDOS between the two or three basis functions cho-sen for the 6s orbital of gold in a LANL2DZ or TZVP basisset, respectively, we will continue to use the value0.036 eV1 for all of them. This can be seen in analogy tothe way the 6s LDOS may be employed for all basis func-tions regardless of their l quantum number without a consid-erable loss of accuracy.83,110,111 However, since this does af-fect the transmission, in particular for systems such as meta-connected benzene see Fig. 10, the possibility of thisparameter needing to be adjusted should be kept in mind.Figure 10 is equivalent to Fig. 8, but includes full and ghosttransmissions for a LDOS value of 0.012 eV1, i.e., the 6sLDOS is distributed evenly among the three basis functions

describing that shell. As to be expected, for the meta-connected benzene both the total and the ghost transmissiongo down compared with the LDOS=0.036 eV1 calcula-tions, although only slightly, and the ghost transmission isstill significant. For the para system, the full transmission isonly shifted, but not diminished.

3. Discussion

One obvious solution to the LDOS problem would be tochoose a minimal basis set on the gold atomshowever, ithas been reported that using a single-zeta basis set for theelectrodes and a double-zeta one with polarization functionsfor the central subsystem instead of having the larger basisset on all atoms may lead to large changes in the calculatedtransport properties.59

Improving the implementation of the WBL approxima-tion described here would involve recalculating the correctLDOS for each basis function every time a new basis set isemployed. This would eradicate the main advantage of theWBL approximation, its computational simplicity while re-taining a surprisingly good agreement with conductivities

obtained using a tight-binding description of the electrodesGreens functions.22,112 Also, even with the smallest reason-able LDOS 0.012 eV1, there is considerable ghost trans-mission in meta benzene see Fig. 10, so correcting thisvalue alone would not solve the problem.

Altogether, the considerations outlined above make itclear that the uncertainty regarding the correct value for theLDOS does not allow for definitive quantitative predictionsto be made from the WBL approach as implemented here. Ofcourse, it may be anticipated that for a reasonable choice ofLDOS values, even when sacrificing some quantitative accu-racy, qualitative conclusions should be affected only negligi-bly.

AuAu

Au

Au

Au

Au

S

Au

Au

Au

Au

Au

AuS

Au

Au

Au

Au

Au

Au

AuAu

Au

Au

Au

S

Au

Au

Au

Au

Au

Au

AuS

Au

Au

Au

Au

Au

Au

-8 -7 -6 -5 -4 -3 -2E / eV

10-4

10-3

10-2

10-1

100

T

full - 0 Au - 0.036full - 3 Au - 0.036full - 0 Au - 0.012

ghost - 0 Au - 0.036ghost - 3 Au - 0.036

ghost - 0 Au - 0.012

-8 -7 -6 -5 -4 -3 -2E / eV

10-4

10-3

10-2

10-1

100

T

full - 0 Au - 0.036full - 3 Au - 0.036full - 0 Au - 0.012

ghost - 0 Au - 0.036ghost - 3 Au - 0.036

ghost - 0 Au - 0.012

(a) (b)

FIG. 10. Influence of the EM size and the LDOS value on transmission functions for a para- and b meta-connected benzene derivatives using Au9 clusters

to mimic the coupling to gold electrodes. The values in the legend refer to the constant LDOS in Eq. 6. KS-DFTBP86, TZVP. Electronic structure program:QCHEM.




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APPENDIX B: TRANSMISSION THROUGH INDIVIDUALCENTRAL SUBSYSTEM MOs

Subsystem MOs are the solutions to the subsystemssecular equation,

HCCC= SCCCC, B1

where HC and SC are the central subsystem blocks of thetotal systems Fock and overlap matrices, respectively, the

entries in CC are the subsystem MO coefficients, C is adiagonal matrix containing the MO energies, and C indicatesthe central subsystem. To make this manuscript self-contained, the procedure employed for calculating the trans-mission through an individual subsystem MO shall be brieflysummarized: the total system is transformed with a Lwdinorthogonalization on the subsystem,

HC = SC1/2HCSC

1/2, B2

while adjusting the elements having indices on both an elec-trode and on the central subsystem accordingly, i.e., the totalHamiltonian and overlap matrix are transformed with a ma-trix which has the electrode blocks equal to the unit matrixand the central block equal to SC1

/2, and the off diagonalblocks are all zero. Then, HC is diagonalized by transformingthe full system with a transformation matrix of the style de-scribed above, but with the HC eigenvectors in the centralblock. Then, all elements with indices not corresponding tothe selected central-subsystem MO are removed from thecentral subblocks and the central system-electrode couplingblocks of the transformed total Fock and overlap matrices,and a transport calculation is carried out for the remainingsystem. Of course, this procedure can also be used to evalu-ate the transmission due to any set of subsystem MOs.

APPENDIX C: COMPUTATIONAL METHODS

Electronic structures of the electrode-molecule-electrodesystems were calculated using a finite cluster approach, i.e.,without employing periodic boundary conditions. Molecularstructures were first optimized for the isolated, twofold nega-tively charged dithiolate molecules using a 6-311G basisset113 and the B3LYP114,115 density functional. To constructthe junctions, the molecules were placed between Au9 clus-ters, mimicking the fcc sites of Au111 surfaces. The sulfur-gold distances were chosen to be 2.48 , as predicted fromKS-DFT calculations for this binding arrangement,116 whilethe gold-gold distances were set to their value in extended

gold crystals 2.88 . Benzene-based molecules includeduCwCu spacer groups on both sides to minimize directinteractions between the hydrogen atoms of the benzenerings and the gold electrode.

Zero-voltage transmission functions were calculated ac-cording to Eq. 1 using a tool written in our laboratory117 forpostprocessing electronic structure calculations. Overlap andFock matrices were obtained from KS-DFT44,45 or HF calcu-lations on molecules using a locally modified version ofQCHEM

118 or ADF.119 The effects of the open-boundary con-ditions and the voltage drop in the molecular junctions wereneglected. The self-energy matrices R and L of the goldelectrodes were calculated within the WBL approximation

see Sec. II, assuming a constant LDOS of 0.036 eV1 ifnot indicated otherwise, which corresponds to the LDOS forelectrons of a given spin quantum number of bulk gold at theFermi energy.84 For the transport calculations, the pure den-sity functional BP86,120,121 as well as the hybrid functionalB3LYP and the HF approximation, was employed. The LosAlamos LANL2DZ effective core potentials122 and thematching basis sets of double-zeta quality were used as

implemented in QCHEM, as well as Ahlrichs SVP, TZVP, andTZVPP basis sets123,124 in combination with Stuttgart effec-tive core potentials for gold atoms which also account forthe scalar relativistic effects.125 SV denotes a split valenceand TZV a triple-zeta split valence basis set, and one and twoPs correspond to one or two sets of polarization functions onall atoms, respectively. While all basis sets used in QCHEMcalculations were of Gaussian type, the Slater-type basis setsof double and triple-zeta quality with none and two sets ofpolarization functions, respectively, were used in the ADFcalculations DZ and TZ2P as implemented in ADF. All ADFcalculations were carried out within the zeroth-order regularapproximation ZORA to describe scalar relativistic effects,

and were using a small frozen core, comprising the 1 s shellof carbon atoms and the 1s through 4d shells of gold atoms.A tight criterion of 108 a.u. for the direct inversion in theiterative subspace error was chosen in all QCHEM calcula-tions, whereas a criterion of 106 a.u. was chosen for thelargest element of the commutator of the Fock matrix and thedensity matrix in the ADF calculations.

Since the choice of the value for the Fermi energy of thegold electrodes within the Landauer approximation is not atall obvious within the approach chosen here, and since itdoes not affect the discussion of parameter dependence ofthe transmission, energy values were not shifted by the Fermienergy. For a general orientation, while the Fermi function

for bulk gold is 5.5 eV,126 the fact that in practical calcu-lations metal clusters of finite size are used to model the bulkelectrodes may shift this value. We restricted the energyrange for which transmission is calculated to the interval of8 to 2 eV, which includes all energy values commonlyaccessible experimentally. It may also be assumed that thisinterval encompasses the region around the Fermi energy forwhich the WBL approximation provides a good description.Lwdin population analyses were carried out by postprocess-ing QCHEM output with ARTAIOS.117

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