The Zero-Voltage Conductance of Nanographenes - American

Article pubs.acs.org/JPCC

The Zero-Voltage Conductance of Nanographenes: Simple Rules and Quantitative Estimates Didier Mayou,† Yongxi Zhou,‡ and Matthias Ernzerhof*,‡,† †

Institut Néel, Bat D, Office D 306, 25 Avenue des Martyrs 38042 BP 166 Grenoble Cedex 9, France Département de Chimie, Université de Montréal, C.P. 6128 Succursale A, Montréal, Québec H3C 3J7, Canada

‡

ABSTRACT: Zero-dimensional graphenes, also called nanographenes (NGs), consist of fragments of graphene with a finite number of hexagons. NGs can be viewed as a subset of the polycyclic aromatic hydrocarbons (PAHs) that continue to attract chemists’ attention. We developed a simple theory for the ballistic electron transport through NGs which combines elements of the electronic structure theory of graphene, intuitive methods for the calculation of the molecular conductance, and chemical concepts such as Kekulé structures. This theory enables one to analyze the relation between the structure of NGs and their conductance. General formulas and rules for the zero-voltage conductance as a function of the contact positions are derived. These formulas and rules require at most simple paper and pencil calculations in applications to systems containing several tens of carbon atoms.

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INTRODUCTION AND SUMMARY OF RESULTS In recent years, graphene has become a widely studied material, and increasingly, zero-dimensional graphene fragments receive attention. Zero-dimensional graphene fragments, which are often referred to as nanographenes (NGs), are finite arrangement of hexagons whose dangling bonds are saturated with hydrogens. This type of system has been known and studied for more than a century and it is a particular example of a polycyclic aromatic hydrocarbon (PAH).1,2 Graphene has various interesting properties, many of them are shared with NGs. In particular the conductance properties of graphene are a focus of attention since this material is seen as a key component of future electronic devices. Given the vast literature on transport in graphene, we cannot provide an overview but merely two recent examples of the literature.3,4 Here, we focus on NG as a component of molecular electronic devices (MEDs).5−9 In the past, simple models of MEDs have been developed10−14 that facilitate a qualitative understanding of structure-conductance relationships. We start from the sourcesink potential (SSP) method11,15 and variations of it,12 and extend on these works to arrive at a qualitative theory for NGs, which facilitates analytic estimates of the zero-voltage conductance. The SSP method is equivalent to the Landauer formula; i.e., it yields identical results11,16 © XXXX American Chemical Society

The conductance of a molecule also depends on the precise nature of the contacts, as discussed for example in the case of graphene constrictions.17 Efficient computational methods for quantum transport, and in particular for treating the contacts, have been developed in.18−22 Here we focus on NG and consider the contacts in the most simplistic way. However, this does not affect the conclusions of this work. We consider prototypical NGs in the Hückel approximation. This means that we describe only the π-electron system generated by the sp2 carbon atoms. Furthermore we consider bipartite molecules. Bipartite molecules are molecules that can be subdivided into two sets ( and ) of atoms, where atoms in  bind only to atoms in  and vice versa. This subdivision is always possible if there are no circular subsystems with an odd number of sp2 carbon atoms. All the NGs fall into the class of bipartite systems. We focus on the conductance (g) per spin channel in the zero-voltage limit, or equivalently the conductance at the Fermi energy (EF), i.e., g(EF) = (e2/h) T(EF), where T(EF) is the transmission probability of the system, e is the electron charge and h is Planck’s constant. Received: December 20, 2012 Revised: March 16, 2013

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dx.doi.org/10.1021/jp3125389 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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captured the attention of various pioneers43,44 of quantum chemistry. NBMOs are zero-energy orbitals found in radicals and they also appear when molecules are discussed and analyzed (chapter 3 of ref 44) in terms of their various fragments. Finally, we summarize the main formulas and rules of our theory so that the reader who wishes to proceed directly to the applications can skip the theory section. All the molecules considered here are assumed to be stable, exhibiting a symmetric spectrum and a finite HOMO/LUMO gap. Since we chose the energy scale such that the Fermi energy (EF) is equal to zero (right in the middle of the HOMO/LUMO gap), the following rules focus on the transmission probability T(E) at E = 0. T(0) is expressed in terms of the real variable s through T (0) = 4s /(1 + s)2

(1)

where s can be calculated for each NG with the help of the following rules. Rule 1: aa- or bb-Connected Nanographenes. Two contacts that are connected to atoms of the same sublattice will always yield s = 0 and T(0) = 0. This result can also be regarded as an interference effect as discussed in the early papers by Baer, Neuhauser, and co-workers.23,33−35 Based on the wavelength of electrons at the Fermi energy, these authors provide concepts for the analysis of the molecular conductance. Rule 2: ab-Connected Nanographenes. If one lead is connected to an a-type atom and the other one to a b-type atom, we delete the b-type connecting atom and proceed to the construction of the resulting defect state DS(b) associated with atom b. DS(b) is localized on the  sublattice. (In the applications presented below, numerous examples of defectorbital constructions are discussed.) The transmission probability T(0) is deduced from eq 1 with the following expression for the variable s,

Figure 1. Illustration of the side-chain effect and the positional dependence of the conductance in benzene. In the upper half of part i, the conjugated molecules are shown and the atom types (a or b) are indicated. In the lower half, contacts are attached in aa configuration, indicated by thick bonds, resulting in vanishing zero-voltage conductance. In part ii, the benzene molecule is shown and connected to the contacts in meta position. Again since this is an aa configuration, the zero-voltage conductance vanishes.

The Hückel model employed in this work has of course its limitations and not all its predictions can be confirmed with more advanced computational methods or by experiment (for a discussion of successes and failures see, e.g., refs 23−26). To validate our predictions, in the second-last paragraph, we perform all-electron, Green’s function calculations combined with Kohn−Sham density functional theory27−32 that show qualitative agreement with predictions of the Hückel model. In the past other simple theories have been provided that establish relationships between molecular structure and molecular conductance, examples include.14,23,26,33−36 Very recently, one of our earlier predictions (side-chain effect37) about the molecular-structure to conductance relationship has been confirmed experimentally.38 Most relevant for the present work are the comprehensive articles by Fowler and coworkers.39−42 In these articles, and in particular in ref 41, various formulas are developed to calculate the conductance of MEDs. A highly remarkable result is a formula for the zerovoltage conductance of a NG in terms of the number of Kekulé structures of the NG and certain fragments of it. The approach described below extends the toolkit of analysis and design aids for the molecular electrician. In particular we will introduce the concept of a defect state (DS) which is the zero-energy orbital obtained after eliminating an atom from the NG. The eliminated atom is an atom a contact is attached to (referred to as connecting atom). Once the first connecting atom is chosen, the corresponding DS is easily calculated and it provides the conductance of the NG for all positions of the second contact. For NGs modified through the addition of a hydrogen atom (or another group) to a carbon atom, that atom is effectively removed from the π-electron system, leading to a DS which again determines the conductance. The DS introduced here in the context of MEDs is closely related to the nonbonding molecular orbital (NBMO) that

2 t 2 (Σl ∈ ad(b)cl) s= βLβR |ca|2

(2)

In this formula, ∑l∈ad(b) cl adds up the coefficients of DS(b) on the atoms that share a bond with the b-type connecting atom, correspondingly, ad(b) denotes the set of atoms that form a bond with the atom b. ca is the coefficient of DS(b) on the a-type connecting atom. βL(R) is the coupling parameter between the left (right) contact and the corresponding connecting atom. t is the coupling parameter between the molecular carbon atoms. Rule 3: ab-Connected Nanographenes with Vanishing Zero-Voltage Conductance. A molecule connected in ab configuration has a vanishing zero-voltage conductance if, and only if, DS(b) has a zero weight on the a-type connecting atom. An equivalent condition is that the molecule obtained by eliminating the a and b connecting atoms has two zero eigenvalues and is therefore a diradical. It is known45 that diradicals (or any molecule with vanishing eigenvalues) are characterized by the fact that they have no Kekulé structure. Kekulé structure refers to an arrangement of bonds where each carbon atom participates in a double bond. Below, we provide examples of such molecules and show that they exhibit an electronic decoupling between their subunits each of which containing a localized DS. Such systems have also been discussed in Dewar’s and Dougherty’s book (p 103) on perturbation molecular orbital theory.44 B



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Rule 4: Eliminating Atoms from the π-Electron System. A molecule can be modified for example if the pz orbital of a certain atom is eliminated from the π-electron system by bond formation to an additional hydrogen atom or another σ-bond forming group. The elimination of a pz orbital of an a-type atom (namely the one labeled a*) results in a DS(a*) which is localized on the  sublattice. If the original NG is ab connected, its conductance depends on the coefficient cb of the b-type connecting atom in DS(a*). If cb = 0, the zerovoltage conductance is that of the initial molecule. If cb ≠ 0 then the zero-voltage conductance of the modified molecule is zero. Equivalent results are obtained if the roles of a- and b-type atoms are exchanged. For an aa-connected molecule, elimination of an a-type pz orbital does not have any consequences and the conductance remains equal to zero. In case a b-type atom is removed, the s variable related to the transmission through eq 1 is obtained according to

respectively, and r is the reflection coefficient. In the context of the Green function approach (see, e.g., refs 46 and 47), the theory developed amounts to taking contact self-energies that are purely imaginary. The Hamiltonian matrix of the system thus consists of the Hamiltonian of the isolated NG (HNG) and the complex potentials added to the diagonal elements of two atoms x and y, H = HNG + Θ xL(r ) + Θ Ry

The reflection coefficient r in this equation is related to the zero-voltage conductance per spin channel via the transmission probability T(E) = 1 − |r(E)|2. r (or equivalently s) is obtained from the secular equation for E = 0, i.e., det(H) = 0.11 For the cases that we consider here, r ∈ [−1, 1] and s ∈ [0, ∞], so that we obtain eq 1 of the introduction, T (0) = 1 − r 2 = 4s /(1 + s)2

2

s=

βL |c a|̃ βR |ca|2

Note that changing r to −r, or equivalently (cf. eq 4) s to 1/s, does not change T(0). A large or small value of s, leads to T(0) ≪ 1. T(0) takes its maximum value T(0) = 1 for s = 1 (s is positive). Having reduced the infinite MED to a molecule to whose Hamilton matrix effective potentials are added, we can focus on these Hamiltonian matrices and work out simple expressions for their zero-energy solutions. An important feature of the systems considered is that they are bipartite, which implies15 that the orbitals of zero energy can be subdivided into two decoupled components, one defined on each sublattice ( and ) of atoms. The two separate components have a simpler structure compared to the parent orbital. Recently, we elaborated15 on this issue and showed how it can be exploited to calculate the zero-voltage conductance. To discuss this point in more detail, we start from the stationary Schrödinger equation for E = 0

(3)

Here cã and ca are the coefficients of the DS(b) on the left and right connecting atom, respectively. This formula shows that T(0) is large for |cã|2 ≈ |ca|2 provided that βL = βR, which we assume in this work. Of course, the rules apply also if the roles of a- and b-type atoms are interchanged. The complete elimination of a pz function from the πelectron system is an extreme case that in most actual system will only be partially realized. Even if the pz forms a bond to another atom or chemical group, the resulting σ bond remains coupled to the π orbitals. However, the predictions made based on the complete elimination of the pz function are confirmed by the all-electron density functional calculations provided in the second to last section. Simple Theory for the Zero-Voltage Conductance of Nanographenes. The rules and formulas described in the introduction are now derived in detail. In MEDs, the molecule is connected to two semi-infinite contacts. Here we use the simplest possible approach and represent the contacts by chains of atoms. To further reduce the complexity of the MED, we employ the SSP method,11,12 which enables one to replace the contacts by complex potentials in a rigorous manner. Since the diagonal elements of the contact Hückel matrices are set to zero, the Fermi energy of the contacts is zero as well (EF = 0). We are only interested in the zero-voltage conductance of the NG (i.e., the conductance at the Fermi energy) and therefore in wave functions of the MED of zero energy. With this restriction, the SSP potentials ΘL(r) and ΘR take a particularly simple form,12 ΘL(r ) = −iβLs ,

ΘR = iβR

1+r s= 1−r

(6)

(7)

HC = 0

with ⎛ HAA + Θ xL(r ) + Θ Ry MAB ⎞ ⎟⎟ H = ⎜⎜ MBA HBB ⎠ ⎝

(8)

and employ the Löwdin partitioning technique48 to obtain an eigenvalue equation involving an effective Hamiltonian Heff A, x y Heff A = HAA + Θ L(r ) + Θ R − MAB

1 MBA HBB

(9)

This effective Hamiltonian acts only on CA, i.e., the projection of the wave function onto . HAA (HBB) is the Hamiltonian matrix of the atoms belonging to  () and MAB is the coupling matrix between the atoms of  and . In this example, the SSPs ΘxL(r) and ΘxR are added to atoms of . An equivalent equation holds for the projection of the wave function on subset . The matrices HAA and HBB in this equation are diagonal with diagonal elements equal to the Fermi energy which is zero. The simple structure of HAA and HBB is what enables us to develop an analytical approach. In particular terms of the type Σ = −MABH−1 BB MBA, appearing on the right-hand-side of eq 9, can be greatly simplified. For Σ to have an eigenvalue of finite (zero) energy, MBA has to have a nonvanishing kernel. We will show that indeed the kernel of

(4) (5)

ΘL is added to the diagonal matrix element of a certain atom (labeled x), which then behaves as if it where coupled to a onedimensional contact (source) with an incoming and partially reflected wave. Similarly, adding ΘR to the diagonal element of an atom labeled y implies that this atom is coupled to a right one-dimensional contact (sink) permitting only an outgoing wave. In eq 4 and eq 5, βL and βR are real negative coupling parameters12 of the molecule to the left and right contact, C



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where Cb denotes the complement of b within , i.e., all the atoms in  except b. hbb is the diagonal element of the b-type connecting atom, and MAb the corresponding coupling matrix 1 to atoms in . To ensure that the term −MA C b H M C bA does

MBA (or variations of MBA) plays the key role for the zerovoltage conductance, giving rise to the DS. In the following sections we distinguish between two different cases (aa, and ab) depending on whether the left contact is connected to an a or b atom, while the right one is attached to an a atom. The bb and ba connections are of course equivalent to aa and ab, respectively. Rule 1: aa-Connected Nanographenes. First we consider the case where the contacts are connected to two atoms of . The Hamiltonian matrix of the system has the general form of eq 8. Using the Löwdin matrix partitioning as announced above, we construct an effective Hamiltonian for the  subsystem at E = 0 a ã Heff A = HAA + Θ L(r ) + Θ R − MAB

1 MBA HBB

CbCb

not diverge, M C bA has to have a nonvanishing kernel. This condition is always satisfied since M C bA is a mapping from an N-dimensional to an (N − 1)-dimensional space. Furthermore, since we suppose that the molecule itself does not have a zero eigenvalue, removing a single atom from it does result in only one zero eigenvalue of the remaining fragment. This is a consequence of the Cauchy interlacing theorem.58 We refer to the kernel of M C bA by the vector CA(b). The Hamiltonian matrix Heff Ab becomes b a Heff Ab = HAA + hbb + Θ L(r ) + Θ R + MAb + MbA

(10)

− 0 × CA (b) ⊗ CtA (b) − ∞ × P( ⊥CA(b))

Since HBB = 0, the coupling matrix MAB (1/HBB)MBA diverges unless MAB, has a nonvanishing kernel, or equivalently, unless the equation

MABC = 0

The term −∞ × P(⊥CA(b)) eliminates all but one (i.e., CA(b)) wave function on , or in other words, the wave function of the MED of zero energy is proportional to CA(b) on . This is expressed by the term 0 × CA(b) ⊗ CtA(b) which projects onto the zero-energy subspace and multiplies the projection by the corresponding eigenvalue. Since the vector CA(b) plays a prominent role in our theory, we refer to is as defect state (DS). This name is motivated by the fact that the elimination of a b-atom leads to a defect in the molecule giving rise to a new orbital of zero energy. We are interested in the device orbital at the Fermi energy; therefore, we focus on states of Heff Ab of zero energy. For such a solution, the determinant of Heff vanishes Ab

(11)

has nontrivial solutions. These solutions contain only atomic coefficients of , they are decoupled from . This would imply that the isolated molecule has eigenvalues equal to zero, a case that we exclude since we consider only stable molecules that have a finite HOMO/LUMO gap. Therefore, we conclude that a ã Heff A = HAA + Θ L(r ) + Θ R − ∞ × PA

(12)

where PA projects onto the entire subspace of wave functions on . This equation has no finite-energy solution if the parameter s in ΘaL(r) is finite. Therefore we conclude that s diverges which means that the reflection is complete (r = 1). Consequently, the aa- and of course also the bb-contact configuration does result in T(0) = 0. Various cases of vanishing conductance discussed in the past fall into this category. For example, some occurrences of the side-chain effect, which has been analyzed extensively in various theoretical articles35,37,40,42,49−51 and which has recently been confirmed experimentally,38 can be explained by invoking rule 1. In the upper half of Figure 1, two examples are presented where stable, conjugated molecules are connected in aa configuration such that the resulting zero-voltage conductance vanishes. However, we stress that not all examples of current suppression through the side-chain effect can be reduced to rule 1. Furthermore, there are theoretical.23,33,34,40,41,52−56 as well as experimental articles57 that discuss the reduced conductance of meta-connected benzene as compared to para- or orthoconnected benzene. Again, in Figure 1, we illustrate that this effect can also be explained through rule 1. As already mentioned in the introduction, rule 1 can be viewed as an interference effect exhibited by electrons at the Fermi energy.23,33−35 Rule 2: ab-Connected Nanographenes. Now we investigate the case where the two contacts are connected to two different sublattices. Starting from eq 6, we construct an effective Hamiltonian for the  sublattice together with the atom b by means of the Löwdin partitioning, b a Heff Ab = HAA + hbb + Θ L(r ) + Θ R + MAb + MbA 1 − MA C b M C bA HC b C b

(14)

det(Heff Ab) = 0

(15)

Elimination of the subspace of infinite energy and representation of Heff Ab in the space spanned by CA(b) and the basis function of b yields a two by two matrix

Heff Ab

⎛ |c |2 Θ (r ) t ∑ cl ⎞⎟ ⎜ a L l ∈ ad(b) ⎟ = ⎜⎜ ⎟ ΘR ⎟ ⎜ t ∑ cl ⎝ l ∈ ad(b) ⎠

(16)

This result is used to obtain an equation for the variable s, from eqs 15 and 16 we get, 2

s=

t 2 (∑l ∈ ad(b) cl) βLβR ca2

This is eq 2 presented in rule 2. As we show below, CA(b) can easily be calculated with paper and pencil. The low bias conductance of the molecule connected through b is therefore formulated in a transparent way. There is also an important link between the DS of a given molecule and the Green’s operator. By definition CA(b) satisfies HNG(b)CA(b) = 0, where HNG(b) is the Hamiltonian matrix of the NG with the b-type connecting atom suppressed. Now if one considers the Hamiltonian matrix of the initial NG then it is easy to see that HNGCA (b) = Cb

(17)

where Cb is a vector whose only nonzero component is on the atom b. The coefficient ca of the a atom the contact is attached

(13) D



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to can be extracted from CA(b) by ca = 1taCA(b), where 1a is a unit basis vector corresponding to the pz function on atom a. Since HNG is a real symmetric matrix ca = 1ta CA (b) = 1at = Cbt

stable, this is not possible. However, it may happen that ca2 = in eq 2. In this case s is infinite, T(0) = 0, and CA(b) is obviously an eigenstate of NG(a,b) with zero energy. Since NG(a, b) contains as many a-type as b-type atoms, there are as many states of zero energy on  as there are on . Therefore the number of zero eigenvalues of NG(a, b) is even. because of the Cauchy interlacing theorem,58 each time one suppresses an atom the maximum increase in the number of zero eigenvalues is one. Consequently, NG(a, b) has either no eigenstate of zero energy or exactly two eigenstates with zero energy. Since CA(b) is an eigenstate of zero energy of NG(a,b) there are two eigenstates of zero energy. The same conclusion can also be drawn from the Coulson and Rushbrooke59 pairing theorem which reveals the symmetry in the spectrum of bipartite molecules. Reciprocally if NG(a, b) has two zero eigenvalues, there is one state CA(a, b) of NG(a, b) defined on , which has zero energy. CA(a, b) can be turned into an eigenstate of NG(b) by setting the coefficient on the additional a-type atom to zero. Obviously this new state CA(b) is also of zero energy and it must be the DS of NG(b) (indeed according to the Cauchy theorem58 NG(b) has only one eigenstate of zero energy). This means that the conductance is zero since the component of CA(b) on the a-type atom is zero. In the next section we show examples of ab-connected NGs that do not conduct. These NGs exhibit a common structure, characterized by two features: (i) The NG can be separated in two parts NGI and NGII, each of which is a stable molecule. (ii) We introduce the notation I for the subset of atoms of NGI that bind to atoms of NGII which in turn form the subset II . NGI and NGII are connected such that I is a subset of  and II is a subset of . We refer to molecules that follow these characteristics as ab prediradicals. ab prediradicals have interesting properties that we state now and prove subsequently. (α) For ab prediradicals, the zero-voltage conductance vanishes if the contact is made to an b-type atom in NGI and a a-type atom in NGII. (β) Furthermore, if we consider the entire NG and the two contacts are both in NGI (NGII), the conductance is the same as if the molecule consisted of NGI (NGII) alone. For the described contact configuration, NGI and NGII are decoupled. (γ) However, if an a-type atom of NGI and a b-type atom of NGII are connected, the conductance is nonzero in general. To prove these statements, we note that HNG is the sum of the Hamiltonian matrices of HI and HII of NGI and NGII, respectively, plus a matrix V = VI,II + VII,I which couples only b-type atoms of NGI to a-type atoms of NGII,

1 Cb HNG

1 1a HNG

(18)

In terms of the Green’s operator G(z) = 1/(z−HNG) this can be written as ca = −1ta G(0)Cb = −CbtG(0)1a

(19)

In addition, from eq 17, we get

∑

HNGCA (b) = Cb = t

l ∈ ad(b)

⇒CbtCb = t 2(

∑

cl (20)

cl)2

l ∈ ad(b)

(21)

Therefore the parameter s for the systems covered by rule 2 is given in terms of Green’s operator of the isolated NG by s= =

CbtCb 1 βLβR (1ta G(0)Cb)2 1 βLβR (1ta G(0)1b )2

(22)

Equation 22 shows that s tends to infinity and T(0) to zero if the Green’s matrix element Gab(0) vanishes. This result will be put to use in the following section (rule 3). Finally we discuss the ab-connected device orbital at E = 0. Since the contacts have been eliminated, we obtain only the projection of the device orbital onto the molecular basis functions. This projected state (CAB) satisfies HCAB = 0

(23)

where H is given in eq 6. Through elimination of the SSPs from H we obtain, HNGCAB = γ Ca + δ Cb

(24)

where γ and δ are complex coefficients. Inversion of the Hamiltonian HNG in eq 24 yields CAB = γ CB(a) + δ CA (b)

(25)

which shows that the molecular part of the device orbital at E = 0 is a linear combination of the DS(a) and DS(b). The coefficients γ and δ can be obtained with elementary means through the theory developed in ref 15. From the component CAB one can, for example, get the repartition of the currents in the molecule and analyze the possible appearance of magnetic moments.52 Rule 3: ab-Connected Nanographenes with Vanishing Zero-Voltage Conductance. For some NGs with contacts in a ab configuration T(0) vanishes. Equation 22 shows that this can occur if 1taG(0)1b = 0 . As we explain now, a vanishing T(0) is related to the number of eigenstates of zero energy of NG(a,b) obtained from the initial NG by eliminating the a-and b-type connecting atoms. Related results are discussed in ref 41. For stable NGs, (∑l∈ad(b)cl)2 ≠ 0. Indeed if this quantity is zero then CA(b) would be an eigenstate of the molecule with zero energy, i.e., HNGCA(b) = 0. Since the NG we consider is

HNG = HI + HII + V

(26)

We then obtain for the Green’s function of the NG at E = 0 GNG(0) =

1 −(HI + HII + V)

= GI (0) ⊕ GII (0) + GI (0)VI , II GII (0) + GII (0)VII , I GI (0) (27)

where GI(II)(0) = (1/−HI(II)). In general, there should be terms in eq 27 to all orders in V. However, they involve contributions of the type VII,IGI(0)VI,II or VI,IIGII(0)VII,I which all vanish since they contain only matrix elements of the Green’s operator GI(0) (GII(0)) corresponding to the frontier I (II ). These E



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NG in eq 10 is still valid and shows that the effective Hamiltonian of the  sublattice diverges at zero energy. Therefore the conductance remains zero. Finally if from an aaconnected NG a b-type atom (b*) is removed, the effective Hamiltonian of the  sublattice diverges for E = 0, except within the subspace defined by CA(b*). Therefore the effective Hamiltonian Heff A reduces to

frontiers consist of atoms of the same sublattice so that the corresponding matrix elements of GI(0) (GII(0)) are zero according to rule 1. Rule 1 can be applied here to NGI and NGII separately, since both are stable molecules, Heff A of NGI and Heff B of NGII diverge meaning that the Green’s matrix elements vanish between two orbitals of  or , respectively. Using eq 22 and eq 27 and the fact that matrix elements of GI(0) (GII(0)) between two orbitals are nonzero only if the two orbitals belong to different sublattices the announced rules follow readily. (α) For b ∈ NGI and a ∈ NGII, 1tbGNG(0)1a = 1tbGI(0)VI,IIGII(0)1a = 0 since I contains only b-type atoms and thus all 1tbGI(0)VI,II elements vanish. (β) For b ∈ NGI and a ∈ NGI, 1tbGNG(0)1a = 1tbGI(0)1a. (γ) Finally, for a ∈ NGI and b ∈ NGII, 1tbGNG(0)1a = 1taGI(0)VI,IIGII(0)1b ≠ 0. In this case, since the Green’s functions of NGI and NGII couple atoms on different sublattices, T(0) does not vanish in general. To conclude, we note that these results could also be derived by analyzing the defect states DS. We elaborate on this issue in the illustration section. Rule 4: Eliminating Atoms from the π-Electron System. There are numerous chemical modifications that can be undergone by NGs and in this paragraph we consider one for which we can find equations for the zero-voltage conductance. We assume that a pz function is excluded from the π system through the attachment of an additional hydrogen atom or another group to the NG. Supposing that the pz function removed belongs to an atom (a*) of , the impact on the conductance is first considered for the ab-connected configuration. Because of the symmetry of the problem, the following considerations can easily be adapted to the case where the pz function removed originates from a b-type atom instead of an a-type one. The conductance depends on the spectrum of the molecule NG(a*,b) obtained from the initial NG by removal of the contact site b and the functionalized site a*. If NG(a*,b) has no zero eigenvalue (or equivalently is not a diradical) then elimination of a* results in a square M C a C b coupling matrix in eq 13 that does not have a finite dimensional kernel. In this case the self-energy term in eq 13 diverges and there is no coupling of the b to any of the a atoms, i.e. T(0) = 0. If NG(a*,b) is a diradical, the conductance does not change upon elimination of the pz orbital. Indeed in that case, as discussed in the derivation of rule 3, the defect state DS(b) of the molecule NG has no weight on the site a* and is insensitive to the removal of site a*. Instead of examining CA(b) we can also focus on CB(a*). The coefficient of a* in the vector CA(b) is obtained by taking the scalar product with a normalized basis function (1a*) on atom a*

2 2 Heff A = |ca| ΘL(r )| + |c a|̃ ΘR

(29)

where ca and cã are the components on the two a-type connecting atoms a (right) and ã (left). The reflection coefficient r and the variable s are obtained from the condition det(Heff A ) = 0 which leads to eq 3 s=

βL |c a|̃ 2 βR |ca|2

If the two contacts are interchanged then s is transformed into 1/s which implies that T(0) (eq 3) remains unchanged.

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ILLUSTRATIONS ACCESSIBLE THROUGH ELEMENTARY CALCULATIONS In this section we present several examples that clarify various aspects of our theory and that demonstrate its usefulness. The first NG that we consider is also the smallest such system, namely benzene. A key element in all the examples considered is the construction of the DS and this is also the first topic

1 Cb H 1 = Cbt 1a * H

ca * = 1ta * CA (b) = 1ta *

∝ CbtCB(a*)

(28) Figure 2. Benzene (i) and naphthalene (ii and iii) with ab contact configuration. The first part of each parentheses indicates which b-type atom is connected to the contact. For the purpose of discussion, b-type atoms are enumerated. The right part shows the subsystem obtained by deleting the connecting b atom, detailing the construction of the DS. In case of a trivial construction, this step is omitted and the DS is shown immediately. The integer numbers associated with the a-type atoms are proportional to the amplitude of the DS.

Therefore we can conclude that if the b-type connecting atom belongs to the DS of NG(a*) the ab-connected NG has a vanishing conductance. If the b-type connecting atom does not belongs to CB(a*) then the conductance of the molecule is that of the initial NG. Next we consider a aa-connected NG with an a-type atom removed (a*). Then the partition used for the aa-connected F



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Figure 3. Coronene with ab contact configuration. In the left part of the figure the connected b-type atom is identified. In the right part, the DS is constructed by first assigning the variable x to the respective atom. Following the enumerated sequence of b atoms, at atom 2 a second unknown (y) has to be introduced. Pursuing the construction of the DS yields an equation relating the coefficients of the atoms binding to atom 11. This equation determines y and leads to the DS depicted in the lower part of the figure.

atoms. We start the construction of the DS at atom 1. Then we introduce a second variable (y) in the intact benzene ring adjacent to atom 2. This additional variable is needed, since we have three carbons connecting to atom 2, but only one condition relating them. Continuing with atom 3 we proceed to atom 4 yielding the equation 2y − x = 0, needed to eliminate y. Unlike in the case of benzene, now we obtain a varying T(0) ranging from 0.36 to 0.85 for contacts on different and on the same hexagon, respectively. For completeness, in Figure 2iii, the β-connected naphthalene is considered. The construction of the DS is uneventful, and follows the sequence indicated in the figure. Apart from one adjacent to the b connecting atom, all the positions are equivalent, yielding T(0) = 0.36. Convex Nanographenes. Next, we consider various examples of NGs whose shape we describe as convex, i.e., a straight line drawn between two peripheral benzene rings lies within the NG. We use the notions convex and linear (see next section) for a qualitative description of NGs rather than as a precisely defined mathematical term. The first such system is coronene (Figure 3) which has 24 carbon atoms. Starting the construction of the DS by assigning the coefficient x to the respective atom in Figure 3, we continue clockwise around the coronene perimeter, following the sequence set by the enumeration of the atoms in . At the first tertiary atom (atom 2), we introduce a second unknown y. We then complete the construction of the DS and arrive at atom 11. To ensure that the sum of the atomic coefficients connected to 11 adds up to zero, the equation 6y − 2x = 0 has to be satisfied, eliminating the unknown y. We normalize the orbital such that x = 3 and thus y = 1. The final DS is shown in the lower part of Figure 3. The variations of the DS and of T(E) around the

discussed below. The DS is an orbital of zero energy, its calculation is equivalent to that of a nonbonding molecular orbital (NBMO) described in an article by Longuet-Higgins.43 Numerous examples and applications of NBMOs can be found in a book by Dewar and Dougherty.44 The Defect States of Small Systems. In Figure 2i, the benzene molecule is shown and the contact position on the  sublattice is indicated through a thick bond. The second contact position is variable and therefore not indicated. According to rule 1, attaching the second contact to any of the enumerated atoms of the  subset results in zero conductance. Contrary to the aa and bb cases, the ab arrangements require a position-dependent discussion. The formula of rule 2 stipulates the calculation of the DS(b) whose coefficients, which are arranged in the vector CA(b), are then employed in eq 1 and eq 2 to determine T(0). DS(b) has nonvanishing contributions only on  and it is determined through the condition that the sum of all the coefficients of atoms that are connected to a given atom of  is zero. Employing the Hückel Hamiltonian of the NG with the b connecting atom removed (NG(b)) this condition reads HNG(b)CA(b) = 0. In the case of benzene it is a trivial task (see Figure 2i) to obtain the DS. Since s is a ratio (see eq 2), the normalization of the DS can be conveniently chosen to result in integer DS coefficients. Supposing for simplicity that βL = βR = t, according to eq 2, the term (∑l∈ad(b)cl)2 has to be of the same magnitude as |ca|2 to result in high conductance. In the case of benzene, all the a positions are equivalent. The value of s is 4, yielding T(EF = 0) = 16/25 = 0.64. In Figure 2ii, we consider naphthalene with one contact in α position. Naphthalene has two tertiary carbon atoms, i.e., carbon atoms that are linked to three other carbon G



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Figure 4. Ovalene with two different ab contact configurations. The respective DSs are constructed following the enumerated sequence of b atoms. Using the symmetry of the fragment, in case i, the construction of the DS is straightforward and requires only the solution of a simple linear equation, 2(x − 2y) −x −y = 0 obtained from the central carbon atom with the number 8. For the nonsymmetric case ii, the construction of the DS is not more involved; the carbon atom 15 yields the equation required to eliminate y.

and the largest one 0.64, as a general trend for this ab contact configuration, T(0) diminished with increasing distance between the contacts. Linear NGs. An example of linear NGs is provided by a member of the polyacene family. In Figure 5i, pentacene is considered. Its DS is obtained quite effortlessly if its construction is commenced at the carbon atom labeled with 1. Connection of the second contact to any of the b-type apex atoms of the upper row yields zero conductance, while the atype apex atoms on the lower row allow for a systematic augmentation of the conductance. If the second contact is attached in para position to the first one, polyacenes of increasing length asymptotically have a transmission probability of one. The transmission probability for an end-to-end connection of the pentacene is 0.11. Changing the position of the b contact from the α to the β position on the first benzene, does not change the general structure of the DS. Pentacene is an example of a linear molecule with zigzag border. It appears interesting to contrast it to a linear system with armchair configuration. Such a molecule (picene) is

perimeter of coronene are quite modest. T takes values between 0.15 and 0.88. Next we consider a system of considerable size, namely ovalene with 32 carbon atoms (Figure 4i). In this case, the construction of the DS can be simplified through the assumption that it exhibits the left-right symmetry of the molecular fragment. This leads to an orbital that contains two unknowns related through 2(x − 2y) − x − y = 0 by the carbon number 8. y = 1 and x = 5 is an integer-valued solution to this equation. The DS indicates that there are again only moderate variations of the conductance by going around the perimeter of the molecule and attaching the second contact to an a-type atom. To be specific, T(0) varies between 0.48 and 0.88, being highest for the pseudo para position, where the two contacts are on opposite carbons. For comparison, we also discuss a nonsymmetric arrangement in Figure 4ii of the b contact. Again, we only have to deal with two variables (x and y) related to each other through the constraint of zero coefficient sum around atom 15. Ignoring nearest neighbor contact configurations, T(0) now tends to be lower. The smallest value is 0.04 H



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Figure 5. Pentacene (i) and picene (ii) with ab contact configurations. The respective DSs are constructed following the enumerated sequence of b atoms. In the left part of parts i and ii, the connected b-type atom is indicated. The resulting DS is depicted on the right. For part ii, the first carbon atom and the even numbered atoms have only two neighbors, thus the signs of the adjacent DS coefficients are inverted. The odd numbered atoms, with the exception of the first one, are tertiary atoms. Once a tertiary atom is reached, the coefficients of two adjacent atoms are known and the third coefficient is simply the negative sum of the other two. In this way the sequence −2= −(1 + 1), 3 = − (−1 − 2), −5 = −(2 + 3), and 8 = −(−3 − 5) is obtained from the tertiary atoms.

segment and the armchair segment. The armchair segment generates a Fibonacci series whose highest element (8) becomes the linear increment in the zigzag segment. It is not too surprising that the end-to-end conductance T(0) = 0.002 reduces further compared to the previous examples. ab Prediradicals: NGs Connected in ab Configuration That Do Not Conduct. An unexpected conductance behavior is found for perylene depicted in Figure 7i. The construction of the DS proceeds along the established lines. In this example, a second variable y has to be introduced and from atoms 7 or 8 we obtain the condition −4y −2x = 2y + x ⇒ −2y = x. The resulting DS vanishes in the lower part of the molecule indicating that T(0) = 0 for any a contact location. Making a connection to an a-type atom in the upper part of the molecule yields a nonvanishing conductance. Next, in Figure 7ii, the contact position in the upper part of perylene is moved from a b- to an a-type atom. The construction of the DS requires the introduction of a third variable z. The variables are linked through the equations 2y − x = 0 and −x + z + 2z + y = 0 read off atoms 2 and 9 respectively. The resulting DS reveals that there is now a finite T(0) for the top-down ab connection. These somewhat surprising findings can be understood in terms of traditional chemical concepts. Removal of a second a-type atom in the lower part of the molecule does not modify the DS of Figure 7. (i). Therefore the substructure of the perylene depicted on the right of Figure 7.(iii) has an even number of atoms but still a doubly degenerate zero-energy solution, representable by wave functions that vanish either in the upper or the lower part of the

depicted in Figure 5ii. As shown in Figure 5ii and discussed in the caption, the sequence of tertiary carbon atoms in picene generates a sequence of DS coefficients F̃n that is related to the Fibonacci numbers Fn = Fn−1 + Fn−2, where F1 = 0 and F2 = 1, through F̃ n = (−1)(n−1)Fn. The Fibonacci sequence shows that the absolute value of the coefficients increases proportional to their previous values, thereby generating an exponential behavior of the DS, resulting in a much stronger localization than in the case of pentacene. The end-to-end transmission probability in this case is very small, i.e., T(0) = 0.02, compared to T(0) = 0.11 for pentacene. Increasing the number of benzene units in the picene molecule would eventually lead to the narrowest possible armchair nanoribbon. It is known (see e.g., ref 60) that, within the Hückel approximation with constant hopping parameters along the system, this ribbon has a finite band gap and thus, there is no device orbital of zero energy. The ballistic transport is therefore off-resonance, in agreement with our findings, yielding an exponential decay of the DS and of T(0). For the pentacene molecule the relation to the infinite zigzag ribbon is less obvious. This is because in the infinite zigzag ribbon, the density of states at the Fermi energy tends to infinity with increasing ribbon length, rendering the contribution of the single DS of zero energy unimportant for the observed conductance behavior. As a final example of this section, in Figure 6, we examine a system that is composed of two linear sections, a zigzag and an armchair segment. The behavior of the combined system is essentially a superposition of the DS structure of the zigzag I



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Figure 6. NG composed of a zigzag and an armchair segment. The construction of the DS follows the enumerated sequence of atoms. Carbon 13 yields the equation −2x − 5y − x − 3y = 0, permitting the elimination of y. As in the case of the pure segments, the DS in the zigzag segment decays linearly and exponentially in the armchair segment. The linear increment in the zigzag segment is increased to 8 by the presence of the armchair segment.

molecule. The subsystem with a and b removed (i.e., NG(a,b)) is therefore a diradical. It is known43,45 that diradicals do not possess Kekulé structures. This sequence of arguments can now be turned around to result in a simple rule for the prediction of vanishing conductance. If by removal of the a and b atoms the contacts are attached to a substructure is obtained that has no Kekulé structure, then the conductance of the initial MED vanishes. This rule is a special case of a general expression for the zero-voltage conductance of NGs in terms of the number of Kekulé structures.41 On the contrary, perylene connected according to Figure 7ii has a nonvanishing conductance from the upper to the lower half of the molecule, and it also has Kekulé structures, such as the one given on the right of Figure 7iii. Perylene is an example of a class of systems that we refer to as ab prediradical. This notion is motivated by the fact that removal of certain a and b atoms leads to a diradical. While the removal of two atoms of the same type always leads to a doubly degenerate orbital of zero energy on the lattice of the other atom type, removal of an a-and a b-type atom usually results in a stable molecule, having no orbitals of zero energy. The reason why perylene can yield a diradical is because it is composed of two stable fragments (naphthalene) that are connected in a particular fashion. We chose another, similar system to explain the general design principle of ab prediradicals. This discussion repeats some of the mathematical derivations of the theory section in a less rigorous but more intuitive fashion. To be

specific we discuss the benzo[a]perylene. Starting with one naphthalene molecule (Figure 8), removal of a b-type atom results in a DS(b) on , while it vanishes on . We refer to this system as fragment NGI. Now, connecting another fragment (NGII) to the atoms of  does not modify the DS of NGI. In particular, we can attach another fragment obtained by removing an atom from the  sublattice in such as way that NGI is connected via b-type atoms to a-type atoms of NGII. This procedure is illustrated in Figure 8, where the NGII is derived from the anthracene molecule. The system obtained in this way is a diradical where the DS in each fragment is preserved upon their connection. Clearly, the example just discussed is a particular case of a general procedure to obtain ab diradicals. These considerations also explain why the ab connection in Figure 7ii is conducting, the DS has nonvanishing contributions on those atoms of the upper naphthalene subunit that are connected to the lower subunit. The construction of diradicals presented above has already been described earlier by Dewar and Dougherty in their book44 on perturbation molecular orbital theory of organic chemistry. In this context, the diradical itself is the system of interest and not the parent prediradical. Chemically Modified NGs: Reducing the Size of the πElectron System. To illustrate rule 4, we consider the pyrene molecule attached to two contacts in aa configuration (Figure 9). As discussed, the aa-contact configuration yields a vanishing zero-voltage conductance. A chemical modification, such as the J



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Figure 7. Perylene with two different ab contact configurations. On the left side of parts i and ii, the connected b-type atom is indicated while the DS is constructed on the right, following the sequence set by the atom labels. While in part i, the DS is localized in the upper part of the perylene, in part ii, the DS covers the entire molecule. Finally, in part iii, the contact a atom as well as the b atom has been removed. For the first connection pattern (i), no Kekulé structure can be found, indicating that the fragment is a diradical. For the second connection pattern (ii), Kekulé structures are found and as we explain in the text, this means that the conductance is nonzero.

following discussion.) Either T(0) is reduced to zero or it is left unchanged, depending on whether the pz orbital of atom b* has a nonzero coefficient in the DS obtained by removal of the atype connecting atom (i.e., DS(a)) or not. Equivalently one can consider whether the contact a has a nonzero coefficient in DS(b*) or not. In the example just discussed, the defect state of the removed orbital on b* spreads over all sites of . Therefore for all ab-contact configurations the conductance is zero.

addition of a hydrogen atom to the starred b atom eliminates the corresponding atomic pz orbital from the π system and results in a DS(b*) which is also provided in Figure 9. As opposed to the other examples there is now a molecular orbital with E = 0 and thus T(0) is evaluated at a resonance energy. The corresponding equation for s (eq 3) contains only the ratio of the coefficients of the two a-type atoms. From Figure 9, we see that perfect transmission (T(0) = 1) can be obtained for several different contact configurations. The transmission probabilities found have to be compared to the unmodified NG for which no aa configuration conducts. It is worth mentioning that every DS discussed in the previous sections can serve as an illustration of rule 4, since a DS can be thought of as the singly occupied orbital of a radical obtained by reducing by one the number of the pz orbitals contributing to the π-electron system. According to rule 4, reduction of the size of the π-electron system can also change the zero-voltage conductance for abcontact configurations. (Note that compared to the theory section, the roles of a- and b-type atoms are interchanged in the

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VERIFYING THE PREDICTIONS WITH ADVANCED ELECTRONIC STRUCTURE CALCULATIONS In this section, we compare some of the predictions of our DS theory with more advanced, all-electron electronic structure calculations. Since these calculations are quite demanding, only a few small examples are considered. We employ the standard method (NEGF-DFT) for the calculation of the molecular conductance. NEGF-DFT combines the nonequilibrium Green’s function (NEGF) formalism46,47 with density functional theory (DFT).27−32 The details of our implementation are presented in previously published articles,25,32 here we give K



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Figure 8. Construction of a diradical derived from benzo[a]perylene. Two stable molecules, NGI and NGII, are converted into fragments (NGI(b) and NGII(a)) exhibiting a DS(b) and DS(a), respectively. The b-type atoms are indicated by filled cycles in NGI(b) and the a-type atoms by open cycles in NGII(a). Combining the two fragments by joining b-type atoms of NGI(b) with a-type atoms of NGII(a) results in a diradical in which the free a-type valence cannot interact with free b-type valence. Therefore two singly occupied molecular orbitals (SOMOs) of zero energy are obtained. Note that the two fragments are joint via atoms that do not contribute to the respective DS.

Figure 9. Pyrene with aa contact configuration. On the left, a typical aa contact configuration is shown, which according to our theory yields a vanishing zero-voltage conductance. Eliminating an atomic pz orbital from  (here we choose the one marked with ∗) results in a DS constructed on the right and depicted on the bottom. The unknown y is determined through the equation 2x + 5y = 0 obtained from atom 7. Since the DS has nonvanishing contributions on all atoms of , a nonvanishing zero-voltage conductance results for any aa contact configuration. However, all abcontact configurations yield T(0) = 0.

codes of the NEGF formalism. All the computations are done with the Perdew−Burke−Ernzerhof hybrid62−64 for exchange and correlation. The basis set employed is LANL2DZ.65,66 The first example that we consider is the perylene molecule discussed above. The contact configurations considered are indicated in Figure 10 adjacent to the corresponding T(E) curves. In agreement with our predictions, the configuration with the b contact in the upper part and a contact in the lower part of the molecule yields vanishing T(0). Only if the a and b contacts are in the upper part and lower part, respectively, is

only a short summary of the technical aspects of the calculations. The infinite system, consisting of the target molecule and the two contacts, is partitioned by means of the Löwdin technique. The contacts are then described by a selfenergy matrix added to the molecular one-particle Green function. This Green function is calculated self-consistently, as prescribed by the Kohn−Sham procedure. We perform these calculations with a modified version of the of quantum chemistry package Gaussian,61 its conventional diagonalization routines for generating the density matrix being replaced by the L



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CONCLUSION Since chemistry established itself as a rigorous physical science in the 19th Century, benzene and its relatives in the family of polycyclic aromatic hydrocarbons (PAH) attracted the attention of chemists. Recently, fullerenes and graphene enriched the repertoire of fused benzene rings. Not surprisingly, finite flakes of graphene, refered to as nanographenes (NGs), increasingly became a focus of research. Furthermore, a modern addition to the range of well-studied molecular properties is ballistic conductance. Here, we combine the various topics and address the problem of the ballistic conductance of NGs. The corresponding molecular electronic devices are infinite systems and this is one reason why the calculation of their conductance is quite challenging. Advanced theories and sophisticated computational methods are available to tackle the transport problem. To add to this, we simplify the theory to the point where quantitative predictions can be made by nonexperts using elementary means. In this sense, our model complements the available tools and it is particularly useful to develop an understanding of some of the fundamental mechanisms involved. Our approach recovers known features of molecular conductors, such as the recently confirmed side-chain effect, and it provides innumerable new predictions about the zerovoltage conductance of NGs. One particularly interesting prediction is the vanishing zero-voltage conductance in certain NGs that we refer to as ab prediradicals. Even though these systems are completely π conjugated, from the conductance point of view they consist of two disconnected parts.

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AUTHOR INFORMATION

Corresponding Author

Figure 10. Transmission probabilities (T(E)) obtained from all electron calculations. Of particular interest for the present work is the region around the Fermi energy (EF). In the upper plot, three contact configurations for perylene are considered: two ba arrangements, where the contacts are attached to a b atom in the upper part of the molecule and to a a atom in the upper or lower part of the molecule. The third arrangement has a contact attached to an a atom in the upper and a b atom in the lower part. In the lower plot, T(E) of pyrene and hydrogenated pyrene is displayed. The aa contact configuration is maintained in all three cases while a hydrogen atom is connected to an a- or b-type atom to study the conductance of the reduced π-electron systems.

*(M.E.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.E. would like to acknowledge the financial support provided by the Université Joseph Fourier, Grenoble, France, as well as the hospitality of the Institut Néel, Grenoble, France. We would like to thank Marc-André Bélanger for a critical reading of the manuscript and for help with the some of the graphics. Furthermore, the financial support through the NSERC is gratefully acknowledged.

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T(0) finite. Also in agreement with our predictions, an ab configuration with both contacts in the upper part yields a finite T(0). Next we consider the pyrene molecule which has also been discussed above. The two contacts are attached in an aa configuration and as expected, Figure 10 confirms that T(0) is zero. A chemical modification consisting of the addition of a hydrogen atom in the indicated b-type position should, according to our theory, result in a finite zero-voltage conductance and this is again confirmed (Figure 10) by the NEGF-DFT calculations. Since the addition of a hydrogen atom leads to spin polarization, the transmission probability of the up and down spin channels has been averaged to yield results that are on the same footing as the Hückel transmission probabilities which are independent of the spin channel. Finally we note that the absolute values of the conductance obtained from our model cannot be directly compared to NEGF-DFT results since in our model the coupling parameters are chosen rather arbitrarily.

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