Aromatic Compounds

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C: Physical Processes in Nanomaterials and Nanostructures

Qualitative Insights into the Transport Properties of Hückel/Möbius (Anti-)Aromatic Compounds: Application to Expanded Porphyrins Thijs Stuyver, Stijn Fias, Paul Geerlings, Frank De Proft, and Mercedes Alonso J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b01424 • Publication Date (Web): 27 Apr 2018 Downloaded from http://pubs.acs.org on April 30, 2018

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The Journal of Physical Chemistry

Qualitative Insights into the Transport Properties of Hückel/Möbius (Anti-)Aromatic Compounds: Application to Expanded Porphyrins Thijs Stuyver,1 ,* Stijn Fias,1 Paul Geerlings,1 Frank De Proft,1 Mercedes Alonso1 1 Algemene Chemie, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium

(Member of the QCMM Ghent − Brussels Alliance Group) ABSTRACT Expanded porphyrins have been recently identified as promising candidates for conductance switching based on aromaticity and molecular topology changes. However, the factors that control electron transport switching across the metal-molecule-metal junction still need to be elucidated. For this reason, the transport properties of Hückel/Möbius (anti-)aromatic compounds are investigated thoroughly in this work in order to gain qualitative understanding into the conductivity of these unique macrocycles. Starting from a polyene model, a simple counting rule is developed to predict the occurrence of quantum interference around the Fermi level at the Hückel level of theory. Next, the different approximations of Hückel theory are lifted, enabling the exploration of the influence of each of these approximations on the transport properties of expanded porphyrins. Along the way, a detailed study on the relationship between the conductance and aromaticity/topology has been undertaken. Even though it has been proposed that the p-conjugated systems of expanded porphyrins can be approximated as polyene macrocycles based on the “annulene model”, it turns out that to whom correspondence should be addressed. Electronic mail: [email protected] * Author

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the distortion induced by the pyrrole rings to the electronic structure of the expanded porphyrins causes the simple counting rule for the prediction of quantum interference developed for polyenes to fail in some specific situations. Nevertheless, our back-of-theenvelope approach enables an intuitive rationalization of most of the transport properties of expanded porphyrins. Our conclusions cast further doubt on the proposed negative relationship between conductance and aromaticity and highlight the importance of the connectivity on determining the shape of the transmission functions of the different states. We hope that the new insights provided here will offer experimentalists a road map towards the design of functional, multi-dimensional electronic switches based on expanded porphyrins. INTRODUCTION Recently, expanded porphyrins have been demonstrated to be good candidates for single-molecule conductance switches upon aromaticity and molecular topology changes.1 Expanded porphyrins can either have a Hückel or Möbius topology and, importantly, several external stimuli are able to reversibly switch these macrocycles from one topology to another.2,3,4 As shown by our recent research, the aromaticity of the p-conjugated system is closely determined by the molecular topology, in agreement with the well-known Hückel and Möbius aromaticity rules.5,6,7,8 Accordingly, topologies with an even number of half-twists in the p-structure (or, more precisely, with an even linking number Lk) should follow the Hückel’s rule for aromaticity, while those with an odd number of half-twists (odd values for Lk) follow the Möbius aromaticity. 9 Protonation and deprotonation of the individual five-membered rings of [4n] p-electron macrocycles induces topology changes enabling the transformation from antiaromatic to aromatic systems. 10,11,12 Furthermore, the [4n+2] and [4n] p-electron systems can be

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easily interconverted by reversible two-electron redox reactions.3,13 As such, expanded porphyrins open up the possibility to study multidimensional switching properties. By assembling penta-, hexa- or heptaphyrins into gold molecular junctions, ratios of conductance up to 103 could be estimated theoretically upon switching the aromaticity and/or topology of the macrocycle.1 These differences in conductance mainly arose from the appearance/disappearance of Quantum Interference (QI) 14,15,16,17,18,19

upon aromaticity switching. Remarkably, the connectivity to the gold

electrodes determines to a large extent the transport properties of the expanded porphyrins. In this study, the orbital rule developed by Yoshizawa and co-workers was shown to be useful in predicting the occurrence of QI in the different states of the switches.14,15,20 Beside the orbital rule, other selection rules for molecular transmission have been proposed, such as the curly arrow approach. 17, 18,19,21 Since the factors that control electron transport switching across the molecular junctions based on expanded porphyrins are not yet completely understood, the transport properties of Hückel/Möbius (anti-)aromatic compounds are investigated thoroughly in this work following a step-by-step approach in order to gain qualitative understanding on the conductance of these challenging systems. Specifically, we focus on hexaphyrins with varying topology and number of p-electrons (Fig. 1) since they provide three-level molecular switches upon redox and topology interconversions with the highest ON/OFF ratio, among the investigated expanded porphyrins.1 First, a simplified polyene model for Hückel and Möbius hexaphyrins is constructed that enables qualitative insights into the influence of both topology and aromaticity switching on the occurrence of quantum interference. Next, additional features are added gradually to the polyene model systems up to the point that the corresponding hexaphyrins are modeled. Along these steps, the transport properties of the different

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systems are determined to understand the influence of each structural modification. This step-by-step approach allow us to understand up to which extent the different qualitative rules to predict quantum interference work for Hückel/Möbius (anti)aromatic systems, and what factors are responsible for the failing of several selection rules when applied to expanded porphyrins. Indeed, we demonstrated that most of the qualitative rules developed for polyenes turn out to have a limited applicability on these p-extended macrocycles.19,21 This is in contradiction to the so-called “annulene model”, which states that the p-conjugated systems of expanded porphyrins can be approximated as annulene/cyclic polyene systems.22,23 According to this model, the macrocyclic aromaticity can be assessed from the number of p-electrons along the conjugation pathway and the molecular topology. In addition, our findings shed more light on the relationship between the magnitude of conductance under small bias and aromaticity when no QI takes place around the Fermi level, a relationship that has attracted significant interest recently. 24,25,26,27,28,29

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Figure 1. Hexaphyrins with Hückel (26H and 28H) and Möbius topologies (26M and 28M) considered in this study. The annulene-type conjugation pathway has been marked in bold and coloured according to the expected aromaticity. The meso-positions regarded as the possible contact positions have been numbered. COMPUTATIONAL DETAILS The initial structures of the Hückel and Möbius topologies of [26] and [28]hexaphyrins were obtained from our previous works, in which an exhaustive conformational analysis was performed for each oxidation state.5 Thiolphenylethynyl linker groups were attached to the different meso-positions of the macrocycles and the resulting geometries were fully optimized at the B3LYP30 ,31 /6-31G(d) level of theory, as implemented in the Gaussian 09 software. 32 The performance of the B3LYP hybrid functional on the geometries and relative conformational energies of expanded porphyrins was assessed in our previous benchmarks from comparison with experiment.5,7,8

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The polyene models were constructed starting from the full optimized hexaphyrin structures (Fig. 2). From these macrocycles, the annulene-type conjugation pathway, denoted in bold in Fig. 1, was cut out and the nitrogen atoms along this path were replaced by carbon atoms. The resulting dangling bonds were taken care of through passivation of the structure with hydrogen atoms, leading to true cyclopolyenes. In order to be able to link the molecules to the gold electrodes for the transmission calculations, thiol (-SH) anchor units connected to the molecule through ethynyl spacers (-C≡C-) were added in the respective contact positions in a next step. The resulting polyene systems were optimized at B3LYP/6-31G(d) level of theory. Following a similar procedure, the N-substituted annulenes were obtained keeping the nitrogen atoms along the main conjugation pathway. All the resulting molecular structures are shown in the SI.

Figure 2. Construction of the polyene and heteropolyene models from the hexaphyrin macrocycle. All transmission calculations were performed using the Non-Equilibrium Green’s Function (NEGF) method combined with DFT as implemented in the Artaios code, 33,34 a postprocessing tool for Gaussian 09. The electrode geometry used in the transmission calculations consists of nine gold atoms per electrode, which are arranged as a six-atom triangular fcc-gold (111)-surface, with a second layer consisting of three gold atoms. The distance between gold atoms in the clusters was set to 2.88 Å. From the final optimized

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molecular structures, the thiol’s hydrogen atoms were removed and the Au9 clusters, were attached in accordance with the methodology presented in a recent study.35 The adsorption site is the middle fcc-hollow site of the first layer. The Au-S distance was set to 2.48 Å.36 For the resulting structures, single-point calculations were performed at the B3LYP/LanL2DZ level of theory, again using the Gaussian 09 software. In the final step, the Hamiltonian and overlap matrices were extracted to carry out the NEGF calculation within the wide-band-limit (WBL) approximation using the post-processing tool Artaios. 37 In the WBL approximation, we used a constant value of 0.036 eV−1 for the local density

of states (LDOS) of the electrode surface. This value was taken from the literature.37 The local transmission plots were also produced with the Artaios code.33 In these plots, the atom-atom local transmission contributions are represented by arrows, with the size of these arrows proportional to the magnitude of the local transmission.38 In order to clearly visualize the preferred path of the current through the molecule, a threshold is usually set, implying that only local transmission contributions that exceed this threshold value are represented by an arrow. Here, this threshold was set to 20% of the biggest local transmission contribution calculated for each plot. By setting such thresholds, i.e. by not drawing numerous (small) atom-atom contributions, the local transmission plots may falsely appear to not conserve the current. To check the validity of the current conservation principle in our local transmission calculations, the current passing through consecutive surfaces perpendicular to the direction of the flux, each separated by five Angström, was calculated for each local transmission plot (see Fig S4S6 in the Supporting Information).

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THEORETICAL BACKGROUND

In a previous work, we investigated the transport properties of planar even alternant

hydrocarbons.39 An alternant hydrocarbon is a conjugated π-electron system, in which all the carbon atoms taking part in the conjugated system can be divided into two classes (e.g., labeled by zeros and stars, as shown for benzene in Fig. 3) in such a way that members of the two classes alternate, i.e., atoms of the same class are never next to one another.40 1 6

2

0

*

*

5 0

0

*

3

4

Figure 3. The carbon atoms of benzene can be divided in two classes (0 and *) of which the members alternate. For alternant hydrocarbons, in general, the simple tight-binding Hamiltonian matrix corresponding to the conjugated system can be written as a block matrix by grouping the rows and columns of the two classes (cf. the benzene example below: 𝛽 set to unity and 𝛼 to zero ). 41,42 0 𝐻=∗ 0 ∗ 0 ∗

𝟏 𝟐 𝟑 𝟒 𝟓 𝟔

0 𝟏 0 1 0 0 0 1

∗ 𝟐 1 0 1 0 0 0

0 𝟑 0 1 0 1 0 0

∗ 𝟒 0 0 1 0 1 0

0 𝟓 0 0 0 1 0 1

∗ 𝟔 1 0 𝟏 0 → 𝐻 = 0 𝟑 0 0 𝟓 0 ∗ 𝟐 ∗ 𝟒 1 ∗ 𝟔 0

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0 𝟏 0 0 0 1 0 1

0 𝟑 0 0 0 1 1 0

0 𝟓 0 0 0 0 1 1

∗ 𝟐 1 1 0 0 0 0

∗ 𝟒 0 1 1 0 0 0

∗ 𝟔 1 0 1 0 0 0

(1)

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It has been demonstrated that the transmission probability close to the Fermi level (E = 𝛼 = 0) for even alternant hydrocarbons connected to contacts at positions r and s in the Source-and-Sink potential method43,44,45,46 can be expressed as39: 𝑇1,3 0 = lim

−4𝛽 ; ∆ 𝐸 Δ13,13 𝐸

7→8 ∆;

𝐸 − 2𝛽; ∆(𝐸)Δ13,13 (𝐸) + 𝛽 C ∆;13,13 (𝐸)

.

(2)

In this expression, Δ(𝐸) denotes the characteristic polynomial of the Hückel/tight-binding Hamiltonian matrix of the π-system of the original molecule incorporated in the molecular electronic device (H). The letters before and after the comma in the subscripts denote, respectively, the rows and columns deleted from the secular determinant that gives rise to this polynomial, so that: Δ 𝐸 = det 𝐸𝟏 − 𝑯 ,

(3)

Δ3,3 𝐸 = det (𝐸𝟏 − 𝑯)3,3 ,

(4)

Δ1,1 𝐸 = det (𝐸𝟏 − 𝑯)1,1 ,

(5)

Δ13,13 𝐸 = det (𝐸𝟏 − 𝑯)13,13 ,

(6)

where 1 is the unit matrix. The proportionality constant 𝛽 can be defined as follows, ; 𝛽JK 𝛽= , 𝛽K

(7)

with 𝛽JK corresponding to the resonance integral between the atom in the molecule at the point of attachment to the contact, and the first atom of the contact, and 𝛽K being the resonance integral between atoms in the contacts. By introducing the following quantity, 1LM 𝑇1,3 0 = lim − 7→8

9

∆13,13 𝐸 , ∆ 𝐸


(8)


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which we called the “relative transmission” and is always a positive number, Eq. 2 can be rewritten as follows, ∆13,13 (0) ∆(0) 0 = ∆ 0 Δ; 0 1 − 2𝛽; 13,13 + 𝛽 C 13,13 ; ∆ 0 ∆ 0 −4𝛽 ;

𝑇1,3

=

(9)

1LM 0 4𝛽; 𝑇1,3

. 1LM 1LM ; 1 + 2𝛽; 𝑇1,3 0 + 𝛽 C 𝑇1,3 (0)

It is straightforward to see that the transmission probability around the Fermi level 𝑇1,3 0 will go to zero (leading to QI) in the limit of the relative transmission 1LM 1LM 𝑇1,3 0 going to zero or infinity. 𝑇1,3 0 in Eq. 8-9 can be demonstrated to be closely

connected to the structure of the considered molecule.39 More specifically, the denominator of Eq. 8, ∆ 0 , is directly linked to the Algebraic Structure Count (ASC) of the molecular graph corresponding to the considered compound and similarly the numerator, ∆13,13 0 , is connected to the ASC of the conjugated system of the compound after atoms r and s have been deleted.47,48 For planar even alternant hydrocarbons (with the exception of systems containing 4n-membered rings) the ASC traces back to the number of Kekulé structures that can be drawn. This can be understood straightforwardly by considering the expression of ∆ 0 for the even alternant hydrocarbon considered. Focusing again on the benzene example (Fig. 3): 0 0 ∆ 0 = det −𝑯 = 0 −1 0 −1

0 0 −1 0 −1 0 0 0 −1 −1 0 0 0 0 0 −1 −1 = 0 −1 0 0 0 0 1 −1 −1 0 0 0 1 0 −1 0 0 0 0

0 0 0 0 1 1

0 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0

(10)

= det 𝑯 . Such a determinant of an n x n matrix can be determined through application of the Leibniz formula as a sum of terms, each of these being a combination of matrix

10


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elements of 𝑯 so that every row and column is selected once and only once in every term,49 Y

det 𝐴 =

𝑠𝑔𝑛( 𝜎)

𝑎X,R(X) .

(11)

XZ[

R∈TU

Here, the sum is computed over all permutations 𝜎 of the set {1,2,3…}. The set of all such permutations is denoted by Sn. For each permutation 𝜎, sgn(𝜎) corresponds to the signature of 𝜎 (+1 or -1), and 𝑎X,R(X) denotes the matrix element in the ith row and 𝜎(𝑖)th column of the determinant. When the upper right (or lower left) determinant in Eq. 10 is calculated this way, then the only terms differing from zero in Eq. 11 will be those for which only matrix elements with value 1 have been selected. Each of the combinations of unit elements in either the upper right or lower left block corresponds to a Kekulé structure, so that the number of non-zero terms that can be obtained per block is equal to the number of Kekulé structures K (Fig. 4).

Figure 4. Kekulé structures of the molecular graph of benzene corresponding to the selection of the double bonds in the matrices above. Since the determinant of a block matrix, such as the one on the right-hand side in Eq. 1, can be expressed as the product of the determinants of its non-zero blocks, one can straightforwardly see that det(H), i.e., Δ(0), will have K2 terms differing from zero.

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Each of these terms will evidently be equal to one since all non-zero matrix elements have been set to unity. If the signature of all these terms is equal (which is generally the case for planar even alternant hydrocarbons), then the determinant at 𝐸 = 0 will be ; proportional to K2.46,50,51,52 A similar reasoning allows Δrs,rs(0) to be linked to 𝐾1,3 , the

number of Kekulé structures which can be drawn for the r,s-deleted molecular graph. Focusing on the benzene molecule, para-connected to the contacts (r=1; s=4), 𝟓 ∆[C,[C = 𝟑 𝟐 𝟔

𝟓 0 0 0 1

𝟑 0 0 1 0

𝟐 0 1 0 0

𝟔 1 0 . 0 0

(12)

The Kekulé structure(s) can again be recognized in the upper right or lower left block of the corresponding Hamiltonian matrix (Fig. 5), and a similar reasoning as for Δ(0) can be followed.

Figure 5. Kekulé structures of the 1,4-deleted molecular graph of benzene, corresponding to the para-connected benzene. The structure is obtained through selection of the double bonds in the matrix above, analogous to Fig. 3. RESULTS AND DISCUSSION Extension of the Hückel Approach towards Anti-aromatic and Möbius Compounds Planar 4n-membered rings (or systems containing such a ring), such as Hückel antiaromatic systems, are the exceptions to the general rule introduced in the theoretical background section. The first explanation of their exceptional behavior dates back to the 1950s, when Dewar and Longuet-Higgins investigated the correspondence between resonance and molecular orbital theory.53 An alternative explanation for this

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behavior was recently given by Movassagh et al.54 Here, we take a slightly different approach and present a proof specific to even-membered cyclic systems, which will facilitate a straightforward expansion from planar systems to Möbius-twisted systems later on. We start by considering the general expression for the upper right or lower left block of a tight-binding Hamiltonian matrix of a cyclic alternant hydrocarbon of dimensions m x m (Eq. 10 for benzene), 1 0 1 1 ⋯ 0 1 ⋮ ⋱ 0 0 ⋯ 0 0

0 1 0 0 0 0 . ⋮ 1 0 1 1

(13)

It is evident that every row and every column of such a block contains exactly 2 non-zero elements since every carbon atom has exactly two neighbors in the conjugated ring system. More specifically, it should be noted that all non-zero elements in a specific row or column are neighbors except for the first row and the final column, i.e., the two non-zero elements are never separated by a zero in between. Taking the determinant of such a block and developing along the first row leads to, 𝟏 1 0

0 1 ⋯ 1 ⋮ ⋱ 0 0 ⋯ 0 0

0 𝟏 0 0 0 0 = (−1)[a[ ⋮ 1 0 1 1 1 ⋯ 1 = ⋮ ⋱ 0 ⋯ 0

1 ⋯ 1 ⋮ ⋱ 0 ⋯ 0

0 0 0 0 ⋮ +(−1)ba[ 1 0 1 1

0 0 0 0 ⋮ +(−1)ba[ 1 0 1 1

1 1 ⋯ 0 1 ⋮ ⋱ 0 0 ⋯ 0 0

1 1 ⋯ 0 1 ⋮ ⋱ 0 0 ⋯ 0 0

0 0 ⋮ 1 1

(14)

0 0 ⋮ , 1 1

where m is the dimension of the square block. The two minors (each of dimensions m-1 x m-1) that appeared on the right side of the previous equation will always be equal to one. This can be understood by considering that the red elements in the minors will always be the only element differing from zero in their respective column (the other

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element in that column has been deleted in the first development step of the original determinant). Further development along the row or column of such a minor that contains only 1 non-zero element in a corner will lead to a single new minor of dimension m-2 x m-2 that will be positively signed (cf. Eq. 11), 𝟏 1 0

0 1 ⋯ 1 ⋮ ⋱ 0 0 ⋯ 0 0

0 0 0 0 0 0 = (−1)[a[ ⋮ 1 0 1 1

1 ⋯ 1 ⋮ ⋱ 0 ⋯ 0

0 0 1 ⋯ 0 0 1 = ⋮ ⋱ ⋮ 0 1 0 ⋯ 0 1 1

0 0 0 0 ⋮ . 1 0 1 1

(15)

The first row of this new minor (dimensions m-2 x m-2) again has only one nonzero element remaining (the green one). Since this is again a corner element, this minor can be developed analogously to Eq. 15 to a new minor, which will have dimensions m-3 x m-3, 𝟏 1 0

0 1 ⋯ 1 ⋮ ⋱ 0 0 ⋯ 0 0

0 0 0 0 0 0 = (−1)[a[ ⋮ 1 0 1 1

1 ⋯ 1 ⋮ ⋱ 0 ⋯ 0

0 0 1 ⋯ 0 0 1 = ⋮ ⋱ ⋮ 0 1 0 ⋯ 0 1 1

0 0 0 0 ⋮ . 1 0 1 1

(16)

This approach can be iterated until the minor is fully developed. The final result of this operation will always be equal to 1. In conclusion, Eq. 14 demonstrates that the dimensions of the square block determine completely whether the determinant of the Hamiltonian matrix of a cyclic system will be zero or not. If m+1 is even, then the values of both minors are added, leading to a value for the determinant of the block being 2. For m+1 to be even, m has to be odd. So, m has to be of the form 2n+1, which leads to dimensions of the entire Hamiltonian matrix of 4n+2 x 4n+2. This corresponds to 4n+2 membered rings, i.e., Hückel aromatic systems. Δ(0) will then evidently be proportional to K2, corresponding to the general behavior for even alternant hydrocarbons. If m+1 is odd, on the other hand, then the values of both minors are subtracted, leading to a value for the

14


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determinant of 0. For m+1 to be odd, m has to be even. This means that m has to be of the form 2n, leading to a Hamiltonian matrix of dimension 4n x 4n. This corresponds to 4n-membered rings, i.e. Hückel anti-aromatic systems, so that Δ(0) will be equal to 0 although two fully paired Kekulé structures can be drawn for these structures.

Next to a planar Hückel topology, cyclic hydrocarbons can also have different p-

conjugation topologies, such as the Möbius topology. 55 , 56 An alternant cyclic hydrocarbon with a Möbius topology is an alternant cyclic hydrocarbon for which the pconjugated system is twisted like a Möbius strip. Whereas the carbon pz-orbitals that constitute the conjugated system are all parallel and in phase for planar Hückel systems, they are slightly tilted for the Möbius systems and one of the orbital overlaps is out-ofphase (Fig. 6).57

Figure 6. The orientation of the pz-orbitals for cyclic polyenes with Hückel topology (left) and Möbius topology (right). In the Hückel method, the tilt between the pz-orbitals of two consecutive carbon atoms can be taken into account by modifying the resonance integrals as follows:58 β θ = β ∗ cos θ = 1 ∗ cos θ = cos θ .

15


(17)


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The angle θ will depend on the size of the ring, but in a first approximation will be uniform along the ring and relatively small for large rings. As such, we can scale β θ back to unity. The out-of-phase overlap, on the other hand, can be taken into account by substituting one β θ with – β θ . This leads to the following general structure of a Hamiltonian matrix of a conjugated cyclic system with a Möbius topology:

𝟎 𝐻=

1 0 1 1 ⋯ 0 1 ⋮ ⋱ 0 0 ⋯ 0 0

0 −1 0 0 0 0 ⋮ 1 0 1 1

1 0 1 1 ⋯ 0 1 ⋮ ⋱ 0 0 ⋯ 0 0

0 −1 0 0 0 0 ⋮ 1 0 1 1 .

(18)

𝟎

Focusing now on a single block, as we did before, and calculating the determinant by development along the first row then leads to the following expression: 𝟏 1 0

0 1 ⋯ 1 ⋮ ⋱ 0 0 ⋯ 0 0

0 −𝟏 0 0 0 0 ⋮ 1 0 1 1

= −1

[a[

1 ⋯ 1 = ⋮ ⋱ 0 ⋯ 0

1 ⋯ 1 ⋮ ⋱ 0 ⋯ 0

0 0 0 0 ⋮ + −1 ∗ −1 1 0 1 1

0 0 0 0 ⋮ + −1 1 0 1 1

ia;

ia[

1 1 ⋯ 0 1 ⋮ ⋱ 0 0 ⋯ 0 0

1 1 ⋯ 0 1 ⋮ ⋱ 0 0 ⋯ 0 0

0 0 ⋮ 1 1

(19)

0 0 ⋮ . 1 1

Comparison of this expression with Eq. 14 leads to the conclusion that Möbius and Hückel planar ring systems have opposite characteristics. Whereas m had to be odd

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for Hückel systems for the determinant of the Hamiltonian matrix being different from zero, in Möbius systems m has to be even. The inverse is also valid; an even m led to a value for the determinant of zero for Hückel (planar) systems, whereas for Möbius systems this will be the case when m is odd. Accordingly, 4n membered Möbius rings are also known as Möbius aromatic rings whereas 4n+2 membered Möbius rings are antiaromatic rings. This difference in behavior is equivalently expressed in terms of the occupation scheme of the MOs for the different systems (Fig. 7).

Figure 7. The occupation scheme of the MOs for Hückel and Möbius structures with 4n+2 (left) and 4n (right). The Fermi level is defined as the middle point between HOMO and LUMO.

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Prediction of Quantum Interference for Cyclic Polyenes in the Simple Model As mentioned above, for a quantum interference to occur close to the Fermi level when contacts are placed in the r and s positions of a molecule, the relative transmission either has to be equal to zero or infinity (Eq. 9). In our simple model, QI will then take place in the following situations for even-membered cyclic polyenes.

For Hückel aromatic [4n+2] p-electrons compounds, the denominator of Eq. 8,

∆ 0 , is proportional to K2; so differing from zero. That means that QI will only occur when the numerator ∆13,13 0 is equal to zero. Since the deleted structure for a planar cycle corresponds to either a single linear polyene or a combination of two polyenes, the ; is valid. As a result, ∆13,13 0 will only be proportionality between ∆13,13 0 and 𝐾1,3

equal to zero when the deleted structure corresponds to a combination of two odd linear polyenes, since no Kekulé structures can be drawn for such a structure. For Hückel anti-aromatic [4n] compounds, the denominator ∆ 0 is equal to zero. As a result, QI will occur whenever the deleted structure consists of one or two even linear polyenes, since ∆13,13 0 will then be equal to a number differing from zero, leading to a relative transmission going to infinity. If the deleted structure corresponds to two odd linear polyenes, then ∆13,13 0 equals zero and the relative transmission will be undefined. Application of the L’Hopital’s rule allows us to conclude that the relative transmission will be equal to a defined number differing from zero so that no QI takes place (cf. Ref. 39). A schematic overview of the quantum interference predictions for planar even-membered cyclic polyenes can be found in Fig. 8a-b.

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Figure 8. Schematic overview of the transport properties for Hückel and Möbius evenmembered cyclic polyenes based on our simple tight-binding Hückel model. The conjugated system is either aromatic or antiaromatic, and two contact groups will always be separated either by an even or odd number of carbon atoms. These two variables fully determine whether quantum interference will occur close to the Fermi level or not. For Möbius aromatic [4n] p-electrons compounds, the denominator ∆ 0 is a number differing from zero. As a result, the numerator has to be equal to zero for a QI to occur at the Fermi level. As for planar cyclic systems, the deleted structures for cycles with Möbius topology will be (a) linear chain(s), although their conjugated system is not planar. Construction of the Hamiltonian matrix of such a deleted structure in the same

19



way as before will lead to a block matrix in which one or two rows or columns have only one non-zero element. The determinant of such a block then has the same form as the left-hand side of Eq. 15. As a result, the same approach as in Eq. 15-16 can be taken to develop this determinant step by step, each time selecting the row or column that only contains a single non-zero element so that every time a single new minor is obtained. If the two chains are even, the entire determinant of the block can be developed this way and will differ from zero, and consequently ∆13,13 0 will be also different from zero, leading to a no QI at the Fermi level. If the two chains are odd, then, just like it was the case for Hückel polyenes, a row or column with only zero elements will appear at a certain point during the development, so the determinant of the block will be zero. As a result, ∆13,13 0 will be zero and QI will occur. For Möbius antiaromatic [4n+2] systems, the reverse is true. The denominator of the relative transmission is again zero for such a compound, so QI will take place when the numerator differs from zero (the relative transmission goes to infinity). This will be the case when the deleted structure consists of chains with an even number of carbon atoms. When the chains contain an odd number of carbon atoms, the relative transmission will be undefined again. Application of the L’Hopital’s rule leads to a defined number for the relative transmission and thus no QI. A schematic overview of the different possibilities described above for Möbius-twisted even-membered cycles can be found in Fig. 8c-d. Therefore, carefully applying our simple tight-binding Hückel model leads to a very interesting result. Consider a 4n+2 cycle able to switch between Hückel and Möbius topology with an even number of carbon atoms separating the two contact positions r and s. In the Hückel topology, a 4n+2 cycle is aromatic and as a result, given that the deleted structure will consist of even polyenes, QI is not expected to occur, as can be

20


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inferred from Fig. 8. By contrast, in the Möbius topology, the same 4n+2 cycle is antiaromatic and, again given that the deleted structure will correspond to even polyenes, QI is predicted to occur (Fig. 8). As such, our simple approach led to the conclusion that QI close to the Fermi level should appear/disappear upon topology switching.

Similar predictions can be made for 4n+2 cycles in which r and s are separated by

an odd number of carbon atoms and also for the different possible positions of r and s on 4n cycles. Importantly, our model predicts the appearance/disappearance of QI upon topology switching for every possible configuration of contacts. It should be noted that this model is extremely approximate; one can now ask the question whether these trends will persist as the crude approximations and assumptions made to get to this model are gradually lifted. This will be the subject of the following sections. The Implications of the Jahn-Teller Effect A first important approximation implicitly assumed in our simple model is the neglect of the Jahn-Teller distortion.59 This distortion has previously been demonstrated to have significant repercussions on the transport properties of π-conjugated systems.60,61 According to the Jahn-Teller theorem, the degeneracy of the frontier orbitals [the Single Occupied Molecular Orbitals (SOMOs)] for the antiaromatic cycles will be lifted through a geometrical distortion. As a result, the bond lengths between the carbon atoms of the conjugated system will no longer be equal, i.e. bond-length alternated structures, leading to unequal resonance integrals β. In this particular situation, the two minors on the right-hand side of Eq. 14 and 19 will no longer cancel each other out. Consequently, ∆ 0 will no longer be equal zero for the antiaromatic

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systems, and QI will only occur for these systems when the number of carbon atoms separating r and s is odd, instead of even as it was concluded based on the simple model without Jahn-Teller. An updated version of the scheme for predicting the occurrence of quantum interferences in even-membered cyclic polyenes with Hückel and Möbius topology can be found in Fig. 9a-d (changes compared to the original scheme in Fig. 8 are denoted in orange).

Figure 9. Schematic overview of the transport properties of both Hückel and Möbius evenmembered cyclic polyenes based on our simple model upon inclusion of the Jahn-Teller effect. Changes compared to Fig. 8 are denoted in orange.

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The predictions of the occurrence of quantum interference in Fig. 9 have been verified using the Non-Equilibrium Green´s Function formalism (NEGF) in combination with Density Functional Theory (DFT), as implemented in the Artaios code (see Computational Methods section).33 The resulting transmission spectra are presented in Fig. 10. As contact positions, the carbon atoms corresponding to the meso-positions indicated in Fig. 1 were taken. For clarity, the number of intermediate carbon atoms along the conjugation path between the contacts is also added between brackets. This number of intermediate carbon atoms was determined in the direction of the shortest bond starting from the first contact position. For 28M, the bond alternation is inverted in the 1,3- and 1,4-connection, leading to an intermediate number of carbon atoms of 9 and 8 respectively.

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Figure 10. The computed transmission spectra for the cyclic polyenes for different configurations of contacts on the Hückel 26H-polyene. The number of intermediate carbon atoms in the direction of the shortest bond, starting from the first contact position, has been placed between parentheses for each configuration.

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Importantly, the plots in Fig. 10 enable the verification of our predictions about the occurrence of quantum interference based on our tight-binding Hückel approach. Furthermore, they illustrate the influence of aromaticity/topology on the magnitude of the transmission probability around the Fermi level when no QI takes place. The main area of interest of the transmission spectrum is the region around the Fermi level, which is indeed the region experimentally accessible through application of a small bias between the contacts and, consequently, determines the conductance of the system. As such, in the analysis of the transport properties below, we focus on this area with the HOMO and LUMO peaks as outer boundaries. It is clear that the transmission probability in this region between the HOMO and LUMO peaks in the transmission spectrum is highest for the Hückel aromatic system (26H-polyene). Whether the two contacts are separated by 8 or 12 carbon atoms does not play any role, the two resulting transmission spectra overlap in this region. This corresponds to a constant conductance for Hückel aromatic systems if a small bias would be applied over the system, irrespective of the distance between the two contacts. This result is in perfect agreement with the results obtained before for bond-equalized cyclic polyenes by Hoffmann et al. and ourselves.39,62 Compared to the spectra without QI close to the Fermi level obtained for 26Hpolyene, the spectrum without QI obtained for the Hückel antiaromatic system (28Hpolyene) exhibits a slightly lower transmission probability in the region between the HOMO and LUMO peak (approximately a ratio of 2). The transmission probability in the region of interest for the Möbius aromatic system (28M-polyene) with 8 carbon atoms between the contacts is similar in magnitude to the probability obtained for the Hückel antiaromatic one (28H-polyene) with a similar number of carbon atoms between the

25



contacts. For this Möbius aromatic system, the transmission probability decreases clearly as the distance between the contacts increases. The Möbius antiaromatic system exhibits a much lower transmission probability in the region of interest than any of the other systems (almost a ratio of 10). It can be also observed that for this system, the transmission probability decreases with an increasing distance between the contacts as well. A decreasing transmission probability around the Fermi level with increasing distance has previously been connected to the presence of bond alternation in the conjugated system.62 Inspection of the geometries of both these systems demonstrates that this is indeed the case (see Fig. S1-S3 in the Supporting Information). For the Möbius antiaromatic system, this bond alternation is the result of the Jahn-Teller effect. The Möbius aromatic system does not undergo a similar Jahn-Teller distortion although bond alternation is still present. This bond alternation arises presumably from the twist in the macrocycle not being uniform across the macrocycle in contrast to the simple model, causing a distortion of the geometry. We note that the results obtained here for the magnitude of the transmission probability around the Fermi level agree with those obtained in Ref. 62 for the expanded porphyrins and cast further doubt on the proposed negative relationship between conductance and aromaticity.24,25,26,27,28,29 The Jahn-Teller distortion not only influences the occurrence of QI and the magnitude of the current when no QI takes place, but also the path the current takes through the molecule. According to Tsuji et al., when no QI takes place, current will travel uniformly through both arms of bond-equalized cyclic polyenes.62 For the Hückel aromatic polyenes, the local transmission plots revealed that indeed the current flow splits upon entering the 26H-polyene and travels equally through both arms and in the same direction (Fig. 11).

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Figure 11. Local transmission plots for Hückel and Möbius cyclic polyenes at the Fermi level (threshold set to 20%). In the antiaromatic compounds, in which Jahn-Teller causes bond alternation, the current travels through one arm only in the direction of the shortest bond, i.e. the bond with most double bond character (Fig. 11), in accordance with previous results by Tsuji et al.62 For the Möbius aromatic compounds, one could naively expect no bond alternation to take place due to the absence of a Jahn-Teller distortion and consequently, the current to travel uniformly through both arms of the macrocycle again. However, as noted above, the twisting along the macrocycle causes a considerable amount of bond alternation, which would result in a preferential path for the current. Fig. 11 confirm this prediction. In conclusion, we demonstrate that our simple model upon inclusion of the JahnTeller effect is able to accurately predict the occurrence of QI for Hückel and Möbius

27



(anti-)aromatic systems. The scheme in Fig. 9 clearly demonstrates that the Jahn-Teller distortion “wipes out” the differences in occurrence of QI between the aromatic and anti-aromatic species predicted in the previous section and the calculated current maps point out that the way the current travels through the molecule is also changed: equal distribution of the current in the case of bond-length equalized structures and a preferential direction in the case of bond-alternated geometries. As a consequence, based on the model upon inclusion of the Jan-Teller effect we do not expect to see quantum interference appearing/disappearing upon changing the topology. However, our recent study on the influence of topological and aromaticity switching on the conductance of expanded porphyrins1 suggest the opposite: when the aromaticity of the expanded porphyrins changed upon redox interconversions of the macrocycle (eg. switching between 26H to 28H), the transport properties also changed drastically. Equally, the change of topology while keeping constant the number of pelectrons also influenced the transport properties and the QI effects changed upon Hückel-Möbius interconversion (eg. 28H to 28M) appearing and disappearing in the transmission plots. How can these apparently contradicting findings be reconciled? To answer this question, we first have to continue down the path we have taken in this section of stripping approximations from our model to get closer to the actual expanded porphyrins studied in Ref. 1. The Influence of Introducing Nitrogen Atoms Along the Conjugation Pathway Our next step involves the introduction of heteroatoms in the Jahn-Teller distorted cyclic polyenes, as shown in Figure 2. So far, the conjugation paths of the cyclic polyenes contained only carbon atoms whereas the annulene-type conjugation pathways of expanded porphyrins also contain nitrogen atoms. About the influence of

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the introduction of heteroatoms in the conjugation path we can be really short. Except for the most elaborate donor-acceptor substitution schemes, substitution barely influences the shape of a QI feature.63,64 Even though regular substitution schemes do not lead to the complete disappearance of QI features, Andrews and co-workers demonstrated that these features can be shifted across the entire HOMO to LUMO range when strong electron-donating and –withdrawing groups are selected.65 Most simple hetero-atomic substitutions however, lead to only slight shifts of the location of QI away from the Fermi level. 66, 67 Based on these findings, the selection rules for the occurrence of QI displayed in Fig. 9 should still be qualitatively valid for Nitrogen-substituted polyenes. Indeed, the transmission spectra computed for the 26H N-substituted polyene shown in Fig. 12 confirm this hypothesis.

Figure 12. The transmission spectra for different connections of contacts on the Hückel 26H N-substituted polyene. The number of intermediate carbon atoms in the direction of the shortest bond, starting from the first contact position, has been placed between parentheses for each configuration.

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From Fig. 12, we can also conclude that the introduction of nitrogen atoms does not influence the behavior of the transmission probability around the Fermi level with increasing distance between the contacts when no QI takes place. The transmission spectra for the systems with the contacts separated by 8 and 12 carbon atoms overlap again in the region between the HOMO and LUMO peak. The Influence of Introducing Five-Membered Rings and the Validity of the Annulene Model We now have reached a point where the conjugation paths for both our model cyclic polyenes and the expanded porphyrins appear to be the same. However, we are still not capable of explaining the changes in the transport properties upon aromaticity/topology switching as found for the real-world hexaphyrins. To arrive at the final understanding, we have to introduce the final feature to the molecule: the fivemembered (pyrrole) rings. These five-membered rings can switch between two forms: the imine and the pyrrole form. According to the “annulene model”, the conjugation path of the expanded porphyrins will run through the upper arm of the five-membered ring in the imine form (Fig. 13), incorporating the pz-orbital of the N-atom. However, when the imine form is converted into the pyrrole form the conjugation path runs through the lower arm of the five-membered ring instead.

Figure 13. A five-membered ring in its imine form (left) and pyrrole form (right). The annulene-type conjugation path for both forms is denoted in red.

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Note that the upper arm of the five-membered ring in Fig. 13 contains 3 atoms, whereas the lower arm contains 4. As such, by changing the five-membered ring from its imine to pyrrole form, the conjugation path can either gain or lose one atom in the expanded porphyrins. Whether the number of atoms of the conjugated system (n) of an expanded porphyrin between contact atoms r and s is odd or even is thus entirely dependent on the protonation pattern of the pyrrole rings in the macrocycle. As such, one could expect changes in the protonation pattern upon topology/aromaticity switching to be the main driver of the appearance/disappearance of QI close to the Fermi level. This reasoning suggests that, in principle, it would be enough to count the number of atoms involved in the conjugation pathway separating the contacts in the expanded porphyrin to predict the occurrence of QI around the Fermi level, similarly as in cyclic polyenes. However, it turns out that the introduction of these five-membered rings has a more profound influence on the transport properties of the macrocycle than simply allowing an alternative conjugation pathway upon protonation. These rings distort the electronic structures compared to the cyclic polyenes, which causes our simple counting rule to be off in some cases. To understand this, we must consider the translation of the expression for the relative transmission (Eq. 8) from characteristic polynomials to Molecular Orbitals (MOs),39 𝑇1,3 0 = −4𝛽

;

Δ13,13 0 Δ 0

= 4𝛽

; k

𝑐1k 𝑐3k 𝜀k

;

.

(20)

In our model up to this point, we have always taken the “annulene model” to be perfect, so that we can take the pseudo-Valence Bond (VB) approach used throughout this text for the conjugation path to estimate the relative transmission. Eq. 20

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demonstrates that the more the electronic structure is distorted from this idealized situation, the worse this approximation will be because the combination of the orbital coefficients may deviate more and more from the idealized polyene case. How bad the situation is will depend on the system and especially the choice of the contact atoms on the macrocycle. Especially in symmetric situations, our counting rule still works fine. Let us consider as an example the hexaphyrins with varying topology and number of pelectrons investigated by us in Ref. 1 (cf. Fig. 1) with the contacts located along the longitudinal axis. The Hückel-aromatic porphyrin(26H) and both Möbius expanded porphyrins (26M and 28M) studied have contacts separated by an even number of atoms, so our model leads to the conclusion that QI should not occur here. For the Hückel antiaromatic compound (28H), the contacts are separated by an odd number of atoms, so we expect QI. This is indeed what is found for the hexaphyrins in Ref. 1 (Fig. 14).

Figure 14. The transmission spectra for the Hückel and Möbius [26] and [28]hexaphyrins with the contacts located along the longitudinal axis (1,4-connected junctions base on Fig. 1).

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For less symmetric systems, our “counting rule”, which worked perfectly for the polyene cycles considered before, is no longer infallible. In Fig. 15a-d, the configurations of contacts are denoted by the meso-positions to which gold electrodes are connected. The corresponding number of intermediate carbon/nitrogen atoms has again been placed between brackets for clarity. A striking example of a deviation of our expectations based on the counting rule is the absence of a QI feature around the Fermi level for 1,5-connected 26H-porphyrin in Fig. 15a. Nevertheless, the transmission probability for this configuration of contacts is still lower than that for the other configurations in the region around the Fermi level.

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Figure 15. The transmission spectra for different configurations of contacts on a) Hückel 26H-porphyrin; b) Möbius 26M-porphyrin; c) Hückel 28H-porphyrin; d) Möbius 28Mporphyrin. The number of intermediate carbon atoms in the direction of the shortest bond, starting from the first contact position, has been placed between parentheses for each configuration. A more reliable, though slightly more complex, approach to predict QI would be to exploit Eq. 20 and consider the MOs of the system studied. Yoshizawa and co-workers came up with an elegant frontier orbital rule, which does a fine job of predicting quantum interference at the Fermi level in systems even when they are not perfectly alternant hydrocarbons anymore.14,15,20 According to their selection rule, QI around the

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Fermi level will occur if the products of the orbital coefficients on contact positions r and ∗ ∗ s of the HOMO (𝐶1nopo 𝐶3nopo ) and the LUMO (𝐶1qrpo 𝐶3qrpo ) have the same sign. As

mentioned in the introduction, this rule was shown by us to be successful in predicting the occurrence of QI around the Fermi level (Fig. 16a-d).1 Nevertheless, the QI which are predicted with this frontier orbital method can sometimes be shifted away from the Fermi level, so that the occurrence of QI cannot always be linked to a reduced conductance under small bias. We note that Tsuji and Yoshizawa recently expanded this frontier orbital rule to be valid for all non-alternant hydrocarbons, not only for those whose electronic structure resembles that of an alternant hydrocarbon.68

Figure 16. HOMO and LUMO orbitals for the Hückel aromatic [26]hexaphyrin (a) and antiaromatic [28]hexaphyrin (b). Symmetry-allowed (red) and symmetry-forbidden (green) connections for electron transmission based on the orbital rule (c and d). A further indication of the profound influence the five-membered rings have on the transport properties can be found in the evolution of the transmission probability in the region between the HOMO and LUMO peak in the transmission spectrum when no QI takes place for the considered systems (Fig. 15). We no longer observe a constant transmission probability with the distance between the contacts for the Hückel aromatic

35



systems (26H) and neither is there a monotonous decrease in the transmission probability with the distance between the contacts for the other systems. We can also observe that the relationship between the conductance and aromaticity are highly dependent on the positions of the contacts, as observed in our previous work. When the contacts are connected along the longitudinal axis of the macrocycle (1,4-connected molecular junction), the Hückel aromatic system will have the highest conductance when a small bias is applied, as is also found in our back-of-theenvelope approach. However, our present analysis also demonstrates that other configurations of the contacts will not necessarily lead to the same conclusions (Fig. 15). CONCLUSIONS In this work, the transport properties of Hückel/Möbius (anti-)aromatic compounds have been investigated in detail. First, we have developed a simple counting rule to predict the occurrence of QI around the Fermi level for cyclic polyenes at the Hückel level of theory. When the contacts in the cycle are separated by an even number of carbon atoms, no QI will take place; when the number of carbon atoms is odd, a QI feature will be present. Gradually, the different approximations assumed in the Hückel model were lifted, enabling us to explore step-by-step the influence of each structural feature on the transport properties of expanded porphyrins. This analysis additionally enabled us to shed some more light into the relationship between aromaticity and conductance. Although the conjugated system of expanded porphyrins can be approximated by polyene macrocycles according to the annulene model, our results indicate that the distortion of the electronic structure of the expanded porphyrins induced by the five-membered rings causes the simple counting rule for the prediction of QI developed for the polyene cycles to fail in some specific situations. Nevertheless,

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our back-of-the-envelope approach enables an intuitive rationalization of most of the transport properties of expanded porphyrins. In those situations for which the counting rule fails to correctly predict QI, the orbital selection rule proposed by Yoshizawa and co-workers works best. Additionally, the configuration of the contacts on both polyenes and expanded porphyrins has a profound influence on the transport properties. These new insights can offer a road map towards the design of functional, multi-dimensional electronic switches based on expanded porphyrins. SUPPLEMENTARY INFORMATION See supplementary material for the bond-lengths for the structures in Fig. 11, a validity check of the conservation of current principle in the local transmission plots and the geometries of the structures studied. ACKNOWLEDGEMENTS T.S. acknowledges the Research Foundation-Flanders (FWO) for a position as research assistant (11ZG615N). S.F. wishes to thank the FWO and the European Union's Horizon 2020 Marie Sklodowska-Curie grant (No 706415) for financially supporting his postdoctoral research at the ALGC group. P.G. and F.D.P. wish to acknowledge the Vrije Universiteit Brussel (VUB) for a Strategic Research Program. F.D.P. also acknowledges the Francqui foundation for a position as Francqui research professor. M. A. thanks the FWO for a postdoctoral fellowship (12F4416N) and the VUB for financial support. Computational resources and services were provided by the Shared ICT Services Centre funded by the Vrije Universiteit Brussel, the Flemish Supercomputer Center (VSC) and FWO.

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Construction of the polyene and heteropolyene models from the hexaphyrin macrocycle.

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Schematic overview of the transport properties for Hückel and Möbius even-membered cyclic polyenes based on our simple tight-binding Hückel model. The conjugated system is either aromatic or antiaromatic, and two contact groups will always be separated either by an even or odd number of carbon atoms. These two variables fully determine whether quantum interference will occur close to the Fermi level or not. 215x240mm (300 x 300 DPI)



Schematic overview of the transport properties of both Hückel and Möbius even-membered cyclic polyenes based on our simple model upon inclusion of the Jahn-Teller effect. Changes compared to Fig. 8 are denoted in orange. 81x83mm (300 x 300 DPI)


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The computed transmission spectra for the cyclic polyenes for different configurations of contacts on the Hückel 26H-polyene. The number of intermediate carbon atoms in the direction of the shortest bond, starting from the first contact position, has been placed between parentheses for each configuration. 170x289mm (300 x 300 DPI)



Local transmission plots for Hückel and Möbius cyclic polyenes at the Fermi level (threshold set to 20%). 129x73mm (300 x 300 DPI)


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The transmission spectra for different connections of contacts on the Hückel 26H N-substituted polyene. The number of intermediate carbon atoms in the direction of the shortest bond, starting from the first contact position, has been placed between parentheses for each configuration.


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The transmission spectra for different configurations of contacts on a) Hückel 26H-porphyrin; b) Möbius 26M-porphyrin; c) Hückel 28H-porphyrin; d) Möbius 28M-porphyrin. The number of intermediate carbon atoms in the direction of the shortest bond, starting from the first contact position, has been placed between parentheses for each configuration.


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Aromatic Compounds

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