Back of the Envelope Selection Rule for Molecular Transmission: A

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Back of the Envelope Selection Rule for Molecular Transmission: A Curly Arrow Approach Thijs Stuyver, Stijn Fias, Frank De Proft, and Paul Geerlings J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b10395 • Publication Date (Web): 03 Nov 2015 Downloaded from http://pubs.acs.org on November 8, 2015

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The Journal of Physical Chemistry

Back of the Envelope Selection Rule for Molecular Transmission: a Curly Arrow Approach T. Stuyver,†,‡,¶ S. Fias,†,¶ F. De Proft,†,¶ and P. Geerlings∗,†,¶ ALGC, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium, Aspirant of the Research Foundation-Flanders (FWO-Vlaanderen), Egmontstraat 5, 1000 Brussels, Belgium., and Member of the QCMM Ghent-Brussels Alliance Group. E-mail: [email protected]

Abstract A new selection rule for transmission under small voltage for molecular electronic devices consisting of alternant hydrocarbons is established, based on curly arrow drawings of the displacement of electrons through the conjugated system of the molecule. If the displacement of electrons from one contact atom to the other can be drawn, transmission is possible. If not, then no transmission will take place. This simple, back of the envelope ansatz allows determination of quantum interferences for a wide range of experimentally studied molecules (linear polyenes, benzenoids, cross-conjugated molecules,...) avoiding any calculation. This new selection rule bridges the conceptual gap between traditional reactivity theory and molecular electronics. ∗

To whom correspondence should be addressed ALGC, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium ‡ Aspirant of the Research Foundation-Flanders (FWO-Vlaanderen), Egmontstraat 5, 1000 Brussels, Belgium. ¶ Member of the QCMM Ghent-Brussels Alliance Group. †

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Introduction Over the past decades, the amount of research in the field of molecular electronics soared. 1 An ever increasing number of experimental and theoretical papers have been published, studying the transport properties of organic molecules containing π-conjugated systems and considering possible applications for the resulting molecular electronic devices. 1–11 Most of the theoretical calculations in these studies were performed at high level of theory, but such calculations do not always lead to simple, e.g. pictorial, insight. In response, the advances in the field have inspired the formulation of several simple rules to rationalize the experimental results and to obtain chemical insight in the relationship between molecular structure and transmission. These rules are based on frontier molecular orbitals, 12–14 graph-theoretical arguments, 15–17 atom-atom polarizabilities, 18 Kekulé structures, 17 Pauling bond orders, 17,19,20 , simple graphical schemes, 21 Hamiltonian matrix inversions, 22 and defect states. 23

In order for these rules to be valid, the following conditions have to be fulfilled: 24 (i) Only a small bias is applied (ii) The Fermi level of the metal contacts is located in between the HOMO and the LUMO of the isolated molecule (iii) A tight-binding (or Hückel) approach is suitable for the studied molecule (iv) The interaction between contacts and molecule is weak.

These conditions are satisfied in most experimental studies in the field of molecular electronics up to date (a full explanation can be found in the appendix). 11,14,25–30 Under these conditions, the expression for the transmission probability close to the Fermi level reduces to 18

T = 4β˜2

2

∆2r,s ∆2


(1)

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where ∆ denotes the characteristic polynomial of the secular determinant of the isolated molecule, the letters before and after the comma in each suffix denote respectively the rows and columns omitted from this initial secular determinant and β˜ is a measure for the in˜ teraction between the contacts and the molecules. Except for the proportionality factor β, the transmission probability only depends on the characteristics of the molecule within the approximation, not on the contacts. The current-voltage relation for molecular electronic devices is then easily obtained starting from the transmission probability 24 2e I= h

Z

µ2

T (E)dE

(2)

µ1

where µ1 and µ2 are the Fermi levels of the left and right contact under (small) bias respectively.

Up to this point, no selection rule has been proposed that links the transmission probability through molecules with the reactivity of those systems. In the present work, we present a new selection rule, based on curly arrow drawings in Kekulé structures. From its conception almost 80 years ago, curly arrow drawings have played a central role in chemical reactivity theory. Every chemist is familiar with these drawings, making this new selection rule very intuitive and accessible and it bridges the gap between traditional reactivity theory and molecular electronics.

In the remainder of this work, we will first derive the new rule and discuss its relations to some other rules that have been proposed. Next, to demonstrate the potency of this new rule, we apply it to a wide range of molecules (octatetraene, benzene, perylene, phenanthro[1,10,9,8opqra]perylene, 5-1,3-propadienyl-1,3,6,8-nonatetraene, 9,10-bis(phenyl-ethynyl)anthracene). These examples were chosen because they are either models for entire classes of molecules 3



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or because they have been the subject of previous research. 5,16,23 Finally we look at some relevant photo-, thermo- and redoxswitches which have been studied in the field of molecular electronics (dimethyldihydropyrene (DHP)/cyclophanediene (CPD), 1,2-di(2-methyl-1naphthyl) perfluorocyclopentene and oligo(phenylene vinylene)quinone). 9,14,25,27,31–33

Theoretical Methods The new selection rule for transmission presented here is based on earlier work on the Pauling (long-) bond order. 34 For all even alternant hydrocarbons (except 4n membered rings), the Pauling bond order between two atoms r and s of a conjugated carbon system can be defined as 19,20,35,36

pprs =

K(G ⊖ rs) K(G)

(3)

where K(G) is the number of Kekulé structures (KS) or perfect matchings for graph G corresponding to the carbon skeleton of a hydrocarbon and K(G ⊖ rs) is the number of Kekulé structures when the vertices corresponding to atom r and atom s are deleted from graph G. Comparison of equation (1) with the alternative expression for the Pauling bond order 36

ppr,s =

∆r,s (0) ∆(0)

(4)

leads to the conclusion that transmission in the weak interaction limit at the Fermi level is proportional to the square of the Pauling bond order. 20

The new selection rule starts from a reinterpretation of equation (3). From this equation, it

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is evident that the Pauling bond order (and thus also the transmission probability) is zero when no Kekulé structure exists for G ⊖ rs. Kekulé structures for G ⊖ rs can be obtained by drawing the displacement of an electron pair in the Kekulé structures of G from atom r to atom s. This way, atom r becomes positively charged and atom s negatively charged, which is equivalent to deleting the two corresponding vertices from the original graph as they are no longer part of the conjugated system. 19 So, if it is impossible to make a curly arrow drawing of the displacement of electrons in any of the Kekulé structures between two atoms, then the Pauling bond order is zero and a quantum interference is present at the Fermi level (transmission is zero).

This new selection rule is closely related to, and can also be regarded as a reinterpretation of, some of the earlier mentioned selection rules, based on graph-theoretical arguments 16 and graphical methods 21 respectively. An analysis of the relation between these rules is given below.

We first focus on the graphical method developed by Markussen and co-workers, which has the same area of application as our rule (even alternant hydrocarbons which are not 4n membered rings as mentioned earlier on). When Kekulé structures are drawn for these systems, all carbon atoms in the π-conjugated system are paired together through double bonds. When it is possible to draw the displacement of electrons from one contact to the other, then new pairs are formed along the (continuous) ’electron path’, all other initial pairs of carbon atoms not belonging to the continuous path are untouched. So whenever it is possible to draw the displacement of electrons from one site to the other in a continuous way, it is also automatically possible to pair all carbon atoms not lying on that path, which is the necessary condition for transmission in the rule of Markussen et al. 21 This understanding also allows us to link these two rules to the graph-theoretical rule developed by Fowler et al. 16 .

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From the work of N. S. Ham 37 (and others 17,19,20,35,36 ), it can be easily demonstrated that for even alternant hydrocarbons, except for 4n membered rings, ∆(0) is equal to the square of the number of Kekulé structures of the original molecule

∆(0) = det(H) = K(G)2

(5)

where H is the Hamiltonian matrix of the π-system in the Hückel approximation. So, for these types of molecules, the denominator of equation (1) always differs from zero. In this case, conduction will only take place if ∆2r,s (0) also differs from zero. ∆2r,s (0) can be decomposed according to the Jacobi/Sylvester determinant identity 38

∆rs,rs =

∆r,r ∆s,s − ∆2r,s ∆

(6)

leading to ∆2r,s = ∆r,r ∆s,s − ∆rs,rs ∆

(7)

where ∆r,r and ∆s,s correspond to the characteristic polynomial of the molecular graphs where respectively atom r and atom s have been deleted, ∆rs,rs corresponds to the characteristic polynomial of the graph where both atoms r and s have been deleted. It is now evident that if Kekulé structures can be drawn for the original molecule, this will not be possible for the molecules when one carbon atom of the π-system is deleted. This means that both ∆r,r (0) and ∆s,s (0) are zero in this case. So for these systems, ∆r,s (0) will only differ from zero if ∆rs,rs (0) differs from zero, or equivalently, if it is possible to draw Kekulé structures for the molecule when the contact atoms have been deleted. As already mentioned, if it is possible to draw the displacement of electrons from one contact to the other, then it is evidently possible to draw Kekulé structures for the resulting contactless molecule. 6


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We should note that Fowler et al. already stated this necessary condition for transmission at the Fermi level in terms of the existence of Kekulé structures for the contactless molecule in the case of nanographenes. 17

It should also be noted that the graph-theoretical rule of Fowler et al. 16 has a broader field of application than our proposed rule or the rule of Markussen et al., 21 since ∆(0) is not forced to differ from zero in theirs. This allows application of the graph-theoretical rule to radicals and 4n-membered rings. Although this broader area of application is definitely an important achievement, we note that these ’special’ types of molecules have attracted only a marginal interest in (especially experimental) molecular electronics, due to their general lack of stability, as was already mentioned by the authors of the rule themselves. 17 So from a practical point of view, our rule and the one developed by Markussen and co-workers is broad enough to be relevant in the field of molecular electronics, but the above stated restrictions should always be kept in mind.

The new selection rule allows the transmission probability around the Fermi level for most hydrocarbons to be predicted in the blink of an eye just by considering Lewis structures. The real virtue of the use of curly arrows to predict transmission probabilities is that it opens up a connection to the widely used concept of resonance effects in reactivity theory. 39 Recently, the linear response kernel, one of the main reactivity indices in conceptual Density Functional Theory (DFT), 40–43 has been used by members of our group to quantify these effects. 43–46 Here we will focus on its Hückel analogue, the atom-atom polarizability, a central concept in chemical reactivity theory. 18,47–49

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Computational Methods Transmission plots were calculated through a tight-binding SSP approach, 15,18,50,51 where the structure of the link between the contacts and the molecule was not made explicit as it does not influence substantially the transport properties of the molecule in general. 52 The presence of the linkers was accounted for by setting the coupling between the attachment sites of the molecule and the gold chains small. As such, βAu−C was set to 0.2, βC−C was set to 1 and βAu−Au was set to 0.6, in accordance with previous calculations. 12,14

A remark to be made is that some of the studied systems are not truly bipartite; they have some impurities that disrupt the perfect symmetry. For example, one of the systems considered has two triple bonds. Since simple Hückel theory does not discriminate between double and triple bonds, we treated the triple bonded carbons in the same way as a double bond and assign it the same α- and β-parameters. This is in accordance with previous work by Fowler et al. 16 Other molecules considered are not entirely planar due to steric features of the two napthyl subunits, thus reducing the overlap between the two adjacent p-orbitals of the central ethylene bridge. To account for this, the resonance integrals between these two orbitals were modified according to the following equation 39

β(θ) = βC−C cos θ

(8)

In the case that hetero-atoms were present, parameters from literature were used (αO = 1, βO = 0.8). 39

The atom-atom polarizability matrices were calculated starting from the molecular orbital expression 18,47–49

8


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πr,s

1 =− π

Z

∞

−∞

∆2r,s (iy) dy ∆2 (iy)

(9)

All programs used were written in house and thoroughly validated.

Validation of the Selection Rule and the Interplay of Transmission with Resonance Effects As already mentioned, we first apply the new selection rule to a wide range of molecules. An ad hoc numbering of the carbon atoms has been used for the calculations (figure 1). For the nomenclature of the compounds, official IUPAC rules were adhered to. Each prediction made, based on the new selection rule, has been verified through Source-and-Sink-Potential method (SSP) calculations. 15–18,23,50,51 Transmission probability matrices at the Fermi level, polarizability matrices and transmission spectra have been calculated for all relevant cases. Those that are not presented in the core of the paper have been added as supporting information. As already mentioned, most of the considered systems have been investigated previously, enabling further validation of our selection rule through comparison with earlier (high-level) calculations and/or experimental data. In the discussions of the systems presented in figure 1, placements of the contacts on carbon atoms belonging to the same set will not be considered as these situations lead to no transmission for even alternant hydrocarbons which are not 4n-membered rings (cfr. law of alternating polarity). 18,22,47 This statement can also evidently be recovered from our new rule since it is impossible to draw the electron displacement in these cases.

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b)

a)

1 4

2 3

1

6

8

6 5

7

2

5

3 4

c)

d)

5

3 4

6

2

4

6

1

9

7

1

13

11

8

10 11

13

19

17

15 14

28

16

18

19

17

15

27

25

23

20 21 22

24

26

e)

9 10

12

16

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12

14

20

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f) 2

1

7

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3

2

4 3

6 5

8 7

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3 4

6

10 11

5

13

11

1

14 7

10 8

15

9 16 22 17 21 18 20 19

12

23

26 24

27

25 30

28 29

Figure 1: List of studied molecules: a) octatetraene b) benzene c) perylene d) phenanthro[1,10,9,8-opqra]perylene e) 5-1,3-propadienyl-1,3,6,8-nonatetraene f) 9,10bis(phenylethynyl)anthracene.

Application of the selection rule to octatetraene leads to the following conclusions (figure 2). The displacement of electrons can only be drawn in one direction. When a first contact is placed on a terminal carbon atom, then all positions of different sets can be reached by the electrons, leading to transmission for every possible position of the second contact atom. When the first contact is placed somewhere else, then the electrons can only be displaced to the left or to the right, depending on the chosen position.

10


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S-Au

S-Au

-

+ S-Au

S-Au S-Au

S-Au

S-Au

+

S-Au

S-Au

S-Au

-

+ S-Au

S-Au S-Au

S-Au

S-Au

+

S-Au

Figure 2: The new selection rule applied to octatetraene.

Calculation of the transmission spectra support these predictions. For example, connection of contacts on the first and eighth carbon atom of the chain leads to the transmission spectrum in figure 3.

Figure 3: Computed transmission spectrum for the 1-8 connection for octatetraene.

From this figure it can be concluded that transmission can take place at the Fermi level, 11



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in agreement with the predictions made through our selection rule. Connection of contacts on the first and seventh carbon atom of the chain on the other hand leads to the transmission spectrum in figure 4.


Here it can be clearly observed that transmission goes to zero when approaching the Fermi level, a quantum interference is present. This again corresponds to our predictions. The spectrum for a 2-3 connection is presented in figure 5.

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Here too, a quantum interference can be observed. A side note to be made is that our selection rule, like every previous selection rule, tells us only something about the transmission probability at and around the Fermi level. Moving further away from E=0 in figure 3, transmission might be possible even though it is impossible to draw the displacement of the electrons in the Lewis structures.

The atom-atom polarizability matrix is presented in table 1.

Table 1: atom-atom polarizability matrix for octatetraene 1 2 3 4 5 6 7 8

1 -0.718 0.361 -0.106 0.199 -0.030 0.157 -0.006 0.144

2 0.361 -0.390 0.061 -0.026 0.008 -0.009 0.001 -0.006

3 -0.106 0.061 -0.509 0.280 -0.053 0.179 -0.009 0.157

4 0.199 -0.026 0.280 -0.453 0.076 -0.053 0.008 -0.030 13

5 -0.030 0.008 -0.053 0.076 -0.453 0.280 -0.026 0.199

6 0.157 -0.009 0.179 -0.053 0.280 -0.509 0.061 -0.106


7 -0.006 0.001 -0.009 0.008 -0.026 0.061 -0.390 0.361

8 0.144 -0.006 0.157 -0.030 0.199 -0.106 0.361 -0.718


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A synthesis of all these results can be found in figure 6. In this figure, the application of the new selection rule can found to the left and a graphical comparison between the atom-atom polarizability and transmission probability around the Fermi level can be found in the middle and to the right. For the polarizability, the first contact atom is denoted by a green circle. Black and red circles on other atoms denote the amount of polarization on the considered atom in response to a unit perturbation at the first contact atom. The black circles correspond to positive and red circles to negative polarizability. For the transmission probability at the Fermi level, the same approach is taken as above. The empty green circles denote the position of the first contact. As the transmission probability is between 0 and 1, no red circles will arise.

S-Au

S-Au

-

+ S-Au

S-Au S-Au

S-Au

S-Au

+

S-Au

S-Au

S-Au

-

+ S-Au

S-Au S-Au

S-Au

S-Au

+

S-Au

Figure 6: Application of the selection rule to octatetraene (left), calculated atom-atom polarizabilities for the considered contact atoms (middle) and calculated transmission probabilities at the Fermi level.

Inspection of the graphics presented in figure 6 allow us to identify 3 different situations (a more in-depth analysis and numerical data can be found in the supporting information):

(i) It is possible to draw the displacement of electrons between the two atoms in the Lewis structure. A positive polarizability exists between the two atoms and transmission takes place at the Fermi level. (ii) It is intrinsically impossible to draw the displacement of electrons between two atoms 14


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because they belong to the same set. The atom-atom polarizability is negative, meaning that no delocalization of electrons upon perturbation takes place between those atoms and neither does a current flow under low bias if the molecule is connected through these atoms in an electrical circuit. (iii) It is impossible to draw the displacement of electrons between the two considered atoms due to structural features of the molecule. A slightly positive polarizability is obtained, but no transmission takes place.

From this analysis, it is easy to explain the operation of some rudimentary switches based on (heat or light triggered) electrocyclic ring-closing reactions of polyenes (figure 7a). The switching properties of these systems were predicted by Hoffmann and co-workers. 22 The new selection rule allows us to see easily that the open forms of both butadiene and hexatriene will not conduct close to the Fermi level while the closed forms will for contacts at positions 2-3 and 2-5 respectively. Later on in this manuscript we will turn to some relevant thermo-, photo- and redoxswitches of which the switching properties have been demonstrated experimentally.

For benzene, a displacement of electrons can be drawn towards ortho- and para-positions, but not to meta-positions (this is true starting from both resonance structures of benzene). So placement of two contacts meta to one another leads to no transmission, while orthoand para-connected benzene will conduct close to the Fermi level (figure 7b). Numerous theoretical and experimental studies have led to the same results. 5,12,13,16,18,19,21–23,27,29

A graphical comparison between the atom-atom polarizability and transmission around the Fermi level for benzene can be found in figure 8a and 8b (same conventions used as before).

This figure demonstrates that transmission is influenced by the relative position of the con-

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Au-S

Au-S

S-Au

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S-Au

+

S-Au

S-Au

-

+

Au-S

S-Au

Au-S

S-Au

+

S-Au

S-Au

S-Au

S-Au

-

+

S-Au

Au-S

S-Au

Au-S

-

+

S-Au

-

S-Au

S-Au

S-Au

+

Au-S

S-Au

Au-S

-

+

S-Au

Au-S

Au-S

(b)

(a)

Figure 7: The new selection rule applied to a) photo/thermoswitches butadiene (up) and hexatriene (down), b) benzene. tacts on the molecule in analogy to the well-known fact that reactivity at a certain site is influenced by the relative position of different substituents (e.g. ortho/para directors in electrophilic aromatic substitution reactions). 39,53,54

(b)

(a)

Figure 8: (a) The atom-atom polarizability and (b) the transmission probability at the Fermi level for benzene.

The placement of the contacts on nanographenes influences the transmission. For perylene, attachment of a contact in positions 2 or 6 makes it impossible to draw an electron displacement to the lower part of the molecule, irrespective of the location of the second contact or starting resonance structure (figure 9a). The central benzene ring acts as a barrier that cannot be crossed. Due to symmetry, the same is true for contacts at positions 15

16


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or 19. This corresponds to previous theoretical results. 23 Other positions of contacts do not give rise to a similar barrier.

Au-S

Au-S

Au-S

Au-S

+

+

-

S-Au

-

S-Au

S-Au

S-Au

S-Au

S-Au

S-Au

+

S-Au

+

-

-

S-Au

S-Au

S-Au

S-Au

S-Au S-Au

S-Au

S-Au

+

+

-

-

S-Au S-Au

S-Au

(a)

S-Au

(b)

Figure 9: The new selection rule applied to a) perylene, b) phenanthro[1,10,9,8opqra]perylene.

For phenanthro[1,10,9,8-opqra]perylene similar results as for perylene are obtained, here the central anthracene acts as a barrier for conduction when the contacts are located at positions 2, 8, 21 or 27 (figure 9b).

From the analogy between perylene and phenanthro[1,10,9,8-opqra]perylene, it can be expected that similar barriers will arise for every possible nanographene which displays a narrowing over its length.

This general prediction of quantum interferences in nanographenes that narrow over their

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length is a nice illustration of the power of this selection rule. While the burden to go through all Kekulé structures makes this rule rather impractical for big nanographenes, some general features of the transport properties of these molecules can now be understood intuitively, which was not the case with previously established rules even though those rules deal more efficiently with these systems.

We also want to stress that our new, simple selection rule only allows prediction of which combinations of contacts will lead to transmission and which ones will not. To get an idea of which combination will lead to the highest transmission, the Pauling bond order rule, 19,20 the graph theoretical rule 17 or the rule established by Ernzerhof et al. should be used. 23

Cross-conjugation can also act as a barrier, which makes it impossible to draw the displacement of electrons from one side of the main chain to the other and so no transmission will take place. To illustrate this, we considered a prototypical cross-conjugated molecule, 5-1,3-propadienyl-1,3,6,8-nonatetraene, in the left part of figure 10.

Au-S

S-Au

Au-S

-

+

Au-S

Au-S

S-Au

-

S-Au

+ S-Au

Figure 10: Application of the selection rule to 5-1,3-propadienyl-1,3,6,8-nonatetraene (left), calculated atom-atom polarizabilities for the considered contact atoms (middle) and calculated transmission probabilities at the Fermi level. Comparing the left and right part of figure 10 (same conventions as before), it can be concluded that the three situations introduced in the discussion of octatetraene, are again present for this molecule. This figure also illustrates that mesomerically active substituents 18


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on one side of the main chain will not influence the reactivity on the other side.

For 9,10- bis(phenylethynyl)anthracene, it is impossible to draw the displacement of electrons out of the first phenyl unit. If a first contact is placed at positions 2 or 6, no transmission will take place from one side of the molecule to the other. When the contact is placed at position 1, then the displacement of electrons to the other side of the molecule can be drawn and, as such, transmission will take place when the second contact is placed at position 28 (figure 11a) in correspondence with previous experimental and theoretical results. 5,16

S-Au

S-Au

+ Au-S

-

S-Au

Au-S

Au-S

Au-S

+

S-Au

+

S-Au

S-Au

-

S-Au

(b) S-Au

(a) Figure 11: The new selection rule applied to (a) 9,10-bis(phenylethynyl)anthracene and (b) bis(phenylethynyl)anthracene with an alternative positioning of the phenylethynyl subunits on the anthracene with contacts kept in place.

A different positioning of one of the phenylethynyl subunits on the anthracene can influence the transmission probability at the Fermi level independently of the positions of the contacts on the terminal benzyl units. Consider for example the situation where the second phenylethynyl subunit of the molecule would have been connected on the 19th or 13th carbon atom, even if the contacts were still located on carbon atom 1 and 28. In this case 19



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the selection rule leads to the conclusion that a quantum interference will be present at the Fermi level (figure 11b).

Photo-, Thermo- and Redoxswitches Now that the selection rule has been established and its potency has been amply demonstrated, we turn to some relevant thermo-, photo- and redoxswitches which have been studied extensively both experimentally and theoretically in the field of molecular electronics (figure 12).

t-Bu

OH

t-Bu

O

HO

Au

N

O

Au

N Au

N N Au

t-Bu t-Bu

F

F

F

Au-S

F

F

F H3C

F

F

F

F

F S-Au

Au-S

F H3C

S-Au S S Au

CH3

CH3

Au

(b)

(a)

Figure 12: a) Photoswitches DHP/CPD (up) and 1,2-di(2-methyl-1-naphthyl) perfluorocyclopentene (down) and b) redoxswitch oligo(phenylene vinylene)quinone.

20


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A first class of these switches are diarylethene photoswitches. This type of switches has attracted serious interest because of their excellent switching properties, thermal stability and fatigue resistance. 9,14,27,31 More precisely the dimethyldihydropyrene (DHP)/cyclophanediene (CPD) and 1,2-di(2-methyl-1-naphthyl)perfluorocyclopentene switches are considered (figure 12a). For the DHP/CPD switch high on/off ratios have been predicted and measured. 9,14 The switching behavior of 1,2-di(2-methyl-1-naphthyl)perfluorocyclopentene has been investigated experimentally and theoretical studies have been performed on its transport properties. 27,32

As the dihedral angles between the ethene plane and benzene ring for the CPD system is estimated to be 60◦ the β-value between the corresponding orbitals was set set to 0.5 in our calculations (equation (8)). A similar approach was taken for the 1,2-di(2-methyl-1naphthyl) perfluorocyclopentene where the dihedral angle between the two naphtyl subunits is estimated to be 80◦ , leading to a β-value of 0.17. 27

From our selection rule, it is easily seen that the closed form of the DHP/CPD switch will conduct around the Fermi level, while the open form will not (figure 13a (up)). Also the conductivity of the 1,2-di(2-methyl-1-naphthyl)perfluorocyclopentene is easily understood by looking at the curly arrow drawings (figure 13a (down)). We can conclude that this system will conduct in its closed form, but not in its open form.

The oligo(phenylene vinylene)quinone redoxswitch is the final example considered. The ratios of conductance between the high- and low-conductivity states for this system has been determined to be over 40 experimentally. 25 This system garnered serious interest for its envisioned applications. 25,33

Our selection rule leads to the conclusion that in the oxidized form, the ketogroups will

21



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act as a barrier, preventing a displacement of electrons towards the upper two carbon atoms of the terminal benzyl subunit (figure 13b). In hydroquinone this barrier is not present, but only one of the two terminal carbon atoms can be reached by the electrons (the one most to the right). As the experimentalists did not discriminate between the two sites and only hovered a probe over the molecule, it could be possible that the measured conductance is lower than what would be achieved if the second contact was entirely attached to the carbon atom most to the right. As a consequence, the ratio of conductance could be higher than the one reported experimentally. t-Bu

t-Bu

-

O

OH

HO

O

+

-

t-Bu

t-Bu

t-Bu

t-Bu

+ +

-

t-Bu

F

t-Bu

F

F F

H3C

F

F

F

Au-S

F

F F

F

S-Au

Au-S

+

-

H3C

-

OH

F

S-Au

O

HO CH3

F

Au-S

CH3

F

F

F

F

F

F

H3C

F

F

F

F

O

Au-S

S-Au

+

F

H3C

-

S-Au

+ CH3

CH3

(a)

(b)

Figure 13: a) Application of the selection rule for photoswitches DHP/CPD (up) and 1,2di(2-methyl-1-naphthyl) perfluorocyclopentene (down) and b) redoxswitch oligo(phenylene vinylene)quinone.

22


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Conclusions In summary, a powerful new selection rule for transmission under small bias for molecular electronic devices consisting of even alternant hydrocarbons (not 4n-membered rings) has been introduced, based on simple, back of the envelope, curly arrow drawings of the displacement of electrons through the conjugated system of the molecule. When it is impossible to draw the displacement of an electronpair from one contact to another, a quantum interference is present at the Fermi level. The new rule demonstrates that transmission around the Fermi level is strongly connected to molecular reactivity. Our knowledge about the proneness of specific carbon atoms to resonance effects can be transferred to the molecular conductivity problem. This rule can also be used to study a wide variety of photo-, thermoand redoxswitches which have aroused great interest in molecular electronics, illustrating its broad applicability.

Supporting information. Numerical results for all systems considered. This material is available free of charge via the internet at http://pubs.acs.org.

Acknowledgement. T.S. acknowledges the Research Foundation - Flanders (FWO) for a position as research assistant (11ZG615N). S. F. wishes to thank the Research Foundation Flanders (FWO) for financial support for his postdoctoral research in the ALGC group. P. G. and F. D. P wishes to acknowledge the VUB for a Strategic Research Program.

Appendix A: Justification of equation (1) In most experimental studies on molecular electronic devices, molecules are introduced between two (Au-)contacts (either through break junctions or scanning tunneling microscopy techniques) and the current is measured between the contacts when a small voltage is applied. 1 As the Fermi level of gold is intermediate between the HOMO and the LUMO of 23



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Page 24 of 33

most organic molecules, 14 the approximation can be made that the (unbiased) transmission probability in the neighborhood of the Fermi level of the isolated molecule will determine the current-voltage characteristics of the molecular electronic device in such measurements. 12 This is evident considering the following equation resulting from the Landauer-Büttiker formalism 55 2e I= h

Z

µ2

T (E, V )dE

(10)

µ1

where µ1 is the Fermi level of the left contact under bias and µ2 is the Fermi level of the right contact under bias.

The total bias is equal to µ1 −µ2 and is centered around the Fermi level of unbiased gold when applied symmetrically (which is, as was already mentioned, located between the HOMO and the LUMO of the isolated molecule). Since only a small bias is applied, the transmission function, T(E,V), will be almost independent of the applied voltage (linear response approximation), 24 showing that the (unbiased) transmission probability values of isolated molecules at and around the Fermi level will indeed be the main contributors to the current when a small voltage is applied.

In order to rationalize the influence of structure on the conductivity of a molecular electronic device, a tight-binding (or Hückel) approach is often taken, 56,57 where the influence of the backbone of the molecule is neglected and all transmission is, as a result, attributed to the π-system. The part of the Hamiltonian matrix of the device corresponding to the molecule then takes the form of an adjacency matrix. 15 It has previously been demonstrated amply that clear trends in the transport properties at Hückel level of theory result in similar trends at higher levels of theory and in experimental data. 11–13,16,20,22,23,27,28

24


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To calculate the transmission probability through a molecule for such a system, two approaches can be taken, the Green’s Function (GF) approach 24,56 or the Source-and-SinkPotential (SSP) method. 23,50,51,58 In both approaches, the solution of the resulting expressions is simplified most by approximating the contacts by simple gold chains, one atom thick. 12,15 Both approaches then lead to the following expression for the transmission probability from one contact to the other 15 (the equivalency between the GF- and SSP-method has recently been demonstrated by Fowler et al.) 59

T (E) = 1 − |r(E)|2 =

˜ r,r ∆ ˜ s,s − ∆ ˜∆ ˜ rs,rs ) 4 sin qL sin qR (∆ ˜ r,r + ∆ ˜ rs,rs |2 ˜ s,s − e−iqL ∆ ˜ − e−iqR ∆ |e−i(qL +qR ) ∆

(11)

where qL and qR are the wave vectors of the incoming and outgoing Bloch waves, ∆ denotes the characteristic polynomial of the secular determinant of the isolated molecule, the letters before and after the comma in each suffix denote respectively the rows and columns omitted from this initial secular determinant and the tilde-signs denote scaling of the polynomials according to

˜ =∆ ∆

˜ s,s = β˜L ∆s,s ∆

˜ r,r = β˜R ∆r,r ∆

˜ rs,rs = β˜L β˜R ∆rs,rs ∆

(12) (13)

where β2 β˜R = rR βR

β2 β˜L = sL βL

25


(14)


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Page 26 of 33

In this expression, βsL and βrR denote the resonance integral between the final (Au-) atom of the contacts and its nearest neighbors in the molecule respectively and βL and βR denote the resonance integral between the atoms in the contacts. β˜L and β˜R in equation (14) express how the strengths of the bonds between the molecule and the contacts are related to those in the wires. 15,18 The higher these factors, the stronger the coupling between the contacts and the molecule.

In most molecular electronic devices synthesized until now, the molecules are poorly connected to the contacts through linkers (e.g. thiol bond), allowing the ’weak interaction limit’-approximation to be made. 15 The combination of this approximation with the previously introduced ones justifies the use of the transmission probability of electrons through molecules at the Fermi level, when no voltage is applied, to be used as a qualitative indication of the transport properties through the molecular electronic devices obtained by incorporating these molecules into a circuit.

This can be easily understood by considering the expression for the transmission probability in the weak interaction limit ∆r,r ∆s,s − ∆∆rs,rs T (0) = 4β˜2 ∆2

(15)

where β˜L = β˜R = β˜ (symmetric connection of the molecules to the contacts). The numerators of equations (15) can be simplified through application of the Jacobi/Sylvester determinant identity 38

∆rs,rs

∆r,r ∆s,s − ∆2r,s = ∆

26


(16)

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leading to a final expression for the transmission probability in the weak interaction limit under small bias (equation (1) in the main text)

T = 4β˜2

∆2r,s ∆2

(17)

In expressions (15) and (1), all terms are only dependent on the structure of the central molecule, the influence of the contacts on the transmission is no longer apparent except for ˜ the proportionality factor β.

References (1) Sun, L.; Diaz-Fernandez, Y. A.; Gschneidtner, T. A.; Westerlund, F.; Lara-Avila, S.; Moth-Poulsen, K. Single-Molecule Electronics: From Chemical Design to Functional Devices. Chem. Soc. Rev. 2014, 43, 7378-7411. (2) Samanta, M.; Tian, W.; Datta, S.; Henderson, J.; Kubiak, C. Electronic Conduction Through Organic Molecules. Phys. Rev. B 1996, 53, R7626. (3) Chen, J.; Reed, M.; Rawlett, A.; Tour, J. Large On-Off Ratios and Negative Differential Resistance in a Molecular Electronic Device. Science 1999, 286, 1550-1552. (4) Postma, H. W. C.; Teepen, T.; Yao, Z.; Grifoni, M.; Dekker, C. Carbon Nanotube Single-Electron Transistors at Room Temperature. Science 2001, 293, 76-79. (5) Mayor, M.; Weber, H. B.; Reichert, J.; Elbing, M.; von Hänisch, C.; Beckmann, D.; Fischer, M. Electric Current Through a Molecular Rod - Relevance of the Position of the Anchor Groups. Angew. Chem. Int. Ed. 2003, 42, 5834-5838. (6) Cuniberti, G.; Fagas, G.; Richter, K. Introducing Molecular Electronics: A Brief Overview ; Springer, 2005. 27



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(7) Green, J. E.; Choi, J. W.; Boukai, A.; Bunimovich, Y.; Johnston-Halperin, E.; Delonno, E.; Luo, Y.; Sheriff, B. A.; Xu, K.; Shin, Y. S. A 160-Kilobit Molecular Electronic Memory Patterned at 1011 Bits Per Square Centimetre. Nature 2007, 445, 414-417. (8) Liu, S.; Zhang, X.; Luo, W.; Wang, Z.; Guo, X.; Steigerwald, M. L.; Fang, X. SingleMolecule Detection of Proteins Using Aptamer-Functionalized Molecular Electronic Devices. Angew. Chemie Int. Ed. 2011, 50, 2496-2502. (9) Roldan, D.; Kaliginedi, V.; Cobo, S.; Kolivoska, V.; Bucher, C.; Hong, W.; Royal, G.; Wandlowski, T. Charge Transport in Photoswitchable Dimethyldihydropyrene-Type Single-Molecule Junctions. J. Am. Chem. Soc. 2013, 135, 5974-5977. (10) Mandado, M.; Ramos-Berdullas, N. Analyzing the Electric Response of Molecular Conductors Using Electron Deformation Orbitals and Occupied-Virtual Electron Transfer. J. Comput. Chem. 2014, 35, 1261-1269. (11) Geng, Y.; Sangtarash, S.; Huang, C.; Sadeghi, H.; Fu, Y.; Hong, W.; Wandlowski, T.; Decurtins, S.; Lambert, C. J.; Liu, S.-X. Magic Ratios for Connectivity-Driven Electrical Conductance of Graphene-Like Molecules. J. Am. Chem. Soc. 2015, 137, 4469-4476. (12) Yoshizawa, K.; Tada, T.; Staykov, A. Orbital Views of the Electron Transport in Molecular Devices. J. Am. Chem. Soc. 2008, 130, 9406-9413. (13) Yoshizawa, K. An Orbital Rule for Electron Transport in Molecules. Acc. Chem. Res. 2012, 45, 1612-1621. (14) Tsuji, Y.; Hoffmann, R. Frontier Orbital Control of Molecular Conductance and its Switching. Angew. Chem. 2014, 126, 4177-4181. (15) Pickup, B. T.; Fowler, P. W. An Analytical Model for Steady-State Currents in Conjugated Systems. Chem. Phys. Lett. 2008, 459, 198-202.

28


Page 28 of 33

Page 29 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(16) Fowler, P. W.; Pickup, B. T.; Todorova, T. Z.; Myrvold, W. A Selection Rule for Molecular Conduction. J. Chem. Phys. 2009, 131, 044104. (17) Fowler, P. W.; Pickup, B. T.; Todorova, T. Z.; Myrvold, W. Conduction in Graphenes. J. Chem. Phys. 2009, 131, 244110. (18) Stuyver, T.; Fias, S.; De Proft, F.; Fowler, P. W.; Geerlings, P. Conduction of Molecular Electronic Devices: Qualitative Insights Through Atom-Atom Polarizabilities. J. Chem. Phys. 2015, 142, 094103. (19) Morikawa, T.; Narita, S.; Klein, D. Molecular Electric Conductance and Long-Bond Structure Counting for Conjugated-Carbon Nano-Structures. J. Chem. Phys. Lett. 2005, 402, 554-558. (20) Stuyver, T.; Fias, S.; De Proft, F.; Geerlings, P. The Relation Between Delocalization, Long Bond Order Structure Count and Transmission: An Application to Molecular Wires. Chem. Phys. Lett. 2015, 630, 51-56. (21) Markussen, T.; Stadler, R.; Thygesen, K. S. The Relation Between Structure and Quantum Interference in Single Molecule Junctions. Nano Lett. 2010, 10, 4260-4265. (22) Tsuji, Y.; Hoffmann, R.; Movassagh, R.; Datta, S. Quantum Interference in Polyenes. J. Chem. Phys. 2014, 141, 224311. (23) Mayou, D.; Zhou, Y.; Ernzerhof, M. The Zero-Voltage Conductance of Nanographenes: Simple Rules and Quantitative Estimates. J. Phys. Chem. C 2013, 117, 7870-7884. (24) Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press, 1997. (25) Tsoi, S.; Griva, I.; Trammell, S. A.; Blum, A. S.; Schnur, J. M.; Lebedev, N. Electrochemically Controlled Conductance Switching in a Single Molecule: Quinone-Modified Oligo (Phenylene Vinylene). ACS Nano 2008, 2, 1289-1295. 29



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(26) Danilov, A.; Kubatkin, S.; Kafanov, S.; Hedeg˚ ard, P.; Stuhr-Hansen, N.; MothPoulsen, K.; Bjørnholm, T. Electronic Transport in Single Molecule Junctions: Control of the Molecule-Electrode Coupling Through Intramolecular Tunneling Barriers. Nano Lett. 2008, 8, 1-5. (27) Tsuji, Y.; Staykov, A.; Yoshizawa, K. Orbital Control of the Conductance Photoswitching in Diarylethene. J. Phys. Chem. C 2009, 113, 21477-21483. (28) Taniguchi, M.; Tsutsui, M.; Mogi, R.; Sugawara, T.; Tsuji, Y.; Yoshizawa, K.; Kawai, T. Dependence of Single-Molecule Conductance on Molecule Junction Symmetry. J. Am. Chem. Soc. 2011, 133, 11426-11429. (29) Arroyo, C. R.; Frisenda, R.; Moth-Poulsen, K.; Seldenthuis, J. S.; Bjørnholm, T.; van der Zant, H. S. Quantum Interference Effects at Room Temperature in OPV-Based Single-Molecule Junctions. Nanoscale Res. Lett. 2013, 8, 1-6. (30) Sangtarash, S.; Huang, C.; Sadeghi, H.; Sorohhov, G.; Hauser, J.; Wandlowski, T.; Hong, W.; Decurtins, S.; Liu, S.-X.; Lambert, C. J. Searching the Hearts of GrapheneLike Molecules for Simplicity, Sensitivity and Logic. J. Am. Chem. Soc. 2015, 137, 11425-11341. (31) Tam, E. S.; Parks, J. J.; Shum, W. W.; Zhong, Y.-W.; Santiago-Berrˇ sos, M. B.; Zheng, X.; Yang, W.; Chan, G. K. L.; Abruna, H. D.; Ralph, D. C. Single-Molecule Conductance of Pyridine-Terminated Dithienylethene Switch Molecules. ACS Nano 2011, 5, 51155123. (32) Irie, M. Diarylethenes for Memories and Switches. Chem. Rev. 2000, 100, 1685-1716. (33) Trammell, S. A.; Seferos, D. S.; Moore, M.; Lowy, D. A.; Bazan, G. C.; Kushmerick, J. G.; Lebedev, N. Rapid Proton-Coupled Electron-Transfer of Hydroquinone Through Phenylenevinylene Bridges. Langmuir 2007, 23, 942-948.

30


Page 30 of 33

Page 31 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(34) Pauling, L. The Nature of the Chemical Bond ; Cornell University Press, 1960. (35) Hosoya, H. Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons. Bull. Chem. Soc. Jpn 1971, 44, 2332-2339. (36) Hosoya, H.; Hosoi, K. Topological Index as Applied to π-Electronic Systems. III. Mathematical Relations Among Various Bond Orders. J. Chem. Phys. 1976, 64, 1065-1073. (37) N. S. Ham Mobile Bond Orders in the Resonance and Molecular Orbital Theories. J. Chem. Phys. 1958, 29, 1229-1231. (38) Sylvester, J. J. XXXVII. On the Relation Between the Minor Determinants of Linearly Equivalent Quadratic Functions. Lond. Edinb. Dubl. Phil. Mag. 1851, 1, 295-305. (39) Streitwieser, A.; Heathcock, C. H.; Kosower, E. M.; Corfield, P. J. Introduction to Organic Chemistry; Macmillan New York, 1992. (40) Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford University Press: Oxford, 1989. (41) Geerlings, P.; De Proft, F.; Langenaeker, W. Conceptual Density Functional Theory. Chem. Rev. 2003, 103, 1793-1874. (42) Fias, S.; Boisdenghien, Z.; Stuyver, T.; Audiffred, M.; Merino, G.; Geerlings, P.; De Proft, F. Analysis of Aromaticity in Planar Metal Systems Using the Linear Response Kernel. J. Phys. Chem. A 2013, 117, 3556-3560. (43) Geerlings, P.; Fias, S.; Boisdenghien, Z.; De Proft, F. Conceptual DFT: Chemistry from the Linear Response Function. Chem. Soc. Rev. 2014, 43, 4989-5008. (44) Sablon, N.; De Proft, F.; Geerlings, P. The Linear Response Kernel: Inductive and Resonance Effects Quantified. J. Phys. Chem. Lett. 2010, 1, 1228-1234. 31



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(45) Sablon, N.; De Proft, F.; Geerlings, P. The Linear Response Kernel of Conceptual DFT as a Measure of Electron Delocalisation. Chem. Phys. Lett. 2010, 498, 192-197. (46) Sablon, N.; De Proft, F.; Sola, M.; Geerlings, P. The Linear Response Kernel of Conceptual DFT as a Measure of Aromaticity. Phys. Chem. Chem. Phys. 2012, 14, 3960-3967. (47) Coulson, C.; Longuet-Higgins, H. C. The Electronic Structure of Conjugated Systems. I. General Theory. Proc. R. Soc. A 1947, 191, 39-60. (48) Coulson, C.; Longuet-Higgins, H. C. The Electronic Structure of Conjugated Systems. II. Unsaturated Hydrocarbons and their Hetero-Derivatives Proc. R. Soc. A 1947, 192, 16-32. (49) Coulson, C.; Longuet-Higgins, H. C. The Electronic Structure of Conjugated Systems. III. Bond Orders in Unsaturated Molecules; IV. Force Constants and Interaction Constants in Unsaturated Hydrocarbons. Proc. R. Soc. A 1948, 193, 447-464. (50) Goyer, F.; Ernzerhof, M.; Zhuang, M. Source and Sink Potentials for the Description of Open Systems with a Stationary Current Passing Through. J. Chem. Phys. 2007, 126, 144104. (51) Ernzerhof, M. A Simple Model of Molecular Electronic Devices and its Analytical Solution. J. Chem. Phys. 2007, 127, 204709. (52) Tsuji, Y.; Staykov, A.; Yoshizawa, K. Orbital views of molecular conductance perturbed by anchor units. J. Am. Chem. Soc. 2011, 133, 5955. (53) Taylor, R. Electrophilic Aromatic Substitution; J. Wiley, 1990. (54) Carey, F. A.; Sundberg, R. J. Advanced Organic Chemistry: Part A: Structure and Mechanisms; Springer Science & Business Media, 2007. (55) Tian, W.; Datta, S.; Hong, S.; Reifenberger, R.; Henderson, J. I.; Kubiak, C. P. Conductance Spectra of Molecular Wires. J. Chem. Phys. 1998, 109, 2874-2882. 32


Page 32 of 33

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(56) Mujica, V.; Kemp, M.; Ratner, M. A. Electron Conduction in Molecular Wires. I. A Scattering Formalism. J. Chem. Phys. 1994, 101, 6849-6855. (57) Mujica, V.; Kemp, M.; Ratner, M. A. Electron Conduction in Molecular Wires. II. Application to Scanning Tunneling Microscopy. J. Chem. Phys. 1994, 101, 6856-6864. (58) Ernzerhof, M. Simple Orbital Theory for the Molecular Electrician. J. Chem. Phys. 2011, 135, 014104. (59) Fowler, P. W.; Pickup, B. T.; Todorova, T. Z. A Graph-Theoretical Model for Ballistic Conduction in Single-Molecule Conductors. Pure Appl. Chem. 2011, 83, 1515-1528.

e-

S

S

e-

X S

S

Figure 14: toc graphic

33


Back of the Envelope Selection Rule for Molecular Transmission: A

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