Supporting Information for the Close Relation Between Quantum

Supporting Information for The close relation between quantum interference in molecular conductance and diradical existence Yuta Tsuji,a Roald Hoffmann,a* Mikkel Strange,b and Gemma C. Solomonb aDepartment of Chemistry and Chemical Biology, Cornell University, Baker Laboratory, Ithaca NY 14853 bNano-Science Center and Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark

*To whom correspondence should be addressed.

E-mail: [email protected]

Table of Contents

S1. Proof of eq. 10 S2

S2. Relation between eq. 9 and eq. 11 S4

S3. Further examples of the relation between QI and diradical existence for linear chain skeletons with N = 6 and 8 S6

S4. Examples of the relation of QI and diradical stability for benzene S8

S5. Atom addition and atom removal which lead to a disjoint diradical S10

S6. A method to generate many hard-zero connections S12

S7. Transmission spectra for p-quinodimethane and benzocyclobutadiene S14

S8. Examples of the relation of QI and diradical stability for butalene and dimethylenecyclobutene S15

S9. An extension of eq. 9 to non-alternant systems S19

S10. Supplementary References S24

S1. Proof of eq. 10 The Hückel Hamiltonian matrix for an augmented molecule, H+rs, can be written as follows:

, [S1] where r < s. Let us calculate the determinant of this matrix, using cofactor expansion. The expansion along N+1th column is easy because all entries other than (r, N+1) are 0. The expansion leads to

. [S2]

The right side is the determinant of a submatrix obtained by deleting the row and column containing 1 (that is, row r and column N+1 in this case) from H+rs; called the minor. The cofactor of the (r, N+1) entry is obtained by multiplying the minor by (-1)N+1+r. It should be noted that throughout this paper we use the convention that the Hückel α is set equal to zero, and that the energy is measured in units of β, the Hückel resonance integral (which is also the transfer integral t of tight-binding theory). The determinants carry appropriate energy units, but these are not shown. th Next consider expansion along N+2 column of the original H+rs matrix. Here note that the N+1th column has already been deleted, so the previous N+2th column has become the new N+1th column. Similarly the previous sth row has become the new s-1th row. The expansion gives

. [S3]

th th We repeat the cofactor expansion with respect to N+1 and N+2 rows of the original H+rs matrix and obtain sequentially the following matrices.

. [S4]

. [S5] Note that (-1)4N+2r+2s-2 = 1. Thus, the right side is equal to the determinant of a matrix obtained by deleting the rth and sth rows and rth and sth columns from the Hamiltonian matrix for the parent molecule. Eq. S5 is the same as eq. 10.

S2. Relation between eq. 9 and eq. 11 Here we use a notation that Coulson and Longuet-Higgins used (S1), namely Δ, which denotes the characteristic polynomial of the secular determinant of the Hückel Hamiltonian matrix for the parent molecule. Δ is defined as follows: Δ(E) = det(H - EI), [S6] where I is the identity matrix. The determinant of H can be obtained by replacing E with E = 0: det(H) = Δ(0). [S7]

Let us introduce another notation, namely Δr,s, which denotes the characteristic polynomial of the secular determinant of H, where row r and column s are deleted. When rows r and s and columns r and s are deleted, such a characteristic polynomial is denoted by Δrs,rs. By using this notation, eq. S5, or eq. 10, can be rewritten as follows:

det(H+rs) = det(H-rs) = Δrs,rs(0). [S8] Note again that we omit the units that these determinants carry. By using the following determinant identity, the Jacobi/Sylvester formula (S2): ∆ ∆ − ∆2 ∆ = r,r s,s r,s , [S9] rs,rs ∆ we can obtain

2 ∆(0)∆rs,rs (0) = −∆r,s (0). [S10]

Note that Δr,r(0) and Δs,s(0) are zero. Since the parent molecule must be a closed-shell even alternant hydrocarbon, the reduced molecules obtained by deleting the rth atom or sth atom, whose determinant can be denoted by Δr,r(0) or Δs,s(0), must be monoradicals. They have a NBMO at E = 0. That is why Δr,r(0) = 0 and Δs,s(0) = 0. Here we introduce another notation, namely det(H)rs, to denote the determinant of H, where the row r and column s are deleted. By using this notation and eqs S7 and S8, eq. S10 can be rewritten as follows:

rs 2 det(H)det(H +rs ) = det(H)det(H −rs ) = −[det(H) ] . [S11] Dividing this equation by [det(H)]2 leads to

2 det(H ) det(H ) det(H)rs  +rs = −rs = −  . [S12] det(H) det(H)  det(H) 

Since Cramer’s rule gives (-1)r+sdet(H)rs/det(H) = -G(r, s) (S2, S3 ), we can reach eq. 9 by a different path. For alternant hydrocarbons without 4n-membered rings, the relation between the

determinant of its Hückel Hamiltonian matrix and the number of Kekulé structures, K, can be described by the following form (S4): det(H) = (-1)N/2K2. [S13] By using this equation, eq. S12 can be rewritten as follows:

2 2  K   K  [G(r, s)]2 =  +rs  =  −rs  , [S14]  K   K 

th th where K+rs is the number of Kekulé structures when two atoms are attached to the r and s th th atoms, and K-rs is the number of Kekulé structures when the r and s atoms are deleted. Given that the conductance between the rth and sth atoms is approximated as proportional to [G(r, s)]2, we derive eq. 12 of the main text.

S3. Further examples of the relation of QI and diradical stability for linear chain skeletons with N = 6 and 8

Fig. S1. Comparison for hexatriene (N = 6) between the conductance when two electrodes are attached as indicated (QI implies a quantum interference feature, a check mark normal transmission), and whether upon the attachment of two carbon atoms one gets a diradical (check) or a closed shell molecule (X).

Fig. S2. Comparison for octatetraene (N = 8) between the conductance when two electrodes are attached as indicated (QI implies a quantum interference feature, a check mark normal transmission), and whether upon the attachment of two carbon atoms one gets a diradical (check) or a closed shell molecule (X).

S4. Examples of the relation of QI and diradical stability for benzene There are three different ways to attach electrodes to benzene: 1,2 (ortho), 1,3 (meta), and 1,4 (para), as shown in Fig. S3. It is experimentally (S5, S6) and theoretically (S7, S8) well- established that the meta-connection leads to QI (easy zero), while the para-connection shows significant transmission (see Fig. S4). Conductance through the ortho-connection is not experimentally observed due to geometrical problems in attaching electrodes, but theoretically is predicted to have comparable transmission to the para connection (S7). The resulting augmented molecules are the quinodimethanes (QDMs). The 1,2 (ortho), 1,3 (meta), and 1,4 (para) cases correspond to o-, m-, and p-QDMs, respectively. They are well-studied molecules in the diradical literature, as many biradicaloid molecules are characterized by the presence of QDM substructures of the three types (S9, S10). The ground spin states for o- and p-QDMs are singlets, whereas m- QDM has a triplet ground state (S11).

Fig. S3. Comparison between the conductance and diradical existence in the process of attaching two carbon atoms to benzene. The numbers of the attached atoms are indicated in red. To distinguish unpaired electrons from asterisks, the unpaired electrons are indicated in blue.

Fig. S4. Computed transmission spectra for 1,2-, 1,3-, and 1,4-connections in benzene.

S5. Atom addition and atom removal which lead to a disjoint diradical The naphthalene case shown in Fig. 6 gave us an example of non-disjoint diradical formation on deletion of atoms. For a linear chain, deleting two atoms between which hard-zero QI occurs always leads to a chain fragmentation into two monoradicals. Examples in the literature for atom removal which leads to a disjoint diradical are few. One such example is found in the open form of a diarylethene, shown in Fig. S5. One of the authors and coworkers (S12, S13) have shown that the 3,10-connection of this molecule shows a large ON/OFF ratio approaching to three orders of magnitude due to the occurrence of QI in the open form. The 3,10-connection is classified as a hard-zero case. The resulting diradicals from atom addition and removal are both disjoint diradicals.

Fig. S5. Photoisomerization of 1,2-di(2-methyl-1-naphthyl)perfluorocyclopentene (top) and atom addition/removal to/of C3 and C10 of the π-conjugated skeleton of the open form (bottom).

Another example is stilbene, which is also a kind of diarylethene. Two of the authors and coworkers (S14) have shown that QI occurs in the 3,3’-connection of electrodes to stilbene, which is classified as a hard-zero case. As shown in Fig. S6, the diradicals resulting from atom addition and removal are both disjoint diradicals.

S10

Fig. S6. Atom addition/removal to/of C3 and C3’ of stilbene.

S11

S6. A method to generate many hard-zero connections In the NBMO of odd alternant hydrocarbons, the carbon atoms can be divided into two groups: starred (denoted by “*”), and unstarred atoms (denoted by “0”). The starred atoms must outnumber the unstarred atoms by one. There are two possible connections between two odd alternant hydrocarbons, which result in a diradical, as shown in Fig. S7. One is the (0-0) linkage that connects two unstarred (or starred) carbons. The other one is the (0-*) linkage that connects a starred atom to an unstarred one. Aoki and Imamura (S15, S16) showed that the 0-0 linkage leads to a disjoint diradical while the 0-* linkage leads to a non-disjoint diradical.

Fig. S7. Schematic representation of the 0-0 linkage (left) and 0-* linkage (right).

Fig. S8 shows examples for odd alternant hydrocarbons (monoradicals). There are various kinds of monoradicals. Hence, 0-0 linkages between any two of them result in a huge number of disjoint diradicals as well as connections that show hard-zero QI.

S12

Fig. S8. Examples for odd alternant hydrocarbons (monoradicals).

Let us take a combination of allyl and benzyl radicals as an example. Fig. S9 shows a possible 0-0 linkage between them. In the resulting diradical, one unpaired electron is located on the starred atoms while the other unpaired electron is located on the unstarred atoms due to the disjoint nature.

Fig. S9. A possible 0-0 linkage between allyl and benzyl radicals.

To annihilate the unpaired electrons and generate a closed-shell molecule, which will be a parent molecule for atom removal and addition, let us attach one atom to a starred atom and another atom to an unstarred atom. An example for such a process is shown in the left of Fig. S10. If electrodes are connected to the attached atoms, hard-zero QI can be observed. In common with Figs S5 and S6, both atom addition and atom removal for the attached atoms lead to disjoint diradicals, as shown on the right of Fig. S10.

Fig. S10. Parent molecule obtained by attaching one atom to a starred atom and another atom to an unstarred one (left). Connection leading to hard-zero QI (middle). Processes of atom addition and atom removal for the connection sites lead to disjoint diradicals (right).

S13

S7. Transmission spectra for p-quinodimethane and benzocyclobutadiene

Fig. S11. Computed transmission spectra for 2,3-, 2,5-, and 2,6-connections in p-quinodimethane.

Fig. S12. Computed transmission spectrum for 3,6-connection in benzocyclobutadiene.

S14

S8. Examples of the relation of QI and diradical stability for butalene and dimethylenecyclobutene derivatives We investigated all possible connection patterns for butalene and dimethylenecyclobutene. Fig. S13 shows the possible substitution patterns on the butalene skeleton. Although the 2,3-case shows QI due to the hard zero in the transmission spectra (see Fig. S14), it has a Kekulé or Lewis structure implying a closed-shell electronic configuration. However, disjoint-type NBMOs exist for this connection (see Fig. S15a). On the other hand, the easy-zero case has a structure with unpaired electrons.

Fig. S13. Comparison between the conductance and diradical existence in the process of attaching two carbon atoms to butalene. The arrows show attachment of electrodes in a molecular transmission experiment (top). The primary contributor in a VB description of the corresponding potential diradical (bottom).

S15

Fig. S14. Computed transmission spectra for 2,3-, 2,5-, and 2,6-connections in butalene.

Fig. S15. Two-fold degenerate NBMOs in a 2,3-augmented butalene (a), and those of the 5,6-CH2- substituted dimethylenecyclobutene (b).

Fig. S16 shows the possible substitution patterns on the dimethylenecyclobutene skeleton. The situation is similar to that of butalene. The 5,6-expanded hydrocarbon case shows a QI feature in the transmission spectra, related to the hard zero (see Fig. S17). Yet the valence structure suggests a closed-shell electronic configuration. Nevertheless, disjoint-type NBMOs exist for this connection (see Fig. S15b). In the easy-zero cases, the structures have unpaired electrons. As for deleting two atoms, between which hard-zero QI occurs, from benzocyclobutadiene, butalene, and dimethylenecyclobutene. A cyclobutadiene always appears.

S16

Fig. S16. Comparison between the conductance and diradical existence in the process of attaching two carbon atoms to dimethylenecyclobutene. The arrows show attachment of electrodes in a molecular transmission experiment (top). The primary contributor in a VB description of the corresponding potential diradical (bottom).

S17

Fig. S17. Computed transmission spectra for 1,2-, 1,5-, 1,6-, 5,5-, and 5,6-connections in dimethylenecyclobutene.

S18

S9. An extension of eq. 9 to non-alternant systems In the first stages of the proof of the correlation between quantum interference and diradical existence, we do not assume that the parent molecule must be an alternant hydrocarbon. Our only assumption is that the parent molecule must contain an even number of carbon atoms and have a closed-shell electronic structure. Since it is probably impossible to generate a π diradical consisting of an odd number of π-conjugated carbon atoms, this assumption is necessary. When we calculate the determinant of eq. 8, that is the first place that it is assumed that the parent molecule must be an alternant hydrocarbon. Suppose this is not so. If the parent molecule is a non-alternant hydrocarbon, the determinant of eq. 8 is G(r, r)G(s, s) – [G(r, s)]2. Therefore, we can rewrite eq. 9 for non-alternant hydrocarbons as

2 det(H +rs ) = det(H){G(r,r)G(s, s)− [G(r, s)] }. [S15] When an r,s-connection leads to QI, G(r, s) = 0, but G(r, r)G(s, s) is not necessarily 0 for non- alternant hydrocarbons. Therefore, det(H+rs) is not necessarily 0 either. Even if det(H+rs) is 0, the resulting molecule is not necessarily a diradical. This is because in non-alternant hydrocarbons the Coulson-Rushbrooke pairing theorem, which ensures that there are at least two NBMOs when the determinant is 0, no longer holds true. Furthermore, there is a situation where det(H+rs) is 0, even if QI does not occur, namely G(r, s) ≠ 0. This happens when G(r, r)G(s, s) = [G(r, s)]2. When r = s, this is always true. Also, this might happen accidentally. Therefore, in non-alternant hydrocarbons the correlation between QI and diradical existence does not always hold true, but, as eq. S15 implies, there is a correlation between the two. Let us look at some examples. Two of us and co-workers (S17) previously showed that in simple odd-membered monocyclic molecules, QI does not occur. Also two of us and co-workers (S18) recently investigated methylenecyclopropene, where QI was not observed. Fulvene may be the smallest non-alternant system which shows QI. Fig. S18 shows some possible substitution patterns on the fulvene skeleton. Fig. S19 shows their transmission spectra. Since in non-alternant hydrocarbons the energy levels are no longer symmetric with respect to E = 0, the transmission spectra are also no longer symmetric. As shown in Fig. S18, we can see a good correlation between QI and diradical existence. In this case G(r, r) = 0 except for r = 6, so det(H+rs) is 0 when QI occurs. However, note that det(H+rs) = 0 does not necessarily ensure that the resulting molecule is a diradical, so in this case the correlation might be accidental. The 6,6-connection is an interesting case. Since G(6, 6) ≠ 0, QI does not occur. Yet its Kekulé structure implies diradical existence. However, this is not so. Although the determinant is 0 due to r = s, there is only one NBMO at E = 0, which corresponds to the LUMO, and the HOMO-LUMO gap is 0.5|β| at the Hückel level. In this case the “normal” Kekulé structure is misleading.

S19

Fig. S18. Comparison between the conductance and diradical existence in the process of attaching two carbon atoms to fulvene. The arrows show attachment of electrodes in a molecular transmission experiment (top). The primary contributor in a VB description of the corresponding potential diradical (bottom).

Fig. S19. Computed transmission spectra for 2,3-, 2,4-, 2,5-, 2,6-, 3,4-, 3,6-, and 6,6-connections in fulvene.

S20

The next example is azulene, whose QI features have recently attracted much attention (S18, S19, S20, S21). Fig. S20 shows the possible substitution patterns on the azulene skeleton; Fig. S21 shows their transmission spectra. Interestingly, all the transmission spectra without QI have the same transmission probability at E = 0. There are five connections which shows QI, but the 1,4-, 1,6-, and 1,8-augmented molecules have a closed-shell Kekulé structure. In the VB description we found five diradical structures, but the 1,3-, 4,6-, and 4,8-augmented molecules are not diradicals at the Hückel level. They have a single NBMO at E = 0 and substantial HOMO- LUMO gaps, ranging from 0.29|β| to 0.36|β|. The 2,5- and 5,7-augmented molecules are classical diradicals with two NBMOs at E = 0. Interestingly, these connections show QI. Except for these two augmented molecules there are no other pure diradicals in this class of molecules. Judging from this limited exploration, even in non-alternant hydrocarbons at least a one- way relationship from diradical to QI, i.e., using diradical characters to predict QI, might be valid. We will report separately further investigation of this phenomenon, using a graphical method.

S21

Fig. S20. Comparison between the conductance and diradical existence in the process of attaching two carbon atoms to azulene. The arrows show attachment of electrodes in a molecular transmission experiment (top). The primary contributor in a VB description of the corresponding potential diradical (bottom).

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Fig. S21. Computed transmission spectra without QI at E = 0 (left) and those with QI at E = 0 (right) in azulene.

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S10. Supplementary References

S1. Coulson C A, Longuet-Higgins HC (1947) The electronic structure of conjugated systems I. general theory. Proc R Soc Lond A 191(1024):39-60.

S2. Stuyver T, Fias S, De Proft F, Fowler PW, Geerlings P (2015) Conduction of molecular electronic devices: Qualitative insights through atom-atom polarizabilities. J Chem Phys 142(9):094103.

S3. Markussen T, Stadler R, Thygesen KS (2010) The relation between structure and quantum interference in single molecule junctions. Nano Lett 10(10):4260-4265.

S4. Graovac A, Gutman I, Trinajstić N, Živković T (1972) Graph theory and molecular orbitals application of Sachs theorem. Theoret chim Acta (Berl) 26(1):67-78.

S5. Kiguchi M, Nakamura H, Takahashi Y, Takahashi T, Ohto T (2010) Effect of anchoring group position on formation and conductance of a single disubstituted benzene molecule bridging Au electrodes: change of conductive molecular orbital and electron pathway. J Phys Chem C 114(50):22254–22261.

S6 . Arroyo CR, et al. (2013) Signatures of quantum interference effects on charge transport through a single benzene ring. Angew Chem 125(11):3234-3237.

S7. Hansen T, Solomon GC, Andrews DQ, Ratner MA (2009) Interfering pathways in benzene: An analytical treatment. J Chem Phys 131(19):194704.

S8. Koga J, Tsuji Y, Yoshizawa K (2012) Orbital control of single-molecule conductance perturbed by π-accepting anchor groups: Cyanide and isocyanide. J Phys Chem C 116(38):20607–20616.

S 9 . Flynn CR, Michl J (1974) π,π-Biradicaloid hydrocarbons: o-Xylylene. Photochemical preparation from 1,4-dihydrophthalazine in rigid glass, electric spectroscopy, and calculations. J Am Chem Soc 96(10):3280-3288.

S10. Shimizu A, Nobusue S, Miyoshi H, Tobe Y (2014) Indenofluorene congeners: Biradicaloids and beyond. Pure Appl Chem 86(4):517–528.

S11. Lahti PM ed. (1999) Magnetic Properties of Organic Materials (Marcel Dekker, New York).

S12. Tsuji Y, Staykov A, Yoshizawa K (2009) Orbital view concept applied on photoswitching systems. Thin Solid Films 518(2):444-447.

S13. Tsuji Y, Staykov A, Yoshizawa K (2009) Orbital control of the conductance photoswitching in diarylethene. J Phys Chem C 113(52):21477-21483.

S14. Pedersen KGL, et al. (2014) Quantum interference in off-resonant transport through single molecules. Phys Rev B 90(12):125413.

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S15. Aoki Y, Imamura A (1992) A simple treatment to design NBMO degenerate systems in alternant and non-alternant hydrocarbons. Theor Chim Acta 84(3):155-180.

S16. Aoki Y, Imamura A (1999) A simple rule to find nondisjoint NBMO degenerate systems for designing high-spin organic molecules. Int J Quant Chem 74(5):491-502.

S17. Movassagh R, Strang G, Tsuji Y, Hoffmann R (2015) The exact form of the green's function of the Hückel (tight binding) model. arXiv preprint arXiv:1407.4780.

S18 . Pedersen KGL, Borges A, Hedegård P, Solomon GC, Strange M (2015) On the illusory connection between cross-conjugation and quantum interference. J Phys Chem C ASAP.

S19. Xia J, Capozzi B, Wei S, Strange M, Batra A, Moreno JR, Amir RJ, Amir E, Solomon GC, Venkataraman L, Campos LM (2014) Breakdown of interference rules in azulene, a nonalternant hydrocarbon. Nano Lett 14(5): 2941–2945.

S20. Stadler R (2015) Comment on “Breakdown of interference rules in azulene, a nonalternant hydrocarbon”. Nano Lett ASAP.

S21. Strange M, Solomon GC, Venkataraman L, Campos LM (2015) Reply to “Comment on ‘Breakdown of interference rules in azulene, a nonalternant hydrocarbon’”. Nano Lett ASAP.

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