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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 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 S2 . [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. S3 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 S4 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. S5 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). S6 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). S7 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. S8 Fig. S4. Computed transmission spectra for 1,2-, 1,3-, and 1,4-connections in benzene. S9 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”).
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