Close Relationships between NMR J-Coupling Alternation (JCA) and

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Quantum Electronic Structure

Close Relationships between NMR J-Coupling Alternation (JCA) and Molecular Properties of Carbon Chains João Paulo Cavalcante Oliveira, and Roberto Rivelino J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b01141 • Publication Date (Web): 07 Feb 2019 Downloaded from http://pubs.acs.org on February 8, 2019

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Journal of Chemical Theory and Computation

Close Relationships between NMR J-Coupling Alternation (JCA) and Molecular Properties of Carbon Chains João P. C. Oliveira†,§,* and Roberto Rivelino†,* †

Instituto de Física, Universidade Federal da Bahia, 40210-340 Salvador, Bahia, Brazil

§

Centro de Ciência e Tecnologia em Energia e Sustentabilidade, Universidade Federal do Recôncavo da Bahia, 44085-132 Feira de Santana, Bahia, Brazil

1 ACS Paragon Plus Environment

Journal of Chemical Theory and Computation 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

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ABSTRACT. We propose a J-coupling alternation (JCA) value that is demonstrated to be a suitable parameter to evaluate the nuclear magnetic resonance (NMR) indirect spin– spin coupling constants (SSCCs) as a function of molecular properties of chains, by increasing their length. As an application, we report a theoretical study of the SSCCs for the interactions between neighbor nuclei in increasingly patterned carbon chains, within density functional theory. First, we examine the J-coupling constants between 1H and 13C nuclei (nJHC) considering the separation distance, as well as between two adjacent

13

C

nuclei (1JCC) considering their relative positions in polyynes and cumulenes. Further, we define and determine JCA in terms of the differences of 1JCC, which is investigated as a function of several molecular properties; e.g., cohesive energy, characteristic vibrational frequency, average polarizability, and energy gap of the systems. We also determine JCA for other types of carbon chains, such as diphenyl-capped-polyynes, polyacetylene and polythiophene. The behavior of JCA as a function of the energy gap may be related to highly -conjugated low band-gap carbon chains. Overall, JCA correlates very well with the electronic properties of these chains, especially with their energy gap, exhibiting positive values for pristine polyyne and polythiophene and negative values for pristine cumulene and plyacetylene. These findings indicate an alternative way to establish an appropriate SSCC descriptor that characterizes the electronic nature of the system, such as the proposed JCA value averaging the whole system, instead of using only the individual J-coupling values to give insights into the properties of large and extended systems.

Keywords: J-Coupling, JCA, DFT, Polyynes, Cumulenes, Polyacetylene, Polythiophene


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1. INTRODUCTION Atomic information from nuclear magnetic resonance (NMR) spectroscopy experiments is extremely important for a global understanding of the electronic structure in molecular systems1-3 as well as in solid state4-6. Along with other NMR properties, indirect spin-spin coupling constants (SSCCs) have long been recognized to give useful values for describing molecular conformations.7 Nowadays, high resolution J-coupling NMR spectra have successfully been employed for studying biological properties of several materials.8-11 As is well-known, indirect SSCCs depend critically on the electronic structure of the bonds between the two nuclei involved in the coupling and are very sensitive to the molecular shape.12 Indeed, the sign and magnitude of these magnetic nuclear coupling constants can unveil features of the electron density around the nuclei. All these findings are appealing to perform detailed studies on the NMR J-couplings in connection with electronic structure for a wide variety of systems. From the theoretical viewpoint, accurate calculations of the SSCCs are also essential to help elucidating chemical structures and have been developed along with ab initio methods over the last 20-30 years13,14, as thoroughly reviewed by Helgaker et al.15 Also, early calculations of SSCCs within density functional theory (DFT) date from that period.16,17 Recently, DFT calculations of NMR properties can predict J-coupling with spectroscopic accuracy for several molecular systems.18-21 In this sense, magnetic properties of large molecules have properly been assessed using DFT12,22 and new computational developments are still ongoing.23-25 Hence, combining calculations of nuclear magnetic properties with DFT appears to consolidate a method of low-to-medium computational cost to investigate large and extended systems starting from accurate electron densities.



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An interesting class of extended materials whose the electronic properties can be assessed using NMR spectroscopy experiments is that of the one-dimensional (1D) linear carbon chains.26-30 These 1D systems may be stabilized by choosing proper termination groups31-35 or confining them inside multiwalled carbon nanotubes,36-38 and have been extensively studied for different purposes.39,40 However, relationships between the molecular properties of these 1D linear carbon chains and their nuclear magnetic properties, such as SSCCs, are scarce in the literature.27 Partly, this is due to the oscillatory dependence of the J-coupling values on nuclei pairs, hampering a direct analysis over the properties of the entire system. Similarly, this problem remains for other types of carbon chains like conjugated low band-gap 1D polymers, as in the cases of polyacetylene41 and polythiophene.42 In this sense, instead of using simple average values of J-couplings to characterize long chains, we define a J-coupling alternation (JCA) value that is useful as a unique scalar parameter and may be easily related with other properties of large molecular systems, as one increases the chain length. The main aim of this paper is to establish useful relationships between scalar Jcoupling constants and electronic properties of different types of carbon chains, which may aid understanding how electron delocalization can modulate NMR indirect couplings. As a practical example, we assess molecular properties of pristine cumulenes (=C=C=) and polyynes (–CC–), containing until 20 carbon atoms in the structure. Additionally, we investigate the behavior of JCA as a function of the energy gap for diphenyl capped polyynes (Ph[n]), including its oxidized form (O=Ph[n])35 and sulfurized analogue (S=Ph[n]), small polyacetylene (H(CH=CH)nH) and polythiophene (PTh) chains. All properties of these systems were calculated within DFT levels of approximation using the oligomer approach, which is a useful method to study the electronic properties of finite polymeric chains.41


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First, we examine the oscillatory behavior of the usual J-coupling constants in polyynes and cumulenes: (i) between 1H and

13

C nuclei considering their separation

distance and (ii) between two 13C nuclei considering their relative positions in the chains. Second, we define a formula to compute JCA values, in terms of the differences of 1JCC for adjacent C–C bonds, and investigate it as a function of properties, such as cohesive energy per atom,34 zero-point-vibrational energy (ZPVE), characteristic vibrational frequency, average polarizability, and energy gap, by increasing the chain length. Third, we show that our JCA definition is directly applicable to other types of conjugated carbon chains, such as polyacetylene and polythiophene, leading to interesting correlations with the calculated energy gap of these systems. Furthermore, a theoretical benchmark study including other electronic structure methods and calculations of NMR properties of related systems, using DFT and different basis sets, as well as additional data and references related to the present calculations are reported in the Supporting Information (SI) associated to this paper.

2. METHODS AND COMPUTATIONAL DETAILS To perform this study we use a good computational compromise between electronic structure and NMR properties. Optimized geometries, electronic structure, harmonic vibrational frequencies, polarizabilities, and NMR properties of cumulenes and polyynes have been calculated using traditional DFT methods43 combined with different Gaussian basis sets, as implemented in the Gaussian09 suite of programs44 and in the EMSL Basis Set Library.45,46 Benchmark calculations (some of them given in the SI) indicate that the B3LYP,47,48 PW9149 and PBE50 functionals describe similarly equilibrium geometries, energetics, and vibrational frequencies of small linear carbon chains, combined with different basis sets (see Tables S1–S4). However, after this analysis, we choose only the



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PBE scheme for optimizing the structures of diphenyl-capped polyynes (and derivatives), polyacetylene, and polythiophene, because of the better computational performance of this functional for these systems. Usually, B3LYP is known to be an accurate scheme for the evaluation, for example, of hyperfine coupling constants.51 However, depending on the convergence criteria, any DFT approach may present failures to describe linear systems or chains as pointed out by Woodcock et al.52 Hence, when accurate energetics are required, it is necessary an appropriate choice of the DFT scheme combined with the basis set to perform systematic calculations (see also Ref.S6 in the SI). For example, some difficulties in obtaining optimized geometries and vibrational spectra using traditional methods (without applying stringent convergence criteria, ultrafine grids, or considering symmetry breaking) are reported in Tables S2–S4 for some linear carbon chains. Most importantly, for our purpose here, PBE has presented a rapid convergence of energy for different types of carbon chains, when combined with the Pople basis sets family.45,46 Moreover, PBE is recognized as a DFT scheme largely utilized for solid-state materials and extended systems.30,32 For the calculations of the indirect SSCCs, we have initially adopted the modified Pople-type basis sets as recommended by Kjær and Sauer.53 In this case, geometry optimizations were performed using B3LYP/6-31+G(d,p), and J-couplings were calculated with B3LYP/6-311+G(d,p)-J and also with B3LYP/aug-cc-pVTZ-J (see basis set nomenclature defined in Refs.53,54) for H2CnH2 (2  n  6). Some results are reported in Tables S5 and S6. As will be discussed in the following, we are interested here in obtaining simultaneously converged structural, vibrational, and NMR properties of large carbon chains (and chains with different end groups), so that the usual levels of DFT for NMR calculations would demand a higher computational cost. On the other hand, the


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PBE level of theory has been successfully employed to address structural, electronic, vibrational, and NMR properties of these 1D systems, as well as of other related lowdimensional systems.32,36,55 Thus, the present proposed method, PBE/6-31G(d), has been found as the best compromise between the computational cost for larger chains and accuracy for both NMR and electronic properties. Importantly, the choice of the level of calculations does not invalidate the systematic analysis we intend to do in this work.

3. RESULTS AND DSCUSSION 3.1. Definition of the J-Coupling Alternation (JCA) Value. As is known1,7, scalar J-couplings contain information about structure and chemical connectivity of molecules. Furthermore, electronic structure modulates these scalar couplings, indicating a direct correlation between them and other molecular properties. Hence, in the case of carbon chains, in which the 1JCC values exhibit an alternating profile, it is more useful to define a JCA value in terms of the differences of 1JCC for adjacent C–C bonds. This is, for a carbon chain of the type −C𝑖 − C𝑖+1 − C𝑖+2 ⋯ C𝑖+𝑘 − C𝑖+𝑘+1 ⋯ C2𝑛 − we set 𝑛 −1

𝑛

𝐽𝐶𝐴 = [

2 𝐽( ∑𝑖=1 𝐶

2𝑖−1 ,𝐶2𝑖

𝑛/2

)

−

2 ∑𝑖=1 𝐽(𝐶2𝑖 ,𝐶2𝑖+1 ) ] 𝑛 −1

(1)

2

As noticed, eq. 1 sums algebraically over all the 1JCC coupling values of the system, including the terminal nuclei, giving an appropriate average alternation value for the SSCCs. We notice that JCA may be defined for distinct atomic chains with different end groups, including for intermolecular chains. Therefore, in the same sense that the bond length alternation (BLA) has been investigated for several chains,35,56 JCA may be used as a unique variable or a function to investigate the behavior of other molecular properties calculated by increasing the length of the chain. We start analyzing the oscillatory behavior of the J-couplings for the carbon chains. Some complementary results are presented in the Supporting Information (e.g., 7 ACS Paragon Plus Environment


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see Figure S2 and S3). Further, in the cases of cumulenes and polyynes, we determine JCA as a function of the number of carbon atoms, cohesive energy per atom (see eq. 2 below),34 characteristic vibrational frequency, ZPVE, average polarizability, and HOMOLUMO energy gap of the chains. 𝐸𝑐𝑜ℎ/𝑎𝑡 = −(𝐸𝑡𝑜𝑡𝑎𝑙 − ∑𝑖 𝑛𝑖 𝐸𝑖 )/ ∑𝑖 𝑛𝑖

(2)

In eq. (2), 𝐸𝑡𝑜𝑡𝑎𝑙 is the total energy of the optimized chain, 𝐸𝑖 is the energy of an isolated atom, and 𝑛𝑖 is the corresponding number of atom of a particular element in the chain. Additionally, we analyze the behavior of JCA as a function of the energy gap for polyynes containing distinct end groups, as well as for polyacetyene and polythiophene small chains. We discuss first the behavior of the J-couplings for polyynes and cumulenes. 3.2. J-Coupling Constants and JCA Correlations for Polyynes and Cumulenes. In Figure 1a (1b), we display all the calculated spin-spin coupling constants, JHC (i. e., 1JHC, 2JHC, …) between 1H and 13C nuclei for polyynes (cumulenes), which is a function of the relative position of the carbon nuclei, with different chain lengths. It is experimentally26 known that for polyynes the JHC coupling constants rapidly decay by increasing the distance between the nuclei. This finding has been also confirmed by Haque et al.27 and is attributable to the localized character of the 1s orbitals of the hydrogen nuclei. As seen in Figure 1a, the most significant JHC couplings in polyynes appear for the interaction with the first and second carbon atoms near the terminal hydrogen atom (1JHC and 2JHC, respectively). However, after the forth nearest neighbor, the long-range (n>4JHC) couplings exhibit small oscillations, i.e., alternating mild negative and positive values (see Table S7) until vanishing completely for very long chains. However, it is important to mention here that large long-range indirect SSCCs are expected for F–F nuclear spin couplings in fluorine-capped polyynes,57 even for distances of few nanometers. Overall, for the present cases, the JHC values are rather similar


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considering polyynes with different lengths, depending only on the relative position between 13C and 1H nuclei. For cumulenes, we notice larger oscillations in the long-range JHC couplings, varying between negative and positive values (see Table S8), as compared to the corresponding values of polyynes, even considering long chains such as C20H4. To obtain the complete JHC profile of cumulenes (Figure 1b), we have considered the average value of the couplings between the two 1H terminal atoms. Now, the second nearest neighbor to the terminal atoms (2JHC) exhibits a negative value (varying from –5.5 Hz in C4H4 to – 11.3 Hz in C20H4), whereas 2JHC in polyynes is almost constant giving an average value of 39.3 Hz. Interestingly, for

n>2

JHC in cumulenes the values pass to strongly oscillate

(varying between 6.3 Hz and –6.2 Hz for the farthest neighbors in C20H4). Moreover, the magnitude of the J-coupling constants of more distant pairs systematically increases with the growing of the chain. For example, the 7JHC value exhibits an increase of ~70% from C8H2 to C20H2. Interestingly, the 9JHC coupling in C14H4 (6.4 Hz) is also slightly larger than 7JHC coupling in C8H4 (5.5 Hz). These findings indicate that the JHC coupling constants profile in cumulenes expresses the effect of their uniform electron density as the chain increases. For completeness of this discussion on the individual values of the Jcouplings, we summarize all the calculated JHC values for polyynes and cumulenes in Tables S7 and S8.



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Figure 1. Calculated nJHC coupling constants for (a) polyynes and (b) cumulenes. The labels C1, C2, C3, …, C20 correspond to the 1st, 2nd, 3rd,…, 20th nearest neighbor 13C nuclei relative to the terminal 1H nucleus, within the PBE/6-31G(d) level.

To obtain a detailed feature of the NMR spectra of the carbon chains and gain more information about the electron distribution in these systems, it is also important to analyze the J-coupling constants for the interaction between two adjacent 13C nuclei (1JCC) at different positions of the chain. In Figure 2a (2b), we depict our calculated 1JCC values for polyynes (cumulenes) as a function of the relative position of the nearest neighbor carbon pairs (i,j). In the case of polyynes, the 1JCC couplings exhibit a zigzag profile, which depends weakly on the size of the chain length. A very similar behavior to this case is also noticed for 1JCC couplings in polyacetylene and polithiophene chains, which are conjugated systems (see Figures S2 and S3). In polyynes, because of the bonding pattern, alternating single and triple bonds, 1JCC couplings of the extremities exhibit the largest values (varying from 231 Hz in C4H2 to 226 Hz in C20H2), since the outermost carbon pairs are of the CC type. On the contrary, the couplings between C–C bonds exhibit the smallest values (varying from 155 Hz in C4H2 to 162 Hz in C20H2) in the 1JCC series. For cumulenes, the 1JCC couplings varies dramatically with the relative position between two adjacent nuclei. In the extremities, the calculated couplings exhibit the smallest values (varying from 113 Hz in C4H2 to 121 Hz in C20H2). The zigzag pattern is now observed only for large chains, although exhibiting smaller oscillations as compared 10 ACS Paragon Plus Environment

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to the behavior the 1JCC series in polyynes (Figure 2a). The typical behavior of these couplings in cumulenes is displayed in Figure 2b. Additionally, all values of 1JCC are given in Table S9 and S10 for polyynes and cumulenes, respectively.

Figure 2. Calculated 1JCC coupling constants for two adjacent carbon pairs (i,j) for (a) polyynes and (b) cumulenes, within the PBE/6-31G(d) level. From the above analysis of the 1JCC series in polyynes and cumulenes (and the corresponding analysis of polyacetylene and polythiophene displayed in the SI), we observe, in general, an alternation of the SSCC values that depends on the chain size and the conjugation along the chains. For this reason, it is more interesting to quantify the alternation degree of the couplings for these structures. A possibility to do that is defining JCA to describe these systems, as given by eq. 1, for the chains exhibiting an alternating profile in the adjacent J-couplings. Here, we obtain a type of average value for couplings between two adjacent carbon nuclei, which allows examining its direct dependence on other molecular properties. As we shall see, this quantity exhibits a better correlation with other molecular properties than when BLA is used.35,56 11 ACS Paragon Plus Environment


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In Figure 3a (3b), we display the behavior of JCA as a function of the number of carbon atoms in the chains of polyyne (cumulene), in comparison with the behavior of the average value of all JHC coupling constants, i.e., JHC. As seen, both JCA and JHC exhibit convergences by growing the chains, with JCA values decreasing for polyynes and increasing for cumulenes. Indeed, JCA exhibits positive values for polyynes and negative values for cumulenes, whereas JHC gives positive values in both cases. For instance, in the cases of long polyynes, such as C20H2, JCA = 52.6 Hz, and long cumulenes, such as C20H4, JCA = –16.5 Hz. It is important to mention here that we have also found that JCA exhibits systematically positive values for polyacetylene and negative values for polythiophene, by increasing the chain length (to be discussed further). This possibly indicates that the sign of JCA can be related to the degree of electron delocalization in highly conjugated carbon chains. We also discuss the behavior of JCA with the energy gap for these cases. Now, we examine more deeply the relationship between JCA and JHC in Figures 3c (3d) plotting the correlation for polyynes (cumulenes). Qualitatively, we notice a fair correlation between these two parameters for these systems, strikingly behaving linearly (with adjusted R-squared of 100%) in the case of cumulenes. In this sense, it is useful to exploit JCA as a characteristic function of the molecular properties of these chains, instead of using, for example, BLA that in many cases does not exhibit a fair correlation35 with the molecular property of interest.


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Figure 3. Profiles of JCA and JHC as a function of the chain length (expressed in terms of the carbon number, n) for (a) polyynes and (b) cumulenes. Correlation functions between JCA and JHC for (c) polyynes and (d) cumulenes. As discussed above, the values of 1JCC (see Figure 2) are significant albeit exhibit profiles with prominent variations for polyynes and cumulenes, as well as for conjugated polymers (see Figures S2 and S3). This hampers to find a direct correlation of these couplings with other molecular properties of the chains. Thus, JCA should play an important role in the direction of correlating a unique SSCC parameter with electronic structure of the carbon chains. Consequently, the alternation of 1JCC, expressed as JCA, may be useful to aid elucidating chemical structures and function of different types of chains. For example, here it is seen that JCA exhibits patterns that allow characterizing different families of carbon chains. Furthermore, the JCA behavior indicates that the interaction between two adjacent

13

C nuclei is closely related to the changes in the

electronic structure of chains caused by the type of terminal atoms ending the structures.



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3.3. JCA as Function of Molecular Properties for Pristine Polyynes and Cumulenes. Now, we discuss the behavior of JCA related to some characteristic molecular properties of polyynes and cumulenes. In Figure 4, we display the dependence of JCA on the cohesive energy per atom, characteristic frequency, and ZPVE of both types of structures. From the plot in Figure 4a, it is possible to infer that the structural stability of polyynes is higher than the structural stability of cumulenes, as we compare structures containing the same number of carbon atoms. We notice that all polyynes structures considered here exhibit cohesive energy per atom larger than the corresponding value for the cumulenes structures, varying from 9.9% for the smallest chain until 4.3% for the largest chain. This finding is expected since very large cumulenes tends to be converted in polyynes by the Peierls distortion effect.58 However, because of the small BLA in cumulenes59, the JCA values increase almost linearly with the increase of the cohesive energy. For example, JCA varies from –60.1 Hz in C4H4, with cohesive energy of 4.76 eV/atom, to –16.5 Hz in C20H4, with cohesive energy of 6.16 eV/atom. On the contrary, JCA decreases with the increase of the cohesive energy for polyynes; i.e., it varies from a value of 76.0 Hz in C4H2, with of 5.23 eV/atom, to a value of 52.6 Hz in C20H2, with 6.42 eV/atom. This result shows that there exists different relationships between the structural stability of these systems and their JCA values, which can be useful to specify each chemically distinct class of carbon chains.


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Figure 4. Profile of JCA as a function of the (a) cohesive energy per atom, (b) lowest vibrational frequency, (c) vibrational frequency of the  stretching mode in polyynes from C8H2 to C20H2, and (d) zero-point vibrational energy. Black squares indicate polyynes and red squares indicate cumulenes, with chains varying from n = 4 to n = 20 carbon atoms.

All the structures of polyynes and cumulenes considered here are found as energy minima at the present level of calculation, which is confirmed by the vibrational analysis. Hence, JCA can be also investigated by analyzing its behavior regarding the characteristic vibrational modes of the systems. In Figure 4b, we plot JCA versus the lowest vibrational frequency by increasing the length of the chain. As already discussed in 3.2, the JCA values of polyynes are positive and decrease with the increasing chain length (Figure 3a), converging to a characteristic value. For this case, JCA also exhibits a fair correlation with the increase of the lowest vibrational frequency, increasing abruptly toward smaller structures. Conversely, the JCA values for cumulenes are negative and increase by growing the chain length (Figure 3b). Now, it exhibits a smooth decreasing behavior with the increase of the lowest vibrational frequency of each cumulene structure. These results indicate that there must be some specific vibrational modes that could be better correlated with JCA for different carbon chains. 15 ACS Paragon Plus Environment


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In the case of polyynes, the highest Raman active vibrational modes27 are denoted as  and  modes, which correspond to C–C bond vibrational stretching. Usually, these modes are in the 2000–2200 cm–1 frequency range and exhibit characteristic signatures of these carbon chains in the Raman spectra.59 In Figure 4c, we display the correlation between JCA and the  stretching modes from C8H2 to C20H2. We notice that as the chain becomes longer these modes undergo a red-shift, which can be related to the conjugation in the polyyne chains. Such an effect is also noticed for linear carbon chains encapsulated in double wall carbon nanotubes.55,60 The lower the red-shift, the greater the delocalization of  electrons and, consequently, the lower the energy gap of the system. This also results in a more well-behaved profile for JCA as a function of the  stretching modes. Within this context and, in addition to the structural stability (Figure 4a), it is also interesting to display the profile of JCA for the dynamic stability of these chains. This can be expressed in terms of the zero-point vibrational energy, depicted in Figure 4d. In this case, JCA exhibits a smooth decay as the ZPVE values increase for the case of polyynes and decays more critically as the ZPVE values decrease in the case of cumulenes. Differently from the structural stability, this behavior now indicates that the bending stiffness of cumulenes is higher than that of polyynes.61 The relationship of JCA and the electronic delocalization can be better understood by analyzing it as function of the calculated HOMO-LUMO gap for these chains, as displayed in Figure 5. In the case of polyynes, we notice that smallest values of JCA are related to the smallest HOMO-LUMO gaps, corresponding to the largest chains (Figure 5a). As an interesting result in the case of cumulenes, we notice again a linear dependence although with JCA decreasing with the increase of the HOMO-LUMO gap. This linearity may be explained by the fact that the degree of electronic delocalization in cumulenes is higher than in polyynes62, exhibiting two non-degenerate, orthogonal π-systems (see 16 ACS Paragon Plus Environment

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Figure 5 in Ref.62), in which only one of them favors a conjugation to the end groups. In Figure 5b, we display the highest occupied molecular orbitals HOMO and HOMO-1 (both MOs were rotated front-on for a better visualization) for C20H2 and C20H4 indicating this fact (see also Figures S4 and S5). Because of the localized bonding pattern in polyynes, the electron density throughout the chain exhibits less uniformity with the increase of the chain, resulting in a nonlinear relationship between JCA and the energy gap. On the contrary, the electron density in cumulenes is more delocalized throughout the chain, leading to a linear behavior between JCA and the energy gap. We analyzed this dependence of JCA on the energy gap in more details in Subsections 3.4 and 3.5.

HOMO HOMO-1 (b) HOMO HOMO-1

Figure 5. (a) Profile of JCA as function of the HOMO-LUMO gap and (b) Kohn-Sham frontier orbitals HOMO and HOMO-1 (both rotated front-on) of C20H2 (top) and C20H4 (bottom). (c) Average static polarizability as a function of the number of carbon atoms and (d) JCA as a function of the average static polarizability for chains varying from n = 4 to n = 20 carbon atoms.

As a complementary information to the analysis of the behavior of JCA with the electronic properties of these carbon chains, it is also useful to investigate the variation



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of JCA with the static polarizability of the systems. As is expected, and recently investigated by Dyck and coworkers63 for polyyne chains, the average static polarizability (𝛼̅) tends to increase as the chain becomes longer, with a significantly large contribution from the longitudinal polarizability. Indeed, this component of static polarizability has been also demonstrated to be strongly gap-dependent for extended systems, such as carbon nanotubes.64 Figure 5c displays the behavior of 𝛼̅ for both types of chains, which agrees with the increase of the chain length. It is also seen that the polarizability of cumulenes increases more quickly for larger chains (varying from 46.79 a.u. in C4H4 to 1111.24 a.u. in C20H4) than the polarizability of polyynes, which increases from 36.89 a.u. in C4H2 to 894.76 a.u. in C20H2, reinforcing that cumulenes possess more polarizable electron densities than polyynes. In Figure 5d, we plot JCA versus our calculated static average polarizability. For both types of carbon chains, we notice again close relationships between JCA and 𝛼̅, which exhibit a decrease with the polarizability increasing for polyynes and an increase accompanying the polarizability increasing for cumulenes. This finding shows that JCA may serve as a good descriptor for the electronic homogeneity between adjacent atomic pairs. It appears that for chains with highly polarizable electron densities JCA is negative, linearly gap-dependent, and converges for characteristic values in long chains, depending on the chemical structure of the system. We examine these findings for linear carbon chains containing different termination groups and small polymer chains in the following. 3.4. Effects of Endcapping Groups on the JCA Values for Carbon Chains. Electronic properties of carbon chains can be tuned by controlling their length and type of end groups. In this direction, it is feasible to obtain, for example, a semiconductor-tometal transition in carbon chains.32-35 To illustrate this effect in connection with the behavior of JCA, we have considered diphenyl-capped polyynes (Ph[n]) and one of their


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oxidized forms (O=Ph[n])35, which can be directly compared to pristine polyynes and cumulenes with the same size. In Figure 6, we display the optimized structures of these systems and summarize the behavior of JCA as a function of the HOMO-LUMO gap, as well as the lowest vibrational frequency (see insets in Figure 6), by increasing the chain length. As expected, and in agreement with Milani et al.,35 our calculated HOMO-LUMO gaps are smaller for larger chains in the case of Ph[n], varying from 2.63 eV (n = 4) to 1.65 eV (n = 12), as illustrated in Figure 6a. In the case of O=Ph[n], the HOMO-LUMO gaps are much smaller and decrease even more as the chain becomes longer, varying from 1.00 eV (n = 4) to 0.71 eV (n = 12), as illustrated in Figure 6b. Here, it is more important to highlight that for both systems (Ph[n] and O=Ph[n]) JCA is negative and exhibits a clear linear behavior with the energy gap, which is similar to the correlation obtained for pristine cumulenes, instead of for pristine polyynes. In fact, the diphenyl substitution at the ends of the chains leads to a formation of a more highly conjugated structure with the electron density behaving in in a similar fashion of pristine cumulenes.62 Moreover, for this type of linear behavior it is appealing to investigate the ratio of JCA to the energy gap (JCA/Eg). For instance, it is worth mentioning that the increasing in the absolute JCA/Eg value is of 15.9 Hz/eV in the case of pristine cumulenes (exhibiting a negative slope), 39.0 Hz/eV in the case of Ph[n], and 154.8 Hz/eV in the case of O=Ph[n], also exhibiting negative slopes. In fact, the linear behavior of JCA with the energy gap appears to be attributable to the effect of the high -conjugation induced by the sp2 end groups, which affects more uniformly the electron densities of the substituted polyynes, which exhibit a cumulenic structure. Furthermore, for this family of compounds, the increasing ratio of JCA to the energy gap may be related to their conducting properties. As is calculated from Figure 6a and 6b, the greater the JCA slope value the more conducting the carbon chain containing sp2-carbon-based end groups. This



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effect is also verified by replacing O with S in O=Ph[n], leading to very low band gap structures,32 for which |JCA/Eg| is as high as 358.3 Hz/eV (see Figure S6).

Figure 6. Linear relationships between JCA and the HOMO-LUMO gap for (a) Ph[n] (adjusted R-squared of 100%) and (b) O=Ph[n] (adjusted R-squared of 100%). Typical optimized structures are displayed in the graphs. The insets display JCA as a function of the lowest vibrational frequency as the chain becomes longer.

Interestingly, the behavior of JCA for both substituted polyynes, i.e., Ph[n] and O=Ph[n], also exhibits a linear correlation with the lowest vibrational frequency calculated for these systems (see insets in Figures 6a and 6b). Now, this lowest-frequencydependence of JCA is different from the behavior obtained for pristine polyynes and cumulenes (as displayed in Figure 4b). Notwithstanding, this is also a consequence of a larger degree of π-electron delocalization along the conjugated systems, which affects more systematically both structural and vibrational properties of the capped carbon 20 ACS Paragon Plus Environment

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chains.35 We notice that the linear correlation between an indirect spin–spin coupling parameter and characteristic vibrational modes, or even the energy gap, of different systems unveils a practical possibility of tailoring properties and functionalities of carbon chains, by employing specific end groups as substituents. In Figure 7, we summarize and compare the behavior of JCA versus the chain length for pristine polyynes and cumulenes and the endcapped linear chains studied here, showing that JCA is quantitatively a good descriptor to characterize linear systems, within a specific family, based on their conducting properties. In the graph, we see that the behavior of JCA is characteristic for each type of carbon chain, assuming well-defined values that are useful to classify the system according to its electron delocalization.

Figure 7. Profile of JCA as a function of the chain length for systems containing different sp2 terminations. The changes in the HOMO-LUMO gap from an insulating to a conducing behavior (vertical arrow) is also illustrated by comparing pristine polyynes (black squares), cumulenes (red circles), Ph[n] (green triangles), and O=Ph[n] (blue triangles).

3.5. JCA versus Energy Gap of Polyacetylene and Polythiophene. In order to understand a little more about the JCA concept and to obtain in-depth information of its behavior with the electronic properties of other polymeric carbon chains, we investigate the growing of small chains of polyacetylene and polythiophene within the oligomer approach. As is known, these systems form -conjugated polymers and are of interest for



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several technological applications in the group of organic conductive polymers.42,65–67 Again, as far as we know, detailed studies on the 13C-NMR SSCCs of these materials are scarce in the literature,41 mainly because of the oscillatory pattern of the individual 1JCC couplings (see Figures S2 and S3), so that the studies are more focused on the determination of chemical shifts.68,69 However, here it is also illustrative to investigate this zigzag behavior of 1JCC in terms of JCA for other polymeric chains, as displayed in Figures 8a and 8b.

Figure 8. Profile of JCA as a function of the chain length extension (a) for cispolyacetylene (red squares) and trans-polyacetylene (black squares), changing with the number of carbon atoms, and (b) for polythiophene, changing with the number of thiophene rings. Relationships between JCA and the HOMO-LUMO gap for increasing chains of (c) polyacetylene and (d) polythiophene. The insets of (c) and (d) display the dependence of the HOMO-LUMO gap on the chain length.

As is expected for these structures (H(CH=CH)nH and PTh), the energy gap decreases rapidly as the chain becomes longer, finding low band-gap states for infinite chains. As a common agreement in many electronic structure calculations reported in the literature,42,65,66 the band gaps for polyacetylene and polythiophene are very small, when


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obtained with usual DFT schemes. Experimentally, the band gap of pristine polyacetylene is in the 1.5–2.5 eV range65 and for pristine polythiophene is in the 2.0–2.5 eV range70, considering infinite chain lengths. The present calculations give energy gap values of 1.05 eV for C20H22 and 1.29 eV for PTh containing 9 thiophene rings, which are in agreement with the literature. However, the goal here is firstly to guarantee the qualitative behavior of the HOMO-LUMO gap for finite chains, by increasing the chain length, and analyzing its correlation with JCA. Hence, in Figures 8c and 8d, we plot JCA as a function of the calculated energy gap for small chains of H(CH=CH)nH and PTh, respectively. In the case of trans-polyacetylene, JCA exhibits positive values, such as in the case of pristine polyynes, and increases as the HOMO-LUMO gap also increases, i.e, for smaller oligomers; whereas for cis-polyacetylene JCA remains practically constant by increasing the chain length (converging to ~19 Hz). This fact results from the small differences in the 1JCC values obtained for the oligomers in the cis configuration for different sizes (see Figure S2), leading to a similar JCA value that is weakly dependent on chain length (Figure 8a). By considering this case, it is seen that the analysis of JCA is interesting since it also contains geometric information about the system in addition to expressing its electronic features. In the case of polythiophene, we compute JCA by increasing the number of thiophene rings in the chain (Figure 8b). The behavior of JCA as a function of the HOMOLUMO gap is displayed in Figure 8d. Similarly to the case of highly conjugated chains (see Figure 6), JCA of polythiophene correlates very well with the energy gap, with a linear fit giving an R-squared of 99.99% for a plot considering up to nine thiophene rings in the chain. De novo, this linearity appears to be related with the electron delocalization along the highly -conjugated chain of polythiophene, which contains an aromatic backbone. For these chains, JCA exhibits negative values, although the slope of the JCA



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plot as a function of the energy gap is now positive and gives a smaller value (3.42 eV/Hz), as compared to the absolute value of JCA/Eg for diphenyl-capped polyynes. However, the structural configuration of polythiophene belongs to a class of carbon chains completely different from those of cumulenes and polyynes containing sp2 terminations. Most importantly for this case, the JCA analysis also plays a fundamental role in the sense of giving insights into the electronic structure of more complexes conjugated polymers, such as polythiophene. For these reasons, we argue that the JCA parameter, to the side of other molecular alternation parameters, is an interesting candidate for the physicochemical characterization of many different types of chains.

4. CONCLUDING REMARKS We have performed a systematic study based on DFT calculations of the NMR indirect SSCCs in connection with electronic structure of finite carbon chains, containing different chemical composition, length extension and terminal groups. Our analysis of the 1JCC coupling profile indicates that it is convenient to define an average indirect scalar parameter, such as the here-proposed JCA, to investigate structural, vibrational, and electronic properties of chains based on a unique J-coupling descriptor. To perform the calculations, we have considered a good compromise between the computational cost for large chains and quality for both NMR and electronic properties. In this sense, we are also thinking in the possibility of treating very large and more complexes systems. Thus, we have defined JCA and investigated its behavior with respect to the properties of polyyne, cumulene, polyacetylene, and polythiophene chains, which are systems of broad and standing interest in the scientific literature. Our results show that JCA exhibits close relationships with several molecular properties investigated for these types of carbon chains. For example, the dependence of


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JCA on the chain length, cohesive energy, HOMO-LUMO gap, static polarizability, ZPVE, and vibrational frequency can unveil important features about these systems. From our analyses, the trends in JCA values allow inferring that chemically different carbon chains exhibit characteristic correlations between energy gap and indirect nuclear spinspin couplings, being useful for identifying their typical electronic behavior. Additionally, JCA exhibits positive values for conjugated chains, such as pristine polyynes and polyacetylene, and negative values for pristine cumulenes and highly conjugated chains, such as polythiophene. Importantly, the effects of sp2 end groups in polyynes lead to an inversion of the sign in JCA, indicating that a higher degree of πelectron delocalization may affect dramatically the behavior of SSCCs in the chains. Finally, this study opens up a new possibility to investigate more deeply the underlying role of indirect SSCCs on several properties of large molecules and extended systems, such as intrinsically conducting polymers, push-pull olefins, -conjugated chromophores, intermolecular hydrogen-bonded chains, linear atomic chains involving other atom types, etc. The concept of JCA is fundamental in these cases since it encompasses information about both geometric aspects and electronic structure of the system. In this sense, we have a key parameter that may be used in the design of materials for a wide range of applications. Results in these directions are ongoing by us and have given evidence that JCA is a promising NMR parameter to establish close relationships with other molecular properties, such as dipole moment, higher-order multipole moments, nonlinear optical properties, and dynamic properties for low-dimensional structures.

ASSOCIATED CONTENT Supporting Information



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Benchmark study of stability and SSCCs for related systems using DFT and wavefunction-based methods combined with different basis sets, as well as complementary J-coupling data and references. This information is available free of charge via the Internet at http://pubs.acs.org

AUTHOR INFORMATION Corresponding Authors *E-mails: [email protected]; [email protected]. ORCID Roberto Rivelino: 0000-0003-2679-1640 João P. C. Oliveira: 0000-0001-8094-5124 Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was partially supported by the Brazilian agencies Conselho Nacional de Desenvolvimento

Científico

e

Tecnológico

(CNPq)

and

Coordenação

de

Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

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