## 4 Discussion
### Determination of the Relative J-Coupling Signs
In this spin system, 15 transitions are theoretically observable, although only 12 to 14 are typically reported in the literature [63; 64]. To illustrate this point, the JCS and high field spectra were simulated for the ABX (C) system in which spins AB were strongly coupled at LF (Figure 10). Each spectrum was simulated with three \(J\)-couplings (\(J_{\rm AB}\), \(J_{\rm AX}\), and \(J_{\rm BX}\)), whose relative signs were changed between each simulation. There are four unique sign combinations for these systems. As Figure 10 demonstrates, at high field (Figure 10a,b), only the relative signs of \(J_{\rm AX}\) and \(J_{\rm BX}\) may be determined, as the black (\(+++\)) and red (\(++-\)) spectra are equivalent, as were the blue (\(+-+\)) and green (\(+--\)) spectra. However, at LF (Figure 10c,d), all three signs may be determined, as each spectrum is distinct for each sign combination. Therefore, for the same molecule analyzed at high field and LF, the LF spectrum allows determination of the relative signs for all coupled nuclei as long as the total spin system contains more than two spins, whereas coupling sign determination at high field requires a strong homonuclear \(J\)-coupling. Figure 10: The \({}^{19}\)F (**a**) and \({}^{1}\)H (**b**) high field NMR spectra of an ABX system, with \(J\)-couplings \(J_{\rm AX}\) = 10 Hz, \(J_{\rm BX}\) = 5 Hz, and \(J_{\rm AC}\) = 7 Hz. Note that the spectra were invariant to the relative sign of \(J_{\rm AB}\) and only depended on the relative sign of \(J_{\rm AX}\) and \(J_{\rm BX}\). In spectra (**c**) and (**d**), the same spin system was in the strong heteronuclear \(J\)-coupling limit, such that the ABX system became the ABC system. Unlike the ABX system, which only has two distinct spectra, the ABC system has four distinct spectra. Therefore, the relative signs of all three \(J\)-couplings may be determined for the ABC system. The absolute signs of the \(J\)-couplings are denoted between the spectra and are consistent between the high field and LF spectra with black (\(+++\)), red (\(++-\)), blue (\(+-+\)), and green (\(+--\)). This extends into larger systems as well, particularly those investigated here, which were 6-spin systems. With larger spin systems of coupled nuclei, the relative signs of the \(J\)-couplings are also determinable. For example, if we investigate the relative signs of 1,4-difluorobenzene, there are five \(J\)-couplings and \(2^{5}\) or 32 possible sign combinations. However, half of the combinations were degenerate and had the same relative sign relation, resulting in 16 unique sign combinations. The 16 unique sign combinations are shown in Figure 11, along with corresponding \(R\)-factors for best fit with the experimentally determined spectra. Clearly the relative signs have a significant impact on the JCS, and must be chosen correctly to accurately simulate the JCS. The sign change of \({}^{4}J_{\rm HH}\) impacted the spectra the least; however, the difference caused by changing the sign clearly impacted the \(R\)-factor. It should be noted that, while the relative signs of these systems can be determined, the absolute signs are not established by the JCS.