## 4 Discussion
### Determination of the Relative J-Coupling Signs  
The \(J\)-couplings of \(1A\)-difluorobenzene are all positive, but for a majority of the fluorobenzenes, the simulated \(J\)-couplings had both positive and negative \(J\)-couplings. Traditionally, weakly coupled spin systems do not impart a sign dependence on the spectrum since the first order perturbation of the \(J\)-coupling results in equal splitting from the central frequency. However, in strongly coupled systems, the relative signs of the \(J\)-couplings are determinable. This has been mathematically developed for high field systems in which the difference in chemical shifts between two environments is on the order of the \(J\)-coupling, i.e., the strong homonuclear \(J\)-coupling limit [60; 61; 62; 63; 64; 65; 66]. The most simplistic of these systems is a strongly coupled AB system, in which A and B are homonuclei. In this case, A and B are both doublets and, as described by [PERSON], the separation between the two middle lines will be \(|J^{2}+(\
u_{\rm A}-\
u_{\rm B})^{2}|^{0.5}-J\), while the separation between the outer two lines is \(|J^{2}+(\
u_{\rm A}-\
u_{\rm B})^{2}|^{0.5}+J\)[67]. This is equivalent at LF to the relation developed for the strongly coupled heteronuclear \(S\)-\(I\) (AB system _via_ Pople notation) system by [PERSON] et al. in which the four strong heteronuclear \(J\)-coupling transitions are given by [14]  
\[v_{1}=v_{I}-\frac{J}{2}+\frac{J^{2}}{4(\
u_{\rm I}-\
u_{\rm S})}, \tag{6}\]  
\[v_{2}=v_{I}+\frac{J}{2}+\frac{J^{2}}{4(\
u_{\rm I}-\
u_{\rm S})}, \tag{7}\]  
\[v_{3}=v_{\rm S}+\frac{J}{2}-\frac{J^{2}}{4(\
u_{\rm I}-\
u_{\rm S})}, \tag{8}\]  
\[v_{4}=v_{\rm S}-\frac{J}{2}-\frac{J^{2}}{4(\
u_{\rm I}-\
u_{\rm S})}, \tag{9}\]  
The two inner lines are the \(\
u_{1}\) and \(\
u_{3}\) resonances, which are separated by  
\[(\
u_{I}-\
u_{\rm S})-J-\frac{J^{2}}{2(\
u_{I}-\
u_{\rm S})}, \tag{10}\]  
while the two outer lines are separated by  
\[(\
u_{I}-\
u_{\rm S})+J+\frac{J^{2}}{2(\
u_{I}-\
u_{\rm S})}, \tag{11}\]  
which is equivalent to the separation obtained by [PERSON]. Because the form of the Hamiltonian for \(J\)-coupling is the same for homonuclear or heteronuclear \(J\)-coupling, the consistency of peak spacing between strongly coupled two spin heteronuclear or homonuclear spins is expected. This gives additional credence to extending the Pople notation into the strong heteronuclear \(J\)-coupling regime, as the strongly coupled heteronuclear and strongly coupled homonuclear system two-spin systems AB and AX give the same analytical result for the transition frequencies. Therefore, the effects observed in the strong homonuclear case will extend into the strong heteronuclear regime. In the strong homonuclear \(J\)-coupling case, the next simplest system is the ABX system, in which two magnetically inequivalent, but strongly coupled protons, weakly couple to a heteronucleus, such as F. In this system, there are three \(J\)-couplings: \(J_{\rm AX}\), \(J_{\rm BX}\), and \(J_{\rm AB}\). This system has been treated by several authors and will not be fully presented here; only the pertinent aspects will be covered. In this system, there are 14 observable transition frequencies [61], which all depend on the term \(C_{m}\), given by  
\[C_{m}=\frac{1}{2}\sqrt{\left[(\
u_{A}-\
u_{B})+m(J_{AX}-J_{BX})\right]^{2}+J_{ AB}^{2}} \tag{12}\]  
Here, \(m\) is the sum of the spins of all X nuclei, and \(m\) can have values of \(\pm 1/2\). In this system, only the relative signs of \(J_{AX}\) and \(J_{\rm BX}\) can be established, since \(C_{m}\) depends on the value \((J_{\rm AX}-J_{\rm BX})\)On the other hand, the \(J_{\rm AB}\) term is always squared and will have the same relation to \(J_{\rm AX}\) and \(J_{\rm AB}\), regardless of the relative sign of \(J_{\rm AB}\). At LF, the same ABX system assumes one or both A and B are in the strongly coupled heteronuclear regime, which transforms into an equivalent ABC system. As was demonstrated by [PERSON], the relative signs of the ABC system can be determined [63].