## 2 Results and Discussion
### Magnetic Properties
On the basis of the crystal structures of **III** and **IV**, the magnetic behavior of both complexes was approximated assuming a 1D antiferromagnetic (AF) chain governed by the exchange coupling Hamiltonian: \(H=-J\Sigma S_{i}S_{i+1}\), where \(J\) is the isotropic interaction parameter. The temperature dependence of the magnetic susceptibility for such a 1D system is adequately described by the Bonner-Fischer approximation [33]:  
\[\chi_{M}=\frac{N\beta^{2}g^{2}}{k_{B}T}\bigg{(}\frac{0.25+0.074975x+0.075235x^{ 2}}{1+0.9931x+0.172135x^{2}+0.757825x^{3}}\bigg{)} \tag{1}\]  
with \(x=|J|/k_{B}T\) and the other terms having their usual meanings. Since the crystal packing for these 1D chains reveals several inter-chain interactions (Figures 3 and 5), a mean field correction was considered using a corrected susceptibility \(\chi_{\rm cor}\), where \(zJ^{\prime}\) corresponds to the interchain interaction parameter [34]:  
\[\chi_{cor}=\frac{\chi_{M}}{1-\frac{zJ^{\prime}\chi_{M}}{Ng^{2}\beta^{2}}} \tag{2}\]  
The increase of \(\chi_{M}\) below 30 K is attributed to paramagnetic monomeric impurities and/or short chains after chain breaking [35, 36] with \(S=1/2\), for which the magnetic susceptibility, \(\chi_{\rm IMP}\), followsthe [PERSON] law. Overall, the susceptibility data were best fitted following the equation: \(\chi_{U}=(1-\rho)\chi_{\mathrm{cor}}+\rho\chi_{\mathrm{IMP}}+\chi_{\mathrm{TIP}}\), where \(\rho\) is the fraction of paramagnetic impurities and \(\chi_{\mathrm{TIP}}\) is the temperature independent paramagnetism (TIP). The parameters obtained from these simulations are listed in Table 1, considering two situations for **IV**, with or without intermolecular interactions, the second situation being the less favorable according to fitting results (Figure 8). Despite the similar nature of the [Cu(\(\mu_{2}\)-NH\({}_{2}\)trz)\({}_{2}\)Cl] polymeric unit in **III** and **IV**, the latter exhibits stronger inter-chain magnetic interactions \(zJ\) (Table 1). The strong AF intra-dimer coupling \(J\) determined in **III** and **IV** is much higher than those of previously reported 1D Cu(II) complexes made of two \(\mu_{2}\)-\(N_{1}\),\(N_{2}\)-NH\({}_{2}\)trz and another bridging atom, _i.e._, N in [Cu(\(\mu_{2}\)-NH\({}_{2}\)trz)\({}_{2}\)(\(\mu_{2}\)-\(N_{3}\)-N,N)]NO\({}_{3}\)[11], or [Cu(\(\mu_{2}\)-NH\({}_{2}\)trz)\({}_{2}\)(\(\mu_{2}\)-NCS-N)]ClO\({}_{4}\)-\(0.5\)H\({}_{2}\)O [12], and O in [Cu(\(\mu_{2}\)-NH\({}_{2}\)trz)\({}_{2}\)(\(\mu_{2}\)-NO\({}_{3}\))]NO\({}_{3}\) (Table 1) [13]. These results allow us to evidence an intermediate case between two structural situations by disclosing the crystal structures of the chain compounds **III** and **IV** which show a new bridge type of formula [Cu(NH\({}_{2}\)trz)\({}_{2}\)Cl]\({}^{+}\). Interestingly, this configuration reveals higher intra-dimer coupling constants compared to the initial case where Cu(II) chains, of formula [Cu(Rtrz)\({}_{3}\)]\({}^{2^{+}}\), made of three Rtrz bridges show very weak \(|J|\) values [27,29], and the third case where neutral Cu(II) chains [Cu(NH\({}_{2}\)trz)Cl\({}_{2}\)], show weak \(J\) values [10,32]. Despite the high \(J\) values of **III** and **IV**, the record ones reported so far for this substance class are hold by [PERSON] _et al._ for [Cu(\(\mu_{2}\)-OH)(\(\mu_{2}\)-XPhtrz)]BF\({}_{4}\cdot\)H\({}_{2}\)O and [Cu(\(\mu_{2}\)-OH) (\(\mu_{2}\)-XPhtrz)(H\({}_{2}\)O)]NO\({}_{3}\) (X = Cl, Br; XPhtrz = \(N\)-[(\(E\))-(4-\(R\)-methylidene)-4\(H\)-1,2,4-triazole-4-amine) which range from \(-\)391(3) to \(-\)608(2) cm\({}^{-1}\)[35]. Ab initio calculations would be useful to appreciate the origin of the magnitude of the coupling constants in these model Cu(II) chain compounds.