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The spin-1/2 alternating Heisenberg chain system Na$_3$Cu$_2$SbO$_6$ featurestwo relevant exchange couplings: $J_{1a}$ within the structural Cu$_2$O$_6$dimers and $J_{1b}$ between the dimers. Motivated by the controversiallydiscussed nature of $J_{1a}$, we perform extensive density-functional-theory(DFT) calculations, including DFT+$U$ and hybrid functionals. Fits to theexperimental magnetic susceptibility using high-temperature series expansionsand quantum Monte Carlo simulations yield the optimal parameters $J_{1a}$ =$-$217 K and $J_{1b}$ = 174 K with the alternation ratio $\alpha =J_{1a}/J_{1b} \simeq$ $-$1.25. For the closely related systemNa$_2$Cu$_2$TeO$_6$, DFT yields substantially enhanced $J_{1b}$, but weaker$J_{1a}$. The comparative analysis renders the buckling of the chains as thekey parameter altering the magnetic coupling regime. Numerical simulation ofthe dispersion relations of the alternating chain model clarify why bothantiferromagnetic and ferrromagnetic $J_{1a}$ can reproduce the experimentalmagnetic susceptibility data.

Microscopic magnetic modeling for the $S$=1/2 alternating chain compounds Na$_3$Cu$_2$SbO$_6$ and Na$_2$Cu$_2$TeO$_6$