  The spin-1/2 alternating Heisenberg chain system Na$_3$Cu$_2$SbO$_6$ features
two relevant exchange couplings: $J_{1a}$ within the structural Cu$_2$O$_6$
dimers and $J_{1b}$ between the dimers. Motivated by the controversially
discussed nature of $J_{1a}$, we perform extensive density-functional-theory
(DFT) calculations, including DFT+$U$ and hybrid functionals. Fits to the
experimental magnetic susceptibility using high-temperature series expansions
and quantum Monte Carlo simulations yield the optimal parameters $J_{1a}$ =
$-$217 K and $J_{1b}$ = 174 K with the alternation ratio $\alpha =
J_{1a}/J_{1b} \simeq$ $-$1.25. For the closely related system
Na$_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 the
key parameter altering the magnetic coupling regime. Numerical simulation of
the dispersion relations of the alternating chain model clarify why both
antiferromagnetic and ferrromagnetic $J_{1a}$ can reproduce the experimental
magnetic susceptibility data.
