Why do Mn and Tc have lower melting points with respect to their immediate neighbors but Re doesn't?

Manganese (Mn) and technetium (Tc) show a sharp decline in their melting point in comparison to the neighboring transition elements, which is unexpected while following the general trend which says the extent of metallic bonding increases with the availability of unpaired electrons. rhenium (Re) however does not show the same tendency despite belonging to the same group.


Upon revisiting their electronic configurations, we can notice technetium is actually d⁶ which could've explained the disparity had technetium shown it instead of rhenium. Somehow manganese and rhenium are both d⁵ yet the sharp decrement is observed in case of manganese (d⁵) and technetium (d⁶) instead. This should suggest melting point isn't directly related to unpaired electrons or unpaired electrons isn't related to extent of metallic bonding, both of which sounds contradictory to the phenomena shown by the rest of the transition elements? What am I missing here?




The melting point depends on how available the valence electrons (which are mostly paired in metallic bonding) are for metallic bonding, and that is not just a function of the number of such electrons formally existing in the atoms.


Most metals aside from alkali metals have at least two valence electrons that are relatively easy to ionize and thus also to delocalize into metallic bonds. Getting three or more metallically bonded electrons per atom is where you begin to run into some resistance, and thus you begin to see differences not only in melting point but also in density.


Let's look at third ionization energies of the relevant elements in Periods 4 and 5. All data are from [Wikipedia](https://en.wikipedia.org/wiki/Molar_ionization_energies_of_the_elements) and are in Joules per mole.


$\ce{Cr}=2987, \color{blue}{\ce{Mn}=3248}, \ce{Fe}=2957$


$\ce{Mo}=2618, \color{blue}{\ce{Tc}=2850}, \ce{Ru}=2747$


The higher ionization energy of the blue elements implies that getting more than two metallically bonding electrons per atom is more difficult for those elements than for their neighbors. With that come the lower melting points and also reduced densities. Manganese at [$7.21$ g/cm³](https://en.wikipedia.org/wiki/Manganese) is well under the average of chromium ([$7.15$](https://en.wikipedia.org/wiki/Chromium)) and iron ([$7.87$](https://en.wikipedia.org/wiki/Iron)).


So why do we see this crest in third ionization energies? Despite the different electron configurations given to the atoms, the $+2$ ions all have the common configuration $\text{[noble gas]}(n-1)d^m$. When $m=5$ for the dication, as with manganese and technetium, the next stage of ionization requires breaking a configuration that has relatively strong exchange-enetgy stabilization.


We cannot extend this comparison to rhenium because the relevant ionization energy data are not available for all the elements in that part of the Periodic Table. But we should expect the $5d$ electrons to be more readily available for metallic bonding with rhenium on relativistic grounds (we do know that rhenium has a lower third ionization energy, $2510$ J/mol, than either manganese or technetium, per the table cited above). The contraction of the $s$ orbitals due to the increased mass-energy of those electrons in heavy elements makes them better at shielding other orbitals. The electrons in those other orbitals are therefore *less* strongly bound to the nucleus. This "reverse inert-pair" effect is prominent with valence $d$ orbitals in the sixth-period transition elements.


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