Why does niobium have a #d^4 s^1# electron configuration but vanadium has a #d^3 s^2# electron configuration?

It follows the normal rules for energy minimization for stability. It is not "weird", except possible from the perspective of numbered shell filling, which is common for all the "transition" elements. The 4d orbital is actually at a higher energy state than the 5s. Hund's Rule dictates the unpaired filling.

See also:
https://www.webelements.com/niobium/
for more properties.

It's not obvious. There are two opposing trends:

ns-(n-1)d ENERGY GAP

• The #3d# orbital of #"V"# is #"2.79 eV"# lower in energy than the #4s# (see Appendix B.9).
• The #4d# orbital of #"Nb"# is only #"1.64 eV"# lower in energy than the #5s# (see Appendix B.9).

For visual comparison, I had graphed data from the above reference:

As a result, one might expect that purely based on energy, at some partway filling point, the fourth valence electron would favor going into the #bb(5s)# to reduce electron repulsion for #"Nb"#.

ORBITAL SIZE

The #4d# orbital is bigger than the #3d# orbital, and the #5s# orbital is bigger than the #4s# orbital.

This would favor putting the fourth valence electron in the #(n-1)d# orbitals more so for #"Nb"# than for #"V"#, because there is less inherent electron repulsion in the bigger orbital.

Due to the first trend favoring putting the fourth valence electron in to the #ns#, I would feel troubled in assuming that it would be one way or the other from orbital size alone, because they are conflicting trends.

CONCLUSION

Apparently, what I'm seeing is that there is some cancellation between:

• having bigger orbitals, reducing electron repulsions and making it more favorable to continue on filling the #d# orbitals with the fourth valence electron.

• having #(n-1)d# orbitals closer in energy to the #ns# than #"V"# does, making it more favorable for the fourth electron to be placed into the #ns# instead of the #d#.

And as it turns out, a #s^1 d^4# configuration is favored over #s^2 d^3#, so it looks like the orbital size trend dominates the energy gap trend for #"Nb"# specifically.