Colorful Solutions

Transition metals (TMs) form coordination complexes (also called complex ions) with ligands. Relevant to the question, TMs have electrons occupying d-orbitals. Ligands are able to form coordinate/dative bonds because they have a lone pair of electrons. These bonds are dative (i.e., both electrons participating in the bond originate from/are donated by the ligand).

As ligands approach the central TM cation, electron repulsion occurs between the lone pair(s) and the d-orbital valence electrons. The effect is that the d-orbital sublevel (comprising all 5 orbitals) is split into one or more higher energy d-orbital group(s) and lower energy d-orbital group(s). The simplest case is for an octahedral geometry (where six ligands form coordinate bonds with the central cation), where there is a two-group d-orbital splitting (see image below).

The energy difference between these groups may correspond to the energy of light in the visible spectrum. This means that when full spectrum white light is transmitted through a solution containing TM coordination complexes, the photons of correct energy are selectively absorbed by electrons occupying lower energy d-orbitals. The transmitted light will appear to be coloured complementary to the colour of photons absorbed by the complexes. Hence, a part of the transmitted light is removed from the incident light.

The degree of splitting, and therefore the colour of the solution depends upon the identity of the transition metal cation (element and charge), the identity or identities of the ligand(s) (single ligand species, mixed ligand complex), hence the number of lone pairs in each ligand (monodentate, bidentate, polydentate, etc. [dentate=tooth(ed)] ), and the solvent used for solution. The spectrochemical series identifies the relative strength of (common) ligands.

Attribution: ​English Wikipedia user YanA (Creative Commons Attribution-Share Alike 3.0 Unported License)

FIY:

• ligands can be neutral (e.g., $H_{2}O$ ) or anions (e.g., $CN^{-}$ )
• d-orbitals and f-orbitals usually undergo field splitting, p-orbitals may, but s-orbitals never split, because...
• electron orbital degeneracy needs to be broken, that is more than two orbitals (multiple quantum states) initially at the same energy in the same sublevel (equal measured energy) need to be split into higher/lower energies relative to one another
• Crystal Field Theory and more recently Ligand Field Theory are used to describe the mechanisms of coordination complexes