What Does the Barycentre Represent in Crystal Field Splitting?

[ampakine]

In this diagram

I can't figure out what that barycentre part of the diagram means. The first section of the diagram represents the energy of the d orbitals before the ligands come into the picture and the 3rd section represents the energy of the d orbitals after the 6 ligands have arranged in an octahedral structure. What does the 2nd section of the graph represent?

On one explanation of CFSE they say this:

If you put an electron into the t2g, like that for Ti3+, then you
stabilize the barycenter of the d orbitals by 0.4 Δo.

but I have no idea what this means. What is the barycentre?

 

[SpectraCat]

Well, barycenter literally means center of mass. In this case, it is just representing the idea that, since energy must be conserved, and you split two states up (doubly degenerate e_g level) and three states down (triply degenerate t_2g level), the barycenter is just the place where the five-fold degenerate energy level would have been in the absence of the splitting, but including the average effect of the crystal field interaction, distributed over 5 orbitals. Does that help?

 

[ampakine]

Ah right, thanks. The only explanation of barycentre I could find were ones relating to astronomy. As for this concept, I've settled for the idea that the middle part of the graph represents the situation in which the ligands form a theoretical spherical charge around the atom as opposed to their charges arranged in an octahedral (or tetrahedral, square planar etc.) structure. In this imaginary spherical distribution of charge each d orbital feels the same amount of repulsion so they remain degenerate.

 

[SpectraCat]

Ah right, thanks. The only explanation of barycentre I could find were ones relating to astronomy. As for this concept, I've settled for the idea that the middle part of the graph represents the situation in which the ligands form a theoretical spherical charge around the atom as opposed to their charges arranged in an octahedral (or tetrahedral, square planar etc.) structure. In this imaginary spherical distribution of charge each d orbital feels the same amount of repulsion so they remain degenerate.

Bingo

 

[LudusRex]

The barycentre, also known as the center of mass, in this context refers to the average energy level of the five d orbitals in a transition metal ion. This is affected by the presence of ligands, which can either raise or lower the energy levels of the d orbitals through their electrostatic interactions with the metal ion.

In the diagram, the second section represents the energy levels of the d orbitals after the ligands have interacted with the metal ion, but before any electrons have been placed in the orbitals. This is known as the crystal field splitting energy (CFSE), which is the difference in energy between the highest energy t2g orbitals and the lowest energy eg orbitals.

The statement about putting an electron into the t2g orbitals stabilizing the barycentre by 0.4 Δo means that placing an electron into one of the t2g orbitals will lower the overall energy of the d orbitals by 0.4 Δo (Delta naught). This is due to the repulsive interactions between the electron and the ligands being partially offset by the attractive interactions between the electron and the metal ion.

Overall, the CFSE plays a crucial role in determining the electronic and magnetic properties of transition metal complexes, and understanding its effects is important in many areas of chemistry, including catalysis and materials science.