Attraction is associated with lowering of energy and repulsion is associated with increase of energy

Since the e_g orbitals are along the axes and the approach of negatively charged ligands is also along the axes, appreciable repulsion occurs between the electrons of e_g orbitals and the negatively charged ligands. As already stated, repulsion is associated with increase of energy, hence e_g orbitals have increased energy. Whereas, the t_2g orbitals are between the axes and the approach of ligands is along the axes, the repulsion between electrons of the t_2g orbitals and the negatively charged ligands is less, and therefore the energy of t₂g orbitals is lower than that of e_g orbitals.

This simple 'rule of thumb' can also be useful in teaching other concepts in ways that are accessible to students. In the same book ^[1], while explaining the linear combination of the atomic orbitals of hydrogen atoms, a plot of energy versus distance between the atoms of Ψ(g) molecular orbital (Figure B) is presented, where g of Ψ(g) stands for gerade. The plot is depicted, but the explanation for the trend of the curve is not explained. The same is the case with another textbook ^[5] written for freshman engineering (non-chemistry discipline) education. This trend can again be explained with the help of our rule of thumb (especially for non-chemistry freshman audience).

Figure B 

Consider Ψ(g) in the above plot. The two 1 s orbitals of different H atoms approach to form a molecular orbital. As the orbitals approach to form a bond, the electron of the H atom of one of the orbitals which was initially under the influence of one nucleus, is now under the influence of two nuclei. This results in more electrostatic attraction and as a consequence lowering of energy (attraction is associated with lowering of energy). The state of affairs is the same with other electron in the other H atom. Thus, the fall in the Ψ(g) curve. After the curve reaches a minimum (here the bond distance is equal to bond length), the energy starts increasing. The reason for this is that if we further push the orbitals (after the minimum point is reached), repulsion between the nuclei commences and as a consequence the energy increases (repulsion is associated with increase in energy).

This thumb rule also explains why Δo (the difference in energy between two levels t_2g and e_g levels in octahedral field) increases on descending a group of transition elements (Table 1).

The above table is again from reference ^[1]. The book only says that Δo (the difference in energy between two levels t_2g and e_g levels in octahedral field) increases on descending a group of transition elements. But it does not explain why it increases down the group. As you go down the group, the distance of d-orbitals from the nucleus increases. In other words the d-orbitals are more available for the approaching negative ligands, hence more repulsion between the electrons of the d-orbital and the negatively charged ligands (a point to be noted is that if the ligands are neutral, the negative end of the dipole molecule is directed towards the metal ion). More repulsion is associated with gain of energy hence the more Ao as one goes down the group. Again this rule of thumb helps in understanding this point.

The splitting of d-orbitals in tetrahedral ligand field is explained well in the book^[1], but again the correct reason for why the t_2g is above the barycenter and e_g below the barycenter is not given explicitly. The splitting is explained as follows (verbatim from J.D.Lee ^[1]) as follows:

A regular tetrahedron is related to a cube, with an atom at the center, and four of the eight corners occupied by ligands. The directions of axes X,Y, and Z point towards the faces of the tetrahedron as shown in Figure C^[1].

Figure C 

The direction of approach of the ligands does not coincide exactly with either the e_g or the t_2g orbitals. The angle between an e_g orbital, central metal and the ligand is half the tetrahedral angle = 109^o28'/2= 54^o44'. The angle between a t_2g orbital, central metal and the ligand is 35^o16'. Thus the t_2g orbitals are nearer to the directions of the ligands than e_g orbitals The approach of the ligands raises the energy of both sets of orbitals, but since t_2g are closest to ligands they are raised most.

The discussion in the foregoing paragraph from J.D. Lee ^[1] can be made more lucid or comprehensible by applying our rule of thumb. Since the t_2g orbitals are closer (Figure D) to the ligands (35^o16') than e_g (Figure E) orbitals (54^o44') the repulsion between the electrons of t_2g orbitals and the ligands is more, hence the energy is raised higher; it does conform to our rule of thumb, repulsion is associated with the gain of energy.

Figure D 

Figure E 

To comprehend this simple rule of thumb, one can visit the basics of physics by considering the following example:

The columbic potential energy (E) of two charged particles 'q₁' and 'q₂' separated by a distance 'r' is given by the equation

E=q1q2/4πε0r where ε0 permittivity of free space and ε0=8.85×10-12C2/Jm.

For oppositely charged particles, that is when there is attraction, E has -sign. As 'r' decreases that is when oppositely charged particles come closer, E becomes more negative in other words energy is lowered. Thus attraction is associated with lowering of energy.

For charges with same sign that is when there is repulsion, E is positive. As 'r' decreases the positive E value increases, in other words energy is increased. Thus repulsion is associated with increase in energy.

Another important point regarding this rule of thumb is that its use is applicable when pure classical model is applied. So this article would find a place for pre-university and freshman students of engineering (non-chemistry discipline).