d Orbitals

d orbitals are a type of atomic orbital, which are regions around the nucleus of an atom where an electron is most likely to be found. They are one of the four main types of orbitals, along with s, p, and f orbitals. Shapes of orbitals are an approximate representation of boundaries in space for finding electrons occupied in that respective orbital. d orbitals are known to have a clover leaf shape or dumbbell inside where electrons can be found. 

SHAPE OF d ORBITAL

The third energy level, when n = 3 is when d orbitals are occupied by electrons according to Aufbau Principle.

As azimuthal quantum number (l) is n-1=3-1=2 .The degeneracy of d orbital is determined by its magnetic quantum number (m). It extends from –l to +l which gives -2, -1, 0, +1, +2. Therefore, the five d orbitals are dxy, dyz, dxz, dx2-y2 and dz2, corresponding to l values -2, -1, 0, +1, +2 respectively.

SHAPE OF DEGENERATE ORBITAL

According to quantum mechanics, there is a good chance of having a non-zero probability of finding an electron anywhere in the space. This is the reason why the shapes of orbitals can never be determined accurately, but are rather given as contour surfaces or boundary regions along which there is a constant probability that we may find an electron.

There are four lobes in the shape of d orbital as there are two nodes in the orbital given by azimuthal quantum number.

INDIVIDUAL SHAPES OF EACH d ORBITAL

• It is known to have doughnut shaped electron cloud around for the orbital dz2 which is symmetrical around Z - axis.

• dx2-y2 shape can be imagined as that of clover leaf with the respective two leaves each directed along X and Y axis.

• dxz, dyz and dxz has dumbbell shape where the lobes are directed at XY, YZ and XZ respectively.

NODAL PLANES

• Out of the five given orbitals, four of them have same shape except for which is dz2. 

• These four (dxz, dyz, dxz, dx2-y2 ) orbitals are arranged in a particular planar fashion and are found to have two intersecting nodal planes in between which are perpendicular to each other.

• The fifth orbital dz2 has a distinct shape even though it is mathematically equivalent to the others.

EFFECT OF SHAPE ON d ORBITAL SPLITTING

d orbital splitting is mainly seen in transition metals where negative charges from the ligands surround the metal in various shapes according to their hybridization. We know that the five d orbitals are degenerate energy-wise in normal conditions where we distribute the negative charges uniformly in all of them. But we observe that there is some electrostatic repulsion between the negative charges from ligand and the electrons in the d orbital, due to which some of the d orbitals become higher in energy than others. This is determined by its geometry such as tetrahedral or octahedral.

OCTAHEDRAL FIELD

The six negative charges from ligands if assumed to be placed at the vertices of an octahedron breaks the degeneracy of these five d orbitals. The dz2 and dx2-y2 orbitals are pointing straight towards the six negative charges located on the XY and Z axes respectively. The energy of an electron in dz2 and dx2-y2 are called as eg orbitals. The other three orbitals-dxy, and dyz, are collectively called as dxz orbitals. These orbitals are oriented at a 45° angle to XZ, YZ and XZ planes. This makes them point between the negative charges from ligand not along them like eg orbitals. Hence, energy of an electron in dxy, dyz, and dxz orbitals are lower than the energy for a spherical uniform distribution of negative charge otherwise.

TETRAHEDRAL FIELD

To understand Tetrahedral field ligand imagine having a cube with transition metal ion at its center and ligands at the four alternate corners. In such case now we can imagine eg orbitals are directed towards face centers and t2g orbitals face the edge centers. As now t2g orbitals are nearer to ligands than eg, they experience higher repulsion and raise to higher energy and eg becomes lower in energy.

Other than octahedral and tetrahedral, other geometries also split the degeneracy of d orbitals with respect to their ligand positions in the geometry, a brief idea of that is given below.