## What is the Shape of the D Orbital?

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.

## How do Orbitals Work?

In the solar system, planets move around the sun in different orbits. Looking at an atom, we see that electrons also revolve around the nucleus in definite energy levels called as orbitals. But drawing an analogy between an orbit and an orbital might not be a perfect analogy, as orbits have defined paths in which planets move around the sun, whereas orbitals can never define a path in which the electrons are revolving around the nucleus of an atom.

To understand this, we need to recall Heisenberg’s Uncertainty Principle which states that it is impossible to determine the position and momentum (or energy) of an electron simultaneously. Hence, all we know about electrons is that they are in a defined energy level in a given atomic orbital, but their position in the orbital cannot be specified accurately.

Each orbital in addition to that is assigned four quantum numbers which defines the orbital shown as follows:

### Principle Quantum number (n)

This gives an idea about the energy by giving the probable distance at which the electron is from the nucleus. As principal quantum number gets higher, the farther is the electron from the nucleus, therefore, the atomic size is bigger.

### Azimuthal Quantum number (l)

It is also called as the orbital angular momentum quantum number as it gives the angular node in each energy level.

1 = (n-1) and each value of l gives information about particular subshell s, p, d and f. This is the main quantum number that plays role in determining the shape of an orbital.

### Magnetic Quantum number (m)

Magnetic quantum number gives the special orientations of each orbital with the subshell. The value of m extends from –l to +l where ‘l’ is azimuthal quantum number. This gives the degeneracy in each subshell.

### The electron spin quantum number (m_{s})

This quantum number describes the direction of spin of an electron and it can only be either +1/2 or -1/2. This is independent of rest of the quantum numbers.

## Types of Orbitals

An energy level or a shell in the atom contains various subshells such as s, p, d and f. These s, p, d and f subshells may further contain orbitals in them.

### s Orbital

The s orbital is spherically shaped and hence the probability of finding an electron from all directions at a given distance is same in s orbital. This is known as spherical symmetry. Each s orbital can hold up to 2 electrons maximum.

### p Orbital

The p orbital is known to have a shape that of two lobes along an axis above and below. It has a degeneracy of 3, hence it can hold up to a maximum of 6 electrons.

### d Orbital

The d orbital has a dumbbell shape and has 5 degenerate orbitals and hence can hold up to a maximum of 10 electrons.

### f Orbital

The f orbital has a complex shape, but when full it resembles a dumbbell like that of d orbital and can a maximum of 14 electrons.

## 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 d_{xy}, d_{yz}, d_{xz}, d_{x}^{2}_{-y}^{2} and d_{z}^{2}, corresponding to l values -2, -1, 0, +1, +2 respectively.

### Shape of degenerate orbitals

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 d
_{z}^{2}which is symmetrical around Z - axis. - d
_{x}^{2}_{-y}^{2}shape can be imagined as that of clover leaf with the respective two leaves each directed along X and Y axis. - d
_{xz}, d_{yz }and d_{xz}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 d
_{z}^{2}. - These four (d
_{xz}, d_{yz}, d_{xz}, d_{x}^{2}_{-y}^{2}) 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 d
_{z}^{2}has a distinct shape even though it is mathematically equivalent to the others.

Orbitals | Corresponding Nodal Planes |

d_{z}^{2} | No nodal plane |

d_{x}^{2}_{-y}^{2} | x-y=0, x+y=0 |

d_{xy} | XZ, YZ |

d_{yz} | XY, ZX |

d_{xz} | XY, YZ |

## 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 d_{z}^{2} and d_{x}^{2}_{-y}^{2} orbitals are pointing straight towards the six negative charges located on the XY and Z axes respectively. The energy of an electron in d_{z}^{2} and d_{x}^{2}_{-y}^{2} are called as e_{g} orbitals. The other three orbitals-d_{xy}, and d_{yz}, are collectively called as d_{xz} 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 e_{g} orbitals. Hence, energy of an electron in d_{xy}, d_{yz}, and d_{xz} 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 e_{g} orbitals are directed towards face centers and t_{2g} orbitals face the edge centers. As now t_{2g} orbitals are nearer to ligands than e_{g}, they experience higher repulsion and raise to higher energy and e_{g} 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.

### Colors Exhibited by Transition metals

As we know transition metals are known to exhibit colors because of their electronic transition from one energy level to other, the splitting of d orbitals in the presence of ligands leads to this energy separation resulting in colored complexes of transition metal compounds.

## Practice Problem

Calculate and give the shapes and names of degenerate orbitals in a d-subshell

Solution

If n=3 for a d subshell, then 1=3-1=2.

As magnetic quantum number (m) gives degeneracy from –l to =l values, the values of m are -2, -1, 0, +1, +2.

Therefore, there are five degenerate orbitals namely d_{xy}, d_{yz}, d_{xz}, d_{x}^{2}_{-y}^{2} and d_{z}^{2}.

## Common Mistakes

Do not mistake the shapes of different d orbitals and the respective planes along which they have nodes.

## Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

- B.Sc. in Chemistry
- Chemical Engineering
- M.Sc in Chemistry
- B.Tech Biochemistry.

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