85. Covalent Bonding(31) – MOT(6)- Symmetry of MOs.


The orbitals can be classified, based on symmetry as – Gerade and Ungerade MOs.


Gerade MOs ⇒ Gerade = Even (German) ⇒They have an inversion center in them, through which if the MO is inverted, results in no sign(+,-) change in the MO i.e there is no change in phase of that orbital.

Ungerade MOsUngerade = Odd (German) ⇒When inverted through an inversion center there is a change in the sign of the MO i.e there is a phase change.


What is an inversion center?

Inversion center is a point in an object, through which when the object is inverted, results in no change in its structure.

e.g. – Sand clock


A sand clock has an inversion center. If the sand clock is rotated through it, there is no change in its structure.

For an object with an inversion center, there exists an identical part at an equal distance on the opposite side of the center(centrosymmetric).


As seen in the above diagram, every point has a similar point on the other side of the inversion center.

After understanding what gerade and ungerade terms mean, let us study how these apply to MOs.

When two s- orbitals overlap, a gerade BMO and an ungerade ABMO are formed.

Gerade MO

In the figure above, when two s- orbitals overlap, a BMO is formed. This orbital when inverted through the inversion center, produces no change in its structure. The signs remain unchanged too i.e there is no change in the phase of that orbital. Thus, this is a gerade orbital. It is denoted by σg.


σ indicates that the MO is symmetric around the internuclear axis. Orbital is formed by the interaction of two s-orbitals or two porbitals. 

An antibonding orbital has ungerade symmetry, as the signs change on inversion. There is a phase change. It is denoted by σ*u.

Ungerade MO
Phase change in an antibonding orbital

Mathematically this phenomenon can be expressed as – 


In post no 33 we have learned that,

An Operator is a symbol for certain mathematical procedure which transforms one function into another function. So, basically it means that we are operating on an expression and changing it to another expression.
∴ Operator(function)= Another function.

Here, P is the operator and it operates inversion on the wave function Ψi (x,y,z) to give the resultant wave function Ψi (-x,-y,-z). This means that the x, y, and z coordinates of the wave function change after the operation. This is shown in the figures above. Whenever a wave function changes its sign, we have a NODEA node is a region in space where the probability of finding an electron is zero or minimal.

Normalization of wave functions.

We know that Ψ denotes the wave function/orbital and Ψ2 denotes the probability of finding the electron. Thus, we have to square the wave functions to find the probability. Thus, Ψ2 is always zero or a positive number (as it is the square of a number). If we square the above equation we get


In the above equation, ‘c’ is a coefficient that can take positive or negative values.  We normalize a wave function as it gives us observable results. Normalizing helps us find the probability of finding an electron in that space. Thus, the normalization of a wave function helps us get tangible results. This is a mathematical process, and thus more details are beyond scope of our study.

We shall see how orbital overlap takes place actually in the next post and then study the symmetry of MOs along with it. Till then

Be a perpetual student of life and keep learning…

Good Day!

References and Further Reading –




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