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When we combine the wave functions of two p_y-orbitals, or any two p orbitals, aligned parallel to each other, then the wave mechanics would produce two banded orbitals, one on each side of the axis, named the π-orbital.

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When we combine the wave functions of two sets of p orbitals align side by side, and parallel to each other, then the wave mechanics would produce a set of δ-looked alike orbitals at right angle to each other. This is named the δ-orbital. For chemical reaction the δ-orbital can be looked upon as two sets of δ-orbitals at right angle to each other.

The Valence Bond theory is not the only possible solution on covalent bond, but I would consider it as the simplest approach and is very useful in explaining the shape and the distance of the bond.

Let us try to construct the molecule of methane gas, CH₄, using the Valence Bond theory. We know that the electrons of the four hydrogen atoms will interact with the valence electrons of the carbon atom to give an octet of electrons for the valence shell of the carbon and two electrons for the outer shell of hydrogen. This will result in three sigma-bonds at right angle to each other (since the p-orbitals of the carbon are at right angle to each other), and a sigma bond from the 2s-orbital of carbon and 1s-orbital of hydrogen with no definite orientation.

Research on methane showed the molecule to be tetrahedral in shape with the carbon atom in the centre. The carbon-hydrogen bonds are all equal in length and equally spaced from each other at angles of slightly over 109�.

HYBRIDISATION OF ATOMIC ORBITALS

The only way to have four identical bonds is to start of with four identical valence orbitals. We asked our mathematician friends to help us combine the 2s- and three 2p- wave functions to see what happen. The mathematicians did a linear combination of the wave functions and informed us that such a combination would give four orbitals equally spaced and at slightly greater than 109� from each other. We took these hybrid valence atomic orbitals of carbon and interact them with the 1s-orbital of hydrogen using the VBT. It gave us the molecular structure of methane shown by experiments. Naturally we name these four hybrid atomic orbitals as sp�-orbitals.

Similarly when we proceed to hybrid one s-orbital with two p-orbitals it produce three sp�-hybrid atomic orbitals. Our mathematician friends inform us that all the orbitals are on the same place at 120� from each other. The remaining p-orbital is perpendicular to the plane.

One s-orbital with one p-orbital will give two sp-hybrid atomic orbitals at 180� from each other. The remaining two p-orbitals will be along the other two axis.

s

+

3p

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sp3

OR

sp2

OR

sp

Hybrid Orbital

Each of the hybrid orbitals should have a small negative orbital in the opposite direction as shown in the insert on the right.

When do we use pure atomic orbitals and when do we use hybrid atomic orbitals? We do not have the answer until we know the structure of the molecule from experiment. That is hybridisation of atomic orbitals is to explain not to predict the types of bonding in the molecule.

MOLECULAR ORBITAL THEORY (MOT)

The other better-known theory on bonding is the Molecular Orbital Theory (MOT).

We have used Wave Mechanics to work out the orbitals for atoms. It is argued that we can do the same for molecules, at least as a proposal. In MOT we consider all the nuclei of the molecule as centers and try to derive the wave functions for all the electrons present in the molecule.

It looks good on paper but too complex in practice. So we start with two proton centres and one electron, or a H₂⁺ ion. Even this prove to be difficult.

We made it even simpler by linearly combining the wave functions of the atomic orbitals of the two hydrogens. In mathematics linear combination means we only consider the process of addition or subtraction, not multiplication or division. This approach is known as the Linear Combination of Atomic Orbitals (LCAO). The molecular orbitals derived are referred to as the LCAO molecular orbitals.

2p	σ*2      π*    π σ2	2p
2s	σ*1 σ1	2s
1s	σ* σ	1s
H	H2+	H

We do a linear combination for each pair of comparable energy levels. That is 1s- wave function with 1s- wave function, 2p- with 2p-, and so on. The mathematical results are the bonding orbital, non-bonding orbitals and anti-bonding orbital show in the diagram. In this computation we assumed that the energy levels of the 2s and 2p orbitals of the hydrogen atom are far enough that they do not interact.

Following the procedure used for configurating the electrons for an atom, we will fill in the electrons for the molecule starting from the lowest energy level.

Let us see how we can apply the results obtained to an oxygen atom. We just fill in the 16 electrons that are in the oxygen molecule.

The electron configuration of the oxygen atom will be,

The next step is to calculate the bond order. That is the total bonding orbitals against the total anti bonding orbitals. In oxygen there will be a sum total of 2 bonding orbital. We say the bond order is 2.

When we consider molecules with three nuclei it is definitely beyond us. So we generally confine the MOT to the bonding that are of interest to us. A good illustration is diborane.

B   B  2p Anti−bonding Non−bonding Bonding   H 1s Molecular Orbitals

We obtain the molecular orbitals by combining the wave functions of a 1s orbital of hydrogen with one 2p orbital from each of two boron atoms. (We leave the mathematics to the experts.) The molecular orbitals computed were: one bonding molecular orbital, one non-bonding molecular orbital, and one anti-bonding molecular orbital.