Linear Combination of Atomic Orbitals

Now we know that electrons exhibit dual nature. When we consider an electron as a particle, we are able to give its address in an atom by a set of quantum numbers n, l, m_l and m_s. When we consider an electron as a wave, we describe it by wave function. A wave function also includes these quantum numbers except m_s, because as a wave we can only define orbital (a space where is the maximum probability of finding an electron).

Wave function has two parts one is Radial function which depends on quantum numbers n and l and the other is Angular wave function which depends on quantum numbers land m_l.

Radial function gives the number of degenerate orbitals for corresponding l value. Otherwise it has no physical meaning. Angular wave function is of much importance because it is used to draw polar diagram of orbital. In LCAO this polar diagrams are used to define overlapping of bonding orbitals.

Angular wave function depends on Polar coordinates r, θ and Φ. Let’s see how these polar coordinates are related to the Cartesian coordinates.

When you try to calculate angular wave function for orbital p_z you will find that the value of cosθ is positive for upper lobe and negative for lower lobe, that’s why in polar diagram upper lobe of p_zorbital has positive sign and lower lobe has negative sign. Similarly, polar diagrams for each type of orbitals have been developed by Schrödinger wave equation.

The signs of polar diagram play an important role in bonding. I’ll give you an example to explain it, suppose two persons extend their right hands forward to shake hands, they can easily and effectively do so. But if one person extends his left hand and the other forwards his right hand, shaking hands will not be efficient. 

Similarly in LCAO when two orbitals get overlapped they have two possibilities, either they are facing similar signs or they are facing opposite signs. When they are facing similar signs, their wave functions add to get more enhanced wave function. It results in increased electron density in between the nuclei which in turn leads to strong bonding of orbitals and formation of bonding molecular orbital.

When they are facing opposite signs, their wave functions get cancelled by each other and results in zero electron density between the nuclei which leads to the formation of antibonding molecular orbital.

Thus each time when two atomic orbitals get combined, they produce one bonding molecular orbital ψ_(g) and one antibonding molecular orbital ψ_(u).

ψ_(g)  = N { ψ_(A)  + ψ_(B) }

ψ_(u)  = N { ψ_(A)  + (- ψ_(B)) } ⋍ N { ψ_(A)  - ψ_(B) }

Bonding molecular orbital's wave function is denoted by ψ_(g); g stands for gerade that means when you rotate the MO about the line joining the two nuclei and then about a line perpendicular to this line, the signs of lobes remain the same. Antibonding molecular orbital's wave function is denoted by ψ_(u); u stands for ungerade that means odd. When you rotate the MO about the line joining the two nuclei, the sign of lobes changes. In short g and u tells us the symmetry of orbitals about its centre.

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How will you define an established person? Someone who has fewer worries, less anxiety and more security in his life is to be considered as an estabilised person. Similarly in the world of atoms stability means less stress and a cool calm life. If a molecular orbital is formed by the overlapping of similar lobes it forms a strong bond by collaboration of atomic orbitals and gets stabilised. Two atomic orbitals which have opposite sign are not compatible with each other and when they get combined they form an antibonding MO. Because of less compatibily of AOs, antibonding MO has much stress and it becomes destabilised.

This stability of MO’s can be measured by amount of their energy. Minimum energy is a sign of stabilization. Stable MO has lesser energy than their parent AOs and unstable MO has more energy. 

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