83.CHEMICAL BONDING (30)- Covalent Bonding(29) – Molecular Orbital Theory(4)- LCAO approximation(2).

 

We have seen in the periodic table series , that the electronic configuration tells us where exactly the electron is. Configuration means order… so electronic configuration gives us the order in which the atomic orbitals are filled.
For e.g.- Electronic configuration of hydrogen atom is 1s1. So , we know that there is only one electron  that resides in the orbital which is closest to the nucleus of the atom in s – orbital. 

In quantum mechanics , the electronic configuration is described by wave functions. In post no 33 we have studied wave function in detail. We know that, this parameter is the mathematical description of sub atomic particles , which behave as waves. So, in MOT too , the wave function will give us information about the electron in that  molecular orbital.

 

How can we explain the phenomenon of bonding and anti bonding molecular orbitals by considering wave functions?

When we are talking about position of a particle (electron in this case) in space, the wave function (Ψ) corresponds to probability amplitude wave function.

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In simple words this means that when we are talking about wave function , we are essentially talking about the amplitude of the electron in that orbital i.e the height of that electron wave.The height of the electron wave gives us an idea of how much region in space can that wave travel  i.e in how much region in space can we hope to find that electron.This is nothing but an orbital, which in quantum mechanics is the region where probability of finding the electron is highest.

We can now relate this to what we learnt in post no 33 ! We can compare the figure of an orbital, we studied there as follows –

831.jpg

 

Now that we have understood how we look at electron waves , lets see what happens to the amplitude of these individual atomic electron waves when they superimpose.

For a bonding molecular orbital , there is CONSTRUCTIVE INTERFERENCE. The amplitudes of two atomic waves add up to give a larger amplitude resultant wave.This means that the probability of finding the electron in the region between the two nuclei increases i.e the probability density (Ψ2) increases.

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Note that the amplitude increases in the region where the two atomic waves overlap.

 

For a anti bonding molecular orbital , there is DESTRUCTIVE INTERFERENCE. The amplitudes of two atomic waves cancel or subtract each other.

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This means that the probability of finding the electron in the region between the two nuclei greatly decreases i.e the probability density (Ψ2) decreases.

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The two orbitals  cancel themselves only at a place where amplitudes of both waves is equal  -the point equidistant from both atoms. This is the point where the amplitude is zero and it is referred to as a node.At a node there is NO PROBABILITY OF FINDING AN ELECTRON .At all other points, one of the AO’s amplitude is larger (in magnitude) than the other one.Thus, there is  a residual amplitude being left behind even after subtraction from destructive interference, which allows an electron to persist there.

For simple diatomic molecules, the mathematical representation for BMOs and ABMOs can be shown as follows –

8394 copy

 

The adjustable or weighting coefficients can be positive or negative depending on the symmetry and energy of an atomic orbital. The coefficients tell us how much the AO is contributing to make the respective MO i.e they give the amount of mixing in the MO. The atomic wavefunctions refer to the wavefunctions of the valence orbital of that atom. As seen above, simple addition and subtraction helps us to derive the MO wavefunction.

To find probability density , we square the wavefunction Ψ. A wavefunction(i.e an orbital cannot be seen as such.The probability density i.e Ψgives us tangible results as it refers to the region where there is probability of finding that electron. Normalizing the probability density gives us numerical results.Tau (τ) in the equations below, is a Ramanujan’s entity and we do not need to earn about it in detail as it is a mathematical concept , beyond our course of study.

8397.jpg

 

For homonuclear diatomic molecules, c12=c2as both atoms contribute equally to form the MO.Thus, we can infer that c1=±c[(2)2 = 4 and (-2)2 = 4 too].

When , c1=cthe wavefunctions add → constructive interference
When , c1= – cthe wavefunctions add → constructive interference

We can expand the equation for diatomic molecules to more atoms(not more than 6 usually) for a multi atomic molecule. The summation (Σ) i.e the addition of all wavefunctions will give the wavefunction of the resultant MO.This can be represented as follows –

8396.jpg

 

Now that we have a clear picture of bonding and anti bonding orbitals , let us start constructing MO diagrams in our next post. Till then,

Be a perpetual student of life and keep learning …

 

 

References and Further Reading –

  1. https://www.youtube.com/watch?v=O192jrR80oo&t=328s
  2. https://www.quora.com/If-antibonding-orbitals-are-formed-as-a-result-of-destructive-interference-then-how-are-electrons-present-in-those-orbitals
  3. https://www.youtube.com/watch?v=1CH6CeORL_g
  4. https://en.wikipedia.org/wiki/Molecular_orbital
  5. https://en.wikipedia.org/wiki/Probability_amplitude

 

 

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