Configuration means order. The electronic configuration of an element describes how electrons are distributed in various atomic orbitals in the atom. Thus, electronic configuration gives us the order in which the atomic orbitals are filled.
e.g.- The electronic configuration of the hydrogen atom is 1s1. This configuration tells us that, there is only one electron that resides in the orbital which is closest to the nucleus of the atom (s- orbital).
In quantum mechanics, the electronic configuration is described by wave functions. A wavefunction is the mathematical description of subatomic particles. At the subatomic level, electrons behave both as particles and as waves. This is called wave-particle duality. MO theory, models electrons as waves and the wave function is the description of an electron in a particular molecular orbital.
How can we explain the phenomenon of bonding and anti-bonding molecular orbitals by considering wave functions?
The wave function (Ψ) that describes the position of a particle (electron in this case) in space, corresponds to the probability amplitude wave function.
In simple words, this means that when we are talking about wave function, we are referring to 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 can the wave travel in space. This is the region where the probability of finding the electron is the highest. This region is nothing but an orbital. Thus, orbitals are regions in space, where the probability of finding electrons is the highest.
We can now relate this to what we learned in post 33! We can compare the figure of an orbital, we studied there as follows –
Now that we have understood the electron waves, let us see what happens to the amplitude of these individual atomic electron waves when they superimpose.
When the waves constructively interfere, we get a bonding molecular orbital. The amplitudes of two atomic waves add up to give a resultant wave, with a larger amplitude. This means that the probability of finding the electron in the region between the two nuclei increases i.e the probability density (Ψ2) increases.
In the figure above, we see two atoms (a and b) with an electron in the 1s orbital. The electron wave in 1sa orbital is shown in red color. The electron wave in 1sb orbital is shown in blue color. These two electron waves superimpose constructively to form a resultant wave (shown in purple color – red+blue). This is the bonding molecular orbital(σ1s). Note that the amplitude of the resultant wave increases in the region where the two atomic waves overlap.
An anti-bonding molecular orbital is formed by DESTRUCTIVE INTERFERENCE. Here, the amplitudes of two atomic waves cancel or subtract each other. 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.
The above figure depicts two electron waves(one red and one blue), which are out-of-phase. When these two waves superimpose, the resultant wave(purple color) has no or very little electron density between the nuclei. This means that there is no or very little probability of finding the electron between the two nuclei. This region is called a ‘node‘.At the node, the amplitude of the resultant wave is zero.
The two orbitals cancel themselves only at a place where the amplitudes of both waves are equal -at the point equidistant from both atoms. 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 –
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 making 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 help us to derive the MO wavefunction.
To find probability density, we square the wavefunction Ψ. A wave function or an orbital is not an observable entity. The probability density i.e Ψ2 gives us tangible results. It refers to the region where there is the probability of finding that electron. Normalizing the probability density gives us numerical results. Tau (τ) in the equations below, is a Ramanujan entity and we do not need to learn about it, as it is a mathematical concept. It is beyond the course of our study.
For homonuclear diatomic molecules, c12=c22 as both atoms contribute equally to form the MO.Thus, we can infer that c1= ±c2 [(2)2 = 4 and (-2)2 = 4 too].
When c1=c2 the wavefunctions add → constructive interference
When c1= – c2 the wavefunctions subtract → destructive 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 –
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 …
Good Day!
References and Further Reading –
- https://www.youtube.com/watch?v=O192jrR80oo&t=328s
- https://www.quora.com/If-antibonding-orbitals-are-formed-as-a-result-of-destructive-interference-then-how-are-electrons-present-in-those-orbitals
- https://www.youtube.com/watch?v=1CH6CeORL_g
- https://en.wikipedia.org/wiki/Molecular_orbital
- https://en.wikipedia.org/wiki/Probability_amplitude
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