Quantum Chemistry || Lect # 9 || Radial Distribution Function || Dr Rizwana

Radial distribution function (RDF) is framed as a probability map for where an electron is likely to be found around a nucleus: it describes the electron density in a spherical region centered on the proton, plotted against the distance r from the nucleus. In this picture, electrons occupy quantized shells and orbitals, and the RDF peaks in the regions where the electron’s wavefunction has the highest probability density. The lecture emphasizes that at r = 0 (zero distance from the nucleus), the probability of finding the electron is zero, so the RDF curves start from the origin rather than showing any electron density at the center.

As the distance from the nucleus increases, the RDF behavior changes systematically with the principal quantum number n. For a given shell, the electron is most likely to be found at the radius where the RDF reaches its maximum peak—interpreted as the most probable region for that electron’s presence. The lecture links this to energy: low-energy electrons remain in lower-energy orbitals rather than “jumping” into higher-energy states, so the RDF peak for a shell corresponds to the dominant location of that electron’s probability density. When moving from n1 to n2 to n3, the most probable radius shifts outward, and the lecture gives a scaling idea consistent with the Bohr-model trend: the characteristic distance for n2 is about four times that for n1, and for n3 about nine times that for n1.

A key comparative point is that the RDF peak height and the spatial spread both evolve with n and orbital type. For higher n values, the electron’s “sphere” expands outward, meaning the probability distribution spreads over a larger region of space. The lecture also compares orbital angular momentum types (s, p, d, f) in terms of where the maximum probability density tends to appear: the highest probability density is associated with s orbitals, followed by p, then d, then f, in the order described. Even when probability density becomes very small between shells or between energy regions, it does not drop to zero everywhere; instead, the distribution shows structured minima and regions of suppressed likelihood.

That structure is attributed to the wave nature of electrons. The RDF is tied to the electron’s wavefunction: by taking the square of the wavefunction (the lecture describes using the “square of that specific electron” and plotting it versus r), one obtains the probability density that produces peaks, troughs, and nodes. The lecture describes “trough” regions as representing the probability suppression between two energy levels, where the probability of finding the electron is almost zero; these correspond to nodes and anti-nodes in the wave picture. Overall, RDF is presented as a direct, visual way to connect quantum wave behavior—peaks, gaps, and nodes—to the spatial likelihood of electron positions around the nucleus, setting up the next discussion on penetration and screening effects.