I know that orbitals are probability distributions. Are electron shells probability distributions too?


3 Answers 3


Short Answer:


Long Answer:

First, strictly speaking, the orbitals themselves in the quantum mechanical sense are not probability distributions. They are eigenfunctions $\Psi_i$ of the Hamiltonian as defined by the time-independent Schroedinger equation $H\Psi_i=E_i\Psi_i$.

The probability distribution function $p(\vec r)$ for electrons is generated by summing the products of the eigenfunctions and their complex conjugates for all occupied eigenfunctions at a given 3-D point $\vec r$ and dividing by the total number $N$ of electrons in the system:

$$ p(\vec r) = \frac{1}{N}\sum_{occ}{\Psi_i^*(\vec r)\Psi_i(\vec r)} $$

Note that $p(\vec r)$ is simply $\frac{1}{N}$ times the electron density function $\rho(\vec r) = \sum_{occ}{\Psi_i^*(\vec r)\Psi_i(\vec r)}$.

To be fair, prior to a detailed introduction to the quantum mechanical description of electronic structure, the distinction between the electronic wavefunction $\Psi(\vec r)$ and the electron density function $\rho(\vec r)$ is typically not clearly established in chemistry instruction, so your having conflated them is quite understandable.

Second, again strictly speaking, electron shells are not probability functions. While Pushkar Soni is correct that the principal quantum number for a given electron approximately indicates the 'electron shell' to which it is assigned, the indistinguishability of electrons (among other factors) in general prohibits unambiguous translation of quantum number information into the real-space function $\rho(\vec r)$. The best we can do is to interpret electron shells as corresponding to certain features of the structure of $\rho(\vec r)$. Offhand, I know of at least two ways of qualitatively identifying 'electron shells' from $\rho(\vec r)$.

First, as Geoff Hutchinson notes, the radial distribution function $r^2\rho(r)$ shows local maxima that can be interpreted as electron shells. Note that $\rho(r)$ in the r.d.f here is a function only of $r$, the distance from an isolated atom/nucleus; the angular dependencies (on $\theta$ and $\varphi$) have been integrated out.

Second, Bader's QTAIM describes how the Laplacian of the electron density, $\nabla^2\rho(\vec r)$, exhibits minima in 3-D space at positions corresponding to the notion of electron shells.

  • $\begingroup$ Good answer. But could you please correct the second sentece in the Long Answer: $\Psi_{i}$ is an eigenfunction of the Hamiltonian. It is also a solution of the time-independent Schrödinger equation, but its not an eigenfunction Schrödinger equation since it is just a nonsense. ;) $\endgroup$
    – Wildcat
    May 28, 2015 at 15:45
  • $\begingroup$ @Wildcat Very good point. Looks like someone else beat me to the edit, though. :-) $\endgroup$
    – hBy2Py
    May 28, 2015 at 21:10

In order to understand you should know the differences between shell, sub-shell and orbital.

  1. Shell represents the main energy level occupied by the electron. It is given by the principal quantum number (n). For example , if n=1, the electron is present in the first main shell, called K-shell. if n=2, the electron is present in the second main shell, called L-shell and so on..

  2. Sub-shell represents the sub-energy level occupied by the electron (as main energy level is considered to consist of a number of energy sub-levels ). it is given by azimuthal quantum number. subshell corresponding to l=0,1,2,3 are represented by s,p,d,and respectively.

  3. Orbitals you know it well.

  4. You may understand it like this-

when n=1 we have only s subshell

when n=2 we have s and p subshell

for n=2 we have s,p and d subshell and so on.

  1. Now, s, p and d and f have certain regions, which we call orbitals. which are probability regions as you know. Here is the link for you,


Must watch the video associated with it named [atomic orbitals and periodic construction]

  1. Last but not the least- we certainly not have electron shells probability distribution. its only orbitals. shells are created by scientists to study the atom systematically but they have physical significance too (i mean here the subshells and orbitals lie). it seems you are not clear with - shell, sub-shell and orbitals. you should first check out what they actually means in order to understand.
  1. Electron shell is well described into the old quantum mechanics developed by Bohr, see Atom Bohr model
  2. Probability distribution is well described into the modern quantum chemistry, see Quantum mechanics, general description

Therefore, both definitions are not comparable among them because each one belongs to different theory conceptions.

  • 5
    $\begingroup$ -1 Uh no. If you take the radial distribution function $r^2 R^2(r)$, you find peaks for each radial wavefunction that correspond somewhat reasonably to the "shell" concept. $\endgroup$ May 28, 2015 at 11:54
  • 1
    $\begingroup$ Even more directly, Bader's QTAIM, which is built off of the electron density $\rho$ -- which is precisely the probability distribution of electrons in a system -- recovers the 'electron shell' phenomenon via the negative Laplacian of the electron density, $-\nabla^2\rho$. $\endgroup$
    – hBy2Py
    May 28, 2015 at 12:48

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