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Electron density, ψ2

The square of the wave function, ψ2, represents electron density at any given point. Since the square of any real number is zero or positive, it is clear that electron density cannot be negative, as expected intuitively.


In many cases, the radial wave function passes through zero. These regions are described as radial nodes, or spherical radial nodes since this describes their shape.

In other cases, the angular wave function passes through zero. These regions are described as angular nodes, or nodal planes in those cases where they are planar. Not all angular nodes are planar: some are conical, for instance.

Radial distribution function, 4πr2ψ2

It is often useful to know the likelihood of finding the electron in an orbital at any given distance away from the nucleus. This enables us to say at what distance from the nucleus the electron is most likely to be found, and also how tightly or loosely the electron is bound in a particular atom. This is expressed by the radial distribution function. For s-orbitals, the radial distribution function is given by multiplying the electron density by 4πr2.

Wave equation, ψ

An orbital is a mathematical function called a wave function that describes an electron in an atom. The wave functions, ψ, of the atomic orbitals can be expressed as the product of a radial wave function, R and an angular wave function, Y. Radial wave functions for a given atom depend only upon the distance, r from the nucleus. Angular wave functions depend only upon direction, and, in effect, describe the shape of an orbital.

ψ = radial function × angular function
= R × Y

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