A node is a region in space where a wavefunction is zero. This implies that the probability of finding an electron at a node is zero. Since a wavefunction can have two components—radial and angular—it can have both radial and angular nodes.
A radial node is a spherical surface where the radial wavefunction of an atomic orbital is zero. The number of radial nodes
in a hydrogenic wavefunction
is given by the formula:
where is the principal quantum number and
is the angular momentum quantum number.
Question
How is eq471 derived?
Answer
When , the associated Laguerre polynomial
. Since the upper summation index of the explicit form of
is
, the polynomial will have
roots if
. In other words,
will be zero (a node) as many times as
.
An angular node is a flat plane or a cone where the angular wavefunction (also known as spherical harmonics) is zero. Using the same logic, the number of angular nodes in
corresponds to the number of roots when
, where
are the associated Legendre polynomials. Since
, the polynomial is of degree
. Therefore,
The total number of nodes in is
The number of radial nodes helps define the spatial distribution of the electron density in orbitals. In other words, they influence the shape and energy of these orbitals, and therefore play a role in chemical bonding between orbitals.