An f-orbital, defined by the quantum number 
, is a region of space around the nucleus where an electron is most likely to be found. The letter ”f” is of spectroscopic origin, standing for ‘fundamental’. f-block elements have either hexagonal close pack, cubic close pack or body centred cubic structures. For non-cubic symmetry systems, we describe them using a general set of orbitals, which are derived in the same way as the d-orbitals. For cubic symmetry systems, they are better described by another set of f-orbitals known as the cubic set, which are derived by taking a different set of linear combinations. Both sets are eigenfunctions of the Schrödinger equation.
One-electron f-orbital wavefunctions can be expressed using the total wavefunction of a hydrogenic atom:
Y_l^{\vert m_l\vert}(\theta,\phi))
where
=N_{n,l}\frac{1}{r}\biggr\(\frac{r}{na_0}\biggr\)^{l+1}e^{-\frac{r}{na_0}}L_{n-l-1}^{2l+1}\biggr\(\frac{2r}{na_0}\biggr\))
, the radial wavefunction, is the radial component of 
.
=N_l^{\vert m_l\vert}P_l^{\vert m_l\vert}(cos\theta)e^{im_l\phi})
, the spherical harmonics, is the angular component of .
^k\frac{(n+l)!}{k!(k+2l+1)!(n-l-1-k)!}\biggr\(\frac{2r}{na_0}\biggr\)^k)
are the associated Laguerre polynomials.
=\frac{1}{2^ll!}(1-cos^2\theta)^{\frac{\vert m_l\vert}{2}}\frac{d^{l+\vert m_l\vert}}{dcos\theta^{l+\vert m_l\vert}}(cos^2\theta-1)^l)
are the associated Legendre polynomials.
!}{2n(n+l)!}\frac{2^{2l+3}}{na_0}})
is the normalisation constant of the radial wavefunction.
!}{(l+\vert m_l\vert)!}\biggr\]^{\frac{1}{2}})
is the normalisation constant of the spherical harmonics.
is the principal quantum number, where 
is the orbital angular momentum quantum number (also known as azimuthal quantum number), where 
and 
.
is the magnetic quantum number, where 
.
General set
The principal quantum number, , is also called a shell. Since 
when , there are seven f-orbitals that are characterised by the set of quantum numbers 
of 
, 
, 
, 
, 
, 
and 
in each shell for 
. Each of the seven f-orbitals has three angular nodes. Consider the set 
. Substituting these values into the explicit formula for Y_l^{\vert m_l\vert}(\theta,\phi))
yields:
Y_3^{0}(\theta,\phi)=r^3(5cos^3\theta-3cos\theta)f(r))
where =\frac{1}{3072\sqrt{5\pi}}\sqrt{\biggr\(\frac{1}{a_0}\biggr\)^9}r^2e^{-\frac{r}{4a_0}})
.
Converting from spherical coordinates to Cartesian coordinates, where 
, gives
-3r^2f(r)\;\;\;\;\;\;\;\;482)
Eq482 is known as the 
orbital. When 
or 
or 
, 
. represents the 
–nodal plane, while 
implies that the angular nodes occur at two conical surfaces with their apices at the origin, extending at )
along the 
-axis.
For the sets 
and 
, we have ^9}r^3e^{-\frac{r}{4a_0}}sin\theta(5cos^2\theta-1) e^{i\phi}})
and ^9}r^3e^{-\frac{r}{4a_0}}sin\theta(5cos^2\theta-1) e^{-i\phi}})
, respectively. These two wavefunctions include the factor 
, which may complicate calculations when they undergo symmetry operations. Therefore, we take linear combinations of these wavefunctions to form simpler wavefunctions. The first linear combination is ^9}r^3e^{-\frac{r}{4a_0}}sin\theta\cos\phi(5cos^2\theta-1)})
, which normalises to ^9}r^3e^{-\frac{r}{4a_0}}sin\theta\cos\phi(5cos^2\theta-1)})
, or equivalently in Cartesian coordinates:
xf(r)\;\;\;\;\;\;\;\;483)
where =\frac{1}{1024\sqrt{30\pi}}\sqrt{\biggr\(\frac{1}{a_0}\biggr\)^9}e^{-\frac{r}{4a_0}})
, 
and .
One angular node occurs at 
or when 
, which represents the 
-nodal-plane. The second and third angular nodes correspond to 
. These nodes are described by two conical surfaces with their apices at the origin, extending at )
along the -axis.
The second normalised linear combination is ^9}r^3e^{-\frac{r}{4a_0}}(5cos^2\theta-1)sin\theta\sin\phi})
, or equivalently,
yf(r)\;\;\;\;\;\;\;\;484)
where 
.
One angular node occurs at 
, which represents the 
-nodal plane. The second and third angular nodes correspond to . These nodes are described by two conical surfaces with their apices at the origin, extending at along the -axis.
For the remaining sets 
, 
, 
and 
, we apply the same logic to give:
Cubic set
Out of the seven cubic f-orbitals, three of them, 
, }})
and 
are the same as the general set of f-orbitals. The other four derived by taking different linear combinations of general set of f -orbitals. They are:
