What are the different kinds of f orbitals?

So there is only one kind of #f# orbitals and that is the #f# orbital. I suppose you mean the different shapes of the #f# orbitals.

So the #f# orbitals have 7 different shapes (since #n=4# and #l=3# resulting in: #-3, -2, -1, 0, 1, 2, 3#) These structures can be seen in this picture:

With the different dimensions. Hope it helps.

They are quite complicated, and can often do combinations of #sigma#, #pi#, #delta#, and even #phi# bonding.

For an introduction into these kinds of bonds:

#sigma# bonds are in every chemical bond. #pi# bonds start showing up in double and triple bonds (e.g. #"O"_2#, #"N"_2#, etc), #delta# bonds start showing up in quadruple bonds (see link), and #phi# bonds aren't seen until a sextuple bond is made (e.g. in #"Mo"_2# or #"W"_2#).

The #4f# orbitals can be separated into three types (here, we use the convention that outer atoms point their #y# axes inwards and #z# axes upwards):

#1)# Two lobes - #sigma# bonding only (#m_l = 0#)

• The #f_(z^3)# (#m_l = 0#) is the only one that only #sigma# bonds. It can bond head-on along the #z# axis.

#2)# Six lobes - #sigma# and #pi# bonding, OR #phi# bonding only (#m_l = -3, +3, -1, +1#)

• The #f_(y(3x^2 - y^2))# (#m_l = -3#) can #sigma# bond along the #x# axes (for example, with a #p_y# orbital) AND #pi# bond along the #y# axes (for example, with a #p_x# orbital, or a #d_(xy)# orbital).

It can alternatively form a #phi# bond (a six-lobed side-on overlap) along the #xy# plane (with another #f_(y(3x^2 - y^2))# orbital in a bimetallic complex).

• The #f_(x(x^2 - 3y^2))# (#m_l = +3#) can #sigma# bond along the #y# axes (for example, with a #p_y# orbital) AND #pi# bond along the #x# axes (for example, with a #p_x# orbital, or a #d_(xy)# orbital).

It can alternatively form a #phi# bond (a six-lobed side-on overlap) along the #xy# plane (with another #f_(x(x^2 - 3y^2))# orbital in a bimetallic complex).

• The #f_(yz^2)# (#m_l = -1#) can form decent #sigma# bonds along the #y# axes, AND/OR #pi# bonds along the #y# AND #z# axes.

It can alternatively form a #phi# bond (a six-lobed side-on overlap) along the #yz# plane (with another #f_(yz^2)# orbital in a bimetallic complex).

• The #f_(xz^2)# (#m_l = +1#) can form decent #sigma# bonds along the #x# axes, AND/OR #pi# bonds along the #x# AND #z# axes.

It can alternatively form a #phi# bond (a six-lobed side-on overlap) along the #xz# plane (with another #f_(xz^2)# orbital in a bimetallic complex).

#3)# Eight lobes - #pi# bonding OR #delta# bonding (#m_l = -2, +2#)

• The #f_(z(x^2 - y^2))# (#m_l = -2#) is for #pi# bonding along ANY of the axes, #x,y#, or #z#. The lobes lie above and below each of the axes, but also along them.

It can alternatively form a #delta# bond with another #f_(z(x^2 - y^2))# orbital in a bimetallic complex.

• The #f_(xyz)# (#m_l = +2#) is for #delta# bonding along ANY of the planes (#xz, yz, xy#) (for example, with #d_(xy)#, #d_(xz)#, or #d_(yz)# orbitals).

It can alternatively form a #pi# bond with another #f_(xyz)# orbital in a bimetallic complex.