Linear Combinations of f Orbitals

absolute value of

m

ℓ

, |

m

ℓ

|:

0

1

2

3

sign of linear combination:

+

-

color key
real and positive	imaginary and positive
real and negative	imaginary and negative
linear combination:	i(4 f +1 +4 f -1 ) = 4 f y 2 z

It is less common to find the

f

atomic orbitals illustrated in chemistry textbooks than the

s

,

p

, and

d

orbitals. Boundary surface pictures of any of these atomic orbitals typically only show the real part of these complex functions and often leave out the sign information as well. The one-electron wavefunctions resulting from the solution of the Schrödinger equation for the hydrogen atom are complex functions except when

m

ℓ

=0

. The real forms of

f

atomic orbitals can be constructed by taking appropriate linear combinations of the complex forms. Here, boundary surfaces of the

f

orbitals are colored to indicate the real and imaginary components as well as the positive and negative signs.