n

m

Short-hand notation

Coefficient (α_nm)

Spatial dependence function

Spherical

Cartesian

1

1

X

α₁₁

$r\cdot \sin \theta \cdot \cos \varphi$

x

1

0

Z

α₁₀

$r\cdot \cos \theta$

z

1

−1

Y

α_1-1

$r\cdot \sin \theta \cdot \sin \varphi$

y

2

2

X²-Y²

α₂₂

$r^2\cdot {\sin }^2\theta \cdot \cos 2\varphi$

$x^2-y^2$

2

1

ZX

α₂₁

$r^2\cdot \sin \theta \cdot \cos \theta \cdot \cos \varphi$

xz

2

0

Z2C

α₂₀

$r^2\cdot \left(3{\cos }^2\theta -1\right)/2$

$z^2-\left(x^2+y^2\right)/2$

2

−1

ZY

α_2-1

$r^2\cdot \sin \theta \cdot \cos \theta \cdot \sin \varphi$

yz

2

−2

XY

α_2-2

$r^2\cdot {\sin }^2\theta \cdot \cos \varphi \cdot \sin \varphi$

xz