This page gives an overview on different orbitals and their shapes for hydrogenic (hydrogen-like single electron atoms) such as H, He⁺, Li²⁺, ... and so on. These pages here provide visualizations for all shapes of 1s- up to the 6d-orbitals, including the more uncommon 4f, 5f (general and cubic sets), and even 5g-orbitals. All representation of atomic orbitals in these sections were calculated from the pure Cartesian wave functions, respectively.

• Quantum Numbers
• Visualizations

All graphics in this section on this page were created using the MolArch⁺ program and POVRAY Persistence of Vision Raytracer. Cartesian wave functions were taken from The Orbitron Gallery of Atomic Orbitals and Molecular Orbitals. Links are provided for still images as well as to 3D-models (Chime and VRML file formats, both require a plugin appropriate for viewing these file types to be installed).

Lobes and Nodes of a 5g-Orbital

Atomic orbitals are representations of the three-dimensional volume and thus the regions in space where electrons are likely to be found around a nucleus. They cannot be observed experimentally, however, the electron density of atoms can be observed experimentally. Since it is impossible to determine the exact position of an electron in an atom, atomic orbitals display the probability of finding an electron at a given position. Orbitals in atoms are characterized by their corresponding Quantum Numbers (QN):

Principal Quantum Number	n	The Principal Quantum Number n = 1, 2, ... describes the size of orbitals. For hydrogenic orbitals, the overall energy and the distance of an orbital from the nucleus depends only on n. Orbitals with the same Principal QN are said to belong to the same shell of an atom (for example the 2s and 2p orbitals, see below); atomic orbitals with n = 1 belong to the "K" shell, n = 2 describes the "L" shell, n = 3 corresponds to the "M" shell, and so on.
Angular Quantum Number	l	The Angular Quantum Number l = 0, ... n-1 (or Azimuthal Quantum Number) describes the shape of orbitals, corresponding to the angular momentum of the state. The s-orbitals (l = 0) are spherical, the p-orbitals (l = 1) are dumbbell shaped, and the d-orbitals (l = 2) feature a clover-leaf like appearance. Orbitals of higher l (l = 2, 3, ...) are the f-, g-, ... orbitals with more complex shapes (for the f-orbitals see below). Orbitals with the same Angular QN belong to the same sub-shell of an atom, but differ in their orientation in space (e.g. the 2px, 2py, and 2pz orbitals, see below). Each of the different angular momentum states takes up to 2(2l + 1) electrons. Historically, the descriptions for the s-, p-, d-, and f-orbitals were derived from a system of categorizing spectral lines as "strong", "principal", "diffuse", or "fundamental". The four types of orbitals were first associated with these spectral line types, the designations "g" ... (with the omission of "j") were added later in alphabetical order.
Magnetic Quantum Number	ml	The Magnetic Quantum Number ml = 0, ±1, ... ±l describes the orientation of orbitals in space. Orbitals of different Magnetic QN have different angular momentum around the z-axis, thus electrons in orbitals of different ml act differently with respect to an external magnetic field applied along the z-axis, e.g. the 2pz (ml = 0) orbital versus the 2px and 2py (ml = ±1) orbitals. This quantum number can be thought of - although somewhat inaccurately - as the quantized projection of the angular momentum vector on the z-axis.
Spin Quantum Number	ms	The Spin Quantum Number s = ±½ describes the spin (direction of rotation, clockwise or counter-clockwise) of the electrons in each orbital. Every orbitals holds a maximum of two electron with opposite spin (Spin Quantum Numbers of opposite sign).

While it is impossible to explicitly solve the Schrödinger equation for multi-electron systems such as "real" atoms (except Hydrogen), the corresponding wave functions and thus the exact solution to the Schrödinger equation can be given for hydrogenic atoms (single electron atoms) such as H, He⁺, Li²⁺, ... The following table provides an overview on the different types and shapes of orbitals from the 1s up to the 6d level (including the less common and less well-known f- and g-orbitals which are not occupied in known atoms in their ground states, but which nevertheless may be of interest in higher and excited states). The orbitals are usually visualized as iso-contour surfaces on the electron density, thus a 90% probability surface displays the three-dimensional volume in which an electron is to be found with a 90% chance. Yellow and blue colors indicate regions of opposite sign of the wave function ψ (the electron density is proportional to ψ²); and the "nodal" planes indicate spatial areas (actually planes, spheres, and cones) were the wave function passes through zero and changes sign.

All graphics below were calculated from the pure Cartesian wave functions of the corresponding orbitals, the images are not scaled relative to each other, so the size of the orbitals may not be directly compared to each other (for orbitals of "real" atoms see the atomic orbitals derived from DFT calculations). Click on the individual images to obtain enlarged visualizations of the orbitals, respectively.

Quantum Numbers

angular QN (l)

l = 0

l = 1

l = 2

principal QN (n )

magnetic QN (ml)

ml = 0

ml = -1, 0, +1

ml = -2, -1, 0, +1, +2

n = 1

s- Orbital

1s

n = 2

s- Orbital

p- Orbitals

2s

2px

2py

2pz

n = 3

s- Orbital

p- Orbitals

d- Orbitals

3s

3px

3py

3pz

3dxy

3dxz

3dyz

3dx2-y2

3dz2

n = 4

s- Orbital

p- Orbitals

d- Orbitals

4s

4px

4py

4pz

4dxy

4dxz

4dyz

4dx2-y2

4dz2

l = 3 ml = -3, ... +3 (general set)

f- Orbitals

4fz3

4fxz2

4fyz2

4fy(3x2-y2)

4fx(x2-3y2)

4fz(x2-y2)

4fxyz

l = 3 ml = -3, ... +3 (cubic set)

f- Orbitals

4fz3

4fy3

4fx3

4fx(z2-y2)

4fy(z2-x2)

4fz(x2-y2)

4fxyz

n = 5

s- Orbital

p- Orbitals

d- Orbitals

5s

5px

5py

5pz

5dxy

5dxz

5dyz

5dx2-y2

5dz2

l = 3 ml = -3, ... +3 (general set)

f- Orbitals

5fz3

5fxz2

5fyz2

5fy(3x2-y2)

5fx(x2-3y2)

5fz(x2-y2)

5fxyz

l = 3 ml = -3, ... +3 (cubic set)

f- Orbitals

5fz3

5fy3

5fx3

5fx(z2-y2)

5fy(z2-x2)

5fz(x2-y2)

5fxyz

l = 4 ml = -4, ... +4

g- Orbitals

5gz4

5gz3x

5gz3y

5gz2xy

5gz2(x2-y2)

5gzx3

5gzy3

5gxy(x2-y2)

5gx4+y4

n = 6

s- Orbital

p- Orbitals

d- Orbitals

6s

6px

6py

6pz

6dxy

6dxz

6dyz

6dx2-y2

6dz2

Note: The f-orbitals are special in as much as two different sets of functions are commonly in use, the general and the cubic set. The latter cubic set may be appropriate for describing atoms in an environment of cubic symmetry. Both sets have three orbitals in common (nf_z³, nf_xyz, and nf_z(x²-y²) with n = 4 (4f), 5 (5f), ...).