Hydrogenic Atomic Orbitals

This page gives an overview on different orbitals and their shapes for hydrogenic (hydrogenlike single electron atoms) such as H, He ^{+}, Li ^{2+}, ... and so on.
These pages here provide visualizations for all shapes of 1s up to the 6dorbitals, including the more uncommon 4f, 5f (general and cubic sets), and even 5gorbitals.
All representation of atomic orbitals in these sections were calculated from the pure Cartesian wave functions, respectively.
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 3Dmodels (Chime and VRML file formats, both require a plugin appropriate for viewing these file types to be installed).


Lobes and Nodes of a 5gOrbital 

Hydrogenic Atomic Orbitals  Quantum Numbers

Atomic orbitals are representations of the threedimensional 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, ... n1 (or Azimuthal Quantum Number) describes the shape of orbitals, corresponding to the angular momentum of the state.
The sorbitals (l = 0) are spherical, the porbitals (l = 1) are dumbbell shaped, and the dorbitals (l = 2) feature a cloverleaf like appearance.
Orbitals of higher l (l = 2, 3, ...) are the f, g, ... orbitals with more complex shapes (for the forbitals see below).
Orbitals with the same Angular QN belong to the same subshell of an atom, but differ in their orientation in space (e.g. the 2p_{x}, 2p_{y}, and 2p_{z} 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 forbitals 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  m_{l} 
The Magnetic Quantum Number m_{l} = 0, ±1, ... ±l describes the orientation of orbitals in space. Orbitals of different Magnetic QN have
different angular momentum around the zaxis, thus electrons in orbitals of different m_{l} act differently with respect to an external magnetic field applied along the zaxis,
e.g. the 2p_{z} (m_{l} = 0) orbital versus the 2p_{x} and 2p_{y} (m_{l} = ±1) orbitals. This quantum number
can be thought of  although somewhat inaccurately  as the quantized projection of the angular momentum vector on the zaxis.

Spin Quantum Number  m_{s} 
The Spin Quantum Number s = ±½ describes the spin (direction of rotation, clockwise or counterclockwise) of the electrons in each orbital.
Every orbitals holds a maximum of two electron with opposite spin (Spin Quantum Numbers of opposite sign).


Hydrogenic Atomic Orbitals  Visualizations

While it is impossible to explicitly solve the Schrödinger equation for multielectron 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 ^{+}, Li ^{2+}, ...
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 wellknown f and gorbitals
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 isocontour surfaces on the electron density, thus a 90% probability surface displays the threedimensional 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 ψ^{2});
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 (m_{l}) 
m_{l} = 0 
m_{l} = 1, 0, +1 
m_{l} = 2, 1, 0, +1, +2 

n = 1 
s Orbital 


1s 

n = 2 
s Orbital 

p Orbitals 




2s 

2p_{x} 
2p_{y} 
2p_{z} 

n = 3 
s Orbital 

p Orbitals 



d Orbitals 






3s 

3p_{x} 
3p_{y} 
3p_{z} 

3d_{xy} 
3d_{xz} 
3d_{yz} 
3d_{x2y2} 
3d_{z2} 

n = 4 
s Orbital 

p Orbitals 



d Orbitals 






4s 

4p_{x} 
4p_{y} 
4p_{z} 

4d_{xy} 
4d_{xz} 
4d_{yz} 
4d_{x2y2} 
4d_{z2} 
l = 3 m_{l} = 3, ... +3 (general set) 
f Orbitals 








4f_{z3} 
4f_{xz2} 
4f_{yz2} 
4f_{y(3x2y2)} 
4f_{x(x23y2)} 
4f_{z(x2y2)} 
4f_{xyz} 
l = 3 m_{l} = 3, ... +3 (cubic set) 
f Orbitals 








4f_{z3} 
4f_{y3} 
4f_{x3} 
4f_{x(z2y2)} 
4f_{y(z2x2)} 
4f_{z(x2y2)} 
4f_{xyz} 

n = 5 
s Orbital 

p Orbitals 



d Orbitals 






5s 

5p_{x} 
5p_{y} 
5p_{z} 

5d_{xy} 
5d_{xz} 
5d_{yz} 
5d_{x2y2} 
5d_{z2} 
l = 3 m_{l} = 3, ... +3 (general set) 
f Orbitals 








5f_{z3} 
5f_{xz2} 
5f_{yz2} 
5f_{y(3x2y2)} 
5f_{x(x23y2)} 
5f_{z(x2y2)} 
5f_{xyz} 
l = 3 m_{l} = 3, ... +3 (cubic set) 
f Orbitals 








5f_{z3} 
5f_{y3} 
5f_{x3} 
5f_{x(z2y2)} 
5f_{y(z2x2)} 
5f_{z(x2y2)} 
5f_{xyz} 
l = 4 m_{l} = 4, ... +4 
g Orbitals 










5g_{z4} 
5g_{z3x} 
5g_{z3y} 
5g_{z2xy} 
5g_{z2(x2y2)} 
5g_{zx3} 
5g_{zy3} 
5g_{xy(x2y2)} 
5g_{x4+y4} 

n = 6 
s Orbital 

p Orbitals 



d Orbitals 






6s 

6p_{x} 
6p_{y} 
6p_{z} 

6d_{xy} 
6d_{xz} 
6d_{yz} 
6d_{x2y2} 
6d_{z2} 

Note: The forbitals 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(x2y2)}
with n = 4 (4f), 5 (5f), ...).
