Editing Atomic orbital (section)

=== Types of orbital ===
[[File:Atomic-orbital-clouds spdf m0.png|thumb|upright=1.5|3D views of some [[Hydrogen-like atom|hydrogen-like]] atomic orbitals showing probability density and phase ('''g''' orbitals and higher not shown)]]
Atomic orbitals can be the hydrogen-like "orbitals" which are exact solutions to the [[Schrödinger equation]] for a [[Hydrogen-like atom|hydrogen-like "atom"]] (i.e., atom with one electron). Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron (i.e., orbitals) but are used as starting points for approximating wave functions that depend on the simultaneous coordinates of all the electrons in an atom or molecule. The [[coordinate system]]s chosen for orbitals are usually [[spherical coordinates]] {{math|(''r'', ''θ'', ''φ'')}} in atoms and [[Cartesian coordinate system|Cartesian]] {{math|(''x'', ''y'', ''z'')}} in polyatomic molecules. The advantage of spherical coordinates here is that an orbital wave function is a product of three factors each dependent on a single coordinate: {{math|1=''ψ''(''r'', ''θ'', ''φ'') = ''R''(''r'') Θ(''θ'') Φ(''φ'')}}. The angular factors of atomic orbitals {{math|1=Θ(''θ'') Φ(''φ'')}} generate s, p, d, etc. functions as [[Spherical harmonics#Real form|real combinations]] of [[spherical harmonics]] {{math|''Y''<sub>''ℓm''</sub>(''θ'', ''φ'')}} (where {{mvar|ℓ}} and {{mvar|m}} are quantum numbers). There are typically three mathematical forms for the radial functions&nbsp;{{math|''R''(''r'')}} which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons:

# The ''hydrogen-like orbitals'' are derived from the exact solutions of the Schrödinger equation for one electron and a nucleus, for a [[hydrogen-like atom]]. The part of the function that depends on distance ''r'' from the nucleus has radial [[node (physics)|nodes]] and decays as <math> e^{-\alpha r} </math>.
# The [[Slater-type orbital]] (STO) is a form without radial nodes but decays from the nucleus as does a hydrogen-like orbital.
# The form of the [[Gaussian orbital|Gaussian type orbital]] (Gaussians) has no radial nodes and decays as <math> e^{-\alpha r^2} </math>.

Although hydrogen-like orbitals are still used as pedagogical tools, the advent of computers has made STOs preferable for atoms and diatomic molecules since combinations of STOs can replace the nodes in hydrogen-like orbitals. Gaussians are typically used in molecules with three or more atoms. Although not as accurate by themselves as STOs, combinations of many Gaussians can attain the accuracy of hydrogen-like orbitals.