Atomic orbital ( $ϕ_{n ℓ m} (r)$ )

For a normalized spatial wavefunction,

⟨ r^{2} ⟩ = \int r^{2} ∣ ψ (r) ∣^{2} d^{3} r

. Its square root is the root-mean-square radius, which differs from the exponential decay length. A spherical hydrogenic donor state with

ψ \propto e^{- r / a}

has

⟨ r^{2} ⟩ = 3 a^{2}

, obtained by dividing the radial integrals of

r^{4} e^{- 2 r / a}

and

r^{2} e^{- 2 r / a}

Radial node ( $n - l - 1$ )

A radial position at which an orbital radial wavefunction vanishes. A hydrogen-like orbital has

n - l - 1

radial nodes;

2 s

has one and

2 p

none.

s orbital ( $ℓ = 0$ )

 1  0

An atomic orbital with orbital angular momentum quantum number zero. Its angular part is spherically symmetric; higher principal shells can have radial nodes.

D orbital ( $l = 2$ )

 1  0

An atomic orbital with orbital angular momentum quantum number

l = 2

. Five angular functions span the subshell; a ligand field can split their energies.

Atomic-orbital nodal plane

A plane on which an atomic orbital wavefunction vanishes. For a real p orbital oriented along

x

, it is the plane

x = 0

. The probability density also vanishes on that plane.

p orbital ( $l = 1$ )

 1  0

An atomic orbital with orbital angular momentum quantum number

l = 1

. Real

p_{x}, p_{y}, p_{z}

combinations have opposite quantum phases in their two lobes and a nodal plane through the atomic nucleus. An orbital phase is not an electric charge.

Orbital hybridization

A change of atomic-orbital basis used to construct directed local bonding orbitals.

s p

s p^{2}

and

s p^{3}

describe idealized combinations; they are model descriptions rather than unique experimental labels for every bonded atom.

Hybrid orbital

A linear combination of atomic orbitals used in a localized description of directional bonding or lone pairs. Hybrid labels are models rather than directly observable fixed atomic states.

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Donald Trump  29 Created 2022-11-02 Updated 2025-04-18

This is a section about Atomic orbital!

Atomic orbital is a very important subject about which there is a lot to say.

For example, this sentence. And then another one.

Barack Obama  11 Created 2022-11-02 Updated 2025-04-18

This is a section about Atomic orbital!

Atomic orbital is a very important subject about which there is a lot to say.

For example, this sentence. And then another one.

Wikipedia Bot  1

An atomic orbital is a mathematical function that describes the wave-like behavior of an electron in an atom. This concept arises from quantum mechanics and is fundamental to understanding the structure of atoms and the arrangement of electrons within them. Here are some key points about atomic orbitals: 1. **Shape and Energy**: Atomic orbitals have distinct shapes (spherical, dumbbell-shaped, etc.) and energy levels. The shape and orientation of an orbital are determined by the quantum numbers that describe it.

Ciro Santilli  40 Updated 2025-07-16

In the case of the Schrödinger equation solution for the hydrogen atom, each orbital is one eigenvector of the solution.

Remember from time-independent Schrödinger equation that the final solution is just the weighted sum of the eigenvector decomposition of the initial state, analogously to solving partial differential equations with the Fourier series.

This is the table that you should have in mind to visualize them: en.wikipedia.org/w/index.php?title=Atomic_orbital&oldid=1022865014#Orbitals_table