Atomic orbital

In quantum mechanics, an atomic orbital ( ) is a function describing the location and wave-like behavior of an electron in an atom. This function describes an electron's charge distribution around the atom's nucleus, and can be used to calculate the probability of finding an electron in a specific region around the nucleus. Each orbital in an atom is characterized by a set of values of three quantum numbers n, ℓ, and mℓ, which respectively correspond to an electron's energy, its orbital angular momentum, and its orbital angular momentum projected along a chosen axis (magnetic quantum number). The orbitals with a well-defined magnetic quantum number are generally complex-valued. Real-valued orbitals can be formed as linear combinations of mℓ and −mℓ orbitals, and are often labeled using associated harmonic polynomials (e.g., xy, x2 − y2) which describe their angular structure. An orbital can be occupied by a maximum of two electrons, each with its own projection of spin m s {\displaystyle m_{s}} . The simple names s orbital, p orbital, d orbital, and f orbital refer to orbitals with angular momentum quantum number ℓ = 0, 1, 2, and 3 respectively. These names, together with their n values, are used to describe electron configurations of atoms. They are derived from description by early spectroscopists of certain series of alkali metal spectroscopic lines as sharp, principal, diffuse, and fundamental. Orbitals for ℓ > 3 continue alphabetically (g, h, i, k, ...), omitting j because some languages do not distinguish between letters "i" and "j". Atomic orbitals are basic building blocks of the atomic orbital model (or electron cloud or wave mechanics model), a modern framework for visualizing submicroscopic behavior of electrons in matter. In this model, the electron cloud of an atom may be seen as being built up (in approximation) in an electron configuration that is a product of simpler hydrogen-like atomic orbitals. The repeating periodicity of blocks of 2, 6, 10, and 14 elements within sections of the periodic table arises naturally from the total number of electrons that occupy a complete set of s, p, d, and f orbitals, respectively, though for higher values of quantum number n, particularly when the atom bears a positive charge, energies of certain sub-shells become very similar and therefore, the order in which they are said to be populated by electrons (e.g., Cr = [Ar]4s13d5 and Cr2+ = [Ar]3d4) can be rationalized only somewhat arbitrarily.

Tables

· Quantum numbers › Complex orbitals

n = 1

Col 1

n = 1

ℓ = 0 (s)

m ℓ = 0 {\displaystyle m_{\ell }=0}

n = 2

Col 1

n = 2

ℓ = 0 (s)

ℓ = 1 (p)

−1, 0, 1

n = 3

Col 1

n = 3

ℓ = 0 (s)

ℓ = 1 (p)

−1, 0, 1

ℓ = 2 (d)

−2, −1, 0, 1, 2

n = 4

Col 1

n = 4

ℓ = 0 (s)

ℓ = 1 (p)

−1, 0, 1

ℓ = 2 (d)

−2, −1, 0, 1, 2

ℓ = 3 (f)

−3, −2, −1, 0, 1, 2, 3

n = 5

Col 1

n = 5

ℓ = 0 (s)

ℓ = 1 (p)

−1, 0, 1

ℓ = 2 (d)

−2, −1, 0, 1, 2

ℓ = 3 (f)

−3, −2, −1, 0, 1, 2, 3

ℓ = 4 (g)

−4, −3, −2, −1, 0, 1, 2, 3, 4

	ℓ = 0 (s)	ℓ = 1 (p)	ℓ = 2 (d)	ℓ = 3 (f)	ℓ = 4 (g)
n = 1	m ℓ = 0 {\displaystyle =0}
n = 2	0	−1, 0, 1
n = 3	0	−1, 0, 1	−2, −1, 0, 1, 2
n = 4	0	−1, 0, 1	−2, −1, 0, 1, 2	−3, −2, −1, 0, 1, 2, 3
n = 5	0	−1, 0, 1	−2, −1, 0, 1, 2	−3, −2, −1, 0, 1, 2, 3	−4, −3, −2, −1, 0, 1, 2, 3, 4

· Quantum numbers › Real orbitals

ℓ = 0 {\displaystyle \ell =0}

Col 1

ℓ = 0 {\displaystyle \ell =0}

ψ m = 0 {\displaystyle \psi _{m=0}}

s {\displaystyle {\text{s}}}

ℓ = 1 {\displaystyle \ell =1}

Col 1

ℓ = 1 {\displaystyle \ell =1}

ψ m = − 1 + ψ m = + 1 {\displaystyle \psi _{m=-1}+\psi _{m=+1}}

p y {\displaystyle {\text{p}}_{y}}

ψ m = 0 {\displaystyle \psi _{m=0}}

p z {\displaystyle {\text{p}}_{z}}

ψ m = − 1 − ψ m = + 1 {\displaystyle \psi _{m=-1}-\psi _{m=+1}}

p x {\displaystyle {\text{p}}_{x}}

ℓ = 2 {\displaystyle \ell =2}

Col 1

ℓ = 2 {\displaystyle \ell =2}

ψ m = − 2 + ψ m = + 2 {\displaystyle \psi _{m=-2}+\psi _{m=+2}}

d x 2 − y 2 {\displaystyle {\text{d}}_{x^{2}-y^{2}}}

ψ m = − 1 + ψ m = + 1 {\displaystyle \psi _{m=-1}+\psi _{m=+1}}

d y z {\displaystyle {\text{d}}_{yz}}

ψ m = 0 {\displaystyle \psi _{m=0}}

d z 2 {\displaystyle {\text{d}}_{z^{2}}}

ψ m = − 1 − ψ m = + 1 {\displaystyle \psi _{m=-1}-\psi _{m=+1}}

d x z {\displaystyle {\text{d}}_{xz}}

ψ m = − 2 − ψ m = + 2 {\displaystyle \psi _{m=-2}-\psi _{m=+2}}

d x y {\displaystyle {\text{d}}_{xy}}

ℓ = 3 {\displaystyle \ell =3}

Col 1

ℓ = 3 {\displaystyle \ell =3}

ψ m = − 3 + ψ m = + 3 {\displaystyle \psi _{m=-3}+\psi _{m=+3}}

f y ( 3 x 2 − y 2 ) {\displaystyle {\text{f}}_{y(3x^{2}-y^{2})}}

ψ m = − 2 + ψ m = + 2 {\displaystyle \psi _{m=-2}+\psi _{m=+2}}

f z ( x 2 − y 2 ) {\displaystyle {\text{f}}_{z(x^{2}-y^{2})}}

ψ m = − 1 + ψ m = + 1 {\displaystyle \psi _{m=-1}+\psi _{m=+1}}

f y z 2 {\displaystyle {\text{f}}_{yz^{2}}}

ψ m = 0 {\displaystyle \psi _{m=0}}

f z 3 {\displaystyle {\text{f}}_{z^{3}}}

ψ m = − 1 − ψ m = + 1 {\displaystyle \psi _{m=-1}-\psi _{m=+1}}

f x z 2 {\displaystyle {\text{f}}_{xz^{2}}}

ψ m = − 2 − ψ m = + 2 {\displaystyle \psi _{m=-2}-\psi _{m=+2}}

f x y z {\displaystyle {\text{f}}_{xyz}}

ψ m = − 3 − ψ m = + 3 {\displaystyle \psi _{m=-3}-\psi _{m=+3}}

f x ( x 2 − 3 y 2 ) {\displaystyle {\text{f}}_{x(x^{2}-3y^{2})}}

	ψ m = − 3 + ψ m = + 3 {\displaystyle \psi _{m=-3}+\psi _{m=+3}}	ψ m = − 2 + ψ m = + 2 {\displaystyle \psi _{m=-2}+\psi _{m=+2}}	ψ m = − 1 + ψ m = + 1 {\displaystyle \psi _{m=-1}+\psi _{m=+1}}	ψ m = 0 {\displaystyle \psi _{m=0}}	ψ m = − 1 − ψ m = + 1 {\displaystyle \psi _{m=-1}-\psi _{m=+1}}	ψ m = − 2 − ψ m = + 2 {\displaystyle \psi _{m=-2}-\psi _{m=+2}}	ψ m = − 3 − ψ m = + 3 {\displaystyle \psi _{m=-3}-\psi _{m=+3}}
ℓ = 0 {\displaystyle \ell =0}				}}
ℓ = 1 {\displaystyle \ell =1}			}_{y}}	}_{z}}	}_{x}}
ℓ = 2 {\displaystyle \ell =2}		d x 2 − }_{x^{2}-y^{2}}}	}_{yz}}	}_{z^{2}}}	}_{xz}}	}_{xy}}
ℓ = 3 {\displaystyle \ell =3}	f y ( 3 x 2 − y 2 ) {\displaystyle {\text{f}}_{y(3x^{2}-y^{2})}}	f z ( x 2 − y 2 ) {\displaystyle {\text{f}}_{z(x^{2}-y^{2})}}	}_{yz^{2}}}	}_{z^{3}}}	}_{xz^{2}}}	}_{xyz}}	f x ( x 2 − 3 y 2 ) {\displaystyle {\text{f}}_{x(x^{2}-3y^{2})}}