Atomic orbitals

Atomic orbitals, i.e. single particle states with well-defined orbital or total angular momentum, are usually the basic building blocks of atomic states. AtomicLevels provides various types and methods to conveniently label the orbitals.

Orbital types

AtomicLevels provides two basic types for labelling atomic orbitals: Orbital and RelativisticOrbital. Stricly speaking, these types do not label orbitals, but groups of orbitals with the same angular symmetry and radial behaviour (i.e. a subshell).

All orbitals are subtypes of AbstractOrbital. Types and methods that work on generic orbitals can dispatch on that.

AtomicLevels.Orbital — Type

struct Orbital{N <: AtomicLevels.MQ} <: AbstractOrbital

Label for an atomic orbital with a principal quantum number n::N and orbital angular momentum ℓ.

The type parameter N has to be such that it can represent a proper principal quantum number (i.e. a subtype of AtomicLevels.MQ).

Properties

The following properties are part of the public API:

.n :: N – principal quantum number $n$
.ℓ :: Int – the orbital angular momentum $\ell$

Constructors

Orbital(n::Int, ℓ::Int)
Orbital(n::Symbol, ℓ::Int)

Construct an orbital label with principal quantum number n and orbital angular momentum ℓ. If the principal quantum number n is an integer, it has to positive and the angular momentum must satisfy 0 <= ℓ < n.

julia> Orbital(1, 0)
1s

julia> Orbital(:K, 2)
Kd

AtomicLevels.RelativisticOrbital — Type

struct RelativisticOrbital{N <: AtomicLevels.MQ} <: AbstractOrbital

Label for an atomic orbital with a principal quantum number n::N and well-defined total angular momentum $j$. The angular component of the orbital is labelled by the $(\ell, j)$ pair, conventionally written as $\ell_j$ (e.g. $p_{3/2}$).

The $\ell$ and $j$ can not be arbitrary, but must satisfy $j = \ell \pm 1/2$. Internally, the $\kappa$ quantum number, which is a unique integer corresponding to every physical $(\ell, j)$ pair, is used to label each allowed pair. When $j = \ell \pm 1/2$, the corresponding $\kappa = \mp(j + 1/2)$.

When printing and parsing RelativisticOrbitals, the notation nℓ and nℓ- is used (e.g. 2p and 2p-), corresponding to the orbitals with $j = \ell + 1/2$ and $j = \ell - 1/2$, respectively.

The type parameter N has to be such that it can represent a proper principal quantum number (i.e. a subtype of AtomicLevels.MQ).

Properties

The following properties are part of the public API:

.n :: N – principal quantum number $n$
.κ :: Int – $\kappa$ quantum number
.ℓ :: Int – the orbital angular momentum label $\ell$
.j :: HalfInteger – total angular momentum $j$

julia> orb = ro"5g-"
5g-

julia> orb.n
5

julia> orb.j
7/2

julia> orb.ℓ
4

Constructors

RelativisticOrbital(n::Integer, κ::Integer)
RelativisticOrbital(n::Symbol, κ::Integer)
RelativisticOrbital(n, ℓ::Integer, j::Real)

Construct an orbital label with the quantum numbers n and κ. If the principal quantum number n is an integer, it has to positive and the orbital angular momentum must satisfy 0 <= ℓ < n. Instead of κ, valid ℓ and j values can also be specified instead.

julia> RelativisticOrbital(1, 0, 1//2)
1s

julia> RelativisticOrbital(2, -1)
2s

julia> RelativisticOrbital(:K, 2, 3//2)
Kd-

AtomicLevels.AbstractOrbital — Type

abstract type AbstractOrbital

Abstract supertype of all orbital types.

Broadcasting

When broadcasting, orbital objects behave like scalars.

The SpinOrbital type can be used to fully qualify all the quantum numbers (that is, also $m_\ell$ and $m_s$) of an Orbital. It represent a since, distinct orbital.

AtomicLevels.SpinOrbital — Type

struct SpinOrbital{O<:Orbital} <: AbstractOrbital

Spin orbitals are fully characterized orbitals, i.e. the projections of all angular momenta are specified.

The string macros @o_str and @ro_str can be used to conveniently construct orbitals, while @os_str, @sos_str, @ros_str, and @rsos_str can be used to construct whole lists of them very easily.

AtomicLevels.@o_str — Macro

@o_str -> Orbital

A string macro to construct an Orbital from the canonical string representation.

julia> o"1s"
1s

julia> o"Fd"
Fd

AtomicLevels.@so_str — Macro

@so_str -> SpinOrbital{<:Orbital}

String macro to quickly construct a non-relativistic spin-orbital; it is similar to @o_str, with the added specification of the magnetic quantum numbers $m_ℓ$ and $m_s$.

Examples

julia> so"1s(0,α)"
1s₀α

julia> so"kd(2,β)"
kd₂β

julia> so"2p(1,0.5)"
2p₁α

julia> so"2p(1,-1/2)"
2p₁β

AtomicLevels.@ro_str — Macro

@ro_str -> RelativisticOrbital

A string macro to construct an RelativisticOrbital from the canonical string representation.

julia> ro"1s"
1s

julia> ro"2p-"
2p-

julia> ro"Kf-"
Kf-

AtomicLevels.@rso_str — Macro

@rso_str -> SpinOrbital{<:RelativisticOrbital}

String macro to quickly construct a relativistic spin-orbital; it is similar to @o_str, with the added specification of the magnetic quantum number $m_j$.

Examples

julia> rso"2p-(1/2)"
2p-(1/2)

julia> rso"2p(-3/2)"
2p(-3/2)

julia> rso"3d(2.5)"
3d(5/2)

AtomicLevels.@os_str — Macro

@os_str -> Vector{Orbital}

Can be used to easily construct a list of Orbitals.

Examples

julia> os"5[d] 6[s-p] k[7-10]"
7-element Vector{Orbital}:
 5d
 6s
 6p
 kk
 kl
 km
 kn

AtomicLevels.@sos_str — Macro

@sos_str -> Vector{<:SpinOrbital{<:Orbital}}

Can be used to easily construct a list of SpinOrbitals.

Examples

julia> sos"3[s-p]"
8-element Vector{SpinOrbital{Orbital{Int64}, Tuple{Int64, HalfIntegers.Half{Int64}}}}:
 3s₀α
 3s₀β
 3p₋₁α
 3p₋₁β
 3p₀α
 3p₀β
 3p₁α
 3p₁β

AtomicLevels.@ros_str — Macro

@ros_str -> Vector{RelativisticOrbital}

Can be used to easily construct a list of RelativisticOrbitals.

Examples

julia> ros"2[s-p] 3[p] k[0-d]"
10-element Vector{RelativisticOrbital}:
 2s
 2p-
 2p
 3p-
 3p
 ks
 kp-
 kp
 kd-
 kd

AtomicLevels.@rsos_str — Macro

@rsos_str -> Vector{<:SpinOrbital{<:RelativisticOrbital}}

Can be used to easily construct a list of SpinOrbitals.

Examples

julia> rsos"3[s-p]"
8-element Vector{SpinOrbital{RelativisticOrbital{Int64}, Tuple{HalfIntegers.Half{Int64}}}}:
 3s(-1/2)
 3s(1/2)
 3p-(-1/2)
 3p-(1/2)
 3p(-3/2)
 3p(-1/2)
 3p(1/2)
 3p(3/2)

Methods

Base.isless — Function

isless(a::Orbital, b::Orbital)

Compares the orbitals a and b to decide which one comes before the other in a configuration.

Examples

julia> o"1s" < o"2s"
true

julia> o"1s" < o"2p"
true

julia> o"ks" < o"2p"
false

AtomicLevels.degeneracy — Function

degeneracy(orbital::Orbital)

Returns the degeneracy of orbital which is 2(2ℓ+1)

Examples

julia> degeneracy(o"1s")
2

julia> degeneracy(o"2p")
6

AtomicLevels.parity — Method

parity(orbital::Orbital)

Returns the parity of orbital, defined as (-1)^ℓ.

Examples

julia> parity(o"1s")
even

julia> parity(o"2p")
odd

AtomicLevels.symmetry — Function

symmetry(orbital::Orbital)

Returns the symmetry for orbital which is simply ℓ.

AtomicLevels.isbound — Function

isbound(::Orbital)

Returns true is the main quantum number is an integer, false otherwise.

julia> isbound(o"1s")
true

julia> isbound(o"ks")
false

AtomicLevels.angular_momenta — Function

angular_momenta(orbital)

Returns the angular momentum quantum numbers of orbital.

Examples

julia> angular_momenta(o"2s")
(0, 1/2)

julia> angular_momenta(o"3d")
(2, 1/2)

angular_momenta(orbital)

Returns the angular momentum quantum numbers of orbital.

Examples

julia> angular_momenta(ro"2p-")
(1/2,)

julia> angular_momenta(ro"3d")
(5/2,)

AtomicLevels.angular_momentum_ranges — Function

angular_momentum_ranges(orbital)

Return the valid ranges within which projections of each of the angular momentum quantum numbers of orbital must fall.

Examples

julia> angular_momentum_ranges(o"2s")
(0:0, -1/2:1/2)

julia> angular_momentum_ranges(o"4f")
(-3:3, -1/2:1/2)

AtomicLevels.spin_orbitals — Function

spin_orbitals(orbital)

Generate all permissible spin-orbitals for a given orbital, e.g. 2p -> 2p ⊗ mℓ = {-1,0,1} ⊗ ms = {α,β}

Examples

julia> spin_orbitals(o"2p")
6-element Vector{SpinOrbital{Orbital{Int64}, Tuple{Int64, HalfIntegers.Half{Int64}}}}:
 2p₋₁α
 2p₋₁β
 2p₀α
 2p₀β
 2p₁α
 2p₁β

julia> spin_orbitals(ro"2p-")
2-element Vector{SpinOrbital{RelativisticOrbital{Int64}, Tuple{HalfIntegers.Half{Int64}}}}:
 2p-(-1/2)
 2p-(1/2)

julia> spin_orbitals(ro"2p")
4-element Vector{SpinOrbital{RelativisticOrbital{Int64}, Tuple{HalfIntegers.Half{Int64}}}}:
 2p(-3/2)
 2p(-1/2)
 2p(1/2)
 2p(3/2)

AtomicLevels.nonrelorbital — Function

nonrelorbital(o)

Return the non-relativistic orbital corresponding to o.

Examples

julia> nonrelorbital(o"2p")
2p

julia> nonrelorbital(ro"2p-")
2p

Index

AtomicLevels.AbstractOrbital
AtomicLevels.Orbital
AtomicLevels.RelativisticOrbital
AtomicLevels.SpinOrbital
AtomicLevels.angular_momenta
AtomicLevels.angular_momentum_ranges
AtomicLevels.degeneracy
AtomicLevels.isbound
AtomicLevels.nonrelorbital
AtomicLevels.parity
AtomicLevels.spin_orbitals
AtomicLevels.symmetry
Base.isless
AtomicLevels.@o_str
AtomicLevels.@os_str
AtomicLevels.@ro_str
AtomicLevels.@ros_str
AtomicLevels.@rso_str
AtomicLevels.@rsos_str
AtomicLevels.@so_str
AtomicLevels.@sos_str