AtomicLevels.jl
AtomicLevels provides a collections of types and methods to facilitate working with atomic states (or, more generally, states with spherical symmetry), both in the relativistic (eigenstates of $J = L + S$) and non-relativistic (eigenstates on $L$ and $S$ separately) frameworks.
The aim is to make sure that the types used to label and store information about atomic states are both efficient and user-friendly at the same time. In addition, it also provides various utility methods, such as generation of a list CSFs corresponding to a given configuration, serialization of orbitals and configurations, methods for introspecting physical quantities etc.
Usage examples
Orbitals
A single orbital can be constructed using string macros
julia> orbitals = o"2s", ro"5f-"(2s, 5f-)
Various methods are provided to look up the properties of the orbitals
julia> for o in orbitals @info "Orbital: $o :: $(typeof(o))" parity(o) degeneracy(o) angular_momenta(o) end┌ Info: Orbital: 2s :: Orbital{Int64} │ parity(o) = even │ degeneracy(o) = 2 └ angular_momenta(o) = (0, 1/2) ┌ Info: Orbital: 5f- :: RelativisticOrbital{Int64} │ parity(o) = odd │ degeneracy(o) = 6 └ angular_momenta(o) = (5/2,)
You can also create a range of orbitals quickly using the @os_str (or @ros_str) string macros
julia> os"5[d] 6[s-p] k[7-10]"7-element Vector{Orbital}: 5d 6s 6p kk kl km kn
Configurations
The ground state of hydrogen and helium.
julia> c"1s",(c"1s2",c"[He]")(1s, (1s², [He]ᶜ))
The ground state configuration of xenon, in relativistic notation.
julia> Xe = rc"[Kr] 5s2 5p6"[Kr]ᶜ 5s² 5p⁴ 5p-²
As we see above, by default, the krypton core is declared “closed”. This is useful for calculations when the core should be frozen. We can “open” it by affixing *.
julia> Xe = c"[Kr]* 5s2 5p6"1s² 2s² 2p⁶ 3s² 3p⁶ 3d¹⁰ 4s² 4p⁶ 5s² 5p⁶
Note that the 5p shell was broken up into 2 5p- electrons and 4 5p electrons. If we are not filling the shell, occupancy of the spin-up and spin-down electrons has to be given separately.
julia> Xe⁺ = rc"[Kr] 5s2 5p-2 5p3"[Kr]ᶜ 5s² 5p-² 5p³
It is also possible to work with “continuum orbitals”, where the main quantum number is replaced by a Symbol.
julia> Xe⁺e = rc"[Kr] 5s2 5p-2 5p3 ks"[Kr]ᶜ 5s² 5p-² 5p³ ks
Excitations
We can easily generate all possible excitations from a reference configuration. If no extra orbitals are specified, only those that are “open” within the reference set will be considered.
julia> excited_configurations(rc"[Kr] 5s2 5p-2 5p3")2-element Vector{Configuration{RelativisticOrbital{Int64}}}: [Kr]ᶜ 5s² 5p-² 5p³ [Kr]ᶜ 5s² 5p- 5p⁴
By appending virtual orbitals, we can generate excitations to configurations beyond those spanned by the reference set.
julia> excited_configurations(rc"[Kr] 5s2 5p-2 5p3", ros"5[d] 6[s-p]"...)64-element Vector{Configuration{RelativisticOrbital{Int64}}}: [Kr]ᶜ 5s² 5p-² 5p³ [Kr]ᶜ 5s 5p-² 5p³ 5d- [Kr]ᶜ 5s 5p-² 5p³ 5d [Kr]ᶜ 5s 5p-² 5p³ 6s [Kr]ᶜ 5s² 5p- 5p⁴ [Kr]ᶜ 5s² 5p- 5p³ 6p- [Kr]ᶜ 5s² 5p- 5p³ 6p [Kr]ᶜ 5s² 5p-² 5p² 6p- [Kr]ᶜ 5s² 5p-² 5p² 6p [Kr]ᶜ 5p-² 5p⁴ 6p- ⋮ [Kr]ᶜ 5s² 5p-² 5p 5d-² [Kr]ᶜ 5s² 5p-² 5p 5d- 5d [Kr]ᶜ 5s² 5p-² 5p 5d- 6s [Kr]ᶜ 5s² 5p-² 5p 5d² [Kr]ᶜ 5s² 5p-² 5p 5d 6s [Kr]ᶜ 5s² 5p-² 5p 6s² [Kr]ᶜ 5s² 5p-² 5p 6p-² [Kr]ᶜ 5s² 5p-² 5p 6p- 6p [Kr]ᶜ 5s² 5p-² 5p 6p²
Again, using the “continuum orbitals”, it is possible to generate the state space accessible via one-photon transitions from the ground state.
julia> Xe⁺e = excited_configurations(rc"[Kr] 5s2 5p6", ros"k[s-d]"..., max_excitations=:singles, keep_parity=false)16-element Vector{Configuration{RelativisticOrbital}}: [Kr]ᶜ 5s² 5p⁴ 5p-² [Kr]ᶜ 5s 5p⁴ 5p-² ks [Kr]ᶜ 5s 5p⁴ 5p-² kp- [Kr]ᶜ 5s 5p⁴ 5p-² kp [Kr]ᶜ 5s 5p⁴ 5p-² kd- [Kr]ᶜ 5s 5p⁴ 5p-² kd [Kr]ᶜ 5s² 5p³ 5p-² ks [Kr]ᶜ 5s² 5p³ 5p-² kp- [Kr]ᶜ 5s² 5p³ 5p-² kp [Kr]ᶜ 5s² 5p³ 5p-² kd- [Kr]ᶜ 5s² 5p³ 5p-² kd [Kr]ᶜ 5s² 5p⁴ 5p- ks [Kr]ᶜ 5s² 5p⁴ 5p- kp- [Kr]ᶜ 5s² 5p⁴ 5p- kp [Kr]ᶜ 5s² 5p⁴ 5p- kd- [Kr]ᶜ 5s² 5p⁴ 5p- kd
We can then query for the bound and continuum orbitals thus.
julia> map(Xe⁺e) do c b = bound(c) num_electrons(b) => b end16-element Vector{Pair{Int64, Configuration{RelativisticOrbital}}}: 44 => [Kr]ᶜ 5s² 5p⁴ 5p-² 43 => [Kr]ᶜ 5s 5p⁴ 5p-² 43 => [Kr]ᶜ 5s 5p⁴ 5p-² 43 => [Kr]ᶜ 5s 5p⁴ 5p-² 43 => [Kr]ᶜ 5s 5p⁴ 5p-² 43 => [Kr]ᶜ 5s 5p⁴ 5p-² 43 => [Kr]ᶜ 5s² 5p³ 5p-² 43 => [Kr]ᶜ 5s² 5p³ 5p-² 43 => [Kr]ᶜ 5s² 5p³ 5p-² 43 => [Kr]ᶜ 5s² 5p³ 5p-² 43 => [Kr]ᶜ 5s² 5p³ 5p-² 43 => [Kr]ᶜ 5s² 5p⁴ 5p- 43 => [Kr]ᶜ 5s² 5p⁴ 5p- 43 => [Kr]ᶜ 5s² 5p⁴ 5p- 43 => [Kr]ᶜ 5s² 5p⁴ 5p- 43 => [Kr]ᶜ 5s² 5p⁴ 5p-julia> map(Xe⁺e) do c b = continuum(c) num_electrons(b) => b end16-element Vector{Pair{Int64, Configuration{RelativisticOrbital}}}: 0 => ∅ 1 => ks 1 => kp- 1 => kp 1 => kd- 1 => kd 1 => ks 1 => kp- 1 => kp 1 => kd- 1 => kd 1 => ks 1 => kp- 1 => kp 1 => kd- 1 => kd
Term symbol calculation
Overview of angular momentum coupling on Wikipedia.
$LS$-coupling. This is done purely non-relativistic, i.e. 2p- is considered equivalent to 2p.
julia> terms(c"1s")1-element Vector{Term}: ²Sjulia> terms(c"[Kr] 5s2 5p5")1-element Vector{Term}: ²Pᵒjulia> terms(c"[Kr] 5s2 5p4 6s 7g")13-element Vector{Term}: ¹D ¹F ¹G ¹H ¹I ³D ³F ³G ³H ³I ⁵F ⁵G ⁵H
$jj$-coupling. $jj$-coupling is implemented slightly differently, it calculates the possible $J$ values resulting from coupling n equivalent electrons in all combinations allowed by the Pauli principle.
julia> intermediate_terms(ro"1s", 1)1-element Vector{IntermediateTerm{HalfIntegers.Half{Int64}, Seniority}}: ₁1/2julia> intermediate_terms(ro"5p", 2)2-element Vector{IntermediateTerm{HalfIntegers.Half{Int64}, Seniority}}: ₀0 ₂2julia> intermediate_terms(ro"7g", 3)10-element Vector{IntermediateTerm{HalfIntegers.Half{Int64}, Seniority}}: ₁9/2 ₃3/2 ₃5/2 ₃7/2 ₃9/2 ₃11/2 ₃13/2 ₃15/2 ₃17/2 ₃21/2
Configuration state functions
CSFs are formed from electronic configurations and their possible term couplings (along with intermediate terms, resulting from unfilled subshells).
julia> sort(vcat(csfs(rc"3s 3p2")..., csfs(rc"3s 3p- 3p")...))7-element Vector{RelativisticCSF{RelativisticOrbital{Int64}, Seniority}}: 3s(₁1/2|1/2) 3p²(₀0|1/2)+ 3s(₁1/2|1/2) 3p-(₁1/2|1) 3p(₁3/2|1/2)+ 3s(₁1/2|1/2) 3p²(₂2|3/2)+ 3s(₁1/2|1/2) 3p-(₁1/2|0) 3p(₁3/2|3/2)+ 3s(₁1/2|1/2) 3p-(₁1/2|1) 3p(₁3/2|3/2)+ 3s(₁1/2|1/2) 3p²(₂2|5/2)+ 3s(₁1/2|1/2) 3p-(₁1/2|1) 3p(₁3/2|5/2)+