AtomicStructure.jl

This library provides structures for representing atoms as linear combinations of single-particle orbitals.

Usage

The radial coordinate is represented using a basis function expansion that fulfils the ContinuumArrays.jl interface. The examples below use CompactBases.jl, but AtomicStructure.jl is not dependent on which basis you choose.

using AtomicStructure
using AtomicLevels
using CompactBases

Grid setup

The grid can be tailored to a specific nucleus, which is why we first decide the nuclear potential to be used, in this case a point charge corresponding to helium:

nucleus = pc"He"

Z = 2 [He]

rₘₐₓ = 300
ρ = 0.25 # Discretization interval
N = ceil(Int, rₘₐₓ/ρ + 1/2)
# Finite-difference scheme that accounts for the boundary condition at
# r = 0
R = StaggeredFiniteDifferences(N, ρ, float(charge(nucleus)))

Radial finite differences basis {Float64} on 0.125..300.125 (formally 0..300.125) with 1201 points spaced by ρ = 0.25

Different atoms

Non-relativistic helium, one configuration state function, the orbitals are automatically initialized to their hydrogenic values:

Atom(R, csfs(c"1s2"), nucleus, verbosity=3)

⎡ Hydrogenic initialization of the orbitals of Atom{Float64,RadialDifferences{Float64,Int64}}(Z = 2 [He]) with 1 CSF
⎢ ⎡ Diagonalizing symmetry ℓ = s, maximum n = 1 => 1 eigenvalues required
⎢ ⎢ Target eigenvalue: ≤ -2.0 Ha
⎢ ⎢ Diagonalizing via arnoldi_shift_invert
⎢ ⎢ Schur values: [0.49955]
⎢ ⎢ Hydrogenic energies [-1.9981975] Ha
⎢ ⎢ Analytic energies   [-2.0000000] Ha
⎢ ⎣ Δ                   [+1.803e-03] Ha
⎢ 
⎢ [ Initial norm of 1s: 0.500000, 1-normalized: 5.000000e-01
⎣ 

Atom{Float64,RadialDifferences{Float64,Int64}}(Z = 2 [He]) with 1 CSF: 1s²(₀¹S|¹S)+

Non-relativistic, with the 2s,2p,3s,3p,3d orbitals as possible correlation orbitals:

Atom(R, csfs(excited_configurations(c"1s2", os"2[s-p]"..., os"3[s-d]"...)), nucleus, verbosity=3)

⎡ Hydrogenic initialization of the orbitals of Atom{Float64,RadialDifferences{Float64,Int64}}(Z = 2 [He]) with 32 CSFs
⎢ ⎡ Diagonalizing symmetry ℓ = s, maximum n = 3 => 3 eigenvalues required
⎢ ⎢ Target eigenvalue: ≤ -2.0 Ha
⎢ ⎢ Diagonalizing via arnoldi_shift_invert
⎢ ⎢ Schur values: [0.49955, 0.285767, 0.264747]
⎢ ⎢ Hydrogenic energies [-1.9981975, -0.5006417, -0.2228092] Ha
⎢ ⎢ Analytic energies   [-2.0000000, -0.5000000, -0.2222222] Ha
⎢ ⎣ Δ                   [+1.803e-03, -6.417e-04, -5.870e-04] Ha
⎢ 
⎢ ⎡ Diagonalizing symmetry ℓ = d, maximum n = 3 => 1 eigenvalues required
⎢ ⎢ Target eigenvalue: ≤ -0.2222222222222222 Ha
⎢ ⎢ Diagonalizing via arnoldi_shift_invert
⎢ ⎢ Schur values: [4.5007]
⎢ ⎢ Hydrogenic energies [-0.2222567] Ha
⎢ ⎢ Analytic energies   [-0.2222222] Ha
⎢ ⎣ Δ                   [-3.448e-05] Ha
⎢ 
⎢ ⎡ Diagonalizing symmetry ℓ = p, maximum n = 3 => 2 eigenvalues required
⎢ ⎢ Target eigenvalue: ≤ -0.5 Ha
⎢ ⎢ Diagonalizing via arnoldi_shift_invert
⎢ ⎢ Schur values: [1.99957, 1.28594]
⎢ ⎢ Hydrogenic energies [-0.4998912, -0.2223581] Ha
⎢ ⎢ Analytic energies   [-0.5000000, -0.2222222] Ha
⎢ ⎣ Δ                   [+1.088e-04, -1.359e-04] Ha
⎢ 
⎢ ⎡ Initial norm of 1s: 0.500000, 1-normalized: 5.000000e-01
⎢ ⎢ Initial norm of 2s: 0.500000, 1-normalized: 5.000000e-01
⎢ ⎢ Initial norm of 2p: 0.500000, 1-normalized: 5.000000e-01
⎢ ⎢ Initial norm of 3s: 0.500000, 1-normalized: 5.000000e-01
⎢ ⎢ Initial norm of 3p: 0.500000, 1-normalized: 5.000000e-01
⎢ ⎣ Initial norm of 3d: 0.500000, 1-normalized: 5.000000e-01
⎣ 

Atom{Float64,RadialDifferences{Float64,Int64}}(Z = 2 [He]) with 32 CSFs:
32-element Array{CSF{Orbital,IntermediateTerm,Term},1}:
 1s²(₀¹S|¹S)+          
 1s(₁²S|²S) 2s(₁²S|¹S)+
 1s(₁²S|²S) 2s(₁²S|³S)+
 1s(₁²S|²S) 3s(₁²S|¹S)+
 1s(₁²S|²S) 3s(₁²S|³S)+
 1s(₁²S|²S) 3d(₁²D|¹D)+
 1s(₁²S|²S) 3d(₁²D|³D)+
 2s²(₀¹S|¹S)+          
 2s(₁²S|²S) 3s(₁²S|¹S)+
 2s(₁²S|²S) 3s(₁²S|³S)+
 2s(₁²S|²S) 3d(₁²D|¹D)+
 ⋮                     
 3s²(₀¹S|¹S)+          
 3s(₁²S|²S) 3d(₁²D|¹D)+
 3s(₁²S|²S) 3d(₁²D|³D)+
 3p²(₀¹S|¹S)+          
 3p²(₂¹D|¹D)+          
 3p²(₂³P|³P)+          
 3d²(₀¹S|¹S)+          
 3d²(₂¹D|¹D)+          
 3d²(₂¹G|¹G)+          
 3d²(₂³P|³P)+          
 3d²(₂³F|³F)+

TODO/Ideas

• [ ] Multiple eigenvalues per symmetry and/or Lagrange multipliers for orthogonality
• [ ] Warn if core modelled by potential differs too much from core of configuration(s) to optimize
  • [ ] Optimize “frozen” orbitals as well
• [ ] Virial theorem V/T = -2 as accuracy indicator
• [ ] Number of eigenvalues required per equation
• [ ] Tabulate bound spectra
• [ ] Extension of atom
  • [ ] onto larger grid
  • [ ] more configurations (e.g. continuum)
• [ ] Evaluate smaller effect [eg. (hyper)fine structure] by inclusion of new operators
  • [ ] via perturbation theory
    • [ ] tests of accuracy using analytic perturbation theory formulas, e.g. hyperfine splitting of Rb in magnetic fields.
  • [ ] reoptimization of orbitals (requires expansion of basis, if not working with spin-orbitals)
• [ ] Generalize atomic operators
  • [ ] AbstractAtomicOperator
  • [ ] DiagonalIntegral
  • [ ] RepulsionIntegral
  • [ ] MultipoleInteraction
  • [ ] Spin–orbit interaction
  • [ ] &c.
• [ ] Generalize notion of energy expression to derive arbitrary equations of motion, either for optimization of orbitals or for time propagation.