Hydrogen molecular ion

[ohno]

Implement the module of :HydrogenMolecularIon. It is thought that this system has an analytically solvable Schrödinger equation (See Wikipedia). But I have not found a closed-form of the solutions. At first, find an appropriate reference from the candidates:

• Burrau(1927) http://publ.royalacademy.dk/books/72/399
• Pauling(1928) https://doi.org/10.1021/cr60018a003
• Stewart(1953) https://doi.org/10.1098/rsta.1953.0014
• Grivet(2002) https://doi.org/10.1021/ed079p127
• Scott(2006) https://doi.org/10.1016/j.chemphys.2005.10.031
• Fernandez(2021) https://doi.org/10.1002/slct.202102509
• Li(2022) https://doi.org/10.1021/acsomega.2c01905

Here is the unfinished code:

# Stewart(1953) https://doi.org/10.1098/rsta.1953.0014
Ψ(λ,μ,φ) = Λ(λ) * M(μ) * Φ(φ)
M(μ;l,m,p) = ?
Λ(λ;R,m,σ,p) = sqrt(λ^2-1)^m * (λ+1)^σ(R,p,m)*exp(-p*λ) * y(ζ(λ))
σ(R,p,m) = ?
ζ(λ) = (λ-1) / (λ+1)