Hydrogen

bohrformula(Z::Int, n::Int)

Hydrogenic energy (in Hartree a.u.) for atom with atomic number Z and principal quantum number n.

\[ E_n = - \frac{Z^2}{2n^2}\]

Example:

julia> Z = 2; n = 4;

julia> bohrformula(Z,n)
-1//8

hydrogenic_reduced_wavefunction(atom::Atom, spinorbit::Spinorbit, grid::CamiDiff.Grid{T}) where T<:Real

Analytic expression for the hydrogenic wavefunction written in the format $Z = \tilde{χ} + i \tilde{χ}^′$, where $\tilde{χ}_{nℓ}(ρ)$ is the reduced radial wavefunction and $\tilde{χ}^′_{nℓ}(ρ)$ its derivative, with $ρ$ the radial distance to the nucleus in a.u.. The expression is evaluated for a given Atom in a given Orbit on a given CamiDiff.Grid. The argument Def completes the definition of the problem.

\[ \tilde{\chi}_{nl}(\rho) =\mathcal{N}_{nl}^{-1/2}(2Z/n)^{l+3/2}\rho^{l+1}e^{-Zρ/n} L_{n-l-1}^{2l+1}(2Z\rho/n)\]

where $L_{n-l-1}^{2l+1}(2Z\rho/n)$ is the generalized Laguerre polynomial CamiMath.generalized_laguerreL and

\[ \mathcal{N}_{nl} = {\displaystyle \int\nolimits _{0}^{\infty}}x^{2l+2}e^{-x} \left[L_{n-l-1}^{2l+1}(x)\right]^{2}dx = \frac{2n\Gamma(n+l+1)}{\Gamma(n-l)}\]

is the norm of the wavefunction.

Example:

julia> codata = castCodata(2022);

julia> atom = castAtom(;Z=1, A=1, Q=0, msg=false);

julia> spinorbit = castSpinorbit(n=25, ℓ=10);

julia> grid = autoGrid(atom, spinorbit, Float64; msg=true);
CamiDiff.Grid created: exponential, Float64, rmax = 3651.58 a.u., N = 1320, h = 0.0075815, r0 = 0.164537

julia> Z = hydrogenic_reduced_wavefunction(atom, spinorbit, grid);
 IOP capture at generalized_laguerre_polynom(35, 21): output converted to BigInt

julia> def = castDef(grid, atom, spinorbit, codata; msg=true);
Def created for ¹H:25n on exponential grid of 1320 points

plot_wavefunction(Z, 1:grid.N, grid, def)

The plot is made using CairomMakie. NB.: plot_wavefunction is not included in the CamiXon package.

reduce_wavefunction(Z::Vector{Complex{T}}, grid::CamiDiff.Grid{T}) where T<:Real

Conversion from the ordinary radial wavefunction $\tilde{R}_{nl}(ρ)$ to the reduced radial wavefuntion

\[ \tilde{\chi}_{nl}(ρ) = ρ \tilde{R}_{nl}(ρ).\]

where $ρ$ is the radial distance to the nucleus in a.u..

Example:

julia> atom = castAtom(Z=1, A=1, Q=0; msg=false);
julia> spinorbit = castSpinorbit(n=1, ℓ=0; msg=false);
julia> grid = autoGrid(atom, spinorbit, Float64);
julia> RH1s_example = [RH1s(atom.Z, grid.r[n]) for n=1:grid.N];
julia> ZH1s_example = reduce_wavefunction(RH1s_example, grid);
julia> ZH1s_generic = hydrogenic_reduced_wavefunction(atom, spinorbit, grid);
julia> @test ZH1s_example ≈ ZH1s_generic
Test Passed

julia> f1 = real(ZH1s_example);
julia> f2 = real(ZH1s_generic);
julia> compare_functions(f1, f2, 1:grid.N, grid)

The plot is made using CairomMakie. NB.: compare_functions is not included in the CamiXon package. ```

restore_wavefunction(Z::Vector{Complex{T}}, atom::Atom, spinorbit::Spinorbit, grid::CamiDiff.Grid{T}) where T<:Real

Conversion from the reduced radial wavefunction $\tilde{\chi}_{nl}(ρ)$ to the ordinary radial wavefuntion $\tilde{R}_{nl}(ρ)$,

\[ \tilde{R}_{nl}(ρ)=\tilde{\chi}_{nl}(ρ)/ρ,\]

where $ρ$ is the radial distance to the nucleus in a.u..

Example:

``` julia> atom = castAtom(Z=1, A=1, Q=0; msg=false); julia> spinorbit = castSpinorbit(n=1, ℓ=0; msg=false); julia> grid = autoGrid(atom, spinorbit, Float64); julia> RH1sexample = [RH1s(atom.Z, grid.r[n]) for n=1:grid.N]; julia> ZH1sexample = reducewavefunction(RH1sexample, grid); julia> RH1sgeneric = restorewavefunction(ZH1s_generic, atom, spinorbit, grid);

julia> @test RH1sexample ≈ RH1sgeneric Test Passed

RH1s(Z::Int, r::T) where T <:Real

Analytic expression for the hydrogenic 1s radial wavefunction and its derivative in the format $Z = \tilde{R} + i \tilde{R}^′$, where

\[ \tilde{R}_{1s}(ρ) = Z^{3/2} 2 e^{-Zρ}\]

is the radial wavefunction and

\[ \tilde{R}^′_{1s}(ρ) = -Z^{5/2} 2 e^{-Zρ}\]

its derivative, with $ρ$ the radial distance to the nucleus in a.u..

Example:

atom = castAtom(Z=1, A=1, Q=0; msg=false);
orbit = castSpinorbit(n=1, ℓ=0; msg=false);
grid = autoGrid(atom, spinorbit, Float64; Nboost=1, msg=false);
def = castDef(grid, atom, spinorbit, codata);

RH1s_example = [RH1s(atom.Z, grid.r[n]) for n=1:grid.N];

plot_wavefunction(RH1s_example, 1:grid.N, grid, def; reduced=false)

The plot is made using CairomMakie. NB.: plot_function is not included in the CamiXon package.

RH2s(Z::Int, r::T) where T <:Real

Analytic expression for the hydrogenic 1s reduced radial wavefunction and its derivative in the format $Z = \tilde{R} + i \tilde{R}^′$, where

\[ \tilde{R}_{2s}(ρ)=\left(Z/2\right)^{3/2}(1-Zρ/2)2e^{-Zρ/2}\]

is the radial wavefunction and

\[ \tilde{R}_{2s}(ρ)=-\left(Z/2\right)^{5/2}(2-Zρ/2)2e^{-Zρ/2}\]

its derivative, with $ρ$ the radial distance to the nucleus in a.u..

Example:

atom = castAtom(Z=1, A=1, Q=0; msg=false);
orbit = castSpinorbit(n=2, ℓ=0; msg=false);
grid = autoGrid(atom, spinorbit, Float64; Nboost=1, msg=false);
def = castDef(grid, atom, spinorbit, codata; msg=false);

RH2s_example = [RH2s(atom.Z, grid.r[n]) for n=1:grid.N];

plot_wavefunction(RH2s_example, 1:grid.N, grid, def; reduced=false)

The plot is made using CairomMakie. NB.: plot_wavefunction is not included in the CamiXon package.

RH2p(Z::Int, r::T) where T <:Real

Analytic expression for the hydrogenic 1s reduced radial wavefunction and its derivative in the format $Z = \tilde{R} + i \tilde{R}^′$, where

\[ \tilde{R}_{2p}(ρ)=\left(Z/2\right)^{3/2}\sqrt{1/3}(Zρ/2)2e^{-Zρ/2}\]

is the radial wavefunction and

\[ \tilde{R}_{2p}(ρ)=\left(Z/2\right)^{3/2}\sqrt{1/3}(1-Zρ/2)2e^{-Zρ/2}\]

its derivative, with $ρ$ the radial distance to the nucleus in a.u..

Example:

atom = castAtom(Z=1, A=1, Q=0; msg=false);
orbit = castSpinorbit(n=2, ℓ=1; msg=false);
grid = autoGrid(atom, spinorbit, Float64; Nboost=1, msg=false);
def = castDef(grid, atom, spinorbit, codata);

RH2p_example = [RH2p(atom.Z, grid.r[n]) for n=1:grid.N];

plot_wavefunction(RH2p_example, 1:grid.N, grid, def; reduced=false)

The plot is made using CairomMakie. NB.: plot_wavefunction is not included in the CamiXon package.

silvera_goldman_triplet(r::T) where T<:Real

Parametrization in Hartree a.u. of the triplet $(^{3}\Sigma_{u}^{+})$ potential of the electronic ground state of $\mathrm{H}_{2}$ (Eh = 219474.6 cm-1),

\[ V_{t}(r)=\mathrm{exp}\left(0.09678-1.10173\thinspace r-0.0394\thinspace r^{2}\right)+F(r)\left(-6.5\thinspace r^{-6}-124\thinspace r^{-8}-3285r^{-10}\thinspace \right)\]

\[ \mathrm{where}\ \ \ \ \ \ \ \ F(r) = \begin{cases} \mathrm{exp}\left[-\left(\frac{10.04}{r}-1\right)^{2}\right] & \mathrm{for}\,\,\,r<10.04\,\mathrm{a.u.}\\ 1 & \mathrm{for}\,\,\,r<10.04\,\mathrm{a.u.} \end{cases}\]

see I.F. Silvera, - Rev. Mod. Phys., 52, 393 (1980).

silvera_goldman_singlet(r::T) where T<:Real

Parametrization in Hartree a.u. of the singlet $(^{1}\Sigma_{g}^{+})$ potential of $\mathrm{H}_{2}$ (Eh = 219474.6 cm-1),

\[ V_{s}(r)=V_{t}(r)-J(r)\]

where $V_{t}$ is the triplet potential (silvera_goldman_triplet) and $J(r)$ is the exchange contribution (silvera_goldman_triplet).

silvera_goldman_exchange(r::T) where T<:Real

Parametrization in Hartree a.u. of the exchange energy difference between singlet $(^{3}\Sigma_{u}^{+})$ and triplet $(^{1}\Sigma_{g}^{+})$ potentials of $\mathrm{H}_{2}$ (Eh = 219474.6 cm-1),

\[ J(r)=\mathrm{exp}\left(-0.288-0.275\thinspace r-0.176\thinspace r^{2}+0.0068\thinspace r^{3}\right)\]

see I.F. Silvera, - Rev. Mod. Phys., 52, 393 (1980).

silvera_goldman_potential(grid::CamiDiff.Grid{T}; S=0) where T<:Real

CamiDiff.Grid representation in Hartree a.u. of the singlet (S=0) and triplet (S=1) potentials of $\mathrm{H}_{2}$,

\[ \mathcal{V}(r)=V_{D}(r)+J(r)\mathbf{s}_{1}\cdot\mathbf{s}_{2},\]

where $\mathbf{S} = \mathbf{s}_{1}+\mathbf{s}_{2}$ and

\[ V_{D}(r)=\frac{1}{4}[V_{s}(r)+3V_{t}(r)] \ \mathrm{and}\ J(r)=V_{t}(r)-V_{s}(r)\]

are known as the direct and exchange contribution, respectively; $V_{s}:$ see silvera_goldman_singlet, $V_{t}:$ see silvera_goldman_triplet, $J(r):$ see silvera_goldman_exchange.

see I.F. Silvera, - Rev. Mod. Phys., 52, 393 (1980).

rotbarrier(grid::CamiDiff.Grid{T}; ℓ=0) where T<:Real

CamiDiff.Grid representation of rotational barrier potential in wavenumber notation,

\[V_{rot}(r) = \frac{\ell(\ell+1)}{r^{2}},\]

where ℓ is the rotational quantum number.