Basis Sets

Hydrogen Orbitals

a0=Quantity["BohrRadius"]/Quantity["Meters"]

ψ[{n_,l_,m_},{r_,θ_,ϕ_}]:=With[{ρ=2 r/(n a0)},Sqrt[(2/(n a0))^3 (n-l-1)!/(2 n (n+l)!)] Exp[-ρ/2] ρ^l LaguerreL[n-l-1,2 l+1,ρ] SphericalHarmonicY[l,m,θ,ϕ]]

DensityPlot3D[(Abs@ψ[{2,1,0},{Sqrt[x^2+y^2+z^2],ArcTan[z,Sqrt[x^2+y^2]],ArcTan[x,y]}])^2,{x,-5 a0,5 a0},{y,-5 a0,5 a0},{z,-5 a0,5 a0},PlotLegends->Automatic]

Radial Part

ψ[{2,0,0},{r,θ,ϕ}]

Plot[ψ[{2,0,0},{r,θ,ϕ}],{r,0, 10 a0}]

Plot[(Abs@ψ[{2,0,0},{r,θ,ϕ}])^2,{r,0, 10 a0}]

ψ[{2,0,0},{r,θ,ϕ}]

ψ[{2,1,0},{r,θ,ϕ}]

Plot[(Abs@ψ[{2,1,0},{r,0,ϕ}])^2,{r,0, 10 a0}]

Angular Part

ψ[{2,1,0},{a0,θ,ϕ}]

PolarPlot[(Abs@ψ[{3,2,0},{a0,θ,ϕ}])^2,{θ,0,2Pi}]

Table[PolarPlot[Simplify[(Abs@ψ[{n,l,m},{a0,θ,ϕ}])^2,Assumptions->ϕ>0],{θ,0,2 Pi}],{n,1,4},{l,0,n-1},{m,-l,l}]

Slater Orbitals (STOs)

a0=a

ψ[{1,0,0},{r,θ,ϕ}]

Table[ψ[{n,0,0},{r,θ,ϕ}],{n,1,5}]

ψ[{1,0,0},{r,θ,ϕ}]

When the nuclei has a charge Z

(((Sqrt[((Z^3)/(a^3))]) (E^(-Z(r/a))))/(Sqrt[π]))

a=1

ψ[{1,0,0},{r,θ,ϕ}]//InputForm

SimplifiedSTO[r_,R_]:=1/(E^(Abs@(r-R))*Sqrt[Pi])

Plot[{SimplifiedSTO[r,0],SimplifiedSTO[r,4],SimplifiedSTO[r,0]+SimplifiedSTO[r,4]},{r,0,10},PlotRange->All]

Plot[(Abs@SimplifiedSTO[r,2])^2,{r,0,10},PlotRange->All]

Gaussian orbitals (GTOs)

SimplifiedGTO[r_,R_]:=1/(E^((Abs@(r-R))^2)*Sqrt[Pi])

Plot[{SimplifiedGTO[r,0],SimplifiedGTO[r,2],SimplifiedGTO[r,0]+SimplifiedGTO[r,2]},{r,0,10},PlotRange->All]

Plane Waves

Plot[Evaluate@Table[Sin[x nx 2Pi/3],{nx,1,4}],{x,0,3}]

L=2;
n=3;
basis=L*LatticeData["SquareLattice","Basis"]//Normal;
points=Tuples[Range[-n,n],2].basis;
lines=Select[Subsets[points,{2}],Norm[#[[1]]-#[[2]]]==L&];
g1=Graphics[{Point[points],Line@lines},Frame->True]

d1=DensityPlot[Sin[x 2 2 Pi/L+y 2 Pi/L ],{x,- n L, n L},{y,- n L, n L},PlotPoints->50]

Show[d1,g1]