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I've been playing around with some d-orbitals and have been trying to view their maximum in two dimensions. At the moment I've progressed to the point where the d__ orbitals are superimposed onto one another, in 2d, using a modification of Jens' answer here but have not managed to obtain the outline. Please pardon the code, its probably a bit convoluted and ever so slightly messy:

I've been playing around with some d-orbitals and have been trying to view their maximum in two dimensions. At the moment I've progressed to the point where the d__ orbitals are superimposed onto one another, in 2d, using a modification of Jens' answer here but have not managed to obtain the outline. Please pardon the code, its probably a bit convoluted and ever so slightly messy:

I've been playing around with some d-orbitals and have been trying to view their maximum in two dimensions. At the moment I've progressed to the point where the d__ orbitals are superimposed onto one another, in 2d, using a modification of Jen's answer here but have not managed to obtain the outline. Please pardon the code, its probably a bit convoluted and ever so slightly messy:

I've been playing around with some d-orbitals and have been trying to view their maximum in two dimensions. At the moment I've progressed to the point where the d__ orbitals are superimposed onto one another, in 2d, using a modification of Jens' answer here but have not managed to obtain the outline. Please pardon the code, its probably a bit convoluted and ever so slightly messy:

{rMin[n_, l_], rMax[n_, l_]} = r /. Simplify[Solve[(l (l + 1))/r^2 - 2/r == -(1/n^2), r], n > 0];

sphericalToCartesian = Thread[{r, \[Theta], \[Phi]} -> {Sqrt[x^2 + y^2 + z^2], ArcCos[z/Sqrt[x^2 + y^2 + z^2]], Arg[x + I y]}];

(*The radial orbitals here are approximated using a Slater-type orbital using Clementi's atomic constants for Fe; see Slater-type orbitals on Wikipedia for further information*)
pimp[n_, l_, m_][r_, \[Theta]_, \[Phi]_] := (2*22.27)^n ((2*22.27)/(2 n)!) (r^(n - 1)) (E^(-22.27 r)) (Im[SphericalHarmonicY[l, m, \[Theta], \[Phi]]] + Im[SphericalHarmonicY[l, -m, \[Theta], \[Phi]]])
pimn[n_, l_, m_][r_, \[Theta]_, \[Phi]_] := (2*22.27)^n ((2*22.27)/(2 n)!) (r^(n - 1)) (E^(-22.27 r)) (Im[SphericalHarmonicY[l, m, \[Theta], \[Phi]]] - Im[SphericalHarmonicY[l, -m, \[Theta], \[Phi]]])
prep[n_, l_, m_][r_, \[Theta]_, \[Phi]_] := (2*22.27)^n ((2*22.27)/(2 n)!) (r^(n - 1)) (E^(-22.27 r)) (Re[SphericalHarmonicY[l, m, \[Theta], \[Phi]]] + Re[SphericalHarmonicY[l, -m, \[Theta], \[Phi]]])
pren[n_, l_, m_][r_, \[Theta]_, \[Phi]_] := (2*22.27)^n ((2*22.27)/(2 n)!) (r^(n - 1)) (E^(-22.27 r)) (Re[SphericalHarmonicY[l, m, \[Theta], \[Phi]]] - Re[SphericalHarmonicY[l, -m, \[Theta], \[Phi]]])

(*To get a 2d plot, I set the earlier evaluation so that x->0*)
plot2dx0[f_, range_, contour_, opt : OptionsPattern[]] := RegionPlot[Evaluate[Abs[f[r, \[Theta], \[Phi]] /. sphericalToCartesian]^2 >contour] /. x -> 0, {y, -range, range}, {z, -range, range}]

(*Plotting the different d-orbitals*)
Show[plot2dx0[prep[3, 2, 0], 0.7, 0.00007], plot2dx0[pimp[3, 2, 1], 0.7, 0.00007], plot2dx0[pimn[3, 2, 2], 0.7, 0.00007], plot2dx0[pren[3, 2, 1], 0.7, 0.00007], plot2dx0[prep[3, 2, 2], 0.7, 0.00007], Frame -> True, FrameLabel -> {"y", "z"}]

(*Output*)

Evaluate[Max[{Abs[prep[3, 2, 0][1.12*r, \[Theta], \[Phi]] /. sphericalToCartesian^2 > 0.0007],Abs[pimp[3, 2, 1][0.8*r, \[Theta], \[Phi]] /. sphericalToCartesian^2 > 0.0007],Abs[pimn[3, 2, 2][1.12*r, \[Theta], \[Phi]] /. sphericalToCartesian^2 > 0.0007],Abs[pren[3, 2, 1][1.12*r, \[Theta], \[Phi]] /. sphericalToCartesian^2 > 0.0007],Abs[prep[3, 2, 2][1.12*r, \[Theta], \[Phi]] /. sphericalToCartesian^2 > 0.0007]}]]

{rMin[n_, l_], rMax[n_, l_]} = r /. Simplify[Solve[(l (l + 1))/r^2 - 2/r == -(1/n^2), r], n > 0];

sphericalToCartesian = Thread[{r, θ, ϕ} -> {Sqrt[x^2 + y^2 + z^2], ArcCos[z/Sqrt[x^2 + y^2 + z^2]], Arg[x + I y]}];

(*The radial orbitals here are approximated using a Slater-type orbital using Clementi's atomic constants for Fe; see Slater-type orbitals on Wikipedia for further information*)
pimp[n_, l_, m_][r_, θ_, ϕ_] := (2*22.27)^n ((2*22.27)/(2 n)!) (r^(n - 1)) (E^(-22.27 r)) (Im[SphericalHarmonicY[l, m, θ, ϕ]] + Im[SphericalHarmonicY[l, -m, θ, ϕ]])
pimn[n_, l_, m_][r_, θ_, ϕ_] := (2*22.27)^n ((2*22.27)/(2 n)!) (r^(n - 1)) (E^(-22.27 r)) (Im[SphericalHarmonicY[l, m, θ, ϕ]] - Im[SphericalHarmonicY[l, -m, θ, ϕ]])
prep[n_, l_, m_][r_, θ_, ϕ_] := (2*22.27)^n ((2*22.27)/(2 n)!) (r^(n - 1)) (E^(-22.27 r)) (Re[SphericalHarmonicY[l, m, θ, ϕ]] + Re[SphericalHarmonicY[l, -m, θ, ϕ]])
pren[n_, l_, m_][r_, θ_, ϕ_] := (2*22.27)^n ((2*22.27)/(2 n)!) (r^(n - 1)) (E^(-22.27 r)) (Re[SphericalHarmonicY[l, m, θ, ϕ]] - Re[SphericalHarmonicY[l, -m, θ, ϕ]])

(*To get a 2d plot, I set the earlier evaluation so that x->0*)
plot2dx0[f_, range_, contour_, opt : OptionsPattern[]] := RegionPlot[Evaluate[Abs[f[r, θ, ϕ] /. sphericalToCartesian]^2 >contour] /. x -> 0, {y, -range, range}, {z, -range, range}]

(*Plotting the different d-orbitals*)
Show[plot2dx0[prep[3, 2, 0], 0.7, 0.00007], plot2dx0[pimp[3, 2, 1], 0.7, 0.00007], plot2dx0[pimn[3, 2, 2], 0.7, 0.00007], plot2dx0[pren[3, 2, 1], 0.7, 0.00007], plot2dx0[prep[3, 2, 2], 0.7, 0.00007], Frame -> True, FrameLabel -> {"y", "z"}]

(*Output*)

Evaluate[Max[{Abs[prep[3, 2, 0][1.12*r, θ, ϕ] /. sphericalToCartesian^2 > 0.0007],Abs[pimp[3, 2, 1][0.8*r, θ, ϕ] /. sphericalToCartesian^2 > 0.0007],Abs[pimn[3, 2, 2][1.12*r, θ, ϕ] /. sphericalToCartesian^2 > 0.0007],Abs[pren[3, 2, 1][1.12*r, θ, ϕ] /. sphericalToCartesian^2 > 0.0007],Abs[prep[3, 2, 2][1.12*r, θ, ϕ] /. sphericalToCartesian^2 > 0.0007]}]]