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I'm visualizing some hydrogen like atomic orbitals. For looking at plane slices of the probability density, the DensityPlot function works well, and with something like:

 DensityPlot[ psi1XYsq[u, v, z], {u, -w, w}, {v, -w, w} ,
  Mesh -> False, Frame -> False, PlotPoints -> 45,
  ColorFunctionScaling -> True, ColorFunction -> "SunsetColors"]
 , {{w, 10}, 1, 20}
 , {z, 1, 20, 1}

I can get a nice plot hybrid orbital sample plot in x y plane

I was hoping that there was something like a DensityPlot3D so that I could visualize these in 3D, but I don't see such a function. I was wondering how DensityPlot be simulated using other plot functions, so that the same idea could be applied to a 3D plot to construct a DensityPlot3D like function?

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You may be interested in heatmap density 3D –  Kuba Sep 15 '13 at 16:33
Or how to visualize 4D function –  Kuba Sep 15 '13 at 16:34
Image3D works, apart from interpolation. A question arises, though: how you would be able to interpret the results? In general, isosurfaces are much more practical than images which look like fuzzy, vague clouds at best, opaque mass with hidden internal structure at worst. –  kirma Sep 15 '13 at 16:47
@kirma "fuzzy, vague clouds" - sounds like an orbital ;) –  Kuba Sep 15 '13 at 16:49
@Kuba, yes, that 4D visualization answer by halmir looks like what I was looking for. –  Peeter Joot Sep 18 '13 at 13:19

3 Answers 3

In an presentation by Markus van Almsick, he gives an solution to visualize atomic orbitals using Image3D.

Radius wave function (hydrogen):

R[n_Integer?Positive, l_Integer?NonNegative, r_] :=
 Block[{ρ = (2 r)/n}, 
   Sqrt[(2/n)^3 (n - l - 1)!/(2 n (n + l)!)] E^(-ρ/2) ρ^l LaguerreL[n - l - 1, 2 l + 1, ρ]] /; l < n

full wave function:

ψ[n_, l_, m_, r_, ϑ_, φ_] := ψ[n, l, m, r, ϑ, φ] = 
  FullSimplify[R[n, l, r] SphericalHarmonicY[l, m, ϑ, φ], {r >= 0, ϑ ∈ Reals, φ ∈ Reals}]

CompileWaveFunction = Compile[{{x, _Real}, {y, _Real}, {z, _Real}},
   Block[{ρ = x^2 + y^2, r, ϑ, φ},
    If[ρ > 0,
     r = Sqrt[ρ + z^2]; ϑ = ArcCos[z/r]; φ = ArcTan[x, y],
     r = Abs[z]; ϑ = π/2 Sign[z]; φ = 0];
   CompilationTarget -> "C"
   ] &;

color function:

colorFunction = (Blend[{
      {0., RGBColor[0.7, 0.8, 1., 0.]}, 
      {0.1, RGBColor[0., 0.7, 0.1, 0.012]}, 
      {0.4, RGBColor[1., 0.1, 0.03169, 0.06723]}, 
      {1., RGBColor[1., 0.95051, 0., 0.10963]}}, #] &)

plot 3p orbital

 {nψ = 
   CompileWaveFunction[ψ[3, 1, 0, r, ϑ, φ]], data, vol},
 data = Table[Abs[nψ[x, y, z]]^2, {z, -20, 20, 0.25}, {y, -20, 20, 0.25}, {x, -20, 20, 0.25}];
 vol = RawArray["Byte", Round[(255/Max[data]) data]];
 Image3D[vol, "Byte", Background -> Black, 
  Method -> {"FastRendering" -> True, "InterpolateValues" -> True}, 
  ColorFunction -> colorFunction, BoxRatios -> 1]

enter image description here

it's quite smooth to rotate the Image3D object

enter image description here

We can also visualize the atomic orbital by plotting the isosurface:

Block[{nψ = CompileWaveFunction[ψ[3, 2, 0, r, ϑ, φ]]},
 ContourPlot3D[Abs[nψ[x, y, z]]^2, {x, -20, 20}, {y, -20, 20}, {z, -20, 20}, 
  PlotPoints -> 15, Contours -> {0.00002}, 
  ColorFunction -> Function[{x, y, z}, ColorData["Rainbow"]@Rescale[Arg[nψ[x, y, z]], {-π, π}]], 
  ColorFunctionScaling -> False, Mesh -> None]

enter image description here

and make a animation showing different isosurfaces

plots = ParallelTable[
   Block[{nψ = CompileWaveFunction[ψ[4, 2, 1, r, ϑ, φ]]}, 
      Abs[nψ[x, y, z]]^2, {x, -20, 20}, {y, -20, 20}, {z, -20, 20}, PlotPoints -> 17, Contours -> {ct}, 
      ColorFunction -> Function[{x, y, z}, ColorData["Rainbow"]@Rescale[Arg[nψ[x, y, z]], {-π, π}]], 
      Boxed -> False, Axes -> False, ColorFunctionScaling -> False, Mesh -> None, 
      ViewPoint -> {0.98, -2.76, 1.7}, ViewVertical -> {-0.004, -0.117, 0.993}]],
   {ct, 0.00003, 0.000015, -0.0000005}

enter image description here

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My preferred method for this kind of thing is projecting each dimension onto a plane and then combining them together. I think MATLAB has similar functionality. Mind you, the answers and comments on my question about projecting are right in pointing out that this will become inefficient for high polygon counts (essentially more PlotPoints) so if you want to Manipulate in a smooth way, you may want to use Texture. See this relevant question for details on that.

Now, I haven't got your function psi1XYsq so I will pick a double gaussian:

E^(-(x^2 + (y - 2)^2 + (z - 3)^2)/10) + E^(-((x + 1)^2 + (y + 2)^2 + (z + 2)^2)/25)

The idea is to use something similar to the function @Jens is using in his answer to How to make a drop-shadow for Graphics3D objects.

Block[{d1, d2, d3, function, options, opacity, x0, y0, z0, min, max},
 {min, max} = {-9, 9};
 {x0, y0, z0} = {0., 0., 0.};
 opacity = 0.9;
 function[x_, y_, z_] := 
  E^(-(x^2 + (y - 2)^2 + (z - 3)^2)/10) + 
   E^(-((x + 1)^2 + (y + 2)^2 + (z + 2)^2)/25);
 options = Sequence @@ {PlotPoints -> 45, Mesh -> None, 
    ColorFunctionScaling -> False, ColorFunction -> "SunsetColors"};
 d1 = First@DensityPlot[function[x, y, z0], {x, min, max}, {y, min, max}, 
     Evaluate@options] /. {x_?AtomQ, y_?AtomQ} -> {x, y, z0};
 d2 = First@DensityPlot[function[x, y0, z], {x, min, max}, {z, min, max}, 
     Evaluate@options] /. {x_?AtomQ, z_?AtomQ} -> {x, y0, z};
 d3 = First@DensityPlot[function[x0, y, z], {y, min, max}, {z, min, max}, 
     Evaluate@options] /. {y_?AtomQ, z_?AtomQ} -> {x0, y, z};
 Show[Graphics3D[{d1, d2, d3}, Lighting -> "Neutral"] /. 
   GraphicsComplex[xx__] -> {Opacity[opacity], GraphicsComplex[xx]}]

3d image

The code is straightforward: min, max define the range for each variable, {x0, y0, z0} define the projection planes, and opacity the Opacity. You will notice I have turned off ColorFunctionScaling so that each slice is bright according to an absolute value and they merge together nicely. If your function is not normalised you may want to normalise it before doing that.

If you can afford lowering the PlotPoints, Manipulate isn't too bad, and you can make animations that look like volumetric rendering (apologies for the 300K gif):

Table[Block[{d1, d2, d3, function, options, opacity, x0, y0, z0, min, 
   {min, max} = {-9, 9};
   {x0, y0, z0} = {0., t, 0.};
   opacity = 0.9;
   function[x_, y_, z_] := 
    E^(-(x^2 + (y - 2)^2 + (z - 3)^2)/10) + 
     E^(-((x + 1)^2 + (y + 2)^2 + (z + 2)^2)/45);
   options = 
    Sequence @@ {PlotPoints -> 25, Mesh -> None, 
      ColorFunctionScaling -> False, 
      ColorFunction -> "SunsetColors"};
   d1 = First@
      DensityPlot[function[x, y, z0], {x, min, max}, {y, min, max}, 
       Evaluate@options] /. {x_?AtomQ, y_?AtomQ} -> {x, y, z0};
   d2 = First@
      DensityPlot[function[x, y0, z], {x, min, max}, {z, min, max}, 
       Evaluate@options] /. {x_?AtomQ, z_?AtomQ} -> {x, y0, z};
   d3 = First@
      DensityPlot[function[x0, y, z], {y, min, max}, {z, min, max}, 
       Evaluate@options] /. {y_?AtomQ, z_?AtomQ} -> {x0, y, z};
   Show[Graphics3D[{d1, d2, d3}, Lighting -> "Neutral"] /. 
     GraphicsComplex[xx__] -> {Opacity[opacity], 
   ], {t, 9, -9, -1.5}];
Export["3d.gif", %];


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What does your {x_?AtomQ, y_?AtomQ} -> {x, y, z0} replacement in this code do? –  Peeter Joot Sep 17 '13 at 22:17
I adds a third dimension to every pair of 2d coordinates of the density plot. –  gpap Sep 17 '13 at 23:46

I recently revisited this, and found that RegionPlot3D is by far the fastest way to plot orbitals, compared to Image3D and ContourPlot3D. I was surprised by the difference, so I thought it's worth posting this.

In addition, I also made the process of choosing the plot parameters automatic, based on simple estimates for the size of the orbital wave function. So you don't have to find the right contour value by trial and error. This automation allows me to plot large numbers of orbitals in one go.

Below, I'm plotting the first ten orbitals in a systematic table, and the whole table takes about as long as doing one or two such plots using ContourPlot3D (depending on what functions I try). There was also no need to compile any of the functions, making the code quite straightforward.

Clear[rMin, rMax, r, θ, ϕ];
{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]}];

ψ[n_, l_, m_][r_, θ_, ϕ_] := 
 Sqrt[ (n - l - 1)!/(n + l)!] E^(-(r/n)) ((2 r)/n)^l 2/
  n^2 LaguerreL[n - l - 1, 2 l + 1, (2 r)/n] SphericalHarmonicY[l, 
   m, θ, ϕ]

plotOrbital[f_, range_, contour_, opt : OptionsPattern[]] := 
  Evaluate[Abs[f[r, θ, ϕ] /. sphericalToCartesian]^2 > 
    contour], {x, -range, range}, {y, -range, range}, {z, -range, 
   range}, opt, Mesh -> False, PlotPoints -> 35, PlotStyle -> Orange, 
  Lighting -> "Neutral", PlotTheme -> "Classic"]

grid = Table[
   Labeled[plotOrbital[ψ[n, l, m], 2 n^2, .05/n^6], 
    Row[{"n = ", n, ", ℓ = ", l, ", m = ", m}]], {n, 1, 
    3}, {l, 0, n - 1}, {m, 0, l}];



Above, the complete hydrogenic orbital wave function is ψ. Given the principal quantum number n, the energy is known. Setting the energy equal to the effective potential yields the classical turning points, {rMin[n_, ℓ_], rMax[n_, ℓ_]}. This is used to determine the required plot size automatically: If the linear dimension r scales with $n^2$, then the volume scales with $n^6$. For a normalized wave function, we can then estimate that the probability density at a typical point will decrease with $1/n^6$. Therefore, I choose the threshold for the equi-probability contour to scale with this factor. Here is a check to verify what I said about the length scale:

Simplify[rMax[n, 0], n > 0]

(* ==> 2 n^2 *)

This is the largest achievable radius because I set the angular momentum l to zero.

In sphericalToCartesian, I write down how to convert to Cartesian coordinates (instead of using built-in functions which differ too much between Mathematica versions).

plotOrbital does the plotting for a given wave function f. In producing the grid of orbitals, I choose the parameters as described above.

Edit: volumetric slices

To show that the RegionPlot approach is also able to provide multiple contours, and to show how these layered contours can give us information about the volume in 3D, here I first combine several outputs of plotOrbital with different contour values in a single Show. The surfaces have been given their own PlotStyle. Then I put the resulting 3D graphics in a Manipulate that allows you to slice through the layers in real time:

With[{n = 3, l = 2, m = 0}, 
 orb = Show[
   Table[plotOrbital[ψ[n, l, m], 2 n^2, c/n^6, 
     PlotStyle -> Directive[Opacity[0.6`], Hue[c]]], {c, 0.3`, 
     0.05`, -0.04`}], PlotRange -> All, BoxRatios -> Automatic]]

With[{n = 3},
  Show[orb, Boxed -> False, Axes -> False, 
   ViewVector -> {{35, -35, 5}, {0, 0, 0}}, ViewAngle -> Pi/4, 
   PlotRange -> {{-#, #}, {y, #}, {-#, #}} &[2 n^2]], {y, -2 n^2, 
   2 n^2}]]


The trick to get slices and hollow spaces between layers in RegionPlot3D is to cut off the plot with a reduced PlotRange. To still keep the object fixed in the view port (instead of moving around to stay centered with the changing PlotRange), I add fixed values for the ViewVector and ViewAngle. This trick using PlotRange doesn't work if you cut the range off directly inside the RegionPlot3D. You have to do it after the fact with Show.

Again, the drawing of multiple layers is very fast using RegionPlot3D. It could even be sped up more using ParallelTable.

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That's fantastic! Indeed RegionPlot3D is much faster. And it also solves the problem of holes in ContourPlot3D. –  xslittlegrass Feb 21 at 22:27

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