Plotting electronic orbitals using Mathematica

As a chemist it is often useful to plot electronic orbitals.  These are used to describe the wave function of electrons in atoms or molecules.  Typically, these are output from electronic structure software in the form of a cube file, first developed by Gaussian.  These files contain volumetric data for a given orbital on a three-dimensional grid.

There exist many applications to visualize cube files, such as VMD or GaussView, but I wanted to take advantage of Mathematica‘s  capability to easily combine graphics, as well as the ability to automate the process in order to efficiently create frames for a movie.

First off, we need a function to extract the data from the cube file.  In the process, we will create the text for an XYZ file, a format also developed by Gaussian.  The function OutForm is used here to mimic the printf function found in other programming languages.

OutForm[num_?NumericQ, width_Integer, ndig_Integer, 
   OptionsPattern[]] :=
  Module[{mant, exp, val},
   {mant, exp} = MantissaExponent[num];
   mant = ToString[NumberForm[mant, {ndig, ndig}]];
   exp = If[Sign[exp] == -1, "-", "+"] <> IntegerString[exp, 10, 2];
   val = mant <> "E" <> exp;
   StringJoin@PadLeft[Characters[val], width, " "]
   ];


ReadCube[cubeFileName_?StringQ] := Module[
   {moltxt, nAtoms, lowerCorner, nx, ny, nz, xstep, ystep, zstep, 
    atoms, desc1, desc2, xyzText, cubeDat, xgrid, ygrid, zgrid, 
    dummy1, dummy2, atomicNumber, atomx, atomy, atomz, tmpString, 
    headerTxt,bohr2angstrom},
   bohr2angstrom = 0.529177249;
   moltxt = OpenRead[cubeFileName];
   desc1 = Read[moltxt, String];
   desc2 = Read[moltxt, String];
   lowerCorner = {0, 0, 0}; 
   {nAtoms, lowerCorner[[1]], lowerCorner[[2]], lowerCorner[[3]]} = 
    Read[moltxt, String] // ImportString[#, "Table"][[1]] &;
   xyzText = ToString[nAtoms] <> "\n";
   xyzText = xyzText <> desc1 <> desc2 <> "\n";
   {nx, xstep, dummy1, dummy2} = 
    Read[moltxt, String] // ImportString[#, "Table"][[1]] &;
   {ny, dummy1, ystep, dummy2} = 
    Read[moltxt, String] // ImportString[#, "Table"][[1]] &;
   {nz, dummy1, dummy2, zstep} = 
    Read[moltxt, String] // ImportString[#, "Table"][[1]] &;
   Do[
    {atomicNumber, dummy1, atomx, atomy, atomz} = 
     Read[moltxt, String] // ImportString[#, "Table"][[1]] &;
    xyzText = If[Sign[lowerCorner[[1]]] == 1,
      xyzText <> ElementData[atomicNumber, "Abbreviation"] <> 
       OutForm[atomx, 17, 7] <> OutForm[atomy, 17, 7] <> 
       OutForm[atomz, 17, 7] <> "\n",
      xyzText <> ElementData[atomicNumber, "Abbreviation"] <> 
       OutForm[bohr2angstrom atomx, 17, 7] <> 
       OutForm[bohr2angstrom atomy, 17, 7] <> 
       OutForm[bohr2angstrom atomz, 17, 7] <> "\n"];
    , {nAtoms}];
   cubeDat = 
    Partition[Partition[ReadList[moltxt, Number, nx ny nz], nz], ny];
   Close[moltxt];
   moltxt = OpenRead[cubeFileName];
   headerTxt = Read[moltxt, Table[String, {2 + 4 + nAtoms}]];
   Close[moltxt];
   headerTxt = StringJoin@Riffle[headerTxt, "\n"];
   xgrid = 
    Range[lowerCorner[[1]], lowerCorner[[1]] + xstep (nx - 1), xstep];
   ygrid = 
    Range[lowerCorner[[2]], lowerCorner[[2]] + ystep (ny - 1), ystep];
   zgrid = 
    Range[lowerCorner[[3]], lowerCorner[[3]] + zstep (nz - 1), zstep];
   {cubeDat, xgrid, ygrid, zgrid, xyzText, headerTxt}
   ];

If you need to create a cube file, then the following function can be used:

WriteCube[cubeFileName_?StringQ, headerTxt_?StringQ, cubeData_] := 
 Module[{stream}, 
  stream = OpenWrite[cubeFileName, FormatType -> FortranForm];
  WriteString[stream, headerTxt, "\n"];
  Map[WriteString[stream, ##, "\n"] & @@ 
     Riffle[ScientificForm[#, {3, 4}, 
         NumberFormat -> (Row[{#1, "E", If[#3 == "", "+00", #3], 
              "\t"}] &), NumberPadding -> {"", "0"}, 
         NumberSigns -> {"-", " "}] & /@ #, "\n", {7, -1, 7}] &, 
   cubeData, {2}];
  Close[stream];]

Next we need the function to plot the orbital,

CubePlot[{cub_, xg_, yg_, zg_, xyz_}, plotopts : OptionsPattern[]] := 
   Module[{xyzplot, bohr2picometer, datarange3D, pr},
    bohr2picometer = 52.9177249;
    datarange3D = 
      bohr2picometer {{xg[[1]], xg[[-1]]}, {yg[[1]], 
         yg[[-1]]}, {zg[[1]], zg[[-1]]}};
    xyzplot = ImportString[xyz, "XYZ"];
    Show[xyzplot, 
     ListContourPlot3D[Transpose[cub, {3, 2, 1}], 
       Evaluate[FilterRules[{plotopts}, Options[ListContourPlot3D]]], 
       Contours -> {-.02, .02}, ContourStyle -> {Blue, Red}, 
       DataRange -> datarange3D, MeshStyle -> Gray, 
       Lighting -> {{"Ambient", White}}], 
       Evaluate[
        FilterRules[{plotopts}, {ViewPoint, ViewVertical, ImageSize}]]]
    ];    

Let’s look at an example. Download the file cys-MO35.cube and place it in your home directory (or anywhere in your $Path). Then, read in the cube file with:

{cubedata,xg,yg,zg,xyz,header}= ReadCube["cys-MO35.cube"];

and plot it via

CubePlot[{cubedata, xg, yg, zg, xyz}]

When I want to create a movie file, I want all the images to have exactly the same ViewAngle, ViewPoint, and ViewCenter.  When you give these options to CubePlot, it feeds them directly to the Show function

vp = {ViewCenter -> {0.5, 0.5, 0.5}, 
   ViewPoint -> {1.072, 0.665, -3.13}, 
   ViewVertical -> {0.443, 0.2477, 1.527}};


CubePlot[{cubedata, xg, yg, zg, xyz}, vp]

Finally, you can also give any options that normally go to ListContourPlot3D

CubePlot[{cubedata, xg, yg, zg, xyz}, vp, 
 ContourStyle -> {Texture[ExampleData[{"ColorTexture", "Vavona"}]], 
   Texture[ExampleData[{"ColorTexture", "Amboyna"}]]}, 
 Contours -> {-.015, .015}] 

Many thanks to Daniel Healion for the ReadCube and WriteCube functions.