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Notebook[{

Cell[CellGroupData[{
Cell["Huckel MO Method - Energies", "Subtitle"],

Cell[TextData[{
  "The Huckel MO method is a set of approximations for estimating the \
energies and shapes of pi MOs found in conjugated organic molecules. The \
energies can be found in two different ways.\n\n1. Write an ",
  StyleBox["n",
    FontSlant->"Italic"],
  " \[Times] ",
  StyleBox["n ",
    FontSlant->"Italic"],
  "square matrix for a conjugated system containing ",
  StyleBox["n",
    FontSlant->"Italic"],
  " carbons. Write",
  " '",
  StyleBox["x",
    FontSlant->"Italic"],
  "' at all diagonal positions. Write '",
  StyleBox["1",
    FontSlant->"Italic"],
  "' at every off-diagonal element ",
  StyleBox["ij",
    FontSlant->"Italic"],
  " if carbons ",
  StyleBox["i",
    FontSlant->"Italic"],
  " and ",
  StyleBox["j",
    FontSlant->"Italic"],
  " are \[Sigma]-bonded. Write '",
  StyleBox["0",
    FontSlant->"Italic"],
  "' at all other off-diagonal elements. The energies are related to values \
of ",
  StyleBox["'x'",
    FontSlant->"Italic"],
  " that make the determinant vanish by\n\n",
  StyleBox["E",
    FontSlant->"Italic"],
  " = \[Alpha] - ",
  StyleBox["x",
    FontSlant->"Italic"],
  " \[Beta]\n\n2. Write the same matrix as in #1, but write '",
  StyleBox["0",
    FontSlant->"Italic"],
  "' at all diagonal positions. The energies correspond to the matrix \
eigenvalues, \[Lambda], by\n\n",
  StyleBox["E",
    FontSlant->"Italic"],
  " = \[Alpha] + \[Lambda] \[Beta]\n"
}], "Text",
  TextAlignment->Left],

Cell[CellGroupData[{

Cell[TextData[{
  "Illustration - energy calculation for allyl cation, ",
  Cell[BoxData[
      \(TraditionalForm\`C\_1 - C\_2 - C\_3\)]]
}], "Subsubtitle"],

Cell[TextData[{
  "Note: To define a matrix, type ",
  StyleBox["SHIFT-CTRL-C",
    FontWeight->"Bold",
    FontVariations->{"Underline"->True}],
  " and fill in the necessary information. Or, select ",
  StyleBox["Create Table/Matrix/Palette",
    FontWeight->"Bold"],
  " from the ",
  StyleBox["Input",
    FontWeight->"Bold"],
  " menu and fill in the information.\n\nMethod 1."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    RowBox[{"m1", " ", "=", " ", 
      RowBox[{"(", GridBox[{
            {"x", "1", "0"},
            {"1", "x", "1"},
            {"0", "1", "x"}
            }], ")"}]}], "\[IndentingNewLine]", \(Solve[0 \[Equal] Det[m1], 
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Cell[BoxData[
    \({{x, 1, 0}, {1, x, 1}, {0, 1, x}}\)], "Output"],

Cell[BoxData[
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Cell["Method 2.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    RowBox[{"m2", " ", "=", " ", 
      RowBox[{"(", GridBox[{
            {"0", "1", "0"},
            {"1", "0", "1"},
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            }], ")"}]}], "\[IndentingNewLine]", \(Eigenvalues[m2]\)}], "Input"],

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Cell[BoxData[
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Cell[TextData[{
  "The eigenvalues convert into these MO energies: ",
  StyleBox["E",
    FontSlant->"Italic"],
  " = \[Alpha] - ",
  Cell[BoxData[
      \(TraditionalForm\`\@2\)]],
  "\[Beta], ",
  "\[Alpha] + ",
  Cell[BoxData[
      \(TraditionalForm\`\@2\)]],
  "\[Beta], \[Alpha]. Because we assume \[Beta] < 0, the first energy \
corresponds to an antibonding MO, the second to a bonding MO, and the third \
to a nonbonding MO.\n"
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Practice Problems", "Subsubtitle"],

Cell[TextData[{
  "Use Method 2 to obtain MO energies for ethylene and butadiene. Make sure \
you convert the eigenvalues into energies.\n\nAre the 4 pi electrons of \
butadiene stabilized compared to the 4 pi electrons found in two isolated \
ethylene molecules? (Note: the Huckel method defines the energy of a pi \
system as the ",
  StyleBox["sum",
    FontVariations->{"Underline"->True}],
  " of the pi electron energies.)\n\nHow do the MO energies of butadiene \
compare to those of ethylene?\n"
}], "Text"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Huckel MO Method - MOs", "Subtitle"],

Cell[TextData[{
  "An MO is defined by the mixing coefficients that indicate how much each AO \
contributes to the MO. The standard method for obtaining these coefficients \
is to substitute the MO energy into the secular equations, and then solve \
these equations.\n\nI haven't succeeded at making this method work in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ". The program gives me the 'trivial' solution only: all coefficients = 0. \
Fortunately, there is another way to get the coefficients (and it's even \
simpler).\n\nThe mixing coefficients are identical to the eigenvectors of the \
'Method 2' matrix defined above. The coefficients obtained in this way do not \
give a normalized MO, so normalization must be carried out in a second step.\n\
"
}], "Text"],

Cell[CellGroupData[{

Cell[TextData[{
  "Illustration - energy & MO calculation for allyl cation, ",
  Cell[BoxData[
      \(TraditionalForm\`C\_1 - C\_2 - C\_3\)]]
}], "Subsubtitle"],

Cell["Method 2.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    RowBox[{"m2", " ", "=", " ", 
      RowBox[{"(", GridBox[{
            {"0", "1", "0"},
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            }], ")"}]}], "\[IndentingNewLine]", \(Eigensystem[m2]\)}], "Input"],

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Cell[BoxData[
    \({{\(-\@2\), \@2, 
        0}, {{1, \(-\@2\), 1}, {1, \@2, 1}, {\(-1\), 0, 1}}}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The output from the ",
  StyleBox["Eigensystem",
    FontWeight->"Bold"],
  " function is complicated and must be read carefully. The first list \
contains the eigenvalues {\[Lambda]1, \[Lambda]2, \[Lambda]3}. The subsequent \
lists contain the eigenvectors corresponding to each of these eigenvalues, \
respectively:\n\n",
  Cell[BoxData[GridBox[{
          {"\[Lambda]", "evector"},
          {\(-\@2\), \({1, \(-\@2\), 1}\)},
          {\(\@2\), \({1, \@2, 1}\)},
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          },
        GridFrame->True]]],
  "\n\nThese results transform into MO energies and ",
  StyleBox["unnormalized",
    FontVariations->{"Underline"->True}],
  " MOs as follows:\n\n",
  Cell[BoxData[GridBox[{
          {"E", "MO"},
          {\(\[Alpha] - \(\(\@2\) \(\[Beta]\)\(\ \)\)\), \(p\_1 - \(\@2\) 
                p\_2 + p\_3\)},
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          {"\[Alpha]", \(p\_1 - p\_3\)}
          },
        GridFrame->True]]],
  "\n\nTo normalize the MOs, recall that the Huckel MO method assumes ",
  StyleBox["no overlap",
    FontVariations->{"Underline"->True}],
  " between different ",
  StyleBox["p",
    FontSlant->"Italic"],
  " orbitals. This greatly simplifies the overlap integral. For example,\n\n\
",
  Cell[BoxData[
      \(TraditionalForm\`1\  = \(\[Integral]\((\[Phi]\_3)\)\^2\  = \ \(\
\[Integral]\(\(N\^2\)(p\_1 - \(\@2\) p\_2 + p\_3)\)\^2\  = \ \(N\^2\)[\
\[Integral]\((p\_1)\)\^2\  + \ 
                2 \(\[Integral]\((p\_2)\)\^2\)\  + \ \
\[Integral]\((p\_3)\)\^2]\)\)\)]],
  "\n\n\t",
  Cell[BoxData[
      \(TraditionalForm\`\(\(=\)\(\ \)\(\(N\^2\)(1\  + \ 2\  + \ 1)\  = \ 
          4\ N\^2\)\)\)]],
  "\n\nWhich gives ",
  StyleBox["N",
    FontSlant->"Italic"],
  " = 1/2 and ",
  Cell[BoxData[
      \(TraditionalForm\`\[Phi]\_3\  = \ \((1/2)\) \((p\_1 - \(\@2\) p\_2 + 
              p\_3)\)\)]],
  "\n\nIn general, the normalization constant for any Huckel MO can be \
obtained as follows:\n\n",
  Cell[BoxData[
      \(TraditionalForm\`\[Phi]\  = \ N \(\[Sum]\(c\_i\) p\_i\)\)]],
  "\n\n",
  Cell[BoxData[
      \(TraditionalForm\`N\  = \ 1/\@\(\[Sum]\ c\_i\^2\)\)]],
  "\n"
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Practice Problems", "Subsubtitle"],

Cell[TextData[{
  "Obtain eigenvectors for ethylene and butadiene. Transform these into \
normalized MOs.\n\n",
  StyleBox["Spartan",
    FontSlant->"Italic"],
  " images of the MOs are shown on the front screen. How do the signs of the \
mixing coefficients correlate with interatomic nodes?"
}], "Text"]
}, Open  ]]
}, Open  ]]
},
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