(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 368167, 8440]*) (*NotebookOutlinePosition[ 369250, 8477]*) (* CellTagsIndexPosition[ 369206, 8473]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["Normal modes of a triangular molecule"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Copyright (c) 1994, Wm. Martin McClain\nwmm@chem.wayne.edu\nDept. Chemistry, \ Wayne State University,\nDetroit, Michigan 48202\n"], "Text", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Intoduction"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "\t\t\t\t\t\t\tThe normal modes of motion of a molecule with N atoms in three \ dimensions are determined by 3N second order coupled differential equations; \ in two dimensions, there are 2N equations. The main mathematical problem is \ to uncouple the equations; if the harmonic approximation is used this can be \ done exactly and the solution then follows in closed form by elementary \ methods. \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tIn this script we work out a two \ dimensional problem using a \"brute force\" method that diagonalizes a \ 2N-by-2N matrix. It is called \"brute force\" because it makes no use of any \ tricks to reduce the size of the matrix. These tricks were important in \ pre-computer days when diagonalization took a major effort, but they are now \ arguably obsolete. Indeed, the brute force method works for asymmetric \ molecules of up to 100 atoms or more, such as small peptides. This script \ shows the simplest possible example, a nice little triangular molecule in two \ dimensions.\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tThis molecule has bond stretching \ forces, but because it is triangular, it does not need specific bond angle \ forces. After you understand this calculation, you will be able make your \ own program for molecules that need bond angle forces (though it is not \ trivial!). Hints will be available in a separate script, if you need them.\n\ \t\t\t\t\t\t\t\n\t\t\t\t\t\t\tIt was at one time thought that triatomic \ oxygen (ozone, O3) was an equilateral triangle, but it is not. In fact, no \ element forms a triatomic molecule that is equilateral in its ground state. \ But this calculation played a central role in determining that ozone is not \ equilateral, and you will see the result that proves this point. The carbon \ atoms in cyclopropane, C3 H6, make an equilateral triangle, and its lower \ frequency vibrations almost obey this model."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Preliminaries needed by the computer"], "Subsection", Evaluatable->False, InitializationCell->True, AspectRatioFixed->True], Cell[TextData["Off[General::spell,General::spell1]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData["<True, AspectRatioFixed->True], Cell[TextData["<True, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["nAvogadro = AvogadroConstant Mole"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ 6.0221367*10^23\ \>", "\<\ 23 6.02214 10\ \>"], "Output", Evaluatable->False, InitializationCell->True, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["cInCmPerSec = SpeedOfLight (100 Second/Meter)"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ 2.99792458*10^10\ \>", "\<\ 10 2.99792 10\ \>"], "Output", Evaluatable->False, InitializationCell->True, AspectRatioFixed->True]}, Open]]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData[ "Step 1: write the potential V in internal coordinates"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Since three points determine a plane, all vibrational motions of a \ triangle are two dimensional. Three internal coordinates determine \ completely the shape of the molecule: We let ", Evaluatable->False, AspectRatioFixed->True], StyleBox["dr12", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" be the", Evaluatable->False, AspectRatioFixed->True], StyleBox[" extension", Evaluatable->False, AspectRatioFixed->True], StyleBox[" from rest length of the bond between atoms ", Evaluatable->False, AspectRatioFixed->True], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", and similarly for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["dr23", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["dr31", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ". In rest position all extensions are zero; as the molecule vibrates, \ they take both positive and negative values. The vibrational potential \ energy, written in ", Evaluatable->False, AspectRatioFixed->True], StyleBox["int", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["ernal coordinates, is", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Vint = kHooke (dr12^2 + dr23^2 +dr31^2);"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Step 2: transform V to Cartesian coordinates"], "Section", Evaluatable->False, PageBreakAbove->True, PageBreakWithin->Automatic, PageBreakBelow->Automatic, AspectRatioFixed->True], Cell[TextData[{ StyleBox["The molecule has atom ", Evaluatable->False, AspectRatioFixed->True], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" at ", Evaluatable->False, AspectRatioFixed->True], StyleBox["{x1,y1}", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", etc., and the bond lengths are taken to be ", Evaluatable->False, AspectRatioFixed->True], StyleBox["2s", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" at rest. Therefore", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "internalToCartesian = \n{dr12->Sqrt[(x1-x2)^2 + (y1-y2)^2]-2s,\n \ dr23->Sqrt[(x2-x3)^2 + (y2-y3)^2]-2s,\n dr31->Sqrt[(x3-x1)^2 + \ (y3-y1)^2]-2s};"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Now we use this replacement rule on the potential energy function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Vint", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" to give ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Vcar", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["the potential energy in terms of cartesian coordinates :", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Vcar = Vint /. internalToCartesian "], "Input", AspectRatioFixed->False], Cell[OutputFormData[ "\<\ kHooke*((-2*s + ((x1 - x2)^2 + (y1 - y2)^2)^(1/2))^2 + (-2*s + ((x2 - x3)^2 + (y2 - y3)^2)^(1/2))^2 + (-2*s + ((-x1 + x3)^2 + (-y1 + y3)^2)^(1/2))^2)\ \>", "\<\ 2 2 2 kHooke ((-2 s + Sqrt[(x1 - x2) + (y1 - y2) ]) + 2 2 2 (-2 s + Sqrt[(x2 - x3) + (y2 - y3) ]) + 2 2 2 (-2 s + Sqrt[(-x1 + x3) + (-y1 + y3) ]) )\ \>"], "Output", Evaluatable->False, AspectRatioFixed->False]}, Open]], Cell[CellGroupData[{Cell[TextData["Vectorize and put in numerical constants"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Rename all Cartesian coordinates as ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z[i]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", with ", Evaluatable->False, AspectRatioFixed->True], StyleBox["i", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" running ", Evaluatable->False, AspectRatioFixed->True], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["6 ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[":", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "toZs = {x1->z[1], y1->z[2],\n x2->z[3], y2->z[4],\n x3->z[5], \ y3->z[6]};"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["The bond length parameter ", Evaluatable->False, AspectRatioFixed->True], StyleBox["s", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " has no influence on the modes or the frequencies, because the problem \ will be solved solely in terms of the partial derivatives of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Vcar", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" wrt the Cartesian coordinates, and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["s", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " disappears as soon as a partial is taken. It is therefore convenient to \ set ", Evaluatable->False, AspectRatioFixed->True], StyleBox["s", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" at this point.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We must now pick a value of the Hooke's Law constant for the three \ equivalent springs. We take a value, learned by experience, which is not \ unreasonable for chemical bonds:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["data = {kHooke->250000 (*dyne/cm*), s->1};"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["VcarN = Vcar /. toZs /. data"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ 250000*((-2 + ((z[1] - z[3])^2 + (z[2] - z[4])^2)^(1/2))^ 2 + (-2 + ((z[3] - z[5])^2 + (z[4] - z[6])^2)^(1/2))^ 2 + (-2 + ((-z[1] + z[5])^2 + (-z[2] + z[6])^2)^ (1/2))^2)\ \>", "\<\ 2 2 2 250000 ((-2 + Sqrt[(z[1] - z[3]) + (z[2] - z[4]) ]) + 2 2 2 (-2 + Sqrt[(z[3] - z[5]) + (z[4] - z[6]) ]) + 2 2 2 (-2 + Sqrt[(-z[1] + z[5]) + (-z[2] + z[6]) ]) )\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Step 3: define a standard equilibrium position"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We need a standard equilibrium position for the triangle, at which we will \ evaluate the partial derivatives. We take it centered on the origin, peak \ pointing up. We number the atoms:\n\n atom 1= peak, atom 2 = left end \ of base, atom 3 = right end of base. \n \n A little geometry \ shows that in this rest position the atom coordinates are"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "rest = {\n\tx1-> 0,\ty1-> +s Sqrt[3.]/2,\n\tx2-> -s,\ty2-> -s Sqrt[3.]/2,\n\ \tx3-> +s,\ty3-> -s Sqrt[3.]/2}/.toZs/.data"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {z[1] -> 0, z[2] -> 0.8660254037844386468, z[3] -> -1, z[4] -> -0.8660254037844386468, z[5] -> 1, z[6] -> -0.8660254037844386468}\ \>", "\<\ {z[1] -> 0, z[2] -> 0.866025, z[3] -> -1, z[4] -> -0.866025, z[5] -> 1, z[6] -> -0.866025}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "As a geometry check, we calculate the rest values of the atom-atom \ distances:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "{ Sqrt[(x1-x2)^2 + (y1-y2)^2],\n Sqrt[(x2-x3)^2 + 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The differential equations are solved. Do not feel \ disappointed that you can not yet see how the molecule moves; that will \ come, but it requires a lot of back transformation. Before we get to that, \ however, let us carry out concretely the program outlined above."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[{ StyleBox["Calculation of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Smat", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold", FontSlant->"Plain"] }], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["It is in fact quite possible to start with ", Evaluatable->False, AspectRatioFixed->True], StyleBox["c", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" (or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["krm", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[") and find ", Evaluatable->False, AspectRatioFixed->True], StyleBox["S", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["D", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[". It is done by the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" command ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Eigensystem", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", plus some clearing up with the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GramSchmidt", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" operator. We multiply matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["krm", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["10^-28", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" for this step, so that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Chop", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" will work correctly :", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["{scaledVals,SmatRawT}=Eigensystem[10^-28 krm]//Chop"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{5.634088477003082723, 2.817032113159937888, 2.817007862476730939, 0, 0, 0}, {{0, 0.574952165637630012, -0.5015535762087078704, -0.2883730462552897782, 0.5015535762087078705, -0.2883730462552897781}, {-0.5761512019156773988, 0, 0.2889744349702766454, 0.5005184034570252594, 0.2889744349702745542, -0.5005184034570264769}, {0, 0.5785418502660499353, 0.4984415815241112425, -0.2901734887151313878, -0.4984415815241084989, -0.2901734887151361396}, {-0.4550008469710104144, 0.3205419547936400413, 0.2656513586169395802, -0.09570671928111616986, 0.2656513586169395801, 0.7347965838836821482}, {0.6122160018129992802, 0.1176701379113694203, -0.2335492624222668117, 0.6045075152195963195, -0.2335492624222668117, -0.3698992482488776158}, {-0.6725139410095702304, -0.2471524463169200665, -0.3971413936456761715, -0.4041624062621258492, -0.3971413936456761713, -0.08860498686917535854}}}\ \>", "\<\ {{5.63409, 2.81703, 2.81701, 0, 0, 0}, {{0, 0.574952, -0.501554, -0.288373, 0.501554, -0.288373}, {-0.576151, 0, 0.288974, 0.500518, 0.288974, -0.500518}, {0, 0.578542, 0.498442, -0.290173, -0.498442, -0.290173}, {-0.455001, 0.320542, 0.265651, -0.0957067, 0.265651, 0.734797}, {0.612216, 0.11767, -0.233549, 0.604508, -0.233549, -0.369899}, {-0.672514, -0.247152, -0.397141, -0.404162, -0.397141, -0.088605}}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox["In ", Evaluatable->False, AspectRatioFixed->True], StyleBox["scaledVals", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ", note that two of the modes are nearly degenerate, differing only in the \ sixth decimal place. If we had not made one atom slightly heavier than the \ the other two, this degeneracy would have been exact. We do have an exact \ degeneracy in the last three eigenvectors, all of which have 0 as their \ eigenvalue. The ratio of the eigenvalues have an interesting simplicity \ about them:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["eValRatios = scaledVals/scaledVals[[3]]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {2.000025826001620343, 1.000008608667206781, 1., 0, 0, 0}\ \>", "\<\ {2.00003, 1.00001, 1., 0, 0, 0}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ "This predicts that equilateral triangular molecules should have two \ fundamental frequencies, with ratio close to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Sqrt[2]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ". If we had not added that extra bit of mass to one oxygen atom, the \ eigenvalue ratios would have been exact integers.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["A digression into reality"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Convert the eigenvalues to vibrational frequencies, in traditional units :"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "freqs = ws = \nSqrt[10^28 scaledVals]/(2 Pi cInCmPerSec) // N \n\ (*reciprocal centimeters*)"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {1260.118130226238675, 891.0361573392212751, 891.0323220471099511, 0, 0, 0}\ \>", "\<\ {1260.12, 891.036, 891.032, 0, 0, 0}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ "The infrared spectrum of ozone can be nicely interpreted in terms of ", Evaluatable->False, AspectRatioFixed->True], StyleBox[" three", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" fundamental frequencies (Herzberg, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["IR and Raman", Evaluatable->False, AspectRatioFixed->True], StyleBox[ ", p. 286). But only two fundamentals are provided by this model. \ Therefore the model is wrong for ozone.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The two strongest Raman lines of cyclopropane (CH2)3 have frequencies of \ 1189 and 866 wavenumbers (Herzberg, ibid., p. 352). "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["(1189/866) // N"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 1.372979214780600462\ \>", "\<\ 1.37298\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox["So their ratio is about 3% lower than the value ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Sqrt[2]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " predicted by a simple model in which the CH2 groups move rigidly \ together like an atom.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "The mode frequencies are the square roots of the eigenvalues. We adopt a \ strange time unit equal to one cycle of the lower frequency vibration:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["{w1,w2,w3,w4,w5,w6} = Sqrt[eValRatios]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {1.414222693214056845, 1.000004304324339787, 1., 0, 0, 0}\ \>", "\<\ {1.41422, 1., 1., 0, 0, 0}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["We now write the solutions for the normal modes:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["normalModes = Array[Q,6];"], "Input", AspectRatioFixed->True], Cell[TextData[ "Qsolns = {\nQ[1] -> A1 Cos[w1 t + p1],\nQ[2] -> A2 Cos[w2 t + p2],\nQ[3] -> \ A3 Cos[w3 t + p3],\nQ[4] -> x0 + vx t,\nQ[5] -> y0 + vy t,\nQ[6] -> a0 + wz \ t};\n"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The arbitrary constants of integration (initial conditions) that \ distinguish all possible motions from each other are the amplitudes (", Evaluatable->False, AspectRatioFixed->True], StyleBox["A1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["A2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["A3", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[") and phases (", Evaluatable->False, AspectRatioFixed->True], StyleBox["p1", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["p2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["p2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[") of the vibrational motions, and the initial positions (", Evaluatable->False, AspectRatioFixed->True], StyleBox["x0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["), and velocities (", Evaluatable->False, AspectRatioFixed->True], StyleBox["vx", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["vy", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["wz", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[") of the uniform motions.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Eigensystem", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " does not automatically orthogonalize degenerate eigenvectors. Test this \ out for yourself, multiplying ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SmatRawT", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " by its own transpose (in both possible multiplication orders). This \ non-orthogonality is contrary to one of our fundamental requirements on \ matrix ", Evaluatable->False, AspectRatioFixed->True], StyleBox["S", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[", so we must cure it with a Gram-Schmidt orthogonalization:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["SmatT = GramSchmidt[SmatRawT]//Chop;"], "Input", AspectRatioFixed->True], Cell[TextData["Smat = Transpose[SmatT];"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Now we check that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Smat", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" has the properties we required of it.\n", Evaluatable->False, AspectRatioFixed->True], StyleBox["(1)", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" Is it unitary?", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Smat.SmatT // Chop"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{1., 0, 0, 0, 0, 0}, {0, 1., 0, 0, 0, 0}, {0, 0, 1., 0, 0, 0}, {0, 0, 0, 1., 0, 0}, {0, 0, 0, 0, 1., 0}, {0, 0, 0, 0, 0, 1.}}\ \>", "\<\ {{1., 0, 0, 0, 0, 0}, {0, 1., 0, 0, 0, 0}, {0, 0, 1., 0, 0, 0}, {0, 0, 0, 1., 0, 0}, {0, 0, 0, 0, 1., 0}, {0, 0, 0, 0, 0, 1.}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ "Yes. It is not necessary to test the other mutiplication order, but you \ may if you like. What general theorem permits me to be confident that the \ other order will also produce a unit matrix? \n\n", Evaluatable->False, AspectRatioFixed->True], StyleBox["(2)", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" Does ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Smat", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" diagonalize ", Evaluatable->False, AspectRatioFixed->True], StyleBox["krm", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" by a similarity transform?", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["SmatT.(10^-28 krm).Smat // Chop;\nMatrixForm[%]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ MatrixForm[{{5.634088477003082724, 0, 0, 0, 0, 0}, {0, 2.817032113159937888, 0, 0, 0, 0}, {0, 0, 2.81700786247673094, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}]\ \>", "\<\ 5.63409 0 0 0 0 0 0 2.81703 0 0 0 0 0 0 2.81701 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["Yes. All is well."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Step 8: pleasing combinations of degenerate modes"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "There is nothing simple or meaningful about the degenerate orthogonal \ vectors created by the Gram-Schmidt procedure. If you want them in a form \ that is easy to visualize, you must recombine them by hand.\n\nWe illustrate \ this by recombining the three zero frequency modes in a meaningful way. \ (Generally, the zero frequency modes are thrown out. But it is good to see \ them dealt with explicitly in one simple example, so here they are.) We lift \ out the last three eigenvectors, as returned by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GramSchmidt", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" :", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["basis0 = Take[SmatT,-3]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{-0.4550008469710106385, 0.320541954793640761, 0.2656513586169403129, -0.09570671928111633612, 0.2656513586169390724, 0.7347965838836815925}, {0.4116972806406613187, 0.4730420668596500937, -0.06815634170913443516, 0.7478749933727185048, -0.06815634170913331267, 0.1952664142950515042}, {-0.5399345987245750174, 0.09057258369703611673, -0.5841899046401015805, 0.1168112948878028057, -0.5841899046401030525, 0.06377043360670083535}}\ \>", "\<\ {{-0.455001, 0.320542, 0.265651, -0.0957067, 0.265651, 0.734797}, {0.411697, 0.473042, -0.0681563, 0.747875, -0.0681563, 0.195266}, {-0.539935, 0.0905726, -0.58419, 0.116811, -0.58419, 0.0637704}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "Legend has it that this little vector space should be spanned by an \ x-translation, a y-translation, and a z-rotation. Let's see if the legend is \ true.\n\nIn Cartesian space, mode vectors representing x-translation amd \ y-translation would be"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "xTransCar = {1,0,1,0,1,0};\nyTransCar = {0,1,0,1,0,1};"], "Input", AspectRatioFixed->True], Cell[TextData[ "But our orthogonalized mode vectors are in rootmass weighted space, and in \ rootmass space the translation vectors are"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["xTransRM = Sqrt[massVec]*xTransCar // Normalize"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {0.5785493364608888041, 0, 0.5767498007284724154, 0, 0.5767498007284724154, 0}\ \>", "\<\ {0.578549, 0, 0.57675, 0, 0.57675, 0}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["yTransRM = Sqrt[massVec]*yTransCar // Normalize"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {0, 0.5785493364608888041, 0, 0.5767498007284724154, 0, 0.5767498007284724154}\ \>", "\<\ {0, 0.578549, 0, 0.57675, 0, 0.57675}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox["These are obviously orthogonal. But are they spanned by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["basis0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox["? To find out, dot them against all three eigenvectors:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["xTCoeffs = Map[Dot[xTransRM,#]&,basis0]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {0.04318829818691525002, 0.1595688755391576582, -0.9862416260019077627}\ \>", "\<\ {0.0431883, 0.159569, -0.986242}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ "Generally, it takes six basis vectors to span a vector with six \ components. But if the legend is true, the three in ", Evaluatable->False, AspectRatioFixed->True], StyleBox["basis0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" should do the job:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["xTCoeffs.basis0 // Chop"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {0.5785493364608888041, 0, 0.5767498007284716267, 0, 0.5767498007284732041, 0}\ \>", "\<\ {0.578549, 0, 0.57675, 0, 0.57675, 0}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox["Yes! Vector ", Evaluatable->False, AspectRatioFixed->True], StyleBox["xTransRM", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" is spanned by ", Evaluatable->False, AspectRatioFixed->True], StyleBox["basis0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["yTCoeffs = Map[Dot[yTransRM,#]&,basis0]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {0.554044287310853006, 0.817634792830895278, 0.1565511841238766443}\ \>", "\<\ {0.554044, 0.817635, 0.156551}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["yTCoeffs.basis0 // Chop"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {0, 0.5785493364608888041, 0, 0.5767498007284727669, 0, 0.5767498007284720641}\ \>", "\<\ {0, 0.578549, 0, 0.57675, 0, 0.57675}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox["Yes again; vector ", Evaluatable->False, AspectRatioFixed->True], StyleBox["yTransRM", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " is also spanned. Now all we have to do is find the vector in this space \ that is orthogonal to both ", Evaluatable->False, AspectRatioFixed->True], StyleBox["xTransRM", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["yTransRM", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ". It should be the legendary rotational displacement mode. \n\nTo get \ the third vector, we will use ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GramSchmidt", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " on the two translation vectors, together with a third vector selected \ from ", Evaluatable->False, AspectRatioFixed->True], StyleBox["basis0", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ". We will get the best accuracy by using one that is already as \ orthogonal as possible to both the translations. Which one is this?", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "{{xTransRM.SmatT[[4]],yTransRM.SmatT[[4]]},\n \ {xTransRM.SmatT[[5]],yTransRM.SmatT[[5]]},\n \ {xTransRM.SmatT[[6]],yTransRM.SmatT[[6]]}}"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{0.04318829818691525002, 0.554044287310853006}, {0.1595688755391576582, 0.817634792830895278}, {-0.9862416260019077627, 0.1565511841238766443}}\ \>", "\<\ {{0.0431883, 0.554044}, {0.159569, 0.817635}, {-0.986242, 0.156551}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox["It looks like ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SmatT[[4]]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" should be the best.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "zeroModesNeat=\nGramSchmidt[{xTransRM,yTransRM,SmatT[[4]]}] // Chop"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {{0.5785493364608888041, 0, 0.5767498007284724154, 0, 0.5767498007284724154, 0}, {0, 0.5785493364608888041, 0, 0.5767498007284724154, 0, 0.5767498007284724154}, {-0.5773477789095459263, 0, 0.2895745902065266302, -0.4994810585005502242, 0.2895745902065251381, 0.4994810585005493556}}\ \>", "\<\ {{0.578549, 0, 0.57675, 0, 0.57675, 0}, {0, 0.578549, 0, 0.57675, 0, 0.57675}, {-0.577348, 0, 0.289575, -0.499481, 0.289575, 0.499481}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ "The last vector could plausibly be parallel to an infinitesimal rotation \ about ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ "; note that the peak atom (#1) is moving purely horizontally, which it \ does when the triangle undergoes an infinitesimal rotation about ", Evaluatable->False, AspectRatioFixed->True], StyleBox["z", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ". We will plot all of these motions later.\n\nNow join these neat zero \ frequency vectors onto the three vibration vectors, to make a basis set that \ spans all possible atomic displacements:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["SmatNeatT = Join[Take[SmatT,3],zeroModesNeat]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{0, 0.574952165637630012, -0.5015535762087078704, -0.2883730462552897782, 0.5015535762087078705, -0.2883730462552897781}, {-0.5761512019156773988, 0, 0.2889744349702762954, 0.5005184034570250582, 0.2889744349702749042, -0.5005184034570266781}, {0, 0.5785418502660499354, 0.4984415815241099229, -0.2901734887151336736, -0.4984415815241098187, -0.2901734887151338539}, {0.5785493364608888041, 0, 0.5767498007284724154, 0, 0.5767498007284724154, 0}, {0, 0.5785493364608888041, 0, 0.5767498007284724154, 0, 0.5767498007284724154}, {-0.5773477789095459263, 0, 0.2895745902065266302, -0.4994810585005502242, 0.2895745902065251381, 0.4994810585005493556}}\ \>", "\<\ {{0, 0.574952, -0.501554, -0.288373, 0.501554, -0.288373}, {-0.576151, 0, 0.288974, 0.500518, 0.288974, -0.500518}, {0, 0.578542, 0.498442, -0.290173, -0.498442, -0.290173}, {0.578549, 0, 0.57675, 0, 0.57675, 0}, {0, 0.578549, 0, 0.57675, 0, 0.57675}, {-0.577348, 0, 0.289575, -0.499481, 0.289575, 0.499481}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["SmatNeat = Transpose[SmatNeatT];"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["At this point the unitarity and the diagonalizing ability of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["SmatNeat", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" should be checked. 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AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " solutions; therefore we implement this as a replacement rule that will \ turn ", Evaluatable->False, AspectRatioFixed->True], StyleBox["q", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["'s into expressions containing ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Q", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["'s:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["qVec = Array[q,6];"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "q2Qgeneral = \n\t\tqVec->SmatNeat.normalModes // Thread;\n\t\t\ Print[ColumnForm[%]]"], "Input", AspectRatioFixed->True], Cell[TextData[ "q[1] -> -0.576151 Q[2] + 0.578549 Q[4] - 0.577348 Q[6]\nq[2] -> 0.574952 \ Q[1] + 0.578542 Q[3] + 0.578549 Q[5]\nq[3] -> -0.501554 Q[1] + 0.288974 Q[2] \ + 0.498442 Q[3] + \n \n 0.57675 Q[4] + 0.289575 Q[6]\nq[4] -> -0.288373 \ Q[1] + 0.500518 Q[2] - 0.290173 Q[3] + \n \n 0.57675 Q[5] - 0.499481 Q[6]\n\ q[5] -> 0.501554 Q[1] + 0.288974 Q[2] - 0.498442 Q[3] + \n \n 0.57675 Q[4] \ + 0.289575 Q[6]\nq[6] -> -0.288373 Q[1] - 0.500518 Q[2] - 0.290173 Q[3] + \n \ \n 0.57675 Q[5] + 0.499481 Q[6]"], "Print", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "We can discard the uniform modes by setting their amplitudes to zero:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["q2Qvibr=\n\t\tq2Qgeneral/.{Q[4]->0,Q[5]->0,Q[6]->0}"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {q[1] -> -0.5761512019156773988*Q[2], q[2] -> 0.574952165637630012*Q[1] + 0.5785418502660499354*Q[3], q[3] -> -0.5015535762087078704*Q[1] + 0.2889744349702762954*Q[2] + 0.4984415815241099229*Q[3] , q[4] -> -0.2883730462552897782*Q[1] + 0.5005184034570250582*Q[2] - 0.2901734887151336736*Q[3] , q[5] -> 0.5015535762087078705*Q[1] + 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