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Weber and G. Hohlneicher, ", StyleBox["Mol. Phys.", FontSlant->"Italic"], " ", StyleBox["101", FontWeight->"Bold"], ", 2125 (2003) [1]. There, it was used to calculate expressions for 141 \ different overlap integrals. The ", StyleBox["I(m,n)", FontSlant->"Italic"], " are useful in the calculation of intensities in vibronic molecular \ spectra. The derivation is based on the method of generating functions [2,3] \ in which a quantity is derived by equating the coefficients of identical \ polynomes occurring in polynomial expansions at both sides of an equation. \ " }], "Text"], Cell["Definition and initialization of variables", "Text", CellDingbat->"\[FilledSquare]", FontFamily->"Helvetica", FontSize->18, FontWeight->"Bold"], Cell[BoxData[ \(\(\(Clear[AMat, EMat, CMat, U, T, Ut, Tt, Ti, Tj, Tk, Tl, U, Ui, Uj, Uk, Ul, BVec, BtVec, rechts, recht, \ rech, \ links, \ Result, \ lhs\ , rhs, \ biggest\ ]\)\(\n\) \)\)], "Input"], Cell[BoxData[{ \(T := {Ti, Tj, Tk, Tl}\), "\n", \(Tt := {Ti, Tj, Tk, Tl}\), "\n", \(U := {Ui, Uj, Uk, Ul}\), "\n", \(Ut := {Ui, Uj, Uk, Ul}\), "\n", \(DVec := {Di, Dj, Dk, Dl}\), "\n", \(BVec := {Bi, Bj, Bk, Bl}\), "\n", \(BtVec := {Bi, Bj, Bk, Bl}\)}], "Input"], Cell[BoxData[{ \(AMat := {{Aii, Aij, Aik, Ail}, {Aij, Ajj, Ajk, Ajl}, {Aik, Ajk, Akk, Akl}, {Ail, Ajl, Akl, All}}\), "\n", \(EMat := {{Eii, Eij, Eik, Eil}, {Eij, Ejj, Ejk, Ejl}, {Eik, Ejk, Ekk, Ekl}, {Eil, Ejl, Ekl, Ell}}\), "\n", \(CMat := {{Cii, Cij, Cik, Cil}, {Cij, Cjj, Cjk, Cjl}, {Cik, Cjk, Ckk, Ckl}, {Cil, Cjl, Ckl, Cll}}\)}], "Input"], Cell[TextData[{ StyleBox["HOWTO calculate an overlap integral", "Section"], "\n", StyleBox["0.)", FontSize->14, FontWeight->"Bold"], " General information: The dummy variables ", StyleBox["U", FontSlant->"Italic"], " and ", StyleBox["T", FontSlant->"Italic"], " are strictly seen 3", StyleBox["N", FontSlant->"Italic"], "-6 dimensional vectors, where ", StyleBox["N", FontSlant->"Italic"], " is the number of nuclei that the molecule possesses. The ", StyleBox["i", FontSlant->"Italic"], "-th element of the vector ", StyleBox["T", FontSlant->"Italic"], " is the dummy variable corresponding to the ", StyleBox["i", FontSlant->"Italic"], "-th normalmode of the initial state. Likewise, the ", StyleBox["j", FontSlant->"Italic"], "-th element of the vector ", StyleBox["U", FontSlant->"Italic"], " ist the dummy variable corresponding to the ", StyleBox["j", FontSlant->"Italic"], "-th normalmode of the final state. The overlap integral ", StyleBox["Imn", FontFamily->"Courier"], "(=", StyleBox["I(m,n)", FontSlant->"Italic"], ") - or more precisely the relative overlap integral ", StyleBox["Imn/Io", FontFamily->"Courier"], " - results from a comparison of the coefficients occurring before \ corresponding polynomes of dummy variables in the expressions \"", StyleBox["links", FontFamily->"Courier"], "\" und \"", StyleBox["rechts", FontFamily->"Courier"], "\", which belong to the proper order. \"Proper order\" is existent, if the \ powers of ", Cell[BoxData[ \(TraditionalForm\`U\_i\)]], " and ", Cell[BoxData[ \(TraditionalForm\`T\_j\)]], " coincide with the quantum numbers of the normalmodes ", StyleBox["i", FontSlant->"Italic"], " and ", StyleBox["j", FontSlant->"Italic"], " of the initial- and the final state.\n", StyleBox["1.)", FontSize->14, FontWeight->"Bold"], " The expression \"", StyleBox["links", FontFamily->"Courier"], "\" is the power series on the left side of Eq.(15) in [1], whose \ coefficients are the Imn. The sums run from 0 ... +\[Infinity], in \ principle, but as one is interested only in transitions between states with \ finite quantum number and as higher summation terms do not give contributions \ to lower orders, the summation over ", StyleBox["j", FontSlant->"Italic"], " can be terminated after the maximum value of the quantum number ", StyleBox["j", FontSlant->"Italic"], ". As an example we look at a simple excitation from the vibrational ground \ state to the 5-th overtone of mode ", StyleBox["j", FontSlant->"Italic"], " in the final state. Here the summation runs from ", StyleBox["j", FontSlant->"Italic"], "=0,.. 5 . All other summations run from 0 to 0, i.e. they drop out. This \ is the reason, why the problem has not to be regarded in the full \ dimensionality of the normalmodes, but only in that dimensions where the \ quantum numbers differ from 0. Hence, the upper setup for ", StyleBox["T", FontFamily->"Courier"], ",", StyleBox["U", FontFamily->"Courier"], ", ", StyleBox["DVec", FontFamily->"Courier"], ", ... is suitable for multidimensional overlap integrals, where at maximum \ 4 modes with quantum numbers unequal 0 are present the initial and the final \ state. For cases with more than 4 different quantum numbers, the vectors and \ matrices must be extended accordingly. From the resulting expression the \ \"proper\" terms (see 0.)) are cut out.\n", StyleBox["2.)", FontSize->14, FontWeight->"Bold"], " The expression \"", StyleBox["rechts", FontFamily->"Courier"], "\" represents the power series expansion of the Exponential function \ occurring on the rhs of Eq.(15) in [1]. Also in this case, we have an \ infinite sum in principle, whose higher terms can be neglected in analogy to \ the considerations under 1.). The sum index ", StyleBox["s", FontSlant->"Italic"], " runs only to a value which is the sum of the maximum quantum numbers, \ which are achieved in every mode of the initial and the final state. As an \ example we look at an excitation from the 2nd overtone of a hotband to the \ third overtone of a mode of the excited state. Then, ", StyleBox["s", FontSlant->"Italic"], " must run up to 3+4=7. Second example: Excitation from the vibrational \ ground state in a combination band ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"\[Chi]", "(", RowBox[{ RowBox[{ RowBox[{"n", FormBox[\(\_j\), "TraditionalForm"]}], "=", "2"}], ",", \(n\_k\)}]}]}], TraditionalForm]]], "=3)). Here, ", StyleBox["s", FontSlant->"Italic"], " runs from 0 ..5. It should be noted that in general terms are created by \ this, which are of higher order in ", Cell[BoxData[ \(TraditionalForm\`U\_i\)]], " or ", Cell[BoxData[ \(TraditionalForm\`T\_j\)]], " as the maximum quantum mumber in mode ", StyleBox["i", FontSlant->"Italic"], " or ", StyleBox["j", FontSlant->"Italic"], ", respectively. This comes from the quadratic expressions ", StyleBox["Ut.CMat.U", FontFamily->"Courier"], " or ", StyleBox["Tt.AMat.T", FontFamily->"Courier"], ", respectively. From the resulting expression for \"", StyleBox["rechts", FontFamily->"Courier"], "\" again only the \"proper\" terms are cut out. \n", StyleBox["3.)", FontSize->14, FontWeight->"Bold"], " When the \"proper\" terms of \"", StyleBox["rechts", FontFamily->"Courier"], "\" and \"", StyleBox["links", FontFamily->"Courier"], "\" are equated, the dummy variables vanish at both sides. Now, the \ equation can be solved for ", StyleBox["Imn", FontFamily->"Courier"], " or ", StyleBox["Imn/Io", FontFamily->"Courier"], ".\n " }], "Text", CellDingbat->"\[FilledSquare]"], Cell["\<\ Here, the search string of the dummy variable has to be specified, \ for which a comparison of coefficients has to be performed. Note, that by \ factorization the dummy variables appear in lexical ordering, i.e. Ti before \ Tj and Tj before Uj. For linear variables use only their Name (e.g. \"Ui\"), \ for higher powers use the \"^\"-symbol (e.g. \"Ui^2\" ). The search string \ must be terminated by a space and a down bar.\ \>", "Text", CellDingbat->"\[FilledSquare]", FontSize->16], Cell[BoxData[ \(text := \(\(Ui^2\ Uj\ _\)\(\n\) \)\)], "Input"], Cell[BoxData[ \(Now\ the\ left\ side\ of\ equation\ \((15)\)\ must\ be\ expanded . \ The\ sums\ have\ to\ run\ only\ over\ those\ quantum\ numbers\ which\ \ are\ different\ from\ zero\ \((e . g . \ n\_i = \(n\_j = \(n\_k = 1\)\), \ n\_l = 2, m\_i = \(m\_j = \(m\_k = \(m\_l = 0\ \[DoubleRightArrow] \[Sum]\+\(\(n\_i\) \(n\_j\) n\_k = \ 0\)\%1\(\[Sum]\+\(n\_l = 0\)\%2\( U\_i\%\(n\_i\)\) \(U\_j\%\(n\_j\)\) \ \(U\_k\%\(n\_k\)\) U\_l\%\(n\_l\)*\((\(\(2\^n\_i\)\_ ... \ \))\)/\((\(n\_i!\) ... )\)^\((1/2)\) Imn\)\)\)\))\)\)], "Text", CellDingbat->"\[FilledSquare]", FontFamily->"Times", FontSize->14], Cell[BoxData[ \(links\ := Sum[Sum[Sum[ Sum[Sum[\ Sum[\n\t\t\t\t\tTm^m\ Tn^n\ Ui^i\ Uj^j\ \ Uk^k\ Ul^ l\ \ \((2^m\ \ 2^n\ 2^i\ 2^j\ 2^ k\ 2^l/\((\(m!\)\ \(n!\)\ \ \(i!\)\ \(j!\)\ \ \(k!\)\ \(l!\))\))\)^\((1/2)\)\ Imn, {i, 0, 2}], {j, 0, 1}], {k, 0, 0}], {l, 0, 0}], {m, 0, 0}], {n, 0, 0}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(Factor[links]\), "\n", \(lhs = \ \(Cases[Expand[links], \ text]\)[\([1]\)]\)}], "Input"], Cell[BoxData[ \(Imn\ \((1 + \@2\ Ui + \@2\ Ui\^2 + \@2\ Uj + 2\ Ui\ Uj + 2\ Ui\^2\ Uj)\)\)], "Output"], Cell[BoxData[ \(2\ Imn\ Ui\^2\ Uj\)], "Output"] }, Open ]], Cell[BoxData[{ \(Now\ the\ right\ side\ of\ \((15)\)\ is\ expanded . \ The\ expansion\ goes\ up\ to\ \ \ s\ = \ \(\(n\_1 + n\_2 + ... \) + n\_N + m\_1 + m\_2 + ... \) + \(\(m\_N . \ By\)\(\ \)\(the\)\(\ \)\(substitution\)\(\ \)\(operator\)\(\ \ \)\ \)\), "\[IndentingNewLine]", \(\(\(''\)\(/.\)\({\ }''\ all\ components\ of\ \ U\ and\ T\ are\ set\ to\ \ zero\ which\ belong\ to\ qunatum\ numbers\ that\ are\ zero . \ As\ example\ look\ at\ fourdimensional\ dummy\ vectors\)\)\ \), "\ \[IndentingNewLine]", \(U = \(\((U\_i, U\_j, U\_k, U\_l)\)\ and\ T = \((T\_i, T\_j, T\_k, T\_l)\)\), \ with\ corresponding\ qunatum\ numbers\ n\_i = \(n\_j = \(n\_k = 1\)\), \ n\_l = 2, m\_i = \(m\_j = \(m\_k = \(m\_l = 0\ \[DoubleRightArrow] \ \[IndentingNewLine]'' /. {Ti \[Rule] 0, Tj -> 0, Tk -> 0, Tl -> 0}''\)\)\)\)}], "Text", CellDingbat->"\[FilledSquare]", FontFamily->"Times", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(rechts = Io\ Sum[1/\(s!\)\ \((\((Tt . AMat . T\ + \ Tt . BVec)\) + \((Ut . CMat . U\ + \ Ut . DVec)\) + \((Ut . EMat . T)\))\)^s, {s, 0, 3}]\ \ /. {Ti \[Rule] 0, Tj -> 0, Tk -> 0, Tl -> 0, Uk \[Rule] 0, \ Ul \[Rule] 0}\)], "Input"], Cell[BoxData[ \(Io\ \((1 + Di\ Ui + Dj\ Uj + Ui\ \((Cii\ Ui + Cij\ Uj)\) + Uj\ \((Cij\ Ui + Cjj\ Uj)\) + 1\/2\ \((Di\ Ui + Dj\ Uj + Ui\ \((Cii\ Ui + Cij\ Uj)\) + Uj\ \((Cij\ \ Ui + Cjj\ Uj)\))\)\^2 + 1\/6\ \((Di\ Ui + Dj\ Uj + Ui\ \((Cii\ Ui + Cij\ Uj)\) + Uj\ \((Cij\ \ Ui + Cjj\ Uj)\))\)\^3)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(Length[Expand[rechts]]\), "\n", \(recht = Cases[Expand[rechts], text]\), "\n", \(Length[Expand[recht]]\), "\n", \(biggest = \(Dimensions[recht]\)[\([1]\)]\), "\n", \(\(\(rhs = Factor[Sum[recht[\([i]\)], {i, 1, \ biggest}]]\)\(\n\) \)\)}], "Input"], Cell[BoxData[ \(56\)], "Output"], Cell[BoxData[ \({2\ Cij\ Di\ Io\ Ui\^2\ Uj, Cii\ Dj\ Io\ Ui\^2\ Uj, 1\/2\ Di\^2\ Dj\ Io\ Ui\^2\ Uj}\)], "Output"], Cell[BoxData[ \(3\)], "Output"], Cell[BoxData[ \(3\)], "Output"], Cell[BoxData[ \(1\/2\ \((4\ Cij\ Di + 2\ Cii\ Dj + Di\^2\ Dj)\)\ Io\ Ui\^2\ Uj\)], "Output"] }, Open ]], Cell["\<\ Solve for the relative overlap integral Imn/Io, print the number of \ terms and simplify the expression:\ \>", "Text", CellDingbat->"\[FilledSquare]", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[{ \(Result = Factor[Imn/Io\ \ *\ rhs/lhs]\), "\n", \(Length[Expand[Result]]\), "\n", \(Final = Simplify[Result]\)}], "Input"], Cell[BoxData[ \(1\/4\ \((4\ Cij\ Di + 2\ Cii\ Dj + Di\^2\ Dj)\)\)], "Output"], Cell[BoxData[ \(3\)], "Output"], Cell[BoxData[ \(1\/4\ \((4\ Cij\ Di + \((2\ Cii + Di\^2)\)\ Dj)\)\)], "Output"] }, Open ]], Cell["\<\ The final expression above holds the value for the relative overlap \ integral Imn/Io. For the calculation of the probability (relative FC factor) \ the squares of the overlap integrals have to be used, i.e. Imn^2/Io^2. In the \ following section the result is printed in Fortran notation. Also a more \ agressive simplification of the routines is performed. \ \>", "Text", CellDingbat->"\[FilledSquare]", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(FortranForm[Final]\)\(\n\) \)\)], "Input"], Cell["(4*Cij*Di + (2*Cii + Di**2)*Dj)/4.", "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ AllerFinalst=FullSimplify[Result]\ \>", "Input"], Cell[BoxData[ \(Cij\ Di + 1\/4\ \((2\ Cii + Di\^2)\)\ Dj\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Literature", "Section"], "\n[1] J.Weber and G.Hohlneicher,\"", StyleBox["Franck-Condon factors for polyatomic molecules\"", FontSlant->"Italic"], ", ", StyleBox["Mol. Phys.", FontSlant->"Italic"], " ", StyleBox["101", FontWeight->"Bold"], ",2125 (2003).\n[2] D. E. Knuth, ", StyleBox["\"", FontSlant->"Italic"], StyleBox["The Art of Computer Programming\"", FontSlant->"Italic"], ", Vol.1, Addison-Wesley (1969).\n[3] R. Courant and D. Hilbert, ", StyleBox["\"Methoden der Mathematischen Physik I\"", FontSlant->"Italic"], ",3rd ed. Vol.1, Springer (1968)." }], "Text", CellDingbat->"\[FilledSquare]"] }, Open ]] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowToolbars->{}, WindowSize->{1019, 636}, WindowMargins->{{75, Automatic}, {Automatic, 168}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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