VibrationalModes computes the vibrational modes within the normal-mode approximation from the mass-weighted Hessian.

The optional numericalgradient keyword is true or false (default).  If false, analytical gradients are employed if available.

If numericalgradient is true, the optional differences keyword selects the order of the finite-difference approximation of the Hessian.  If the keyword is set to "second" (default), then the Hessian is computed by second-order finite differences, and if the keyword is set to "first", then the Hessian is computed by first-order finite differences.  

The optional hessianreturn keyword can be set to true to return the Hessian.  By default it is set to false.

The optional isotopes keyword can be assigned to a set or list of atomic isotopes to specify the isotopes to be used for the atomic masses.  If an isotope of an element is not set, then the average atomic mass is employed.  

The output of the procedure is a Maple sequence.  The first element is a Vector of the vibrational wavenumbers in reciprocal centimeters.  The second element is a square Matrix of the normal modes.  The modes are returned as the columns of the Matrix.  If the hessianreturn keyword is set to true, the Hessian matrix is returned as the third element in the sequence.

$\mathrm{with}\left(\mathrm{QuantumChemistry}\right)\:$

$\mathrm{molecule} ≔ \left[ \left[''H''\, 0\, 0\, -0.55\right]\, \left[''F''\, 0\, 0\, \+0.55\right] \right]\;$

$\mathrm{molecule}≔\left[\left[''H''\,0\,0\,\mathrm{-0.55000000}\right]\,\left[''F''\,0\,0\,0.55000000\right]\right]$

$\mathrm{molecule2}\,\mathrm{output} ≔ \mathrm{GeometryOptimization}\left(\mathrm{molecule}\,\'\mathrm{HartreeFock}\'\right)\:$

$\mathrm{molecule2}\;$

$\left[\left[''H''\,0\,0\,\mathrm{-0.55000000}\right]\,\left[''F''\,\mathrm{-6.19215306}{10}^{\mathrm{-12}}\,\mathrm{-5.84276514}{10}^{\mathrm{-12}}\,0.40546311\right]\right]$

$\mathrm{emodes}\,\mathrm{vmodes} ≔ \mathrm{VibrationalModes}\left(\mathrm{molecule2}\,\'\mathrm{HartreeFock}\'\right)\;$

$\mathrm{emodes},\mathrm{vmodes}≔\left[\begin{array}{c}4458.49660987\end{array}\right],\left[\begin{array}{c}3.98478924{10}^{\mathrm{-8}}\\ \mathrm{-2.55052735}{10}^{\mathrm{-8}}\\ \mathrm{-0.97448389}\\ 2.29458291{10}^{\mathrm{-8}}\\ 1.53101910{10}^{\mathrm{-8}}\\ 0.22445747\end{array}\right]$

$\mathrm{Isotopes}\left(F\right)\;$

$\left[F_{14}\,F_{15}\,F_{16}\,F_{17}\,F_{18}\,F_{19}\,F_{20}\,F_{21}\,F_{22}\,F_{23}\,F_{24}\,F_{25}\,F_{26}\,F_{27}\,F_{28}\,F_{29}\right]$

$\mathrm{Isotopes}\left(H\right)\;$

$\left[H_1\,H_2\,H_3\,H_4\,H_5\,H_6\right]$

$\mathrm{emodes}\,\mathrm{vmodes} ≔ \mathrm{VibrationalModes}\left(\mathrm{molecule2}\,\'\mathrm{HartreeFock}\'\, \mathrm{isotopes}\=\left\{F\left[14\right]\,H\left[1\right]\right\}\right)\;$

$\mathrm{emodes},\mathrm{vmodes}≔\left[\begin{array}{c}4498.26759674\end{array}\right],\left[\begin{array}{c}3.80385774{10}^{\mathrm{-8}}\\ \mathrm{-2.63756531}{10}^{\mathrm{-8}}\\ \mathrm{-0.96592317}\\ 2.54820018{10}^{\mathrm{-8}}\\ 1.76546610{10}^{\mathrm{-8}}\\ 0.25882894\end{array}\right]$