## 3 Application  
Example calculations have been performed by generating multiple internal coordinate geometric data sets for four molecules, water (H\({}_{2}\)O) [12], formaldehyde (H\({}_{2}\)CO) [13; 14], vinyl alcohol (C\({}_{2}\)H\({}_{4}\)O)[15; 16] and vitamin C [ascorbic acid (C\({}_{6}\)H\({}_{8}\)O\({}_{6}\))] [17; 18]. By implementing _MolecGeom_ PES sizes ranging from as few as 200 to more than 20,000 geometries have been defined. In order to generate molecule-specific geometric output, it is imperative to note that reference bond lengths and bond angles must be specified. In the case of water, these correspond to the equilibrium bond lengths (OH\({}_{1}\) and OH\({}_{2}\)) of 1.81 atomic units (au; 1 au = 0.529177 A) and the equilibrium bond angle (HOH) of 104.5\({}^{\circ}\). These are used as a starting point to select a range of geometric distortions of 0.005 au, above and below the range start of 1.80 au and the range end of 1.82 au. The angle of the range start is 103.5\({}^{\circ}\) and the range end is at 105.5\({}^{\circ}\), and increments are calculated every 0.100\({}^{\circ}\) in the vicinity of equilibrium values. Applying these parameters for water, the _MolecGeom_ code generates a 525-geometry surface. In the case of formaldehyde, the equilibrium CO, CH\({}_{1}\), and CH\({}_{2}\) bond lengths are 2.30, 2.10, and 2.10 au, along with two HCO and the H\({}_{2}\)C dihedral wagging bond angles of 120.0\({}^{\circ}\), 120.0\({}^{\circ}\) and 180.0\({}^{\circ}\). These are then used as a starting point to select a range of numbers that lie around these equilibrium values. The range start for the CO calculation is 2.20 au and the range end is 2.40 au, with an incrementation of 0.09 au. The range start for the first and second CH stretch is 2.00 au and the range end is set at 2.20 au, with the same 0.09 au incrementation. Next, the distortions of the two HCO angles were calculated. Their range start begins at 115.0\({}^{\circ}\) and the range ends at 130.0\({}^{\circ}\), with 5.0\({}^{\circ}\) increments. The wagging angle range start begins at 170.0\({}^{\circ}\) and the range end is set to 190.0\({}^{\circ}\), with 5.0\({}^{\circ}\) incrementations. The _MolecGeom_ code calculated a new geometry with every incrementation within each range start and end, thus creating a comprehensive geometric list for formaldehyde containing 2160 geometries for follow-on implementation. After importing the respective data sets to a compute engine, calculations are carried out to generate the energies corresponding to these PESs of water and formaldehyde. That is, these calculations produce energies for all the _MolecGeom_-generated geometric points for each of these polyatomic molecules. These PESs are then ported into a modified version of _Survibtm_[6] to calculate the fundamental vibrational frequencies shown in Tables 1 and 2 below. For reference, the fundamental visual vibrations of water and formaldehyde are shown in Figures 1 and 2, respectively. In Table 1, the calculated value of 3654 cm\({}^{-1}\) denotes the symmetric stretch of the water molecule, meaning the O-H bonds are stretched equally. The following value of 1595 cm\({}^{-1}\) corresponds to the bending motion that occurs where the water molecule bond angle opens and closes. The last value of 3753 cm\({}^{-1}\) describes the asymmetric stretching of the molecule, where one O-H bond is contracted and the other O-H bond is stretched and vice versa. Furthermore, Table 2 contains the six fundamental transition frequencies corresponding to the motions depicted in Figure 2 for formaldehyde, as shown above. These are \begin{table}
\begin{tabular}{c c c c} \hline Calculated & **3654** & **1594** & **3753** \\ \hline Experimental & 3657 & 1595 & 3756 \\ \hline Error & 0.08\% & 0.00\% & 0.08\% \\ \hline \end{tabular}
\end{table} 
Table 1: Water fundamental vibrational frequencies (cm\({}^{-1}\)) from a 125-geometry surface with comparisons with observed values [19].