\[E^{\varepsilon}v=G+\sum_{i}\;\omega_{i}\;(v_{i}+\frac{1}{2})+\sum_{j}\;i\; \leq\;j\;x_{ij}\;(v_{i}+\frac{1}{2})\;(v_{j}+\frac{1}{2}) \tag{7}\]Equation (7) defines the vibrational energy for any asymmetric top (_a_) molecule (moments of inertia \(A_{e}\
eq B_{e}\
eq C_{e}\)) expressed in the terms of harmonic (\(\omega_{i}\)) and anharmonic (\(\omega_{ij}\)) spectroscopic constants. An analogous derivation, using RSPT the energy levels of a symmetric top (_s_) molecule (\(A_{e}=B_{e}\
eq C_{e}\) or \(A_{e}\
eq B_{e}=C_{e}\)), results in the term energy formula [2]. \[ \begin{split} E^{s}v=G+\sum\omega_{r}\ (v_{r}+\frac{1}{2})+\sum \omega_{t}\ (v_{t}+1)+\sum x_{r}\ (v_{r}+\frac{1}{2})\ (v_{r}+\frac{1}{2})+\sum x_{r}\ (v_{r}+\frac{1}{2})\\ (v_{r}+1)+\sum x_{tt^{\prime}}\ (v_{t}+1)\ (v_{t}+1)\ \sum g_{tt^{ \prime}2}+\sum g_{tt^{\prime}l_{e}^{\prime}}\end{split}  \tag{8}\]  
In Equation (8) the subscripts \(r\) and \(s\) refer to nondegenerate modes of vibration and \(t\) and \(t^{\prime}\) refer to degenerate ones. The so-called Dunham constant \(G\) in Equations (7) and (8) is an additive constant, also defined algebraically in terms of the \(\omega_{i}\). \(k_{ijk}\), and \(k_{ijkl}\) constants, that does not depend on vibrational or rotational quantum numbers [2]. Energies \(E_{vl}\) for a rotational level \(J_{v}\)B within vibrational level \(v\) are given by  
\[E_{vl}=B_{v}\ J_{v}\ B\ (J_{v}\ B+1) \tag{9}\]  
For an asymmetric top molecule, the rotational constant along the principal axis \(B\) is \(B^{a}_{v}=B_{eq}-\sum x_{r}^{B}\ (v_{r}+\frac{1}{2})\). For a symmetric top molecule, the rotational constant along the principal axis \(B\) is  
\[B^{s}_{v}=B_{eq}-\sum_{r}\ \alpha_{r}^{B}\ (v_{r}+\frac{1}{2})-\sum_{t}\ \alpha_{t}^{B}\ (v_{t}+1)\pm\sum_{t}\ q_{t}\ l_{t} \tag{10}\]  
where the rotational constants \(B_{v}\) are defined in terms of vibration-rotation coupling constants \(\alpha^{B}\), the doubling constant \(q_{t}\) and the vibrational angular momentum quantum number \(l_{t}\). [2] Equations defining rotational levels \(J_{v}\ A\) and \(J_{v}\ C\) with respect to the \(A\) and \(C\) principal moment of inertia axes are parallel to Equations (9) and (10). Finally, the term energy \(T\ (v_{J})\) for a state defined by vibrational quantum numbers \(vi\), \(i=1\), (3N-6) and corresponding rotational quantum numbers \(J_{v}\) is given by  
\[T\ (v_{J})=E_{v}+E_{vl} \tag{11}\]  
Thus, the frequency \(v\) of any spectral line corresponding to a change \(v^{\prime}\) to \(v\) in vibrational and \(J^{\prime}\) to \(J\) rotational sets of quantum numbers can be calculated as a term energy difference by using Equation (11), that is:  
\[v=T\ (v^{\prime}J^{\prime})\ -T\ (v_{J}) \tag{12}\]  
It is noted that the constant G, also known the lead Dunham constant Y\({}_{00}\), in Equations (7) and (8) cancels when Equation (12) is used to calculate a line frequency, since it does not depend on vibrational or rotational quantum numbers. Based on the RSPT formalism outlined above, a general-purpose code _Survibtm_ has been published for the vibrational-rotational analysis of polyatomic molecules [6]. _Survibtm_ was developed for fitting multi-dimensional potential energy and property surfaces and calculating vibrational-rotational spectra, line intensities and property expectation values of symmetric and asymmetric top polyatomic molecules. The code performs seven main types of calculations, all based on providing PESs and corresponding surfaces for molecular properties. These include (1) determining the equilibrium (stationary) point and saddle point geometries; (2) calculating internal coordinate force fields based on power series expansions in internal coordinates; (3) performing normal mode analyses; (4) calculating spectroscopic constants based on RSPT to second order; (5) calculating expectation values of properties; and (6) calculating transition dipole moments and line intensities [3]. _Survibtm_ is interfaced with _MolecGeom_ and employed in this study to analyze a given PES.