Theta Calculations

Abstract

The method for calculating these theta values is derived from Demaison, et al. J. Phys. Chem. A 2011, 115, 14078–14091.

For planar molecules, com-pac calculates three estimates of the angle (\(\theta\)) between the parent species' principal axes and an isotopologue's principal axes. These are labelled \(\theta_7\), \(\theta_8\), and \(\theta_9\), corresponding to the three equations described in the introduction of Demaison, et al.

Two coordinate frames

The inertia matrix of a molecule depends on the coordinate system in which the atomic positions are expressed. com-pac works with two frames:

Parent frame: the principal axes frame of the first (parent) isotopologue.
Isotopologue frame: the principal axes frame specific to each isotopologue.

In the parent frame, only the parent isotopologue is expected to have a fully diagonal interia matrix. The other isotopologues when represented in the parent frame will almost certainly have non-zero off-diagonal elements. When the inertia matrix for an isotopologue is calculated while in its respective frame, only then is the inertia matrix guaranteed to be fully diagonal.

In principle, the difference of an isotopologue's inertia matrix in the parent frame and in its own frame is correlated with degree of rotation between the two frames. The \(\theta\) values are supposed to quantify that rotation.

Note

This calculation is only performed for planar molecules, i.e. those for which the \(I^e_{ac}\) and \(I^e_{bc}\) elements of the parent inertia matrix are zero. It's not clear from the theory how a corresponding calculation would be done for non-planar molecules.

The three estimates

Warning

The following calculations are based on the developer's interpretation of the paper and may be incorrect!

For each isotopologue, the following quantities are used:

\(I^e_{aa}\), \(I^e_{bb}\), \(I^e_{ab}\) — elements of the isotopologue's inertia matrix evaluated in the parent frame
\(I^e_a\), \(I^e_b\) — principal moments of inertia from the isotopologue's own principal axes frame (i.e., the diagonal elements of its diagonalised inertia matrix)

All three \(\theta\) values are reported in degrees.

\(\theta_7\)

The following equation is derived from Equation 7 of Demaison et al.

\[ \theta_7 = \frac{1}{2} \arctan\!\left(\frac{2\, I^e_{ab}}{I^e_{aa} - I^e_{bb}}\right) \]

\(\theta_8\)

The following equation is derived from Equation 8 of Demaison et al.

\[ \theta_8 = \frac{1}{2} \arccos\!\left(\frac{I^e_{aa} - I^e_{bb}}{I^e_a - I^e_b}\right) \]

\(\theta_9\)

The following equation is derived from Equation 9 of Demaison et al.

\[ \theta_9 = \frac{1}{2} \arcsin\!\left(\frac{2\, I^e_{ab}}{I^e_a - I^e_b}\right) \]

where \(I^e_{ab}\) is the off-diagonal element of the inertia matrix evaluated in the parent frame.

Note

It's implied that \(I^e_{ab}\) is the value for the isotopologue's inertia when in the parent frame, since in the isotopologue frame the value is guaranteed to be zero.