Molecular Momenta of Inertia - A General Approach

If we imagine a molecule in motion to be fixed in one point, the total (classical) angular momentum J is described with

Here, r_i is a vector representing the radius and v_i the velocity of the i^th atom relative to that point. As r_i is constant for molecules we assume to be rigid, it is solely velocity v_i that rules the rotation of this object. Using v_i = ω ·r_i  we obtain

If we proceed our calculation aiming on the angular momentum respective the three directions of the Cartesian coordinate system, we obtain J_x as x componente of the angular momentum.

Similar equations provide the angular momenta respective the y and z component of J; (r_i² = x_i² + y_i² + z_i²). The following set of formulas represent a concise description of the angular momenta in each direction of the Cartesian coordinate system

Here, the folling expressions for the momenta of inertia have been used

To calculate the principle momenta of inertia, the so-called principle axes transformation is performed, i.e. we look for values for which the determinante becomes zero

The three solutions for I (emerging an third-order equation) represent the principle momenta of inertia I_A, I_B, I_C.

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