In other words, the geometry used for vibrational analysis must be optimized at the same level of theory and with the same basis set that the second derivatives were generated with. Vibrational analysis, as it’s descibed in most texts and implemented in Gaussian, is valid only when the first derivatives of the energy with respect to displacement of the atoms are zero. There is an important point worth mentioning before starting. I will add some subscripts to indicate which coordinate system the matrix is in. I will try to stick close to the notation used in “Molecular Vibrations” by Wilson, Decius and Cross. I’ll outline the general polyatomic case, leaving out details for dealing with frozen atoms, hindered rotors, and the like. In this section, I’ll describe exactly how frequencies, force constants, normal modes and reduced mass are calculated in Gaussian, starting with the Hessian, or second derivative matrix. Diatomics are simply treated the same way as polyatomics, rather than using a different coordinate system.
The vibrational analysis of polyatomics in Gaussian is not different from that described in “Molecular Vibrations” by Wilson, Decius and Cross. This differs from the coordinate system used in most texts, where a unit step of one is used for the change in interatomic distance (in a diatomic). So why is the reduced mass different in Gaussian? The short answer is that Gaussian uses a coordinate system where the normalized cartesian displacement is one unit.
One of the most commonly asked questions about Gaussian is “What is the definition of reduced mass that Gaussian uses, and why is is different than what I calculate for diatomics by hand?” The purpose of this document is to describe how Gaussian calculates the reduced mass, frequencies, force constants, and normal coordinates which are printed out at the end of a frequency calculation.
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