![]() If you have optimized to a transition state, or to a higher order saddle point, then there will be some negative frequencies which may be listed before the “zero frequency” modes. In general, the frequencies for for rotation and translation modes should be close to zero. The output for water HF/3-21G* looks like this: Full mass-weighted force constant matrix: Gaussian converts them to cm, then prints out the 3 N (up to 9) lowest. The roots of the eigenvalues are the fundamental frequencies of the molecule. The eigenvectors, which are the normal modes, are discarded they will be calculated again after the rotation and translation modes are separated out. Where, and so on, are the mass weighted cartesian coordinates.Ī copy of is diagonalized, yielding a set of 3 N eigenvectors and 3 Neigenvalues. The first thing that Gaussian does with these force constants is to convert them to mass weighted cartesian coordinates (MWC). The refers to the fact that the derivatives are taken at the equilibrium positions of the atoms, and that the first derivatives are zero. This is a matrix ( N is the number of atoms), where are used for the displacements in cartesian coordinates. ![]() We start with the Hessian matrix, which holds the second partial derivatives of the potential V with respect to displacement of the atoms in cartesian coordinates (CART): This is seldom a large problem, since the frequencies of the other modes, like the stretching mode, are are still useful. Using a single reference method will yield different frequencies for the and vibrations, while a multireference method shows the cylindrical symmetry you might expect. One case that comes to mind is molecules which are in a ground state. For example, a single reference method, such as Hartree-Fock (HF) theory is not capable of describing a molecule that needs a multireference method. (There are certain exceptions, such as analysis along an IRC, where the non-zero derivative can be projected out.) For example, calculating frequencies using HF/6-31g* on MP2/6-31G* geometries is not well defined.Īnother point that is sometimes overlooked is that frequency calculations need to be performed with a method suitable for describing the particular molecule being studied. Analysis at transition states and higher order saddle points is also valid. 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. Calculate reduced mass, force constants and cartesian displacements.Transform the Hessian to internal coordinates and diagonalize.Generate coordinates in the rotating and translating frame.Determine the principal axes of inertia.Mass weight the Hessian and diagonalize.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. Get PDF file of this paper (you may need to Right-Click this link to download it).
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