Rajini said:
Dear all,
Any quantum mechanical software (e.g., Gaussian03) can compute the fundamental frequencies (3N-6) and force constant for each frequency..
I don't understand exactly about force constant..??
For a particular mode of vibration what does this force constant mean?
For a molecular with 2 atoms we can tell force constant is between two atoms..but for polyatomic molecules is there any good way to understand??
thanks
The vibrational motions of polyatomic molecules are more complicated than those of diatomic molecules, but the idea of the force constant is the same. I have written up a description below, which uses the concept of a force constant in two distinct ways: to describe a single chemical bond connecting two atoms, and to describe a harmonic normal mode of the entire molecule. Both definitions are consistent, and are correlated to the associated harmonic motion through the angular frequency, which is defined as:
\omega=\sqrt{\frac{k}{\mu}}
where k is the force constant and mu is the reduced mass.
For any system of N atoms, you can keep track of the motion of the molecule by plotting the displacements of each of the atoms along the x,y and z Cartesian axes, describing a system with 3N overall degrees of freedom (DoF's). Now, if there are chemical bonds connecting those atoms in some pattern to form a molecule, there will be correlations between the motions of the atoms. You can think of each bond as a localized spring connecting two atoms, with an associated force constant that is proportional to the strength of the bond. If you were to average all the motions of the molecule for a long time, you would find the there is some "average" configuration, for which you can define a center of mass (CoM) in the usual way.
You can then describe the translation of the system through space in terms of the translation of its center of mass, using up 3 DoF's. You could also describe the overall rotation of the system around its COM, using up 3 DoF's (or just 2 if the molecule has a linear structure). The remaining 3N-6 DoF's (or 3N-5 for a linear molecule) then describe the vibrational motions of the molecule around it's averge configuration. The key point here is that this set of 3N-6 DoF's is *complete*, that it, you can use it to represent *any* motion of the molecule, therefore it represents a complete basis for the motions of the molecule. This means that you can set up a system of 3N-6 linear equations in terms of the masses of the atoms and force constants of the individual bonds, and solve it to obtain the orthogonal basis of harmonic normal modes for the molecule, which also form a complete set. These normal modes are more "natural" for describing the molecular motions, because they are orthogonal ... if you excite a vibration of one of them (say by absorption of an infrared photon of the appropriate frequency), then to a good approximation, the vibrational energy stays in that mode, and does not get transferred to the other modes. The motions described by local vibrational of chemical bonds are *not* orthogonal in general .. ff you were to excite the motion of a single bond between two atoms, the energy would rapidly (over a few vibrational periods) start to leak out into the other bonds in the molecule. This is why vibrational spectroscopy (infrared and Raman) measures the fundamental frequencies of the normal modes of the molecules.
Ok ... that got a little long-winded .. hope it's helpful.
