F.c. analysis

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

SpectraCat
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.

Hi,
So force constant calculated for a normal mode by such quantum calculations can be said as...force constant of that particular normal mode instead of just saying it as simply force constant between bonds??
or
average force constant?
Can Raman or IR measure overtones??
Also can you also elaborate reduced mass for polyatomic molecules??
thanks

SpectraCat
Hi,
So force constant calculated for a normal mode by such quantum calculations can be said as...force constant of that particular normal mode instead of just saying it as simply force constant between bonds??

Yes

Can Raman or IR measure overtones??

In principle yes, there is always some anharmonicity of the vibrational potential, which allows for non-zero intensity of overtones. However in practice, this intensity is often too small to be measured.

Also can you also elaborate reduced mass for polyatomic molecules??
thanks

This is very tricky to do ... the way it is handled in normal mode analysis is to define a system of mass-weighted coordinates ... these automatically include the contributions of the different masses to the system. The reduced masses can then be calculated once the normal modes are known, but there is no easy way to write them in a closed form. If you are really interested in the details of all of this, I suggest you have a look at http://www.gaussian.com/g_whitepap/vib.htm" [Broken] from the Gaussian website.

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Hi spectracat,
actually why i ask this question?
I have a phonon spectrum..which displays only the vibrations of iron nucleus..from that spectrum i can get the force constant of iron with neighboring atoms/ligands..
and
i have another spectrum of a molecular which is calculated using gaussian03..i just want to compare the force constants of experimentally obtained ones with calculated ones..
By using visualizing software i can roughly predict the pure iron vibrations..so i guess if everything is correct then the f.c should be same for that vibration..??
if not i can assume that the calculated ones may deviate from experimentally obtained f.c due to non pure iron vibrations..
so
is there any other suggestions??
another small question..normally overtones should appear roughly at two times the fundamental frequency?? is that correct
thanks

SpectraCat
Hi spectracat,
actually why i ask this question?
I have a phonon spectrum..which displays only the vibrations of iron nucleus..from that spectrum i can get the force constant of iron with neighboring atoms/ligands..

Ok, I am not really that familiar with phonon spectra ... I assume that you can extract force constants from them *if* you know the reduced mass of the phonon, which seems to be to be a highly non-trivial quantity to obtain.

i have another spectrum of a molecular which is calculated using gaussian03..i just want to compare the force constants of experimentally obtained ones with calculated ones..
By using visualizing software i can roughly predict the pure iron vibrations..so i guess if everything is correct then the f.c should be same for that vibration..??

This is a good question ... and I have to say that I don't really know. I would be very cautious drawing parallels between force constants calculated for molecules to those measured for solid-state phonons. The reason for this is that phonons are global normal mode excitations of a lattice, whereas molecular normal modes have much more localized character. So, the force constants for a vibration between a group of atoms in a molecular calculation may not be directly comparable to the force constants for phonons in a solid state lattice.

Having said that, I should think that the molecular force constants could be used as inputs into a calculation to derive the phonon frequency spectrum. This is not my field, so I can't really help you much ... perhaps ZapperZ or another condensed phase specialist could jump in here?

if not i can assume that the calculated ones may deviate from experimentally obtained f.c due to non pure iron vibrations..

Yes ... but not necessarily for that reason .. see above

another small question..normally overtones should appear roughly at two times the fundamental frequency?? is that correct
thanks

Yes, that is correct. Since overtones arise from anharmonic effects, there will be a small red shift of the overtone .. that is, it should appear at a frequency that is slightly less than twice the fundamental.