Say CO2. It has wavenumber value around 2300 cm-1 in ATR-FTIR spectra. What is the theoretical method to calculate wavenumber for a specific bond in a specific vibrational mode?
IR has a theoretical basis. You can predict the IR spectrum empoying theoretical methods (ab initio)
The best way to calculate a vibrational mode is to perform what we call the normal coordinate analysis, employing classic physics to solve the vibrational motin of the nuclei in a specific molecular system.
Starts with the point group of the chemical system, and them you have to apply the simmetry elements to obtain the the symmetry of the normal vibrations, and then to build the particular internal coordinate for each one of the normal vibrations. After thatm you can obtain in matrix notation the kinectic and the potential energies of the molecule, which is useful to finally calculate tyhe F and G matrix to solve the secular equation, where it can be obtained the force constants for each one of the bands you can see in the vibrational spectrum (or infrared or Raman) of the system.
Of course, this is a method which is based on the experimental data, and you need to have the cristallographic data as well to solve G and F matrix.
Have a look on F.A. Cotton, Chemical applications of Group Theory, second edition, 1971, J. Wiley, which is the version I have, but there is a modern one.
Of course, wavenumbers can be calculated. For example, you can use a simplified formula from classical mechanics: the vibration frequency f of an oscillator formed by 2 masses m and M linked with each other with a spring of rigidity k reads: f = 1/(2 pi) *(k/mu)^(1/2), where the reduced mass mu is mu = (mM)/(m+M). k is close to 500, 1000 and 1500 N/m for a simple, double and triple bond, respectively. In the case of a C-H bond, for example, considering the mass of C and H atoms (in kg), you'll find f = 9.09E13 Hz, i.e. 3030 cm-1, instead of 2950 cm-1 found experimentally for the CH2 group.
Alain
Thanks Dr. Celzard for your input.
It works for all kinds of bonds. But there are different vibrational modes of a same bond. Like CO2 has stretching (symmetric and antisymmetric) and bending (twisting, bending). And these different modes have different wave-numbers of the same molecule.
Does your said method give these different wave-numbers for different modes?
And also can we quote this method in a paper?
yes
ϖ= (1/2πC)√(K/μ) ϖ - wave number
C- speed of light
K- force constant
μ- reduced mass
you need to know force constant
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