# Can anyone help me with this question on vibrational modes of free base porphyrin (C20H14N4)?

4 a. The structure of free-base porphyrin $$(\ce{H2P}$$, chemical formula: $$\ce{C20H14N4})$$ is shown below. Given that this molecule belongs to the $$D_\mathrm{2h}$$ point group (character table given above) identify the symmetries (irreps) for all the vibrational modes of this planar polyatomic molecule;

(i) Identify the number of vibrational modes in $$\ce{H2P}$$. [3 marks]
(ii) Work out the reducible representation for the vibrational modes. [6 marks]
(iii) Use the reduction formula to identify the (irreps) for all the vibrational modes. [6 marks]
(iv) Identify which of the vibrational modes are Raman-active and which are IR-active. [4 marks]

In particular I'm getting stuck with figuring out the unmoved atoms when applying symmetry operations. I know i need to calculate gamma vib in order to use the reduction formula but I cannot picture how to apply the symmetry operations to calculate unmoved atoms for each symmetry class (aside from E) because it is a more complicated structure than I'm used to dealing with (h20, benzene etc)

• One thing to think about in the question: is the free base porphyrin really D2h symmetric? – Martin - マーチン May 13 at 22:48

If you rotate the molecule by $$45^\circ$$, the rotation axes will fall on the cartesian axes $$x, y, x$$, and the mirror planes will be at $$xy, yz, xz$$.
Now, you can see that the NH groups are unaffected by rotation about $$y$$ or reflection on $$yz$$, the deprotonated N groups by rotation about $$x$$ or reflection on $$xz$$, and all the atoms by reflection on $$xy$$. You can ignore the position of the double bonds because it is an aromatic system.