D4h Symmetry Group: 9 Normal Modes of Vibration

[Rajini]

Hello all,
In a D4h symmetry group we have 5(3)-6=9 normal mode of vibrations.
Normally in books they show only 7. Because 2 of that 7 doubly degenerate E_u modes. And i know the how it vibrates (picture shown in book). But does anyone know how their degenerate partners vibrate ? Is there some rare book, which show all these ?
Thanks

 

[SpectraCat]

Hello all,
In a D4h symmetry group we have 5(3)-6=9 normal mode of vibrations.
Normally in books they show only 7. Because 2 of that 7 doubly degenerate E_u modes. And i know the how it vibrates (picture shown in book). But does anyone know how their degenerate partners vibrate ? Is there some rare book, which show all these ?
Thanks

First of all, your count of the vibrational modes is only correct for a 5 atom molecule or complex ... there are plenty of molecules that can have D_4h symmetry with more than 5 atoms. For example, if an octahedral molecule undergoes a Jahn-Teller distortion, it will generally have D_4h symmetry.

Note that although the modes are degenerate, they are also orthogonal. So typically you will have two indistinguishable motions along two perpendicular axes, or within two perpendicular planes. The canonical example is for a linear molecule, where you have two degenerate bending modes. If you take the molecule to define the z-axis, then the degenerate bending motions are in the xz and yz planes.

Anyway, for your 5 atom case I guess we are talking about a square-planar configuration. In that case, at least one of the degenerate modes is going to be the in-plane degenerate bend. For that one, label the atoms A,B,C,D going around the central atom in a clockwise fashion. Now, one of the pair of modes will have A & B going "to the right" when C & D are moving "to the left", and vice versa. The other mode will have atoms B & C moving "up", while A & D are moving down. I have to say that I cannot think of the other degenerate mode for your case right now, but hopefully what I have written here will help you to figure it out.

 

[Rajini]

Hello Spectracat,

Yes as you said it is a planar molecule. One atom (Fe) in center and four atoms (O) makes a square around it. Obviously it is in D4h symmetry. For this type i found normal mode of vibration from Nakamoto book. In that book i can see all normal mode of vibration of the D4h symmetry. But the partners for the two doubly degenerate modes (Eu) is not given.
Anyway tomorrow i'll upload a hand-drawn picture tomorrow.
What you meant by orthogonal ? just give me some insight related to these Eu modes.
thanks

 

[Rajini]

Hello,
i am now clear. Main fact is that the degenerate modes can be decomposed into two.
Information with picture:
1. Chao-Yang Hsu and Milton Orchin, Journal of chemical Education, Vol. 51, pp. 725-729 (1974) and
2. Inorganic Chemistry, Vol. 3, 1368-1373 (1964).
I prefer 1.
In picture v8 and v9 are partners for v7 and v6, respectively

 

[pmacias]

for bringing up this topic about the D4h symmetry group and its normal modes of vibration. I can provide some insights into this topic.

Firstly, the D4h symmetry group has a total of 9 normal modes of vibration, as you correctly mentioned. These modes correspond to the different ways in which a molecule or a crystal can vibrate. These vibrations are important because they provide information about the structure and properties of the molecule or crystal.

In some cases, it is possible for two modes to have the same frequency and therefore be degenerate. This is what happens with the doubly degenerate Eu modes in the D4h symmetry group. These modes have the same frequency, but they vibrate in different directions. For example, one mode might vibrate in the x-direction while the other vibrates in the y-direction. This is why they are called degenerate partners.

As for your question about whether there is a book that shows all 9 normal modes of vibration, I cannot say for sure. However, there are certainly resources available that can provide this information. One such resource is a group theory table, which lists all the possible normal modes of vibration for different symmetry groups, including D4h. These tables can be found in textbooks or online.

In addition, there are computer programs and simulations that can show the vibrations of molecules or crystals in 3D, including their degenerate partners. These can be very helpful in visualizing and understanding the vibrations.

In conclusion, the D4h symmetry group has 9 normal modes of vibration, with 2 of them being doubly degenerate Eu modes. While there may not be a specific book that shows all 9 modes, there are resources available that can provide this information, such as group theory tables and computer simulations. I hope this helps clarify the topic for you.