vibration mode analysis: NH3

Feedback welcome: yeschowh@snu.ac.kr :)

Partially inspired by this page.

Vibrational Modes By Irreducible Representations: Case of the NH3 Molecule

If you are not familiar on this kind of topic, I recommend reading H2O case first. :)

Regular representation of C₃ᵥ Point Group

You can easily find the character table of C3v point group by searching online. 

While we could directly derive the character table from the multiplication table (or regular representation), let's focus on understanding the underlying concepts first.

Instead, let's find out what is multiplication table and regular representation and why it is sometimes useful.

Multiplication table:

(Group) multiplication table is defined as follows, I think instead of long explanation, direct demonstration for C3v point group would be much intuitive explanation.

Regular representation:

The multiplication table provides additional insight into the group structure. As shown in the figure, we can construct one matrix representation from the multiplication table. We call this as regular representation.

I won’t delve into a formal proof of why this matrix representation satisfies the group’s structure, (since I do not know).

However, such matrix representations are particularly useful for less common groups where standard tables are not available—though that’s not the case for C3v.

For the ones who will not believe my story, I prepared a code block (of julia), which can compare the multiplication result with above table. Regular representation case? I remained the most fun part for you!

using LinearAlgebra, Rotations

# Define the symmetry operations as 3x3 matrices

E = Matrix{Float64}(I, 3, 3)

  # E: Identity

θ1 = 2π / 3;  R1 = Matrix{Float64}(RotZ(θ1));

  # R1: 120 degrees rotation about z-axis

θ2 = 4π / 3;  R2 = Matrix{Float64}(RotZ(θ2));

  # R2: 240 degrees rotation about z-axis

n1 = [1, 0, 0];  r1 = E - 2 * (n1 * n1');

  # r1: Reflection across the yz-plane

n2 = RotZ(2π/3) * n1;  r2 = E - 2 * (n2 * n2')

  # r2: Reflection across a plane rotated 120 degrees from yz-plane

n3 = RotZ(4π/3) * n1;  r3 = E - 2 * (n3 * n3')

  # r3: Reflection across a plane rotated 240 degrees from yz-plane

# Example: Test some multiplications

res = r1 * r2;  res - R1

Long story, but anyway, useful point of regular representation is that this contains the whole irrep for the given group.

Regular representation contains the 1d irrep for 1 time, (if exists) 2d irrep for 2 times, (if exists) 3d irrep for 3 times, etc.

(Again, I don't know the detailed derivation, but) this provide one mysterious identity of character table.

• square of character table of column of E operation is the same with group order |G|.

Since regular representation (1) contains n-dim irrep for n times and (2) matrix dimension is |G|.

For C3v case, 1* 1 + 1 * 1 + 2 * 2 = 6.

So in principle, if you know multiplication table, you already have all irrep inside your regular representation.

Of course, reducing this reducible representation into block-diagonal irrep is another problem.

I heard that there is a general scheme to reduce large reducible representation into block diagonal irrep, this paper, but I haven't tried this yet, I will update it after trying it.

Vibrational Degree of Freedom from Displacement Representation

Actually, remaining part is fundamentally the same with previous H2O case, except for this case, not just assigning +1 / 0 / -1 for coordinate transformation for x, y and z.

Let's summarize the process, again without formal proof.

(1) Check the number of atoms position unchanged, N

(2) For coordinate axis, calculate trace of the operation, tr(R)

Character of displacement representation is given as N * tr(R).

For instance, character of C3 operation in displacement representation is calculated as follows:

Also, displacement representation contains 3 translational and 3 rotational motion in its representation.

We need to identify which representation corresponds to translation or rotation.

Actually, this part is already done for many cases by commonly available tables.

If you searched character table for C3v, you may have noticed that there are some letters given right to the table.

Identification of translation and rotation motion from character table

It is convenient to start with vector and pseudovector to deal with translation and rotation part from displacement representation.

Translation could be considered as vector, with respect to its response on translation, rotation, and reflection.

On contrary, rotation could be considered as pseudovector, with respect to its response on translation, rotation, and reflection.

Translation part for (x, y, z), vector, respect to symmetry operation of C3v leads to the following form.

Rotation part for (Rx, Ry, Rz), pseudovector, respect to symmetry operation of C3v leads to the following form.

We can arrange the whole character table for (x, y, z) and (Rx, Ry, Rz) as follows.

It seems like that combined result for (x, y) and (Rx, Ry) responds like irrep E.

Frankly speaking, I did not fully understand on this process.

I just understand this as:

(1) If character table obtained from this process contains non-integer value, such as +1/2 or -1/2, then it is mixed with another degree of freedom, which means just tracking x or y is not enough in this case.

(2) This could be interpreted as these two degree of freedom could not be decomposed into two separate 1D irrep, which means (x, y) or (Rx, Ry) combined term could be mapped as 2D irrep E, in this case.

This seems quite arbitrary, but still we can get a rough estimate on how those translation / rotation could be mapped to one irrep.

There would be much more smart way to convey this, but I am not sure right now. I just learned this empirical way..

If you do, please let me know! :)

I spent too much on regular representation blah blah too much. Let's not do how those eigen-mode are obtained.

I assume that would not be much different from H2O case.

Just for check, below 6 modes are vibration mode of NH3 molecule.