H2O vibration and symmetry

Although, in many cases in condensed matter system, lattice vibration is rather on focus, but let's start from molecular case first.

Inspired by this video.

Vibrational Modes By Irreducible Representations: Case of the H2O Molecule

When molecules vibrate, they do so in specific, quantized patterns called normal modes. These modes are not random but are tightly governed by the molecule’s symmetry — a concept beautifully described using group theory. Let’s explore how this works using one of the most familiar molecules: water (H₂O).

The Setup: Symmetry and Motion

Water is a bent molecule with an angle of approximately 104.5° between its hydrogen atoms. It belongs to the C2v point group, a symmetry group that includes:

The identity operation (E)
A 2-fold rotation around the principal axis (C₂)
Two vertical mirror planes which includes prinical axis (σᵥ and σ'ᵥ).

Water has 3 atoms, each with 3 degrees of freedom (x, y, z), totaling 3 x 3 = 9 degrees of freedom. From these:

3 modes correspond to translational motion,
3 modes correspond to rotational motion,
and the remaining 3 are vibrational modes.

Character table: In a nutshell

Before diving into the details of molecular vibrations, it's often a smart move to start with the character table.

Think of it as a cheat sheet — a compact summary that encodes how symmetry operations “behave” in the language of linear algebra. Technically, it's a table built from the traces (or “characters”) of matrices that represent each symmetry operation.

Now, we haven't formally defined what a representation is yet — but for now, imagine it's just a matrix that faithfully captures how each symmetry operation interacts with the others. It’s like assigning a specific dance move to each symmetry operator while keeping the rhythm of the whole choreography intact.

Our goal is to find all the irreducible representations (irreps) of the C2v point group — the fundamental building blocks of this symmetry group.

(Yes, I know — we just dropped another undefined term: irreducible. Don’t worry! For now, just think of it as a kind of matrix that can’t be “broken down” into smaller, simpler blocks. It’s the most atomic form of a representation — clean, indivisible, and powerful.)

Why the Character Table is Kind of a Big Deal (at Least for C2v)

So here it is — the character table for the C₂v point group. A neat little grid with a few +1 and –1.
But somehow… this tiny table tells us everything about the molecule’s symmetry behavior. How?!

Let’s take a look:

There are 4 symmetry operations → group order = 4.

Now, here’s something beautiful (and a little nerdy):

The sum of the squares of all irreducible representation dimensions equals the group order.

And here’s another trick up its sleeve:

Each row in this table is orthogonal to the others — in the dot product sense.

So we already know all irreps are one-dimensional and fully account for the group. No room for surprises — this is the complete picture.

But you might wonder — why is this one little table so powerful? Isn’t it just a list of numbers?

Actually, this table encodes everything we need to understand how vibrations, orbitals, or even electronic transitions behave under symmetry.

For point group C2v, the character table looks like this: (You can find it on google, keyword as "C2v character table" etc.)

(You may obtain this table from multiplication table, but let's forget about how you can derive this now.)

Long introduction to character table, let's dive into how character table could be used to identify to find out the symmetry of vibration mode.

How to decompose displacement representation into (1) translation, (2) rotation, and (3) vibration.

Before we jump into the math, let’s step back for a moment —
How does a molecule actually move in 3D space?

It can translate along x, y, and z.
It can also rotate about those same axes (or equivalently, in the xy, yz, and zx planes).
And finally, it can vibrate — meaning, atoms can jiggle relative to one another without moving the entire molecule.

So, for a molecule with N atoms, the total number of degrees of freedom is 3N.
Out of these:

3 are taken by pure translation (motion of the whole molecule),
3 are for rotation (unless it’s a linear molecule — then only 2),
and the rest belong to vibration — the real stars of our analysis.

🔹 Vibrational modes for a non-linear molecule: 3N−6

🔹 Vibrational modes for a linear molecule: 3N−5

So for a non-linear molecule like H₂O, which has 3 atoms, we expect: 3N−6=3 vibrational modes

The good news? We don’t need to calculate 9 degrees of freedom from scratch — just figure out which 3 modes belong to vibration.

Here's how it goes:

In a Nutshell: The Game Plan

Start by building a representation that captures all the ways the atoms can move (including translation and rotation).

This will be a 3N×3N matrix-based representation.

Decompose this representation into a sum of irreducible representations
Identify which parts of this decomposition correspond to translation and rotation
Whatever remains is what we’re really after: The irreps that describe vibrational modes.

Let's Build a Displacement Representation (Example: H₂O in C2v)

Now that we know what we’re looking for —
Let’s actually build the displacement representation for a real molecule.

We’ll take our favorite: H₂O — the bent little molecule with a big role in everything.

Step 1: Know Your Ingredients

H₂O has 3 atoms → 3N=9 total degrees of freedom.
Each atom can move in x, y, z directions.
So we’re building a 9-dimensional reducible representation.

But we don’t need to write full 9×9 matrices. Instead, we just care about their traces — the characters.

For each symmetry operation in C2v (E, C₂, σᵥ(xz), σᵥ'(yz)), we just need to calculate:

How many of the 9 displacement directions (1) stay the same, (2) get flipped, or moved elsewhere.

Let’s go one by one.

✅ E (Identity):

Nothing moves, nothing flips.

All 9 directions (3 per atom) are invariant

Total trace = 9

✅ C₂ (180° rotation about z-axis):

O stays at the center → its x, y, z all flip according to C₂(z):
- x → –x → contributes –1
- y → –y → –1
- z → z → +1,

Total from O: –1 –1 +1 = –1

The two H atoms switch positions

→ their displacements don't map back onto themselves
→ Contribute 0 for all 6 directions.

Total trace = –1

✅ σᵥ(xz): Mirror in xz-plane

O stays put:
- x → x, z → z → +1 each
- y → –y → –1
  → Total from O: +1 –1 +1 = +1
Two H atoms flip across the mirror → switch places
→ Their displacements do not map back → contribute 0

Total trace = +1

✅ σᵥ′(yz): Mirror in yz-plane

O stays put:
- x → –x → –1
- y → y → +1, z → z → +1
  → O contributes: +1 + 1 -1 = +1
Both H atoms are in the mirror plane, so they don’t move:
- Their y and z stay the same → each contributes +1
- x flips → each contributes –1

→ (each) H contributes: +1 + 1 -1 = +1

So from 2 H atoms:

→ (+1–1+1)×2 = 2( +1 –1 +1 ) × 2 = 2(+1–1+1)×2 = 2

Total trace = 1 (from O) + 2 = 3

Time to Decompose: What Symmetry Do the Vibrations Have?

Alright, we’ve built our character table for displacement representation of H2O:

but that’s just a big, total sum of all possible motions. (including translation / rotation)

We need to break it down. This is where group theory magic happens:

We’ll decompose this reducible representation into irreducible pieces.

Step 1: Use the Projection Formula (Simpler Than It Sounds)

For each irrep Γi, the coefficient ai (how many times it appears) is given by:

Although the formula looks relatively complicated, it's nothing but a vector projection to some orthogonal basis sets. (I mean this works like such way)

So explicit calculation of this formula leads to...

A₁ :A₁ appears 3 times.

🔹 A₂: A₂ appears 1 time.

🔹 B₁: B₁ appears 3 times.

🔹 B₂: B₂ appears 2 times.

Step 2: Subtract Translations and Rotations

Remember, 3 of these modes are translations, and 3 are rotations. For H2O (C2v), we know:

Translations: x (B₂), y (B₁), z (A₁)
Rotations: Rx (B₁), Ry (B₂), Rz (A₂)

(If you may searched character table from website, you may notice that x / y translation & rotation maps to different irrep. This is because I used different coordinate system from their. I used coordinate system such that yz-plane includes the H2O plane.)

You may wonder, how do we know each translation (rotation) maps to this symmetry.

This may be non-trivial question for general systems, however "in this system" it could be mapped w/o paper & pencil.

Consider a displacement along x, then consider how this displacement changes respect to each operation.

E(x) => +x / C2z(x) => -x / Mxz(x) => +x / Myz(x) => -x.

This tells us that translation mode along x maps to B2.

So, our displacement representation is composed of:

And among 9 modes, we can nest out translation / rotation.

🎶 And There You Have 3 Vibration Mode!

For H₂O, the vibrational modes are:

2 A₁:
- One is the symmetric stretch (both H’s in and out together),
- One is the bending mode (the angle between H atoms opens and closes).
1 B₁:
- The asymmetric stretch (one H in, one H out).

🧩 So, how do you know the B₁ mode is asymmetric stretching?

We've already decomposed H2O’s vibrational modes as:

Since, I have my trial-error notes on explicit obtaining B1 mode, let's find out for this mode.

Consider the character table of B1 mode, since it is a 1D representation, interpretation is rather straight-forward.

🧮 Direct Calculation on the 9D Displacement Vector

Direct calculation on 9D vector to each operation leads to following processes.

It could be quite overload for complicated systems, but if you understand the principle, you can implement this by programming language quite intuitively. (Although I did not.)

If we apply C2 rotation respect to Z-axis, then it should flip sign.

If we apply mirror respect to YZ-plane, then it should "not" flip their sign.

If we apply mirror respect to XZ-plane, then it should flip their sign.

In summary, B1 mode should have corresponding constraints in its vibration.

With this, we can remove the redundant degree of freedom for B1 mode, which shows that it is anti-stretching mode.

Of course, we cannot obtain the condition of exactly which direction Hydrogen atom oscillates, it can be directly toward oxygen atom and we may expect this, but this should be derived from specific model Hamiltonian.

But anyway, we can derive the symmetry of oscillation mode with following technical details.

If you ask what about above 2D representation? This seems not generalized easily..

Although there is a tons of documentation related on this topic, I haven't searched yet.

Actually, I don't know exactly. I am on a still learning step.. sorry for the poor help.

Maybe you can get some information from another documentation on NH3 molecule, on this website.

If you have any suggestion in this documentation, please let me know by my e-mail or so.

e-mail: yeschowh@snu.ac.kr

Thank you for the reading!

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