Point Groups

From Operations to Groups

In the previous section we catalogued individual symmetry operations — rotations, reflections, inversion, and improper rotations. A crystal does not possess just one of these in isolation; it possesses a complete set that is closed under composition. This set is a point group: a mathematical group whose operations all leave at least one point in space unmoved.

The point group is the "fingerprint" of a crystal's macroscopic symmetry. It determines which physical properties are allowed (piezoelectricity, optical activity, second-harmonic generation) and, computationally, how much VASP can reduce the Brillouin zone sampling. There are exactly 32 crystallographic point groups, and every crystal belongs to one of them.

Intuition: Think of a point group as the complete "instruction manual" of symmetries for a crystal. If you know the point group, you know every rotation, reflection, and inversion that maps the crystal onto itself — and therefore every physical property that symmetry forbids or allows.

What Is a Point Group?

A point group is a set of symmetry operations $G = \{g_1, g_2, \ldots, g_n\}$ with the following properties:

1. Every operation $g_i$ leaves at least one point fixed (the origin).
2. The set is closed under composition: if $g_i, g_j \in G$, then $g_i \cdot g_j \in G$.
3. The identity $E$ is in the set.
4. Every operation has an inverse in the set: for each $g_i$, there exists $g_i^{-1} \in G$ such that $g_i \cdot g_i^{-1} = E$.

The number of operations in the group is called the order of the group, denoted $|G|$. The order ranges from 1 (for the trivial group $C_1$ containing only $E$) to 48 (for the full cubic group $O_h$).

Critically, point groups do not include translations. They describe the symmetry of the shape of the unit cell and its contents, not the infinite periodic arrangement. (Translations are included when we move from point groups to space groups in a later section.)

The 32 Crystallographic Point Groups

The crystallographic restriction theorem (only 1, 2, 3, 4, 6-fold rotations) limits the number of possible point groups in three dimensions to exactly 32. These are distributed across the 7 crystal systems as follows:

The total is $2 + 3 + 3 + 7 + 5 + 7 + 5 = 32$.

Schoenflies vs Hermann-Mauguin Notation

Two notation systems coexist in crystallography, and you will encounter both in textbooks and VASP output:

Schoenflies Notation

Preferred in molecular spectroscopy and chemistry. It uses letter-number combinations:

Hermann-Mauguin (International) Notation

Preferred in crystallography and used by the International Tables. It lists the symmetry elements along specific directions:

Correspondence for Common Groups

Group Axioms

A point group is a group in the mathematical sense. This means it satisfies four axioms. Let us verify each one using the zinc blende point group $T_d$ as our running example.

Axiom 1: Closure

For any two operations $g_i, g_j \in G$, the product $g_i \cdot g_j$ is also in $G$. In other words, combining any two symmetry operations of the crystal yields another symmetry operation of the crystal.

Axiom 2: Associativity

For any three operations, the grouping does not matter:

This follows automatically because matrix multiplication is associative.

Axiom 3: Identity

There exists an element $E \in G$ such that $E \cdot g = g \cdot E = g$ for all $g \in G$. This is the identity operation (do nothing).

Axiom 4: Inverse

For every operation $g \in G$, there exists $g^{-1} \in G$ such that $g \cdot g^{-1} = g^{-1} \cdot g = E$. For rotations, the inverse is the rotation by the negative angle. For reflections and inversion, the operation is its own inverse.

T_d: The Point Group of Zinc Blende

The zinc blende structure of CdSe belongs to point group $T_d$ (Hermann-Mauguin: $\bar{4}3m$). This is one of the most important point groups in semiconductor physics, shared by GaAs, InP, ZnS, and many other III-V and II-VI compounds.

The 24 Operations of $T_d$

$T_d$ has order 24, with operations organized into 5 conjugacy classes:

Total: $1 + 8 + 3 + 6 + 6 = 24$ operations.

Understanding Each Class

8$C_3$ rotations: A cube has 4 body diagonals (from corner to opposite corner). Each diagonal is a 3-fold axis, and each admits two non-trivial rotations ($C_3$ and $C_3^2$): $4 \times 2 = 8$.

3$C_2$ rotations: Three 2-fold axes pass through the centres of opposite faces of the cube (along x, y, z).

6$S_4$ improper rotations: The same three axes as $C_2$, but now the operations are $S_4$ and $S_4^3$: $3 \times 2 = 6$.

6$\sigma_d$ mirror planes: Six dihedral mirror planes, each containing one face diagonal and one body diagonal of the cube.

Character Table of $T_d$

The character table encodes how each irreducible representation transforms under the symmetry operations. This table is essential for analysing electronic states and vibrational modes:

Key features of this character table:

• The A1 representation is totally symmetric — it transforms like a scalar.
• The T2 representation is 3-dimensional and transforms like the coordinates $(x, y, z)$. Electronic p-orbitals at the $\Gamma$ point of zinc blende transform as $T_2$.
• The sum of squares of dimensions equals the group order: $1^2 + 1^2 + 2^2 + 3^2 + 3^2 = 24$.

Physical Consequences of Symmetry

The point group directly determines which physical properties a crystal can exhibit. Neumann's principle states: the symmetry of any physical property must include the symmetry of the point group of the crystal.

Piezoelectricity

A crystal is piezoelectric only if its point group lacks an inversion centre. Of the 32 point groups, 21 are non-centrosymmetric, and 20 of those are piezoelectric (the exception is $O$ = 432, which has too much rotational symmetry).

Zinc blende ($T_d$) is non-centrosymmetric and therefore piezoelectric. This property is exploited in CdSe quantum dot-based sensors and actuators.

Optical Activity

Optical activity (rotation of the plane of polarised light) requires the crystal to be chiral — it must lack both mirror planes and an inversion centre. Only 11 of the 32 point groups are chiral:

Note that $T_d$ is not chiral (it has mirror planes), so zinc blende CdSe is not optically active despite being non-centrosymmetric.

Second-Harmonic Generation (SHG)

Like piezoelectricity, second-harmonic generation requires the absence of inversion symmetry. The SHG tensor $d_{ijk}$ transforms as a third-rank tensor, and its non-zero components are dictated by the point group.

Symmetry-Allowed Tensor Components

The point group constrains the form of physical property tensors:

VASP Connection: Point Groups in Practice

VASP identifies the point group of your structure automatically and uses it to reduce computational cost. Here is how point groups affect your calculations.

k-Point Reduction

The point group maps some k-points in the Brillouin zone onto others. VASP exploits this to compute only the irreducible k-points and then reconstructs the full result using symmetry. The reduction factor can be dramatic:

For a $10 \times 10 \times 10$ k-point grid (1000 points), the $T_d$ symmetry of zinc blende reduces this to roughly 47 irreducible k-points — a 21x speedup.

Reading the Point Group from OUTCAR

1# Search for symmetry information in OUTCAR
2$ grep "point group" OUTCAR
3
4 Found point group: T_d
5 The equivalent point group in H-M notation is: -43m
6
7 The point group operations are:
8  E    8C3  3C2  6S4  6sigma_d
9
10 KPOINTS: Automatic mesh
11 Irreducible k-points:   47 out of 1000

Symmetry and INCAR Settings

1# Standard INCAR for zinc blende CdSe
2SYSTEM  = CdSe zinc blende
3ISYM    = 2       # Use symmetry (PAW recommended)
4SYMPREC = 1E-5    # Symmetry tolerance in angstroms
5
6# For band structure calculations along high-symmetry paths:
7# Symmetry is automatically considered for k-path selection
8
9# When symmetry is broken (e.g., by a defect):
10# ISYM = 0      # Disable symmetry
11# This increases the number of k-points computed

Symmetry-Broken Calculations

When you introduce a defect (e.g., substituting one Cd with Mn in Mn:CdSe), the point group symmetry is reduced. VASP will detect fewer symmetry operations, and more k-points will be needed:

Verifying Symmetry Before Submission

Use spglib to check the point group before running VASP, so you can estimate the computational cost:

1import spglib
2import numpy as np
3
4# CdSe zinc blende
5lattice = np.diag([6.077, 6.077, 6.077])
6positions = [
7    [0.000, 0.000, 0.000],
8    [0.500, 0.500, 0.000],
9    [0.500, 0.000, 0.500],
10    [0.000, 0.500, 0.500],
11    [0.250, 0.250, 0.250],
12    [0.750, 0.750, 0.250],
13    [0.750, 0.250, 0.750],
14    [0.250, 0.750, 0.750],
15]
16numbers = [48, 48, 48, 48, 34, 34, 34, 34]
17
18cell = (lattice, positions, numbers)
19dataset = spglib.get_symmetry_dataset(cell)
20
21print(f"Space group:  {dataset['international']} ({dataset['number']})")
22print(f"Point group:  {dataset['pointgroup']}")
23print(f"Symmetry ops: {len(dataset['rotations'])}")
24print(f"Hall symbol:  {dataset['hall']}")
25
26# Estimate k-point reduction
27nkpts_full = 10 * 10 * 10
28nkpts_irr = nkpts_full // len(dataset['rotations'])
29print(f"Estimated irreducible k-points: ~{nkpts_irr} out of {nkpts_full}")
30
31# Output:
32# Space group:  F-43m (216)
33# Point group:  -43m
34# Symmetry ops: 24
35# Hall symbol:  F -4 2 3
36# Estimated irreducible k-points: ~41 out of 1000

Summary

Point groups classify the macroscopic symmetry of crystals into exactly 32 possible groups. Each group is a collection of symmetry operations satisfying the four group axioms, and each determines the physical properties a crystal can exhibit.

1. A point group is a set of symmetry operations that leave at least one point fixed and satisfy closure, associativity, identity, and inverse axioms.
2. There are exactly 32 crystallographic point groups, distributed across the 7 crystal systems.
3. Two notation systems coexist: Schoenflies (used in physics and chemistry) and Hermann-Mauguin (used in crystallography and VASP).
4. Zinc blende CdSe has point group $T_d$ (order 24) with 5 conjugacy classes: $E, 8C_3, 3C_2, 6S_4, 6\sigma_d$.
5. Physical consequences: The point group determines whether a crystal can be piezoelectric, optically active, or exhibit second-harmonic generation — all before any calculation.
6. Computational impact: VASP uses the point group to reduce k-point sampling, potentially giving speedups of up to 48x for high-symmetry structures.
7. Defects break symmetry: Doping reduces the point group and increases computational cost. Always verify symmetry with spglib before submission.

Looking Ahead: Point groups describe the symmetry at a single point. To capture the full symmetry of a crystal — including translational periodicity — we need space groups. In the next section, we will combine point groups with lattice translations (and screw axes and glide planes) to arrive at the 230 space groups that classify all possible crystal symmetries.