Space Groups

Introduction: From Point Symmetry to Full Space Symmetry

In the previous section, we explored point groups — the set of symmetry operations that leave at least one point in space unmoved. Point groups describe the symmetry of an isolated molecule or the external shape of a crystal. But a crystal is not just a single motif: it is an infinite periodic arrangement of atoms. To fully describe the symmetry of this periodic arrangement, we need a richer mathematical framework that combines point symmetry with translational symmetry.

This leads us to space groups, the complete classification of all possible symmetries of a three-dimensional periodic crystal. A space group tells you every possible way atoms can be mapped onto each other through any combination of rotations, reflections, inversions, and translations — including exotic operations like screw axes and glide planes that combine rotation or reflection with a fractional translation.

The Big Picture: Point groups classify the symmetry of shapes. Space groups classify the symmetry of infinite periodic patterns. Every crystal belongs to exactly one of 230 space groups, and this classification determines which atomic arrangements are possible, which diffraction peaks appear, and how electronic bands behave.

What Is a Space Group?

A space group is the full set of symmetry operations — called isometries — that map an infinite crystal structure onto itself. Formally, each operation can be written using Seitz notation as:

where $R$ is a point-symmetry operation (rotation, reflection, inversion, or rotoinversion) and $\mathbf{t}$ is a translation vector. When the operation acts on a point $\mathbf{r}$, it produces:

Every space group contains three essential ingredients:

1. A Bravais lattice: The set of all pure translations $\{E \mid \mathbf{T}\}$ where $\mathbf{T} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3$ with integer $n_i$. This defines the periodicity.
2. A point group: The set of rotational and reflective symmetries (with translations removed). This is the space group's "point group part."
3. Additional symmetry elements: Screw axes and glide planes that combine point-symmetry operations with fractional translations.

The 230 Space Groups

One of the triumphs of 19th-century mathematical crystallography was the independent derivation by Fedorov (1891), Schoenflies (1891), and Barlow (1894) that there are exactly 230 distinct space groups in three dimensions. This number arises from systematically combining:

• 14 Bravais lattices (defining the translation symmetry)
• 32 crystallographic point groups (defining the rotational/reflective symmetry)
• All possible screw axes and glide planes compatible with each combination

The 230 space groups are organized hierarchically:

The distribution of the 230 space groups across crystal systems is uneven:

Symmorphic vs Non-Symmorphic Space Groups

Space groups divide into two fundamental categories based on whether they contain symmetry operations that combine point symmetry with fractional translations.

Symmorphic Space Groups

A space group is symmorphic if every symmetry operation can be written as a pure point-group operation $\{R \mid \mathbf{0}\}$ combined with a pure lattice translation $\{E \mid \mathbf{T}\}$. In other words, the translation part $\mathbf{t}$ in $\{R \mid \mathbf{t}\}$ is always a full lattice vector (or zero). There are 73 symmorphic space groups.

Examples include:

• $P\bar{1}$ — primitive triclinic with inversion only
• $Pm\bar{3}m$ — primitive cubic with full octahedral symmetry (e.g., simple perovskites)
• $F\bar{4}3m$ — face-centered cubic with tetrahedral symmetry (our zinc blende CdSe)

Non-Symmorphic Space Groups

A space group is non-symmorphic if it contains at least one operation where the translation part $\mathbf{t}$ is a fractional lattice vector — i.e., a screw axis or glide plane. There are 157 non-symmorphic space groups.

Examples include:

• $P2_1/c$ — the most common space group in organic crystals, with a 2₁ screw axis and a c-glide plane
• $P6_3/mmc$ — hexagonal, with a 6₃ screw axis (e.g., wurtzite structures)
• $Fd\bar{3}m$ — diamond cubic, with d-glide planes (e.g., Si, Ge, diamond)

Screw Axes

A screw axis is a symmetry operation that combines a rotation by $2\pi/n$ around an axis with a translation of $p/n$ of the lattice repeat distance along that axis. It is denoted $n_p$, where $n$ is the rotation order and $p$ is the number of fractional translation steps.

The Seitz notation for a screw axis along the $c$-axis is:

After $n$ applications of the screw operation, the rotation returns to the identity and the total translation is $p \cdot \mathbf{c}$ — always an integer number of lattice translations.

Screw Axis Types

Chirality of Screw Axes

Screw axes with $p$ and $n - p$ form enantiomorphic (mirror-image) pairs:

• $3_1$ and $3_2$ are left- and right-handed helices
• $4_1$ and $4_3$ are enantiomorphic; $4_2$ is achiral (self-mirror)
• $6_1$ and $6_5$ are enantiomorphic, as are $6_2$ and $6_4$; $6_3$ is achiral

Glide Planes

A glide plane combines a reflection through a plane with a translation parallel to that plane by a fraction of the lattice vector. In Seitz notation:

where $\sigma$ is a mirror reflection and $\mathbf{t}_g$ is a fractional translation parallel to the mirror plane. Applying the glide operation twice gives a pure lattice translation.

Types of Glide Planes

The d-glide (diamond glide) is special because it involves a quarter translation and only appears in face-centered or body-centered lattices. The diamond structure (space group $Fd\bar{3}m$) is named after this glide.

International Tables Notation

The Hermann-Mauguin notation (also called International notation) is the standard way to write space group symbols. Every symbol has a systematic structure:

Reading a Space Group Symbol

A space group symbol consists of:

1. Lattice letter (first character): indicates the centering type of the conventional unit cell.
2. Symmetry elements (remaining characters): describe the symmetry operations along specific crystallographic directions, which depend on the crystal system.

Lattice Letters

Symmetry Symbols in Different Crystal Systems

The positions after the lattice letter refer to specific crystallographic directions, which vary by crystal system:

Common Symmetry Symbols

F43m: The Space Group of Zinc Blende CdSe

The zinc blende (sphalerite) structure of CdSe belongs to space group F43m, number 216 in the International Tables. Let us decode each part of this symbol to understand the full symmetry of our target material.

Decoding the Symbol

Point Group and Crystal System

The point group of $F\bar{4}3m$ is $\bar{4}3m$ (Schoenflies: $T_d$), which is the tetrahedral symmetry group. This is a subgroup of the full octahedral group $O_h$. The key distinction from the diamond structure ($Fd\bar{3}m$, point group $m\bar{3}m$) is that zinc blende lacks an inversion center because the two sublattices are occupied by different atoms (Cd and Se).

The $T_d$ point group has 24 symmetry operations:

The Structure in Detail

The zinc blende structure can be described as two interpenetrating FCC lattices, offset by $(\frac{1}{4}, \frac{1}{4}, \frac{1}{4})$ along the body diagonal:

• Cd sublattice: FCC at $(0, 0, 0)$ and face-centered equivalents
• Se sublattice: FCC at $(\frac{1}{4}, \frac{1}{4}, \frac{1}{4})$ and face-centered equivalents

Each Cd atom is tetrahedrally coordinated by 4 Se atoms, and vice versa. The tetrahedral coordination is a direct consequence of the $T_d$ site symmetry.

Equivalent Positions

In any space group, the symmetry operations map each point in the unit cell to a set of equivalent positions. These positions are classified as general or special.

General Positions

A general position $(x, y, z)$ is one that does not lie on any symmetry element (rotation axis, mirror plane, or inversion center). The number of equivalent general positions equals the order of the point group multiplied by the number of lattice points per cell.

For $F\bar{4}3m$, the general position has multiplicity 96 (24 point-group operations × 4 face-centered lattice points). Placing a single atom at a general position $(x, y, z)$ generates 96 symmetry-equivalent atoms in the conventional unit cell.

Special Positions

A special position lies on one or more symmetry elements, so some symmetry operations map it back onto itself rather than generating a new position. This reduces the multiplicity (the number of equivalent points generated).

The special positions of a space group are labeled with Wyckoff letters (a, b, c, ...) in order of increasing multiplicity, starting from the position with the highest site symmetry. We will explore Wyckoff positions in detail in the next section.

VASP Connection

In VASP, space-group symmetry plays a central role in determining the efficiency and accuracy of calculations. VASP automatically detects the symmetry of your input structure and uses it to reduce the computational effort.

Symmetry in the POSCAR File

The POSCAR file defines the crystal structure. VASP infers the space group from the atomic positions and lattice vectors. Here is a POSCAR for zinc blende CdSe:

1CdSe zinc blende (F-43m, #216)
26.052
3  0.0  0.5  0.5
4  0.5  0.0  0.5
5  0.5  0.5  0.0
6Cd Se
71  1
8Direct
9  0.000  0.000  0.000   ! Cd at 4a Wyckoff position
10  0.250  0.250  0.250   ! Se at 4c Wyckoff position

Note that this POSCAR uses the primitive FCC cell (with rhombohedral lattice vectors), which contains only 2 atoms instead of the 8 atoms in the conventional cubic cell. VASP works most efficiently with primitive cells.

Symmetry Detection: SYMPREC

VASP uses the SYMPREC parameter (in INCAR) to determine how closely atoms must match ideal symmetric positions. This is a distance tolerance in angstroms:

1# INCAR settings for symmetry
2SYMPREC = 1E-5    # Default: tight tolerance (recommended for high-symmetry structures)
3ISYM = 2          # Use symmetry (default)
4                  # ISYM = 0 switches symmetry off completely
5                  # ISYM = -1 also off (used for some magnetic calculations)

Reading Symmetry from OUTCAR

After a VASP calculation, the OUTCAR file reports the detected symmetry. Here is what you would see for an ideal zinc blende CdSe structure:

1Symmetry group of crystal:          F-43M
2 Subroutine PRICEL returns:
3 Original cell was transformed to a primitive cell.
4 Found 24 symmetry operations.
5
6 The static configuration has the point symmetry T_d .
7 The point group associated with its reciprocal lattice has the symmetry O_h.
8
9 Isometries found:  24  point symmetry operations
10  Number of symmetry adapted k-points:    10
11  (out of    256 k-points in the full Brillouin zone)

Key things to check in the OUTCAR:

1. Space group name matches your expected structure (F-43M for zinc blende)
2. Number of symmetry operations is correct (24 for $T_d$)
3. k-point reduction shows how many irreducible k-points remain after symmetry (here 10 out of 256 — a 25.6x speedup)

Checking Symmetry with VASPKIT or Spglib

You can verify the space group of your POSCAR externally using tools like spglib (via Python):

1import spglib
2import numpy as np
3
4# Zinc blende CdSe (primitive FCC cell)
5lattice = 6.052 * np.array([
6    [0.0, 0.5, 0.5],
7    [0.5, 0.0, 0.5],
8    [0.5, 0.5, 0.0]
9])
10
11positions = [
12    [0.000, 0.000, 0.000],  # Cd
13    [0.250, 0.250, 0.250],  # Se
14]
15
16numbers = [48, 34]  # Cd = 48, Se = 34
17
18cell = (lattice, positions, numbers)
19dataset = spglib.get_symmetry_dataset(cell, symprec=1e-5)
20
21print(f"Space group: {dataset['international']}")
22print(f"Space group number: {dataset['number']}")
23print(f"Point group: {dataset['pointgroup']}")
24print(f"Number of symmetry operations: {len(dataset['rotations'])}")
25# Output:
26# Space group: F-43m
27# Space group number: 216
28# Point group: -43m
29# Number of symmetry operations: 24

Summary

Space groups provide the complete symmetry classification of three-dimensional crystal structures, unifying translational periodicity with point-group symmetry.

Key Takeaways

1. Space groups combine a Bravais lattice, a point group, and possible screw axes and glide planes into a single mathematical group describing all symmetry operations of a crystal.
2. There are exactly 230 space groups in three dimensions, distributed across 7 crystal systems and 32 point groups.
3. Symmorphic space groups (73 total) contain only pure point-group operations and lattice translations. Non-symmorphic groups (157 total) also include screw axes and/or glide planes.
4. Screw axes ($n_p$) combine rotation by $2\pi/n$ with a translation of $p/n$ along the axis.
5. Glide planes (a, b, c, n, d) combine reflection with a fractional translation parallel to the mirror plane.
6. CdSe zinc blende belongs to F43m (No. 216): face-centered cubic, tetrahedral point group $T_d$, no inversion center.
7. In VASP, symmetry is auto-detected and used to reduce k-point sampling. The SYMPREC parameter controls the detection tolerance, and OUTCAR reports the detected space group and number of symmetry operations.

Coming Next: In Wyckoff Positions, we will examine exactly where atoms can sit within the unit cell of a given space group, how site symmetry constrains atomic environments, and how doping changes the symmetry classification.