Symmetry Operations

What Symmetry Means for Crystals

A crystal looks the same after certain geometric transformations — rotations, reflections, inversions, and combinations thereof. Each such transformation is called a symmetry operation. The full set of symmetry operations that maps a crystal onto itself defines the crystal's symmetry group, which in turn dictates its physical properties: optical activity, piezoelectricity, allowed vibrational modes, and even the electronic band structure.

Computationally, symmetry is a powerful tool for efficiency. If a crystal has 48 symmetry operations, VASP can reduce the number of k-points in a Brillouin zone calculation by up to a factor of 48, cutting the computational cost dramatically.

Intuition: A symmetry operation is any action that, if performed on the crystal while you look away, leaves you unable to tell anything happened. The crystal is indistinguishable before and after the operation.

The Identity Operation (E)

The simplest symmetry operation is the identity, denoted $E$ (from the German Einheit, meaning unity). It does nothing — every atom stays where it is:

While trivial, the identity is mathematically essential. It serves as the identity element of the symmetry group, satisfying $E \cdot g = g \cdot E = g$ for every operation $g$. Every crystal, no matter how asymmetric, possesses at least this one symmetry operation.

In matrix form, the identity is the $3 \times 3$ identity matrix:

Rotation Operations

A proper rotation $C_n$ rotates the crystal by $\frac{2\pi}{n}$ radians (or $\frac{360^\circ}{n}$) about an axis. The integer $n$ is called the order of the rotation.

Repeated application of $C_n$ generates a series of operations: $C_n, C_n^2, C_n^3, \ldots, C_n^n = E$. For example, $C_4$ generates the set $\{C_4, C_4^2 = C_2, C_4^3, C_4^4 = E\}$.

Rotation Matrices About the z-Axis

A rotation by angle $\theta$ about the $z$-axis has the matrix:

The specific matrices for the crystallographic rotations about z are:

$C_2$ (180 deg rotation about z)

$C_3$ (120 deg rotation about z)

$C_4$ (90 deg rotation about z)

$C_6$ (60 deg rotation about z)

The Crystallographic Restriction Theorem

Not every rotation is compatible with a periodic lattice. The crystallographic restriction theorem states that only 1-fold, 2-fold, 3-fold, 4-fold, and 6-fold rotations are compatible with translational periodicity in three dimensions.

Why No 5-Fold or 7-Fold?

The proof is elegant. Consider a rotation $C_n$ acting on a lattice point $\mathbf{R}$. Because the lattice is periodic, the rotated point must also be a lattice point. In matrix form, when the rotation is expressed in the lattice basis (fractional coordinates), all matrix elements must be integers. The trace of the rotation matrix satisfies:

Since $\cos(2\pi/n)$ must yield an integer trace, only the values $\cos(2\pi/n) \in \{-1, -\frac{1}{2}, 0, \frac{1}{2}, 1\}$ are allowed, giving $n \in \{1, 2, 3, 4, 6\}$.

Reflection (Mirror Planes)

A reflection (denoted $\sigma$) maps every point to its mirror image through a plane. Mirror planes are classified by their orientation relative to the principal rotation axis:

The matrix for reflection through the $xy$-plane (a horizontal mirror $\sigma_h$) is:

Similarly, reflection through the $xz$-plane flips the $y$-coordinate:

A reflection has determinant $-1$ and is its own inverse: $\sigma^2 = E$. Physically, applying a mirror reflection twice returns every atom to its original position.

Inversion

The inversion operation (denoted $i$) maps every point $\mathbf{r}$ to $-\mathbf{r}$ through a centre of symmetry (called the inversion centre or centre of symmetry):

The matrix representation is simply:

Like reflection, inversion is its own inverse: $i^2 = E$. The determinant of the inversion matrix is $(-1)^3 = -1$, classifying it as an improper operation.

Physical Significance

The presence or absence of inversion symmetry has profound physical consequences:

• Centrosymmetric crystals (with inversion) cannot exhibit piezoelectricity or second-harmonic generation.
• Non-centrosymmetric crystals (without inversion) can be piezoelectric. Zinc blende CdSe is a classic example — it lacks inversion symmetry and is therefore piezoelectric.
• In X-ray diffraction, Friedel's law ($|F(hkl)| = |F(\bar{h}\bar{k}\bar{l})|$) holds exactly only for centrosymmetric crystals.

Improper Rotations

An improper rotation $S_n$ is a compound operation: a rotation $C_n$ followed by a reflection $\sigma_h$ through the plane perpendicular to the rotation axis:

Special cases connect to operations we already know:

The matrix for $S_4$ about the z-axis is:

The determinant is $-1$, confirming that $S_4$ is an improper operation.

Matrix Representation

Every symmetry operation can be represented as a $3 \times 3$ matrix $\mathbf{W}$ acting on position vectors. For a point symmetry operation (no translational component), the transformed position is:

The matrices $\mathbf{W}$ for all crystallographic point symmetry operations share these properties:

• Orthogonal: $\mathbf{W}^T \mathbf{W} = \mathbf{I}$ (lengths and angles are preserved)
• Determinant: $\det(\mathbf{W}) = +1$ for proper rotations, $\det(\mathbf{W}) = -1$ for improper operations
• Integer entries in fractional basis: When expressed in the lattice vector basis, the matrix elements are integers (0 or ±1 for cubic systems)

Complete Set for the Cubic System

Here is a summary of key symmetry matrices in Cartesian coordinates:

1import numpy as np
2
3# Define fundamental symmetry matrices
4E = np.eye(3)                          # Identity
5i_inv = -np.eye(3)                     # Inversion
6C2z = np.diag([-1, -1, 1])            # C2 about z
7C4z = np.array([[0,-1,0],[1,0,0],[0,0,1]], dtype=float)  # C4 about z
8sigma_h = np.diag([1, 1, -1])         # Mirror in xy-plane
9sigma_v = np.diag([1, -1, 1])         # Mirror in xz-plane
10S4z = sigma_h @ C4z                    # Improper rotation S4 about z
11
12# Verify properties
13for name, W in [("E", E), ("i", i_inv), ("C2z", C2z), ("C4z", C4z),
14                ("sigma_h", sigma_h), ("S4z", S4z)]:
15    det = np.linalg.det(W)
16    orth = np.allclose(W.T @ W, np.eye(3))
17    print(f"{name:8s}: det = {det:+.0f}, orthogonal = {orth}")

Combining Operations

Symmetry operations can be composed (applied one after another) by multiplying their matrices. If operation $A$ is applied first and then operation $B$, the combined operation is:

Note the order: $B \cdot A$ means "apply $A$ first, then $B$." Matrix multiplication is generally not commutative, so $A \cdot B \neq B \cdot A$ in general.

Example: Two C4 Rotations

Applying $C_4$ about z twice should give $C_2$ about z:

Group Multiplication Table

For a small set of operations, we can build a group multiplication table (also called a Cayley table) that shows the result of every pairwise composition. Here is the table for the group $\{E, C_2, \sigma_v, \sigma_v'\}$ (the point group $C_{2v}$):

This table satisfies all four group axioms: closure (every product is in the set), associativity, identity (E), and inverses (each element appears exactly once in every row and column).

VASP Connection: Symmetry Detection

VASP automatically detects symmetry operations at the start of every calculation. The key parameter controlling this behaviour is the ISYM tag in the INCAR file:

INCAR Example

1# INCAR: Symmetry settings
2ISYM   = 2    # Use symmetry (recommended for PAW)
3SYMPREC = 1E-5  # Tolerance for symmetry detection (angstroms)
4
5# For defect calculations where symmetry is broken:
6# ISYM = 0   # Turn off symmetry completely

Reading Symmetry from OUTCAR

After a VASP run, the detected symmetry operations are printed near the top of the OUTCAR file. Look for lines like:

1Found 24 space group operations (of which 24 are symmorphic)
2
3 irot       det(A)        alpha          n_x          n_y          n_z
4    1     1.000000     0.000000     1.000000     0.000000     0.000000
5    2     1.000000   120.000000     0.577350     0.577350     0.577350
6    3     1.000000   240.000000     0.577350     0.577350     0.577350
7   ...

Each line gives the determinant (proper vs improper), the rotation angle, and the rotation axis. This information tells you exactly which point group VASP has identified for your structure.

1import spglib
2import numpy as np
3
4# CdSe zinc blende structure
5lattice = np.diag([6.077, 6.077, 6.077])
6positions = [
7    [0.000, 0.000, 0.000],  # Cd
8    [0.500, 0.500, 0.000],  # Cd
9    [0.500, 0.000, 0.500],  # Cd
10    [0.000, 0.500, 0.500],  # Cd
11    [0.250, 0.250, 0.250],  # Se
12    [0.750, 0.750, 0.250],  # Se
13    [0.750, 0.250, 0.750],  # Se
14    [0.250, 0.750, 0.750],  # Se
15]
16numbers = [48, 48, 48, 48, 34, 34, 34, 34]  # Cd=48, Se=34
17
18cell = (lattice, positions, numbers)
19sym = spglib.get_symmetry(cell)
20print(f"Number of symmetry operations: {len(sym['rotations'])}")
21# Output: 24 (for zinc blende Td symmetry)
22
23spg = spglib.get_spacegroup(cell)
24print(f"Space group: {spg}")
25# Output: F-43m (216)

Summary

Symmetry operations are the building blocks of crystal symmetry. Every operation that maps a crystal onto itself can be expressed as a $3 \times 3$ orthogonal matrix, and the collection of all such operations forms a mathematical group.

1. Identity (E): Does nothing; the mandatory trivial symmetry.
2. Proper rotations ($C_n$): Rotation by $2\pi/n$; only $n = 1, 2, 3, 4, 6$ are compatible with lattice periodicity.
3. Reflections ($\sigma$): Mirror planes; classified as horizontal, vertical, or dihedral.
4. Inversion (i): Maps $\mathbf{r} \to -\mathbf{r}$; its presence determines centrosymmetry and rules out piezoelectricity.
5. Improper rotations ($S_n$): Compound rotation + reflection; special cases include $S_1 = \sigma$ and $S_2 = i$.
6. Matrix representation: All operations are orthogonal matrices with det = ±1, enabling computational manipulation.
7. VASP uses symmetry (ISYM tag) to reduce computational cost by exploiting equivalent k-points and equivalent atoms.

Looking Ahead: Individual symmetry operations are ingredients. In the next section, we will combine them into point groups — complete sets of operations that describe the macroscopic symmetry of a crystal. The 32 crystallographic point groups classify every crystal into one of 7 crystal systems.