# Crystal symmetries

## Theory

In this discussion we will only describe the situation for a monoatomic crystal $\mathcal C \subset \mathbb R^3$, the extension being easy. A symmetry of the crystal is an orthogonal matrix $\widetilde{S}$ and a real-space vector $\widetilde{\tau}$ such that

The symmetries where $\widetilde{S} = 1$ and $\widetilde{\tau}$ is a lattice vector are always assumed and ignored in the following.

We can define a corresponding unitary operator $U$ on $L^2(\mathbb R^3)$ with action

We assume that the atomic potentials are radial and that any self-consistent potential also respects this symmetry, so that $U$ commutes with the Hamiltonian.

This operator acts on a plane-wave as

where we set

(these equations being also valid in reduced coordinates).

It follows that the Fourier transform satisfies

In particular, if $e^{ik\cdot x} u_{k}(x)$ is an eigenfunction, then by decomposing $u_k$ over plane-waves $e^{i G \cdot x}$ one can see that $e^{i(S^T k) \cdot x} (U u_k)(x)$ is also an eigenfunction: we can choose

This is used to reduce the computations needed. For a uniform sampling of the Brillouin zone (the *reducible $k$-points*), one can find a reduced set of $k$-points (the *irreducible $k$-points*) such that the eigenvectors at the reducible $k$-points can be deduced from those at the irreducible $k$-points.

## Example

Let us demonstrate this in practice. We consider silicon, setup appropriately in the `lattice`

and `atoms`

objects as in Tutorial and to reach a fast execution, we take a small `Ecut`

of `5`

and a `[4, 4, 4]`

Monkhorst-Pack grid. First we perform the DFT calculation disabling symmetry handling

```
model = model_LDA(lattice, atoms)
basis_nosym = PlaneWaveBasis(model, Ecut; kgrid=kgrid, use_symmetry=false)
scfres_nosym = @time self_consistent_field(basis_nosym, tol=1e-8)
```

n Energy Eₙ-Eₙ₋₁ ρout-ρin Diag --- --------------- --------- -------- ---- 1 -7.868374256427 NaN 1.87e-01 2.9 2 -7.872765700546 -4.39e-03 2.86e-02 1.9 3 -7.872865497888 -9.98e-05 2.51e-03 1.0 4 -7.872877843188 -1.23e-05 7.28e-04 2.0 5 -7.872878168872 -3.26e-07 2.20e-04 1.2 6 -7.872878211156 -4.23e-08 1.45e-05 1.2 7 -7.872878213110 -1.95e-09 3.38e-06 2.3 4.071594 seconds (1.35 M allocations: 889.175 MiB, 2.89% gc time)

and then redo it using symmetry (the default):

```
basis_sym = PlaneWaveBasis(model, Ecut; kgrid=kgrid)
scfres_sym = @time self_consistent_field(basis_sym, tol=1e-8)
```

n Energy Eₙ-Eₙ₋₁ ρout-ρin Diag --- --------------- --------- -------- ---- 1 -7.868299000393 NaN 1.88e-01 3.7 2 -7.872766007754 -4.47e-03 2.87e-02 1.8 3 -7.872865586317 -9.96e-05 2.61e-03 1.0 4 -7.872877871689 -1.23e-05 7.10e-04 2.0 5 -7.872878174807 -3.03e-07 2.12e-04 1.2 6 -7.872878211189 -3.64e-08 1.60e-05 1.3 7 -7.872878213108 -1.92e-09 3.16e-06 2.2 0.802379 seconds (229.52 k allocations: 159.750 MiB, 2.97% gc time)

Clearly both yield the same energy but the version employing symmetry is faster, since less $k$-points are explicitly treated:

`(length(basis_sym.kpoints), length(basis_nosym.kpoints))`

(10, 64)

Both SCFs would even agree in the convergence history if exact diagonalization was used for the eigensolver in each step of both SCFs. But since DFTK adjusts this `diagtol`

value adaptively during the SCF to increase performance, a slightly different history is obtained. Try adding the keyword argument `determine_diagtol=(args...; kwargs...) -> 1e-8`

in each SCF call to fix the diagonalization tolerance to be `1e-8`

for all SCF steps, which will result in an almost identical convergence history.

We can also explicitly verify both methods to yield the same density:

```
(norm(scfres_sym.ρ.real - scfres_nosym.ρ.real),
norm(values(scfres_sym.energies) .- values(scfres_nosym.energies)))
```

(4.853878957531699e-6, 2.991888977660011e-6)

To demonstrate the mapping between `k`

-points due to symmetry, we pick an arbitrary `k`

-point in the irreducible Brillouin zone:

```
ikpt_irred = 2
kpt_irred_coord = basis_sym.kpoints[ikpt_irred].coordinate
basis_sym.ksymops[ikpt_irred]
```

6-element Array{Tuple{StaticArrays.SArray{Tuple{3,3},Int64,2,9},StaticArrays.SArray{Tuple{3},Float64,1,3}},1}: ([1 0 0; 0 1 0; 0 0 1], [0.0, 0.0, 0.0]) ([0 0 1; 1 0 0; 0 1 0], [0.0, 0.0, 0.0]) ([0 1 0; 0 0 1; 1 0 0], [0.0, 0.0, 0.0]) ([0 0 -1; 0 -1 0; -1 0 0], [0.0, 0.0, 0.0]) ([0 -1 0; -1 0 0; 0 0 -1], [0.0, 0.0, 0.0]) ([-1 0 0; 0 0 -1; 0 -1 0], [0.0, 0.0, 0.0])

This is a list of all symmetries operations $(S, \tau)$ that can be used to map this irreducible $k$-point to reducible $k$-points. Let's pick the third symmetry operation of this $k$-point and check.

```
S, τ = basis_sym.ksymops[ikpt_irred][3]
kpt_red_coord = S * basis_sym.kpoints[ikpt_irred].coordinate
ikpt_red = findfirst(kcoord -> kcoord ≈ kpt_red_coord,
[k.coordinate for k in basis_nosym.kpoints])
[scfres_sym.eigenvalues[ikpt_irred] scfres_nosym.eigenvalues[ikpt_red]]
```

7×2 Array{Float64,2}: -0.12437 -0.12437 0.0951517 0.095152 0.176058 0.176058 0.231686 0.231686 0.366275 0.366276 0.397683 0.397683 0.405286 0.405285