Support for phonons in PairwisePotentials

src/structure.jl

@@ -52,7 +52,7 @@ function estimate_integer_lattice_bounds(M::AbstractMatrix{T}, δ, shift=zeros(3
     # then xi = <ei, M^-1 Mx> = <M^-T ei, Mx> <= ||M^-T ei|| δ.
     inv_lattice_t = compute_inverse_lattice(M')
     xlims = [norm(inv_lattice_t[:, i]) * δ + shift[i] for i in 1:3]
-    
+
     # Round up, unless exactly zero (in which case keep it zero in
     # order to just have one x vector for 1D or 2D systems)
     xlims = [xlim == 0 ? 0 : ceil(Int, xlim .- tol) for xlim in xlims]

src/terms/pairwise.jl

@@ -1,3 +1,15 @@
+"""
+Complex-analytic extension of `LinearAlgebra.norm(x)` from real to complex inputs.
+Needed for phonons as we want to perform a matrix-vector product `f'(x)·h`, where `f` is
+a real-to-real function and `h` a complex vector. To do this using automatic
+differentiation, we can extend analytically f to accept complex inputs, then differentiate
+`t -> f(x+t·h)`. This will fail if non-analytic functions like norm are used for complex
+inputs, and therefore we have to redefine it.
+"""
+function norm_cplx(x)
+    sqrt(sum(x.^2))
+end
+
 struct PairwisePotential
     V
     params
@@ -10,8 +22,8 @@ Lennard—Jones terms.
 The potential is dependent on the distance between to atomic positions and the pairwise
 atomic types:
 For a distance `d` between to atoms `A` and `B`, the potential is `V(d, params[(A, B)])`.
-The parameters `max_radius` is of `100` by default, and gives the maximum (reduced) distance
-between nuclei for which we consider interactions.
+The parameters `max_radius` is of `100` by default, and gives the maximum distance (in
+Cartesian coordinates) between nuclei for which we consider interactions.
 """
 function PairwisePotential(V, params; max_radius=100)
     params = Dict(minmax(key[1], key[2]) => value for (key, value) in params)
@@ -56,11 +68,22 @@ end
 
 # This could be factorised with Ewald, but the use of `symbols` would slow down the
 # computationally intensive Ewald sums. So we leave it as it for now.
+# `q` is the phonon `q`-point (`Vec3`), and `ph_disp` a list of `Vec3` displacements to
+# compute the Fourier transform of the force constant matrix.
+# Computes the local energy and forces on the atoms of the reference unit cell 0, for an
+# infinite array of atoms at positions r_{iR} = positions[i] + R + ph_disp[i]*e^{iq·R}.
 function energy_pairwise(lattice, symbols, positions, V, params;
-                         max_radius=100, forces=nothing)
-    T = eltype(lattice)
+                         max_radius=100, forces=nothing, ph_disp=nothing, q=nothing)
+    isnothing(ph_disp) && @assert isnothing(q)
     @assert length(symbols) == length(positions)
 
+    T = eltype(positions[1])
+    if !isnothing(ph_disp)
+        @assert !isnothing(q) && !isnothing(forces)
+        T = promote_type(complex(T), eltype(ph_disp[1]))
+        @assert size(ph_disp) == size(positions)
+    end
+
     if forces !== nothing
         @assert size(forces) == size(positions)
         forces_pairwise = copy(forces)
@@ -89,22 +112,29 @@ function energy_pairwise(lattice, symbols, positions, V, params;
             R == zero(R) && i == j && continue
             ai, aj = minmax(symbols[i], symbols[j])
             param_ij = params[(ai, aj)]
-            Δr = lattice * (positions[i] - positions[j] - R)
-            dist = norm(Δr)
+            ti = positions[i]
+            tj = positions[j] + R
+            if !isnothing(ph_disp)
+                ti += ph_disp[i] # * cis2pi(dot(q, zeros(3))) === 1
+                                 #  as we use the forces at the nuclei in the unit cell
+                tj += ph_disp[j] * cis2pi(dot(q, R))
+            end
+            Δr = lattice * (ti .- tj)
+            dist = norm_cplx(Δr)
             energy_contribution = V(dist, param_ij)
             sum_pairwise += energy_contribution
             if forces !== nothing
-                # We use ForwardDiff for the forces
-                dE_ddist = ForwardDiff.derivative(d -> V(d, param_ij), dist)
+                dE_ddist = ForwardDiff.derivative(real(zero(eltype(dist)))) do ε
+                    V(dist + ε, param_ij)
+                end
                 dE_dti = lattice' * dE_ddist / dist * Δr
                 forces_pairwise[i] -= dE_dti
-                forces_pairwise[j] += dE_dti
             end
         end # i,j
     end # R
     energy = sum_pairwise / 2  # Divide by 2 (because of double counting)
     if forces !== nothing
-        forces .= forces_pairwise ./ 2
+        forces .= forces_pairwise
     end
     energy
 end

src/workarounds/forwarddiff_rules.jl

@@ -274,3 +274,26 @@ function Smearing.occupation(S::Smearing.FermiDirac, d::ForwardDiff.Dual{T}) whe
     end
     ForwardDiff.Dual{T}(Smearing.occupation(S, x), ∂occ * ForwardDiff.partials(d))
 end
+
+# Fix for https://github.com/JuliaDiff/ForwardDiff.jl/issues/514
+function Base.:^(x::Complex{ForwardDiff.Dual{T,V,N}}, y::Complex{ForwardDiff.Dual{T,V,N}}) where {T,V,N}
+    xx = complex(ForwardDiff.value(real(x)), ForwardDiff.value(imag(x)))
+    yy = complex(ForwardDiff.value(real(y)), ForwardDiff.value(imag(y)))
+    dx = complex.(ForwardDiff.partials(real(x)), ForwardDiff.partials(imag(x)))
+    dy = complex.(ForwardDiff.partials(real(y)), ForwardDiff.partials(imag(y)))
+
+    expv = xx^yy
+    ∂expv∂x = yy * xx^(yy-1)
+    ∂expv∂y = log(xx) * expv
+    dxexpv = ∂expv∂x * dx
+    if iszero(xx) && ForwardDiff.isconstant(real(y)) && ForwardDiff.isconstant(imag(y)) && imag(y) === zero(imag(y)) && real(y) > 0
+        dexpv = zero(expv)
+    elseif iszero(xx)
+        throw(DomainError(x, "mantissa cannot be zero for complex exponentiation"))
+    else
+        dyexpv = ∂expv∂y * dy
+        dexpv = dxexpv + dyexpv
+    end
+    complex(ForwardDiff.Dual{T,V,N}(real(expv), ForwardDiff.Partials{N,V}(tuple(real(dexpv)...))),
+            ForwardDiff.Dual{T,V,N}(imag(expv), ForwardDiff.Partials{N,V}(tuple(imag(dexpv)...))))
+end

test/phonons.jl

@@ -0,0 +1,114 @@
+# TODO: Temporary, explanations too dry. To be changed with proper phonon computations.
+using Test
+using DFTK
+using LinearAlgebra
+using ForwardDiff
+using StaticArrays
+
+# Convert back and forth between Vec3 and columnwise matrix
+fold(x)   = hcat(x...)
+unfold(x) = Vec3.(eachcol(x))
+
+function prepare_system(; n_scell=1)
+    positions = [[0.,0,0]]
+    for i in 1:n_scell-1
+        push!(positions, i*ones(3)/n_scell)
+    end
+
+    a = 5. * length(positions)
+    lattice = a * [[1 0 0.]; [0 0 0.]; [0 0 0.]]
+
+    s = DFTK.compute_inverse_lattice(lattice)
+    directions = [[s * [i==j,0,0] for i in 1:n_scell] for j in 1:n_scell]
+
+    params = Dict((:X, :X) => (; ε=1, σ=a / length(positions) /2^(1/6)))
+    V(x, p) = 4*p.ε * ((p.σ/x)^12 - (p.σ/x)^6)
+
+    (positions=positions, lattice=lattice, directions=directions, params=params, V=V)
+end
+
+# Compute phonons for a one-dimensional pairwise potential for a set of `q = 0` using
+# supercell method
+function test_supercell_q0(; n_scell=1, max_radius=1e3)
+    blob = prepare_system(; n_scell)
+    positions = blob.positions
+    lattice = blob.lattice
+    directions = blob.directions
+    params = blob.params
+    V = blob.V
+
+    s = DFTK.compute_inverse_lattice(lattice)
+    n_atoms = length(positions)
+
+    directions = [reshape(vcat([[i==j, 0.0, 0.0] for i in 1:n_atoms]...), 3, :) for j in 1:n_atoms]
+
+    Φ = Array{eltype(positions[1])}(undef, length(directions), n_atoms)
+    for (i, direction) in enumerate(directions)
+        Φ[i, :] = - ForwardDiff.derivative(0.0) do ε
+            new_positions = unfold(fold(positions) .+ ε .* s * direction)
+            forces = zeros(Vec3{complex(eltype(ε))}, length(positions))
+            DFTK.energy_pairwise(lattice, [:X for _ in positions], new_positions, V, params;
+                                 forces, max_radius)
+            [(s * f)[1] for f in forces]
+        end
+    end
+    sqrt.(abs.(eigvals(Φ)))
+end
+
+# Compute phonons for a one-dimensional pairwise potential for a set of `q`-points
+function test_ph_disp(; n_scell=1, max_radius=1e3, n_points=2)
+    blob = prepare_system(; n_scell)
+    positions = blob.positions
+    lattice = blob.lattice
+    directions = blob.directions
+    params = blob.params
+    V = blob.V
+
+    pairwise_ph = (q, d; forces=nothing) ->
+                     DFTK.energy_pairwise(lattice, [:X for _ in positions],
+                                          positions, V, params; q=[q, 0, 0],
+                                          ph_disp=d, forces, max_radius)
+
+    ph_bands = []
+    qs = -1/2:1/n_points:1/2
+    for q in qs
+        as = ComplexF64[]
+        for d in directions
+            res = -ForwardDiff.derivative(0.0) do ε
+                forces = zeros(Vec3{complex(eltype(ε))}, length(positions))
+                pairwise_ph(q, ε*d; forces)
+                [DFTK.compute_inverse_lattice(lattice)' *  f for f in forces]
+            end
+            [push!(as, r[1]) for r in res]
+        end
+        M = reshape(as, length(positions), :)
+        @assert ≈(norm(imag.(eigvals(M))), 0.0, atol=1e-8)
+        push!(ph_bands, sqrt.(abs.(real(eigvals(M)))))
+    end
+    return ph_bands
+end
+
+@testset "Phonon consistency" begin
+    max_radius = 1e3
+    tolerance = 1e-6
+    n_points = 10
+
+    ph_bands = []
+    for n_scell in [1, 2, 3]
+        push!(ph_bands, test_ph_disp(; n_scell, max_radius, n_points))
+    end
+
+    # Recover the same extremum for the system whatever case we test
+    for n_scell in [2, 3]
+        @test ≈(minimum(fold(ph_bands[1])), minimum(fold(ph_bands[n_scell])), atol=tolerance)
+        @test ≈(maximum(fold(ph_bands[1])), maximum(fold(ph_bands[n_scell])), atol=tolerance)
+    end
+
+    # Test consistency between supercell method at `q = 0` and direct `q`-points computations
+    for n_scell in [1, 2, 3]
+        r_q0 = test_supercell_q0(; n_scell, max_radius)
+        @assert length(r_q0) == n_scell
+        ph_band_q0 = ph_bands[n_scell][n_points÷2+1]
+        @test norm(r_q0 - ph_band_q0) < tolerance
+    end
+end

test/runtests.jl

@@ -125,5 +125,10 @@ Random.seed!(0)
         include("forwarddiff.jl")
     end
 
+    # Phonons
+    if "all" in TAGS
+        include("phonons.jl")
+    end
+
     ("example" in TAGS) && include("runexamples.jl")
 end