DFT in practice : Part I. Ersen Mete

DFT in practice : Part I

plane wave expansion & the Brillouin zone integration

Ersen Mete

Department of Physics Balıkesir University, Balıkesir - Turkey

Outline

Plane wave expansion

Kohn-Sham equations for periodic systems Potential terms in plane waves

Brillouin zone integration ~

k-point sampling & Monkhorst-Pack grid Integration in irreducible Brillouin zone Smearing methods

Kohn-Sham equations to be solved self-consistently

h −1 2∇ 2_{+ V} eff(~r) i ψi(~r) = εiψi(~r) Veff(~r) = υext(~r) + Z d~r0 n(~r 0₎ |~r0_{− ~r|} | {z } Hartree potential + δEXC[n] δn(~r) | {z } Exchange−correlation potential n(~r) =X i |ψi(~r)|2

If the exchange-correlation functional is exact the total energy and the density will be exact.

However, Kohn-sham single particle wavefunctions and eigenenergies do not correspond to any exact physical value.

Basis set representation of Kohn-Sham orbitals

Atomic Orbitals (AO)

ψα(~r) = ψα(r )Y`m(θ, ϕ) where ψα(r ) =

 e−αr2

Gaussian e−αr Slater

molecular structures : slightly distorted atoms small basis sets can give good results _X

easy to represent vacuum regions _X

basis functions are attached to nuclear positions×

non-orthogonal×

Basis set representation of Kohn-Sham orbitals

Plane waves

ψ~_k(~r) = ei~k·~r

orthogonal _X periodic X(solids)

×(finite systems) =⇒ PBC with supercell approach practical for Fourier transfom and computation of matrix elements_X atomic wave functions require large number of plane waves ×

Bravais lattice, primitive cell & Brillouin zone

Direct lattice : a =~a1,~a2,~a3

Reciprocal lattice derives from confinement and Fourier analysis of periodic functions : ei~bi·~aj _{= δ}

ij =⇒ ~bi· ~aj = 2π`(`=integer)

Reciprocal latt. : b = 2π(_ea)−1=~b1, ~b2, ~b3



Cell volume : Ωcell = det(a)

BZ volume : ΩBZ = det(b) = (2π)3/Ωcell

Direct lattice vectors : ~T (n1, n2, n3) = n1~a1+ n2~a2+ n3~a3≡ ~Tn

Reciprocal latt. vectors : ~G (m1, m2, m3) = m1~b1+ m2~b2+ m3~b3≡ ~Gm

1st BZ (Wigner-Seitz cell) : boundaries are the bisecting planes of ~G vectors where Bragg scattering occurs.

Born-von Karman supercell approach

point defect slab molecule

ψ_{i ,~k}(~r + Nj~aj) = ψi ,~k(~r)

Plane wave expansion of the single particle wavefunctions

Bloch’s theorem

For a periodic Hamiltonian : ψ_{i ,~k}(~r) = ei~k·~r√1

Ncell

u_{i ,~k}(~r) nψ_{i ,~k}(~r + ~aj) = ei~k·~ajψi ,~k(~r))

o

where ~k is in the first BZ and periodic u_{i ,~k}(~r) can be expanded in a Fourier series : u_{i ,~k}(~r) = √1 Ωcell X m ci ,m(~k)ei ~Gm·~r n u_{i ,~k}(~r + ~aj) = u_{i ,~k}(~r)) o Equivalently, ψ_{i ,~k}(~r) =X m ci ,m 1 √ Ωe i (~k+~Gm)·~r _≡X ~q ci ,~q|~qi (~q = ~k + ~Gm)

Fourier representation of the Kohn-Sham equations

For a crystal,

Implications of Bloch states

ψ_{i ,~k} is a superposition of PWs with ~q wavevectors which differ by ~

Gm to maintain the PBC.

~

k is well-defined crystal momentum which is conserved. Potential has the lattice periodicity as u_{i ,~k}(~r).

Veff(~r) =

X

m

Veff(~Gm)ei ~Gm·~r

Only ~Gmvectors are allowed in the Fourier expansion.

The non-zero matrix elements are, h~q0|Veff|~qi = X m V (~Gm)h~q0|ei ~Gm·~r|~qi = X m Veff(~Gm)δ_~q0_−~q,~G m

Substituting these Bloch states into the Kohn-Sham equations, multiplying by h~q0| from the left and integrating over ~r gives a set of matrix equations for any given ~k.

Z d~rh~q0| ×    h −1 2∇ 2_{+ V} eff(~r) i X ~q ci ,m|~qi = εi X ~q ci ,m|~qi    X m0  1 2|~k + ~Gm| 2_δ m,m0+ V_eff(~G_m− ~G_m0)  ci ,m0 = ε_ic_{i ,m}

Matrix diagonalization needed, but still easier to solve.

Eigenvectors and eigenenergies for each ~k are independent in the 1st BZ.

In the large Ω = NcellΩcell limit, ~k points become a dense continuum

Hartree potential in plane waves

VH(~G ) = 1 Ωcell Z d~re−i ~G ·~r Z d~r0 P ~ G0n(~G0)ei ~G 0_·~r0 |~r − ~r0_| ! = 1 Ωcell X ~ G0 n(~G0) Z d~rei (~G0−~G )·~r Z d~ue i ~G0_·~u |~u| = 1 Ωcell X ~ G0 n(~G0)Ωcellδ~G ,~G02π Z ∞ 0 Z 1 −1 eiG0u cos θ u u 2_{d (cos θ)du} = 2πn(~G ) Z ∞ 0 eiGu− e−iGu iG du = 4π n(~G ) G Z ∞ 0 sin(Gu)du = 4πn(~G ) G η→0lim Z ∞ 0 e−ηusin(Gu)du = 4πn(~G ) G η→0lim G η2_{+ G}2 = 4π n(~G ) G2 (G 6= 0)

Exchange-correlation potential in plane waves

VXC(~G ) = 1 Ωcell Z d~re−i ~G ·~r ∂ ∂n n n(~r)xc([n],~r) o = 1 Ωcell Z d~re−i ~G ·~r    X ~ G0 xc([n],~G0)ei ~G 0_·~r + X ~ G ,~G ” n(~G ”)ei ~G ”·~r∂xc([n],~G 0₎ ∂n e i ~G0·~r    = 1 Ωcell    X ~ G0 xc([n],~G0) Z d~rei (~G0−~G )·~r+X ~ G ,~G ” n(~G ”)∂xc([n],~G 0₎ ∂n Z d~rei (~G ”−~G +~G0)·~r    = 1 Ωcell    X ~ G0 xc([n],~G0)Ωcellδ~G ,~G0+ X ~ G ,~G ” n(~G ”)∂xc([n],~G 0₎ ∂n ΩcellδG ”,~~ G −~G0    = xc([n],~G ) + X ~ G0 n(~G − ~G0)∂xc([n],~G 0₎ ∂n

Fourier representation of E

H

and E

XC

terms

By similar arguments, EH= 1 2 Z Z d~rd~r0n(~r)n(~r 0₎ |~r − ~r0_| = 2πΩcell X ~ G 6=0 n(~G )2 G2 EXC = Z d~rn(~r)xc(~r) = Ωcell X ~ G n(~G )xc(~G )

External potential in terms of structure and form factors

Ionic potential as a superposition of isolated atomic potentials

υext(~r) = nsp X κ=1 nκ X j =1 X ~ T Vκ(~r − ~τκ,j− ~T )

nsp species, nκ atoms at ~τκ,j for κ, the set of translation vectors ~T

Vext(~G ) =

1 Ω

Z

Ω

d~rυext(~r)e−i ~G ·~r=

1 Ω nsp X κ=1 nκ X j =1 Z Ω

d~uVκ(~u)e−i ~G ·(~u+~τκ,j)X

~ T e−i ~G ·~T | {z } Ncell = 1 Ωcell nsp X κ=1 Z Ω d~uVκ(~u)e−i ~G ·~u | {z } Ωκ_Vκ_{(~G )} nκ X j =1 e−i ~G ·~τκ,j | {z } Sκ_{(~G )} ≡ nsp X κ=1 Ωκ Ωcell Sκ(~G )Vκ(~G )

Form factor for a spherically symmetric ionic potential

Vκ(~G ) = 1 Ωκ Z ∞ 0 Z 1 −1 Z 2π 0 Vκ(r )e−i ~G ·~rr2d φ d (cos θ) dr = 2π Ωκ Z Vκ(r )e −iGr (cos θ) −iGr    1 -1 r2dr = 2π Ωκ Z ∞ 0 eiGr− e-iGr iGr r 2_dr = 4π Ωκ Z ∞ 0 j0(|~G |r )Vκ(r )r2dr = Vκ(|~G |)

Cutoff : finite basis set

Computationally, a complete expansion in terms of infinitely many plane waves is not possible.

The coefficients, cm(~k), for the lowest orbitals decrease exponentially

with increasing PW kinetic energy (~k + ~Gm)2/2.

A cutoff energy value, Ecut,

determines the number of PWs (Npw)

in the expansion, satisfying,

(~k+ ~Gm)2

2 < Ecut (PW sphere) −→

Npw is a discontinuous function of the PW kinetic energy cutoff.

Basis set size depends only on the computational cell size and the cutoff energy value.

Electron density in the plane wave basis

n(~r) = 1 Nk occ X ~_k,i n_{i ,~k}(~r) where n_{i ,~k}(~r) = |ψ_{i ,~k}(~r)|2

Then, in terms of Bloch states,

n_{i ,~k}(~r) = 1 Ω X m,m0 ci ,m∗ (~k)ci ,m0(~k)ei (~Gm0−~Gm)·~r and n_{i ,~G}(~r) = 1 Ω X m c_{i ,m}∗ (~k)ci ,m”(~k)

Brillouin zone integration

Properties like the electron density, total energy, etc. can be evaluated by intergration over ~k inside the BZ.

for the ith _{band of a function f} i(~k) 1 ΩBZ Z BZ d~k fi(~k) =⇒ ¯fi = 1 Nk X ~k fi(~k) Example : a 2D square lattice 25 ~k-points in the BZ ¯ fi = fi(~k1) + . . . + fi(~k25)

~

_{k-point sampling : uniform vs non-uniform}

Moreno and Soler [Phys.Rev.B 45,24] :

A mesh with uniformly distributed ~k-points is preferred.

Example : A rectangular lattice

They have nearly the same number of ~k-points. (a) : _{Isotropic sampling X}

(b) : _{finer sampling vertically X} poor sampling horizontally × −→

Monkhorst-Pack grid

A uniform mesh of ~k-points can be generated by Monkhorst-Pack procedure ~_k n1,n2,n3= 3 X 1 2ni− Ni− 1 2Ni ~ Gi

where Ni is the number of ~k-points in each direction and ni = 1, . . . , Ni.

Example : 4×4×1 MP grid for 2D square lattice ~ G1= ~b1= πaˆkx, ~G2= ~b2= πakˆy, ~G3= 0 ~_k1,1= -3 8, -3 8
_~ k1,3= -3₈,1₈
~_k 1,2= -38, -1 8
_~ k1,4= -38, 3 8

Shifted-grids

Example : a 2D square lattice

Centered on Γ Centered around Γ

=⇒

25 ~k-points [shifted by (1/8,1/8,0)] 16 ~k-points

Time-reversal symmetry

Kramer’s theorem

Let the hamiltonian, H, be invariant under time-reversal, THT−1= H (in the absence of magnetic fields!)

then Hψ = εψ

THψ = HT ψ = εT ψ

T ψ ≡ ψ∗(~r, −t) is also a solution of the same Schr¨odinger equation with the same eigenvalue.

The solutions ψ and T ψ are orthogonal, hψ|T ψi = 0 Bloch states, ψ_{i ,-~k} and ψ∗

i ,~k satisfy the condition ψ(~r + ~Tn) = e

i~k·~Tn_ψ(~r)

=⇒ ε_{i ,-~k} = ε_{i ,~k}

Lattice symmetry and the Irreducible Brillouin Zone

We use symmetry of the Bravais lattice to reduce the calculation to a summation over ~k inside the IBZ.

Example : IBZ of the 2D square lattice Special ~k-points :

Γ ≡ (0, 0) : full symmetry of the point group. X ≡ 1₂, 0
: E , C2, σx, σy

M ≡ 1₂,1₂
: full symmetry of the point group.

Then, we can unfold the IBZ by the symmetry operations to get the solution for the full BZ.

Preserve symmetry

Example : Hexagonal cell

Shift the ~k-point mesh to preserve hexagonal symmetry! In this case, even meshes break the symmetry A mesh centered on Γ preserves the symmetry.

Integration in the IBZ

Generate a uniform mesh in reciprocal space Shift the mesh if required

Employ all symmetry operations of the Bravais lattice to each of the ~

k-points

Select the ~k-points which fall into the IBZ

Calculate weights, ω~_k, for each of the selected ~k-points

ω~_k =

number of symmetry connected ~k-points total number of ~k-points in the BZ IBZ integration is then given by

¯ fi = IBZ X ~ k ω~kfi(~k)

Example : 4×4×1 MP grid for 2D square lattice 4×4×1 MP grid = 16 ~k-points in the BZ 4 equivalent ~k4,4= 3₈,3₈
=⇒ wk =1₄

4 equivalent ~k3,3= 1₈,3₈
=⇒ wk =1₄

8 equivalent ~k4,3= 3₈,1₈
=⇒ wk =1₂

Then, the BZ integration reduces to, ¯

Smearing methods : fractional occupation numbers

Fermi level : the energy of the highest occupied band.

IBZ integration over the filled states : ¯fi = IBZ

X

~_k

ω~kfi(~k)θ(εi(~k) − εF)

For insulators and semiconductors, DOS goes to zero smoothly before the gap.

For metals, the resolution of the step function at the Fermi level is very difficult in plane waves.

Trick is to replace sharp θ(εi(~k) − εF) function with a smoother

Fermi-Dirac smearing

In order to overcome the discontinuity of the functions at the Fermi level,

f {εi(~k)} =

1 e(εi(~k)−εF)/σ_{+ 1}

where σ = kBT

T −→ 0 : Fermi-Dirac distribution approaches to the step function.

Finite temperature T introduces entropy to the system of non-interacting particles,

S (f ) = −[f ln f + (1 − f ) ln(1 − f )] and the Free energy is given by,

F = E −X

i

σS (fi)

Drawback : reduced occupancies below εF are not compensated new

Gaussian smearing

An approximate step function is obtained by integration of a Gaussian-approximated delta function.

f {(ε_i~k)} = 1 2  1 − erf εi~k − εF σ 

Smearing parameter, σ, has no physical interpretation. Entropy and the free energy cannot be written in terms of f .

S ε − εF σ  = 1 2√πexp " − ε − εF σ 2#

Method of Methfessel-Paxton

To overcome the drawback introduced by the Fermi-Dirac smearing,

Expand the step function in a complete set of orthogonal Hermite functions. (Hermite polynomials multiplied by Gaussians)

δ(x ) ≈ DN= N X n AnH2ne−x 2 → (H2n: δ(x ) is even) θ(x ) ≈ SN = 1 − Z ∞ −∞ DN(t)dt S0= 1

2(1 − erf(x )) → (Gaussian smearing) SN = S0(x ) + N X n=1 AnH2n−1(x )e−x 2

Marzari-Vanderbilt : cold smearing

Amends the negative occupation numbers introduced by Methfessel-Paxton.

The delta function is approximated by a Gaussian multiplied by a first order polynomial, ˜ δ(x ) = √1 πe −[x−(1/√2]2_{(2 −}√_{2x )} where x = εi~k − εF σ

Linear tetrahedron method

Divide the BZ into tetrahedra Interpolate the function Xi

within these tetrahedra

Integrate the interpolated function to obtain ¯Xi