The Plane-Wave Pseudopotential Method

CECAM Tutorial on Simulating Matter at the Nanoscale..., 13 Nov. 2006

Outline

• Solution of the DFT problem: Self-consistency, global minimization

• Crystals: periodicity, direct and reciprocal lattice, unit cell, Brillouin Zone

• Plane waves (PW) basis set

• Pseudopotentials (PP)

• PW+PP technicalities

How to find the DFT ground state of a system ?

• Solving the Kohn-Sham equations self-consistently

(T + V + VH(n) + Vxc(n))ψi = εiψi

with charge density n(r) =∑

i

fi|ψi(r)|2, i = combined k and band index,

fi = occupancy of states, and orthonormality constraints 〈ψi|ψj〉 = δij

• By constrained minimization of the energy functional

E[ψi] =∑

i

fi〈ψi|T + V |ψi〉 + EH +Exc(n) + Eion−ion

with charge density and orthonormality constraints as above, i.e. minimize:

E[ψi, Λij] = E[ψi] − Λij (〈ψi|ψj〉 − δij)

where the Λij are Lagrange multipliers

Self-consistency

Starting from some guess of the input charge density nin(r):

nin −→ (VH + Vxc)(nin) −→ ψi(r) −→ nout(r) =∑

i

fiψ∗i (r)ψi(r)

Simply re-inserting nout as nin will almost invariably not converge. Reason: Inorder to converge, such procedure must reduce the error at each step, but thelow-frequency (small-G) components of the error will likely not be reduced in typicalcondensed-matter systems.

Simple mixing algorithm:

nnew = αnout + (1 − α)nin, 0 < α < 1

Guaranteed to converge if α is small enough. More sophisticated algorithms (Anderson,Broyden) use the input and output of several preceding steps to determine the nextoptimal input combination.

Exchange-correlation potential

The exchange-correlation potential and energy in LDA are relatively simple functionsof the local charge density at point r:

Vxc(r) = vxc(n(r)), Exc[n] =∫exc(n(r))n(r)dr

They are easily calculated in r-space. Gradient-corrected functionals depend on thegradient of the charge density as well. Slightly more complex to calculate.

Diagonalization of the Kohn-Sham Hamiltonian

Whatever approach one chooses, the basic building block of a DFT calculation is theevaluation of the Kohn-Sham Hamiltonian HKS. How to solve Hψ = εψ ? Expandψ in some suitable basis set φi as

ψ(r) =∑

i

ciφi(r).

For an orthonormal basis set, solve

∑j

(Hij − εδij)cj = 0

where the matrix elements Hij = 〈φi|H|φj〉. For a non-orthonormal basis set, solve:

∑j

(Hij − εSij)cj

where Sij = 〈φi|φj〉 (overlap matrix).

Total energy

Once convergence is reached, the total energy of the system may be calculated:

E =∑

i

fi〈ψi|T + V |ψi〉 +EH +Exc[n] +Eion−ion

where

EH =e2

2

∫n(r)n(r′)|r − r′| drdr

′

Exc[n] =∫exc(n(r))n(r)dr

Eion−ion =e2

2

′∑µ,ν

ZµZν

| Rµ − Rν |

(the primed sum excludes terms with Rµ − Rν = 0)

Equivalent expression for the energy, using Kohn-Sham eigenvalues:

E =∑

i

fiεi − EH +∫

(exc(n) − Vxc(n))n(r)dr + Eion−ion

Forces, Structural Optimization, Dynamics etc

The total energy depends on atomic positions Rµ into the unit cell and on the lattice.Forces acting on atoms are the derivatives of the total energy wrt atomic positions:

Fµ = − ∂E

∂Rµ=

∑i

fi〈ψi| ∂V∂Rµ

|ψi〉

The Hellmann-Feynman theorem holds in DFT due to its variational character!

Once forces are calculated, one can perform structural optimization, moleculardynamics, etc.

Periodicity

Let us consider the case of the infinite perfect crystals, having translation symmetry.A perfect crystal is described in terms of

• a unit cell that is periodically repeated

• a basis of atomic positions di in the unit cell

• a lattice of translation vectors R = n1R1 + n2R2 + n3R3

• a reciprocal lattice of vectors G such that G · R = 2πl, with l integer.Such conditions hold if G = m1G1 +m2G2 +m3G3 with Gi · Rj = 2πδij.

Non periodic systems ?

What about e.g. defects in crystals, surfaces, alloys, amorphous materials, liquids,molecules, clusters? none of these has perfect periodicity. One can use supercells,introducing an artificial periodicity.

The supercell geometry is dictated by the type of system under investigation:

• Molecules, clusters:the supercell must allow a minimum distance of at least a few A (∼ 6) betweenthe closest atoms in different periodic replica.

• Defects in crystals:the supercell is commensurate with the perfect crystal cell. The distance betweenperiodic replica of the defect must be “big enough” to minimize spurious defect-defect interactions.

• Surfaces:slab geometry. The number of layers of the materials must be “big enough” tohave “bulk behaviour” in the furthest layer from the surface. The number ofempty layers must be “big enough” to have minimal interactions between layers indifferent regions.

• Alloys, amorphous materials, liquids:the supercell must be “big enough” to give a reasonable description of physicalproperties.

Band Structure

The one-electron states ψ(r) of a perfect crystal Hamiltonian H = T+V are describedby a band index i and a wave vector k.

It is convenient to consider the thermodynamic limit: a slab of crystal composed ofN = N1N2N3 unit cells, N → ∞, obeying Periodic Boundary Conditions:

ψ(r + N1R1) = ψ(r + N2R2) = ψ(r + N3R3) = ψ(r).

There are N wave vectors k in the unit cell of the reciprocal lattice, called theBrillouin Zone. The one-electron states (energy bands) can be written as

ψi,k(r) = eik·rui,k(r)

where ui,k(r) is translationally invariant:

ui,k(r + R) = ui,k(r).

Most popular basis sets:

• Localized basis sets:Bloch sums φk =

∑R exp(−ik · R)φ(r− R) of functions centered on atoms

– Linear Combinations of Atomic Orbitals (LCAO)– Gaussian-type Orbitals (GTO)– Linearized Muffin-Tin Orbitals (LMTO)

• Delocalized basis sets:

– Plane Waves (PW)

• Mixed basis sets:

– Linearized Augmented Plane Waves (LAPW)– Projector Augmented Plane Waves (PAW)

PW basis set

A PW basis set for states of wave vector k is defined as

〈r|k + G〉 =1√NΩ

ei(k+G)·r,h2

2m|k + G|2 ≤ Ecut

Ω = unit cell volume, NΩ = crystal volume, Ecut = cutoff on the kinetic energy ofPWs (in order to have a finite number of PWs!). The PW basis set is complete forEcut → ∞ and orthonormal: 〈k + G|k + G′〉 = δGG′

The components on a PW basis set are the Fourier transform:

|ψi〉 =∑G

ci,k+G|k + G〉

ci,k+G = 〈k + G|ψi〉 =1√NΩ

∫ψi(r)e−i(k+G)·rdr = ψi(k + G).

Hamiltonian diagonalization

The solution of HKSψi = εiψi at fixed potential:

∑G′

(HKS(k + G,k + G′) − εiδG,G′

)ci,k+G′ = 0

requires the diagonalization of the matrix HKS, whose matrix elements are:

HKS(k + G,k + G′) = 〈k + G|HKS|k +G′〉

Matrix element of the Hamiltonian in PWs

• Kinetic energy term:

〈k + G|T |k + G′〉 =h2

2m(k + G)2δGG′

• Hartree and exchange correlation terms:

〈k + G|VH|k + G′〉 = VH(G − G′) = 4πe2n(G − G′)|G − G′|2 (G = G′!)

and〈k + G|Vxc|k + G′〉 = Vxc(G − G′)

where

f(G) =1Ω

∫f(r)e−iG·rdr

is the Fourier Transform of f(r).

• External ionic potential:

V ≡ V (r) =∑

µ∈cell

∑R

Vµ(r − dµ − R)

〈k + G|V |k + G′〉 =1

NΩ

∫e−i(k+G)·r ∑

µ

∑R

Vµ(r− dµ − R)ei(k+G′)·rdr

=∑

µ

Sµ(G− G′)Vµ(G − G′)

where Sµ(G) is the structure factor, containing geometrical information:

Sµ(G) =type µ∑di∈cell

e−iG·di

If the potential has spherical symmetry, V (r) = V (r), then:

V (G) = V (G) =4πΩ

∫r2V (r)

sin(Gr)G

dr.

Advantages and disadvantages of various basis sets

• Localized basis sets:

+ fast convergence with respect to basis set size (just a few functions per atomneeded)

+ no problem with finite systems– difficult to evaluate convergence quality (no systematic way to improve

convergence)– difficult to use (two- and three-center integrals)– difficult to calculate forces (Pulay forces)

• Plane Waves:

– slow convergence with respect to basis set size (many more PWs than localizedfunctions needed)

– require supercells with finite systems+ easy to evaluate convergence quality (just increase cutoff)+ easy to use (Fourier transform)+ easy to calculate forces (no Pulay forces)

Charge density

The calculation of the charge density

n(r) =∑i,k

fi,kψ∗i,k(r)ψi,k(r)

requires in principle a sum over an infinite (N ) number of k-points in the BrillouinZone. As a matter of fact, for insulators just a few “special” points (i.e. a uniformgrid in most cases) are required. For metals, a much denser grid is usually required.

Charge density in reciprocal space:

n(G) =1Ω

∫n(r)e−iG·rdr =

∑G′

∑i,k

fi,kc∗i,k+G′ci,k+G+G′

This is NOT how n(G) is calculated, in practical calculations

The need for Pseudopotentials

Are PWs a practical basis set for electronic structure calculations? Not really! Fromelementary Fourier analysis: length scale δ −→ Fourier components up to q ∼ 2π/δ.In a solid, this means ∼ 4π(2π/δ)3/3ΩBZ PWs (ΩBZ = volume of the BrillouinZone).

Estimate for diamond: 1s wavefunction has δ 0.1 a.u., Ω = (2π)3/(a3

0/4) with lattice parametera0 = 6.74 a.u. −→ 250, 000 PWs!

0 0.5 1 1.5 2 2.5 3 3.5 4r (a.u.)

−1

0

1

2

R(r

)

C 1s2 2s2 2p2

1s2s2p

Need to:

• get rid of core states

• get rid of orthogonality wiggles close to the nucleus

Solution: Pseudopotentials (PP). A smooth effective potential that reproduces theeffect of the nucleus plus core electrons on valence electrons.