Quantum-ESPRESSO: The SCF Loop and Some Relevant Input Parameters Sandro Scandolo ICTP (most slides courtesy of Shobhana Narasimhan)

Quantum-ESPRESSO:The SCF Loop and

Some Relevant Input Parameters

Sandro Scandolo ICTP

(most slides courtesy of Shobhana Narasimhan)

ASESMA-20102

www.quantum-espresso.org

ASESMA-20103

Quantum-ESPRESSO: the project

• Quantum ESPRESSO stands for opEn Source Package for Research in Electronic Structure, Simulation, and Optimization. It is freely available to researchers around the world under the terms of the GNU General Public License.

• Quantum ESPRESSO is an initiative of the DEMOCRITOS National Simulation Center (Trieste) and SISSA (Trieste), in collaboration with the CINECA National Supercomputing Center in Bologna, the Ecole Polytechnique Fédérale de Lausanne, Princeton University, the Massachusetts Institute of Technology, and Oxford University. Courses on modern electronic-structure theory with hands-on tutorials on the Quantum ESPRESSO codes are offered on a regular basis in developed as well as developing countries, in collaboration with the Abdus Salam International Centre for Theoretical Physics in Trieste.

ASESMA-20104

What can QE do (I)

• # Ground-state calculations:

• * Self-consistent total energies, forces, stresses;• * Electronic minimization with iterative diagonalization techniques, damped-dynamics, conjugate-gradients;• * Kohn-Sham orbitals;• * Gamma-point and k-point sampling, and a variety of broadening schemes (Fermi-Dirac, Gaussian, Methfessel-Paxton, and Marzari-Vanderbilt);• * Separable norm-conserving and ultrasoft (Vanderbilt) pseudo-potentials, PAW (Projector Augmented Waves);• * Several exchange-correlation functionals: from LDA to generalized-gradient corrections (PW91, PBE, B88-P86, BLYP) to meta-GGA, exact exchange

(HF) and hybrid functionals (PBE0, B3LYP, HSE);• * Hubbard U (LDA+U);• * Berry's phase polarization;• * Spin-orbit coupling and noncollinear magnetism;• * Maximally-localized Wannier functions using WANNIER90 code.

• # Structural Optimization:

• * GDIIS with quasi-Newton BFGS preconditioning;• * Damped dynamics;• * Ionic conjugate-gradients minimization;• * Projected velocity Verlet;• * Transition states and minimum energy paths:• o Born-Oppenheimer nudged elastic band;• o Born-Oppenheimer string dynamics;

• # Ab-initio molecular dynamics

• * Car-Parrinello Molecular Dynamics;• o Microcanonical (Verlet) dynamics;• o Isothermal (canonical) dynamics - Nosé-Hoover thermostats and chains;• o Isoenthalpic, variable cell dynamics (Parrinello-Rahman);• o Constrained dynamics;•

ASESMA-20105

What can QE do (II)

• * Born-Oppenheimer Molecular Dynamics;• o Microcanonical (Verlet) dynamics;• o Isothermal (canonical) dynamics - Anderson, Berendsen thermostats;• o Isoenthalpic, variable cell dynamics (Parrinello-Rahman);• o Constrained dynamics;• o Ensemble-DFT dynamics (for metals/fractional occupations);

• # Response properties (density-functional perturbation theory):

• * Phonon frequencies and eigenvectors at any wavevector;• * Full phonon dispersions; inter-atomic force constants in real space;• * Translational and rotational acoustic sum rules;• * Effective charges and dielectric tensors;• * Electron-phonon interactions;• * Third-order anharmonic phonon lifetimes;• * Infrared and (non-resonant) Raman cross-sections;• * EPR and NMR chemical shifts using the GIPAW method;

• # Spectroscopic properties:

• * K- and L1-edge X-ray Absorption Spectra (XAS)

• * Electronic excitations with Many-Body Perturbation Theory using YAMBO code.

• # Quantum Transport:

• * Ballistic Transport using PWCOND module.

• * Coherent Transport from Maximally Localized Wannier Functions using WanT code.

ASESMA-20106

QE platforms

Platforms:

Runs on almost every conceivable current architecture from large parallel machines (IBM SP and BlueGene, Cray XT, Altix, Nec SX) to workstations (HP, IBM, SUN, Intel, AMD) and single PCs running Linux, Windows, Mac OS-X, including clusters of 32-bit or 64-bit Intel or AMD processors with various connectivity (gigabit ethernet, myrinet, infiniband...). Fully exploits math libraries such as MKL for Intel CPUs, ACML for AMD CPUs, ESSL for IBM machines.

ASESMA-20107

How to thank the authors

ASESMA-20108

Useful information about input variables

ASESMA-20109

INPUT_PW.html

ASESMA-201010

The Kohn-Sham problem

Want to solve the Kohn-Sham equations:

Note that self-consistent solution necessary, as H depends on solution:

Convention:

)()()]([)]([)(2

1 2 rrrrr iiiXCHnuc nVnVV

H

Hrni )(}{

1 eme

ASESMA-201011

Self-consistent Iterative Solution

Vnuc known/constructed

Initial guess n(r)

Calculate VH[n] & VXC[n]

Veff(r)= Vnuc(r) + VH (r) + VXC (r)

Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)

Calculate new n(r) = i|i(r)|2

Self-consistent?

Problem solved! Can now calculate energy, forces, etc.

Generate new n(r)

Yes

No

How to solve theKohn-Sham eqns.for a set of fixednuclear (ionic)positions.

ASESMA-201012

Plane Waves & Periodic Systems

• For a periodic system:

• The plane waves that appear in this expansion can be represented as a grid in k-space:

rGk

GGkk r

)(

,

1)( iec

where G = reciprocallattice vector

kx

ky • Only true for periodic systems that grid is discrete.

• In principle, still need infinite number of plane waves.

ASESMA-201013

Truncating the Plane Wave Expansion

• In practice, the contribution from higher Fourier components (large |k+G|) is small.

• So truncate the expansion at some value of |k+G|.

• Traditional to express this cut-off in energy units:

cutEm

2

|| 22 Gk

kx

E=Ecut

ky

Input parameter ecutwfc

Step 0: Defining the (periodic) system

Namelist ‘SYSTEM’

ASESMA-201015

How to Specify the System

• All periodic systems can be specified by a Bravais Lattice and an atomic basis.

+ =

ASESMA-201016

How to Specify the Bravais Lattice / Unit Cell

- Gives the type of Bravais lattice (SC, BCC, Hex, etc.)

Input parameter ibrav

Input parameters {celldm(i)}

- Give the lengths [& directions, if necessary] of the lattice vectors a1, a2, a3

• Note that one can choose a non-primitive unit cell

(e.g., 4 atom SC cell for FCC structure).

a1

a2

ASESMA-201017

Atoms Within Unit Cell – How many, where?

Input parameter nat

- Initial positions of atoms (may vary when “relax” done).-Can choose to give in units of lattice vectors (“crystal”) or in Cartesian units (“alat” or “bohr” or “angstrom”)

- Number of atoms in the unit cell

Input parameter ntyp

- Number of types of atoms

FIELD ATOMIC_POSITIONS

ASESMA-201018

Step 1: Obtaining Vnuc

Vnuc known/constructed

Initial guess n(r)

Calculate VH[n] & VXC[n]

Veff(r)= Vnuc(r) + VH (r) + VXC (r)

Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)

Calculate new n(r) = i|i(r)|2

Self-consistent?

Problem solved! Can now calculate energy, forces, etc.

Generate new n(r)

Yes

No

ASESMA-201019

Nuclear Potential

• Electrons experience a Coulomb potential due to the nuclei.

• This has a known and simple form:

• But this leads to computational problems!

r

ZVnuc

ASESMA-201020

Problem for Plane-Wave Basis

Core wavefunctions: sharply peaked near nucleus.

High Fourier components present

i.e., need large Ecut

Valence wavefunctions: lots of wiggles near nucleus.

ASESMA-201021

Solutions for Plane-Wave Basis

Core wavefunctions: sharply peaked near nucleus.

High Fourier components present

i.e., need large Ecut

Valence wavefunctions: lots of wiggles near nucleus.

Don’t solve for the core electrons!

Remove wiggles from valence electrons.

ASESMA-201022

Pseudopotentials for Quantum Espresso - 1

• Go to http://www.quantum-espresso.org; Click on “PSEUDO”

ASESMA-201023

Pseudopotentials for Quantum Espresso - 2

• Click on element for which pseudopotential wanted.

ASESMA-201024

Pseudopotentials for Quantum Espresso - 3

Pseudopotential’s name gives information about :

• type of exchange-correlation functional

• type of pseudopotential

• e.g.:

ASESMA-201025

Element & Vion info for Quantum Espresso

NOTE

• Should have same exchange-correlation functional for all pseudopentials.

• ecutwfc, ecutrho depend on type of pseudopotentials used (should test for system & property of interest).

ATOMIC_SPECIESBa 137.327 Ba.pbe-nsp-van.UPFTi 47.867 Ti.pbe-sp-van_ak.UPFO 15.999 O.pbe-van_ak.UPF

ASESMA-201026

Step 2: Initial Guess for n(r)

Vion known/constructed

Initial guess n(r)

Calculate VH[n] & VXC[n]

Veff(r)= Vion(r) + VH (r) + VXC (r)

Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)

Calculate new n(r) = i|i(r)|2

Self-consistent?

Problem solved! Can now calculate energy, forces, etc.

Generate new n(r)

Yes

No

ASESMA-201027

Initial Choice of n(r)

Various possible choices, e.g.,:

Superpositions of atomic densities.

Converged n(r) from a closely related calculation

(e.g., one where ionic positions slightly different).

Approximate n(r) , e.g., from solving problem in a smaller/different basis.

Random numbers.

Input parameter startingwfc

ASESMA-201028

Initial Choice of n(r)

ASESMA-201029

Step 3: VH & VXC

Vion known/constructed

Initial guess n(r)

Calculate VH[n] & VXC[n]

Veff(r)= Vion(r) + VH (r) + VXC (r)

Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)

Calculate new n(r) = i|i(r)|2

Self-consistent?

Problem solved! Can now calculate energy, forces, etc.

Generate new n(r)

Yes

No

ASESMA-201030

Exchange-Correlation Potential

• VXC EXC/n contains all the many-body information.• Known [numerically, from Quantum Monte Carlo ;

various analytical approximations] for homogeneous electron gas.

• Local Density Approximation:

Exc[n] = n(r) VxcHOM[n(r)] dr

-surprisingly successful!

• Generalized Gradient Approximation(s): Include terms involving gradients of n(r)

Replace

pz

pw91, pbe

(in name of pseudopotential)

(in name of pseudopotential)

ASESMA-201031

Step 4: Potential & Hamiltonian

Vion known/constructed

Initial guess n(r)

Calculate VH[n] & VXC[n]

Veff(r)= Vion(r) + VH (r) + VXC (r)

Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)

Calculate new n(r) = i|i(r)|2

Self-consistent?

Problem solved! Can now calculate energy, forces, etc.

Generate new n(r)

Yes

No

ASESMA-201032

Kohn-Sham equations in plane wave basis

• Eigenvalue equation is now:

• Matrix elements are:

• Ionic potential given by:

ASESMA-201033

Step 5: Diagonalization

Vion known/constructed

Initial guess n(r)

Calculate VH[n] & VXC[n]

Veff(r)= Vion(r) + VH (r) + VXC (r)

Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)

Calculate new n(r) = i|i(r)|2

Self-consistent?

Problem solved! Can now calculate energy, forces, etc.

Generate new n(r)

Yes

No

Expensive!

ASESMA-201034

Exact Diagonalization is Expensive!

Have to diagonalize (find eigenvalues & eigenfunctions of) H k+G,k+G’

Typically, NPW > 100 x number of atoms in unit cell.

Expensive to store H matrix: NPW2 elements to be stored.

Expensive (CPU time) to diagonalize matrix exactly, ~ NPW

3 operations required.

Note, NPW >> Nb = number of bands required = Ne/2 or a little more (for metals).

So ok to determine just lowest few eigenvalues.

ASESMA-201035

Iterative Diagonalization

• Can recast diagonalization as a minimization problem.

• Then use well-established techniques for iterative minimization by searching in the space of solutions,

e.g., Conjugate Gradient.

• Another popular iterative diagonalization technique is the Davidson algorithm.

Input parameter diagonalization

Input parameter nbnd

-which algorithm used for iterative diagonalization

-how many eigenvalues computed

ASESMA-201036

Step 6: New Charge Density

Vion known/constructed

Initial guess n(r)

Calculate VH[n] & VXC[n]

Veff(r)= Vion(r) + VH (r) + VXC (r)

Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)

Calculate new n(r) = i|i(r)|2

Self-consistent?

Problem solved! Can now calculate energy, forces, etc.

Generate new n(r)

Yes

No

ASESMA-201037

0Brillouin Zone Sums

Many quantities (e.g., n, Etot) involve sums over k. In principle, need infinite number of k’s. In practice, sum over a finite number: BZ “Sampling”. Number needed depends on band structure. Typically need more k’s for metals. Need to test convergence wrt k-point sampling.

k

F

BZ

wPN

Pk

kk

k)(1

ASESMA-201038

0

nk1, nk2, nk3, k1, k2, k3

K_POINTS { tpiba | automatic | crystal | gamma }

Types of k-point meshes

Special Points: [Chadi & Cohen] Points designed to give quick convergence for particular

crystal structures. Monkhorst-Pack: Equally spaced mesh in reciprocal space. May be centred on origin [‘non-shifted’] or not [‘shifted’]

b1

1st BZb2

If ‘automatic’, use M-P mesh:

nk1=nk2=4

shift

ASESMA-201039

Step 7: Test for Convergence

Vion known/constructed

Initial guess n(r)

Calculate VH[n] & VXC[n]

Veff(r)= Vion(r) + VH (r) + VXC (r)

Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)

Calculate new n(r) = i|i(r)|2

Self-consistent?

Problem solved! Can now calculate energy, forces, etc.

Generate new n(r)

Yes

No

ASESMA-201040

How To Decide If Converged?

• Check for self-consistency. Could compare:

• New and old wavefunctions / charge densities.

• New and old total energies.

• Compare with energy estimated using Harris-Foulkes functional.

Input parameter conv_thr

Input parameter electron_maxstep

-Maximum number of scf steps performed

-Typically OK to use 1.e-08

ASESMA-201041

Step 8: Mixing

Vion known/constructed

Initial guess n(r)

Calculate VH[n] & VXC[n]

Veff(r)= Vion(r) + VH (r) + VXC (r)

Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)

Calculate new n(r) = i|i(r)|2

Self-consistent?

Problem solved! Can now calculate energy, forces, etc.

Generate new n(r)

Yes

No

Can take a longtime to reachself-consistency!

ASESMA-201042

Mixing - 1

Iterations n of self-consistent cycle:

Successive approximations to density:

nin(n) nout(n) nin(n+1).

nout(n) fed directly as nin(n+1) ?? No, usually doesn’t converge.

Need to mix, take some combination of input and output densities (may include information from several previous iterations).

Goal is to achieve self consistency (nout = nin ) in as few iterations as possible.

ASESMA-201043

Mixing - 2

Simplest prescription = linear mixing:

nin(n+1) = nout(n) + (1-) nin(n).

There exist more sophisticated prescriptions (Broyden mixing, modified Broyden mixing of various kinds…)

Input parameter mixing_mode

Input parameter mixing_beta

-Typically use value between 0.1 & 0.7 (depends on type of system)

ASESMA-201044

Mixing - 3

ASESMA-201045

Output Quantities

Vion known/constructed

Initial guess n(r)

Calculate VH[n] & VXC[n]

Veff(r)= Vion(r) + VH (r) + VXC (r)

Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)

Calculate new n(r) = i|i(r)|2

Self-consistent?

Problem solved! Can now calculate energy, forces, etc.

Generate new n(r)

Yes

No

ASESMA-201046

Output Quantities

(Converged) Diagonalization i, i

Strictly speaking, only n(r) & Etot are “correct”.

* n

To get Etot: Sum over eigenvalues, correct for double-counting of Hartree & XC terms, add ion-ion interactions.

Very useful quantity! Can use to get structures, heats of formation, adsorption

energies, diffusion barriers, activation energies, elastic moduli, vibrational frequencies,…

ASESMA-201047

Geometry Optimization Using Etot

Simplest case: only have to vary one degree of freedom

- e.g., structure of diatomic molecule

- e.g., lattice constant of a cubic (SC, BCC, FCC) crystal Can just look for minimum in binding curve:

ASESMA-201048

Forces

Need for geometry optimization and molecular dynamics. Could get as finite differences of total energy - too expensive! Use force (Hellmann-Feynman) theorem:

- Want to calculate the force on ion I:

- Get three terms:

When is an eigenstate,

-Substitute this...

ASESMA-201049

Forces (contd.)

• The force is now given by

• Note that we can now calculate the force from a calculation at ONE configuration alone – huge savings in time.

• If the basis depends upon ionic positions (not true for plane waves), would have extra terms = Pulay forces.

• should be exact eigenstate, i.e., scf well-converged!

0

0

Input parameter tprnfor

ASESMA-201050

An Outer Loop: Ionic Relaxation

Forces =0?

Move

ions

Structure Optimized!

Inner SCF loopfor electroniciterations

Outer loopfor ioniciterations

ASESMA-201051

Geometry Optimization With Forces

• Especially useful for optimizing internal degrees of freedom, surface relaxation, etc.

• Choice of algorithms for ionic relaxation, e.g., steepest descent, BFGS.

0

calculation = ‘relax’

NAMELIST &IONS

Input parameter ion_dynamics

ASESMA-201052

Structure of the input file

ASESMA-201053

Input file: namelists

ASESMA-201054

Input file: namelists

ASESMA-201055

Input file: input_cards

ASESMA-201056

Input file: input_cards

ASESMA-201057

Input file: a simple example