Quantum-ESPRESSO: The SCF Loop and Some Relevant Input Parameters

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Quantum-ESPRESSOThe SCF Loop and Some Relevant
Input Parameters

• Sandro Scandolo
• ICTP
• (most slides courtesy of Shobhana Narasimhan)

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www.quantum-espresso.org
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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.

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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 (LDAU)
• 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

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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

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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.

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How to thank the authors
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Useful information about input variables
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INPUT_PW.html
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The Kohn-Sham problem

• Want to solve the Kohn-Sham equations
• Note that self-consistent solution necessary, as
  H depends on solution
• Convention

H
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Self-consistent Iterative Solution
How to solve the Kohn-Sham eqns. for a set of
fixed nuclear (ionic) positions.
Yes
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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

where G reciprocal lattice vector
ky

• Only true for periodic systems that grid is
  discrete.
• In principle, still need infinite number of plane
  waves.

kx
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Truncating the Plane Wave Expansion

• In practice, the contribution from higher Fourier
  components (large kG) is small.
• So truncate the expansion at some value of kG.
• Traditional to express this cut-off in energy
  units

ky
Input parameter ecutwfc
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Step 0 Defining the (periodic) system
Namelist SYSTEM
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How to Specify the System

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

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How to Specify the Bravais Lattice / Unit Cell
Input parameter ibrav

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

Input parameters celldm(i)
- Give the lengths directions, if
necessary of the lattice vectors a1, a2, a3
a2
a1

• Note that one can choose a non-primitive unit
  cell
• (e.g., 4 atom SC cell for FCC structure).

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Atoms Within Unit Cell How many, where?
Input parameter nat
- Number of atoms in the unit cell
Input parameter ntyp
- Number of types of atoms
FIELD ATOMIC_POSITIONS

• - 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)

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Step 1 Obtaining Vnuc
Yes
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Nuclear Potential

• Electrons experience a Coulomb potential due to
  the nuclei.
• This has a known and simple form
• But this leads to computational problems!

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Problem for Plane-Wave Basis

• Core wavefunctions sharply peaked near
  nucleus.

Valence wavefunctions lots of wiggles near
nucleus.
High Fourier components present i.e., need large
Ecut ?
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Solutions for Plane-Wave Basis

• Core wavefunctions sharply peaked near
  nucleus.

Valence wavefunctions lots of wiggles near
nucleus.
High Fourier components present i.e., need large
Ecut ?
Dont solve for the core electrons!
Remove wiggles from valence electrons.
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Pseudopotentials for Quantum Espresso - 1

• Go to http//www.quantum-espresso.org Click on
  PSEUDO

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Pseudopotentials for Quantum Espresso - 2

• Click on element for which pseudopotential wanted.

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Pseudopotentials for Quantum Espresso - 3

• Pseudopotentials name gives information about
• type of exchange-correlation functional
• type of pseudopotential
• e.g.

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Element Vion info for Quantum Espresso
ATOMIC_SPECIES Ba 137.327 Ba.pbe-nsp-van.UPF Ti
47.867 Ti.pbe-sp-van_ak.UPF O 15.999
O.pbe-van_ak.UPF

• 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).

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Step 2 Initial Guess for n(r)
Yes
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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
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Initial Choice of n(r)
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Step 3 VH VXC
Yes
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Exchange-Correlation Potential

• VXC ? dEXC/dn contains all the many-body
  information.
• Known numerically, from Quantum Monte Carlo
  various analytical approximations for
  homogeneous electron gas.
• Local Density Approximation
• Excn ? n(r) VxcHOMn(r) dr
• -surprisingly successful!
• Generalized Gradient Approximation(s) Include
  terms involving gradients of n(r)

Replace
pz
(in name of pseudopotential)
pw91, pbe
(in name of pseudopotential)
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Step 4 Potential Hamiltonian
Yes
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Kohn-Sham equations in plane wave basis

• Eigenvalue equation is now
• Matrix elements are
• Ionic potential given by

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Step 5 Diagonalization
Expensive! ?
Yes
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Exact Diagonalization is Expensive!

• Have to diagonalize (find eigenvalues
  eigenfunctions of) H kG,kG
• Typically, NPW gt 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,
• NPW3 operations required.
• Note, NPW gtgt Nb number of bands required
  Ne/2 or a little more (for metals).
• So ok to determine just lowest few eigenvalues.

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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
-which algorithm used for iterative
diagonalization
Input parameter nbnd
-how many eigenvalues computed
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Step 6 New Charge Density
Yes
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Brillouin Zone Sums
0

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

eF
e
k
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Types of k-point meshes
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• 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

K_POINTS tpiba automatic crystal gamma
1st BZ
If automatic, use M-P mesh
nk1, nk2, nk3, k1, k2, k3
b1
shift
nk1nk24
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Step 7 Test for Convergence
Yes
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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
-Typically OK to use 1.e-08
Input parameter electron_maxstep
-Maximum number of scf steps performed
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Step 8 Mixing
Can take a long time to reach self-consistency!
?
Yes
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Mixing - 1

• Iterations n of self-consistent cycle
• Successive approximations to density
• nin(n) ? nout(n) ? nin(n1).
• nout(n) fed directly as nin(n1) ?? No, usually
  doesnt 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.

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Mixing - 2

• Simplest prescription linear mixing
• nin(n1) b nout(n) (1-b) 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)

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Mixing - 3
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Output Quantities
Yes
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Output Quantities

• (Converged) Diagonalization ? ei, yi
• Strictly speaking, only n(r) Etot are
  correct.
• yy ? 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,

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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

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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...

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Forces (contd.)
0

• 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!

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Input parameter tprnfor
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An Outer Loop Ionic Relaxation
Inner SCF loop for electronic iterations
Move ions
Outer loop for ionic iterations
Forces 0?
Structure Optimized!
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Geometry Optimization With Forces
0

• Especially useful for optimizing internal degrees
  of freedom, surface relaxation, etc.
• Choice of algorithms for ionic relaxation, e.g.,
  steepest descent, BFGS.

calculation relax
NAMELIST IONS
Input parameter ion_dynamics
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Structure of the input file
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Input file namelists
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Input file namelists
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Input file input_cards
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Input file input_cards
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Input file a simple example