Quantum-ESPRESSO:
The SCF Loop and
Some Relevant Input Parameters
Sandro Scandolo
ICTP
(most slides courtesy of Shobhana Narasimhan)
www.quantum-espresso.org
2
ASESMA-2010
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.
3
ASESMA-2010
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;
4
ASESMA-2010
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.
5
ASESMA-2010
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.
6
ASESMA-2010
How to thank the authors
7
ASESMA-2010
Useful information about input variables
8
ASESMA-2010
INPUT_PW.html
9
ASESMA-2010
The Kohn-Sham problem
 Want to solve the Kohn-Sham equations:
 1 2




V
(
r
)

V
[
n
(
r
)]

V
[
n
(
r
)]
nuc
H
XC
 2
 i (r)   i i (r)
H
 Note that self-consistent solution necessary, as H
depends on solution:
{ i }  n(r )  H
 Convention:
e    me  1
10
ASESMA-2010
Self-consistent Iterative Solution
Vnuc known/constructed
Initial guess n(r)
Calculate VH[n] & VXC[n]
Generate
new
n(r)
Veff(r)= Vnuc(r) + VH (r) + VXC (r)
How to solve the
Kohn-Sham eqns.
for a set of fixed
nuclear (ionic)
positions.
Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)
Calculate new n(r) = i|i(r)|2
No
Self-consistent?
Yes
Problem solved! Can now calculate energy, forces, etc.
11
ASESMA-2010
Plane Waves & Periodic Systems
• For a periodic system:
where G = reciprocal
1
 (r)   c e
lattice vector

• The plane waves that appear in this expansion can
i ( k  G )r
k
k ,G
G
be represented as a grid in k-space:
ky
• Only true for periodic
systems that grid is
discrete.
• In principle, still need
kx
infinite number of
plane waves.
12
ASESMA-2010
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:
ky
 |k G |
 Ecut
2m
2
2
E=Ecut
kx
Input parameter ecutwfc
13
ASESMA-2010
Step 0:
Defining the (periodic) system
Namelist ‘SYSTEM’
How to Specify the System
• All periodic systems can be specified by a Bravais
Lattice and an atomic basis.
+
=
15
ASESMA-2010
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).
16
ASESMA-2010
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”)
17
ASESMA-2010
Step 1: Obtaining Vnuc
Vnuc known/constructed
Initial guess n(r)
Calculate VH[n] & VXC[n]
Generate
new
n(r)
Veff(r)= Vnuc(r) + VH (r) + VXC (r)
Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)
Calculate new n(r) = i|i(r)|2
No
Self-consistent?
Yes
Problem solved! Can now calculate energy, forces, etc.
18
ASESMA-2010
Nuclear Potential
• Electrons experience a Coulomb potential due to the
nuclei.
• This has a known and simple form:
Vnuc
Z

r
• But this leads to computational problems!
19
ASESMA-2010
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

20
ASESMA-2010
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
Don’t solve for the
core electrons!

Remove wiggles from
valence electrons.
21
ASESMA-2010
Pseudopotentials for Quantum Espresso - 1
•
Go to http://www.quantum-espresso.org; Click on “PSEUDO”
22
ASESMA-2010
Pseudopotentials for Quantum Espresso - 2
• Click on element for which pseudopotential wanted.
23
ASESMA-2010
Pseudopotentials for Quantum Espresso - 3
Pseudopotential’s name
gives information about :
• type of exchangecorrelation functional
• type of pseudopotential
• e.g.:
24
ASESMA-2010
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).
25
ASESMA-2010
Step 2: Initial Guess for n(r)
Vion known/constructed
Initial guess n(r)
Calculate VH[n] & VXC[n]
Generate
new
n(r)
Veff(r)= Vion(r) + VH (r) + VXC (r)
Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)
Calculate new n(r) = i|i(r)|2
No
Self-consistent?
Yes
Problem solved! Can now calculate energy, forces, etc.
26
ASESMA-2010
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
27
ASESMA-2010
Initial Choice of n(r)
28
ASESMA-2010
Step 3: VH & VXC
Vion known/constructed
Initial guess n(r)
Calculate VH[n] & VXC[n]
Generate
new
n(r)
Veff(r)= Vion(r) + VH (r) + VXC (r)
Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)
Calculate new n(r) = i|i(r)|2
No
Self-consistent?
Yes
Problem solved! Can now calculate energy, forces, etc.
29
ASESMA-2010
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:
Exc[n] =  n(r) VxcHOM[n(r)] dr
-surprisingly successful!
pz (in name of pseudopotential)
Replace
• Generalized Gradient Approximation(s): Include
terms involving gradients of n(r)
pw91, pbe (in name of pseudopotential)
30
ASESMA-2010
Step 4: Potential & Hamiltonian
Vion known/constructed
Initial guess n(r)
Calculate VH[n] & VXC[n]
Generate
new
n(r)
Veff(r)= Vion(r) + VH (r) + VXC (r)
Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)
Calculate new n(r) = i|i(r)|2
No
Self-consistent?
Yes
Problem solved! Can now calculate energy, forces, etc.
31
ASESMA-2010
Kohn-Sham equations in plane wave basis
• Eigenvalue equation is now:
• Matrix elements are:
• Ionic potential given by:
32
ASESMA-2010
Step 5: Diagonalization
Vion known/constructed
Initial guess n(r)
Calculate VH[n] & VXC[n]
Generate
new
n(r)
Veff(r)= Vion(r) + VH (r) + VXC (r)
Expensive!
Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)

Calculate new n(r) = i|i(r)|2
No
Self-consistent?
Yes
Problem solved! Can now calculate energy, forces, etc.
33
ASESMA-2010
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,
~ NPW3 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.
34
ASESMA-2010
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
35
ASESMA-2010
Step 6: New Charge Density
Vion known/constructed
Initial guess n(r)
Calculate VH[n] & VXC[n]
Generate
new
n(r)
Veff(r)= Vion(r) + VH (r) + VXC (r)
Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)
Calculate new n(r) = i|i(r)|2
No
Self-consistent?
Yes
Problem solved! Can now calculate energy, forces, etc.
36
ASESMA-2010
Brillouin Zone Sums






0
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.
F

1
P 
Nk
 P(k)w
kBZ
k
k
37
ASESMA-2010
Types of k-point meshes
0
 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’]
1st
BZ
b2 K_POINTS { tpiba | automatic | crystal | gamma }
If ‘automatic’, use M-P mesh:
nk1, nk2, nk3, k1, k2, k3
b1
shift
nk1=nk2=4
38
ASESMA-2010
Step 7: Test for Convergence
Vion known/constructed
Initial guess n(r)
Calculate VH[n] & VXC[n]
Generate
new
n(r)
Veff(r)= Vion(r) + VH (r) + VXC (r)
Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)
Calculate new n(r) = i|i(r)|2
No
Self-consistent?
Yes
Problem solved! Can now calculate energy, forces, etc.
39
ASESMA-2010
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
40
ASESMA-2010
Step 8: Mixing
Can take a long
time to reach
self-consistency!

Vion known/constructed
Initial guess n(r)
Calculate VH[n] & VXC[n]
Generate
new
n(r)
Veff(r)= Vion(r) + VH (r) + VXC (r)
Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)
Calculate new n(r) = i|i(r)|2
No
Self-consistent?
Yes
Problem solved! Can now calculate energy, forces, etc.
41
ASESMA-2010
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.
42
ASESMA-2010
Mixing - 2
 Simplest prescription = linear mixing:
nin(n+1) = 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)
43
ASESMA-2010
Mixing - 3
44
ASESMA-2010
Output Quantities
Vion known/constructed
Initial guess n(r)
Calculate VH[n] & VXC[n]
Generate
new
n(r)
Veff(r)= Vion(r) + VH (r) + VXC (r)
Hi(r) = [-1/22 + Veff(r)] i(r) = i i(r)
Calculate new n(r) = i|i(r)|2
No
Self-consistent?
Yes
Problem solved! Can now calculate energy, forces, etc.
45
ASESMA-2010
Output Quantities
 (Converged) Diagonalization  i, i
 Strictly speaking, only n(r) & Etot are “correct”.
  *  n
 To get Etot: Sum over eigenvalues, correct for doublecounting 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,…
46
ASESMA-2010
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:
47
ASESMA-2010
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...
48
ASESMA-2010
Forces (contd.)
0
• The force is now given by
0
• 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!
Input parameter tprnfor
49
ASESMA-2010
An Outer Loop: Ionic Relaxation
Inner SCF loop
for electronic
iterations
Move
ions
Outer loop
for ionic
iterations
Forces =0?
Structure Optimized!
50
ASESMA-2010
Geometry Optimization With Forces
0
• Especially useful for optimizing internal degrees of
freedom, surface relaxation, etc.
calculation = ‘relax’
• Choice of algorithms for ionic relaxation, e.g., steepest
descent, BFGS.
NAMELIST &IONS
Input parameter ion_dynamics
51
ASESMA-2010
Structure of the input file
52
ASESMA-2010
Input file: namelists
53
ASESMA-2010
Input file: namelists
54
ASESMA-2010
Input file: input_cards
55
ASESMA-2010
Input file: input_cards
56
ASESMA-2010
Input file: a simple example
57
ASESMA-2010