Convergence – how to get a good answer!mijp1/teaching/grad_FPMM/...Cost of higher tolerance is...

Convergence – how to get a good answer!

Matt Probert Condensed Matter Dynamics Group Department of Physics, University of York, U.K. http://www-users.york.ac.uk/~mijp1

Research results

n  First-principles methods may be used for subtle, elegant and accurate computer experiments and insight into the structure and behaviour of matter.

Results of First-Principles Simulations

• Results of First-Principles

Simulations

• Synopsis

Convergence

Structural Calculations

Surfaces

Defect calculations

Summary

Convergence - CASTEP Workshop: Oxford 2015 2 / 29

First-principles methods may be used for

subtle, elegant and accurate computer ex-

periments and insight into the structure

and behaviour of matter.

Structure of phase III of solid hydrogen

Chris J. Pickard and Richard J. Needs Na-

ture Physics 3, 473 - 476 (2007)

Structure of phase III of solid hydrogen Chris J. Pickard and Richard J. Needs Nature Physics 3, 473 - 476 (2007)

Research results

n  First-principles methods may be used for subtle, elegant and accurate computer experiments and insight into the structure and behaviour of matter.

Results of First-Principles Simulations


Simulations

• Synopsis

Convergence


Surfaces

Defect calculations

Summary


First principles results may be worthless nonsense

… or worthless nonsense!

Overview of lecture

n  Approximations and convergence

n  CASTEP details

n  Examples

Approximations and convergence

Approximations

n  Distinguish physical approximations n  Born-Oppenheimer Approximation n  Level of Theory and XC functional

n  and convergable, numerical approximations n  basis-set size n  Integral evaluation cutoffs n  numerical approximations - FFT grid n  Iterative schemes: number of iterations and

exit criteria (tolerances) n  system size

Approximations

n  Scientific integrity and reproducibility: n  All methods should agree answer to (for

example) “What is lattice constant of silicon in LDA approximation” if sufficiently well-converged.

n  No ab-initio calculation is ever fully-converged! n  Need to appreciate how to acceptable level of

convergence for given level of theory

Basis set size

n  Basis set is fundamental approximation to get correct shape of wavefunction.

n  How accurate does it need to be? n  Variational principle guarantees we get an

upper bound on true ground state E0

n  With energy error n  With plane waves can increase size and

accuracy by adding waves with higher G n  Much harder with Gaussian basis – need to

add ad hoc ‘diffuse’ or ‘polarization’ functions

�

n

=(x

n

� xn�1)

T (rf (xn

)�rf (xn�1))

(rf (xn

)�rf (xn�1))

T (rf (xn

)�rf (xn�1))

E

cut

=~22m

|Gmax

|2

�

GG

0 =�⇢ (G)

�⇢ (G0)

�

GG

0 ' ↵ |G|2 �GG

0

|G|2 +G

2o

E (V ) = E0 +9V0B0

16

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V

◆ 23

� 1

#3

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V

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#9=

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B0 = �V

@P

@V

��P=0

V0 B

00 =

@B

@P

��P=0

E = E0 +1

2k (x� x0)

2

E = E0 +L

2k�

2

� =

0

@�

xx

�

xy

�

xz

�

yx

�

yy

�

yx

�

zx

�

zy

�

zz

1

A

�E ⇠ |� |2

2

Properties

n  Fortunately well converged properties may frequently be computed using an incomplete basis.

n  Size of planewave basis set governed by single parameter: n cut_off_energy (in energy units) or basis_precision=COARSE/MEDIUM/FINE in CASTEP .param file

n  NB Though E is monotonic in Ecut it is not necessarily regular.

Error cancellation

n  Consider energetics of simple chemical reaction: MgO(s) + H2O(g) → Mg(OH)2,(s)

n  Reaction energy computed as

n  What if compute at Ecut=500 & 4000 eV? n  Change in energy of each component: ΔE(MgO)

= -0.021 eV, ΔE(H2O) = -0.566 eV, ΔE(Mg(OH)2) = -0.562 eV, error in ΔE = -0.030 eV

n  i.e. errors cancel in final result if same Ecut etc n  Energy differences converge faster than E

�n =(xn � xn�1)

T (rf (xn)�rf (xn�1))

(rf (xn)�rf (xn�1))T (rf (xn)�rf (xn�1))

Ecut =~22m

|Gmax|2

�GG0 =�⇢ (G)

�⇢ (G0)

�GG0 ' ↵ |G|2 �GG0

|G|2 +G

2o

E (V ) = E0 +9V0B0

16

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B0 = �V

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E = E0 +1

2k (x� x0)

2

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� =

0

@�xx �xy �xz

�yx �yy �yx

�zx �zy �zz

1

A

�E ⇠ |� |2

4E = Eproduct �X

Ereactants = EMg(OH)2 � E (EMgO + EH2O)

2

Planewave convergence

n  Properties of a plane-wave basis. n  Cutoff determines highest representable

spatial Fourier component of density. n  Energy differences converge faster than total

energies. Electron density varies most rapidly near nuclei, where min. effect on bonding

n  Ecut (and Gmax) depend on each element n  Simulation cutoff is max of all elements in calc

n  Required cutoff is system-independent but not property-independent.

Pseudopotentials

n  With efficient pseudopotentials rate of convergence with cutoff is not always smooth. n  Can get plateaus, etc n  Sometimes cause is over-optimization of the

pseudopotential at too low a desired cutoff. n  In which case choose criterion for convergence

energy to be plateau values. n  Absolute energy convergence is rarely

desirable. Force and stress convergence is much more useful criterion.

FFT grid

n  FFT grid should be large enough to handle all G-vectors of density ρ(r), within cutoff: n  |G| ≤ 2|Gmax| n  Guaranteed to avoid ”aliasing” errors in

case of LDA and pseudopotentials without additional core-charge density.

n  LDA can often use 1.5Gmax or 1.75Gmax with little error penalty – faster and less RAM

n  GGA functionals have higher Fourier components, and so use 1.75Gmax - 2Gmax

FFT grid and ultrasoft n  CASTEP can use ultrasoft pseudopotentials

n  See next week for more – faster and less RAM n  Finer grid needed for USP augmentation

charges or core-charge densities. n  CASTEP has a second, finer grid for core/

augmentation charges n  Set by parameter fine_grid_scale

(multiplier of Gmax ) or fine_gmax (inverse length Gfine) – keep Gmax for orbitals.

n  fine_gmax is transferable to other systems but fine_grid_scale is not.

Force convergence

n  CASTEP can do MD or geom opt with forces

Force and Stress


Simulations

• Synopsis

Convergence

• Approximations and

Convergence

• Convergence with basis

set

• Error Cancellation

• Plane-wave cutoff

convergence/testing

• Pseudopotentials and

cutoff behaviour

• FFT Grid parameters

• Force and Stress

• Iterative Tolerances

• K-point convergence

• Strategies for

convergence testing

• Automation of

convergence testing

• Golden rules of

convergence testing


Surfaces

Defect calculations

Summary


Frequently need to find crystal structure at mechanical equilibrium.

• Given guessed or exptl. initial structure, seek local minimum of

Born-Oppenheimer energy surface generated by K-S functional.

• Energy minimum implies forces are zero but not vice versa.

• CASTEP provides for “geometry op-

timization” using quasi-newton meth-

ods.

• Need to worry about accuracy of

forces.

• Forces usually converge at lower cut-

off than total energy because density

in region of nucleus unimportant.400 500 600 700 800 900

Ecut (eV)-0.1

-0.08-0.06-0.04-0.02

00.020.040.060.08

0.1

Conv

erge

nce

erro

r∆E (eV)∆F (eV/A)∆P/1000 (GPa)∆P/100 (inc FSBC)

Cut-off convergence testcubic ZrO2

• different physical quantities converge at different rates.

• Always test convergence specifically of quantities of importance to your

planned calculation

n  Forces usually converge at lower Ecut than total energy as nuclear region unimportant

n  Pressure/stress different again

n  Test the convergence of each property!!!

Iterative solvers

n  Parameter elec_energy_tol specifies when SCF energy converged. n  Optimizer also exits if max_SCF_cycles

reached n  i.e. you must always check that it really did

find the ground state!

n  How accurate does SCF convergence need to be?

Solver tolerances

n  Energetics: same accuracy of result. n  Geometry/MD: need much smaller (tighter) elec_energy_tol to converge forces. n  Cost of higher tolerance is only a few

additional SCF iterations. n  Or use elec_force_tol to exit SCF only

when force convergence criterion satisfied. n  Inaccurate forces are common cause of

geometry optimization failure.

SCF convergence

n  Example: E is converged in 5-6 SCF cycles n  Force is converged in 12 SCF cycles …

Iterative Tolerances


Simulations

• Synopsis

Convergence


Convergence


set



convergence/testing


cutoff behaviour





• Strategies for

convergence testing

• Automation of

convergence testing

• Golden rules of

convergence testing


Surfaces

Defect calculations

Summary


• How to control the iterative solvers?

• Parameter elec energy tol specifies

when SCF energy converged.

• Optimizer also exits if max cycles reached –

always check that it really did converge.

• How accurate does SCF convergence need

to be?

• Energetics: same accuracy of result.

• Geometry/MD: much smaller energy toler-

ance needed to converge forces.

• Cost of higher tolerance is only a few addi-

tional SCF iterations.

• Or use elec force tol to exit SCF only

when force convergence criterion satisfied.

• Inaccurate forces are common cause of geometry optimization failure.

Iterative Tolerances


Simulations

• Synopsis

Convergence


Convergence


set



convergence/testing


cutoff behaviour





• Strategies for

convergence testing

• Automation of

convergence testing

• Golden rules of

convergence testing


Surfaces

Defect calculations

Summary


• How to control the iterative solvers?

• Parameter elec energy tol specifies

when SCF energy converged.

• Optimizer also exits if max cycles reached –

always check that it really did converge.

• How accurate does SCF convergence need

to be?

• Energetics: same accuracy of result.

• Geometry/MD: much smaller energy toler-

ance needed to converge forces.

• Cost of higher tolerance is only a few addi-

tional SCF iterations.

• Or use elec force tol to exit SCF only

when force convergence criterion satisfied.

• Inaccurate forces are common cause of geometry optimization failure.

Quality of Forces/Stresses

n  Can only find a good structure if have reliable forces/stresses

n  Stresses converge more slowly than energies as increase number of plane waves n  CONVERGENCE!

n  Should also check degree of SCF convergence elec_energy_tol (fine quality ≤ 10-6 eV/atom)

can also set elec_force_tol – useful with DM Stresses converge slower than forces!

K-point convergence

n  Brillouin Zone sampling is very important n  Usually set in <seed>.cell with keyword kpoint_mp_grid p q r or kpoint_mp_spacing X with optional offset kpoint_mp_offset. n  Convergence is not variational and

frequently oscillates. n  Finite-temperature smearing can accelerate

convergence, but must extrapolate result back to T=0K

K-point convergence of Al

n  Even simple metals like Al require dense k-point meshes in primitive cell

n  Metals require more k-points than insulators due to occupancy discontinuity at E=EF

n  Can get some error cancellation with insulators but only if identical cells

K-point convergence


Simulations

• Synopsis

Convergence


Convergence


set



convergence/testing


cutoff behaviour





• Strategies for

convergence testing

• Automation of

convergence testing

• Golden rules of

convergence testing


Surfaces

Defect calculations

Summary


Brillouin Zone sampling is another important convergence issue. Usually set in

<seed>.cell with keyword kpoint mp grid p q r with optional offset

kpoint mp offset.

• Convergence is not variational and fre-

quently oscillates.

• Even simple metals like Alneed dense

meshes for primitive cell.

• Finite-temperature smearing can acceler-

ate convergence, but must extrapolate the

result back to 0K.

• In case of insulators some k-point error

cancellation occurs but only between iden-

tical simulation cells.

Consequently comparative phase stabil-

ity energetics and surface energetics fre-

quently demand high degree of k-point

convergence.0 5 10 15 20 25 30

Monkhorst-Pack q-56.42

-56.41

-56.4

-56.39

-56.38

-56.37

-56.36

Ener

gy (e

V)

-56.42

-56.41

-56.4

-56.39

-56.38

-56.37

-56.36

Ener

gy (e

V)

Al k-point convergence testSmear=0.08, a0=4.04975, PBE, Ec=160eV

Strategies for convergence testing

Strategies for testing convergence

n  Tests should be simple and fast n  Make use of transferability of cut_off_energy

and grid_scale and kpoint_mp_spacing or use length scaling for k-point grids convergence

n  Test on representative small systems n  Test Ecut with coarse k-point grid n  Test k-point grids with coarse Ecut

n  Work with fixed geometry as optimization depends on details of convergence

n  Accuracy of forces a good proxy for phonon convergence

Automation of convergence tests

n  Recently added castepconv to the standard set of CASTEP tools n  A suite of Python programs to automate

convergence testing n  Can do cutoff energy, k-point grid and

fine_gmax n  And graph the results

Golden rules of convergence test

n  Change one parameter at a time n  Do not use too high an Ecut – are you

converging augmentation charges? Check fine_gmax instead

n  Use forces & stresses as proxies for other more expensive properties (e.g. phonons)

n  Write your convergence criteria in paper n  Convergence is when parameter of interest

stops changing, NOT when run out of resources or when agree with experiment!

Structural calculations

Modelling molecules

n  CASTEP has 3D periodic boundary conditions n  Surround molecule by vacuum space n  Periodic array …

Solids, molecules and surfaces


Simulations

• Synopsis

Convergence


• Solids, molecules and

surfaces

• Variable Volume

Calculations

• Variable Volume

Calculations -II

Surfaces

Defect calculations

Summary


Sometimes we need to compute a non-periodic system with a PBC code.

• Surround molecule by vacuum space to model using periodic code.

• Similar trick used to construct slab for surfaces.

Modelling surfaces

n  Ditto – add vacuum to separate surfaces

n  But how much vacuum? Must converge it! n  And for slab, need enough layers to get to

bulk-like environment n  Another convergence parameter!

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Dipolar surfaces

n  Beware polar materials and dipole surfaces n  Surface energy does not converge with slab

thickness! n  Use same cell for bulk and slab where possible

to get k-point error cancellation n  If not possible then must use absolute

converged k-point set as no error cancellation n  Can use dipole_correction=static/self-consistent with dipole_dir=x/y/z/all to correct electrostatics

Variable cell calculations

n  Can get equilibrium lattice parameter from min E with vol or from where stress=0 n  Stress is automated with geom opt but

converges less well than min E with Ecut

Structural Calculations 17 / 29

Solids, molecules and surfaces

Sometimes we need to compute a non-periodic system with a PBC code.

• Surround molecule by vacuum space to model using periodic code.

• Similar trick used to construct slab for surfaces.

• Must include enough vacuum space that periodic images do not interact.

• To model surface, slab should be thick enough that centre is “bulk-like”.

• Beware of dipolar surfaces. Surface energy does not converge with slab thickness.

• When calculating surface energy, try to use same cell for bulk and slab to maximise error cancellation of k-point set.

• Sometimes need to compare dissimilar cells – must use absolutely converged k-point set as no error cancellation.


Variable Volume Calculations

• Two ways to evaluate equilibrium lattice parameter - minimum of energy or zero of stress.

• Stress converges less well than energy minimum with cutoff.

• Incomplete basis error in stress approximated by Pulay stress correction.

• “Jagged” E vs V curve due to discreteness of NPW. Can be corrected using Francis-Payne method (J. Phys. Conden. Matt 2, 4395

(1990))

• Finite-size basis corrections for energy and stress automatically computed by CASTEP.


9

Finite Basis Correction

n  “Jagged” E vs V curve due to change in number of plane waves when change volume n  Gmin changes with real-space cell size n  Correct using Francis-Payne method (J. Phys.

Conden. Matt 2, 4395 (1990)) n  Use finite_basis_corr=MANUAL/AUTO

with additional finite_basis_npoints at finite_basis_spacing values of Ecut

n  Or specify basis_dE_dlogE instead.

Number of plane waves

n  Two possibilities when changing cell size: n  fixed basis size (fixed_NPW=true)

n Number of plane-waves NPW is constant n Cell expansion lowers Gmax and K.E. of each

plane wave, and therefore lowers effective Ecut. n Easy to code but easy to get erroneous results. n Need very well-converged cutoff for success.

n  fixed cutoff => physically correct n  reset basis every time change cell parameters n  vary NPW so as to keep Gmax and Ecut fixed

Surface modelling

n  Surfaces can be modelled as a slab cleaved from bulk crystal. Can calculate n  Surface energy or free energy n  Surface reconstructions n  Energies of steps n  Adsorption energies and structures of

adsorbates n  Surface chemical reaction energies.

Choice of surface

n  Choice of slab: usually need to make both surfaces identical.

n  Surfaces related by symmetry operation are more easily geometry optimised.

n  Simulation cell should not be optimised. (use fix_all_cell=T in .cell file ). n  In-plane cell parameters usually fixed at

values from relaxed bulk crystal. n  Perpendicular direction has extra vacuum

Surface convergence

n  Esurf is very sensitive to convergence of total energies. n  Sometime get k-point error

cancellation of slab vs bulk if use non-primitive bulk cell with same in-plane vectors as slab. (not CaCO3 shown).

n  Only 1 k-point in perp to slab. n  Must converge slab thickness

and vacuum gap

Surface Energy calculations

• Surface free energy defined as

Esurf = (Eslab − Ebulk)/A

A is total area of both surfaces.

• Esurf is sensitive quantity requiring well-converged total energies.

• Can sometimes gain some k-point error cancellation between slab and bulk

calcs by using non-primitive bulk cell with same in-plane vectors as slab. (not

CaCO3 101̄4 shown).

• Sometimes need to compare dissimilar cells – must use absolutely converged

k-point set as no error cancellation.

• Need only 1 k-point in direction perpendicular to slab. Any dispersion in bands

is error due to insufficient vacuum gap - no point in calculating accurately!

• Need to test convergence with both slab thickness and vacuum gap


12

101_

4

Polarization in slabs

n  Electric dipoles perpendicular to surface raise theoretical difficulties n  Energy does not converge with slab thickness n  Three types of polar surface:


• Electric dipoles perpendicular to surface raise theoretical difficulties

• Energy does not converge with slab thickness.

• P.W. Tasker (Surf. Sci 87, 315 (1979) described 3 classes of surface.

• In classical charge model, Type III unstable and must always reconstruct.

• In ab initio calculation, surfaces can instead become metallic.

• Polar molecules on surface also raise dipole-dipole problems.

• Can sometimes use double surface with inversion symmetry.


Defect calculations 26 / 29

Defect models

Point- and extended- defects may be modelled using supercell approach. Several tricky convergence issues with supercells, to converge

defect-defect interaction to zero. See M. I. J. Probert and M. C. Payne Phys. Rev. B67 075204 (2003).

• Only need to converge energy to a few meV, but still need accuracy in forces to correctly describe strain relaxation.

• Strain relaxation. Local strain around defect decreases as 1/R.

Can model long-range strain relaxation using classical models if suitable potential exists.

• Charged defects can be modelled using periodic interaction correction terms (M. Leslie and M. Gillan, J. Phys. Cond. Mat. 18, 973

(1985))

• Corrections for higher multipoles also available G. Makov and M. C. Payne Phys. Rev. B51, 4104 (1995), L. N. Kantorovich Phys.

Rev. B60, 15476 (1999)).


13


n  In classical charge model, Type III unstable and must always reconstruct.

n  In ab initio calculation, surfaces can instead become metallic.

n  Polar molecules on surface also raise dipole-dipole problems. n  Use CASTEP dipole correction scheme

n  Can sometimes use double surface with inversion symmetry n  Dipoles cancel.

Defect calculations

n  Can model defects by supercell approach: n  Some tricky convergence issues to get defect-

defect interaction to zero. n  Only need to converge energy to a few meV,

but need accurate forces to get relaxation OK. n  Local strain around defect decays as 1/R. Can

use classical models if know suitable pot n  Charged defects can be modelled using

periodic interaction correction terms. Also correction terms for higher order multipoles.

Example: Si(100) Surface Reconstruction

Si(100) supercell with vacuum gap

… with added hydrogen passivation

9 layers of Silicon and 7 Ǻ vacuum

Convergence Tests

n  Converge cut-off energy → 370 eV n  Converge k-point sampling → 9 k-points n  Converge number of bulk layers → 9 layers n  Converge vacuum gap → 9 Å n  only then see asymmetric dimerisation:

()=best lit. value

Si(100) – The Movie

Si(100) – BFGS Progress

n  Initial slow decrease in energy due to surface layer compression. n  Then small barrier to dimer formation overcome around iteration 14. n  Then rapid energy drop due to dimerisation. n  Final barrier to asymmetric dimerisation overcome around iteration 24.

Summary

Summary n  First principles materials modelling can give

highly accurate property values. n  Need to know how convergence error

depends on basis set, kpoints, grid, etc. n  Degree of convergence required depends on

scientific question asked … n  An over-converged calculation is costly. n  An under-converged calculation is useless. n Be systematic: use automated scripts

such as castepconv

References n  Anne E Mattson et al, “Designing meaningful density

functional theory calculation in materials science - a primer” Model. Sim. Mater. Sci Eng. 13 R1-R31 (2005)

n  M Leslie and M Gillan, “The energy and elastic dipole tensor of defects in ionic crystals calculated by the supercell method”, J. Phys. Cond. Mat. 18, 973 (1985)

n  G Makov and MC Payne, “Periodic boundary conditions in ab initio calculations”, Phys. Rev. B 51, 4014 (1995)

n  MIJ Probert and MC Payne, “Improving the convergence of defect calculations in supercells: An ab initio study of the neutral silicon vacancy”, Phys. Rev. B 67 075204 (2003)

Convergence – how to get a good answer!mijp1/teaching/grad_FPMM/...Cost of higher tolerance is...

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