DFT with plane waves,pseudopotentials

Tutoriel CPMD/CP2KParis, 6–9 avril 2010

François-Xavier Coudert

Basis setsIn quantum chemistry, the base objects are the wavefunctions:

... which we represent as linear combinations of basis functions:

This family has to be finite, limited in sizeIt maps a continuous problem to linear algebra:✓ Wavefunctions are vectors

✓ Operators are matrices

The basis set should allow enough flexibility to representthe “real” wavefunctions

Atomic basis setsAlso called “localized basis sets”:centered on nuclei (or sometimes ghost atoms)Slater-types or Gaussian functionsMatches the “linear combination of atomic orbitals” ansatz;also include “polarization” and “diffuse” orbitals

Thus, choices are:✓ Function type✓ Number of orbitals per atom✓ Including polarisation/diffuse orbitals (and how many)?

Atomic basis sets — Pros and cons

★ Correspond to chemical picture★ Describe well the atoms, even with few orbitals★ No so much harder to describe inner electrons

★ Easily tunable★ No implicit periodicity

★ Non-orthogonal★ Depend on atomic positions★ Basis set superposition error★ So many parameters... how do you optimize?

Plane-waves basis setsWe choose the basis functions as plane waves:

G is a vector of the reciprocal lattice.For an orthorhombic unit cell:

Infinite sum would be the Fourier series of the wavefunction

How many basis functions do we have?Ruled by one parameter, Gmax

1 Plane wave basis set

1.1 Definition

Basis set {!!} of plane waves: (! is the volume of the box)

!!("r ) =1!!

ei "G!·"r (1)

The wavefunctions are expanded as

#i("r ) =1!!

!

"G

ci( "G )ei "G·"r (2)

For an orthorhombic box with lengths Lx, Ly and Lz, the wavevectors"G are

"G = i ·2$

Lx· "x + j ·

2$

Ly· "y + k ·

2$

Lz· "z ; with i, j, k " Z

2

1 Plane wave basis set

1.1 Definition

Basis set {!!} of plane waves: (! is the volume of the box)

!!("r ) =1!!

ei "G!·"r (1)

The wavefunctions are expanded as

#i("r ) =1!!

!

"G

ci( "G )ei "G·"r (2)

For an orthorhombic box with lengths Lx, Ly and Lz, the wavevectors"G are

"G = i ·2$

Lx· "x + j ·

2$

Ly· "y + k ·

2$

Lz· "z ; with i, j, k " Z

2

1 Plane wave basis set

1.1 Definition

Basis set {!!} of plane waves: (! is the volume of the box)

!!("r ) =1!!

ei "G!·"r (1)

The wavefunctions are expanded as

#i("r ) =1!!

!

"G

ci( "G )ei "G·"r (2)

For an orthorhombic box with lengths Lx, Ly and Lz, the wavevectors"G are

"G = i ·2$

Lx· "x + j ·

2$

Ly· "y + k ·

2$

Lz· "z ; with i, j, k " Z

2

Plane-waves sets — Pros and cons

★ Orthogonal★ Independent of atomic positions (no Pulay forces)★ Improving is easy: increase the cutoff★ Easy to use on any atomic type (no basis set optimization)★ Use of Fast Fourier Transform

★ Implicit periodicity★ All-or-nothing description (no spot favored)

★ Large number of basis functions needed★ How do you get back chemical information?★ Inner wavefunctions vary too rapidly:

Pseudopotentials are needed

Plane-waves basis setsNumber of plane waves used: described by an energy cutoff

!!"#$%"%&'(&)*+,-*(&*.$/&%01/2%*3/4(4

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! "#$%&%!!""'()*"

! +,!##"$"

%

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"#$%&'!#(!#..%,1'6!78!9!+1/2$'!345

!!-$"!&-

#""&-

""#

:'.1,-#.0$!$0))1.'5

./01233!'()*#

#

4''% ./01233" (#

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! "#$%&%!!""'()*"

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%

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!!-$"!&-

#""&-

""#

:'.1,-#.0$!$0))1.'5

./01233!'()*#

#

4''% ./01233" (#

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! "#$%&%!!""'()*"

! +,!##"$"

%

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!!-$"!&-

#""&-

""#

:'.1,-#.0$!$0))1.'5

./01233!'()*#

#

4''% ./01233" (#

1.2 Size of the basis set

The size of the basis set is determined by a cut-o! of the kinetic

energy associated to ! !G("r ) = 1!!

ei !G·!r

!1

2"2!!G("r ) =

1

2#"G#2!!G("r )

Ekin =1

2#"G#2 < Ecut

2!L

(2#)3

!NPW $

4

3#G3

max =4

32

3

2 E3

2

cut

NPW $1

2#2

23

2

3!E

3

2

cut %1

2#2!E

3

2

cut

6

kinetic energy associated with each plane wave

1.2 Size of the basis set

The size of the basis set is determined by a cut-o! of the kinetic

energy associated to ! !G("r ) = 1!!

ei !G·!r

!1

2"2!!G("r ) =

1

2# "G#2!!G("r )

Ekin =1

2# "G#2 < Ecut

2!L

(2#)3

!NPW $

4

3#G3

max =4

32

3

2 E3

2

cut

NPW $1

2#2

23

2

3!E

3

2

cut %1

2#2!E

3

2

cut

6

Plane-waves basis setsExample of convergence

silicon bulk

Courtesy F. Bruneval

Plane-waves basis setsExample of convergence

dihydrogen molecule

Courtesy F. Bruneval

CPMD 3.13.2 Tutorial 33

4.1.4 Other Output Files

Apart from the console output, our CPMD run created a few other files. Most importantly therestart file RESTART.1 and its companion file LATEST. The restart file contains the final stateof the system when the program terminated. The restart file is binary and – due to the wayhow FORTRAN stores binary data – not generally compatible between different machines andcompilers. Most FORTRAN compilers have flags that affect the way how binary files are encodedand those one may be able to compile CPMD so that it can read data from a different platform,but that in turn implies, that binary files can be incompatible between two different compilationson the same machine with the same compiler, if different settings for the encoding were used.The restart file is needed for other calculations (see RESTART), which need a converged wave-function as a starting point. The file GEOMETRY.xyz contains the coordinates of the atoms inAngstrom and in a format, that can be read in by many molecular visualization programs. TheGEOMETRY contains coordinates and velocities/forces in atomic units.

4.2 Choosing the Plane Wave Cutoff

After getting familiar with the input and output of CPMD, we now have to run a test on the va-lidity of the pseudopotential and its cutoff requirements for the subsequent calculations (geometryoptimization and molecular dynamics). For that purpose we make a copy of the input file fromthe wavefunction optimization and add the keyword PRINT FORCES ON to the &CPMD

section and then re-run the wavefunction optimization for a series of wavefunction cutoff valuesranging from 10ryd up to 200-500ryd (depending on the available memory. Take note of the valuesof total energy, and force for each cutoff value. Since the convergence behavior can be differentfor each pseudopotential (and each property), we exchange the pseudopotential file as well usingH BHS LDA.psp as replacement. When plotting the relative errors from those calculations, we geta graph similar to the following:

0

10

20

30

40

50

40025015010070503020

Error in %

Cutoff in Ryd

Force MT�

�

�

�

� � � � � � � � � � � � �

�Force BHS

+

+

+

+

+ + + + + + + + + + + + +

+Bond Energy MTBond Energy BHS

Energy and forces have a the same convergence trend, but for the forces we have a not as regularbehavior in the relative error. Next we see that for both pseudopotentials, the relative error in theforce reaches some kind of plateau starting at 50ryd, with the Martins-Troullier (MT) potentialbehaving a bit better than the BHS-type pseudopotential parameters. For problems requiring evenmore accurate forces, a choice of 100-120ryd would be better. Please note, however, that theseerrors only document the convergence behavior of the pseudopotential with respect to the basisset size (= cutoff), they do not include systematic error contributions, e.g. the error from usingDFT altogether in or from the choice of pseudization cutoff radius (which can be chose to makea “softer” pseudopotential at the expense of the transferability of the pseudopotential). Typical

k-point samplingBloch’s theorem:eigenstates in a system under a periodic potential U(r) can be written as

Operators such as the Haminltonian areblock-diagonal by k-point

Important to describe waves and correlation lengths longer than unit cell: solids (esp. conductors / semi-conductors)

What we described before: only the Γ point is sampled (k = 0)Use supercell to compensate for this

Reminder: Block theorem for periodic systems

!i("r;"k) = ei!k·!r ui("r;"k)

ui("r;"k) = ui("r + "L;"k)

! point approximation: "k = "0 is the only point considered in the Brillouin

zone

at the ! point:

!i("r; !) = ui("r;"k) =1!"

!

!G

ci( "G )ei !G·!r

use of a supercell method to compensate !-point approximation

4

k-point samplingExample for silicon bulk

Example: silicon

Si8: Si64:

Point !: ESi = 3.90 A.U. Point !: ESi = 3.943 A.U.

4 points k: ESi = 3.948 A.U. 4 points k: ESi = 3.947 A.U.

32 points k: ESi = 3.951 A.U.

5

Fourier transformsWe use an auxiliary grid in real space for some calculations

Discrete Fourier Transform:

Moving from a grid in real space: x = {0, 1, ..., N} * L/Nto a grid in reciprocal space: G = {–N/2+1, ..., N/2} * 2π/LTransform is inversible if both have the same number of points.

If N is a product of small prime numbers, use FFT (N log N)instead of DFT (N2)

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! !! "#$"

%! !" ""#" ! !

%! !" "#"

&'&

$ ! "(#" ! !

! !! "

#"

%!

$!#)&

"(#" ! !

# ! !!#" +"#$%&'&!,()%"&%!-%*+#,(%-

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6

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%! !""#&'((" *&'( * %! !""++

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Fourier transformsExample of the use of Fourier transforms:calculating the Hartree potential

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'! "! # "

% !# "(% !""

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"

(#$ !""##*(%!" "

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Avoiding PBC?How to describe an isolated systemwith a plane-waves basis set?

Replicate the box, leave lots of space

Or truncate the Coulombic interactionusing a nearest-neighbor convention

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104 User’s Guide CPMD 3.13.2

-1 -0,5 0 0,5 1Reaction Coordinate

0

1

2

3

Ener

gy [

kcal

/ m

ol ]

8 A10 A15 A20 A25 A

0

12

24

36

CPU

tim

e [ h

]

(a) Rcut = 25 Ry, Symm. = 1

-1 -0,5 0 0,5 1Reaction Coordinate

0

1

2

3

Ener

gy [

kcal

/ m

ol ]

8 A10 A15 A20 A25 A

0

12

24

36

CPU

tim

e [ h

]

(b) Rcut = 45 Ry, Symm. = 1

Figure 7: Potential energy profile along reaction pathway with 25 Ry cutoff 7(a), and 45 Ry cutoff7(b) with different box size. The noise in vacuum areas results in the artifact observed for large

box with small cutoff. Functional: BLYP, pseudopotentials: Vanderbilt.

- SURFACE and POLYMER are available and should be safe to use (MORTENSEN is

default for SURFACE and POLYMER)

- If you do an isolated system calculation, your cell has to be cubic, if you use POLYMERcell dimensions b and c have to be equal.

Finally, for many systems using a large enough cell and periodic boundary conditions is also an

option. In general, the computed properties of molecules should be independent of the scheme

used (either pbc or isolated box) except in difficult cases such as charged molecules, where the

calculation in an isolated box is recommended. The PBC calculation is always cheaper for a box of

the same size, so for a neutral molecule such as water molecule you would save time and memory

by not using SYMMETRY 0.

9.4.1 Choosing Supercell Dimensions and Wavefunction Cutoff in Practice

The accuracy of isolated system calculations depends a lot on the correct choice of both the

supercell dimensions and the plane wave cutoff value for given set of pseudopotentials. This may

be accomplished by choosing some relevant parameter of the system under consideration (e.g.

total energy or the energy difference between important configurations) and looking on how does

it depend on the box size and the cutoff value.

Since this is a two-dimensional optimization in the CPMD program parameter space you should

perform the benchmark in a systematic way. One approach, that applies to isolated systems, is to

start with the recommended 3 A spacing between the outermost atoms and the boundary of the

box (when using the default setting of POISSON SOLVER HOCKNEY and run the tests for

different cutoff values. These would be e.g. 20, 25, 30, and 35 Ry for Vanderbilt pseudopotentials

and 60, 70, 80, and 90 Ry for Troullier-Martins. A good choice is the minimum value at which the

control parameter of the system does not change any further. Then by decreasing box dimensions

(in 1 A steps) in each direction try to minimize it by running the benchmarks for different cutoffvalues and observing the control parameter. Simulating lots of empty space does not teach us a lot

but impacts the CPU time significantly. At the same time one has to take into account that the

molecule may change shape or fluctuate around. If the system has no initial rotation or translation,

the molecule should stay in place, but due to numerical errors, this is not always the case. Check

out the keyword SUBTRACT for one method to cope with it.

Another issue is that large vacuum areas tend to produce “noise”, particularly with gradient cor-

rected functionals (see Figure 7). In principle this can be reduced by increasing the wavefunction

cutoff, but the value required to completely remove the noise may be ridiculously large. Alterna-

This potential is also periodic and corresponds to the

nearest-neighbour convention

For a positive charge at x = 0.3, periodically replicated (L = 2) in 1D:

-4 -2 0 2 40

The nearest-image convention can be used for describing an isolated

system; if its extend is less than L, there is no interaction with this

system and its images.

24

PseudopotentialsLength scales of the core and valence electron orbitals:

Rapidly varying functions (core states) require high PW cutoffGoal of cutoff is to “smooth things at the core”

bonding region bonding region

PseudopotentialsCore electrons do not participate in bonding, excitations, conductivityDo not treat the core electrons explicitly any moreInstead of the nuclei Coulombic potential,use a core “nuclear + core electrons” potential,Valence electrons will “feel” this pseudopotential

Eigenvalues/energies of the one-particle levels [ in a.u. ~ 27 eV ]

Deep core qualitatively: n is the principal q. number

Valence different, strongly modified by e-e interactions ~ a few eVs

One-particle eigenvalues for carbon C (Z=6) and copper Cu (Z=29)

2p E_1s= - 0.5 Valence 4s E_4s = - 0.2

2s E_2s= - 0.7 3d E_3d = - 0.5

1s E_1s= - 11.0 3p E_3p = - 3.5

3s E_3s = - 5.0

Core 2p E_2p = -35.

2s E_2s = - 41.

1s E_1s = - 392.

Note: different energy scales in core vs. valence

!"#$%&'()*%+,-%"./0"

Atom with nuclear charge Z:

energies of one-aprticle states

En!"Z2 #$2 n2%,

C (Z=6) Cu (Z=29)

One-electron eigenstates energies (in a.u.)

PseudopotentialsPseudopotentials are nonlocal operators, i.e. they are not simple functions Vps(r)Because they represent core states, they have s, p, d... components(we says “channels”: they represent how they act on each symmetry-adapated projection of the valence electrons)

Dictionary and notations

!"#$%&'()*%+,-%"./0"

In condensed matter physics: pseudopotentials or PPs In quantum chemistry : effective core potentials or ECPs

- radial pseudopotential function for a given l-symmetry channel - outside the core will be just - Z_eff/r = - (Z-Z_core)/r projection operator onto a given |lm> state -> nonlocal!!!

- number of different occupied channels -> number of nonlocal projection operators

vl !r "

#lm$ %lm#

Vps&ion'(lvl !r "(

m#lm$ %lm#'(

l'0

lmax

)vl !r "&vloc*(m#lm$ %lm#+vloc !r "

vloc !r "

lmax

PseudopotentialsThe valence states will thus be different around r = 0Beyond a certain radius (rcore), valence states should be identical to all-electron results Pseudopotentials

Schematically

Ersen Mete DFT in practice : Part II

PseudopotentialsThe valence states will thus be different around r = 0Beyond a certain radius (rcore), valence states should be identical to all-electron results

One-particle construction of PP:

norm-conservation

!"#$%&'()*%+,-%"./0"

!cklm "

- beyond certain r_core the atom and each valence state should look

exactly like if the core electrons were present

- one-particle properties have to be conserved: charge outside

r_core has to be the same as pseudo-charge

- optimize any variational parameters ,

eg, minimize the energy or local energy variance (Umrigar et al,'88)

R#$r1 , r2 , ... , rN%

2s state 2p (both similar)

(pseudo orbital is nodeless:

becomes the ground state)

r_core

Example of PPs for C atom (Lester et al, JCP 2003)

!"#$%&'()*%+,-%"./0"

Pseudopotentials — Pros and cons★ Smaller energy, smaller cutoff★ Smaller number of electrons/wavefunctions to calculate

(think lanthanides)★ Can include scalar relativistic effects

(think lanthanides again)★ Can be extended to include spin-orbit interactions (heavy nuclei)

★ For atomic basis sets: use basis set optimized together with PP

★ If your core is too fat, you describe your system badly(polarization, core relaxation)

★ Choice of core radius★ Core–valence overlap might become larger, e.g. at high pressure★ Cost of Ylm projection operators (evaluated in real space)

Pseudopotentials — accuracyExample: how do you choose a good core for an Fe atom?

Another example: Na = [He] 2s2 2p6 3s1 ; and not Na = [Ne] 3s1

Comparison efficiency/accuracy for Fe atom

!"#$%&'()*%+,-%"./0"

Fe atom -> [Ne] 3s^23p^63d^64s^2= [Ar] 3d^64s^2

all-electron [Ne]-core [Ar]-core

E_HF [au] -1262.444 -123.114 -21.387

E_VMC[au] -1263.20(2) -123.708(2) -21.660(1)

[au] ~ 50 1.5 0.16

efficiency = 0.02 2.1 125

valence errors 0 < 0.1 eV ~ 0.5 eV !!!

Ne-core PPs represent the best compromise for QMC: high accuracy,

acceptable efficiency

!VMC2

1!2Tdecorr

Courtesy L. Mitas, NCSU

Pseudopotentials: summaryThere are many families of pseudopotentials:✓ norm-conserving (conservative choice)✓ ultrasoft (lower energy cutoff)

Keep in mind:✓ pseudopotentials are technical, difficult topics✓ only an auxiliary concept

But...✓ they save a lot of CPU time✓ allow calculations otherwise impossible

They can mess up your calculation for good...(but all might still look unsuspicious)