VASP-lecture-Hybrids functionals LDA GFGA.pdf

VASP: Hybrid functionals
University of Vienna,
Faculty of Physics and Center for Computational Materials Science,
Vienna, Austria

Outline
● Introduction: DFT, HF
● Hybrid functionals
● Assessing hybrid functionals
● Returning to direct optimization

The Many-Body Schrödinger equation
For instance, many-body WF storage demands are prohibitive:
A solution: map onto “one-electron” theory:
5 electrons on a 10×10×10 grid ~ 10 PetaBytes !

Hohenberg-Kohn-Sham DFT
Map onto “one-electron” theory:
Total energy is a functional of the density:
The density is computed using the one-electron orbitals:
The one-electron orbitals are the solutions of the Kohn-Sham equation:
BUT:
In practice: Exchange-correlation functionals are modeled on the uniform electron
gas (Monte Carlo calculations): e.g. the Local Density Approximation (LDA).

Hartree-Fock
Slater determinant
For instance
Pauli exclusion principle:
Roothaan equations:
Orbital dependent
non-local potential
Compare to DFT:
Per definition: no electronic correlation

Need to go beyond Hartree-Fock?
Simplest possible metal: Homogeneouselectron gas
G. Brocks, AQM lecture notes 2004
At the Fermi energy: - DoS -> 0
- vg -> infinity
HF for the homogeneouselectron gas:
Singularity for
FT
Problem:
Solution: introducescreening
HEG exact in PBE0 and HSE03, not B3LYP
J. Paier, , M. Marsman, and G. Kresse, J. Chem. Phys. 127, 024103 2007

Need to go beyond DFT and Hartree-Fock?
(More) accurate treatment of
electronic correlation needed for,
e.g:
• Band gaps (optical properties
• Total energy differenceswith
chemical accuracy
(1 kcal/mol ≈ 40 meV)
• Atomization and formation energies
• Reaction barriers
• Van der Waals interactions

Hartree-Fock/DFT hybrid functionals
Definition: Exchange-correlation functionals that admix a certain amount
of Fock exchange to (a part of) a local or semi-local density functional.
• Present a definite improvement over the (semi)-local density
functional description of the propertiesof molecular systems.
• Some hybrid functionals yield an improved description of structural,
electronic, and thermo-chemical properties of small/medium gap
solid state systems.

Hybrid functionals: PBE0, HSE
PBE0:
HSE03:
non-empirical: justified using the adibatic connection formula.
J. Perdew, M. Ernzerhof, and K. Burke, J. Chem. Phys. 105, 9982 (1996).
Decomposed Coulomb kernel (Savin et al.):
semiemperical: μ is chosen to yield an optimal description of the atomization
energies of the molecues in Pople’s G2-1 test set.
J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003).

Hybrid functionals: B3LYP
B3LYP:
semiempirical: coefficientschosen to reproduceexperimental atomization
energies, electron and proton affinities and ionization potentials of the
molecules in Pople’s G2 test set and their atomic constituents.
A. D. Becke, J. Chem. Phys. 98, 5648 (1993).
M. J. Frisch et al., GAUSSIAN03 Rev. C.02, Gaussian Inc., Wallingford, CT 2004.

Computational aspects
with
• FFT overlap density to reciprocalspace
• Division by Laplace operator and FFT to real space
• Evaluate
N.B.: for all combinations of k, q, n, and m.

Effort:
• Bulk:
• Molecular systems:
Computational aspects: Scaling

Computational aspects: Downsampling
Convergenceof 𝐸𝑋
𝐻𝐹 w.r.t. the BZ sampling use to represent 𝑉𝑋
𝐻𝐹
Example: Al (fcc)
Short(er) range in real space ⟹ Reduced BZ sampling

L
2L
Downsampling (cont.)
1 band
2 k-points
2 bands
1 k-point
Unit cells
1
n
k-points
n
1
⟺

Unit cells
1
n
k-points
n
1
⟺
Downsampling (cont.)
Assume a maximum interaction range 𝑅 = 𝑚𝐿, then a supercell of twice this
size, i.e., 2𝑚 unit cells, correctly incorporatesall interactions using only the
gamma-point.
This is equivalent to the description one obtains using a single unit cell and an
equidistant 2𝑚 sampling of the 1st BZ.
Ergo: reducing the range of the Fock exchange interaction in the HSE functional
allows for the representation of the Fock potential on a coarser grid of k-points.

The HSE Fock exchange energy:
The representation of the correspondingshort-rangeFock potential in
reciprocal space:
Full q-grid:
Downsampled q-grid:
Down-sampling (cont.)

Atomization energies of small molecules
Subset of G2-1 test set: Deviation w.r.t. experiment, in [kcal/mol]
Significant improvementof atomization energies

Lattice constants: PBE, PBE0, HSE03, and B3LYP
M. Marsman, et al.,
J. Phys. Cond. Matt. 20 (2008) 064201
MRE: Mean relative error (%)
MARE: Mean absolute relative error

Solid state systems: Atomization energies

Solid state systems: Atomization energies
• Hybrid functionals overestimate the exchange splitting in d-elements:
leads to an increased stability of the spin-polarized atom.
• B3LYP overestimates 𝐸𝑥𝑐 of localized electrons.
• B3LYP fails to describe “free electron like” behaviour:
LYP underestimatesthe correlation energy in itinerant systems
(does not attain HEG limit!).
J. Paier, M. Marsman, and G. Kresse, J. Chem. Phys. 127, 024103 (2007).

Solid state systems: Heats of formation
M. Marsman, J. Paier, A. Stroppa and G. Kresse, J. Phys. Cond. Matt. 20 (2008) 064201

Solid state systems: Band gaps

Transition-metal monoxides: MnO
Comparison between PBE0 d-projected
DOS (bottom) of both Mn sites, together
with the experimental (top) inverse
photoemission data, and the difference
between on- and off-resonance photo-
Emission spectra.
C. Franchini, V. Bayer, R. Podloucky, J. Paier, and G. Kresse,
Phys. Rev. B 72, 045132 (2005).

Transition-metal monoxides (cont.)
Lattice constant
Magnetic
moment Band gap

CO adsorption on d-metal surfaces
• DFT incorrectly predicts that CO prefersthe
hollow site: P. Feibelman et al.,
J. Phys. Chem. B 105, 4018 (2001)
• The error is relatively large.
Best DFT/PBE calculations:
• 4 layers, 𝑐 2 × 4 , Θ = 0.25 ML, asymmetric
setup, 10 Å vacuum.

CO adsorption on d-metal surfaces (cont. I)

CO adsorption on d-metal surfaces (cont. II)
Hybrid functionals reducethe tendency to stabilize adsorption at the
hollow sites w.r.t. the top site.
Reduced CO 2𝜋∗ ⟷ metal-d interaction
• Improved description of the CO LUMO (2𝜋∗
) w.r.t. the Fermi level
(shifted upwards).
• Downshift of the metal d-band center of gravity in Cu(111).
• But: Overestimation of the metal d-bandwidth.
A. Stroppa, K. Termentzidis, J. Paier, G. Kresse, and J. Hafner, Phys. Rev. B 76, 195440 (2007).
A. Stroppa and G. Kresse, New Journal of Physics 10, 063020 (2008).

Conclusions

Conclusions (cont.)
Solid state systems:
• The PBE0 and HSE hybrid functionals providean improved description of the
structural (lattice constants and bulk moduli) and electronic (band gap)
properties of systems with a small/medium sized band gap.
• PBE0, HSE, and B3LYP atomization energies are in overall worseagreement
with experiment than those obtained using the semi-local PBE density
functional, in case of B3LYP even drastically so.
This is mainly due to a worsedescription of metallic systems.
• CO adsorption on d-metal (111) surfaces: hybrid functionals reduce the
tendency to stabilize adsorption at the hollow sites w.r.t. the top site.

INCAR Tags and links
Hybrid functionals and Hartree-Fock
The VASP manual chapter on hybrid functionals and Hartree-Fock.
LHFCALC Switch on Hybrid and Hartree-Fock type calculations.
HFSCREEN Specifies the range separating parameter in HSE functionals.
ENCUTFOCK Specifies the FFT grids used in the HF routines.
GGA-tag Override the type of density functional specified in the POTCAR.
NKRED, NKREDX, NKREDY, NKREDZ
Down-sampling the k-point mesh in the representation of the Fock potential.
AEXX, AGGAX, AGGAC and ALDAC
The fractions of Fock-exchange, gradient corrections to the exchange
and correlation, and the fraction of LDA correlation.
INCAR settings:
PBE0: LHFCALC = .TRUE.
HSE06:† LHFCALC = .TRUE. ; HFSCREEN = 0.2 (with PBE POTCAR, or GGA = PE).
B3LYP: LHFCALC = .TRUE. ; GGA = B3 ; AEXX = 0.2 ; AGGAX = 0.72 ; AGGAC = 0.81 ; ALDAC = 0.19
Hartree-Fock: LHFCALC = .TRUE. ; AEXX = 1.0 ; ALDAC = 0.0 ; AGGAC = 0
†A. V. Krukau , O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J. Chem. Phys. 125, 224106 (2006).

Band structure with VASP
(A) Standard procedure (DFT)
① Create SC charge density (CHGCAR) using a uniform k-point mesh
(for instance 8x8x8)
② Perform a non-SC calculation (ICHARG=11) using the pre-converged
charge density (CHGCAR) from step 1, and a new KPOINTS file with
the suitable high-symmetry k-points and number of intersections.
KPOINTS
10 k-points along G-X
10 ! 10 intersections
Line-mode
cart
0 0 0 ! G
0 0 1 ! X
③ plot with p4v

(B) Additional zero-weight k-points procedure (HF)
① Standard HF/DFT+HF using a conventional (uniform) k-point mesh
② (i) cp IBZKPT KPOINTS (explicit list of k-points used in step (1)
(ii) add desired k-points with zero weight:
KPOINTS
Automatically generated mesh
18
Reciprocal lattice
0.00000000000000 0.00000000000000 0.00000000000000 1
0.25000000000000 0.00000000000000 0.00000000000000 8
0.50000000000000 0.00000000000000 0.00000000000000 4
0.25000000000000 0.25000000000000 0.00000000000000 6
0.50000000000000 0.25000000000000 0.00000000000000 24
-0.25000000000000 0.25000000000000 0.00000000000000 12
0.50000000000000 0.50000000000000 0.00000000000000 3
-0.25000000000000 0.50000000000000 0.25000000000000 6
0.00000000 0.00000000 0.00000000 0.000
0.00000000 0.05555556 0.05555556 0.000
0.00000000 0.11111111 0.11111111 0.000
0.00000000 0.16666667 0.16666667 0.000
0.00000000 0.22222222 0.22222222 0.000
0.00000000 0.27777778 0.27777778 0.000
0.00000000 0.33333333 0.33333333 0.000
0.00000000 0.38888889 0.38888889 0.000
0.00000000 0.44444444 0.44444444 0.000
0.00000000 0.50000000 0.50000000 0.000
③ Plot with p4v (remove k-points of uniform mesh)

(C) Interpolation using MLWFs via VASP2WANNIER90 (DFT, HF, DFT+HF, GW)
① Standard DFT calculation using a conventional (uniform) k-point mesh. Include TAG
LWANNIER90=.TRUE. to generate a default wannier90.win file.
② Suitably modify wannier90.win according to the instructions given in the WANNIER90
manual, and run VASP again (DFT/HF/HF+DFT/GW) in order to generate the inputs file
for the wannier90 interpolation (wannier90.eig, wannier90.amn, wannier90.mmn)
③ Run WANNIER90 (wannier90.x wannier90) in order to generate the MLWFs. Input files:
wannier90.win, wannier90.eig, wannier90.amn, and wannier90.mmn.
④ Run WANNIER90 in Band structure mode (add band structure flags in wannier90.win).
Input files: wannier90.win, wannier90.chk, and wannier90.eig.
⑤ Plot using xmgrace or gnuplot.
VASP2WANNIER90:
C. Franchini, R. Kovácik, M. Marsman, S.S. Murthy, J. He, C. Ederer, G. Kresse, J. Phys.: Condens. Matter 24, 235602 (2012).
(Compile with -DVASP2WANNIER90, see http://cms.mpi.univie.ac.at/wiki/index.php/LWANNIER90)

Wannier90.win for Si
num_wann=8
num_bands=8
Begin Projections
Si:sp3
End Projections
dis_froz_max=9
dis_num_iter=1000
guiding_centres=true
# BS flags restart = plot
bands_plot = true
begin kpoint_path
G 0.00000 0.00000 0.0000 X 0.50000 0.00000 0.5000
end kpoint_path
bands_num_points 40
bands_plot_format gnuplot xmgrace
begin unit_cell_cart
2.7150000 2.7150000 0.0000000
0.0000000 2.7150000 2.7150000
2.7150000 0.0000000 2.7150000
end unit_cell_cart
begin atoms_cart
Si 0.0000000 0.0000000 0.0000000
Si 1.3575000 1.3575000 1.3575000
end atoms_cart
mp_grid = 4 4 4
begin kpoints
0.0000000 0.0000000 0.0000000
0.2500000 0.0000000 0.0000000
...
-0.2500000 0.2500000 -0.5000000
end kpoints

Return to direct optimization: Why?
Pure DFT functionals depend only in the density
DFT-Hartree-Fock hybrid functionals depend explicitly on the orbitals
So density-mixing will not always work (reliably).
Unfortunately we know direct optimization schemes are proneto charge sloshing
for metallic and small-gap systems.

Mixed scheme
• The gradient of the wave functionsis given by
where
• A search direction towardsthe groundstatew.r.t. unitary transformations
between the orbitals within the subspace spanned by the orbitals can be
found fromperturbation theory:
But this is exactly the term that is proneto charge sloshing!
• Solution: Use density mixing to determine the optimal unitary transformation
matrix 𝐔𝑛𝑚.

Optimal subspace rotation
• Define a Hamilton matrix:
where
• Determine the subspace rotation matrix V that diagonalizes the Hamiltonian.
• Recompute the (partial) occupancies ⟶ {f’}
• The transformed orbitals:
and partial occupancies {f’} define a new charge density 𝜌′
• Mix 𝜌 and 𝜌′
• and iterate the above until a stable point is found ⟶ 𝜌𝑠𝑐
• The optimal subspace rotation is then given by :
where
• N.B.: we do not update the orbital dependentpart of the Hamiltonian

The full mixed scheme
The iterative optimization of the orbitals cycles through the following steps:
1. Construct the Hamiltonian H, from the currentorbitals and partial
occupancies, and calculate 𝐇ȁ ۧ
𝝍
2. Inner loop: determine the self-consistentHamiltonian 𝐇𝑠𝑐, defining
the preconditioned direction for the optimal subspace rotation U.
3. Minimization along the preconditioned search direction defined by
and by U, and by a gradient acting on the partial occupancies.
For instance by means of a conjugate-gradientalgorithm.
This loop is repeated until the change in the freeenergy from one iteration to
the next dropsbelow the required convergencethreshold ∆𝐸thr (usually 10−4
eV).

Example: fcc Fe
The convergence behaviour of HSE03 calculations using the improved direct optimization procedure (solid lines)
and a standard conjugate gradient algorithm (dotted lines). Calculations on single, double, four times repeated cells
are marked with circles, diamonds, and squares, respectively.

ALGO The INCAR tag that sets which algorithm is used for the electronic
minimization.
IALGO and LDIAG Same as above, but more to choose from(ALGO is the preferred
tag). For the direct optimizers (ALGO =All | Damped) LDIAG = .TRUE.
switches on the density mixer in the determination of the subspace rotation
matrix.
Mixing tags The VASP manual chapter on the settings for the density mixer.
TIME Time step in the direct optimization method ALGO = Damped, and trial time
step for the conjugate gradient direct optimizer ALGO = All.
Electronic optimization Lecture from the VASP workshop in Vienna (2003).

Hartree-Fock within the PAW formalism
In principle we would want to follow the original scheme:
• Solve the non-spinpolarized spherical atom within the
(scalar relativistic) Hartree-Fock approximation.
• Compute all-electron partial waves.
• Pseudize .. etc etc …
Unfortunately this is altready problematic in the second step, solving
with
⟹ numerically unstable for unbound partial waves with 𝜖 < 0.

The HF-PAW method (cont.)
A pragmatic solution:
• Keep the partial waves and projector functions( ۧ
ȁ𝜙𝑖 , ൿ
ȁ ෨
𝜙𝑖 , and ۧ
ȁ ෤
𝑝𝑖 ) obtained
with DFT.
• Orthogonalize the all-electron partial waves ۧ
ȁ𝜙𝑖 with respect to the Hartree-
Fock corestates ۧ
ȁ𝜙𝑖 → ۧ
ȁ𝜙′𝑖 .
• Compute the PAW strength parameters and augmentation charges using ۧ
ȁ𝜙𝑖
and the Hartree-Fock core states:
and
The most important step in this scheme is:

Bulk equilibrium volumes for GaAs and Si:
3s and 3p levels in the spherical Si atom.
All energies in eV.

The remaining discrepancies are related to the factthat the one-center expansions of
the wave functions inside the PAW spheres are not complete enough (particularly
further away from the nucleus)
This causes problems for at least two reasons:
1. The core-valence exchange interaction is computed between the Hartree-Fock core
states 𝜓𝑐
HF
and the one-center expansion of 𝜓 only.
2. The point where the Hartree potentials arising from the all-electron and pseudo ionic
cores match moves into regions where the one-center expansions are not good. This is
bound to happen to some degree since we adjust:
and leave fixed.

LRHFATM = .TRUE. Changes the DFT AE core charge density to a Hartree-
Fock one.

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