Pseudopotentials:
From Ultrahard to Ultrasoft
Shaw and Harrison, Phys Rev. 163, 604 (1967).
Johan M. Carlsson
Theory department
Fritz-Haber-Institut der Max-Planck-Gesellschaft
Faradayweg 4-6, D14195 Berlin
email: johanc@fhi-berlin.mpg.de
Outline
• What is a pseudo potential?
• Historical pseudo potentials
• Norm conserving pseudo potentials
• Ultrasoft pseudo potentials
• How to generate an (ultrasoft) pseudo potential
• Testing your pseudo potential
What is a pseudo potential?
The solid state Hamiltonian contains contributions from the
ion-ion, ion-electron and electron-electron interaction:
H=Tkin+VII +VeI + VH + VXC
The Born-Oppenheimer approximation states that we can
consider the motion of the electrons and ions independently.
The electron-ion interaction can then be seen as the
electrons are interacting with the rigid ions.
The valence electrons determine the chemical bonding of
atoms into molecules and crystals.
The atom
2p
E (eV)
2s
Valence electrons
Vpot
1s
Ze2
=−
4πε0r
Core electrons
r (Å)
Core electrons are tightly bound to the core.
Valence electrons experience the core as screened by the core
electrons, i.e. an effect potential -> pseudo potential
What is a pseudo potential?
The core electrons
screen the nucleus, such
that the valence
electrons only
experience an effective
pseudo potential, which
is much smoother than
the bare nuclear
Coulumb potential.
I.e. the pseudo potential
mimics how the valence
electrons experience the
world.
VPS
VAE
r (au)
Why do we use pseudo potentials?
Most of the chemistry and physics of materials is
determined by the valence electrons.
Advantages:
More efficient calculations: Smoother potentials decrease
the number of expansion coefficients for the Kohn-Sham
orbitals
Disadvantages:
The wavefunctions inside the pseudo potential region
deviate from the true behaviour.
Less degrees of freedom, since the core electrons are
frozen into the pseudo potential
History of Pseudo potentials:
Hard core pseudo potentials
Shaw and Harrison, Phys Rev. 163, 604 (1967).
Early attempts: Cut off the Coulomb potential and insert a
repulsive core.
Ok for analytical calculations, but bad if you are expanding
your wavefunctions in a basis set.
Empirical pseudo potentials
• Generated by fitting to experimental data in
order to reproduce particular properties, such
as:
• Band structure
• Bulk properties
• Work well for the systems they are designed
for.
• But they cannot be used for other systems:
They are not transferable
Ab-initio pseudo potentials
Generate the pseudo potentials based on a ab-initio allelectron atomic calculation. The goal is to have a
consistent procedure where the pseudo potential
generation and the solid state calculation keep the same
level of accuracy.
Use physical criteria for the generation of the pseudo
potentials to maintain as much of the properties of the
all-electron calculation, such that the wave function of the
valence electrons are unchanged by the introduction of
the pseudo potential.
Ab-initio pseudo potentials
1)
The pseudo wave functions should be identical to the all-electron wave
functions outside the cut-off radius.
ψ AE (r) = φ PsP (r),r > rc
2)
The eigenvalues should be conserved.
∈ iAE =∈ iPsP
3)
The total charge of each pseudo wave function should equal the charge of
the all-electron wave function => Norm-conservation criteria:
rc
∫
0
4)
ψiAE (r)
2
rc
dr = ∫
φiPsP (r)
2
dr
0
The scattering properties of the pseudo potential should be conserved.
d
d
ln{ψ AE (r)}
= ln{φ PsP (r)}
r=rlog D
r=rlog D
dr
dr
D. R. Hamann, M. Schlüter, and C. Chang, Phys. Rev. Lett. 43, 1494 (1979).
Norm conservation
Gauss law:
n(r′)dr′
V (r ) = ∫
r − r′
implies that the potential outside a
charge distribution only depends on
the charge inside the distribution, but
not the shape.
I.e. the electrons outside the core
region would not notice if the charge
distribution is replaced by a smoother
one inside the core region as long as
the total charge is not changed.
The norm conserving pseudo
potential
The norm conserving pseudo potentials are usually divided into
a local potential, that can be described by a radial function, Vloc ( r ) and
a non-local part.
VNL = ∑ χ l Vl χ l
l
The non-local part is the deviation from the
all-electron potential inside the core region.
Carbon
Vloc
The projectors for each angular momenta l,
are defined
VAE
⎛ h 2∇ 2
⎞
+ Vloc (r )− ∈i ⎟⎟ φi ps
χ l = ⎜⎜ −
⎝ 2m
⎠
The total pseudo potential can be written as:
r (au)
V ps = Vloc (r ) + ∑ χ l Vl χ l
l
L. Kleinmann and D. M. Bylander, Phys. Rev. Lett. 48, 1425 (1982).
Non-linear core-correction:
The poor mans all-electron method
The exchange-correlation energy is a non-linear functional of the electron
density. There may be regions where the core and valence charge overlap, such
that the electron density is higher than just the valence density. The exchangecorrelation energy for the valence electrons is then underestimated.
ρ (r )
rψ (r )
Solution: The non-linear core correction: Add a fraction of the core charge to the
valence charge in the region where the core and valence charge overlap.
r (au)
r (au)
S. G. Louie, S. Froyen, and M. L. Cohen, Phys. Rev. B 26, 1738 (1982).
Ultrasoft pseudo potentials
D. Vanderbildt, Phys. Rev. B 41, 7892 (1990):
ψAE4s
Release the norm conservation criteria to obtain
smoother pseudo wave functions. This can be
done by splitting the pseudo wave functions into
two parts:
Ultrasoft valence wave function that do not fulfill
the norm conservation criteria:
φUS4s
φιUS(r)
Plus a core augmentation charge (charge deficit in
the core region):
*
*
Qnm (r) = ψ AE n (r)ψ AE m (r) − φ US n (r)φ US m (r)
Q4s (r)
Zinc 4s
The ultrasoft pseudo potential
The ultrasoft pseudopotential takes the form:
0
V US = Vloc (r) + ∑ Dnm
βnI βmI
nmI
The β-projector functions are defined through:
βn = ∑
m
χm
χ m φn
And they are strictly localized inside the cut-off region for the wave functions
since the χ-functions are defined through
χ n = (∈ n −Tˆ − Vloc ) φ n
And D is defined by
0
Dnm
= φ n χ m + ∈ m qnm
The scattering properties of the pseudo potential can be improved by using
more than one β-projector function per angular momentum channel.
Ultrasoft charge density
Necessary to use a generalized eigenvalue formalism.
Introduce the overlap operator S:
Sˆ = 1 + ∑ qnm βnI βmI
nmI
Where
rc
qnm = ∫ Qnm (r )dr
0
The charge density is now defined through:
2
I
n(r) = ∑ φi* (r)Sˆ φi (r) =∑ [φi (r) + ∑ Qnm
(r) φi βnI βmI φi
i
i
nmI
]
Kohn-Sham formalism
The ultrasoft pseudo potential
0
βnI βmI
V US = Vloc (r) + ∑ Dnm
nmI
is determined by Vloc(r),
D0
nm,
Qnm(r) and βn (r).
The Kohn-Sham equations take the form:
H φiUS =∈ i Sˆ φiUS
Where H has an additional non-local term:
−h 2∇ 2
I
+V eff(r) + ∑ Dnm
βnI βmI
H=
2m
nmI
And the D-function needs to be updated during the self consistent calculation
according to:
I
0
I
Dnm
= Dnm
+ ∫ Veff (r )Qnm
dr
The major approximation is that the augmentation charge is not updated in the
self-consistent calculation.
Ultrasoft vs Norm conserving
pseudo potentials
Disadvantage: Additional terms in the Kohn-Sham formalism
More operations per computational cycle.
Advantage:
Less plane waves required for the ultrasoft pseudo wavefunctions
The reduction of the basis set dominate for larger systems.
Generating
Ultrasoft pseudo potentials
Atomic all-electron calculation Õ wavefunctions ψAE and eigenvalues ∈AE
Choose the cut-off radius and construct the pseudo wavefunctions: φUS
Zinc
3p
4s
rψ
3d
r (au)
The augmentation charge
The Q-functions for the
augmentation charge is
generated as the difference
between the all-electron and the
pseudo wave functions:
*
Zinc 4s
ψAE4s
φUS4s
-
*
Qnm (r) = ψ AE n (r)ψ AE m (r) − φ US n (r)φ US m (r)
=
Q4s (r)
The Beta-projector functions
The Beta-projector functions:
χm
βn = ∑
m χ m φn
Define the non-local part of the pseudo potential.
Wave functions
Beta functions
φp1
βs1
φp2
φs1
φs2
r (au)
βp2
r (au)
βs2
βp1
The Beta-projector functions
Select the number of Beta-projector functions and the eigenvalues to optimize
the scattering properties of the pseudo potential
Zinc:
One Beta per angular momentum channel
Zinc:
Two Betas per angular momentum channel
d
d
p
p
s
s
Energy (au)
Energy (au)
Adding a second Beta function per angular momentum channel
improves the scattering properties over a longer energy interval.
The local potential
The local potential Vloc, is finally obtained by inverting the Schrödinger equation.
Choose the cut-off radius and the angular momentum for the local potential.
Carbon
Vloc
VAE
r (au)
The local potential is important for the scattering of higher angular momentum
components, that are not explicitly taken care of by the Beta-projectors.
Fine tuning the pseudo potential
The parts of the pseudo potential is finally smoothened.
This process is controled by cut-off parameters pc and qc.
Be careful not to over optimize the smoothening process
to obtain a very soft potential. This may lead to that very
high fourier component occur.
Is it necessary to use the non-linear core correction?
Real space Q-functions
Fourier transformed Q-functions
qc
r (au)
q (au)
Testing the pseudo potentials
First step: Checking with the generator program:
Do the eigenvalues agree with the all-electron eigenvalues?
Are there any ghost states?
The scattering properties: Do the logarithmic derivatives of the pseudo
potential agree with the all-electron potential?
Carbon
f
d
d
ln{φ PsP (r)}
r= rlog D
dr
s
p
Energy (au)
Convergence Test
ΔE (eV/atom)
C_00PBEOpt.uspcc
C_00PBE.usp
Ecut (eV)
Check that the pseudo potential really converges.
Molecules
Test the bond length, binding energy and vibration frequency.
Ebind(eV/molecule)
rC_00PBEOpt.uspcc
rC_00PBE.usp
C_00PBE.usp
C_00PBEOpt.uspcc
O2 molecule
Ebind
d(Å) (eV)
Vib
freq
(cm-1)
C_00PBE.usp
1.24 5.67
1602
C_00PBEOpt.uspcc
1.22 6.05
1576
All-electron*
1.22 6.24
Experiment*
1.21 5.12
d (Å)
Be careful to not choose a too large cut-off radius.
*Peier et. al., J.Chem.Phys.122, 234102 (2005).
Optimize the pseudo potential to reproduce the all-electron value,
not the experimental result!
Bulk properties
a0=equilibrium lattice parameter and V0 equilibrium volume
Ecoh=cohesive energy
B0=Bulk modulus= V d2E
dV2
The bulk properties can be determined using the Murnaghan
equation of state:
B′
⎡
⎛V0 ⎞ ⎤
B0V
V0
E(V ) = E0 +
⎢B′(1− ) −1+ ⎜ ⎟ ⎥
⎝V ⎠ ⎦
V
B′( B′ −1) ⎣
E0=total energy at equilibrium lattice constant, B0=Bulk modulus,
B’ =first derivative of B0 with respect to pressure
Murnaghan, Proc. Nat. Acad. Sci. USA 30, 244 (1944)
Bulk properties of diamond
Test the lattice parameter, cohesive energy and Bulk modulus.
B′
⎡
⎛V0 ⎞ ⎤
B0V
V0
E(V ) = E0 +
⎢B′(1− ) −1+ ⎜ ⎟ ⎥
⎝V ⎠ ⎦
V
B′( B′ −1) ⎣
Ecoh(eV/molecule)
a(Å)
Ecoh
(eV)
C_00PBE.usp
C_00PBEOpt.uspcc
a (Å)
Bulk
modulus
(GPa)
C_00PBE.usp
3.567 7.58
420
C_00PBEOpt.u
spcc
3.571 7.66
423
All-electron*
3.572
433
Experiment+
3.57
7.37
*Kunc et al., Hi. Pres. Res. 24, 101 (2004).
+C. Kittel, Introduction to Solid State Physics.
442
Summary
The pseudo potentials mimics how the valence electrons
experience the screened core potential.
The pseudo potential makes the calculations more efficient.
Before you start a calculation:
TEST YOUR PSEUDO POTENTIAL!!!!!
Being able to generate your pseudo potantial gives you full
control over your calculations
References
D. R. Hamann, M. Schlüter, and C. Chang, Phys. Rev.
Lett. 43, 1494 (1979).
L. Kleinmann and D. M. Bylander, Phys. Rev. Lett. 48,
1425 (1982).
S. G. Louie, S. Froyen, and M. L. Cohen, Phys. Rev. B 26,
1738 (1982).
D. Vanderbildt, Phys. Rev. B 41, 7892 (1990).
K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D.
Vanderbildt, Phys. Rev. B 47, 10142 (1993).