Electronic Structure Calculations, Density Functional

Transcription

Tutoriel BigDFT
G RENOBLE
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Electronic Structure Calculations,
Density Functional Theory and
its Modern Implementations
Solving
Basis sets
Plane waves
Thierry Deutsch
Direct Minimisation
Diagonalisation
Gaussians
Real space
L_Sim - CEA Grenoble
Conclusion
19 October 2011
Laboratoire de Simulation Atomistique
http://inac.cea.fr/L_Sim
T. Deutsch
Outline
1
Review of Atomistic calculations
Introduction
Atomistic Methods
2
Density Functional Theory (quick view)
Kohn-Sham formalism
Pseudopotential
Solving Kohn-Sham equations
3
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space basis sets
4
Conclusion
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Review
Introduction
Atomistic Methods
Review of Atomistic simulations
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Review of Atomistic Simulations
Theory – Experiment – Simulation
Hardware – Computers
Algorithms
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
Atomistic Simulations
Force fields (interatomic potentials)
Tight Binding Methods
Hartree-Fock
Density Functional Theory
Configuration interactions
Quantum Monte-Carlo
T. Deutsch
Review of Atomistic Simulations
Theory – Experiment – Simulation
Hardware – Computers
Algorithms
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
Force fields (interatomic potentials)
Hartree-Fock
Configuration interactions
Quantum Monte-Carlo
T. Deutsch
Algorithms
Search of efficient algorithms:
Fast Fourier Transform (FFT) : TCPU ∝ n log (n) ;
Order Natoms methods
Review
Introduction
Atomistic Methods
Electrostatic (Hartree) potential of an electronic density ρ(r ):
DFT
Kohn-Sham
Pseudopotential
ρ(r 0 ) 0
dr
V |r − r 0 |
Z
V (r ) =
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
Mesh: n = 100 × 100 × 100 = 106
Direct: n2 = 1012 : ∝ 100 s
FFT: n log (n) ∝ 106 : ∝ 0.1 ms
T. Deutsch
Algorithms
Search of efficient algorithms:
Fast Fourier Transform (FFT) : TCPU ∝ n log (n) ;
Order Natoms methods
Review
Introduction
Atomistic Methods
Electrostatic (Hartree) potential of an electronic density ρ(r ):
DFT
Kohn-Sham
Pseudopotential
ρ(r 0 ) 0
dr
V |r − r 0 |
Z
V (r ) =
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
Mesh: n = 100 × 100 × 100 = 106
Direct: n2 = 1012 : ∝ 100 s
FFT: n log (n) ∝ 106 : ∝ 0.1 ms
T. Deutsch
Two intrinsic difficulties for numerical atomistic simulations,
related to complexity:
Interactions The way that atoms interact is known:
i } ∂Ψ
∂t = H Ψ
Review
Introduction
Atomistic Methods
H ψ = E0 ψ
Exploration of the configuration space
(ART, minima hopping, spline search)
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
i } ∂Ψ
∂t = H Ψ
Review
Introduction
Atomistic Methods
H ψ = E0 ψ
DFT
Kohn-Sham
E
pot
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
R1
Gaussians
Real space
Conclusion
T. Deutsch
i } ∂Ψ
∂t = H Ψ
Review
Introduction
Atomistic Methods
H ψ = E0 ψ
DFT
Kohn-Sham
E
pot
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
R1
Gaussians
Real space
Conclusion
R2
T. Deutsch
i } ∂Ψ
∂t = H Ψ
Review
Introduction
Atomistic Methods
H ψ = E0 ψ
DFT
Kohn-Sham
E
pot
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
R1
Gaussians
Real space
R3
Conclusion
R2
T. Deutsch
i } ∂Ψ
∂t = H Ψ
Review
Introduction
Atomistic Methods
H ψ = E0 ψ
DFT
Kohn-Sham
E
pot
Pseudopotential
R1000
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
R1
Gaussians
Real space
Conclusion
R41
R3
Rn
R2
T. Deutsch
Goals of Atomistic Methods
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
Criteria
Goals
General (Generality)
all atoms of the periodic table
all kinds of bonds (transition state)
Precision
geometry
atomic positions
lengths of bonds
angles of bonds
energy of bonds
vibration frequencies
dipolar moments
UV and visible spectra
photo-emission
magnetic moments
System size
Time scale
±0.001 angström
±0.001 angström
±1 degree
0.001 eV/atom (0.1 kJ/mol)
1 GHz
0.01 Debye (1 Debye = 3.336 × 10−30 C.m
0.01 eV
0.01 eV
0.01 µB
104 atoms for ab initio methods
106 atoms for parametrized methods
1s
T. Deutsch
Criteria for classification
3 criteria
Review
Introduction
Atomistic Methods
P
G
1→ Generality
(elements, alloys)
DFT
Kohn-Sham
2→ Precision (∆r , ∆E)
Pseudopotential
Solving
3→ System size (N, ∆t)
Basis sets
Plane waves
S
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Quantum Mechanics
Schrödinger equation for N electrons in an
external potential Vext (r ):
antisymmetric Ψ (r1 , r2 , . . . , rN ) = −Ψ (r2 , r1 , . . . , rN )
Review
~2
Introduction
Atomistic Methods
DFT
H
= −
2m
Kohn-Sham
N
∑ ∇2ri +
i
1
∑
1
4πε0 i =
6 j |ri − rj |
N
+ ∑ Vext (ri )
i
Pseudopotential
Solving
Basis sets
H Ψ (r1 , r2 , . . . , rN ) = E Ψ (r1 , r2 , . . . , rN )
Plane waves
Direct Minimisation
Diagonalisation
Ion-electron interactions:
Gaussians
Real space
Conclusion
Vext (r ) = −
1
∑
Zα
4πε0 α |Rα − r |
T. Deutsch
Hartree-Fock and Post Hartree-Fock
We build the most simple antisymmetric wavefunction for N
electrons from N wavefunctions at 1 electron (Slater determinant):
φ1 (x1 )
φ2 (x1 )
Φ (x1 , . . . , xN ) = ..
.
φN (x1 )
Review
Introduction
...
...
..
.
...
..
.
φN (xN ) φ1 (xN )
φ2 (xN )
The Hamiltonian are exact.
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Many Slater determinants: Configuration interaction, MP2, coupled
clusters, . . .
P
G
P
G
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
S
S
T. Deutsch
Hartree-Fock and Post Hartree-Fock
We build the most simple antisymmetric wavefunction for N
electrons from N wavefunctions at 1 electron (Slater determinant):
φ1 (x1 )
φ2 (x1 )
Φ (x1 , . . . , xN ) = ..
.
φN (x1 )
Review
Introduction
...
...
..
.
...
..
.
φN (xN ) φ1 (xN )
φ2 (xN )
The Hamiltonian are exact.
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Many Slater determinants: Configuration interaction, MP2, coupled
clusters, . . .
P
G
P
G
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
S
S
T. Deutsch
Quantum Monte Carlo
Very complicated wavefunctions for N electrons (Jastrow factor):
φ1 (x1 )
..
Ψ (x1 , . . . , xN ) = .
φN (x1 )
Review
...
..
.
...
φ1 (xN ) ..
∏ J (rij )
.
φN (xN ) i 6=j
Introduction
Atomistic Methods
DFT
Integration of the total energy: E =
Kohn-Sham
< Ψ|H |Ψ >
< Ψ|Ψ >
Use of Monte-Carlo methods for the integration.
Pseudopotential
Solving
Basis sets
P
G
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
S
Diffusive Monte Carlo
(fixed node, sign problem)
T. Deutsch
Density Functional Theory (DFT)
Theorem: The total energy of the ground state in an external
potential Vext is a functional of the electronic density ρ(r ):
Z
Vext (r )ρ(r )dr
E [ρ(r )] = F [ρ(r )] +
r
Review
Variational principles: E [ρ0 (r )] ≤ E [ρ(r )]
∀ ρ(r )
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Kohn-Sham Equation: (approximation)
Having a one-electron hamiltonian in a mean field.
P
G
1 2
− 2 ∇ + Veff (r ) ψi = εi ψi
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
occupied
ρ(r ) =
∑
|ψi (r )|2
S
i
T. Deutsch
Density Functional Theory (DFT)
Theorem: The total energy of the ground state in an external
potential Vext is a functional of the electronic density ρ(r ):
Z
Vext (r )ρ(r )dr
E [ρ(r )] = F [ρ(r )] +
r
Review
Variational principles: E [ρ0 (r )] ≤ E [ρ(r )]
∀ ρ(r )
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Kohn-Sham Equation: (approximation)
P
G
1 2
− 2 ∇ + Veff (r ) ψi = εi ψi
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
occupied
ρ(r ) =
∑
|ψi (r )|2
S
i
T. Deutsch
Semi-empirical quantum methods
We solve a one-electron parametrized hamiltonian
(Slater-Koster parameter):
H | ψi >= εi | ψi >
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
occupied
Band term:
Eband =
εi
P
G
ions
Plane waves
Direct Minimisation
∑
i
Basis sets
Repulsive term:
Erep =
Diagonalisation
∑ f (RI,J )
I ,J
Gaussians
Real space
Conclusion
Etotal = Erep + Eband
S
Mainly used for electronic properties (106 atoms) or transport
T. Deutsch
Choice of Atomistic Methods
Chemistry
P
G
Review
Introduction
Atomistic Methods
Compromise between
accuracy, system size and
CPU time.
DFT
Kohn-Sham
Physics
S
S
P
G
P
G
P
G
Force Fields
Pseudopotential
Solving
Basis sets
Plane waves
Tight Binding
S
Hartree-Fock
S
Direct Minimisation
Diagonalisation
Ab initio (DFT)
Real space
Conclusion
Configuration
Interactions
Quantum Monte-Carlo
P
G
Gaussians
S
P
G
S
T. Deutsch
Chemistry
P
G
Review
Introduction
Atomistic Methods
Compromise between
CPU time.
DFT
Kohn-Sham
Physics
S
S
P
G
P
G
P
G
Force Fields
Pseudopotential
Solving
Basis sets
Plane waves
Tight Binding
S
Hartree-Fock
S
Direct Minimisation
Diagonalisation
Ab initio (DFT)
Real space
Conclusion
Configuration
Interactions
Quantum Monte-Carlo
P
G
Gaussians
S
P
G
S
T. Deutsch
Chemistry
P
G
Review
Introduction
Atomistic Methods
Compromise between
CPU time.
DFT
Kohn-Sham
Physics
S
S
P
G
P
G
P
G
Force Fields
Pseudopotential
Solving
Basis sets
Plane waves
Tight Binding
S
Hartree-Fock
S
Direct Minimisation
Diagonalisation
Ab initio (DFT)
Real space
Conclusion
Configuration
Interactions
Quantum Monte-Carlo
P
G
Gaussians
S
P
G
S
T. Deutsch
Chemistry
P
G
Review
Introduction
Atomistic Methods
Compromise between
CPU time.
DFT
Kohn-Sham
Physics
S
S
P
G
P
G
P
G
Force Fields
Pseudopotential
Solving
Basis sets
Plane waves
Tight Binding
S
Hartree-Fock
S
Direct Minimisation
Diagonalisation
Ab initio (DFT)
Real space
Conclusion
Configuration
Interactions
Quantum Monte-Carlo
P
G
Gaussians
S
P
G
S
T. Deutsch
Chemistry
P
G
Review
Introduction
Atomistic Methods
Compromise between
CPU time.
DFT
Kohn-Sham
Physics
S
S
P
G
P
G
P
G
Force Fields
Pseudopotential
Solving
Basis sets
Plane waves
Tight Binding
S
Hartree-Fock
S
Direct Minimisation
Diagonalisation
Ab initio (DFT)
Real space
Conclusion
Configuration
Interactions
Quantum Monte-Carlo
P
G
Gaussians
S
P
G
S
T. Deutsch
Chemistry
P
G
Review
Introduction
Atomistic Methods
Compromise between
CPU time.
DFT
Kohn-Sham
Physics
S
S
P
G
P
G
P
G
Force Fields
Pseudopotential
Solving
Basis sets
Plane waves
Tight Binding
S
Hartree-Fock
S
Direct Minimisation
Diagonalisation
Ab initio (DFT)
Real space
Conclusion
Configuration
Interactions
Quantum Monte-Carlo
P
G
Gaussians
S
P
G
S
T. Deutsch
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham equations
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Ab initio calculations with DFT
Several advantages
4 Ab initio: No
adjustable parameters
4 DFT: Quantum
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
mechanical
(fundamental)
treatment
Main limitations
8 Approximated approach
8 Requires high computer
power, limited to few
hundreds atoms in most
cases
Wide range of applications: nanoscience, biology, materials
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Quantum Mechanics
Schrödinger equation for N electrons in an
external potential Vext (r ):
antisymmetric Ψ (r1 , r2 , . . . , rN ) = −Ψ (r2 , r1 , . . . , rN )
Review
~2
Introduction
Atomistic Methods
DFT
H
= −
2m
Kohn-Sham
N
∑ ∇2ri +
i
1
∑
1
4πε0 i =
6 j |ri − rj |
N
+ ∑ Vext (ri )
i
Pseudopotential
Solving
Basis sets
H Ψ (r1 , r2 , . . . , rN ) = E Ψ (r1 , r2 , . . . , rN )
Plane waves
Direct Minimisation
Diagonalisation
Ion-electron interactions:
Gaussians
Real space
Conclusion
Vext (r ) = −
1
∑
Zα
4πε0 α |Rα − r |
T. Deutsch
The Hohenberg-Kohn theorem
Schrödinger equation
N
1 2
1
H = ∑ − ∇ri + Vext (ri , {R }) + ∑
i =1
1
2 i 6=j |ri − rj |
2
Very difficult to solve for more than two electrons!
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
The fundamental variable of the problem is however not the
wavefunction, but the electronic density
Z
ρ(r ) = N dr2 · · · drN ψ∗ (r , r2 , · · · , rN )ψ(r , r2 , · · · , rN )
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Hohenberg-Kohn theorem (1964)
Real space
Conclusion
The ground state density ρ(r ) of a many-electron system
uniquely determines (up to a constant) the external potential .
The external potential is a functional of ρ: Vext = Vext [ρ]
T. Deutsch
The Hohenberg-Kohn theorem
Schrödinger equation
N
1 2
1
H = ∑ − ∇ri + Vext (ri , {R }) + ∑
i =1
1
2 i 6=j |ri − rj |
2
Very difficult to solve for more than two electrons!
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
The fundamental variable of the problem is however not the
wavefunction, but the electronic density
Z
ρ(r ) = N dr2 · · · drN ψ∗ (r , r2 , · · · , rN )ψ(r , r2 , · · · , rN )
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Hohenberg-Kohn theorem (1964)
Real space
Conclusion
The ground state density ρ(r ) of a many-electron system
uniquely determines (up to a constant) the external potential .
The external potential is a functional of ρ: Vext = Vext [ρ]
T. Deutsch
Kohn-Sham formalism
H-K theorem: E is an unknown functional of the density
E = E [ρ] → Density Functional Theory
Kohn-Sham approach
Review
Introduction
Mapping of an interacting many-electron system into a
system with independent particles moving into an effective
potential.
Atomistic Methods
DFT
Kohn-Sham
Find a set of orthonormal orbitals Ψi (r) that minimizes:
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
E =−
1
N
Z
∗
2
Ψi (r)∇ Ψi (r)dr +
∑
2 i =1
1
Z
2
ρ(r)VH (r)dr
Diagonalisation
Z
Gaussians
+ Exc [ρ(r)] +
Real space
Conclusion
Vext (r)ρ(r)dr
N
ρ(r) = ∑ Ψ∗i (r)Ψi (r)
∇2 VH (r) = −4πρ(r)
i =1
T. Deutsch
Kohn-Sham formalism
Kohn-Sham approach
Review
Introduction
potential.
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
E =−
1
N
Z
∗
2
∑
2 i =1
1
Z
2
ρ(r)VH (r)dr
Diagonalisation
Z
Gaussians
+ Exc [ρ(r)] +
Real space
Conclusion
Vext (r)ρ(r)dr
N
ρ(r) = ∑ Ψ∗i (r)Ψi (r)
∇2 VH (r) = −4πρ(r)
i =1
T. Deutsch
Kohn-Sham formalism
Kohn-Sham approach
Review
Introduction
potential.
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
E =−
1
N
Z
∗
2
∑
2 i =1
1
Z
2
ρ(r)VH (r)dr
Diagonalisation
Z
Gaussians
+ Exc [ρ(r)] +
Real space
Conclusion
Vext (r)ρ(r)dr
N
ρ(r) = ∑ Ψ∗i (r)Ψi (r)
∇2 VH (r) = −4πρ(r)
i =1
T. Deutsch
Performing a DFT calculation
A self-consistent equation
ρ(r) = ∑ Ψ∗i (r)Ψi (r), where |ψi i satisfies
i
1 2
− ∇ + VH [ρ] + Vxc [ρ] + Vext + Vpseudo |ψi i = Ei |ψi i ,
2
Review
Introduction
Atomistic Methods
(Kohn-Sham) DFT “Ingredients”
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
An XC potential, functional of the density
several approximations exists (LDA,GGA,. . . )
A choice of the pseudopotential (if not all-electrons)
(norm conserving, ultrasoft, PAW,. . . )
A basis set for expressing the |ψi i
An (iterative) algorithm for finding the wavefunctions |ψi i
A (good) computer. . .
T. Deutsch
Exchange-Correlation Energy
Exc [ρ] = Kexact [ρ] − KKS [ρ] + Ve−e exact [ρ] − VHartree [ρ]
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
KKS [ρ]: Kinetic energy for a non-interaction electron gas
DFT is an exact reformulation of the many-body problem
Kohn-Sham formalism is exact (mapping of an
interacting electron system into a non-interacting system
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
Many types:
Local Density Approximation (LDA)
Generalized gradient approximation (GGA), meta-GGA,
Hybrid functional (a part of exchange energy)
Random Phase Approximation (RPA), . . .
T. Deutsch
Exchange-Correlation Energy
Exc [ρ] = Kexact [ρ] − KKS [ρ] + Ve−e exact [ρ] − VHartree [ρ]
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
KKS [ρ]: Kinetic energy for a non-interaction electron gas
DFT is an exact reformulation of the many-body problem
Kohn-Sham formalism is exact (mapping of an
interacting electron system into a non-interacting system
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
Many types:
Generalized gradient approximation (GGA), meta-GGA,
Hybrid functional (a part of exchange energy)
Random Phase Approximation (RPA), . . .
T. Deutsch
Assumption: For each point r in space with a density ρ(r ), the
exchange and correlation energy at that point is the same as for an
uniform electron gas with that density:
Z
LDA
ρ(r )εxc (ρ)dr
=
Exc
V
Review
LDA
(r )
Vxc
=
LDA
∂Exc
∂ρ(r )
=
εxc (ρ) + ρ(r )
Introduction
Atomistic Methods
DFT
Kohn-Sham
∂εxc (ρ)
∂ρ(r )
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
εxc (ρ): Exchange-correlation energy for a uniform electron gas of
1/3 R
LDA
density (analytic):
Exc
= − 43 π3
ρ(r )4/3 dr
Correlation energy (interpolated analytically): Based on
Quantum Monte Carlo calculations (Ceperley and Alder (1980))
for the exchange and correlation of the electron gas.
Based on uniform gas of electron: Good cancellation of error
T. Deutsch
Generalized Gradient Approximations (GGA)
Review
Introduction
Atomistic Methods
Use the gradient of the density to have a more accurate exchange
energy
Many generalised gradient approximations: Perdew (1985), Lee,
Yang and Parr (1988), Perdew and Wang (1991), PBE
(1996),HCTH (1998), ...
Perdew and Yue (1986) functional (which is relatively simple):
1/3 Z
3 3
GGA
ρ(r )4/3 F (s)dr
[ρ] = −
Exc
4 π
DFT
Kohn-Sham
Pseudopotential
Solving
s=
|∇ρ(r )|
2kF (ρ(r ))
kF = ((3π2 ρ(r ))1/3
Basis sets
Plane waves
Direct Minimisation
F (s) = (1 + 1.296s2 + 14s4 + 0.2s6 )1/15
Diagonalisation
Gaussians
Real space
Conclusion
LDA: Overbind (energies too high) underestimate bond
lengths
GGA: In general better energies specially PBE
T. Deutsch
Generalized Gradient Approximations (GGA)
Review
Introduction
Atomistic Methods
Use the gradient of the density to have a more accurate exchange
energy
Many generalised gradient approximations: Perdew (1985), Lee,
Yang and Parr (1988), Perdew and Wang (1991), PBE
(1996),HCTH (1998), ...
Perdew and Yue (1986) functional (which is relatively simple):
1/3 Z
3 3
GGA
ρ(r )4/3 F (s)dr
[ρ] = −
Exc
4 π
DFT
Kohn-Sham
Pseudopotential
Solving
s=
|∇ρ(r )|
2kF (ρ(r ))
kF = ((3π2 ρ(r ))1/3
Basis sets
Plane waves
Direct Minimisation
F (s) = (1 + 1.296s2 + 14s4 + 0.2s6 )1/15
Diagonalisation
Gaussians
Real space
Conclusion
LDA: Overbind (energies too high) underestimate bond
lengths
GGA: In general better energies specially PBE
T. Deutsch
Hybrid functionals
Add a small part of exchange interaction (25% to 50%)
Review
Introduction
EExchange = −
Atomistic Methods
DFT
Kohn-Sham
φ∗i (r)φj (r)φi (r0 )φ∗j (r0 )
drdr0
|r − r0 |
V
Z
∑
i <j
Many hybrid functionals: B3LYP, HSE, . . .
Pseudopotential
Solving
The gap is (really) better and also cohesion energies
Basis sets
Plane waves
Direct Minimisation
Very time-consuming (10)
Diagonalisation
Gaussians
Not the right physics and sometimes not so accurate. . .
Real space
Conclusion
T. Deutsch
Hybrid functionals
Add a small part of exchange interaction (25% to 50%)
Review
Introduction
EExchange = −
Atomistic Methods
DFT
Kohn-Sham
φ∗i (r)φj (r)φi (r0 )φ∗j (r0 )
drdr0
|r − r0 |
V
Z
∑
i <j
Many hybrid functionals: B3LYP, HSE, . . .
Pseudopotential
Solving
The gap is (really) better and also cohesion energies
Basis sets
Plane waves
Direct Minimisation
Very time-consuming (10)
Diagonalisation
Gaussians
Not the right physics and sometimes not so accurate. . .
Real space
Conclusion
T. Deutsch
i
1 2
2
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Pseudopotentials → smoothening of wavefunctions
For chemical properties only the valence electrons are
relevant:
Eliminate the chemically
inactive core electrons
Review
Vext (r ) = −
1
∑
Zα
4πε0 α |Rα − r |
Reduce the number of electron
orbitals
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
The pseudo-wavefunctions of
the valence electrons are
smooth
Eliminate the rapid variations of
the valence wavefunction in the
core region
Real space
Conclusion
A reasonable approximation
The pseudopotential approximation is less severe than the
approximate nature of the exchange correlation functional
T. Deutsch
Non-Local Pseudopotentials
V PP (r , r 0 )
=
∑ Ylm (r )Vl (r )δr ,r Ylm (r 0 )
0
l ,m
where Yl ,m are spherical harmonics.
Review
VNL
=
∑ | φIlm > Vl <φIlm |
I ,l ,m
Introduction
Atomistic Methods
DFT
where I is the index of atom, l and m and quantum numbers.
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Use ab initio atomic wavefunctions to obtain
pseudopotentials.
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
The (potential) energy contribution due to the valence charge
density must be substracted away to give an ionic
pseudopotential:
Vlion
,ps (r ) = Vl ,ps (r ) + (VHartree [ρ(r )] + Vxc [ρ(r )])free atom
T. Deutsch
Gaussian type separable Pseudopotentials (HGH)
Local part
Vloc (r )
−Zion
=
r
r
"
1
erf √
+ exp −
2
2rloc
(
C1 + C2
Review
r
rloc
2
+ C3
r
2 #
rloc
4
r
+ C4
rloc
r
6 )
rloc
Introduction
Atomistic Methods
DFT
Nonlocal (separable) part H (~r ,~r 0 )
Kohn-Sham
2
Pseudopotential
Solving
H sep (~r ,~r 0 )
=
Basis sets
∑ ∑ Ys,m (r̂ ) pis (r ) his pis (r 0 ) Ys∗,m (rˆ0 )
i =1 m
Plane waves
+
Direct Minimisation
Diagonalisation
∑ Yp,m (r̂ ) p1p (r ) h1p p1p (r 0 ) Yp∗,m (rˆ0 )
m
Gaussians
Real space
Conclusion
p1l (r )
=
√
2
rle
l + 23
rl
− 12 ( rr
l
)2
q
Γ(l + 32 )
p2l (r )
=
√
2
r l +2 e
l+ 7
rl 2
− 12 ( rr )2
l
q
Γ(l + 72 )
T. Deutsch
i
1 2
2
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Basis sets for electronic structure calculation
How can we express the Kohn-Sham wavefunctions?
Plane Waves
4 Localization in Fourier space, efficient preconditioning
4 Systematic convergence properties
Review
Introduction
Atomistic Methods
8 No localization in real space. Empty regions must be
“filled” with PW. Non adaptive
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Gaussians, Slater type Orbitals
4 Real space localized, well suited for molecules and
other open structures
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
4 Small number of basis functions for moderate accuracy
8 Many different recipes for generating basis sets
8 Over-completeness before convergence.
Non systematic basis set.
T. Deutsch
Basis sets for electronic structure? Wavelets
A basis set both adaptive and systematic, real space based
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Wavelet basis sets
Localized both in real and
in Fourier space
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Allow for adaptivity
(for internal electrons)
Systematic basis set
Real space
Conclusion
T. Deutsch
i
1 2
2
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Kohn-Sham Equations: Operators
Apply different operators
Review
Introduction
1 2
− ∇ + Veff (r )(r ) ψi = εi ψi
2
Atomistic Methods
Kohn-Sham
Veff (r ) = Vext (r ) +
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
ρ(r 0 ) 0
dr + µxc (r )
V |r − r 0 |
Z
DFT
E [ρ] can be expressed by the orthonormalized states of one
particule: ψi (r ) with the fractional occupancy number fi
( 0 ≤ f i ≤ 1) :
Real space
Conclusion
ρ(r ) =
∑ fi |ψi (r )|2
i
T. Deutsch
Kohn-Sham Equations: Computing Energies
Calculate different integrals
E [ρ] = K [ρ] + U [ρ]
Review
K [ρ] = −
Introduction
Atomistic Methods
1 ~2
2 me
Z
∑
i
V
dr ψ∗i ∇2 ψi
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Z
U [ρ] =
V
dr Vext (r ) ρ(r )+
Direct Minimisation
Gaussians
Z
2 V
|
Plane waves
Diagonalisation
1
ρ(r )ρ(r 0 )
+
E [ρ]
| xc{z }
|r − r 0 |
{z
} exchange−correlation
dr dr 0
Hartree
We minimise with the variables ψi (r ) and fi
Real space
Conclusion
Z
with the constraint
V
dr ρ(r ) = Nel .
T. Deutsch
KS Equations: Self-Consistent Field
−
Set of self-consistent equations:
1 ~2
2 me
2
∇ + Veff
ψi = εi ψi
with an effective potential:
Review
Introduction
Atomistic Methods
Z
V
DFT
|
Kohn-Sham
ρ(r 0 )
|r − r 0 |
{z
}
dr 0
Hartree
+
δExc
δρ(r )
| {z }
exchange−correlation
Pseudopotential
Solving
Basis sets
and:
Plane waves
Direct Minimisation
ρ(r ) = ∑i fi |ψi (r )|2
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
−
1 ~2
2 me
2
∇ + Veff
ψi = εi ψi
Review
Introduction
Atomistic Methods
Z
V
DFT
|
Kohn-Sham
ρ(r 0 )
|r − r 0 |
{z
}
dr 0
Hartree
+
δExc
δρ(r )
| {z }
Pseudopotential
Solving
Basis sets
and:
Plane waves
Direct Minimisation
ρ(r ) = ∑i fi |ψi (r )|2
Diagonalisation
Gaussians
Real space
Conclusion
Poisson Equation:
∆VHartree = ρ
2
2
2
(Laplacian: ∆ = ∂∂x 2 + ∂∂y 2 + ∂∂z 2 )
Real Mesh (1003 = 106 ): 106 × 106 = 1012 evaluations !
T. Deutsch
−
1 ~2
2 me
2
∇ + Veff
ψi = εi ψi
Review
Introduction
Atomistic Methods
Z
V
DFT
|
Kohn-Sham
ρ(r 0 )
|r − r 0 |
{z
}
dr 0
Hartree
+
δExc
δρ(r )
| {z }
Pseudopotential
Solving
Basis sets
and:
Plane waves
Direct Minimisation
ρ(r ) = ∑i fi |ψi (r )|2
Diagonalisation
Gaussians
Real space
Conclusion
Poisson Equation:
∆VHartree = ρ
2
2
2
(Laplacian: ∆ = ∂∂x 2 + ∂∂y 2 + ∂∂z 2 )
T. Deutsch
−
1 ~2
2 me
2
∇ + Veff
ψi = εi ψi
Review
Introduction
Atomistic Methods
Z
V
DFT
|
Kohn-Sham
ρ(r 0 )
|r − r 0 |
{z
}
dr 0
Hartree
+
δExc
δρ(r )
| {z }
Pseudopotential
Solving
Basis sets
and:
Plane waves
Direct Minimisation
ρ(r ) = ∑i fi |ψi (r )|2
Diagonalisation
Gaussians
Real space
Conclusion
Poisson Equation:
∆VHartree = ρ
2
2
2
(Laplacian: ∆ = ∂∂x 2 + ∂∂y 2 + ∂∂z 2 )
T. Deutsch
−
1 ~2
2 me
2
∇ + Veff
ψi = εi ψi
Review
Introduction
Atomistic Methods
Z
V
DFT
|
Kohn-Sham
ρ(r 0 )
|r − r 0 |
{z
}
dr 0
Hartree
+
δExc
δρ(r )
| {z }
Pseudopotential
Solving
Basis sets
and:
Plane waves
Direct Minimisation
ρ(r ) = ∑i fi |ψi (r )|2
Diagonalisation
Gaussians
Real space
Conclusion
Poisson Equation:
∆VHartree = ρ
2
2
2
(Laplacian: ∆ = ∂∂x 2 + ∂∂y 2 + ∂∂z 2 )
T. Deutsch
−
1 ~2
2 me
2
∇ + Veff
ψi = εi ψi
Review
Introduction
Atomistic Methods
Z
V
DFT
|
Kohn-Sham
ρ(r 0 )
|r − r 0 |
{z
}
dr 0
Hartree
+
δExc
δρ(r )
| {z }
Pseudopotential
Solving
Basis sets
and:
Plane waves
Direct Minimisation
ρ(r ) = ∑i fi |ψi (r )|2
Diagonalisation
Gaussians
Real space
Conclusion
Poisson Equation:
∆VHartree = ρ
2
2
2
(Laplacian: ∆ = ∂∂x 2 + ∂∂y 2 + ∂∂z 2 )
T. Deutsch
Ab Initio Methods: DFT
Review
Introduction
Atomistic Methods
DFT
h
−
1 2
∇ +V (r )
2
i
+µxc (r ) ψki= εki ψki
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Review
Introduction
Atomistic Methods
DFT
h
−
1 2
∇ +V (r )
2
i
Kohn-Sham
Pseudopotential
Solving
Basis sets
atomic orbitals
Gaussians
plane waves
Slater
augmented
numerical
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
real space
finite difference
Wavelet
T. Deutsch
pseudopotential
self-consistent (SCF)
norm-conserving
PAW
all electrons
Harris-Foulkes functional
Review
Introduction
Atomistic Methods
DFT
h
−
1 2
∇ +V (r )
2
i
Kohn-Sham
Pseudopotential
Solving
Basis sets
atomic orbitals
Gaussians
plane waves
Slater
augmented
numerical
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
real space
finite difference
Wavelet
T. Deutsch
pseudopotential
norm-conserving
PAW
all electrons
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
h
−
1 2
∇ +V (r )
2
i
periodic
Solving
Basis sets
non-periodic
atomic orbitals
Gaussians
plane waves
Slater
augmented
numerical
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
real space
finite difference
Wavelet
T. Deutsch
pseudopotential
norm-conserving
PAW
all electrons
Hybrid functionals
beyond LDA
GW method
LDA,GGA
LDA+U
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
h
−
1 2
∇ +V (r )
2
i
periodic
Solving
Basis sets
non-periodic
atomic orbitals
Gaussians
plane waves
Slater
augmented
numerical
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
real space
finite difference
Wavelet
T. Deutsch
pseudopotential
norm-conserving
PAW
all electrons
Hybrid functionals
beyond LDA
GW
LDA,GGA
LDA+U
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
h
−
1 2
∇ +V (r )
2
i
periodic
Solving
Basis sets
non-periodic
atomic orbitals
Gaussians
plane waves
Slater
augmented
numerical
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
non-spin polarized
Conclusion
real space
non-collinear
finite difference
spin polarized
Wavelet
collinear
T. Deutsch
pseudopotential
norm-conserving
PAW
all electrons
Hybrid functionals
relativistic
non-relativistic
beyond LDA
GW
LDA,GGA
LDA+U
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
h
−
1 2
∇ +V (r )
2
i
periodic
Solving
Basis sets
non-periodic
atomic orbitals
Gaussians
plane waves
Slater
augmented
numerical
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
non-spin polarized
Conclusion
real space
non-collinear
finite difference
spin polarized
Wavelet
collinear
T. Deutsch
pseudopotential
norm-conserving
PAW
all electrons
Hybrid functionals
relativistic
non-relativistic
beyond LDA
GW
LDA,GGA
LDA+U
Review
Introduction
Atomistic Methods
DFT
h
−
1 2
∇ +V (r )
2
Kohn-Sham
i
periodic
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
non-periodic
N 3 scaling
atomic orbitals
Gaussians
plane waves
Slater
augmented
numerical
Diagonalisation
Gaussians
Real space
O (N ) methods
non-spin polarized
Conclusion
real space
non-collinear
finite difference
spin polarized
Wavelet
collinear
T. Deutsch
List of ab initio Codes
Plane Waves
ABINIT — Louvain-la-Neuve — http://www.abinit.org
CPMD — Zurich, Lugano — http://www.cpmd.org
PWSCF — Italy — http://www.pwscf.org
VASP — Vienna — http://cms.mpi.univie.ac.at/vasp
Gaussian
Review
Introduction
Atomistic Methods
DFT
Gaussian — http://www.gaussian.com
DeMon — http://www.demon-software.com
CP2K — http://cp2k.berlios.de
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Siesta — Madrid —
http://www.uam.es/departamentos/ciencias/fismateriac/siesta
(numerical basis sets)
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
Wien — Vienna — http://www.wien2k.at (FPLAPW, all
electrons)
Real space basis set
ONETEP — http://www.onetep.soton.ac.uk
BigDFT — http://inac.cea.fr/L_Sim/BigDFT
T. Deutsch
Basis sets
Review
Introduction
Atomistic Methods
DFT
Minimisation of the electronic density
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Plane Wave basis sets
Natural basis set for electronic structure calculations in
periodic solids.
Review
1
Introduction
√ exp(iG · r)
Atomistic Methods
V
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
where V is the volume of the periodicity volume and G is a
multiple of the reciprocal lattice vectors, i.e. there are
integers l1 , l2 , l3 such that G = l1 b1 + l2 b2 + l3 b3 .
Gaussians
Real space
Conclusion
T. Deutsch
Plane Wave basis sets
Matrix elements of the kinetic energy part of the
Hamiltonian calculated analytically
Z
1 2
exp(−iG · r) − ∇ exp(iG0 · r) = G2 δG,G0
2
Review
Introduction
Atomistic Methods
DFT
The potential energy part obtained numerically by
calculating the wave-functions in real space
Kohn-Sham
Pseudopotential
Ψ(r) = ∑ exp(iG · r)
Solving
G
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
and then integrating numerically in real space.
The transformation on a grid in real space is done with
the help of the Fast Fourier transformation at a cost of
N log2 (N ) operations for a basis set of N plane waves.
T. Deutsch
Basis sets: Planes Waves
Operator approach: no need to set up Hamiltonian
matrix
FFT allows for quasi linear M log(M ) scaling where M is
number of plane waves
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Localization in Fourier space allows for efficient
preconditioning techniques. Hence number of iterations
independent of M
Systematic convergence propertie
No localization in real space. Empty regions have to be
filled with plane waves. Only supercell approach.
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
O(N) scaling hard to achieve
Resolution cannot adaptively be refined around the
nucleus. All electron calculations impossible, hard
pseudopotentials difficult.
T. Deutsch
Operator approach: no need to set up Hamiltonian
matrix
FFT allows for quasi linear M log(M ) scaling where M is
number of plane waves
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Localization in Fourier space allows for efficient
preconditioning techniques. Hence number of iterations
independent of M
Systematic convergence propertie
No localization in real space. Empty regions have to be
filled with plane waves. Only supercell approach.
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
O(N) scaling hard to achieve
Resolution cannot adaptively be refined around the
nucleus. All electron calculations impossible, hard
pseudopotentials difficult.
T. Deutsch
pseudopotential
norm-conserving
PAW
all electrons
Hybrid functionals
relativistic
non-relativistic
Review
Introduction
Atomistic Methods
DFT
h
beyond LDA
GW
LDA,GGA
LDA+U
i
1 2
− 2 ∇ +V (r ) +µxc (r ) ψki= εki ψki
periodic
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
non-periodic
N 3 scaling
Direct Minimisation
Diagonalisation
Gaussians
O (N ) methods
Real space
atomic orbitals
Gaussians
plane waves
Slater
augmented
numerical
non-spin polarized
Conclusion
real space
non-collinear
finite difference
spin polarized
Wavelet
collinear
T. Deutsch
Plane waves: List of Codes
PAW, linear response, GW, many features, good //
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
USP, good //, QM/MM with Gromacs
USP, linear response, //
robust, fast, hybrid functional, GW, //
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Direct Minimisation: Flowchart
(
)
ψj = ∑
j
cG eiG.r
G
Basis (NG ): 0, . . . , G
Orthonormalized
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
(
)
ψj = ∑
j
cG eiG.r
Basis (NG ): 0, . . . , G
G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
Review
j
Basis (23 NG ): 0, . . . , 2G
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
(
)
ψj = ∑
j
cG eiG.r
Basis (NG ): 0, . . . , G
G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
j
Review
Basis (23 NG ): 0, . . . , 2G
Introduction
Atomistic Methods
FFT
DFT
Kohn-Sham
Pseudopotential
Solving
Vxc [ρ(r )]
−G2 VH (G) = ρ(G)
VNL ({ψj })
Veffective
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
(
)
ψj = ∑
j
cG eiG.r
Basis (NG ): 0, . . . , G
G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
j
Review
Basis (23 NG ): 0, . . . , 2G
Introduction
Atomistic Methods
FFT
DFT
Kohn-Sham
Pseudopotential
Vxc [ρ(r )]
Solving
−G2 VH (G) = ρ(G)
VNL ({ψj })
Veffective
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
1 2 j
2 G cG
Kinetic Term
Gaussians
Real space
Conclusion
T. Deutsch
(
)
ψj = ∑
j
cG eiG.r
Basis (NG ): 0, . . . , G
G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
j
Review
Basis (23 NG ): 0, . . . , 2G
Introduction
Atomistic Methods
FFT
DFT
Kohn-Sham
Pseudopotential
Vxc [ρ(r )]
Solving
−G2 VH (G) = ρ(G)
VNL ({ψj })
Veffective
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
1 2 j
2 G cG
FFT
δcGj = −
∂Etotal
+ Λjl cGl
∂cj∗ (G) ∑
l
Conclusion
Kinetic Term
Λjl =< ψj |H |ψl >
T. Deutsch
(
)
ψj = ∑
j
cG eiG.r
Basis (NG ): 0, . . . , G
G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
j
Review
Basis (23 NG ): 0, . . . , 2G
Introduction
Atomistic Methods
FFT
DFT
Kohn-Sham
Pseudopotential
Vxc [ρ(r )]
Solving
−G2 VH (G) = ρ(G)
VNL ({ψj })
Veffective
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
1 2 j
2 G cG
FFT
δcGj = −
Kinetic Term
∂Etotal
+ Λjl cGl
∂cj∗ (G) ∑
l
Conclusion
new ,j
cG
=
j
j
cG + hstep δcG
Steepest Descent,
Conjugate Gradient,
Direct Inversion of the Iterative Subspace
T. Deutsch
(
)
ψj = ∑
j
cG eiG.r
Basis (NG ): 0, . . . , G
G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
j
Review
Basis (23 NG ): 0, . . . , 2G
Introduction
Atomistic Methods
FFT
DFT
Kohn-Sham
Pseudopotential
−G2 VH (G) = ρ(G)
Vxc [ρ(r )]
Solving
VNL ({ψj })
Veffective
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
1 2 j
2 G cG
FFT
δcGj = −
Kinetic Term
∂Etotal
+ Λjl cGl
∂cj∗ (G) ∑
l
j
Stop when δcG small
new ,j
cG
=
j
j
cG + hstep δcG
Steepest Descent,
Conjugate Gradient,
Direct Inversion of the Iterative Subspace
T. Deutsch
Direct Minimization: List of Codes
Plane Waves
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Gaussian
CP2K — http://cp2k.berlios.de
Real space basis set
PARATEC — http://www.nersc.gov/projects/paratec
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Diagonalisation Scheme: Flowchart
)
(
ψj = ∑
j
cG eiG.r
, {fj }
G
Basis (NG ): 0, . . . , G
Orthonormalized
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
)
(
ψj = ∑
j
cG eiG.r
, {fj }
G
Basis (NG ): 0, . . . , G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
j
Review
Basis (23 NG ): 0, . . . , 2G
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
)
(
ψj = ∑
j
cG eiG.r
, {fj }
Basis (NG ): 0, . . . , G
G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
j
Review
Basis (23 NG ): 0, . . . , 2G
Introduction
Atomistic Methods
FFT
DFT
Kohn-Sham
Pseudopotential
Solving
Vxc [ρ(r )]
−G2 VH (G) = ρ(G)
VNL ({ψj })
Veffective
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
)
(
ψj = ∑
j
cG eiG.r
, {fj }
Basis (NG ): 0, . . . , G
G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
j
Review
Basis (23 NG ): 0, . . . , 2G
Introduction
Atomistic Methods
FFT
DFT
Kohn-Sham
Pseudopotential
Vxc [ρ(r )]
Solving
−G2 VH (G) = ρ(G)
VNL ({ψj })
Veffective
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
1 2 j
2 G cG
Kinetic Term
Gaussians
Real space
Conclusion
T. Deutsch
)
(
ψj = ∑
j
cG eiG.r
, {fj }
Basis (NG ): 0, . . . , G
G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
j
Review
Basis (23 NG ): 0, . . . , 2G
Introduction
Atomistic Methods
FFT
DFT
Kohn-Sham
Pseudopotential
Vxc [ρ(r )]
Solving
−G2 VH (G) = ρ(G)
VNL ({ψj })
Veffective
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
1 2 j
2 G cG
Kinetic Term
FFT
1 2
− 2 ∇ + Veff (r ) ψj = εj ψj
Conclusion
T. Deutsch
)
(
ψj = ∑
j
cG eiG.r
, {fj }
Basis (NG ): 0, . . . , G
G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
Basis (23 NG ): 0, . . . , 2G
j
Review
Introduction
Atomistic Methods
FFT
DFT
Kohn-Sham
Pseudopotential
Vxc [ρ(r )]
Solving
−G2 VH (G) = ρ(G)
VNL ({ψj })
Veffective
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
1 2 j
2 G cG
Kinetic Term
FFT
1 2
− 2 ∇ + Veff (r ) ψj = εj ψj
Conclusion
ρnew (r ) = αρout + (1 − α)ρin
ρout (r ) = ∑ fj |ψj (r )|2
Mixing Density:
Anderson,
Broyden,DIIS
j
T. Deutsch
)
(
ψj = ∑
j
cG eiG.r
, {fj }
Basis (NG ): 0, . . . , G
G
Orthonormalized
inv FFT
ρ(r ) = ∑ fj |ψj (r )|2
Basis (23 NG ): 0, . . . , 2G
j
Review
Introduction
Atomistic Methods
FFT
DFT
Kohn-Sham
Pseudopotential
Vxc [ρ(r )]
Solving
−G2 VH (G) = ρ(G)
VNL ({ψj })
Veffective
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
1 2 j
2 G cG
Kinetic Term
FFT
1 2
− 2 ∇ + Veff (r ) ψj = εj ψj
Conclusion
Stop when |ρnew − ρin | small
ρnew (r ) = αρout + (1 − α)ρin
ρout (r ) = ∑ fj |ψj (r )|2
Mixing Density:
Anderson,
Broyden,DIIS
j
T. Deutsch
Iterative diagonalization
Standard diagonalization routines, such as found in software
packages like LAPACK, are designed to find all the
eigenvalues and eigenvectors of a matrix.
Review
Scale like n3 , where n is the dimension of the matrix
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
We need to calculate only a small number of eigenvalues and
eigenvectors of matrices of a very large dimension, another
type of diagonalization, namely iterative diagonalization is
used:
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
Lanczos, Davidson, RM-DIIS, Conjugate Gradient, LOBPCG,
...
T. Deutsch
Diagonalization scheme: List of Codes
Diagonalization: Conjugate-Gradient, LOBPCG
Mixing: Broyden, Pulay, Anderson, others based on
dielectric constant
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Diagonalization: Lanczos, Davidson
Mixing: Broyden, Pulay, Anderson
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Diagonalization: RM-DIIS
Real space
Conclusion
T. Deutsch
Gaussians
Atom centered Gaussians are the most popular basis
functions for electronic structure calculations in the quantum
chemistry community.
Review
Introduction
Atomistic Methods
DFT
(x − Xi )l1 (y − Yi )l2 (z − Zi )l3
× exp −α((x − Xi )2 + (y − Yi )2 + (z − Zi )2 )
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
The different functions associated to Ri = (Xi , Yi , Zi ) differ by
their extent, specified by α and by the exponents l1 , l2 , l3 .
Because the Gaussians have qualitatively the shape of
atomic orbitals one needs for moderate accuracy only a small
number of basis function per atom
(typically 10 to 20 per atom).
T. Deutsch
Basis sets: Gaussians
Real space localization makes them well suited for
molecules and other open structures.
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Kinetic and overlap matrix elements can be calculated
analytically.
Good description of core electrons.
Small number of basis functions necessary for moderate
accuracy
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
No systematic convergence. Over-completeness before
convergence. This leads also to problems with O(N).
Many different recipes for generating basis sets.
Real space
Conclusion
Pulay forces have been implemented.
T. Deutsch
Real space localization makes them well suited for
molecules and other open structures.
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Kinetic and overlap matrix elements can be calculated
analytically.
Good description of core electrons.
Small number of basis functions necessary for moderate
accuracy
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
No systematic convergence. Over-completeness before
convergence. This leads also to problems with O(N).
Many different recipes for generating basis sets.
Real space
Conclusion
Pulay forces have been implemented.
T. Deutsch
pseudopotential
norm-conserving
PAW
all electrons
Hybrid functionals
relativistic
non-relativistic
Review
Introduction
Atomistic Methods
DFT
h
beyond LDA
GW
LDA,GGA
LDA+U
i
1 2
k k
k
− 2 ∇ +V (r ) +µxc (r ) ψ=
i εi S ψi
periodic
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
non-periodic
N 3 scaling
Direct Minimisation
Diagonalisation
Gaussians
O (N ) methods
Real space
atomic orbitals
Gaussians
plane waves
Slater
augmented
numerical
non-spin polarized
Conclusion
real space
non-collinear
finite difference
spin polarized
Wavelet
collinear
T. Deutsch
Real space basis set: wavelets (BigDFT)
pseudopotential
norm-conserving
PAW
all electrons
Hybrid functionals
relativistic
non-relativistic
Review
Introduction
Atomistic Methods
DFT
h
beyond LDA
GW method
LDA,GGA
LDA+U
i
1 2
− 2 ∇ +V (r ) +Vxc (r ) ψki= εki ψki
periodic
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
non-periodic
N 3 scaling
Direct Minimisation
Diagonalisation
Gaussians
O (N ) methods
Real space
atomic orbitals
Gaussians
plane waves
Slater
augmented
numerical
non-spin polarized
Conclusion
real space
non-collinear
finite difference
spin polarized
Wavelet
collinear
T. Deutsch
Real space: List of Codes
sinc function, O(N) method, inside Material Studio,
good //
Review
Introduction
wavelets, good //
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
finite difference, good //
OCTOPUS —http://www.tddft.org
finite difference, TD-DFT
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
GPAW — https://wiki.fysik.dtu.dk/gpaw
Grid-based projector-augmented wave method,
finite difference
T. Deutsch
good //
Review
Introduction
wavelets, good //
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
finite difference
T. Deutsch
good //
Review
Introduction
wavelets, good //
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
finite difference
T. Deutsch
good //
Review
Introduction
wavelets, good //
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
finite difference
T. Deutsch
good //
Review
Introduction
wavelets, good //
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
finite difference
T. Deutsch
Review
Introduction
Conclusion
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Conclusion
T. Deutsch
Conclusion
Functionals: Mixed between physics and chemistry
approach
QM/MM (Quantum Mechanics/Molecular Modeling)
Review
Introduction
Atomistic Methods
DFT
Kohn-Sham
Pseudopotential
Solving
Basis sets
Plane waves
Direct Minimisation
Diagonalisation
Gaussians
Real space
Multi-scale approach (more than 1000 atoms feasible for
a better parametrization)
Order N (real space basis set as wavelets)
Numerical experience (high performance computing):
One-day simulation
Better exploration of atomic configurations
Molecular Dynamics (4s per step for 32 water
molecules)
Conclusion
T. Deutsch

Electronic Structure Calculations, Density Functional

Transcription

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