Theory and Practice of Density Functional...
Transcript of Theory and Practice of Density Functional...
http://www.pt.tu-clausthal.de/atp/
Peter E. BlöchlInstitute for Theoretical Physics
Clausthal University of Technology, Germany
Theory and Practice of Density Functional Theory
1
Tuesday, October 4, 11
Standard model of Solid State Physics
• particles: nuclei and electrons• interaction: electrostatics • equations of motion:
– Schrödinger equation: – Poisson equation:
2
i~@t| i = H| i
r2�(~r) =1
✏0⇢(~r)
but: exponential wall of many-particle physicsproblem: quantum mechanics + interaction
Tuesday, October 4, 11
Density functional theorymaps interacting electrons onto non-interacting quasi-electrons in an effective potential
• one-particle Schrödinger equation
• limited to ground state properties• based on an exact theorem
– but: implementation requires approximations
3
✓� ~22m
~r2 + veff (~r)
◆ n(~r) = n(~r)✏n
Tuesday, October 4, 11
4
McKee et al.,Science 300,1726 (2003)
Ojima,Yoshimura, Ueda PRB65, 75408 (2004)
Hu et al. PRB65,75408(2002)
Norga et al. MRS Proc. 2004
gate
n!doped n!doped
p!doped
Metal!Oxide Field!Effect Transistor (MOSFET)
silicon waver
source drain
channel
gate oxide
p!doped
Appetizer: high-K oxides for new transistorsFörst, Ashman, Schwarz, Blöchl, Nature 53, 427 (2004)
Ashman, Först, Schwarz, Blöchl, PRB 69, 75309 (2004)Först, Ashman, Blöchl, German patent DE10303875
Tuesday, October 4, 11
• Minimize total energy functional:
• Solve Kohn-Sham equations:
Total energy functional
5
� ~22m
~r2 + veff
(~r)
� n
(~r) = n
(~r)✏n
and veff
(~r) =
Zd3r0
e2[n(~r0) + Z(~r0)]
4⇡✏0|~r � ~r0|| {z }vHartree(~r)
+�E
xc
�n(~r)| {z }µxc(~r)
E[| n
i, fn
] =X
n
fn
h n
| ~p2
2m|
n
i| {z }
Ekin
+1
2
Zd3r
Zd3r0
e2[n(~r) + Z(~r)][n(~r0) + Z(~r0)]
4⇡✏0
|~r � ~r0|| {z }EHartree
+ Exc
[n(~r)]| {z }Exc
�X
n,m
⇤n,m
(h n
| m
i � �n,m
)| {z }
constraint
with n(~r) =X
n
fn
| (~r)|2
| {z }electron density
Z(~r) = �X
j
Zj
�(~r � ~Rj
)
| {z }charge density of nuclei in electron charges
Tuesday, October 4, 11
• Minimize total energy functional:
• Solve Kohn-Sham equations:
6
� ~22m
~r2 + veff
(~r)
� n
(~r) = n
(~r)✏n
and veff
(~r) =
Zd3r0
e2[n(~r0) + Z(~r0)]
4⇡✏0|~r � ~r0|| {z }vHartree(~r)
+�E
xc
�n(~r)| {z }µxc(~r)
E[| n
i, fn
] =X
n
fn
h n
| ~p2
2m|
n
i| {z }
Ekin
+1
2
Zd3r
Zd3r0
e2[n(~r) + Z(~r)][n(~r0) + Z(~r0)]
4⇡✏0
|~r � ~r0|| {z }EHartree
+ Exc
[n(~r)]| {z }Exc
�X
n,m
⇤n,m
(h n
| m
i � �n,m
)| {z }
constraint
with n(~r) =X
n
fn
| (~r)|2
| {z }electron density
Z(~r) = �X
j
Zj
�(~r � ~Rj
)
| {z }charge density of nuclei in electron charges
Total energy functional
Tuesday, October 4, 11
• Minimize total energy functional:
• Solve Kohn-Sham equations:
7
� ~22m
~r2 + veff
(~r)
� n
(~r) = n
(~r)✏n
and veff
(~r) =
Zd3r0
e2[n(~r0) + Z(~r0)]
4⇡✏0|~r � ~r0|| {z }vHartree(~r)
+�E
xc
�n(~r)| {z }µxc(~r)
E[| n
i, fn
] =X
n
fn
h n
| ~p2
2m|
n
i| {z }
Ekin
+1
2
Zd3r
Zd3r0
e2[n(~r) + Z(~r)][n(~r0) + Z(~r0)]
4⇡✏0
|~r � ~r0|| {z }EHartree
+ Exc
[n(~r)]| {z }Exc
�X
n,m
⇤n,m
(h n
| m
i � �n,m
)| {z }
constraint
with n(~r) =X
n
fn
| (~r)|2
| {z }electron density
Z(~r) = �X
j
Zj
�(~r � ~Rj
)
| {z }charge density of nuclei in electron charges
Total energy functional
Tuesday, October 4, 11
Self-consistency loop
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� ~22m
~r2 + veff
(~r)
� n
(~r) = n
(~r)✏n
and veff
(~r) =
Zd3r0
e2[n(~r0) + Z(~r0)]
4⇡✏0|~r � ~r0|| {z }vHartree(~r)
+�E
xc
�n(~r)| {z }µxc(~r)
E[| n
i, fn
] =X
n
fn
h n
| ~p2
2m|
n
i| {z }
Ekin
+1
2
Zd3r
Zd3r0
e2[n(~r) + Z(~r)][n(~r0) + Z(~r0)]
4⇡✏0
|~r � ~r0|| {z }EHartree
+ Exc
[n(~r)]| {z }Exc
�X
n,m
⇤n,m
(h n
| m
i � �n,m
)| {z }
constraint
with n(~r) =X
n
fn
| (~r)|2
| {z }electron density
Z(~r) = �X
j
Zj
�(~r � ~Rj
)
| {z }charge density of nuclei in electron charges
veff
(⌅r) = vext
(⌅r) +
Zd3r0
e2n(1)(⌅r0)
4⇤⇥0|⌅r � ⌅r0|+
�Exc
[n(1)]
�n(1)(⌅r)
�~22me
⇤r2 + veff (⇤r)� �n
�⇥n(⇤x) = 0
n(1)(⇥r) =X
n
fnX
�
�⇤n(⇥x)�n(⇥x)
Tuesday, October 4, 11
• Paradigm shift: – eigenvalue problems minimize energy
• Access to the molecular dynamics
Car-Parrinello method
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Car, Parrinello, PRL 55, 2471(1985)
Tuesday, October 4, 11
E0
=µ
minn(~r)
⇢FW [n] +
Zd3r v
ext
(~r)n(~r)� µ
✓Zd3r n(~r)�N
◆
| {z }charge constraint
�
=µ
minn(~r)
⇢Ts
[n]| {z }F
0[n]
+
Zd3r v
ext
(~r)n(~r) + FW [n]� F 0[n]| {z }EH+Exc
�µ
✓Zd3r n(~r)�N
◆
| {z }charge constraint
�
Levy’s constrained searchIs there a density functional?constrained search: proof of existence by presenting a construction principle1. sort all fermionic many-particle wave functions
according to their density
2. For each density find
a. the wave function with lowest kinetic energy and
b. the wave function with lowest kinetic and interaction energy
This defines two universal density functionals, F0[n] and FW[n]
3. determine energy and density of the ground state for a given external potential by minimization over all densities
- potential energy depends only on the density
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W
00 0
0
W W W
>
|!">
|!#>|!$>
|!%>
F [n4(r)]F [n1(r)]
F [n1(r)]
F [n2(r)]
F [n2(r)] F [n3(r)]
F 3(r)]
F [n4(r)]
[n
n1(r) n2(r) n3(r) n4(r)
|!&>
|!'> |!(>
|!)
FW [n] =�W
(~r),EW
min| i
h |T + W | i| {z }kinetic + interaction
+
Zd3r �W (~r)
✓h |n(~r)| i � n(~r)
◆
| {z }density constraint
�EW
✓h | i � 1
◆
| {z }norm constraint
F 0[n] =�0
(~r),E0
min| i
h |T | i| {z }kinetic
+
Zd3r �0(~r)
✓h |n(~r)| i � n(~r)
◆
| {z }density constraint
�E0
✓h | i � 1
◆
| {z }norm constraint
Tuesday, October 4, 11
n(r1)h(r1,r2)
|r2−r1|0
n2n(2)(r2,r1)
n(2)(r2,r1)
0
Exchange-correlation hole• interaction energy
• two-particle density– n(2)(r,r’)= probability density of finding one electron
at r and another at r’ times N(N-1) (number of pairs)
• exchange correlation hole hxc(r,r’)
• properties of the exchange correlation hole– each electron interacts with N-1 other electrons:
hole integrates to -1 electron (Charge sum rule)
– two electrons with the same spin cannot be at the same position (Pauli principle)
– the hole vanishes at large distances (short-sightedness)
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Ee�e =1
2
Zd3r
Zd3r0
e2n(2)(~r, ~r0)
4⇡✏0|~r � ~r0|
n(2)(~r, ~r0) = n(1)(~r)n(1)(~r0) + n(1)(~r)hxc
(~r0,~r)
Ee�e
=1
2
Zd3r
Zd3r0
e2n(1)(~r)n(1)(~r0)
4⇡✏0|~r � ~r0|| {z }Hartree energy
+
Zd3r n(1)(~r) ·
"1
2
Zd3r0
e2hxc
(~r, ~r0)
4⇡✏0|~r � ~r0|
#
| {z }potential energy of exchange and correlation
Tuesday, October 4, 11
Model for Exc
• properties of the exchange correlation hole
– each electron interacts with N-1 other electrons: hole integrates to -1 electron (Charge sum rule)
– two electrons with the same spin cannot be at the same position (Pauli principle)
– the hole vanishes at large distances (short-sightedness)
• Model:
– homogeneously charged sphere
For a free electron gas not much worse than a Hartree Fock calculation
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(r0)=φ−e
φ(r)−e
h(r,r0)r0
4 0πεe2 n(r0)2π
g (r,r0)n(r)
r
r
r
3334
exchange-correlation energy per electron
Tuesday, October 4, 11
Exc
=
Z 1
0d�
Zd3r n(~r)
1
2
Zd3r0
e2h�
(~r, ~r0)
4⇡✏0|~r � ~r0|| {z }dE/d�
=
Zd3r n(~r)
1
2
Zd3r0
e2R 10 d�h
�
(~r, ~r0)
4⇡✏0|~r � ~r0|
H(�) = H0 + �W
Exc
[n(1)] = FW [n(1)]� F 0[n(1)]� EHartree
Correlation energy
• Exchange energy uses hole of non-interacting electrons
• Coulomb repulsion pushes electrons away
• deformation of wave function raises kinetic energy
•adiabatic connection:– switch the interaction slowly on and
integrate opposing “force”
– never evaluate kinetic energy explicitly
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Coulomb hole
Correlation energy is due to the balance between energy gain by lowering the
Coulomb repulsion and kinetic energy cost
h (r
r0
r
0,r)h (r
h (r0,r)
!1
0
0
!1
r
h (r
Tuesday, October 4, 11
Jacobs ladder to heaven
14
5. Exact: Constrained search
4. Hybrid functionals: Include exact (nonlocal) exchange: left-right correlations
3. Meta-GGA: use kinetic energy density to estimate the flexibility of the electron gas
2. GGA: asymmetry of the xc-hole favors surfaces and thus weakens bonds
1. LDA: xc-hole from free electron gas: strong overbinding
Tuesday, October 4, 11
Local density approximation (LDA)
• Local density aproximation (LDA) imports the “exact” exchange correlation hole of a free electron gas
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0 0
r−r00−0.3a 0−0.2a 00.1a0−0.1a r0
r−r000.3a00.2a00.1a
r−r0−1.5a0 −1.0a0 −0.5a0 r0 00.5a
00.5a 01.0a 01.5a 02.0a
Exact
LSD
R
R
Exact
LSD
LSD
ExactLSD
Exact
R
Exc
=
Zd3r n(~r)✏
xc
(n(~r))
from free electron gas
local
Why does it work?Sum rules!
• charge sum rule exact• only spherical average matters• only first moment of radial hole matters
✏xc
=1
2
4⇡e2
4⇡✏0
Z 1
0dr r
⌧h(~r0,~r0 + ~r)
�
angle
1 = 4⇡
Z 1
0dr r2
⌧h(~r0,~r0 + ~r)
�
angle
Tuesday, October 4, 11
Generalized gradient approximation (GGA)
• an electron far away from an atom sees a positive ion
• the exchange hole is entirely on the atom.
• the LDA hole is centered on the electron and smeared out over a wide region.
• LDA overestimates the exchange energy at the surface
• GGA stabilizes the surfaces (overbinding reduced)
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✏xc
(r ! 1) = �1
2
e2
4⇡✏r�1
| {z }GGA
<< A exp(��r)| {z }LDA
reduced gradient is largest in the tails
GGA correction stabilizes tails
binding energies are dramatically improved
DFT became of interest to chemists
rs| n|Δ
rsrs r0
n
−h(r0,r)
r
n reduced gradient x=|�n|/n4/3
Tuesday, October 4, 11
Hybrid functionals• adiabatic connection
• approximate integral by weighted sum of the values at the end-points of the interval
– Hartree-Fock exchange introduces nonlocality
– finite band gaps for Mott insulators
– correct dissociation limit of bonds
– related to GW, LDA+U, etc.
17
Exc
=
Z 1
0d�
Zd3r n(~r)
1
2
Zd3r0
e2h�
(~r, ~r0)
4⇡✏0|~r � ~r0|| {z }dE/d�
=
Zd3r n(~r)
1
2
Zd3r0
e2R 10 d�h
�
(~r, ~r0)
4⇡✏0|~r � ~r0|
H(�) = H0 + �W
d3r 4 0|r−r’|πεe2h (r,r’)λ
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Hartree−Fock(hole of non−interacting electrons)
xcε
0 1 λ
λ=0 no interaction:exchange
λ=1 full interaction
λ-average (area or dashed line):exchange-correlation
Exc =
Zd� Uxc(�)
⇡ 1
4Uxc(� = 0) +
3
4Uxc(� = 1)
=1
4UHFxc| {z }
EHFx
�1
4UGGAxc| {z }
EGGAx
+1
4UGGAxc (� = 0) +
3
4UGGAxc (� = 1)
| {z }EGGA
xc
= EGGAxc +
1
4
✓EHF
x � EGGAx
◆
Tuesday, October 4, 11
Band-gap problem
• GGA’s underestimate band gaps– in silicon 0.7 eV instead
of 1.17 eV
• Many transition metal oxides are metals instead of insulators!
• Problem: energy of average density instead of average of energies
18
e=
g
LDA
Eg
correct
dN
dE
E
N
E
10!1 2
Tuesday, October 4, 11
Electronic structure methods:how to solve the Kohn-Sham equations
19
pseudopotentialsHamann-Bachelet-Schlueter, Troulier-Martins, Kleinman-Bylander, Vanderbilt ultrasoft
linear methodslinear augmented muffin tin-orbital (LMTO) method linear augmented plane wave (LAPW) method
projector augmented wave method
augmented wave methodsKorringa Kohn Rostocker (KKR), augmented plane wave (APW) method
Tuesday, October 4, 11
atomic valence wave functions (s and d) of Pt
r
What is the problem?
1. wave function oscillates strongly in the atomic region (Coulomb singularity)
2. wave function needs to be very flexible in the bonding region and the tails (Chemistry)
3. most electrons (core) are “irrelevant”
4. relativistic effects
5. tiny but finite nucleus
20
K.Sc
hwar
z Ta
lk/W
WW
Fro
m A
PW t
o LA
PW t
o (L
)APW
+lo
Tuesday, October 4, 11
Strategies
Pseudopotentials
21
• Chop off singular potential• node-less wave functions• no core states• no information on inner electrons• transferability problems
Augmented waves
• start with envelope function• replace incorrect shape with
atomic partial waves• complex basisset• retain full information on wave
functions and potential
LMTO, ASW, LAPW, APW, KKR, PAW, (all-electron)
Hamann-Bachelet-Schlüter, Kerker, Kleinman-Bylander, Troullier Martins, Ultrasoft,etc
Tuesday, October 4, 11
PAW augmentation
22
| i|{z}all-electron
=
pseudoz}|{| i +
P↵ |�↵ihp↵| ˜ iz}|{| 1i|{z}
1-center, all-el.
�
1-center, pseudoz}|{| 1i|{z}
P↵ |˜�↵ihp↵| ˜ i
+ -=
= -+
Tuesday, October 4, 11
PAW transformation theory
23
| i = | i+X
↵
⇣|�↵i � |�↵i
⌘hp↵| i
plane waves atomic partial wave
auxiliarypartial wave
projector function
plane waves
T = 1 +X
R
SR SR =X
↵
⇣|�↵i � |�↵i
⌘hp↵|
all-electronz}|{| i|{z}true
= T
pseudoz}|{| i|{z}
auxiliary
with
0 1 2 3r [a0]
-1
0
1
2
ps
d
Fe
Tuesday, October 4, 11
PAW transformation theory
24
| i = | i+X
↵
⇣|�↵i � |�↵i
⌘hp↵| i
plane waves atomic partial wave
auxiliarypartial wave
projector function
plane waves
T = 1 +X
R
SR SR =X
↵
⇣|�↵i � |�↵i
⌘hp↵|
all-electronz}|{| i|{z}true
= T
pseudoz}|{| i|{z}
auxiliary
with
Tuesday, October 4, 11
=
+
PAW augmentation
25
| i|{z}all-electron
=
pseudoz}|{| i +
P↵ |�↵ihp↵| ˜ iz}|{| 1i|{z}
1-center, all-el.
�
1-center, pseudoz}|{| 1i|{z}
P↵ |˜�↵ihp↵| ˜ i
= + -
Tuesday, October 4, 11
E =X
n
h n
|�~22m
e
~r2| n
i+Z
d3r v(~r)n(~r)
+1
2
Zd3r
Zd3r0
e2[n(~r) + Z(~r)][n(~r0) + Z(~r0)]
4⇡✏0|~r � ~r0| + Exc
[n(~r)]
E1R
=X
i,j2R
Di,j
h�j
|�~22m
e
~r2|�i
i+Nc,RX
n2R
h�c
n
|�~22m
e
~r2|�c
n
i
+1
2
Zd3r
Zd3r0
e2[n1(~r) + Z(~r)][n1(~r0) + Z(~r0)]
4⇡✏0|~r � ~r0| + Exc
[n1(~r)]
E1R
=X
i,j2R
Di,j
h�j
|�~22m
e
~r2|�i
i+Z
d3r v(~r)n1(~r)
+1
2
Zd3r
Zd3r0
e2[n1(~r) + Z(~r)][n1(~r0) + Z(~r0)]
4⇡✏0|~r � ~r0| + Exc
[n1(~r)]
PAW total energy
26
E0[{| ni, Rj}] = E +X
R
⇣E1
R � E1R
⌘Also, the total energy is written as sum of• an extended plane wave part• two one-center expansions for each site
“smart zero”
contains pseudized core(non-linear core correction)
Compensation density:
Tuesday, October 4, 11
Accuracy
• VASP: [Paier,Hirschl, Marsman, Kresse JCP122,234102 (1005)]
• GPAW: [Carsten Rostgaard]
27
All-electron calculations with PAW (continued)
PBE atomization energies relative to Gaussian:
Tuesday, October 4, 11
The End
28Tuesday, October 4, 11