Download Basics of density functional perturbation theory

CECAM Tutorial: “Dynamical, dielectric and magnetic properties of solids with ABINIT”
Lyon, 12-16 may 2014
Basics of density functional
perturbation theory
Gian-Marco Rignanese
European Theoretical Spectroscopy Facility
Institute of Condensed Matter and Nanosciences,
B-1348 Louvain-la-Neuve, Belgique
Born-Oppenheimer approximation
• Quantum treatment for electrons →
Kohn-Sham equation
Eel [r] = Â yv |T +Vext | yv ⇥ + EHxc [r]
v

r(r) = Â yv⇤ (r)yv (r)
v
1 2
— +Vext (r) +VHxc (r) yi (r) = ei yi (r)
2
• Classical treatment for nuclei →
unit cell
a
Fka
=
atom
∂ Etot
∂ Raka
direction
(α=1,2,3)
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Newton equation
d 2 Raka
= Mk
dt 2
{R}
Etot = Eion + Eel
2
Harmonic approximation
Etot
a
Rκα
= Raα + τκα + uaκα
cell
equilibrium
atomic
position
displacement
u
Etot ({u})
(0)
= Etot +
1
∑ ′∑′ ′ 2
aκα a κ α
CECAM Tutorial, Lyon, 12-16 may 2014
!
∂ 2 Etot
′
a
a
∂ uκα ∂ uκ ′ α ′
"
a′
a
uκα uκ ′ α ′
3
Phonons
• The matrix of interatomic force!constants (IFCs)
" is defined as
Cκα ,κ ′ α ′ (a, a′ ) =
∂ 2 Etot
′
a
a
∂ uκα ∂ uκ ′ α ′
• Its Fourier transform (using translational invariance)
C̃κα ,κ ′ α ′ (q) = ∑ Cκα ,κ ′ α ′ (0, a′ )eiq·Ra′
a′
allows one to compute phonon frequencies and eigenvectors as
solutions of the following generalized eigenvalue problem:
2
C̃ka,k 0 a 0 (q)Umq (k 0 a 0 ) = Mk wmq
Umq (ka)
Â
0 0
ka
phonon
displacement
pattern
CECAM Tutorial, Lyon, 12-16 may 2014
masses
square of
phonon
frequencies
4
Example: Diamond
1400
1200
z
-1
Frequency(cm)
1000
L
U
€
x
800
K
X
W
y
600
400
200
0
€
K,U
X
€
L
Theory vs. Experiment
[X. Gonze, GMR, and R. Caracas, Z. Kristallogr. 220, 458 (2000)]
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5
Total energy derivatives
• In fact, many physical properties are derivatives of the total
energy (or a suitable thermodynamic potential) with respect to
external perturbations.
• Possible perturbations include:
★ atomic
displacements,
★ expansion or contraction of the primitive cell,
★ homogeneous external field (electric or magnetic),
★ alchemical change
★ ...
CECAM Tutorial, Lyon, 12-16 may 2014
6
Total energy derivatives
• Related derivatives of the total energy (Eel + Eion)
★ 1st
order:
★ 2nd order:
★ 3rd
order:
forces, stress, dipole moment, ...
dynamical matrix, elastic constants, dielectric
susceptibility, Born effective charge tensors,
piezoelectricity, internal strains
non-linear dielectric susceptibility, phonon-phonon
interaction, Grüneisen parameters, ...
• Further properties can be obtained by integration over phononic
degrees of freedom (e.g., entropy, thermal expansion, ...)
CECAM Tutorial, Lyon, 12-16 may 2014
7
Total energy derivatives
• These derivatives can be obtained from
★ direct
approaches:
■ finite differences (e.g. frozen phonons)
■ molecular-dynamics spectral analysis methods
★ perturbative approaches
• The former have a series of limitations (problems with
commensurability, difficulty to decouple the responses to
perturbations of different wavelength, ...). On the other hand, the
latter show a lot of flexibility which makes it very attractive for
practical calculations.
CECAM Tutorial, Lyon, 12-16 may 2014
8
Outline
Perturbation Theory
Density Functional Perturbation Theory
Atomic displacements and
homogeneous electric field
CECAM Tutorial, Lyon, 12-16 may 2014
9
Outline
Perturbation Theory
Density Functional Perturbation Theory
Atomic displacements and
homogeneous electric field
CECAM Tutorial, Lyon, 12-16 may 2014
10
Reference and perturbed systems
• Let us assume that all the solutions are known for a reference
system for which the one-body Schrödinger equation is:
E
E
(0)
(0)
(0)
H (0) yi
= ei yi
with the normalization condition:
D
E
(0)
(0)
yi yi
=1
• Let us now introduce a perturbation of the external potential
characterized by a small parameter λ:
(0)
(1)
(2)
Vext (λ ) = Vext + λ Vext + λ 2Vext + · · ·
known at all orders.
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11
Reference and perturbed systems
• We now want to solve the perturbed Schrödinger equation:
H(l )|yi (l ) = ei (l )|yi (l )
with the normalization condition:
yi (l )|yi (l )⇥ = 1
• Idea: all the quantities (X=H, εi, ψi) are written as a perturbation
series with respect to the parameter λ :
X(λ ) = X (0) + λ X (1) + λ 2 X (2) + λ 3 X (3) + · · ·
with X
(n)
1 dnX
=
n! dl n
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l =0
12
Expansion of the Schrödinger equation
• Starting from:
and inserting:
H(l )|yi (l )⇥ = ei (l )|yi (l )⇥
l
H(! ) = H (0) + ! H (1) + ! 2 H (2) + . . .
(0)
yi (l ) = yi
(0)
ei (l ) = ei
(1)
+ l yi
(1)
+ l ei
+ l 2 yi
(2)
+ l 2 ei
(2)
+...
+...
we get:
⌘
⌘⇣
(0)
(1)
(2)
H (0) + ! H (1) + ! 2 H (2) + . . . |"i + ! |"i + ! 2 |"i + . . . =
⇣
⌘⇣
⌘
(0)
(1)
(2)
(0)
(1)
(2)
#i + ! #i + ! 2 #i + . . . |"i + ! |"i + ! 2 |"i + . . .
⇣
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13
Expansion of the Schrödinger equation
• This can be rewritten as:
(0)
H (0) |yi
setting λ=0
⇣
⌘
(1)
(0)
+l H (0) |yi + H (1) |yi
⇣
⌘ deriving w.r.t. λ
(2)
(1)
(0)
+l 2 H (0) |yi + H (1) |yi + H (2) |yi
and setting λ=0
+ ... =
deriving twice
(0)
(0)
ei |yi
⇣
⌘
(0) (1)
(1) (0)
+l ei |yi + ei |yi
⇣
⌘
(0)
(2)
(1)
(1)
(2)
(0)
+l 2 ei |yi + ei |yi + ei |yi
w.r.t. λ and
setting λ=0
+ ...
CECAM Tutorial, Lyon, 12-16 may 2014
14
Expansion of the Schrödinger equation
• Finally, we have that:
(0)
= "i |!i
(1)
+ H (1) |!i
(2)
+ H (1) |!i
H (0) |!i
H (0) |!i
H (0) |!i
(0)
(0)
(0)
0th order
(0)
= "i |!i
(1)
(2)
"i |!i
(0)
(1)
+ "i |!i
+ H (2) |!i
(0)
=
+ "i |!i
+ "i |!i
(1)
(1)
(1)
(2)
(0)
(0)
1st order
2nd order
···
CECAM Tutorial, Lyon, 12-16 may 2014
15
Expansion of the normalization condition
• Starting from:
and inserting:
⇥yi (l )|yi (l )⇤ = 1
(0)
yi (l ) = yi
(1)
+ l yi
l
+ l 2 yi
(2)
+...
we get:
(0) (0)
yi |yi ⇥
⇣
⌘
(0) (1)
(1) (0)
+l yi |yi ⇥ + yi |yi ⇥
⇣
(0) (2)
(1) (1)
2
yi |yi ⇥ + yi |yi ⇥ +
+l
+ ... = 1
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(2) (0)
yi |yi ⇥
⌘
16
Expansion of the normalization condition
• Finally, we have that
(0)
(0)
(0)
(1)
(1)
(0)
(0)
(2)
(1)
(1)
yi |yi ⇥ = 1
0th order
yi |yi ⇥ + yi |yi ⇥ = 0
1st order
(2)
(0)
yi |yi ⇥ + yi |yi ⇥ + yi |yi ⇥ = 0
2nd order
···
CECAM Tutorial, Lyon, 12-16 may 2014
17
1st order corrections to the energies
• Starting from the 1st order of the Schrödinger equation
(1)
H (0) |yi
(0)
+ H (1) |yi
(0)
(0)
and premultiplying by yi | , we get:
(0)
(1)
(0)
(0)
(1)
= ei |yi
(0)
(0)
(1)
(0)
+ ei |yi
(1)
(1)
(0)
(0)
yi |H (0) |yi ⇥ + yi |H (1) |yi ⇥ = ei yi |yi ⇥ + ei yi |yi ⇥
| {z }
|
{z
}
(0)
(0) (1)
1
ei yi |yi ⇥
and thus, finally, we have the Hellman-Feynman theorem:
(1)
ei
•
(0)
(0)
= yi |H (1) |yi ⇥
The 0th order wavefunctions are thus the only required ingredient
to obtain the 1st order corrections to the energies.
CECAM Tutorial, Lyon, 12-16 may 2014
18
2nd order corrections to the energies
• Starting from the 2nd order of the Schrödinger equation
(2)
(1) (1)
(2) (0)
H |!i + H |!i + H |!i =
(0) (2)
(1) (1)
(2) (0)
"i |!i + "i |!i + "i |!i
(0)
and premultiplying by yi | , we get:
(0)
(0) (2)
ei yi |yi ⇥
(0)
|
{z
}
(0)
(2)
(0)
(1)
(0)
(0)
!i |H (0) |!i ⇥ + !i |H (1) |!i ⇥ + !i |H (2) |!i ⇥ =
(0)
"i
(0)
(2)
(1)
!i |!i ⇥ + "i
and thus, finally, we have:
(2)
!i
=
(0)
(1)
(2)
!i |!i ⇥ + "i
(0)
(0)
(2) (0)
⇥"i |H |"i ⇤ + ⇥"i |H (1)
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(0)
(0)
!i |!i ⇥
| {z }
1
(1) (1)
!i |"i ⇤
19
2nd order corrections to the energies
• Since the energies are real, we can write that:
(2)
!i
=
(0)
(0)
(2) (0)
⇥"i |H |"i ⇤ + ⇥"i |H (1)
(0)
(1)
(2) (0)
⇥"i |H |"i ⇤ + ⇥"i |H (1)
=
or, combining both equalities:
(2)
(0)
=⇥"i |H (2) |"i ⇤
⌘
1 ⇣ (0) (1)
(1) (1)
(1)
(1) (0)
!i |"i ⇤ + ⇥"i |H (1) !i |"i ⇤
+
⇥"i |H
2
Using the expansion of the normalization condition at 1st order,
!i
•
(0)
(1) (1)
!i |"i ⇤
(1) (0)
!i |"i ⇤
we can finally write that: ⇣
⌘
1
(2)
(0)
(0)
(0)
(1)
(1)
(0)
!i = "i |H (2) |"i ⇥ +
"i |H (1) |"i ⇥ + "i |H (1) |"i ⇥
2
• To obtain the 2nd order corrections to the energies, the only
required ingredients are the 0th and 1st order wavefunctions.
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20
1st order corrections to the wavefunctions
• The 1st order of the Schrödinger equation
H
(0)
can be rewritten
⇣
H (0)
•
(1) (0)
+ ei |yi
+H
(1)
gathering the terms containing |yi :
⌘
⇣
⌘
(0)
(1)
(1)
(0)
ei
|yi ⇥ =
H (1) ei
|yi ⇥
(1)
|yi
(1)
(0)
|yi
(0) (1)
= ei |yi
producing the so-called Sternheimer equation.
(1)
In order to get |yi one would like to invert the
(H (0) ei(0) )
(0)
operator, but it cannot be done as such since ei is an eigenvalue
of H (0). The problem can be solved by expressing the 1st order
wavefunction as a linear combination of the 0th order ones:
(1)
|yi
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= Â ci j |y j
(1)
(0)
j
21
1st order corrections to the wavefunctions
• We separate the 0th order wavefunctions into two subsets:
★ those
(0) (0)
associated to ei(0) :
j I if H (0) |y (0)
⇥
=
e
j
i |y j ⇥
(0)
(just |yi if the energy is non-degenerate)
★ those that belong to the subspace that is orthogonal: j ∈ I⊥
|yi ⇤ = Â ci j |y j ⇤ = Â ci j |y j ⇤ +
(1)
(1)
(1)
(0)
(0)
j I
j
Â
(1)
j I⇥
equation can thus be rewritten as:
• The Sternheimer
⌘
⇣
⌘
⇣
H (0)
(0)
ei
|yi ⌅ = Â ci j
(1)
(1)
j
=
Â
j⇥I ⇤
=
CECAM Tutorial, Lyon, 12-16 may 2014
⇣
H (0)
⇣
(1)
(0)
ci j e j
H (1)
(1)
ei
(0)
ei
⌘
(0)
|y j ⌅
(0)
ei
(0)
ci j |y j ⇤
⌘
(0)
|y j ⌅
(0)
|yi ⌅
22
1st order corrections to the wavefunctions
•
Premultiplying by yk(0) | with k ∈ I⊥, we get:
⌘
⇣
(0) (0)
(0)
(1) (0)
(0)
(1)
(0)
(1)
⇧
=
⌅y
|H
e
e
⌅y
|y
c
e
 ij j
j
i |yi ⇧
i
k
k
| {z }
j⇥I ⇤
(0) (0)
⇤yk |yi ⌅ = 0
δkj
since k I ⇥
⌘
⇣
(0)
(0)
(1)
(0)
(0)
= ⇥yk |H (1) |yi ⇤
cik ek
ei
and, thus:
(1)
ci j
=
1
(0)
ei
(0)
(0)
(1)
⇤
⌅y
|H
|y
⇧
for
j
⇥
I
j
i
(0)
ej
CECAM Tutorial, Lyon, 12-16 may 2014
23
1st order corrections to the wavefunctions
• Premultiplying by yk(0) | with k ∈ I, we get:
Â
j⇥I ⇤
(1)
ci j
⇣
(0)
ej
(0)
ei
⌘
(0)
(0)
(0)
(1)
|
(1)
ek dki
{z
0
=
}
(0)
ei |yi ⇧
⌅yk |y j ⇧ = ⌅yk |H (1)
(0)
(0)
⇥yk |yi ⇤ = dki
since k
(0)
(0)
yk |H (1) |yi ⇥
I
• For k = i, it is nothing but the Hellmann-Feynman theorem.
(1)
•
But, it does not provide any information on the ci j for j 2 I.
In fact, there is a gauge freedom that allows to choose them
equal to zero.
CECAM Tutorial, Lyon, 12-16 may 2014
24
1st order corrections to the wavefunctions
• Finally, we can write the so-called sum over states expression:
(1)
|yi ⇧ =
•
Â
(0)
|y j ⇧ (0)
ei
j⇥I ⇤
1
(0)
(0)
(1)
⌅y
|H
|yi ⇧
j
(0)
ej
which requires the knowledge of all the 0th order wavefunctions
and energies.
Instead, if we define the projector PI ? onto the subspace I ? by:
PI ⇥ =
Â
j I⇥
(0)
(0)
|y j ⌅⇤y j |
we can rewrite the Sterheimer equation in that subspace:
⇣
⌘
(0)
(1)
(0)
(1) (0)
PI ⇥ H
ei
PI ⇥ |yi ⇤ = PI ⇥ H |yi ⇤
CECAM Tutorial, Lyon, 12-16 may 2014
25
1st order corrections to the wavefunctions
• In this form, the singularity has disappeared and it can thus be
inverted:
(1)
PI ⇥ |yi ⇤ =
h
⇣
PI ⇥ H
(0)
(0)
ei
⌘
PI ⇥
i
1
H
(1)
(0)
|yi ⇤
and defining the Green’s function in the subspace I ? as:
h ⇣
⌘ i 1
GI ? (e) = PI ? e H (0) PI ?
we can write:
(1)
(0)
(0)
PI |yi ⇥ = GI (ei )H (1) |yi ⇥
• This is the Green's function technique for dealing with the
Sternheimer equation.
CECAM Tutorial, Lyon, 12-16 may 2014
26
2nd order corrections to the energies
• The sum over states expression for the 1st order wavefunctions:
(1)
|yi ⇧ =
Â
(0)
|y j ⇧ (0)
ei
j⇥I ⇤
1
(0)
(0)
(1)
⌅y
|H
|yi ⇧
j
(0)
ej
can be inserted in the 2nd order corrections to the energies:
⇣
⌘
1
(2)
(0)
(0)
(0)
(1)
(1)
(0)
!i = "i |H (2) |"i ⇥ +
"i |H (1) |"i ⇥ + "i |H (1) |"i ⇥
2
leading to:
(2)
!i
(0)
(0)
=⌅"i |H (2) |"i ⇧
+
(0)
(0)
⌅"i |H (1) |" j ⇧ (0)
!i
j⇥I ⇤
#
CECAM Tutorial, Lyon, 12-16 may 2014
1
(0)
(0)
(1)
"
|H
|"i ⇧
⌅
j
(0)
!j
27
2nd order corrections to the energies
• Alternatively, we can write that:
⇤!i (" )|H(" ) #i (" )|!i (" )⌅ = 0 ⇥"
and the perturbation expansion at the 2nd order gives:
(0)
⇥!i |H (0)
(0)
+⇥!i |H (1)
(0)
+⇥!i |H (2)
(0)
(2)
"i |!i ⇤
(1) (1)
(1)
"i |!i ⇤+⇥!i |H (0)
(2) (0)
(1)
"i |!i ⇤+⇥!i |H (1)
(0)
(1)
"i |!i ⇤
(1) (0)
(2)
"i |!i ⇤+⇥!i |H (0)
(0)
(0)
"i |!i ⇤ = 0
• It can be shown that the sum of the terms in a row or in column
vanishes! Getting rid of the first row and the last column, we get
another expression for the 2nd order corrections to the energies:
(2)
!i
(0)
(1)
(0) (1)
(2) (0)
(0)
=⇥"i |H |"i ⇤ + ⇥"i |H
!i |"i ⇤
(0)
(1) (1)
(1)
(1) (0)
(1)
(1)
+ ⇥"i |H
!i |"i ⇤ + ⇥"i |H
!i |"i ⇤
CECAM Tutorial, Lyon, 12-16 may 2014
28
2nd order corrections to the energies
• Actually, a number of other expressions exist for the 2nd order
•
corrections to the energies.
However, it can be demonstrated that this expression is
variational in the sense that the 2nd order corrections to the
(1)
!
energies can be obtained by minimizing it with respect to i :
n
! (2) = min ⇥" (0) |H (2) |" (0) ⇤ + ⇥" (1) |H (0) ! (0) |" (1) ⇤
i
i
i
i
i
i
(1)
"i
(0)
+⇥"i |H (1)
under the constraint that:
(0)
(1) (1)
(1)
!i |"i ⇤ + ⇥"i |H (1)
(1)
(1)
(1) (0)
!i |"i ⇤
o
(0)
!i |!i ⇥ + !i |!i ⇥ = 0
CECAM Tutorial, Lyon, 12-16 may 2014
29
3rd order corrections to the energies
• Starting from the 3rd order of the Schrödinger equation
(3)
H (0) |!i
(2)
+ H (1) |!i
(0)
(3)
"i |!i
(1)
+ H (2) |!i
(1)
(2)
+ "i |!i
(0)
+ H (3) |!i
(2)
(1)
+ "i |!i
and premultiplying by yi(0) | , we get:
(0)
(0) (3)
!i "i |"i ⇥
|
{z
}
(0)
(0)
(0) (3)
(1) (2)
!i |H |!i ⇥ + !i |H |!i ⇥
+
=
(3)
(0)
+ "i |!i
(0)
(0)
(2) (1)
(3) (0)
!i |H |!i ⇥ + !i |H |!i ⇥
(0)
(0) (3)
(1)
(0) (2)
= "i !i |!i ⇥ + "i !i |!i ⇥
(2)
(0) (1)
(3)
(0) (0)
+ "i !i |!i ⇥ + "i !i |!i ⇥
CECAM Tutorial, Lyon, 12-16 may 2014
|
{z
1
}
30
3rd order corrections to the energies
• Finally, we we can write:
(3)
!i
(0)
(0)
=⇥"i |H (3) |"i ⇤
(0)
+ ⇥"i |H (2)
(0)
+ ⇥"i |H (1)
(2)
(1)
(1)
(2)
!i |"i ⇤
!i |"i ⇤
• This expression of the 3rd order corrections to the energies
requires to know the wavefunctions up to the 2nd order.
CECAM Tutorial, Lyon, 12-16 may 2014
31
3rd order corrections to the energies
• Alternatively, we can write that:
⇤!i (" )|H(" ) #i (" )|!i (" )⌅ = 0 ⇥"
and the perturbation expansion at the 3rd order gives:
(0)
⇥!i |H (0)
(0)
+⇥!i |H (1)
(0)
+⇥!i |H (2)
(0)
+⇥!i |H (3)
(0)
(3)
"i |!i ⇤
(1) (2)
(1)
"i |!i ⇤+⇥!i |H (0)
(2) (1)
(1)
"i |!i ⇤+⇥!i |H (1)
(3) (0)
(1)
"i |!i ⇤+⇥!i |H (2)
(0)
(2)
"i |!i ⇤
(1) (1)
(2)
"i |!i ⇤+⇥!i |H (0)
(2) (0)
(2)
"i |!i ⇤+⇥!i |H (1)
(0)
(1)
"i |!i ⇤
(1) (0)
(3)
"i |!i ⇤+⇥!i |H (0)
(0)
(0)
"i |!i ⇤ = 0
• Again, the sum of the terms in a row or in column vanishes. So,
getting rid of the first two rows and the last two columns, we get
another expression that does not require to know the 2nd order
wavefunctions :
(3)
(0)
(0)
(1)
(1) (1)
!i =⇥"i |H (3) |"i ⇤ + ⇥"i |H (1) !i |"i ⇤
(0)
(1)
(1)
(0)
+ ⇥"i |H (2) |"i ⇤ + ⇥"i |H (2) |"i ⇤
CECAM Tutorial, Lyon, 12-16 may 2014
32
Summary
• There are 4 different methods to get the 1st order wavefunctions:
★ solving
the Sternheimer equation directly, complemented by a
condition derived from the normalization requirement
★ using
the Green’s function technique
★ exploiting
the sum over states expression
the constrained functional for the 2nd order
corrections to the energies
★ minimizing
• With these 1st order wavefunctions, both the 2nd and 3rd order
corrections to the energies can be obtained.
• More generally, the nth order wavefunctions give access to the
(2n)th and (2n+1)th order energy [“2n+1” theorem].
CECAM Tutorial, Lyon, 12-16 may 2014
33
Outline
Perturbation Theory
Density Functional Perturbation Theory
Atomic displacements and
homogeneous electric field
CECAM Tutorial, Lyon, 12-16 may 2014
34
Reference system
• In DFT, one needs to minimize the electronic energy functional:
D
E
(0)
(0)
(0)
(0)
(0)
Eel [r ] = Â yi T +Vext yi
+ EHxc [r (0) ]
Ne
i=1
in which the electronic density is given by:
i⇤
Ne h
(0)
(0)
r (0) (r) = Â yi (r) yi (r)
under the constraint that:
D
i=1
(0)
yi
(0)
yj
E
= di j
equation:
• Alternatively,Eonecan solve the reference Shrödinger
E
E
1 2
(0)
(0)
(0)
(0)
(0)
= ei yi
H
=
— +Vext +VHxc yi
2
where the Hartree and exchange correlation potential is:
(0)
(0) ]
[r
d
E
(0)
Hxc
VHxc (r) =
d r(r)
(0)
(0)
yi
CECAM Tutorial, Lyon, 12-16 may 2014
35
Perturbed system
• The electronic energy functional to be minimized is:
Ne
Eel [r(l )] = Â yi (l ) |T +Vext (l )| yi (l )⇥ + EHxc [r(l )]
i=1
in which the electronic density is given by:
Ne
r(r; l ) = Â yi⇤ (r; l )yi (r; l )
under the constraint that:
i=1
↵
yi (l )| y j (l ) = di j
• Alternatively, the Shrödinger equation to be solved is:
1 2
H(l ) |yi (l )⇥ =
— +Vext (l ) +VHxc (l ) |yi (l )⇥ = ei (l ) |yi (l )⇥
2
where the Hartree and exchange correlation potential is:
d EHxc [r(l )]
VHxc (r; l ) =
d r(r)
CECAM Tutorial, Lyon, 12-16 may 2014
36
1st order of perturbation theory in DFT
• For the energy, it can be shown that:
D
E d
(0)
(0)
= Â yi (T +Vext )(1) yi
EHxc [r (0) ]
+
dl
l =0
i=1
This is the equivalent of the Hellman-Feynman theorem for
density-functional formalism.
(1)
Eel
Ne
the constraint leads to:
• For the wavefunctions,
D
E D
E
(0)
yi
(1)
yj
CECAM Tutorial, Lyon, 12-16 may 2014
+
(1)
yi
(0)
yj
=0
37
2nd order energy in DFPT
• For the energy, it can be shown that:
hD
E D
Ei
(2)
(1)
(0)
(0)
(1)
Eel = Â yi (T +Vext )(1) yi
+ yi (T +Vext )(1) yi
Ne
i=1
Ne
hD
E D
(0)
(0)
(1)
+ Â yi (T +Vext )(2) yi
+ yi (H
i=1
1
+
2
+
with
Z
Z Z
(1)
ei )(0) yi
d 2 EHxc [r (0) ] (1)
(1) 0
0
r
(r)r
(r
)drdr
d r(r)d r(r0 )
d EHxc [r (0) ]
dl d r(r)
l =0
2
1
d
(0)
E
[r
]
r (1) (r)dr +
Hxc
2
2 dl
h
i⇤
⌘
⇣h
i⇤
(0)
(0)
(1)
(1)
r (1) (r) = Â yi (r) yi (r) + yi (r) yi (r)
Ei
l =0
Ne
i=1
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38
1st order wavefunctions in DFPT
• The 1st order wavefunctions can be obtained by minimizing:
hn
o n
oi
(2)
(2)
(0)
(1)
Eel = Eel
yi
, yi
n
o
D
E
(1)
with respect to yi
under the constraint: yi(0) y (1)
=0
j
• These can also be obtained by solving the Sternheimer equation:
⇣
Ĥ (0)
(0)
ei
⌘
(1)
|yi ⇥ =
H (1) = (T +Vext )(1) +
(1)
ei
=
CECAM Tutorial, Lyon, 12-16 may 2014
Z
⇣
Ĥ (1)
(1)
ei
⌘
(0)
|yi ⇥
d 2 EHxc [r (0) ] (1) 0 0
r (r )dr
0
d r(r)d r(r )
(0) (1) (0)
yi |H |yi ⇥
39
Higher orders in DFPT
• More generally, it easy to show that since there is a variational
principle for the 0th order energy:
hn
oi
(0)
(0)
(0)
Eel = Eel
yi
non-variational expression can be obtained for higher orders:
oi
hn
(1)
(1)
(0)
Eel = Eel
yi
hn
o n
oi
(2)
(2)
(0)
(1)
Eel = Eel
yi
, yi
hn
o n
o n
oi
(3)
(3)
(0)
(1)
(2)
Eel = Eel
yi
, yi
, yi
•
···
But, this is not the best that can be done!
CECAM Tutorial, Lyon, 12-16 may 2014
40
Higher orders in DFPT
• We assume that all wavefunctions are known up to (n-1)th order:
(0)
(1)
n 1 (n 1)
<n
n
+ O(l n )
ȳi = yi + O(l ) = yi + l yi + · · · + l yi
• The variational property of the energy functional implies that:
Eel [{ytrial + O(h)}] = Eel [{ytrial }] + O(h 2 )
• Taking {ytrial } = yi<n and h = l n , we see that:
the wavefunctions are known up to (n-1)th order,
the energy is know up to (2n-1)th order;
★ if
the wavefunctions are known up to nth order,
the energy is know up to (2n+1)th order.
★ if
• This is the “2n+1” theorem.
CECAM Tutorial, Lyon, 12-16 may 2014
41
Higher orders in DFPT
• Since the variational principle is also an extremal principle
[the error is either > 0 → minimal principle, or
< 0 → maximal principle], the leading missing
term is also of definite sign (it is also an extremal principle):
oi
hn
(0)
=
yi
hn
oi
(1)
(1)
(0)
Eel = Eel
yi
hn
o n
oi
(2)
(2)
(0)
(1)
Eel = Eel
yi
, yi
hn
o n
oi
(3)
(3)
(0)
(1)
Eel = Eel
yi
, yi
hn
o n
o n
oi
(4)
(4)
(0)
(1)
(2)
Eel = Eel
yi
, yi
, yi
hn
o n
o n
oi
(5)
(5)
(0)
(1)
(2)
Eel = Eel
yi
, yi
, yi
(0)
Eel
(0)
Eel
CECAM
· · · Tutorial, Lyon, 12-16 may 2014
n
o
(0)
variational w.r.t. yi
n
o
(1)
variational w.r.t. yi
n
o
(2)
variational w.r.t. yi
42
Mixed derivatives in DFPT
• Similar expressions exist for mixed derivatives (related to two
different perturbations j1 and j2):
n
o
(0)
Eelj1 j2 = Eelj1 j2 yi ; yij1 ; yij2
• The extremal principle is lost but the expression is stationary:
★ the
error is proportional to the product of errors made in the
1st order quantities for the first and second perturbations;
★ if these errors are small, their product will be much smaller;
★ however, the sign of the error is undetermined, unlike for the
variational expressions.
CECAM Tutorial, Lyon, 12-16 may 2014
43
Order of calculation in DFPT
(0)
(0)
1. Ground state calculation: Vext
! yi and r (0)
2. FOR EACH pertubation j1 DO
(0)
use yi
j1
Vext
!
and r (0)
yij1
and r
j1
ENDDO
minimize 2nd order energy
solve Sternheimer equation
3. FOR EACH pertubation pair {j1, j2} DO
j1
j2
j1 j2
y
and
y
E
determine
using both i
i (stationary expression)
j1
y
using just i
ENDDO
4. Post-processing to get the physical properties from E j1 j2
CECAM Tutorial, Lyon, 12-16 may 2014
44
Outline
Perturbation Theory
Density Functional Perturbation Theory
Atomic displacements and
homogeneous electric field
CECAM Tutorial, Lyon, 12-16 may 2014
45
Perturbations of the periodic solid
• Let us consider the case where the reference system is periodic:
(0)
(0)
Vext (r + Ra ) = Vext (r)
• It can be shown that if the perturbation is characterized by a
wavevector q such that:
(1)
iq·Ra (1)
Vext (r + Ra ) = e
Vext (r)
all the responses, at linear order, will also be characterized by q:
r (1) (r + Ra ) = eiq·Ra r (1) (r)
yi,k,q (r + Ra ) = eiq·Ra yi,k,q (r)
(1)
(1)
···
CECAM Tutorial, Lyon, 12-16 may 2014
46
Perturbations of the periodic solid
• We define related periodic quantities:
r̄ (1) (r) = e
iq·r (1)
r
(r)
ui,k,q (r) = (Ne W0 )1/2 e
(1)
i(k+q)·R
(1)
yi,k,q (r)
• In the equations of DFPT, only these periodic quantities appear:
the phases e
iq·r
and e
i(k+q)·R
can be factorized.
• The treatment of perturbations incommensurate with the
•
unperturbed system periodicity is mapped onto the original
periodic system.
This is interesting for atomic displacements but more
importantly for electric fields.
CECAM Tutorial, Lyon, 12-16 may 2014
47
Electronic dielectric permittivity tensor
• The dielectric permittivity tensor is the coefficient of
proportionality between the macroscopic displacement field and
the macroscopic electric field, in the linear regime:
Dmac,a = Â eaa 0 Emac,a 0
a0
•
∂ Dmac,a
∂ Pmac,a
eaa 0 =
= daa 0 + 4p
∂ Emac,a 0
∂ Emac,a 0
At high frequencies of the applied field, the dielectric
permittivity tensor only includes a contribution from the
electronic polarization:
4π Eα∗ Eα ′
∞
εαα ′ = δαα ′ −
2Eel
Ω0
1
′
∞
∑′ q̂α εαα ′ q̂α = ε −1
αα
G=0,G′ =0 (q → 0)
unit vector
CECAM Tutorial, Lyon, 12-16 may 2014
48
Treatment of homogeneous electric fields
• When the perturbation is an electric field, we have:
(1)
Vext (r) = E · r
which breaks the periodic boundary conditions.
of the energy:
• To obtain the 2nd order derivative
Z
Ea⇤ Ea 0
2Eel
=
r Ea 0 (r)ra dr
the following matrix elements need to be computed:
D
E
D
E
(0)
(0)
uc,k ra 0 uEv,ka and uEc,ka ra 0 uv,k
conduction
valence
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49
Treatment of homogeneous electric fields
• These matrix elements can be determined writing that:
⇣
(0)
ei,k
⌘D
E D
(0)
(0)
(0)
(0)
(0)
e j,k ui,k ra u j,k = ui,k Hkk ra
D
(0)
= ui,k
(0)
(0)
ra Hkk u j,k
∂ Hkk (0) E
i
u j,k
∂ ka
(0)
E
which leads to the Sternheimer equation:
(0)
⇣
⌘
E
E
∂
H
(0)
(0)
(0)
(0)
Pc Hkk e j,k Pc ra u j,k = Pc i kk u j,k
∂ ka
⇣
⌘
(0)
(1)
(0)
PI ⇥ H (0) ei
PI ⇥ |yi ⇤ = PI ⇥ H (1) |yi ⇤
DDK
CECAM Tutorial, Lyon, 12-16 may 2014
perturbation
50
Born effective charge tensor
• It is defined as the proportionality coefficient relating at linear
order, the polarization per unit cell, created along the direction α,
and the displacement along the direction α’ of the atoms
belonging to the sublattice κ:
!
!
∂ Pmac,α !!
∂ Fκα ′ !!
∗
Zκαα ′ = Ω0
=
!
∂ uκα ′ (q = 0) Eα =0
∂ Eα !u ′ =0
κα
• It also describes the linear relation between the force in the
•
direction α’ on an atom κ and the macroscopic electric field
Both can be connected to the mixed 2nd order derivative of the
energy with respect to uκα’ and Eα
∗
• Sum rule: ∑ Zκαα
′ =0
κ
CECAM Tutorial, Lyon, 12-16 may 2014
51
Born effective charge tensor
• Model system:
-q(r) 𝒫(r) +q(r)
★ diatomic
molecule:
r
★ dipole moment related to static charge q(r): 𝒫(r)= q(r) r
• Atomic polar charge Z*(r) such that ∂𝒫(r)= Z*(r)∂r
★ purely
covalent case:
★ purely ionic case:
★ mixed ionic-covalent:
q(r)
৪ZҶҨ*(r)
)৻*
৪‫ܖ‬Ҩ )৻*
Q
r
r
Q
CECAM Tutorial, Lyon, 12-16 may 2014
q(r)=0=Z*(r)
q(r)=Q ≠ 0 ⇒Z*(r)=Q
∂𝒫(r)= q(r)∂r + ∂q(r) r
ಘ৺)৻*
Ҩ
৪ )৻* > ৺)৻* , ৻
ಘ৻
ಘ৺)৻*
Ҩ
৪౿ಀ )৻* > ৺)৻*ಂ౿ಀ , ৻౿
ಘ৻ಀ
52
Static dielectric permittivity tensor
• The mode oscillator strength tensor is defined as
Sm,αα ′ =
!
∗
∗
Z
U
∑ καβ mq=0 (κβ )
κβ
"!
′
∗
U
(
Z
κβ
′
′
∑ κα β mq=0 )
κβ ′
"
• The macroscopic static (low-frequency) dielectric permittivity
tensor is calculated by adding the ionic contribution to the
electronic dielectric permittivity tensor:
εαα ′ (ω ) =
0
εαα
′
=
∞
εαα
′
∞
εαα
′
CECAM Tutorial, Lyon, 12-16 may 2014
Sm,αα ′
4π
+
2
2
Ω0 ∑
m ωm − ω
Sm,αα ′
4π
+
Ω0 ∑
ωm2
m
53
LO-TO splitting
• The macroscopic electric field that accompanies the collective
atomic displacements at q → 0 can be treated separately:
NA
C̃κα ,κ ′ α ′ (q → 0) = C̃κα ,κ ′ α ′ (q = 0) + C̃κα
,κ ′ α ′ (q → 0)
where the nonanalytical, direction-dependent term is:
!
"!
"
′ ′ ∗
∗
q
Z
qβ Zκ ′ β ′ α ′
∑
∑
γ
β
β
κβ
α
4
π
NA
C̃κα
,κ ′ α ′ (q → 0) =
Ω0
∑β β ′ qβ εβ∞β ′ qβ′
• The transverse modes are common to both C
̃ matrices but the
longitudinal ones may be different, the frequencies are related by
′
′ qα Sm,αα ′ qα
π
4
∑
αα
ωm2 (q → 0) = ωm2 (q = 0) +
∞ q′
Ω0 ∑αα ′ qα εαα
′ α
CECAM Tutorial, Lyon, 12-16 may 2014
54
Example 1: Zircon (phonons)
6
1200
LDA
PBE
PBEsol
AM05
WC
HTBS
2
0
800
Relative error (%)
Theoretical frequency (cm−1)
1000
4
600
400
2
4
6
8
LDA
PBE
PBEsol
AM05
WC
HTBS
10
200
12
14
0
0
200
400
600
800
1000
−1
Experimental frequency (cm )
CECAM Tutorial, Lyon, 12-16 may 2014
1200
0
200
400
600
800
1000
1200
1
Experimental frequency (cm )
55
Example 1: Zircon (Born effective charges)
Nominal:
M → +4
Si → +4
O → -2
• M : anomalously large (esp. Z⊥)
• Si : smaller deviations (↗ and ↘)
→
→
PbZrO3, ZrO2
SiO2 (α-quartz or stishovite)
→ SiO2 stishovite or TiO2 rutile
• O : strong anisotropy
↗ in the y-z plane (plane of the M-O bonds) (rem: ≠ from SiO2 α-quartz ↘ in the x direction
Interpretation:
• mixed ionic-covalent bonding
where 2 ↘ components)
• closer to stishovite than α-quartz in agreement with naive bond counting
for O atoms
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56
Example 1: Zircon (dielectric properties)
* are the largest for the lowest and highest frequency modes
• Sm and Zm
• due to the frequency factor, it is the lowest frequency mode that
contributes the most to ε0
* are smaller for HfSiO ← mass difference and Born effective charges
• Sm and Zm
4
• this effect can be compensated by a lower frequency for hafnon [e.g. A2u(1)]
CECAM Tutorial, Lyon, 12-16 may 2014
57
Example 2: Copper (thermodynamics)
0
0.5
0
0
(b)
200
400
800
600
1000
T (K)
600
800
1000
LDA
PBE
PBEsol
AM05
WC
HTBS
Expt.
B0 (MBar)
)
1.6
1
5
(10
400
T (K)
LDA
PBE
PBEsol
AM05
WC
HTBS
Expt.
2
200
(c) 1.8
3
2.5
0
1.5
1
1.4
1.2
0.5
1
0
0
200
400
800
600
1000
0
200
400
(c) 1.8
800
1000
300
LDA
PBE
PBEsol
AM05
WC
HTBS
Expt.
250
Frequency (cm-1 )
1.6
B0 (MBar)
600
T (K)
T (K)
1.4
1.2
200
150
100
50
0
1
0
200
400
X
Γ
600
W
800
(K)
CECAM Tutorial, Lyon, T12-16
may 2014
L
Γ
1000
K
X
58
•
•
•
•
•
•
•
!
!
S. Baroni, P. Giannozzi & A. Testa, Phys. Rev. Lett. 58, 1861 (1987)
X. Gonze & J.-P. Vigneron, Phys. Rev. B 39, 13120 (1989)
X. Gonze, Phys. Rev. A 52 , 1096 (1995)
S. de Gironcoli, Phys. Rev. B 51, 6773 (1995)
X. Gonze, Phys. Rev. B. 55, 10337 (1997)
X. Gonze & C. Lee, Phys. Rev. B. 55, 10355 (1997)
S. Baroni, S. de Gironcoli, A. Dal Corso, P. Giannozzi,
Rev. Mod. Phys. 73, 515 (2001)
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59