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Institute of Condensed Matter and Nanosciences,B-1348 Louvain-la-Neuve, Belgique
European Theoretical Spectroscopy Facility
Gian-Marco Rignanese
CECAM Tutorial: “Dynamical, dielectric and magnetic properties of solids with ABINIT”Lyon, 12-16 may 2014
Basics of density functional perturbation theory
CECAM Tutorial, Lyon, 12-16 may 2014
• Quantum treatment for electrons → Kohn-Sham equation
• Classical treatment for nuclei → Newton equation
Born-Oppenheimer approximation
2
atom direction (α=1,2,3)
unit cell
1
2—2 +Vext(r)+VHxc(r)
yi(r) = eiyi(r)
F
a
ka = ∂E
tot
∂R
a
ka
R
= Mkd
2R
a
kadt
2
E
tot
= E
ion
+E
el
Eel [r] = Âvyv |T +Vext |yv+EHxc[r]
r(r) = Âv
yv (r)yv(r)
CECAM Tutorial, Lyon, 12-16 may 2014
Raκα
= Raα+ τκα +u
aκα
Etot (u) = E(0)tot + ∑
aκα∑
a′κ ′α ′
1
2
!
∂ 2Etot
∂uaκα ∂ua′
κ ′α ′
"
uaκα u
a′
κ ′α ′
Harmonic approximation
3
u
Etot
cell equilibrium atomic
position
displacement
CECAM Tutorial, Lyon, 12-16 may 2014
• The matrix of interatomic force constants (IFCs) is defined as
• Its Fourier transform (using translational invariance)allows one to compute phonon frequencies and eigenvectors as solutions of the following generalized eigenvalue problem:
Phonons
4
phonon displacement
pattern
masses square of phonon
frequencies
Cκα ,κ ′α ′(a,a′) =
!
∂ 2Etot
∂uaκα ∂ua′
κ ′α ′
"
Cκα ,κ ′α ′(q) = ∑a′
Cκα ,κ ′α ′(0,a′)eiq·Ra′
Âk 0a 0
Cka,k 0a 0(q)Umq(k 0a 0) = Mk w2mqUmq(ka)
CECAM Tutorial, Lyon, 12-16 may 2014
Example: Diamond
5
Theory vs. Experiment [X. Gonze, GMR, and R. Caracas, Z. Kristallogr. 220, 458 (2000)]
CECAM Tutorial, Lyon, 12-16 may 2014
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 ...
6
CECAM Tutorial, Lyon, 12-16 may 2014
Total energy derivatives
• Related derivatives of the total energy (Eel + Eion) 1st order: forces, stress, dipole moment, ... 2nd order: dynamical matrix, elastic constants, dielectric
susceptibility, Born effective charge tensors, piezoelectricity, internal strains
3rd order: 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, ...)
7
CECAM Tutorial, Lyon, 12-16 may 2014
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.
8
CECAM Tutorial, Lyon, 12-16 may 2014
OutlinePerturbation Theory
Density Functional Perturbation Theory
Atomic displacements and homogeneous electric field
9
CECAM Tutorial, Lyon, 12-16 may 2014
OutlinePerturbation Theory
Density Functional Perturbation Theory
Atomic displacements and homogeneous electric field
10
CECAM Tutorial, Lyon, 12-16 may 2014
• Let us assume that all the solutions are known for a reference system for which the one-body Schrödinger equation is:
with the normalization condition:
• Let us now introduce a perturbation of the external potential characterized by a small parameter λ:
known at all orders.
Reference and perturbed systems
11
H(0)y(0)
i
E= e(0)i
y(0)i
E
Vext(λ ) =V(0)ext +λV
(1)ext +λ 2
V(2)ext + · · ·
Dy(0)
i
y(0)i
E= 1
CECAM Tutorial, Lyon, 12-16 may 2014
• We now want to solve the perturbed Schrödinger equation:
with the normalization condition:
• Idea: all the quantities (X=H, εi, ψi) are written as a perturbation series with respect to the parameter λ :
Reference and perturbed systems
12
X(λ ) = X(0) +λX
(1) +λ 2X(2) +λ 3
X(3) + · · ·
with X (n) =1n!
dnXdl n
l=0
H(l )|yi(l )= ei(l )|yi(l )
yi(l )|yi(l )= 1
CECAM Tutorial, Lyon, 12-16 may 2014
• Starting from:
and inserting:
we get:
Expansion of the Schrödinger equation
13
H(l )|yi(l )= ei(l )|yi(l ) l
yi(l ) = y(0)i +ly(1)
i +l 2y(2)i + . . .
ei(l ) = e(0)i +le(1)i +l 2e(2)i + . . .
H(! ) = H(0) +!H(1) +! 2H(2) + . . .
H(0) +!H(1) +! 2H(2) + . . .
|"(0)
i +! |"(1)i +! 2|"(2)
i + . . .=
#(0)i +!#(1)i +! 2#(2)i + . . .
|"(0)
i +! |"(1)i +! 2|"(2)
i + . . .
CECAM Tutorial, Lyon, 12-16 may 2014
• This can be rewritten as:
Expansion of the Schrödinger equation
14
setting λ=0
deriving w.r.t. λ and setting λ=0
deriving twice w.r.t. λ and setting λ=0
H(0)|y(0)i
+l
H(0)|y(1)i +H(1)|y(0)
i
+l 2
H(0)|y(2)i +H(1)|y(1)
i +H(2)|y(0)i
+ . . . =
e(0)i |y(0)i
+l
e(0)i |y(1)i + e(1)i |y(0)
i
+l 2
e(0)i |y(2)i + e(1)i |y(1)
i + e(2)i |y(0)i
+ . . .
CECAM Tutorial, Lyon, 12-16 may 2014
• Finally, we have that:
Expansion of the Schrödinger equation
15
0th order
1st order
2nd order
H(0)|!(0)i = "(0)i |!(0)
i
H(0)|!(1)i +H(1)|!(0)
i = "(0)i |!(1)i + "(1)i |!(0)
i
H(0)|!(2)i +H(1)|!(1)
i +H(2)|!(0)i =
"(0)i |!(2)i + "(1)i |!(1)
i + "(2)i |!(0)i
· · ·
CECAM Tutorial, Lyon, 12-16 may 2014
• Starting from:
and inserting:
we get:
Expansion of the normalization condition
16
yi(l )|yi(l )= 1 l
yi(l ) = y(0)i +ly(1)
i +l 2y(2)i + . . .
y(0)i |y(0)
i +l
y(0)
i |y(1)i + y(1)
i |y(0)i
+l 2y(0)
i |y(2)i + y(1)
i |y(1)i + y(2)
i |y(0)i
+ . . . = 1
CECAM Tutorial, Lyon, 12-16 may 2014
• Finally, we have that
Expansion of the normalization condition
17
0th order
1st order
2nd order
y(0)i |y(0)
i = 1
y(0)i |y(1)
i + y(1)i |y(0)
i = 0
y(0)i |y(2)
i + y(1)i |y(1)
i + y(2)i |y(0)
i = 0
· · ·
CECAM Tutorial, Lyon, 12-16 may 2014
• Starting from the 1st order of the Schrödinger equation
and premultiplying by , we get:
and thus, finally, we have the Hellman-Feynman theorem:
• The 0th order wavefunctions are thus the only required ingredient to obtain the 1st order corrections to the energies.
1st order corrections to the energies
18
H(0)|y(1)i +H(1)|y(0)
i = e(0)i |y(1)i + e(1)i |y(0)
i
y(0)i |
y(0)i |H(0)|y(1)
i + y(0)i |H(1)|y(0)
i = e(0)i y(0)i |y(1)
i + e(1)i y(0)i |y(0)
i
e(1)i = y(0)i |H(1)|y(0)
i
| z e(0)i y(0)
i |y(1)i
| z 1
CECAM Tutorial, Lyon, 12-16 may 2014
• Starting from the 2nd order of the Schrödinger equation
and premultiplying by , we get:
and thus, finally, we have:
2nd order corrections to the energies
19
y(0)i |
| z 1
| z e(0)i y(0)
i |y(2)i
H(0)|!(2)i +H(1)|!(1)
i +H(2)|!(0)i =
"(0)i |!(2)i + "(1)i |!(1)
i + "(2)i |!(0)i
!(0)i |H(0)|!(2)
i + !(0)i |H(1)|!(1)
i + !(0)i |H(2)|!(0)
i =
"(0)i !(0)i |!(2)
i + "(1)i !(0)i |!(1)
i + "(2)i !(0)i |!(0)
i
!(2)i = "(0)i |H(2)|"(0)
i + "(0)i |H(1) !(1)i |"(1)
i
CECAM Tutorial, Lyon, 12-16 may 2014
• Since the energies are real, we can write that:
or, combining both equalities:
• Using the expansion of the normalization condition at 1st order, we can finally write that:
• To obtain the 2nd order corrections to the energies, the only required ingredients are the 0th and 1st order wavefunctions.
2nd order corrections to the energies
20
!(2)i = "(0)i |H(2)|"(0)
i + "(0)i |H(1) !(1)i |"(1)
i
= "(0)i |H(2)|"(0)
i + "(1)i |H(1) !(1)i |"(0)
i
!(2)i ="(0)i |H(2)|"(0)
i
+12
"(0)
i |H(1) !(1)i |"(1)i + "(1)
i |H(1) !(1)i |"(0)i
!(2)i = "(0)i |H(2)|"(0)
i + 12
"(0)
i |H(1)|"(1)i + "(1)
i |H(1)|"(0)i
CECAM Tutorial, Lyon, 12-16 may 2014
• The 1st order of the Schrödinger equation
can be rewritten gathering the terms containing :
producing the so-called Sternheimer equation.
• In order to get one would like to invert theoperator, but it cannot be done as such since is an eigenvalue of . The problem can be solved by expressing the 1st order wavefunction as a linear combination of the 0th order ones:
1st order corrections to the wavefunctions
21
H(0)|y(1)i +H(1)|y(0)
i = e(0)i |y(1)i + e(1)i |y(0)
i
H(0) e(0)i
|y(1)
i =
H(1) e(1)i
|y(0)
i |y(1)
i
|y(1)i (H(0) e(0)i )
e(0)i
H(0)
|y(1)i = Â
jc(1)i j |y(0)
j
CECAM Tutorial, Lyon, 12-16 may 2014
• We separate the 0th order wavefunctions into two subsets: those associated to :
(just if the energy is non-degenerate) those that belong to the subspace that is orthogonal: j ∈ I⊥
• The Sternheimer equation can thus be rewritten as:
1st order corrections to the wavefunctions
22
e(0)i|y(0)
i j I if H(0)|y(0)
j = e(0)i |y(0)j
|y(1)i = Â
jc(1)i j |y(0)
j = ÂjI
c(1)i j |y(0)j + Â
jIc(1)i j |y(0)
j
H(0) e(0)i
|y(1)
i = Âj
c(1)i j
H(0) e(0)i
|y(0)
j
= ÂjI
c(1)i j
e(0)j e(0)i
|y(0)
j
=
H(1) e(1)i
|y(0)
i
CECAM Tutorial, Lyon, 12-16 may 2014
• Premultiplying by with k ∈ I⊥, we get:
and, thus:
ÂjI
c(1)i j
e(0)j e(0)i
y(0)
k |y(0)j =y(0)
k |H(1) e(1)i |y(0)i
y(0)k |
1st order corrections to the wavefunctions
23
| z δkj y(0)
k |y(0)i = 0
since k I
c(1)ik
e(0)k e(0)i
=y(0)
k |H(1)|y(0)i
c(1)i j =1
e(0)i e(0)j
y(0)j |H(1)|y(0)
i for j I
CECAM Tutorial, Lyon, 12-16 may 2014
• Premultiplying by with k ∈ I, we get:
• For k = i, it is nothing but the Hellmann-Feynman theorem.But, it does not provide any information on the .
• In fact, there is a gauge freedom that allows to choose them equal to zero.
ÂjI
c(1)i j
e(0)j e(0)i
y(0)
k |y(0)j =y(0)
k |H(1) e(1)i |y(0)i
y(0)k |
1st order corrections to the wavefunctions
24
| z 0 y(0)
k |y(0)i = dki
since k I
c(1)i j for j 2 I
e(1)k dki = y(0)k |H(1)|y(0)
i
CECAM Tutorial, Lyon, 12-16 may 2014
• Finally, we can write the so-called sum over states expression:
which requires the knowledge of all the 0th order wavefunctions and energies.
• Instead, if we define the projector
we can rewrite the Sterheimer equation in that subspace:
1st order corrections to the wavefunctions
25
PI? onto the subspace I? by:
PI = ÂjI
|y(0)j y(0)
j |
PI
H(0) e(0)i
PI |y
(1)i =PIH(1)|y(0)
i
|y(1)i = Â
jI|y(0)
j 1
e(0)i e(0)j
y(0)j |H(1)|y(0)
i
CECAM Tutorial, Lyon, 12-16 may 2014
• In this form, the singularity has disappeared and it can thus be inverted:
and defining the Green’s function in the subspace as:
we can write:
• This is the Green's function technique for dealing with the Sternheimer equation.
1st order corrections to the wavefunctions
26
PI |y(1)i =
hPI
H(0) e(0)i
PI
i1H(1)|y(0)
i
I?
PI |y(1)i = GI(e
(0)i )H(1)|y(0)
i
GI?(e) =hPI?
e H(0)
PI?
i1
CECAM Tutorial, Lyon, 12-16 may 2014
• The sum over states expression for the 1st order wavefunctions:
can be inserted in the 2nd order corrections to the energies:
leading to:
|y(1)i = Â
jI|y(0)
j 1
e(0)i e(0)j
y(0)j |H(1)|y(0)
i
2nd order corrections to the energies
27
!(2)i = "(0)i |H(2)|"(0)
i + 12
"(0)
i |H(1)|"(1)i + "(1)
i |H(1)|"(0)i
!(2)i ="(0)i |H(2)|"(0)
i
+ #jI
"(0)i |H(1)|"(0)
j 1!(0)i !(0)j
"(0)j |H(1)|"(0)
i
CECAM Tutorial, Lyon, 12-16 may 2014
• Alternatively, we can write that:
and the perturbation expansion at the 2nd order gives:
• 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:
2nd order corrections to the energies
28
!i(" )|H(" ) #i(" )|!i(" )= 0 "
!(0)i |H(0) "(0)i |!(2)
i +!(0)
i |H(1) "(1)i |!(1)i +!(1)
i |H(0) "(0)i |!(1)i
+!(0)i |H(2) "(2)i |!(0)
i +!(1)i |H(1) "(1)i |!(0)
i +!(2)i |H(0) "(0)i |!(0)
i = 0
!(2)i ="(0)i |H(2)|"(0)
i + "(1)i |H(0) !(0)i |"(1)
i
+ "(0)i |H(1) !(1)i |"(1)
i + "(1)i |H(1) !(1)i |"(0)
i
CECAM Tutorial, Lyon, 12-16 may 2014
• 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 energies can be obtained by minimizing it with respect to :
under the constraint that:
2nd order corrections to the energies
29
!(1)i
!(2)i =min"(1)i
n
"(0)i |H(2)|"(0)
i + "(1)i |H(0) !(0)i |"(1)
i
+"(0)i |H(1) !(1)i |"(1)
i + "(1)i |H(1) !(1)i |"(0)
i o
!(0)i |!(1)
i + !(1)i |!(0)
i = 0
CECAM Tutorial, Lyon, 12-16 may 2014
• Starting from the 3rd order of the Schrödinger equation
and premultiplying by , we get:
3rd order corrections to the energies
30
y(0)i |
| z 1
| z !(0)i "(0)
i |"(3)i
H(0)|!(3)i +H(1)|!(2)
i +H(2)|!(1)i +H(3)|!(0)
i =
"(0)i |!(3)i + "(1)i |!(2)
i + "(2)i |!(1)i + "(3)i |!(0)
i
!(0)i |H(0)|!(3)
i + !(0)i |H(1)|!(2)
i
+ !(0)i |H(2)|!(1)
i + !(0)i |H(3)|!(0)
i
= "(0)i !(0)i |!(3)
i + "(1)i !(0)i |!(2)
i
+ "(2)i !(0)i |!(1)
i + "(3)i !(0)i |!(0)
i
CECAM Tutorial, Lyon, 12-16 may 2014
• Finally, we we can write:
• This expression of the 3rd order corrections to the energies requires to know the wavefunctions up to the 2nd order.
3rd order corrections to the energies
31
!(3)i ="(0)i |H(3)|"(0)
i
+ "(0)i |H(2) !(2)i |"(1)
i
+ "(0)i |H(1) !(1)i |"(2)
i
CECAM Tutorial, Lyon, 12-16 may 2014
• Alternatively, we can write that:
and the perturbation expansion at the 3rd order gives:
• 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 :
3rd order corrections to the energies
32
!i(" )|H(" ) #i(" )|!i(" )= 0 "
!(0)i |H(0) "(0)i |!(3)
i +!(0)
i |H(1) "(1)i |!(2)i +!(1)
i |H(0) "(0)i |!(2)i
+!(0)i |H(2) "(2)i |!(1)
i +!(1)i |H(1) "(1)i |!(1)
i +!(2)i |H(0) "(0)i |!(1)
i +!(0)
i |H(3) "(3)i |!(0)i +!(1)
i |H(2) "(2)i |!(0)i +!(2)
i |H(1) "(1)i |!(0)i +!(3)
i |H(0) "(0)i |!(0)i = 0
!(3)i ="(0)i |H(3)|"(0)
i + "(1)i |H(1) !(1)i |"(1)
i
+ "(0)i |H(2)|"(1)
i + "(1)i |H(2)|"(0)
i
CECAM Tutorial, Lyon, 12-16 may 2014
• 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 minimizing the constrained functional for the 2nd order
corrections to the energies
• 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].
Summary
33
CECAM Tutorial, Lyon, 12-16 may 2014
OutlinePerturbation Theory
Density Functional Perturbation Theory
Atomic displacements and homogeneous electric field
34
CECAM Tutorial, Lyon, 12-16 may 2014
• In DFT, one needs to minimize the electronic energy functional:
in which the electronic density is given by:
under the constraint that:
• Alternatively, one can solve the reference Shrödinger equation:
where the Hartree and exchange correlation potential is:
Reference system
35
V (0)Hxc(r) =
dE(0)Hxc[r(0)]
dr(r)
r(0)(r) =Ne
Âi=1
hy(0)
i (r)i
y(0)i (r)
Eel [r(0)] =Ne
Âi=1
Dy(0)
i
T +V (0)ext
y(0)i
E+E(0)
Hxc[r(0)]
Dy(0)
i
y(0)j
E= di j
H(0)y(0)
i
E=
1
2—2 +V (0)
ext +V (0)Hxc
y(0)i
E= e(0)i
y(0)i
E
CECAM Tutorial, Lyon, 12-16 may 2014
• The electronic energy functional to be minimized is:
in which the electronic density is given by:
under the constraint that:
• Alternatively, the Shrödinger equation to be solved is:
where the Hartree and exchange correlation potential is:
Perturbed system
36
r(r;l ) =Ne
Âi=1
yi (r;l )yi(r;l )
yi(l )| y j(l )↵= di j
H(l ) |yi(l )=1
2—2 +Vext(l )+VHxc(l )
|yi(l )= ei(l ) |yi(l )
VHxc(r;l ) = dEHxc[r(l )]dr(r)
Eel [r(l )] =Ne
Âi=1
yi(l ) |T +Vext(l )|yi(l )+EHxc[r(l )]
CECAM Tutorial, Lyon, 12-16 may 2014
1st order of perturbation theory in DFT
• For the energy, it can be shown that:
This is the equivalent of the Hellman-Feynman theorem for density-functional formalism.
• For the wavefunctions, the constraint leads to:
37
Dy(0)
i
y(1)j
E+D
y(1)i
y(0)j
E= 0
E(1)el =
Ne
Âi=1
Dy(0)
i
(T +Vext)(1)y(0)
i
E+
ddl
EHxc[r(0)]l=0
CECAM Tutorial, Lyon, 12-16 may 2014
2nd order energy in DFPT
• For the energy, it can be shown that:
with
38
E(2)el =
Ne
Âi=1
hDy(1)
i
(T +Vext)(1)y(0)
i
E+D
y(0)i
(T +Vext)(1)y(1)
i
Ei
+Ne
Âi=1
hDy(0)
i
(T +Vext)(2)y(0)
i
E+D
y(1)i
(H ei)(0)y(1)
i
Ei
+12
Z Z d 2EHxc[r(0)]
dr(r)dr(r0)r(1)(r)r(1)(r0)drdr0
+Z d
dlEHxc[r(0)]
dr(r)
l=0
r(1)(r)dr+ 12
d2
dl 2 EHxc[r(0)]l=0
r(1)(r) =Ne
Âi=1
hy(1)
i (r)i
y(0)i (r)+
hy(0)
i (r)i
y(1)i (r)
CECAM Tutorial, Lyon, 12-16 may 2014
1st order wavefunctions in DFPT
• The 1st order wavefunctions can be obtained by minimizing:
with respect to under the constraint:
• These can also be obtained by solving the Sternheimer equation:
39
E(2)el = E(2)
el
hn
y(0)i
o
,n
y(1)i
oi
n
y(1)i
o Dy(0)
i
y(1)j
E= 0
H(0) e(0)i
|y(1)
i =
H(1) e(1)i
|y(0)
i
H(1) = (T +Vext)(1) +
Z d 2EHxc[r(0)]
dr(r)dr(r0)r(1)(r0)dr0
e(1)i = y(0)i |H(1)|y(0)
i
CECAM Tutorial, Lyon, 12-16 may 2014
Higher orders in DFPT
• More generally, it easy to show that since there is a variational principle for the 0th order energy:
non-variational expression can be obtained for higher orders:
• But, this is not the best that can be done!
40
E(0)el = E(0)
el
hn
y(0)i
oi
E(1)el = E(1)
el
hn
y(0)i
oi
E(2)el = E(2)
el
hn
y(0)i
o
,n
y(1)i
oi
E(3)el = E(3)
el
hn
y(0)i
o
,n
y(1)i
o
,n
y(2)i
oi
· · ·
CECAM Tutorial, Lyon, 12-16 may 2014
• We assume that all wavefunctions are known up to (n-1)th order:
• The variational property of the energy functional implies that:
• Taking , we see that:
if 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.
• This is the “2n+1” theorem.
Higher orders in DFPT
41
yi = y<ni +O(l n) = y(0)
i +ly(1)i + · · ·+l n1y(n1)
i +O(l n)
ytrial=
y<ni
and h = l nEel [ytrial +O(h)] = Eel [ytrial]+O(h2)
CECAM Tutorial, Lyon, 12-16 may 2014
• 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):
Higher orders in DFPT
42
E(0)el = E(0)
el
hn
y(0)i
oi
variational w.r.t.
n
y(0)i
o
E(1)el = E(1)
el
hn
y(0)i
oi
E(2)el = E(2)
el
hn
y(0)i
o
,n
y(1)i
oi
variational w.r.t.
n
y(1)i
o
E(3)el = E(3)
el
hn
y(0)i
o
,n
y(1)i
oi
E(4)el = E(4)
el
hn
y(0)i
o
,n
y(1)i
o
,n
y(2)i
oi
variational w.r.t.
n
y(2)i
o
E(5)el = E(5)
el
hn
y(0)i
o
,n
y(1)i
o
,n
y(2)i
oi
· · ·
CECAM Tutorial, Lyon, 12-16 may 2014
• Similar expressions exist for mixed derivatives (related to two different perturbations j1 and j2):
• 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.
Mixed derivatives in DFPT
43
E j1 j2el = E j1 j2
el
n
y(0)i ;y j1
i ;y j2i
o
CECAM Tutorial, Lyon, 12-16 may 2014
1. Ground state calculation:
2. FOR EACH pertubation j1 DO
use
ENDDO
3. FOR EACH pertubation pair j1, j2 DO
determine using both (stationary expression)
using just ENDDO
4. Post-processing to get the physical properties from
Order of calculation in DFPT
44
V
(0)ext
! y(0)i
and r(0)
y(0)i and r(0)
V
j1ext
! y j1i
and r j1
E j1 j2
minimize 2nd order energy solve Sternheimer equation
E j1 j2
y j1i
y j1i and y j2
i
CECAM Tutorial, Lyon, 12-16 may 2014
OutlinePerturbation Theory
Density Functional Perturbation Theory
Atomic displacements and homogeneous electric field
45
CECAM Tutorial, Lyon, 12-16 may 2014
Perturbations of the periodic solid
• Let us consider the case where the reference system is periodic:
• It can be shown that if the perturbation is characterized by a wavevector q such that:
all the responses, at linear order, will also be characterized by q:
46
V
(0)ext
(r+Ra
) =V
(0)ext
(r)
V
(1)ext
(r+Ra
) = e
iq·Ra
V
(1)ext
(r)
r(1)(r+Ra) = eiq·Rar(1)(r)
y(1)i,k,q(r+Ra) = eiq·Ray(1)
i,k,q(r)
· · ·
CECAM Tutorial, Lyon, 12-16 may 2014
Perturbations of the periodic solid
• We define related periodic quantities:
• In the equations of DFPT, only these periodic quantities appear: the phases 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.
47
r(1)(r) = eiq·rr(1)(r)
u(1)i,k,q(r) = (Ne W0)1/2ei(k+q)·Ry(1)
i,k,q(r)
eiq·r and ei(k+q)·R
CECAM Tutorial, Lyon, 12-16 may 2014
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:
• At high frequencies of the applied field, the dielectric permittivity tensor only includes a contribution from the electronic polarization:
48unit vector
Dmac,a = Âa 0
eaa 0Emac,a 0
ε∞αα ′ = δαα ′ −
4π
Ω0
2EE ∗
α Eα ′
el
∑αα ′
qα ε∞αα ′ q
′
α =1
ε−1
G=0,G′=0(q → 0)
eaa 0 =∂Dmac,a∂Emac,a 0
= daa 0 +4p∂Pmac,a∂Emac,a 0
CECAM Tutorial, Lyon, 12-16 may 2014
Treatment of homogeneous electric fields
• When the perturbation is an electric field, we have:
which breaks the periodic boundary conditions.
• To obtain the 2nd order derivative of the energy:
the following matrix elements need to be computed:
49
2EE a Ea 0
el =Z
rEa 0 (r)ra dr
Du(0)c,k
ra 0
uEav,k
Eand
DuEa
c,k
ra 0
u(0)v,k
E
conduction valence
V
(1)ext
(r) = E · r
CECAM Tutorial, Lyon, 12-16 may 2014
Treatment of homogeneous electric fields
• These matrix elements can be determined writing that:
which leads to the Sternheimer equation:
50
e(0)i,k e(0)j,k
Du(0)i,k
ra
u(0)j,k
E=D
u(0)i,k
H(0)kk ra ra H(0)
kk
u(0)j,k
E
=D
u(0)i,k
i∂H(0)
kk∂ka
u(0)j,k
E
Pc
H(0)
kk e(0)j,k
Pc ra
u(0)j,k
E=Pc i
∂H(0)kk
∂ka
u(0)j,k
E
PI
H(0) e(0)i
PI |y
(1)i =PIH(1)|y(0)
i
DDK perturbation
CECAM Tutorial, Lyon, 12-16 may 2014
• 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:
• 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:
Born effective charge tensor
51
Z∗
καα ′ = Ω0
∂Pmac,α
∂uκα ′(q = 0)
!
!
!
!
Eα=0
=∂Fκα ′
∂Eα
!
!
!
!
uκα ′=0
∑κ
Z∗
καα ′ = 0
CECAM Tutorial, Lyon, 12-16 may 2014
Born effective charge tensor
• Model system: diatomic molecule: 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: q(r)=0=Z*(r) purely ionic case: q(r)=Q ≠ 0 ⇒Z*(r)=Q mixed ionic-covalent: ∂𝒫(r)= q(r)∂r + ∂q(r) r
52
+q(r)-q(r)r
𝒫(r)
q(r)
r
Q
r
Z*(r) Q
CECAM Tutorial, Lyon, 12-16 may 2014
Static dielectric permittivity tensor
• The mode oscillator strength tensor is defined as
• The macroscopic static (low-frequency) dielectric permittivity tensor is calculated by adding the ionic contribution to the electronic dielectric permittivity tensor:
53
Sm,αα ′ =
!
∑κβ
Z∗
καβU∗
mq=0(κβ )
"!
∑κβ ′
Z∗
κα ′β ′Umq=0(κβ ′)
"
εαα ′(ω) = ε∞αα ′ +
4π
Ω0∑m
Sm,αα ′
ω2m −ω2
ε0
αα ′ = ε∞αα ′ +
4π
Ω0∑m
Sm,αα ′
ω2m
CECAM Tutorial, Lyon, 12-16 may 2014
LO-TO splitting
• The macroscopic electric field that accompanies the collective atomic displacements at q → 0 can be treated separately:
where the nonanalytical, direction-dependent term is:
• The transverse modes are common to both C matrices but the longitudinal ones may be different, the frequencies are related by
54
Cκα ,κ ′α ′(q → 0) = Cκα ,κ ′α ′(q = 0)+CNA
κα ,κ ′α ′(q → 0)
CNA
κα ,κ ′α ′(q → 0) =4π
Ω0
!
∑β qγ Z∗
κβα
"!
∑′
β q′β Z∗
κ ′β ′α ′
"
∑ββ ′ qβ ε∞ββ ′q
′
β
ω2m(q → 0) = ω2
m(q = 0)+4π
Ω0
∑αα ′ qα Sm,αα ′q′α
∑αα ′ qα ε∞αα ′q′α
CECAM Tutorial, Lyon, 12-16 may 2014
0 200 400 600 800 1000 1200
14
12
10
8
6
4
2
0
2
4
6
Experimental frequency (cm1)
Re
lative
err
or
(%)
LDA
PBE
PBEsol
AM05
WC
HTBS
0 200 400 600 800 1000 12000
200
400
600
800
1000
1200
Experimental frequency (cm−1)
Theo
retic
al fr
eque
ncy
(cm−1
)
LDAPBEPBEsolAM05WCHTBS
Example 1: Zircon (phonons)
55
CECAM Tutorial, Lyon, 12-16 may 2014
Example 1: Zircon (Born effective charges)
56
• M : anomalously large (esp. Z⊥) → PbZrO3, ZrO2
• Si : smaller deviations ( and ) → SiO2 (α-quartz or stishovite)
• O : strong anisotropy → SiO2 stishovite or TiO2 rutile in the y-z plane (plane of the M-O bonds) (rem: ≠ from SiO2 α-quartz in the x direction where 2 components)
Interpretation:• mixed ionic-covalent bonding
• closer to stishovite than α-quartz in agreement with naive bond counting for O atoms
Nominal: M → +4 Si → +4 O → -2
CECAM Tutorial, Lyon, 12-16 may 2014
Example 1: Zircon (dielectric properties)
57
• due to the frequency factor, it is the lowest frequency mode that contributes the most to ε0
• Sm and Zm are the largest for the lowest and highest frequency modes*
• this effect can be compensated by a lower frequency for hafnon [e.g. A2u(1)]
• Sm and Zm are smaller for HfSiO4 ← mass difference and Born effective charges*
CECAM Tutorial, Lyon, 12-16 may 2014
Freq
uenc
y (c
m-1
)
0
50
100
150
200
250
300
Γ X W L Γ K X
Example 2: Copper (thermodynamics)
58
0 200 400 600 800 1000
T (K)
0
0.5
1
1.5
(%)
LDA
PBE
PBEsol
AM05
WC
HTBS
Expt.
(a)
0 200 400 600 800 1000
T (K)
0
0.5
1
1.5
2
2.5
3
(10
51 )
LDA
PBE
PBEsol
AM05
WC
HTBS
Expt.
(b)
0 200 400 600 800 1000
T (K)
1
1.2
1.4
1.6
1.8
B0
(M
Bar)
LDA
PBE
PBEsol
AM05
WC
HTBS
Expt.
(c)
0 200 400 600 800 1000
T (K)
0
0.5
1
1.5
(%)
LDA
PBE
PBEsol
AM05
WC
HTBS
Expt.
(a)
0 200 400 600 800 1000
T (K)
0
0.5
1
1.5
2
2.5
3
(10
51 )
LDA
PBE
PBEsol
AM05
WC
HTBS
Expt.
(b)
0 200 400 600 800 1000
T (K)
1
1.2
1.4
1.6
1.8
B0
(M
Bar)
LDA
PBE
PBEsol
AM05
WC
HTBS
Expt.
(c)
CECAM Tutorial, Lyon, 12-16 may 2014
!!
• 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)
59