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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

2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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Page 1: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 2: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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)

Page 3: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 4: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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)

Page 5: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

CECAM Tutorial, Lyon, 12-16 may 2014

Example: Diamond

5

Theory vs. Experiment [X. Gonze, GMR, and R. Caracas, Z. Kristallogr. 220, 458 (2000)]

Page 6: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 7: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 8: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 9: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

CECAM Tutorial, Lyon, 12-16 may 2014

OutlinePerturbation Theory

Density Functional Perturbation Theory

Atomic displacements and homogeneous electric field

9

Page 10: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

CECAM Tutorial, Lyon, 12-16 may 2014

OutlinePerturbation Theory

Density Functional Perturbation Theory

Atomic displacements and homogeneous electric field

10

Page 11: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 12: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 13: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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 + . . .

Page 14: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

+ . . .

Page 15: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

· · ·

Page 16: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 17: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

· · ·

Page 18: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 19: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 20: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 21: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 22: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 23: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 24: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 25: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 26: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 27: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 28: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 29: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 30: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

Page 31: 2014 L017 CECAMabinit14.sciencesconf.org/conference/abinit14/... · Institute of Condensed Matter and Nanosciences, B-1348 Louvain-la-Neuve, Belgique European Theoretical Spectroscopy

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

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• 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

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• 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

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OutlinePerturbation Theory

Density Functional Perturbation Theory

Atomic displacements and homogeneous electric field

34

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• 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

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• 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 )]

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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

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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)

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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

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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

· · ·

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• 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)

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• 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

· · ·

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• 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

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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

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OutlinePerturbation Theory

Density Functional Perturbation Theory

Atomic displacements and homogeneous electric field

45

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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)

· · ·

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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

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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 ′ −

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

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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

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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

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• 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

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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

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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 ′ +

Ω0∑m

Sm,αα ′

ω2m −ω2

ε0

αα ′ = ε∞αα ′ +

Ω0∑m

Sm,αα ′

ω2m

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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

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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

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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

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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*

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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)

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CECAM Tutorial, Lyon, 12-16 may 2014

!!

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