Density functional perturbation theory for lattice dynamics …...Density functional perturbation theory for lattice dynamics with ultrasoft pseudopotentials and PAW Andrea Dal Corso

DFPT with US-PPsDFPT with PAW

Grid and images

Density functional perturbation theory forlattice dynamics with ultrasoftpseudopotentials and PAW

Andrea Dal Corso

SISSA and DEMOCRITOSTrieste (Italy)

Andrea Dal Corso Density functional perturbation theory


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Outline

1 DFPT with US-PPs

2 DFPT with PAW

3 Grid and images



Grid and images

US-PPs Hamiltonian[−1

2∇2 + VNL +

∫d3r Vσeff (r)K (r)

]|ψiσ〉 = εiσS|ψiσ〉,

Vσeff (r) = Vloc(r) +∫

d3r1ρ(r1)|r− r1|

+ Vσxc(r),

VNL(r1, r2) =∑Inm

D(0)γ(I)nm βγ(I)n (r1 − RI)β

∗γ(I)m (r2 − RI),

ρσ(r) =∑

i

θ̃F ,iσ〈ψiσ|K (r)|ψiσ〉,

K (r; r1, r2) = δ(r− r1)δ(r− r2)+

∑Inm

Qγ(I)nm (r− RI)βγ(I)n (r1 − RI)β

∗γ(I)m (r2 − RI).



Grid and images

US-PPs - PerturbationAtoms move:

RI = R` + ds → R` + ds + u(`, s)

Vloc , VNL, K , the overlap matrix

S(r1, r2) = δ(r1 − r2) +∑Inm

qγ(I)nm βγ(I)n (r1 − RI)β

∗γ(I)m (r2 − RI),

and the ortogonality constraint:

〈ψiσ|S|ψjσ〉 = δij ,

depend on the perturbation. Calling λ the perturbation, we have

〈dψiσdλ|S|ψjσ〉+ 〈ψiσ|S|

dψjσdλ〉 = −〈ψiσ|

∂S∂λ|ψjσ〉,



Grid and images

US-PPs - Linear system

[−1

2∇2 + VσKS + Q

σ − εiσS]|∆̃µψiσ〉 = −P†c,iσ

[dVσKSdµ

− εiσ∂S∂µ

]|ψiσ〉,

with

P†c,iσ =

θ̃F ,iσ −∑j

βiσ,jσS|ψjσ〉〈ψjσ|

.and

VσKS(r1, r2) = VNL(r1, r2) +∫

d3r V σeff (r)K (r; r1, r2).



Grid and images

US-PPs - First derivatives of the KS potential

dVσKS(r1, r2)dµ

=∂VσKS(r1, r2)

∂µ+

∫d3r

dVσHxc(r)dµ

K (r; r1, r2).

∂VσKS(r1, r2)∂λ

=∂VNL(r1, r2)

∂λ+

∫d3r

∂Vloc(r)∂λ

K (r; r1, r2)

+

∫d3r V σeff (r)

∂K (r; r1, r2)∂λ

.



Grid and images

US-PPs - Second derivatives of the energy

d2F (1)totdµdλ

=∑iσ

θ̃F ,iσ〈ψiσ|

[∂2VσKS∂µ∂λ

− εiσ∂2S∂µ∂λ

]|ψiσ〉,

d2F (2)totdµdλ

= 2 Re∑iσ

〈∆µψiσ|[∂VσKS∂λ

− εiσ∂S∂λ

]|ψiσ〉,

d2F (3)totdµdλ

=∑σ

∫d3r

dVσHxc(r)dµ

∆λρσ(r),

d2F (4)totdµdλ

= −∑iσ

{〈δµψiσ|

[∂VσKS∂λ

− εiσ∂S∂λ

]|ψiσ〉+ (µ↔ λ)

}.



Grid and images

US-PPs - Second partial derivatives of the KSpotential

∂2VσKS(r1, r2)∂µ∂λ

=∂2VNL(r1, r2)

∂µ∂λ

+

∫d3r

∂2Vloc(r)∂µ∂λ

K (r; r1, r2)

+

∫d3r V σeff (r)

∂2K (r; r1, r2)∂µ∂λ

+

[∫d3r

∂Vloc(r)∂λ

∂K (r; r1, r2)∂µ

+ (λ↔ µ)].

A. Dal Corso, Phys. Rev. B 64, 235118 (2001).



Grid and images

The dynamical matrix

The dynamical matrix is:

Φαβ(q, s, s′

)=

1N

∑``′

e−iq·R`d2Ftot

duα(`, s)duβ(`′, s′)eiq·R`′ ,

So we take µ→ uα(`, s) and λ→ uβ(`′, s′).In a periodic solid the index i on the wavefunctions becomes aBloch vector and a band index k, v .



Grid and images

US - PPs Changes summaryCompute ∆µρσ(r). drho.f90, incdrhous.f90,compute_drhous.f90.

Compute d2F (4)tot

dµdλ . drho.f90, compute_nldyn.f90.

Add the contributions to ∂2VσKS(r1,r2)∂µ∂λ . dvanqq.f90,

dynmat_us.f90, addusdynmat.f90.Add the contributions to ∂V

σKS(r1,r2)∂µ . dvqpsi_us_only.f90.

Add the contributions to dVσKS(r1,r2)

dµ . newdq.f90,adddvscf.f90.Add the augmentation charge to the induced charge.addusddens.f90.Compute d

2F (3)totdµdλ . drhodvus.f90.

Add the contributions to d2F (2)tot

dµdλ . drhodvnl.f90.



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PAW - Hamiltonian[−1

2∇2 + VNL +

∫d3r Vσeff (r)K (r)

]|ψiσ〉 = εiσS|ψiσ〉,

VNL(r1, r2) =∑Inm

(D1,σI,nm − D̃

1,σI,nm

)βγ(I)n (r1 − RI)β

∗γ(I)m (r2 − RI),

ρσ(r) =∑

i

θ̃F ,iσ〈ψiσ|K (r)|ψiσ〉,

K (r; r1, r2) = δ(r− r1)δ(r− r2)+

∑Inm

Qγ(I)nm (r− RI)βγ(I)n (r1 − RI)β

∗γ(I)m (r2 − RI).

L. Paulatto, G. Fratesi, and S. de Gironcoli, unpublished.Andrea Dal Corso Density functional perturbation theory


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PAW - Hamiltonian

D1,σI,mn =∫

ΩI

d3r ΦI,AEm (r)(−12∇2)ΦI,AEn (r)

+

∫ΩI

d3r ΦI,AEm (r)ΦI,AEn (r)V

I,σeff (r),

D̃1,σI,mn =∫

ΩI

d3r ΦI,PSm (r)(−12∇2)ΦI,PSn (r)

+

∫ΩI

d3r ΦI,PSm (r)ΦI,PSn (r)Ṽ

I,σeff (r) +

∫ΩI

d3r QI,mn(r)ṼI,σeff (r)

D1,σI,mn and D̃1,σI,mn depend on the atomic positions through V

I,σeff (r)

and Ṽ I,σeff (r), respectively.Andrea Dal Corso Density functional perturbation theory


Grid and images

PAW - Linear system

[Hσ + Qσ − εiσS] |∆̃µψiσ〉 = −P†c,iσ

[dHσ

dµ− εiσ

∂S∂µ

]|ψiσ〉,

dHσ

dµ=

dVσKSdµ

+∑I,mn

∆D1,σ,µI,mn |βIm〉〈βIn|,

where ∆D1,σ,µI,mn =(

dD1,σI,mndµ −

dD̃1,σI,mndµ

).

dD1,σI,mndµ

=∑σ1

∫ΩI

d3r ΦI,AEm (r)ΦI,AEn (r)

dV I,σeffdρ1,Iσ1

dρ1,Iσ1dµ

.



Grid and images

PAW - Second derivatives of the energy

Terms (1), (2), (4) as with US-PPs, only the term (3) has a PAWcontribution:

d2E (3)totdµdλ

=d2E (3)UStot

dµdλ+∑σ

∑I,mn

∆D1,σ,µI,mn bσ,λI,mn.

A. Dal Corso, Phys. Rev. B 81, 075123 (2010).



Grid and images

PAW - Changes summary

Save bσ,λI,mn when computed. drho.f90, addusddens.f90.

Compute ∆D1,σ,µI,mn inside the spheres. PAW_dpotential,PAW_dusymmetrize.Add the contributions to dH

σ

dµ newdq.f90.

Add the PAW contribution to d2F (3)tot

dµdλ . drhodvus.f90.



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Phonon parallelization: grid, images

Parallelization modes of QE:G-vectors.bands.k-points.

Additional parallelization of phonon:q-vectors.Irreducible representations.

Actually this is implemented using grid techniques: one qpoint per run or one irrep per run.Another possibility is to use images. The total number ofprocessors is split into several groups (images) each imagerunning an independent copy of ph.x.



Grid and images

Phonon parallelization: grid, images

Problems with the grid:It requires complex scripts to coordinate and collect theresults of different runs.

Problems with images:Images do not communicate among themselves, becausedifferent runs are independent.Load balacing is difficult.Final results need to be collected running ph.x anothertime.



Grid and images

The future: thermo_pw

thermo_pw solves two of these problems:

Images can communicate through a master-slavesapproach via MPI calls.The code can run in a synchronous and asynchronousmode. It can collect the final results automatically.The code can mix calls to pw.x and ph.x so that it ispossible for instance to optimize the structure beforecalling ph.x, or call ph.x for several geometries andcompute anharmonic properties.



Grid and images

Asynchronous parallelization via MPI routinesBoth master and slaves compute all the tasks to do (forinstance all the irreps and q points) and assign a number toeach task.Master:

1 During initialization calls a nonblocking receive of theready variable from all the slaves (mpi_irecv).

2 Tests if some slave has sent the ready variable(mpi_test).

3 If not, it continues its work. If a slave has sent its readyvariable it sends (with a blocking send) to the slave thenumber of the next task to do (mpi_send) or the no_worknumber if there is no more work to do.

4 Finally makes another nonblocking receive of the readyvariable from the slave that has received the work to doand continues its work.



Grid and images

Asynchronous parallelization via MPI routines

Slave:1 Sends (with a blocking send) the ready variable to the

master (mpi_send).2 Receives (with a blocking receive) the number of the task

to do. When it receives it, it starts to do its work or exit if thetask number corresponds to no_work (mpi_receive).

3 When it finishes its task it restarts from [1]To coordinate the work it is sufficient to initialize the masterdoing [1] at the beginning of the asynchronous work and thatthe master calls as often as possible a routine that executes [2],[3], [4] (for instance after each scf step). The most often themaster calls this routine the shorter is the inactivity interval ofthe slaves.



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Bibliography

1 S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett. 58,1861 (1987); P. Giannozzi, S. de Gironcoli, P. Pavone, andS. Baroni, Phys. Rev. B 43, 7231 (1991).

2 S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi,Rev. Mod. Phys. 73, 515 (2001).

3 (DFPT + US) A. Dal Corso, Phys. Rev. B 64, 235118(2001).

4 (DFPT + PAW) A. Dal Corso, Phys. Rev. B 81, 075123(2010).

5 (GRID) R. di Meo, A. Dal Corso, P. Giannozzi, and S.Cozzini, ICTP lecture notes 24, 163 (2009).



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Bibliography

6 (spin-orbit US) A. Dal Corso and A. Mosca Conte, Phys.Rev. B 71, 115106 (2005).

7 (dfpt spin-orbit US) A. Dal Corso, Phys. Rev. B 76, 054308(2007).

8 (spin-orbit PAW) A. Dal Corso, Phys. Rev. B 82, 075116(2010).

9 (lsda and σ-gga) A. Dal Corso and S. de Gironcoli, Phys.Rev. B 62, 273 (2000).

10 (nlcc) A. Dal Corso, S. Baroni, R. Resta, and S. deGironcoli, Phys. Rev. B 47, 3588 (1993).

11 (US dielectric constant) J. Tóbik and A. Dal Corso, J.Chem. Phys. 120, 9934 (2004).



Grid and images

phq_init.f90, compute_becalp.f90phq_init.f90 computes the products becp

〈βIm|ψkvσ〉 =1√N

eik·R`βsmkvσ,

and alphap

∂〈βIm|∂uα(`, s)

|ψkvσ〉 =1√N

eik·R`αsαmkvσ .

compute_becalp.f90 computes the products becq

〈βIm|ψk+qvσ〉 =1√N

ei(k+q)·R`βsmk+qvσ,

and alpq

∂〈βIm|∂uα(`, s)

|ψk+qvσ〉 =1√N

ei(k+q)·R`αsαmk+qvσ.



Grid and images

compute_weights.f90, compute_alphasum.f90,compute_becsum_ph.f90

compute_weights.f90 computes the weights (saved in wgg)in the definition of (i → kv , j → k + qv ′)

|δµψiσ〉 =∑

j

[θ̃F ,iσθiσ,jσ + θ̃F ,jσθjσ,iσ

]|ψjσ〉〈ψjσ|

∂S∂µ|ψiσ〉.

compute_alphasum.f90 computes:

csασnm =1N

∑kv

θ̃F ,kvσ

[α∗sαnkvσ β

smkvσ + β

∗snkvσα

sαmkvσ

].

compute_becsum_ph.f90 computes:

bσsnm =1N

∑kv

θ̃F ,kvσβ∗snkvσβ

smkvσ,



Grid and images

drho.f90

This routine computes the fourth part of the dynamical matrix:

d2F (4)totdµdλ

= −∑iσ

{〈δµψiσ|

[∂VσKS∂λ

− εiσ∂S∂λ

]|ψiσ〉+ (µ↔ λ)

}.

and the change of the charge due to the displacement of theaugmentation charge:

∆µρσ(r) = −∑

i

〈ψiσ|K (r)|δµψiσ〉+∑

i

θ̃F ,iσ〈ψiσ|∂K (r)∂µ|ψiσ〉

for all the modes and saves it on disk. This is done with thehelp of several routines.



Grid and images

drho.f90

Actually the quantity that is needed is the charge induced by aphonon perturbation of wavevector q:

∆usα(q)ρσ(r) =∑`

eiq·R`∆uα(`,s)ρσ(r)

so we define:

δusα(q)ψkvσ(r) =∑`

eiq·R`δuα(`,s)ψkvσ(r)



Grid and images

compute_drhous.f90

This is a driver that for each k point calls incdrhous.f90 toaccumulate two quantities needed to compute ∆usα(q)ρσ(r).

∆usα(q)ρσ(r) =∑kv

ψ∗kvσ(r)δusα(q)ψkvσ(r)

+∑kv

∑Inm

Qγ(I)nm (r− RI)〈ψkvσ|βIn〉〈βIm|δusα(q)ψkvσ〉

+ . . .

The first term is accumulated by incdrhous.f90 directly,while the second is accumulated by addusdbec.f90



Grid and images

incdrhous.f90This routine accumulates a part of ∆usα(q)ρσ(r)

∆usα(q)ρσ(r) =∑kv

ψ∗kvσ(r)δusα(q)ψkvσ(r) + . . .

The rest is calculated by drho.f90 calling addusddens.f90with iflag=1.

∆usα(q)ρσ(r) = · · ·+∑kv

∑Inm

Qγ(I)nm (r− RI)〈ψkvσ|βIn〉〈βIm|δusα(q)ψkvσ〉

+∑`

eiq·R`∑snm

[Qγ(I)nm (r− RI)csασnm

+∂Qγ(I)nm (r− RI)∂uα(`, s)

bσsnm

],



Grid and images

addusddens.f90

When (iflag=1) this routine receives in dbecsum

dusα(q)σs1nm =2N

∑kv

β∗s1nkvσ δusα(q)βs1mkvσ ,

where

〈βIm|δusα(q)ψkvσ〉 =1√N

ei(k+q)·R`δusα(q)βs1mkvσ

and implements the expression written above.



Grid and images

dvanqq.f90This routine computes four integrals.

1Iuα(`,s)σnm =∫

d3r V σeff (r)∂Qγ(s)nm (r− RI)∂uα(`, s)

,

2Iuα(`,s)Inm =∫

d3r∂Vloc(r)∂uα(`, s)

Qγ(I)nm (r− RI),

4Iuα(`,s)βσnm =∫

d3r Vσeff (r)∂2Qγ(s)nm (r− RI)∂uα(`, s)∂uβ(`, s)

,

5Iuα(`,s)uβ(`′,s′)

nm =

∫d3r

∂Vloc(r)∂uα(`, s)

∂Qγ(s′)

nm (r− RI′)∂uβ(`′, s′)

.



Grid and images

dvanqq.f90

In previous equation:

1Iuα(`,s)σnm =1I∗sασnm

2Iuα(`,s)Inm =1N

∑q

e−iq·(R`−R`1 ) 2Isαqs1nm

4Iuα(`,s)βσnm =4 Isαβσnm

5Iuα(`,s)uβ(`1,s1)nm =1N

∑q

eiq·(R`−R`1 ) 5Iss1αβqnm



Grid and images

compute_nldyn.f90

This routine computes

d2F (4)totdµdλ

= −∑iσ

{〈δµψiσ|

[∂V̄σKS∂λ

− εiσ∂S∂λ

]|ψiσ〉+ (µ↔ λ)

}.

Note that the term computed by drho.f90 should have ∂VσKS

∂λ .Here the bar on VσKS indicates that a part is not calculated bythis routine but in drho.f90. This part is:

−∑iσ

∫d3r

dVloc(r)dλ

ψiσ(r)δµψ∗iσ(r),



Grid and images

compute_nldyn.f90

We have

d2F (4)totdµdλ

= −∑iσ

∑Inm

{Deff ,Iσnm 〈δµψiσ|

∂βIn∂λ〉〈βIm|ψiσ〉

+ Deff ,Iσnm 〈δµψiσ|βIn〉〈∂βIm∂λ|ψiσ〉

+ 2IλInm〈δµψiσ|βIn〉〈βIm|ψiσ〉

+ 1IλσInm〈δµψiσ|βIn〉〈βIm|ψiσ〉+ (µ↔ λ)},

where we defined Deff ,Iσmn = DIσmn − εiσqImn.



Grid and images

compute_nldyn.f90

In terms of the quantities defined above we can write:

〈δusα(q)ψkvσ| =∑

l

e−iqR`〈δuα(`,s)ψkvσ|

=∑v ′

wkvσ,k+qv ′σ∑mn

qsmn

(β∗snkvσα

sαmk+qv ′σ

+ α∗sαnkvσ βsmk+qv ′σ

)〈ψk+qv ′σ|

=∑v ′

A∗usα(q)kv ′vσ 〈ψk+qv ′σ|



Grid and images

compute_nldyn.f90

So that the expression calculated by this routine is:

Φαβ(q, s, s′

)= − 1

N

∑kσ

∑vv ′

∑nm

{A∗usα(q)kv ′vσ D

eff ,s′σnm α

∗s′βnk+qv ′σβ

s′mkvσ

+ A∗usα(q)kv ′vσ Deff ,s′σnm β

∗s′nk+qv ′σα

s′βmkvσ

+ A∗usα(q)kv ′vσ∑s1

2Is′βq

s1nm β∗s1nk+qv ′σβ

s1mkvσ

+ A∗usα(q)kv ′vσ1Is

′βσnm β

∗s′nk+qv ′σβ

s′mkvσ

}+ h.c.,

The hermitean conjugate (h.c.) is added in drho.f90.



Grid and images

dynmat_us.f90

This routine computes directly a part of the term:

d2F (1)totdµdλ

=∑iσ

θ̃F ,iσ〈ψiσ|

[∂2VσKS∂µ∂λ

− εiσ∂2S∂µ∂λ

]|ψiσ〉,

in particular the part similar to the norm conserving potentialthat corresponds to:


=∂2VNL(r1, r2)

∂µ∂λ+

∫d3r

∂2Vloc(r)∂µ∂λ

K (r; r1, r2) + . . .

and calls addusdynmat.f90 to compute the rest.



Grid and images

addusdynmat.f90

This routine computes the rest of the term d2F (1)tot

dµdλ . In particularthe part that corresponds to the terms:


= · · ·+∫

d3r V σeff (r)∂2K (r; r1, r2)

∂µ∂λ

+

[∫d3r

∂Vloc(r)∂λ

∂K (r; r1, r2)∂µ

+ (λ↔ µ)].

that can be written in terms of the four integrals calculated bydvanqq.f90



Grid and images

addusdynmat.f90

The two terms calculated by this routine can be written in termsof the previous four integrals and of the quantities computed bycompute_alphasum.f90 and compute_becsum.f90.

Φ(1b)αβ (q, s, s

′) = δss′

{∑σ

∑nm

4Isαβσnm bsσnm

+

[∑σ

∑nm

1I∗sασnm csβσnm +

(α↔ β

)]},

Φ(1c)αβ (q, s, s

′) =

{[∑σ

∑nm

5Iss′αβq

nm bs′σnm +

∑σ

∑nm

2I∗sαqs′nm cs′βσnm

]+h.c.

}.



Grid and images

dvqpsi_us_only.f90

Computes a part of the right-hand side of the linear system:[dVσNLdµ

− εiσ∂S∂µ

]|ψiσ〉

The contribution of the local potential is calculated indvqpsi_us.f90, the contribution of dV

σHxc(r)dµ is calculated in

solve_linter.f90 while an additional US part is calculatedin adddvscf.f90.



Grid and images

dvqpsi_us_only.f90

This routine computes the following term:

[∂VσNL∂µ

− εiσ∂S∂µ

]|ψiσ〉 =

∑Imn

{Deff ,Iσnm |


+ Deff ,Iσnm |βIn〉〈∂βIm∂λ|ψiσ〉

+ 2IλInm|βIn〉〈βIm|ψiσ〉

+ 1IλσInm|βIn〉〈βIm|ψiσ〉

}



Grid and images

newdq.f90

This routine computes the following integral:

3IµσInm =∫

d3rdVσHxc(r)

dµQγ(I)nm (r− RI),

and is called by solve_linter.f90 after computing a newestimate of dV

σHxc(r)dµ . Defining:

3Iusα(q)σI1nm =∑`

eiq·R` 3Iuα(`,s)σI1nm

we have3Iusα(q)σI1nm = e

iq·R`1 3Iusα(q)σs1nm



Grid and images

adddvscf.f90This routine computes the part of the right-hand side of thelinear system∫

d3rdVσHxc(r)

dµ[K (r; r1, r2)− δ(r− r1)δ(r− r2)] |ψiσ〉 =

Using the integral 3IµσInm computed by newdq.f90. Expandingwe have:

=∑Imn

3IµσInm|βIm〉〈βIn|ψiσ〉

and the implemented term is:∑s1

∑nm

3Iusα(q)σs1nm βγ(s1)n (k + q + G)e−i(k+q+G)·τs1β

s1mkvσ



Grid and images

incdrhoscf.f90, addusdbec.f90

incdrhoscf.f90 accumulates for each k

2 Re∑

i

ψiσ(r)∗∆̃µψiσ(r)

as in the norm conserving case, while addusdbec.f90accumulates the term

ausα(q)σs1nm =2N

∑kv

β∗s1nkvσ ∆̃usα(q)βs1mkvσ ,

that is used by addusddens.f90 to calculate theaugmentation part.



Grid and images

addusddens.f90

This routine (called with iflag=0) computes:

dρσ(r)dµ

= 2 Re∑

i

〈ψiσ| [K (r)− 1] |∆̃µψiσ〉+ ...

= 2∑kv

∑Inm

Qγ(I)nm (r− RI)〈ψkvσ|βIn〉〈βIm|∆̃usα(q)ψkvσ〉+ ...

using the quantity accumulated by addusdbec.f90. Readsfrom disk ∆µρσ(r) and adds it to

dρσ(r)dµ .



Grid and images

drhodvus.f90

After computing the induced potential this routine computes theterm of the dynamical matrix that is obtained from:

d2F (3)totdµdλ

=∑σ

∫d3r

dVσHxc(r)dµ

∆λρσ(r),

where ∆λρσ(r) is read from disk. In the dynamical matrix thisterm is actually:

Φ(3)αβ (q, s, s

′) =∑σ

∫Ω

d3rdV ∗σHxc(r)dusα(q)

∆us′β(q)ρσ(r),



Grid and images

drhodv.f90This routine needs a few generalizations. It must calculate

d2F (2)totdµdλ

= 2 Re∑iσ

〈∆µψiσ|[∂VσKS∂λ

− εiσ∂S∂λ

]|ψiσ〉,

which is a term similar to that computed bycompute_nldyn.f90 with 〈∆µψiσ| instead of 〈δµψiσ|. Thecontribution of the local potential is similar to the normconserving case and calculated in drhodvloc.f90. We needthe two products becpq:

〈βI1m|∆usα(q)ψkvσ〉 =1√N

ei(k+q)·R`1 ∆usα(q)βs1mkvσ

and dalpq

∂〈βI′m|∂us′β(q)

|∆usα(q)ψkvσ〉 =1√N

eik·R`′ ∆usα(q)αs′βm

kvσ .



Grid and images

drhodvnl.f90

In analogy with compute_nldyn.f90

d2F (4)totdµdλ

= 2∑iσ

∑Imn

{Deff ,Iσnm 〈∆µψiσ|


+ Deff ,Iσnm 〈∆µψiσ|βIn〉〈∂βIm∂λ|ψiσ〉

+ 2IλInm〈∆µψiσ|βIn〉〈βIm|ψiσ〉

+ 1IλσInm〈∆µψiσ|βIn〉〈βIm|ψiσ〉}



Grid and images

drhodvnl.f90

In analogy with compute_nldyn.f90

Φ(2)αβ

(q, s, s′

)=

2N

∑kvσ

∑mn

{Deff ,s

′σnm (∆

usα(q)α∗s′βn

kvσ )βs′mkvσ

+ Deff ,s′σ

nm (∆usα(q)β∗s

′nkvσ )α

s′βmkvσ

+∑s1

2Is′βq

s1nm (∆usα(q)β∗s1nkvσ )β

s1mkvσ

+ 1Is′βσ

nm (∆usα(q)β∗s

′nkvσ )β

s′mkvσ

},

where we used the time reversal symmetry.



Grid and images

PAW_dusymmetrize, PAW_dpotential

In solve_linter.f90:

dρI,σmndµ

= 2Re∑

i

〈ψi,σ|βIm〉〈βIn|∆µψi,σ〉+ bσ,µI,mn.

becsumort is added to the first term contained in dbecsum.dρI,σmndµ is symmetrized in PAW_dusymmetrize.

dD1,σI,mndµ

=∑σ1

∫ΩI

d3r ΦI,AEm (r)ΦI,AEn (r)

dV I,σeffdρ1,Iσ1

dρ1,Iσ1dµ

.

anddD̃1,σI,mn

dµ are calculated by PAW_dpotential. These routinesare in PAW_onecenter.f90 and PAW_symmetry.f90.



Grid and images

PAW - drhodvus.f90

This routine is called after the calculation of ∆D1,σ,µI,mn . It has it inthe variable int3_paw and computes

d2E (3)totdµdλ

=d2E (3)UStot

dµdλ+∑σ

∑I,mn

∆D1,σ,µI,mn bσ,λI,mn.


DFPT with US-PPsDFPT with PAWGrid and images

Density functional perturbation theory for lattice dynamics …...Density functional perturbation theory for lattice dynamics with ultrasoft pseudopotentials and PAW Andrea Dal Corso

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