Computer lab: Density functional perturbation theory. theory for lattice dynamics

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1 Computer lab: density functional perturbation theory for lattice dynamics SISSA and DEMOCRITOS Trieste (Italy)

2 Outline 1 The dynamical matrix

3 Dynamical matrix We want to write a small computer program that calculates the dynamical matrix of a solid: D sαs β(q) = + Ω Ω d 3 2 V loc (r) r u sα(q) u s β(q) ρ(r) I 1 ( ) ( ) d 3 V loc (r) ρ(r) r I 2 u sα (q) u s β(q) + D Ewald sαs β (q) I 3. We need three routines: dynmat0, drhodv and d2ionq to calculate I 1, I 2 and I 3 respectively.

ρ(r) I 1 ( ) ( ) d 3 V loc (r) ρ(r) r I 2 u sα (q) u s β(q) + D Ewald sαs β (q) I 3.

4 dynmat0 I 1 can be calculated using the charge density only. It is calculated in reciprocal space. Writing: V loc (r) = µ,s = vloc s (k)eik r e ik Rµ e ik ds e ik uµ,s, µ,s k where vloc s (k) = 1 V d 3 r vloc s (r)e ik r is the Fourier transform of (r), we have that: v s loc v s loc (r R µ d s u µ,s ) 2 V loc (r) u sα(q) u s β(q) = δ s,s ṽloc s (G)e ig ds G α G β e ig r, where ṽloc s (G) = 1 Ω d 3 r vloc s (r)e ig r. G

vloc s (r)e ik r is the Fourier transform of (r), we have that: v s loc v s loc (r R µ d s u µ,s ) 2 V

5 dynmat0 - I Writing the charge density in Fourier series: ρ(r) = G ρ(g)e ig r, I 1 is: I 1 = δ s,s Ω G ρ(g)ṽ s loc (G) e ig ds G α G β. dynmat0 calculates this sum using the fact that ṽloc s (G) is real and this integral is also real. So I 1 = δ s,s Ω ] ṽloc s (G)G αg β [Rρ(G)cos(G d s ) Iρ(G)sin(G d s ). G

dynmat0 calculates this sum using the fact that ṽloc s (G) is real and this

6 dynmat0 - II Note that we will calculate the phonon frequencies of Si using the Appelbaum and Hamann pseudo-potential so: Ωṽ s loc where k 2 (k) = e 4α { 4πZ ve 2 k 2 + ( π α Z v = 4 v 1 = Ha v 2 = Ha α = /(a.u.) 2 ) 3 [ 2 v 1 + v ( )]} 2 3 α 2 k 2 4α

where k 2 (k) = e 4α { 4πZ ve 2 k 2 + ( π α Z v = 4 v 1 = 3.

7 To calculate I 2 we need the periodic part of the bare perturbation V loc (r) u sα(q). We have: V loc (r) u sα (q) = µ e iq Rµ V loc(r) u µsα = i G ṽ s loc (q + G)(q + G) αe i(q+g) ds e i(q+g) r. Therefore the periodic part has Fourier components: V loc (q + G) = iṽloc s u sα (q) (q + G)(q + G) αe i(q+g) ds. This expression is calculated by the routine compute_dvloc.

8 Charge density response Then we need the lattice-periodic part of the induced charge density which is: ρ(r) u s β(q) = 2 kv u kv (r)pk+q c u kv (r) u s β(q). The routine incdrhoscf calculates this expression after the calculation of Pc k+q u kv (r) u s β (q). Note that if each band is occupied by 2 electrons, in the nonmagnetic case, we need another factor of 2 for spin degeneracy. The sum over k is done as in the calculation of the charge density.

The routine incdrhoscf calculates this expression after the calculation of Pc k+q u kv (r) u s β (q).

9 P k+q c u kv (r) u s β (q) is the solution of the linear system: [H k+q + Q ɛ kv ] P k+q c u kv (r) u s β(q) = Pk+q c V KS (r) u s β(q) u kv(r). This linear system is solved by an iterative conjugate gradient algorithm that requires a routine that applies the left hand side to an arbitrary function and a routine that computes the right hand side. The left hand side is applied by the routine ch_psi_all, while the right hand side is calculated from the induced Kohn and Sham potential.

This linear system is solved by an iterative conjugate gradient algorithm that requires a routine that applies the left

10 ch_psi_all - II The operator Q = v α u k+qv u k+qv vanishes when applied to Pc k+q so it will not change the solution. If α > max(ɛ kv ɛ k+qv ) it makes the operator H k+q + Q ɛ kv nonsingular and the linear system well defined. We take α = 2(max(ɛ kv ) min(ɛ kv )) and the same α is used for all k points.

If α > max(ɛ kv ɛ k+qv ) it makes the operator H k+q + Q ɛ kv nonsingular

11 is given by: V KS (r) u s β(q) = V loc (r) u s β(q) + d 3 r 1 r) ρ(r ) r r eiq(r u s β(q) + V xc(r) ρ(r) ρ u s β(q). The first term is calculated by compute_dvloc, while the other two terms are calculated by dv_of_drho.

12 dv_of_drho The induced exchange and correlation potential is calculated in real space. Vxc(r) ρ is calculated at the beginning of the run by phq_init and dmxc by numerical differentiation: [ ] V xc (r) = V xc (r, ρ + ) V xc (r, ρ ) /2, ρ and the induced exchange and correlation potential is calculated in real space: V xc (r) ρ ρ(r) u s β(q).

differentiation: [ ] V xc (r) = V xc (r, ρ + ) V xc (r, ρ ) /2, ρ and the induced

13 dv_of_drho - II The induced Hartree potential is calculated in reciprocal space, and a Fourier transform is later used to calculate it in real space. We have: d 3 r 1 r) ρ(r ) r r eiq(r u s β(q) = e iq r ρ(q + G) d 3 r 1 ei(q+g) r u s G β(q) r r = e ig r ρ(q + G) u s G β(q) 4π 1 q + G 2, ρ(q+g) where u s β (q) is calculated by a Fourier transform of ρ(r) u s β (q).

We have: d 3 r 1 r) ρ(r ) r r eiq(r u s β(q) = e iq r ρ(q + G) d 3 r 1 ei(q+g) r u s G β(q)

14 Putting all together: solve_linter We need a driver to solve the self-consistent linear system. The driver is solve_linter. This routine allocates space for the V loc (r) input and output induced potential, apply u s β (q) (using dvqpsi) and the input induced potential to u kv (r), and apply the projector Pc k+q to prepare the right hand side of the linear system. Then it calls cgsolve_all to solve the linear system and incdrhoscf to calculate the contribution of the k point to the induced charge density. These steps are repeated for all k points and then dv_of_drho is used to calculate the output induced potential. Finally a mixing routine checks if self-consistency has been achieved. If not the mixing routine provides the input induced potential for the next iteration.

prepare the right hand side of the linear system. Then it calls cgsolve_all to solve the linear system and incdrhoscf to calculate the contribution of the k point to the induced charge density.

15 drhodv After the self-consistent solution of the linear system we obtain ρ(r) the self-consistent induced charge density u s β (q). After a V Fourier transform of loc (q+g) V u sα(q) we get loc (r) u and sα(q) I 2 = ( ) ( ) Ω V loc (r i ) ρ(r i ), N 1 N 2 N 3 u sα (q) u s β(q) i where the sum is over all the points of the real space mesh and N 1, N 2 and N 3 are the dimensions of the mesh.

After a V Fourier transform of loc (q+g) V u sα(q) we get loc (r) u and sα(q) I 2 = ( ) ( ) Ω V

16 d2ionq Rev. Mod. Phys. 73, 515 (2001).

17 dyndia Finally we need a driver for the diagonalization of the dynamical matrix. The driver is dyndia. It divides the dynamical matrix by the masses, it forces it to be exactly Hermitian, it calls the routine cdiagh to diagonalize it and writes on output the frequencies. Imaginary frequencies are written with a minus sign.

It divides the dynamical matrix by the masses, it forces it to be exactly

18 Bibliography 1 S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett. 58, 1861 (1987); P. Giannozzi, S. de Gironcoli, P. Pavone, and S. 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 A. Dal Corso, Phys. Rev. B 64, (2001). 4 A. Dal Corso, Phys. Rev. B 76, (2007). 5 A. Dal Corso, Phys. Rev. B 81, (2010).

de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001). 3 A. Dal Corso, Phys.

= N 2 = 3π2 n = k 3 F. The kinetic energy of the uniform system is given by: 4πk 2 dk h2 k 2 2m. (2π) 3 0

= N 2 = 3π2 n = k 3 F. The kinetic energy of the uniform system is given by: 4πk 2 dk h2 k 2 2m. (2π) 3 0 Chapter 1 Thomas-Fermi Theory The Thomas-Fermi theory provides a functional form for the kinetic energy of a non-interacting electron gas in some known external potential V (r) (usually due to impurities)