Tutorial on DFPT and TD-DFPT: calculations of phonons and absorption spectra

Transcription

1 Tutorial on DFPT and TD-DFPT: calculations of phonons and absorption spectra Iurii Timrov SISSA Scuola Internazionale Superiore di Studi Avanzati, Trieste Italy Computer modelling of materials at the nanoscale The University of Tokyo, Japan 25 April 2014

2 Outline This afternoon Density functional perturbation theory (DFPT) Time-dependent density functional perturbation theory (TDDFPT) Introduction Exercise 1: Phonon calculations at Γ Exercise 2: Phonon dispersion Introduction Exercise 1: Calculation of independent particle spectrum Exercise 2: turbodavidson Exercise 3: turbolanczos

3 Part 1 : Calculation of phonons

4 Phonons Let us consider a unit cell with atoms. index of an atom in the unit cell is the Cartesian index is the point in the Bravais-lattice, identifying the position of a given unit cell is the number of unit cells in the crystal is the -component of the displacement of s-th atom variables Interatomic Force Constants :

5 Phonons Fourier transformation: Essence of the Bloch theorem: Important concept: We can perform calculations of Interatomic Force Constants for each q independently!!!

6 Phonons Important concept: Instead of we need only Interatomic Force Constants!!! Sampling theorem The number of q points is equal to the number of R points at which Interatomic Force Constants are computed.

7 Phonons Normal mode frequencies,, and eigenvectors, determined by the secular equation: are where is the dynamical matrix. Interatomic Force Constants (IFC) Diagonalization of dynamical matrix gives phonon modes at q.

8 Exercise 1: Phonon calculations at Γ Go to the directory with the input files: cd Part1_phonons/exercise1 In this directory you will find: File describing how to do the exercise README Si.scf.in Input file for the SCF calculation Si.ph.in Si.vbc.UPF Pseudopotential of silicon reference Directory with the reference results out Input file for the phonon calculation at Γ Directory for temporary files

9 Exercise 1: Phonon calculations at Γ Step 1. Perform an SCF calculation for silicon at the equilibrium structure using the code pw.x

10 Exercise 1: Phonon calculations at Γ Step 2. Perform a phonon calculation at Γ using the code ph.x The same prefix as in the SCF calculation Threshold for self-consistency Atomic mass Directory for temporary files File containing the dynamical matrix Coordinates of the q point in units of 2*pi/a in Cartesian framework

11 Exercise 1: Phonon calculations at Γ Dynamical matrix file Si.dyn :

12 Exercise 1: Phonon calculations at Γ Acoustic sum rule at Γ Because of numerical inaccuracies the interatomic force constants do not strictly satisfy the acoustic sum rule (ASR). However, the ASR can be imposed using the code dynmat.x. The input is Si.dynmat.in : File containing the dynamical matrix A way to impose the acoustic sum rule

13 Exercise 1: Phonon calculations at Γ The program dynmat.x produces the file dynmat.out which contains the new frequencies:

14 Exercise 2: Phonon dispersion Go to the directory with the input files: cd Part1_phonons/exercise2 In this directory you will find: README File describing how to do the exercise Si.scf.in Input file for the SCF calculation Si.ph.in Input file for the phonon calculation on a q-grid Si.q2r.in Input file for calculation of Interatomic Force Constants Si.matdyn.in Input file for Fourier Interpolation for various q points Si.plotband.in Input file for plotting a phonon dispersion Si.vbc.UPF Pseudopotential of silicon reference Directory with the reference results out Directory for temporary files

15 Exercise 2: Phonon dispersion Step 1. Perform an SCF calculation for silicon at the equilibrium structure using the code pw.x Step 2. Perform a phonon calculation on a uniform grid of q points using the code ph.x Option for the calculation on a grid Uniform grid of q points

16 Exercise 2: Phonon dispersion The phonon code ph.x generates the files of the dynamical matrices on the specified grid of q points. The files are Si.dyn1, Si.dyn2,..., Si.dyn8. The file Si.dyn0 contains the list of the inequivalent q points (8, in this case).

17 Exercise 2: Phonon dispersion Step 3. Calculation of the Interatomic Force Constants (IFC) using the code q2r.x Fourier transforms of IFC's : are Cartesian components, and Fourier transforms of IFC's on a grid of q points nq1 x nq2 x nq3 in reciprocal space are atomic indices. IFC's in a supercell nq1 x nq2 x nq3 in real space

18 Exercise 2: Phonon dispersion Input file Si.q2r.in : Dynamical matrices from the phonon calculation A way to impose the acoustic sum rule Output file of the interatomic force constants The denser the grid of q points, the larger the vectors R for which the Interatomic Force Constants are calculated! To perform the calculation:

19 Exercise 2: Phonon dispersion Step 4. Calculate phonons at generic q' points using IFC by means of the code matdyn.x IFC's on a grid in real space Fourier interpolation Fourier transforms of IFC's at generic q' point in reciprocal space Input file Si.matdyn.in : Acoustic sum rule Atomic mass File with IFC's Output file with the frequencies Number of q points Coordinates of q points

20 Exercise 2: Phonon dispersion Step 5. Plot the phonon dispersion using the code plotband.x and gnuplot. Input file Si.plotband.in : Input file with the frequencies at various q' Range of frequencies for a visualization Output file with frequencies which will be used for plot Plot of the dispersion (we will produce another one) Fermi level (needed only for band structure plot) Frequency step on the plot freq.ps Reference frequency on the plot freq.ps Use gnuplot and the file plot_dispersion.gnu in order to plot the phonon dispersion of silicon (experimental_data.dat). You will obtain a postscript file phonon_dispersion.eps which you can visualize.

21 Exercise 2: Phonon dispersion Phonon dispersion of silicon (file phonon_dispersion.eps):

22 Exercise 2: Phonon dispersion How to determine whether the quality of the Fourier interpolation is satisfactory? Compare with the direct calculation (no interpolation)!

23 Exercise 2: Phonon dispersion Comparison of the phonon dispersion computed using the Fourier interpolation with the direct calculation at several q points. The q-grid 4x4x4 is very satisfactory for the Fourier interpolation for silicon!

24 Exercise 2: Phonon dispersion The agreement of an ab initio calculation of the phonon dispersion using the Fourier interpolation on a q-grid 4x4x4 is excellent with the experimental data!

25 Exercise 2: Phonon dispersion The Fourier interpolation works if the Interatomic Force Constants (IFC's) are known on a sufficiently large supercell, i.e. on a large enough grid of q points in the phonon calculation. There are cases when the IFC's are long range and the Fourier interpolation does not work properly: When there are Kohn anomalies in metals. In this case the dynamical matrices are not a smooth function of q and the IFC's are long range. In polar insulators where the atomic displacements generate long range electrostatic interactions and the dynamical matrix is not analitical for q 0. However, this case can be dealt with by calculating the Born effective charges and the dielectric tensor of the material.

26 Part 2 : Calculation of absorption spectra

27 Independent Particle Approximation The simplest approximation: Independet Particle Approximation (IPA) which allows us to describe single-particle excitations. Fermi's golden rule The transition probability per unit time from occupied states states reads: to empty is the external potential induced by the electric field and are the eigenvalues and the eigenfunctions of the ground-state Kohn-Sham equation Absorption coefficient:

28 Exercise 1: Calculation of absorption spectrum of benzene in IPA Go to the directory with the input files: cd Part2_absorption_spectra/exercise1 In this directory you will find: README File describing how to do the exercise Benzene.scf.in Input file for the SCF calculation Benzene.epsilon.in Input file for a calculation of spectrum H.vbc.UPF Pseudopotential of hydrogen C.vbc.UPF Pseudopotential of carbon plot_spectrum.gnu Script to plot spectrum using gnuplot reference Directory with the reference results out Directory for temporary files

29 Exercise 1: Calculation of absorption spectrum of benzene in IPA Step 1. Perform an SCF calculation for benzene at the equilibrium structure using the code pw.x

30 Exercise 1: Calculation of absorption spectrum of benzene in IPA Step 2. Perform a calculation of the absorption spectrum for benzene in the independent particle approximation using the code epsilon.x Input file for the calculation of spectrum Type of the calculation The same prefix as in the SCF calculation Directory for temporary files Type of smearing of the spectrum The value of the smearing in ev Minimum value of frequencies for a plot in ev Maximum value of frequencies for a plot in ev Number of points between wmin and wmax

31 Exercise 1: Calculation of absorption spectrum of benzene in IPA The code epsilon.x produces 4 files: epsr.dat Real part of the response epsi.dat Imaginary part of the response (this is what we need) eels.dat Electron energy loss spectrum ieps.dat Response computed on the imaginary axis of frequency The content of epsi.dat looks like: Use gnuplot and the script plot_spectrum.gnu in order to plot the absorption spectrum of benzene Benzene_spectrum.eps

32 Exercise 1: Calculation of absorption spectrum of benzene in IPA Absorption spectrum of benzene in the Independent Particle Approximation (file Benzene_spectrum.eps):

33 turbo-davidson program for calculation of absorption spectra The turbo_davidson.x program allows us to calculate absorption spectra of molecules using linear-response time-dependent density functional theory (TDDFT). The interactions of electrons (Hartree and Exchange-Correlation effects) are taken into account fully ab initio and self-consistently. The electronic transitions from occupied to empty states can be analyzed by selecting a frequency range in which the transitions are expected to occur. However, the overall absorption spectrum in a wide frequency range cannot be calculated at once. Several calculations are required. Xiao-Chuan Ge's PhD thesis Seeing colors with TDDFT

34 turbo-davidson program for calculation of absorption spectra Linear-response TDDFPT equations: Rewrite these equations in the matrix form: Casida's equations where and interaction term Davidson algorithm is used (the same algorithm as in the ground state SCF calculation) to solve the Casida's equations and to obtain the eigenvalues which are used for a calculation of the absorption coefficient.

35 Exercise 2: Calculation of absorption spectrum of benzene using turbo_davidson.x Go to the directory with the input files: cd Part2_absorption_spectra/exercise2 In this directory you will find: README File describing how to do the exercise Benzene.scf.in Input file for the SCF calculation Benzene.davidson.in Input file for a Davidson calculation of the eigenvalues Benzene.tddfpt_pp.in Input file for a postprocessing calculation of the spectrum H.vbc.UPF Pseudopotential of hydrogen C.vbc.UPF Pseudopotential of carbon plot_spectrum.gnu Script to plot spectrum using gnuplot reference Directory with the reference results out Directory for temporary files

36 Exercise 2: Calculation of absorption spectrum of benzene using turbo_davidson.x Step 1. Perform an SCF calculation for benzene at the equilibrium structure using the code pw.x Step 2. Perform the Davidson calculation without the interaction using the code turbo_davidson.x The same prefix as in the SCF calculation Directory for temporary files Switch off the interaction Number of eigenvalues to be calculated Number of initial vectors Maximum number of basis allowed for the sub-basis Convergence threshold Minimum value of frequencies for a plot in Ry Maximum value of frequencies for a plot in Ry Frequency step in Ry Lorentzian broadening parameter in Ry Reference frequency in Ry where the peak is expected Number of occupied states to be considered Number of empty states to be considered

37 Exercise 2: Calculation of absorption spectrum of benzene using turbo_davidson.x The code turbo_davidson.x produces a file Benzene-dft.eigen : Step 3. Perform a spectrum calculation using the postprocessing code turbo_spectrum.x and using the eigenvalues computed in the previous step. The input file Benzene.tddfpt_pp.in reads: The same prefix as in the SCF calculation Directory for temporary files Type of previous calculation The value of Lorenzian smearing in Ry Minimum value of frequencies for a plot in Ry Maximum value of frequencies for a plot in Ry Frequency step in Ry Frequency with Davidson eigenvalues

38 Exercise 2: Calculation of absorption spectrum of benzene using turbo_davidson.x The code turbo_spectrum.x produces a file Benzene.plot which can be used for plotting the absorption spectrum : Step 4. Plot the spectrum using gnuplot and the script plot_spectrum.gnu Since the interaction was switched off (if_dft_spectrum=.true.), you should obtain the same spectrum as the one obtained using the epsilon.x code in the exercise1. The script plot_spectrum.gnu will do such a comparison, and it will produce the file Benzene_spectrum.eps which you can visualize.

39 Exercise 2: Calculation of absorption spectrum of benzene using turbo_davidson.x Comparison of the absorption spectrum of benzene computed in the Independent Particle Approximation using turbo_davidson.x and epsilon.x (file Benzene_spectrum.eps):

40 Exercise 2: Calculation of absorption spectrum of benzene using turbo_davidson.x Now switch on the interaction! Make the following modifications in the input files: In the file Benzene.davidson.in In the file Benzene.tddfpt_pp.in set eign_file = 'Benzene.eigen' set if_dft_spectrum =.false. In plot_spectrum.gnu change the title to: 'turbo-davidson.x (interacting electrons)' Once these modifications are done, repeat steps 2, 3, and 4: Step 2. Step 3. Step 4. gnuplot load 'plot_dispersion.gnu' Note! The calculation will be a bit too long: 22 minutes using 2 cores. Therefore, let us see the output files in the directory reference.

41 Exercise 2: Calculation of absorption spectrum of benzene using turbo_davidson.x Comparison of the absorption spectrum of benzene computed using turbo_davidson.x with interaction and using epsilon.x in the Independent Particle Approximation (file Benzene_spectrum.eps): Interaction opens the gap and blue-shifts the peak

42 turbo-lanczos program for calculation of absorption spectra The turbo_lanczos.x program allows us to calculate absorption spectra of molecules using linear-response time-dependent density functional perturbation theory (TDDFPT) without computing empty states! The interactions of electrons (Hartree and Exchange-Correlation effects) are taken into account fully ab initio and self-consistently. The electronic transitions from occupied to empty states cannot be analyzed (use turbo_davidson.x for this purpose). The overall absorption spectrum in a wide frequency range can be calculated at once! This is a very big advantage of the turbo_lanczos.x code! Dario Rocca's PhD thesis TDDFT: New algorithms with applications to molecular spectra

43 turbo-lanczos program for calculation of absorption spectra Linear-response TDDFPT equations: These equations are solved using the Liouville-Lanczos approach: Absorption coefficient is computed as: Lanczos algorithm is used to solve recursively quantum Liouville equation in the standard batch representation. This allows us to avoid inversions and multiplications of large matrices.

44 Exercise 3: Calculation of absorption spectrum of benzene using turbo_lanczos.x Go to the directory with the input files: cd Part2_absorption_spectra/exercise3 In this directory you will find: README File describing how to do the exercise Benzene.scf.in Input file for the SCF calculation Benzene.lanczos.in Input file to perform Lanczos recursions Benzene.tddfpt_pp.in Input file for a postprocessing calculation of spectrum H.vbc.UPF Pseudopotential of hydrogen C.vbc.UPF Pseudopotential of carbon plot_spectrum.gnu Script to plot spectrum using gnuplot reference Directory with the reference results out Directory for temporary files

45 Exercise 3: Calculation of absorption spectrum of benzene using turbo_lanczos.x Step 1. Perform an SCF calculation for benzene at the equilibrium structure using the code pw.x Step 2. Perform Lanczos recursions using the code turbo_lanczos.x The input file is Benzene.lanczos.in : The same prefix as in the SCF calculation Directory for temporary files The code writes restart files every restart_step iterations Restart iterations after previous calculation Number of Lanczos iterations Polarization direction of incoming light, 1=x

46 Exercise 3: Calculation of absorption spectrum of benzene using turbo_lanczos.x Note! The calculation will be a bit too long: 35 minutes using 2 cores. Therefore, let us see the output files in the directory reference. In the output file Benzene.lanczos.out there is information about each Lanczos iteration : In the directory out, which contains temporary files, there is a file Benzene.beta_gamma_z.1 which contains the information about Lanczos coefficients:

47 Exercise 3: Calculation of absorption spectrum of benzene using turbo_lanczos.x Step 3. Perform a spectrum calculation using the postprocessing code turbo_spectrum.x and using the Lanczos coefficients computed in the previous step. The input file Benzene.tddfpt_pp.in reads: The same prefix as in the SCF calculation Directory for temporary files Number of calculated Lanczos coefficient Number of extrapolated Lanczos coefficient Type of extrapolation (bi-constant) The value of Lorenzian smearing in Ry Minimum value of frequencies for a plot in Ry Maximum value of frequencies for a plot in Ry Frequency step in Ry Polarization direction (same as in turbo_lanczos.x)

48 Exercise 3: Calculation of absorption spectrum of benzene using turbo_lanczos.x The code turbo_spectrum.x produces a file Benzene.plot which can be used for plotting the absorption spectrum : Step 4. Plot the spectrum using gnuplot and the script plot_spectrum.gnu You should obtain the same excitation peak in the spectrum as was obtained using the code turbo_davidson.x in the exercise2 including interaction. The script plot_spectrum.gnu will do such a comparison, and it will produce the file Benzene_spectrum.eps which you can visualize.

49 Exercise 3: Calculation of absorption spectrum of benzene using turbo_lanczos.x Comparison of the absorption spectrum of benzene computed using turbo_lanczos.x and using turbo_davidson.x both including the interation (file Benzene_spectrum.eps):

50 Exercise 3: Calculation of absorption spectrum of benzene using turbo_lanczos.x turbo_lanczos.x allows us to obtain the absorption spectrum in a wide frequency range just by repeating a postprocessing calculation using turbo_spectrum.x in a larger frequency range. This cannot be done with turbo_davidson.x

51 Comparison of the absorption spectra of benzene and naphthalene In naphthalene the pi-orbitals are more delocalized than in benzene, which leads to smaller energy gap, and which hence red-shifts the peaks.

52 Comparison of the absorption spectra of benzene using local and non-local XC functionals Hybrid exchange-correlation functionals open the energy gap and blue-shift the peaks (SCF: input_dft='pbe0', Lanczos: d0psi_rs=.true. and the molecule must be centered in the supercell)

53 Exercise 4: Convergence of the absorption spectrum of Na2 Go to the directory with the input files: cd Part2_absorption_spectra/exercise4 Study the convergence of the absorption spectrum of Na2 with respect to: The number of Lanczos iterations itermax = 50, 100, 150,... without using the extrapolation extrapolation = 'no' How many Lanczos iterations is needed to converge the spectrum when the extrapolation is used extrapolation = 'osc'? The size of the supercell celldm(1) = 20, 30, 40,...

54 Exercise 4: Convergence of the absorption spectrum of Na2

55 Exercise 4: Convergence of the absorption spectrum of Na2 Behaviour of Lanczos coefficients

56 Exercise 4: Convergence of the absorption spectrum of Na2 Extrapolation of the Lanczos chains In the Liuoville-Lanczos approach, the absorption coefficient is computed as: where is the tridiagonal matrix composed of the Lanczos coefficients. Extrapolation

57 Exercise 4: Convergence of the absorption spectrum of Na2