# Exercises¶

## Getting started¶

Welcome to SALMON Exercises!

In these exercises, we explain the use of SALMON from the very beginning, taking a few samples that cover applications of SALMON in several directions. We assume that you are in the computational environment of UNIX/Linux OS. First you need to download and install SALMON in your computational environment. If you have not yet done it, do it following the instruction, download and Install and Run.

As described in Install and Run, you are required to prepare at least an input file and pseudopotential files to run SALMON. In the following, we present input files for several sample calculations and provide a brief explanation of the input keywords that appear in the input files. You may modify the input files to execute for your own calculations. Pseudopotential files of elements that appear in the samples are also attached. We also present explanations of main output files.

We present 10 exercises.

First 3 exercises (Exercise-1 ~ 3) are for an isolated molecule, acetylene C2H2. If you are interested in learning electron dynamics calculations in isolated systems, please look into these exercises. In SALMON, we usually calculate the ground state solution first using a static density functional theory (DFT). This is illustrated in Exercise-1. After finishing the ground state calculation, two exercises of electron dynamics calculations based on time-dependent density functional theory (TDDFT) are prepared. Exercise-2 illustrates the calculation of linear optical responses in real time, obtaining polarizability and photoabsorption of the molecule. Exercise-3 illustrates the calculation of electron dynamics in the molecule under a pulsed electric field.

Next 3 exercises (Exercise-4 ~ 6) are for a crystalline solid, silicon. If you are interested in learning electron dynamics calculations in extended periodic systems, please look into these exercises. Exercise-4 illustrates the ground state calculation of the crystalline silicon based on DFT. Exercise-5 illustrates the calculation of linear response properties of the crystalline silicon based on TDDFT to obtain the dielectric function. Exercise-6 illustrates the calculation of electron dynamics in the crystalline silicon induced by a pulsed electric field.

Exercise-7 is for a simultaneous calculation of the propagation of a pulsed light and electronic motion in a bulk silicon, coupling Maxwell equations for the electromagnetic fields of the pulsed light and the electron dynamics in the unit cells based on TDDFT. This calculation is quite time-consuming and is recommended to execute using massively parallel supercomputers. Exercise-7 illustrates the calculation of a linearly polarized pulsed light irradiating normally on a surface of a bulk silicon.

Next 2 exercises (Exercise-8 ~ 9) are for geometry optimization based on DFT and Ehrenfest molecular dynamics based on TDDFT for an isolated molecule, acetylene C2H2. Exercise-8 illustrates the geometry optimization in the ground state. Exercise-9 illustrates the Ehrenfest molecular dynamics induced by a pulsed electric field.

Exercise-10 is for a macroscopic light propagation through a metallic nanosphere solving Maxwell equations. The optical response of the nanosphere is described by a dielectric function. Finite-Difference Time-Domain (FDTD) method is used to calculated the three-dimensional light propagation.

Input files of exercises are included in SALMON, in the directory SALMON/samples/exercise_##_<description>/.

## C2H2 (isolated molecules)¶

### Exercise-1: Ground state of C2H2 molecule¶

In this exercise, we learn the calculation of the ground state of acetylene (C2H2) molecule, solving the static Kohn-Sham equation. This exercise will be useful to learn how to set up calculations in SALMON for any isolated systems such as molecules and nanoparticles.

Acetylene molecule is a linear chain molecule composed of two Carbon atoms and two Hydrogen atoms.

In SALMON, we use a three-dimensional (3D) uniform grid system to express physical quantities such as electron orbitals.

#### Input files¶

To run the code, following files in the directory SALMON/samples/exercise_01_C2H2_gs/ are used:

 file name description C2H2_gs.inp input file that contains input keywords and their values C_rps.dat pseodupotential file for carbon atom H_rps.dat pseudopotential file for hydrogen atom

Pseudopotential files are needed for two elements, Carbon (C) and Hydrogen (H). The pseudopoential depends on the angular momentum, and looks as follows (for Carbon).

In the input file C2H2_gs.inp, input keywords are specified. Most of them are mandatory to execute the ground state calculation. This will help you to prepare an input file for other systems that you want to calculate. A complete list of the input keywords that can be used in the input file can be found in List of input keywords.

!########################################################################################!
! Excercise 01: Ground state of C2H2 molecule                                            !
!----------------------------------------------------------------------------------------!
! * The detail of this excercise is explained in our manual(see chapter: 'Exercises').   !
!   The manual can be obtained from: https://salmon-tddft.jp/documents.html              !
! * Input format consists of group of keywords like:                                     !
!     &group                                                                             !
!       input keyword = xxx                                                              !
!     /                                                                                  !
!   (see chapter: 'List of input keywords' in the manual)                                !
!----------------------------------------------------------------------------------------!
! * Conversion from unit_system = 'a.u.' to 'A_eV_fs':                                   !
!   Length: 1 [a.u.] = 0.52917721067    [Angstrom]                                       !
!   Energy: 1 [a.u.] = 27.21138505      [eV]                                             !
!   Time  : 1 [a.u.] = 0.02418884326505 [fs]                                             !
!########################################################################################!

&calculation
!type of theory
theory = 'dft'
/

theory specifies which theoretical method is used in the calculation.
&control
!common name of output files
sysname = 'C2H2'
/

sysname is a prefix for filenames of output files.
&units
!units used in input and output files
unit_system = 'A_eV_fs'
/

unit_system specifies which unit system is used in the input and output files.
&system
!periodic boundary condition
yn_periodic = 'n'

!number of elements, atoms, electrons and states(orbitals)
nelem  = 2
natom  = 4
nelec  = 10
nstate = 6
/

yn_periodic specifies whether or not periodic boundary condition is applied.
nelem is the number of elements in the system.
natom is the number of atoms in the system.
nelec is the number of electrons in the system.
nstate is the number of orbitals that are used in the calculation.
&pseudo
!name of input pseudo potential file
file_pseudo(1) = './C_rps.dat'
file_pseudo(2) = './H_rps.dat'

!atomic number of element
izatom(1) = 6
izatom(2) = 1

!angular momentum of pseudopotential that will be treated as local
lloc_ps(1) = 1
lloc_ps(2) = 0
!--- Caution ---------------------------------------!
! Indices must correspond to those in &atomic_coor. !
!---------------------------------------------------!
/

file_pseudo(n) specifies the filename of the pseudopotential file of the n-th element.
izatom(n) is the atomic number of the n-th element.
lloc_ps(n) specifies which angular momentum component is chosen as the local potential for the n-th element.
&functional
!functional('PZ' is Perdew-Zunger LDA: Phys. Rev. B 23, 5048 (1981).)
xc = 'PZ'
/

xc specifies the exchange-correlation potential to be used in the calculation.
&rgrid
!spatial grid spacing(x,y,z)
dl(1:3) = 0.25d0, 0.25d0, 0.25d0

!number of spatial grids(x,y,z)
num_rgrid(1:3) = 64, 64, 64
/

dl(i) specifies the spatial grid spacing in i-th direction.
num_rgrid(i) specifies the number of grid points in i-th direction.
&scf
!maximum number of scf iteration and threshold of convergence
nscf      = 300
threshold = 1.0d-9
/

nscf specifies the maximum number of SCF iterations.
threshold specifies the threshold to judge the convergence.
&analysis
!output of all orbitals, density,
!density of states, projected density of states,
!and electron localization function
yn_out_psi  = 'y'
yn_out_dns  = 'y'
yn_out_dos  = 'y'
yn_out_pdos = 'y'
yn_out_elf  = 'y'
/

yn_out_psi, yn_out_dns, yn_out_dos, yn_out_pdos, yn_out_elf specify output files that are generated after the calculation.
&atomic_coor
!cartesian atomic coodinates
'C'    0.000000    0.000000    0.599672  1
'H'    0.000000    0.000000    1.662257  2
'C'    0.000000    0.000000   -0.599672  1
'H'    0.000000    0.000000   -1.662257  2
!--- Format ---------------------------------------------------!
! 'symbol' x y z index(correspond to that of pseudo potential) !
!--------------------------------------------------------------!
/

&atomic_coor specifies spatial coordinates of atoms.

#### Execusion¶

In a multiprocess environment, calculation will be executed as:

$mpiexec -n NPROC salmon < C2H2_gs.inp > C2H2_gs.out  where NPROC is the number of MPI processes. A standard output will be stored in the file C2H2_gs.out. #### Output files¶ After the calculation, following output files and a directory are created in the directory that you run the code in addition to the standard output file,  name description C2H2_info.data information on ground state solution C2H2_eigen.data orbital energies C2H2_k.data k-point distribution (for isolated systems, only gamma point is described) data_for_restart directory where files used in the real-time calculation are contained psi_ob1.cube, psi_ob2.cube, ... electron orbitals dns.cube a cube file for electron density dos.data density of states pdos1.data, pdos2.data, ... projected density of states elf.cube electron localization function (ELF) PS_C_KY_n.dat information on pseodupotential file for carbon atom PS_H_KY_n.dat information on pseodupotential file for hydrogen atom You may download the above files (zipped file, except for the directory data_for_restart) from: We first explain the standard output file. In the beginning of the file, input variables used in the calculation are shown. ############################################################################## # SALMON: Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience # # Version 2.0.1 # ############################################################################## Libxc: [disabled] theory= dft use of real value orbitals = T ====== MPI distribution: nproc_k : 1 nproc_ob : 1 nproc_rgrid : 1 1 2 OpenMP parallelization: number of threads : 256 .........  After that, the SCF loop starts. At each iteration step, the total energy as well as orbital energies and some other quantities are displayed. ----------------------------------------------- iter= 1 Total Energy= -197.59254070 Gap= -20.17834599 Vh iter= 234 1 -29.9707 2 -28.3380 3 -13.0123 4 5.8457 5 -9.9213 6 -14.3326 iter and int_x|rho_i(x)-rho_i-1(x)|dx/nelec = 1 0.31853198E+00 Ne= 10.0000000000000 ----------------------------------------------- iter= 2 Total Energy= -280.97950515 Gap= -9.59770609 Vh iter= 247 1 -17.4334 2 -24.4941 3 -20.1872 4 0.8020 5 -3.4058 6 -8.7957 iter and int_x|rho_i(x)-rho_i-1(x)|dx/nelec = 2 0.54493263E+00 Ne= 10.0000000000000 ----------------------------------------------- iter= 3 Total Energy= -295.67034640 Gap= -6.90359156 Vh iter= 229 1 -16.0251 2 -19.7759 3 -17.6765 4 -0.9015 5 -2.9323 6 -7.8050 iter and int_x|rho_i(x)-rho_i-1(x)|dx/nelec = 3 0.13010987E+00 Ne= 10.0000000000000  When the convergence criterion is satisfied, the SCF calculation ends. ----------------------------------------------- iter= 162 Total Energy= -339.69525272 Gap= 6.78870999 Vh iter= 1 1 -18.4106 2 -13.9966 3 -12.4163 4 -7.3386 5 -7.3386 6 -0.5498 iter and int_x|rho_i(x)-rho_i-1(x)|dx/nelec = 162 0.50237787E-08 Ne= 9.99999999999999 ----------------------------------------------- iter= 163 Total Energy= -339.69525269 Gap= 6.78870999 Vh iter= 1 1 -18.4106 2 -13.9966 3 -12.4163 4 -7.3386 5 -7.3386 6 -0.5498 iter and int_x|rho_i(x)-rho_i-1(x)|dx/nelec = 163 0.69880308E-09 Ne= 9.99999999999999 #GS converged at 164 : 0.69880308E-09  Next, the force acting on ions and some other information related to orbital energies are shown. ===== force ===== 1 -0.33652081E-05 0.16854696E-04 -0.59496450E+00 2 -0.59222259E-06 0.24915590E-05 0.57651725E+00 3 -0.37839836E-05 0.20304090E-04 0.59493028E+00 4 -0.86779607E-06 0.39560274E-05 -0.57651738E+00 orbital energy information------------------------------- Lowest occupied orbital -0.676576619015730 Highest occupied orbital (HOMO) -0.269686750876529 Lowest unoccupied orbital (LUMO) -2.020624936948345E-002 Highest unoccupied orbital -2.020624936948345E-002 HOMO-LUMO gap 0.249480501507045 Physicaly upper bound of eps(omega) 0.656370369646246 --------------------------------------------------------- Lowest occupied orbital[eV] -18.4105868958642 Highest occupied orbital (HOMO)[eV] -7.33855002098465 Lowest unoccupied orbital (LUMO)[eV] -0.549840032009334 Highest unoccupied orbital[eV] -0.549840032009334 HOMO-LUMO gap[eV] 6.78870998897532 Physicaly upper bound of eps(omega)[eV] 17.8607468638548 --------------------------------------------------------- writing restart data... writing completed.  In the directory data_for_restart, files that will be used in the next-step time evolution calculations are stored. Other output files include following information. C2H2_info.data Calculated orbital and total energies as well as parameters specified in the input file are shown. C2H2_eigen.data Orbital energies. #esp: single-particle energies (eigen energies) #occ: occupation numbers, io: orbital index # 1:io, 2:esp[eV], 3:occ  C2H2_k.data k-point distribution(for isolated systems, only gamma point is described). # ik: k-point index # kx,ky,kz: Reduced coordinate of k-points # wk: Weight of k-point # 1:ik[none] 2:kx[none] 3:ky[none] 4:kz[none] 5:wk[none] # coefficients (2*pi/a [a.u.]) in kx, ky, kz  psi_ob1.cube, psi_ob2.cube, ... Cube files for electron orbitals. The number in the filename indicates the index of the orbital. Atomic unit is adopted in all cube files. dns.cube A cube file for electron density. dos.data A file for density of states. The units used in this file are affected by the input parameter, unit_system in &unit. elf.cube A cube file for electron localization function (ELF). We show several image that are created from the output files. • Highest occupied molecular orbital (HOMO) The output files psi_ob1.cube, psi_ob2.cube, ... are used to create the image. • Electron density The output files dns.cube, ... are used to create the image. • Electron localization function The output files elf.cube, ... are used to create the image. ### Exercise-2: Polarizability and photoabsorption of C2H2 molecule¶ In this exercise, we learn the linear response calculation in the acetylene (C2H2) molecule, solving the time-dependent Kohn-Sham equation. The linear response calculation provides the polarizability and the oscillator strength distribution of the molecule. This exercise should be carried out after finishing the ground state calculation that was explained in Exercise-1. Polarizability $$\alpha_{\mu \nu}(t)$$ is the basic quantity that characterizes optical responses of molecules and nano-particles, where $$\mu, \nu$$ indicate Cartesian components, $$\mu, \nu = x,y,z$$. The polarizability $$\alpha_{\mu \nu}(t)$$ relates the $$\mu$$ component of the electric dipole moment at time $$t$$, $$p_{\mu}(t)$$, with the $$\nu$$ component of the electric field at time $$t'$$, $$p_{\mu}(t) = \sum_{\nu=x,y,z} \alpha_{\mu \nu}(t-t') E_{\nu}(t').$$ We introduce a frequency-dependent polarizability by the time-frequency Fourier transformation of the polarizability, $$\tilde \alpha_{\mu \nu}(\omega) = \int dt e^{i\omega t} \alpha_{\mu \nu}(t).$$ The imaginary part of the frequency-dependent polarizability is related to the photoabsorption cross section $$\sigma(\omega)$$ by $$\sigma(\omega) = \frac{4\pi \omega}{c} \frac{1}{3} \sum_{\mu=x,y,z} {\rm Im} \tilde \alpha_{\mu \mu}(\omega).$$ The photoabsorption cross section is also related to the oscillator strength distribution by $$\sigma(\omega) = \frac{2\pi^2 e^2}{mc} \frac{df(\omega)}{d\omega}.$$ In SALMON, the polarizability is calculated in time domain. First the ground state orbital $$\phi_i(\mathbf{r})$$ that satisfies the Kohn-Sham equation, $$H_{\rm KS} \phi_i(\mathbf{r}) = \epsilon_i \phi_i(\mathbf{r}),$$ is prepared. Then an impulsive force given by the potential $$V_{\rm ext}(\mathbf{r},t) = I \delta(t) z,$$ is applied to all electrons in the C2H2 molecule along the molecular axis which we take $$z$$ axis. $$I$$ is the magnitude of the impulse, and $$\delta(t)$$ is the Dirac's delta function. The orbital is distorted by the impulsive force at $$t=0$$. Immediately after the impulse is applied, the orbital becomes $$\psi_i(\mathbf{r},t=0_+) = e^{iIz/\hbar} \phi_i(\mathbf{r}).$$ After the impulsive force is applied at $$t=0$$, a time evolution calculation is carried out without any external fields, $$i\hbar \frac{\partial}{\partial t} \psi_i(\mathbf{r},t) = H_{\rm KS}(t) \psi_i(\mathbf{r},t).$$ During the time evolution, the electric dipole moment given by $$p_z(t) = \int d\mathbf{r} (-ez) \sum_i \vert \psi_i(\mathbf{r},t) \vert^2,$$ is monitored. After the time evolution calculation, a time-frequency Fourier transformation is carried out for the electric dipole moment to obtain the frequency-dependent polarizability by $$\tilde \alpha_{zz}(\omega) = - \frac{e}{I} \int dt e^{i\omega t} p_z(t).$$ #### Input files¶ To run the code, following files are necessary:  file name description C2H2_response.inp input file that contains input keywords and their values C_rps.dat pseodupotential file for carbon atom H_rps.dat pseudopotential file for hydrogen atom restart directory created in the ground state calculation (rename the directory from data_for_restart to restart) First three files are prepared in the directory SALMON/samples/exercise_02_C2H2_lr/. The file C2H2_rt_response.inp that contains input keywords and their values. The pseudopotential files should be the same as those used in the ground state calculation. In the directory restart, those files created in the ground state calculation and stored in the directory data_for_restart are included. Therefore, copy the directory as cp -R data_for_restart restart if you calculate at the same directory as you did the ground state calculation. In the input file C2H2_rt_response.inp, input keywords are specified. Most of them are mandatory to execute the linear response calculation. This will help you to prepare the input file for other systems that you want to calculate. A complete list of the input keywords that can be used in the input file can be found in List of input keywords. !########################################################################################! ! Excercise 02: Polarizability and photoabsorption of C2H2 molecule ! !----------------------------------------------------------------------------------------! ! * The detail of this excercise is explained in our manual(see chapter: 'Exercises'). ! ! The manual can be obtained from: https://salmon-tddft.jp/documents.html ! ! * Input format consists of group of keywords like: ! ! &group ! ! input keyword = xxx ! ! / ! ! (see chapter: 'List of input keywords' in the manual) ! !----------------------------------------------------------------------------------------! ! * Conversion from unit_system = 'a.u.' to 'A_eV_fs': ! ! Length: 1 [a.u.] = 0.52917721067 [Angstrom] ! ! Energy: 1 [a.u.] = 27.21138505 [eV] ! ! Time : 1 [a.u.] = 0.02418884326505 [fs] ! !----------------------------------------------------------------------------------------! ! * Copy the ground state data directory('data_for_restart') (or make symbolic link) ! ! calculated in 'samples/exercise_01_C2H2_gs/' and rename the directory to 'restart/' ! ! in the current directory. ! !########################################################################################! &calculation !type of theory theory = 'tddft_response' /  theory specifies which theoretical method is used in the calculation. &control !common name of output files sysname = 'C2H2' /  sysname is a prefix for filenames of output files. &units !units used in input and output files unit_system = 'A_eV_fs' /  unit_system specifies which unit system is used in the input and output files. &system !periodic boundary condition yn_periodic = 'n' !number of elements, atoms, electrons and states(orbitals) nelem = 2 natom = 4 nelec = 10 nstate = 6 /  yn_periodic specifies whether or not periodic boundary condition is applied. nelem is the number of elements in the system. natom is the number of atoms in the system. nelec is the number of electrons in the system. nstate is the number of orbitals that are used in the calculation. &pseudo !name of input pseudo potential file file_pseudo(1) = './C_rps.dat' file_pseudo(2) = './H_rps.dat' !atomic number of element izatom(1) = 6 izatom(2) = 1 !angular momentum of pseudopotential that will be treated as local lloc_ps(1) = 1 lloc_ps(2) = 0 !--- Caution ---------------------------------------! ! Indices must correspond to those in &atomic_coor. ! !---------------------------------------------------! /  file_pseudo(n) specifies the filename of the pseudopotential file of the n-th element. izatom(n) is the atomic number of the n-th element. lloc_ps(n) specifies which angular momentum component is chosen as the local potential for the n-th element. &functional !functional('PZ' is Perdew-Zunger LDA: Phys. Rev. B 23, 5048 (1981).) xc = 'PZ' /  xc specifies the exchange-correlation potential to be used in the calculation. &rgrid !spatial grid spacing(x,y,z) dl(1:3) = 0.25d0, 0.25d0, 0.25d0 !number of spatial grids(x,y,z) num_rgrid(1:3) = 64, 64, 64 /  dl(i) specifies the spatial grid spacing in i-th direction. num_rgrid(i) specifies the number of grid points in i-th direction. &tgrid !time step size and number of time grids(steps) dt = 1.25d-3 nt = 5000 /  dt specifies the time step. nt is the number of time steps for the time propagation. &emfield !envelope shape of the incident pulse('impulse': impulsive field) ae_shape1 = 'impulse' !polarization unit vector(real part) for the incident pulse(x,y,z) epdir_re1(1:3) = 0.0d0, 0.0d0, 1.0d0 !--- Caution ---------------------------------------------------------! ! Definition of the incident pulse is written in: ! ! https://www.sciencedirect.com/science/article/pii/S0010465518303412 ! !---------------------------------------------------------------------! /  ae_shape1 specifies the envelope of the field. For a linear response calculation, as_shape1='impulse' is used. It indicates that a weak impulsive perturbation is applied at $$t=0$$. epdir_re1(i) specifies the i-th component of the real part of the polarization unit vector. &analysis !energy grid size and number of energy grids for output files de = 1.0d-2 nenergy = 3000 /  de specifies the energy grid size for frequency-domain analysis. nenergy specifies the number of energy grid points for frequency-domain analysis. &atomic_coor !cartesian atomic coodinates 'C' 0.000000 0.000000 0.599672 1 'H' 0.000000 0.000000 1.662257 2 'C' 0.000000 0.000000 -0.599672 1 'H' 0.000000 0.000000 -1.662257 2 !--- Format ---------------------------------------------------! ! 'symbol' x y z index(correspond to that of pseudo potential) ! !--------------------------------------------------------------! /  &atomic_coor specifies spatial coordinates of atoms. #### Execusion¶ Before execusion, remember to copy the directory restart that is created in the ground state calculation as data_for_restart in the present directory. In a multiprocess environment, calculation will be executed as: $ mpiexec -n NPROC salmon < C2H2_rt_response.inp > C2H2_rt_response.out


where NPROC is the number of MPI processes. A standard output will be stored in the file C2H2_rt_response.out.

#### Output files¶

After the calculation, following output files are created in the directory that you run the code in addition to the standard output file,

 file name description C2H2_response.data polarizability and oscillator strength distribution as functions of energy C2H2_rt.data components of change of dipole moment (electrons/plus definition) and total dipole moment (electrons/minus + ions/plus) as functions of time C2H2_rt_energy.data total energy and electronic excitation energy as functions of time PS_C_KY_n.dat information on pseodupotential file for carbon atom PS_H_KY_n.dat information on pseodupotential file for hydrogen atom

We first explain the standard output file. In the beginning of the file, input variables used in the calculation are shown.

##############################################################################
# SALMON: Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
#
#                             Version 2.0.1
##############################################################################
Libxc: [disabled]
theory= tddft_response

Total time step      =        5000
Time step[fs]        =  1.250000000000000E-003
Energy range         =        3000
Energy resolution[eV]=  1.000000000000000E-002
Field strength[a.u.] =  1.000000000000000E-002
use of real value orbitals =  F
======
.........


After that, the time evolution loop starts. At every 10 iteration steps, the time, dipole moments in three Cartesian directions, the total number of electrons, the total energy, and the number of iterations solving the Poisson equation are displayed.

 time-step    time[fs]                           Dipole moment(xyz)[A]      electrons  Total energy[eV]    iterVh
#----------------------------------------------------------------------
10    0.01250000 -0.56521137E-07 -0.28812833E-07 -0.25558983E-01    10.00000000     -339.68150366   34
20    0.02500000 -0.19835467E-06 -0.10147641E-06 -0.45169126E-01     9.99999999     -339.68147442   49
30    0.03750000 -0.37937911E-06 -0.19537418E-06 -0.57843871E-01     9.99999999     -339.68146891   45
40    0.05000000 -0.56465010E-06 -0.29324906E-06 -0.64072126E-01     9.99999999     -339.68146804   38
50    0.06250000 -0.73343753E-06 -0.38431758E-06 -0.65208422E-01     9.99999999     -339.68146679   25
60    0.07500000 -0.87559727E-06 -0.46276791E-06 -0.62464066E-01     9.99999999     -339.68146321   35
70    0.08750000 -0.98769124E-06 -0.52594670E-06 -0.56740338E-01     9.99999998     -339.68145535   20
80    0.10000000 -0.10701350E-05 -0.57309375E-06 -0.48483747E-01     9.99999998     -339.68144840   40
90    0.11250000 -0.11253992E-05 -0.60455485E-06 -0.38296037E-01     9.99999998     -339.68144186   21


Explanations of other output files are given below:

C2H2_rt.data

Results of time evolution calculation for vector potential, electric field, and dipole moment. In the first several lines, explanations of included data are given.

# Real time calculation:
# Ac_ext: External vector potential field
# E_ext: External electric field
# Ac_tot: Total vector potential field
# E_tot: Total electric field
# ddm_e: Change of dipole moment (electrons/plus definition)
# dm: Total dipole moment (electrons/minus + ions/plus)
# 1:Time[fs] 2:Ac_ext_x[fs*V/Angstrom] 3:Ac_ext_y[fs*V/Angstrom] 4:Ac_ext_z[fs*V/Angstrom]
# 5:E_ext_x[V/Angstrom] 6:E_ext_y[V/Angstrom] 7:E_ext_z[V/Angstrom]
# 8:Ac_tot_x[fs*V/Angstrom] 9:Ac_tot_y[fs*V/Angstrom] 10:Ac_tot_z[fs*V/Angstrom]
# 11:E_tot_x[V/Angstrom] 12:E_tot_y[V/Angstrom] 13:E_tot_z[V/Angstrom]
# 14:ddm_e_x[Angstrom] 15:ddm_e_y[Angstrom] 16:ddm_e_z[Angstrom] 17:dm_x[Angstrom]
# 18:dm_y[Angstrom] 19:dm_z[Angstrom]


Using first column (time in femtosecond) and 19th column (dipole moment in $$z$$ direction), the following graph can be drawn.

The dipole moment shows oscillations in femtosecond time scale that reflec electronic excitations.

C2H2_response.data

Time-frequency Fourier transformation of the dipole moment gives the polarizability and the strength function.

# Fourier-transform spectra:
# alpha: Polarizability
# df/dE: Strength function
# 1:Energy[eV] 2:Re(alpha_x)[Augstrom^2/V] 3:Re(alpha_y)[Augstrom^2/V]
# 4:Re(alpha_z)[Augstrom^2/V] 5:Im(alpha_x)[Augstrom^2/V] 6:Im(alpha_y)[Augstrom^2/V]
# 7:Im(alpha_z)[Augstrom^2/V] 8:df_x/dE[none] 9:df_y/dE[none] 10:df_z/dE[none]


Using first column (energy in electron-volt) and 10th column (oscillator strength distribution in $$z$$ direction), the following graph can be drawn.

There appears many peaks above the HOMO-LUMO gap energy. The strong excitation appears at around 9.3 eV.

C2H2_rt_energy.data

Energies are stored as functions of time.

# Real time calculation:
# Eall: Total energy
# Eall0: Initial energy
# 1:Time[fs] 2:Eall[eV] 3:Eall-Eall0[eV]


Eall and Eall-Eall0 are total energy and electronic excitation energy, respectively.

### Exercise-3: Electron dynamics in C2H2 molecule under a pulsed electric field¶

In this exercise, we learn the calculation of the electron dynamics in the acetylene (C2H2) molecule under a pulsed electric field, solving the time-dependent Kohn-Sham equation. As outputs of the calculation, such quantities as the total energy and the electric dipole moment of the system as functions of time are calculated. This tutorial should be carried out after finishing the ground state calculation that was explained in Exercise-1.

In the calculation, a pulsed electric field specified by the following vector potential will be used,

$$A(t) = - \frac{E_0}{\omega} \hat z \cos^2 \frac{\pi}{T} \left( t - \frac{T}{2} \right) \sin \omega \left( t - \frac{T}{2} \right), \hspace{5mm} (0 < t < T).$$

The electric field is given by $$E(t) = -(1/c)(dA(t)/dt)$$. The parameters that characterize the pulsed field such as the amplitude $$E_0$$, frequency $$\omega$$, pulse duration $$T$$, polarization direction $$\hat z$$, are specified in the input file. In the time dependent Kohn-Sham equation, the external field is included as the scalar potential, $$V_{\rm ext}(\mathbf{r},t) = eE(t)z$$.

#### Input files¶

To run the code, following files are necessary:

 file name description C2H2_rt_pulse.inp input file that contain input keywords and their values. C_rps.dat pseodupotential file for carbon H_rps.dat pseudopotential file for hydrogen restart directory created in the ground state calculation (rename the directory from data_for_restart to restart)

First three files are prepared in the directory SALMON/samples/exercise_03_C2H2_rt/. The file C2H2_rt_pulse.inp that contains input keywords and their values. The pseudopotential files should be the same as those used in the ground state calculation. In the directory restart, those files created in the ground state calculation and stored in the directory data_for_restart are included. Therefore, copy the directory as cp -R data_for_restart restart if you calculate at the same directory as you did the ground state calculation.

In the input file C2H2_rt_pulse.inp, input keywords are specified. Most of them are mandatory to execute the calculation of electron dynamics induced by a pulsed electric field. This will help you to prepare the input file for other systems and other pulsed electric fields that you want to calculate. A complete list of the input keywords that can be used in the input file can be found in List of input keywords.

!########################################################################################!
! Excercise 03:  Electron dynamics in C2H2 molecule under a pulsed electric field        !
!----------------------------------------------------------------------------------------!
! * The detail of this excercise is explained in our manual(see chapter: 'Exercises').   !
!   The manual can be obtained from: https://salmon-tddft.jp/documents.html              !
! * Input format consists of group of keywords like:                                     !
!     &group                                                                             !
!       input keyword = xxx                                                              !
!     /                                                                                  !
!   (see chapter: 'List of input keywords' in the manual)                                !
!----------------------------------------------------------------------------------------!
! * Conversion from unit_system = 'a.u.' to 'A_eV_fs':                                   !
!   Length: 1 [a.u.] = 0.52917721067    [Angstrom]                                       !
!   Energy: 1 [a.u.] = 27.21138505      [eV]                                             !
!   Time  : 1 [a.u.] = 0.02418884326505 [fs]                                             !
!----------------------------------------------------------------------------------------!
! * Copy the ground state data directory('data_for_restart') (or make symbolic link)     !
!   calculated in 'samples/exercise_01_C2H2_gs/' and rename the directory to 'restart/'  !
!   in the current directory.                                                            !
!########################################################################################!

&calculation
!type of theory
theory = 'tddft_pulse'
/

theory specifies which theoretical method is used in the calculation.
&control
!common name of output files
sysname = 'C2H2'
/

sysname is a prefix for filenames of output files.
&units
!units used in input and output files
unit_system = 'A_eV_fs'
/

unit_system specifies which unit system is used in the input and output files.
&system
!periodic boundary condition
yn_periodic = 'n'

!number of elements, atoms, electrons and states(orbitals)
nelem  = 2
natom  = 4
nelec  = 10
nstate = 6
/

yn_periodic specifies whether or not periodic boundary condition is applied.
nelem is the number of elements in the system.
natom is the number of atoms in the system.
nelec is the number of electrons in the system.
nstate is the number of orbitals that are used in the calculation.
&pseudo
!name of input pseudo potential file
file_pseudo(1) = './C_rps.dat'
file_pseudo(2) = './H_rps.dat'

!atomic number of element
izatom(1) = 6
izatom(2) = 1

!angular momentum of pseudopotential that will be treated as local
lloc_ps(1) = 1
lloc_ps(2) = 0
!--- Caution ---------------------------------------!
! Indices must correspond to those in &atomic_coor. !
!---------------------------------------------------!
/

file_pseudo(n) specifies the filename of the pseudopotential file of the n-th element.
izatom(n) is the atomic number of the n-th element.
lloc_ps(n) specifies which angular momentum component is chosen as the local potential for the n-th element.
&functional
!functional('PZ' is Perdew-Zunger LDA: Phys. Rev. B 23, 5048 (1981).)
xc = 'PZ'
/

xc specifies the exchange-correlation potential to be used in the calculation.
&rgrid
!spatial grid spacing(x,y,z)
dl(1:3) = 0.25d0, 0.25d0, 0.25d0

!number of spatial grids(x,y,z)
num_rgrid(1:3) = 64, 64, 64
/

dl(i) specifies the spatial grid spacing in i-th direction.
num_rgrid(i) specifies the number of grid points in i-th direction.
&tgrid
!time step size and number of time grids(steps)
dt = 1.25d-3
nt = 5000
/

dt specifies the time step.
nt is the number of time steps for the time propagation.
&emfield
!envelope shape of the incident pulse('Ecos2': cos^2 type envelope for scalar potential)
ae_shape1 = 'Acos2'

!peak intensity(W/cm^2) of the incident pulse
I_wcm2_1 = 5.00d13

!duration of the incident pulse
tw1 = 6.00d0

!mean photon energy(average frequency multiplied by the Planck constant) of the incident pulse
omega1 = 3.10d0

!polarization unit vector(real part) for the incident pulse(x,y,z)
epdir_re1(1:3) = 0.00d0, 0.00d0, 1.00d0
!--- Caution ---------------------------------------------------------!
! Definition of the incident pulse is written in:                     !
! https://www.sciencedirect.com/science/article/pii/S0010465518303412 !
!---------------------------------------------------------------------!
/

ae_shape1 specifies the envelope of the field.
I_wcm2_1 specify the intensity of the pulse in unit of W/cm2.
tw1 specifies the duration of the pulse.
omega1 specifies the mean photon energy of the pulse.
epdir_re1(i) specifies the i-th component of the real part of the polarization unit vector.
&analysis
!energy grid size and number of energy grids for output files
de      = 1.0d-2
nenergy = 10000
/

de specifies the energy grid size for frequency-domain analysis.
nenergy specifies the number of energy grid points for frequency-domain analysis.
&atomic_coor
!cartesian atomic coodinates
'C'    0.000000    0.000000    0.599672  1
'H'    0.000000    0.000000    1.662257  2
'C'    0.000000    0.000000   -0.599672  1
'H'    0.000000    0.000000   -1.662257  2
!--- Format ---------------------------------------------------!
! 'symbol' x y z index(correspond to that of pseudo potential) !
!--------------------------------------------------------------!
/

&atomic_coor specifies spatial coordinates of atoms.

#### Execusion¶

Before execusion, remember to copy the directory restart that is created in the ground state calculation as data_for_restart in the present directory. In a multiprocess environment, calculation will be executed as:

$mpiexec -n NPROC salmon < C2H2_rt_pulse.inp > C2H2_rt_pulse.out  where NPROC is the number of MPI processes. A standard output will be stored in the file C2H2_rt_pulse.out. #### Output files¶ After the calculation, following output files are created in the directory that you run the code in addition to the standard output file,  file name description C2H2_pulse.data time-frequency Fourier transform of dipole moment C2H2_rt.data components of change of dipole moment (electrons/plus definition) and total dipole moment (electrons/minus + ions/plus) as functions of time C2H2_rt_energy.data total energy and electronic excitation energy as functions of time PS_C_KY_n.dat information on pseodupotential file for carbon atom PS_H_KY_n.dat information on pseodupotential file for hydrogen atom You may download the above files (zipped file) from: We first explain the standard output file. In the beginning of the file, input variables used in the calculation are shown. ############################################################################## # SALMON: Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience # # Version 2.0.1 ############################################################################## Libxc: [disabled] theory= tddft_pulse Total time step = 5000 Time step[fs] = 1.250000000000000E-003 Energy range = 10000 Energy resolution[eV]= 1.000000000000000E-002 Laser frequency = 3.10[eV] Pulse width of laser= 6.00000000[fs] Laser intensity = 0.50000000E+14[W/cm^2] use of real value orbitals = F ====== ........  After that, the time evolution loop starts. At every 10 iteration steps, the time, dipole moments in three Cartesian directions, the total number of electrons, the total energy, and the number of iterations solving the Poisson equation are displayed.  time-step time[fs] Dipole moment(xyz)[A] electrons Total energy[eV] iterVh #---------------------------------------------------------------------- 10 0.01250000 -0.57275542E-07 -0.29197105E-07 -0.74600728E-06 10.00000000 -339.69524047 1 20 0.02500000 -0.20616352E-06 -0.10537273E-06 -0.10256205E-04 10.00000000 -339.69524348 1 30 0.03750000 -0.40063325E-06 -0.20597522E-06 -0.47397133E-04 10.00000000 -339.69524090 3 40 0.05000000 -0.59093535E-06 -0.30630513E-06 -0.13774845E-03 10.00000000 -339.69524287 1 50 0.06250000 -0.75588343E-06 -0.39552925E-06 -0.31097825E-03 10.00000000 -339.69523949 5 60 0.07500000 -0.89221538E-06 -0.47142217E-06 -0.59735355E-03 10.00000000 -339.69523784 11 70 0.08750000 -0.99769538E-06 -0.53192187E-06 -0.10253308E-02 10.00000000 -339.69523285 5 80 0.10000000 -0.10738281E-05 -0.57676878E-06 -0.16195168E-02 9.99999999 -339.69522482 19 90 0.11250000 -0.11250289E-05 -0.60722757E-06 -0.23985719E-02 9.99999999 -339.69521092 2  Explanations of other output files are given below: C2H2_rt.data Results of time evolution calculation for vector potential, electric field, and dipole moment. In the first several lines, explanations of data included data are given. # Real time calculation: # Ac_ext: External vector potential field # E_ext: External electric field # Ac_tot: Total vector potential field # E_tot: Total electric field # ddm_e: Change of dipole moment (electrons/plus definition) # dm: Total dipole moment (electrons/minus + ions/plus) # 1:Time[fs] 2:Ac_ext_x[fs*V/Angstrom] 3:Ac_ext_y[fs*V/Angstrom] 4:Ac_ext_z[fs*V/Angstrom] # 5:E_ext_x[V/Angstrom] 6:E_ext_y[V/Angstrom] 7:E_ext_z[V/Angstrom] # 8:Ac_tot_x[fs*V/Angstrom] 9:Ac_tot_y[fs*V/Angstrom] 10:Ac_tot_z[fs*V/Angstrom] # 11:E_tot_x[V/Angstrom] 12:E_tot_y[V/Angstrom] 13:E_tot_z[V/Angstrom] # 14:ddm_e_x[Angstrom] 15:ddm_e_y[Angstrom] 16:ddm_e_z[Angstrom] 17:dm_x[Angstrom] # 18:dm_y[Angstrom] 19:dm_z[Angstrom]  The applied electric field is drawn using the first column (time in femtosecond) and the 7th column (electric field in $$z$$ direction in Volt per Angstrom). The induced dipole moment is drawn using the first column (time in femtosecond) and 19th column (dipole moment in $$z$$ direction). It shows an oscillation similar to the applied electric field. However, the response is not linear since the applied electric field is rather strong. C2H2_pulse.data Time-frequency Fourier transformation of the dipole moment. In the first several lines, explanations of data included data are given. # Fourier-transform spectra: # energy: Frequency # dm: Dopile moment # 1:energy[eV] 2:Re(dm_x)[fs*Angstrom] 3:Re(dm_y)[fs*Angstrom] 4:Re(dm_z)[fs*Angstrom] # 5:Im(dm_x)[fs*Angstrom] 6:Im(dm_y)[fs*Angstrom] 7:Im(dm_z)[fs*Angstrom] # 8:|dm_x|^2[fs^2*Angstrom^2] 9:|dm_y|^2[fs^2*Angstrom^2] 10:|dm_z|^2[fs^2*Angstrom^2]  The spectrum of the induced dipole moment, $$|d(\omega)|^2$$ is shown in logarithmic scale as a function of the energy, $$\hbar \omega$$. High harmonic generations are visible in the spectrum. C2H2_rt_energy.data Energies are stored as functions of time. In the first several lines, explanations of data included data are given. # Real time calculation: # Eall: Total energy # Eall0: Initial energy # 1:Time[fs] 2:Eall[eV] 3:Eall-Eall0[eV]  Eall and Eall-Eall0 are total energy and electronic excitation energy, respectively. The figure below shows the electronic excitation energy as a function of time, using the first column (time in femtosecond) and the 3rd column (Eall-Eall0). Although the frequency is below the HOMO-LUMO gap energy, electronic excitations take place because of nonlinear absorption process. #### Additional exercise¶ If we change parameters of the applied electric field, we find a drastic change in the electronic excitations. In the example below, we increase the intensity from I_wcm2_1 = 5.00d13 to I_wcm2_1 = 1.00d12 and changes the frequency from omega1 = 3.10d0 to omega1 = 9.28d0. The new frequency corresponds to the resonant excitation energy seen in the linear response analysis shown in in Exercise-2. The change in the input file is shown below. &emfield !envelope shape of the incident pulse('Ecos2': cos^2 type envelope for scalar potential) ae_shape1 = 'Acos2' !peak intensity(W/cm^2) of the incident pulse I_wcm2_1 = 1.00d12 !duration of the incident pulse tw1 = 6.00d0 !mean photon energy(average frequency multiplied by the Planck constant) of the incident pulse omega1 = 9.28d0 !polarization unit vector(real part) for the incident pulse(x,y,z) epdir_re1(1:3) = 0.00d0, 0.00d0, 1.00d0  The applied electric field shows a rapid oscillation. The induced dipole moment also shows a rapid oscillation and does not decrease even though the electric field decreases. This is because the frequency of the applied electric field coincides with the excitation energy of the molecule. The electronic excitation energy also shows a monotonic increase. Although the strength of the applied electric field is much smaller than the previous case, the amount of the excitation energy is larger, again due to the resonant excitation. ## Crystalline silicon (periodic solids)¶ ### Exercise-4: Ground state of crystalline silicon¶ In this exercise, we learn the the ground state calculation of the crystalline silicon that has a diamond structure. A cubic unit cell that contains eight silicon atoms is adopted in the calculation. This exercise will be useful to learn how to set up calculations in SALMON for any periodic systems such as crystalline solid. #### Input files¶ To run the code, following files in the directory SALMON/samples/exercise_04_bulkSi_gs/ are used:  file name description Si_gs.inp input file that contains input keywords and their values Si_rps.dat pseodupotential file for silicon atom In the input file Si_gs.inp, input keywords are specified. Most of them are mandatory to execute the ground state calculation. This will help you to prepare an input file for other systems that you want to calculate. A complete list of the input keywords that can be used in the input file can be found in List of input keywords. !########################################################################################! ! Excercise 04: Ground state of crystalline silicon(periodic solids) ! !----------------------------------------------------------------------------------------! ! * The detail of this excercise is explained in our manual(see chapter: 'Exercises'). ! ! The manual can be obtained from: https://salmon-tddft.jp/documents.html ! ! * Input format consists of group of keywords like: ! ! &group ! ! input keyword = xxx ! ! / ! ! (see chapter: 'List of input keywords' in the manual) ! !----------------------------------------------------------------------------------------! ! * Conversion from unit_system = 'a.u.' to 'A_eV_fs': ! ! Length: 1 [a.u.] = 0.52917721067 [Angstrom] ! ! Energy: 1 [a.u.] = 27.21138505 [eV] ! ! Time : 1 [a.u.] = 0.02418884326505 [fs] ! !########################################################################################! &calculation !type of theory theory = 'dft' /  theory specifies which theoretical method is used in the calculation. &control !common name of output files sysname = 'Si' /  sysname is a prefix for filenames of output files. &units !units used in input and output files unit_system = 'A_eV_fs' /  unit_system specifies which unit system is used in the input and output files. &system !periodic boundary condition yn_periodic = 'y' !grid box size(x,y,z) al(1:3) = 5.43d0, 5.43d0, 5.43d0 !number of elements, atoms, electrons and states(bands) nelem = 1 natom = 8 nelec = 32 nstate = 32 /  yn_periodic specifies whether or not periodic boundary condition is applied. al(i) specifies the side length of the unit cell. nelem is the number of elements in the system. natom is the number of atoms in the system. nelec is the number of electrons in the system. nstate is the number of orbitals that are used in the calculation. &pseudo !name of input pseudo potential file file_pseudo(1) = './Si_rps.dat' !atomic number of element izatom(1) = 14 !angular momentum of pseudopotential that will be treated as local lloc_ps(1) = 2 !--- Caution -------------------------------------------! ! Index must correspond to those in &atomic_red_coor. ! !-------------------------------------------------------! /  file_pseudo(n) specifies the filename of the pseudopotential file of the n-th element. izatom(n) is the atomic number of the n-th element. lloc_ps(n) specifies which angular momentum component is chosen as the local potential for the n-th element. &functional !functional('PZ' is Perdew-Zunger LDA: Phys. Rev. B 23, 5048 (1981).) xc = 'PZ' /  xc specifies the exchange-correlation potential to be used in the calculation. &rgrid !number of spatial grids(x,y,z) num_rgrid(1:3) = 12, 12, 12 /  num_rgrid(i) specifies the number of real-space grid point in i-th direction. &kgrid !number of k-points(x,y,z) num_kgrid(1:3) = 4, 4, 4 /  num_kgrid(i) specifies the number of k-points for i-th direction discretizing the Brillouin zone. &scf !maximum number of scf iteration and threshold of convergence nscf = 300 threshold = 1.0d-9 /  nscf specifies the maximum number of SCF iterations. threshold specifies the threshold to judge the convergence. &atomic_red_coor !cartesian atomic reduced coodinates 'Si' .0 .0 .0 1 'Si' .25 .25 .25 1 'Si' .5 .0 .5 1 'Si' .0 .5 .5 1 'Si' .5 .5 .0 1 'Si' .75 .25 .75 1 'Si' .25 .75 .75 1 'Si' .75 .75 .25 1 !--- Format ---------------------------------------------------! ! 'symbol' x y z index(correspond to that of pseudo potential) ! !--------------------------------------------------------------! /  &atomic_red_coor specifies spatial coordinates of atoms in reduced coordinate system. #### Execusion¶ In a multiprocess environment, calculation will be executed as: $ mpiexec -n NPROC salmon < Si_gs.inp > Si_gs.out


where NPROC is the number of MPI processes. A standard output will be stored in the file Si_gs.out.

#### Output files¶

After the calculation, following output files and a directory are created in the directory that you run the code in addition to the standard output file,

 name description Si_info.data information on ground state solution Si_eigen.data energy eigenvalues of orbitals Si_k.data k-point distribution PS_Si_KY_n.dat information on pseodupotential file for silicon atom data_for_restart directory where files used in the real-time calculation are contained
You may download the above files (zipped file, except for the directory data_for_restart) from:

We first explain the standard output file. In the beginning of the file, input variables used in the calculation are shown.

##############################################################################
# SALMON: Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
#
#                             Version 2.0.1
##############################################################################
Libxc: [disabled]
theory= dft
use of real value orbitals =  F
r-space parallelization: off
======
MPI distribution:
nproc_k     :          16
nproc_ob    :           1
nproc_rgrid :           1           1           1
OpenMP parallelization:
.........


After that, the SCF loop starts. At each iteration step, the total energy as well as orbital energies and some other quantities are displayed.

-----------------------------------------------
iter=     1     Total Energy=       314.78493406     Gap=   -95.88543131
k=           1
1        37.5762      2        63.8589      3        58.1850      4        43.0042
5        61.5347      6        29.5604      7        41.5986      8        39.3545
9        48.5641     10        68.0003     11        75.5196     12        85.4113
..........
21        94.1224     22        53.0821     23        72.0170     24        46.7797
25        88.6077     26        98.2698     27        42.8071     28        65.0812
29        60.3648     30        39.6787     31        83.5629     32        62.7365

iter and int_x|rho_i(x)-rho_i-1(x)|dx/nelec        =      1 0.49478519E+00
Ne=   32.0000000000000
-----------------------------------------------
iter=     2     Total Energy=        62.72724688     Gap=   -77.31200657
k=           1
1        14.4913      2        32.6869      3        30.3561      4        20.6816
5        30.3907      6        16.9184      7        22.2967      8        18.5338
9        29.0117     10        41.9687     11        42.3490     12        54.6262
..........


When the convergence criterion is satisfied, the SCF calculation ends.

 iter=    60     Total Energy=      -850.76385275     Gap=     1.06020364
k=           1
1        -3.7745      2        -3.0158      3        -3.0158      4        -3.0158
5        -0.4300      6        -0.4300      7        -0.4300      8         0.3765
9         3.9530     10         3.9530     11         3.9530     12         4.6110
..........
21         9.6233     22         9.6233     23         9.6956     24         9.9111
25        11.0259     26        11.0259     27        11.4165     28        11.5976
29        11.9826     30        11.9887     31        12.0967     32        12.3585

iter and int_x|rho_i(x)-rho_i-1(x)|dx/nelec        =     60 0.77889300E-09
Ne=   32.0000000000000
#GS converged at    61  : 0.77889300E-09
===== force =====
1  0.60775985E-08  0.15425240E-07 -0.22474791E-07
2 -0.10689345E-06  0.88233132E-07  0.35122981E-09
3  0.39762202E-07 -0.23921918E-07  0.11855231E-07
4 -0.79441825E-07 -0.28978042E-07 -0.34109698E-07
5  0.37990526E-07  0.67211638E-08  0.20384753E-07
6  0.96418986E-07 -0.70404285E-07  0.10198912E-06
7  0.16145540E-07  0.30561301E-07 -0.63738382E-07
8  0.26042178E-07  0.30977639E-07 -0.40587816E-07
band information-----------------------------------------
Bottom of VB -0.194818046940532
Top of VB  0.216611832367042
Bottom of CB  0.255573599266334
Top of CB  0.533770712688357
Fundamental gap  3.896176689929157E-002
BG between same k-point  3.896176691206812E-002
Physicaly upper bound of CB for DOS  0.453918744010958
Physicaly upper bound of eps(omega)  0.609598295602846
---------------------------------------------------------
Bottom of VB[eV]  -5.30126888998779
Top of VB[eV]   5.89430797692564
Bottom of CB[eV]   6.95451161825061
Top of CB[eV]   14.5246403913758
Fundamental gap[eV]   1.06020364132497
BG between same k-point[eV]   1.06020364167264
Physicaly upper bound of CB for DOS[eV]   12.3517577246945
Physicaly upper bound of eps(omega)[eV]   16.5880139474728
---------------------------------------------------------
writing restart data...
writing completed.


In the directory data_for_restart, files that will be used in the next-step time evolution calculations are stored.

Other output files include following information.

Si_info.data

Orbital and total energies as well as parameters specified in the input file.

Total number of iteration =           60

Number of states =           32
Number of electrons =           32

Total energy (eV) =   -850.763852754463
1-particle energies (eV)
1        -3.7745      2        -3.0158      3        -3.0158      4        -3.0158
5        -0.4300      6        -0.4300      7        -0.4300      8         0.3765
9         3.9530     10         3.9530     11         3.9530     12         4.6110


Si_eigen.data

Orbital energies.

#esp: single-particle energies (eigen energies)
#occ: occupation numbers, io: orbital index
# 1:io, 2:esp[eV], 3:occ
k=     1,  spin=     1
1  -0.3774501171245852E+001   0.2000000000000000E+001
2  -0.3015778973884847E+001   0.2000000000000000E+001
3  -0.3015778969794385E+001   0.2000000000000000E+001


Si_k.data

Data of k-points.

# k-point distribution
# ik: k-point index
# kx,ky,kz: Reduced coordinate of k-points
# wk: Weight of k-point
# 1:ik[none] 2:kx[none] 3:ky[none] 4:kz[none] 5:wk[none]
1 -0.375000000000000E+000 -0.375000000000000E+000 -0.375000000000000E+000  0.156250000000000E-001
2 -0.125000000000000E+000 -0.375000000000000E+000 -0.375000000000000E+000  0.156250000000000E-001
3  0.125000000000000E+000 -0.375000000000000E+000 -0.375000000000000E+000  0.156250000000000E-001


### Exercise-5: Dielectric function of crystalline silicon¶

In this exercise, we learn the linear response calculation of the crystalline silicon. A cubic unit cell that contains eight silicon atoms is used in the calculation. This exercise should be carried out after finishing the ground state calculation that was explained in Exercise-4.

In this exercise, we calculate a dielectric function of silicon as a final object. We first summarize definitions of relevant quantities. We introduce a conductivity in time domain, $$\sigma_{\mu \nu}(t)$$, where $$\mu, \nu$$ indicate Cartesian components, $$\mu, \nu = x,y,z$$. It relates the applied electric field $$E_{\nu}(t)$$ with the induced current density averaged over the unit cell volume, $$J_{\mu}(t)$$,

$$J_{\mu}(t) = \sum_{\nu=x,y,z} \int dt' \sigma_{\mu \nu}(t-t') E_{\nu}(t').$$

Integrating the current density over time, we obtain the polarization density as a functioon of time,

$$P_{\mu}(t) = \int^t dt' J_{\mu}(t').$$

Then, the dielectric function is introduced by

$$D_{\mu}(t) = E_{\mu}(t)+4\pi P_{\mu}(t) = \sum_{\nu} \int^t dt' \epsilon_{\mu \nu}(t-t') E_{\nu}(t').$$

Frequency-dependent dielectric function $$\epsilon_{\mu \nu}(\omega)$$ is obtained from $$\epsilon_{\mu \nu}(t)$$ by taking time-frequency Fourier transformation.

In SALMON, the dielectric function is calculated in the following way. First the ground state Bloch orbitals $$u_{n{\bf k}}({\bf r})$$ that satisfies the Kohn-Sham equation,

$$H_{\bf k} u_{n{\bf k}}({\bf r}) = \epsilon_{n{\bf k}}({\bf r}),$$

is calculated. Then an impulsive force characterized by the magnitude of the impulse $$I$$ is applied to all electrons in $$z$$ direction. This is equivalent to shift the wave vector by $${\bf k} \rightarrow {\bf k} + I/\hbar \hat z$$, where $$\hat z$$ is a unit vector in $$z$$ direction. We make a time evolution calculation with the shifted wave vector as

$$i\hbar \frac{\partial}{\partial t} u_{n{\bf k}}({\bf r},t) = H_{{\bf k} + I/\hbar \hat z}(t) u_{n{\bf k}}({\bf r},t).$$

During the time evolution, the electric current density given by

$${\bf J}(t) = \frac{-e}{m \Omega} \int d{\bf r} u_{n{\bf k}}^* \left( -i\hbar\nabla + \hbar {\bf k} + I \hat z \right) u_{n{\bf k}} + \delta {\bf J}(t).$$

is monitored, where $$\Omega$$ is the volume of the unit cell and $$\delta {\bf J}(t)$$ is a current component coming from nonlocal pseudopootential.

After the time evolution calculation, a time-frequency Fourier transformation is carried out for the electric current density to obtain the frequency-dependent conductivity by

$$\tilde \sigma_{zz}(\omega) = -\frac{e}{I} \int dt e^{i\omega t} J_z(t).$$

The dielectric function and the conductivity is related in frequency representation by

$$\epsilon_{\mu \nu}(\omega) = \delta_{\mu \nu} + \frac{4\pi i \sigma_{\mu \nu}(\omega)}{\omega}.$$

We note that the dielectric function of a crystalline silicon is isotropic, $$\epsilon_{\mu \nu} = \delta_{\mu \nu} \epsilon(\omega)$$.

#### Input files¶

To run the code, following files are necessary:

 file name description C2H2_response.inp input file that contains input keywords and their values Si_rps.dat pseodupotential file for silicon atom restart directory created in the ground state calculation (rename the directory from data_for_restart to restart)

First two files are prepared in the directory SALMON/samples/exercise_05_bulkSi_lr/. The file Si_rt_response.inp contains input keywords and their values. The pseudoopotential file should be the same as that used in the ground state calculation. In the directory restart, those files created in the ground state calculation and stored in the directory data_for_restart are included. Therefore, coopy the directory as cp -R data_for_restart restart if you calculate at the same directory as you did the ground state calculation.

In the input file Si_rt_response.inp, input keywords are specified. Most of them are mandatory to execute the linear response calculation. This will help you to prepare the input file for other systems that you want to calculate. A complete list of the input keywords that can be used in the input file can be found in List of input keywords.

!########################################################################################!
! Excercise 05: Dielectric function of crystalline silicon                               !
!----------------------------------------------------------------------------------------!
! * The detail of this excercise is explained in our manual(see chapter: 'Exercises').   !
!   The manual can be obtained from: https://salmon-tddft.jp/documents.html              !
! * Input format consists of group of keywords like:                                     !
!     &group                                                                             !
!       input keyword = xxx                                                              !
!     /                                                                                  !
!   (see chapter: 'List of input keywords' in the manual)                                !
!----------------------------------------------------------------------------------------!
! * Conversion from unit_system = 'a.u.' to 'A_eV_fs':                                   !
!   Length: 1 [a.u.] = 0.52917721067    [Angstrom]                                       !
!   Energy: 1 [a.u.] = 27.21138505      [eV]                                             !
!   Time  : 1 [a.u.] = 0.02418884326505 [fs]                                             !
!----------------------------------------------------------------------------------------!
! * Copy the ground state data directory('data_for_restart') (or make symbolic link)     !
!   calculated in 'samples/exercise_04_bulkSi_gs/' and rename the directory to 'restart/'!
!   in the current directory.                                                            !
!########################################################################################!

&calculation
!type of theory
theory = 'tddft_response'
/

theory specifies which theoretical method is used in the calculation.
&control
!common name of output files
sysname = 'Si'
/

sysname is a prefix for filenames of output files.
&units
!units used in input and output files
unit_system = 'A_eV_fs'
/

unit_system specifies which unit system is used in the input and output files.
&system
!periodic boundary condition
yn_periodic = 'y'

!grid box size(x,y,z)
al(1:3) = 5.43d0, 5.43d0, 5.43d0

!number of elements, atoms, electrons and states(bands)
nelem  = 1
natom  = 8
nelec  = 32
nstate = 32
/

yn_periodic specifies whether or not periodic boundary condition is applied.
al(i) specifies the side length of the unit cell.
nelem is the number of elements in the system.
natom is the number of atoms in the system.
nelec is the number of electrons in the system.
nstate is the number of orbitals that are used in the calculation.
&pseudo
!name of input pseudo potential file
file_pseudo(1) = './Si_rps.dat'

!atomic number of element
izatom(1) = 14

!angular momentum of pseudopotential that will be treated as local
lloc_ps(1) = 2
!--- Caution -------------------------------------------!
! Index must correspond to those in &atomic_red_coor.   !
!-------------------------------------------------------!
/

file_pseudo(n) specifies the filename of the pseudopotential file of the n-th element.
izatom(n) is the atomic number of the n-th element.
lloc_ps(n) specifies which angular momentum component is chosen as the local potential for the n-th element.
&functional
!functional('PZ' is Perdew-Zunger LDA: Phys. Rev. B 23, 5048 (1981).)
xc = 'PZ'
/

xc specifies the exchange-correlation potential to be used in the calculation.
&rgrid
!number of spatial grids(x,y,z)
num_rgrid(1:3) = 12, 12, 12
/

num_rgrid(i) specifies the number of real-space grid point in i-th direction.
&kgrid
!number of k-points(x,y,z)
num_kgrid(1:3) = 4, 4, 4
/

num_kgrid(i) specifies the number of k-points for i-th direction discretizing the Brillouin zone.
&tgrid
!time step size and number of time grids(steps)
dt = 0.002d0
nt = 6000
/

dt specifies the time step.
nt is the number of time steps for the time propagation.
&emfield
!envelope shape of the incident pulse('impulse': impulsive field)
ae_shape1 = 'impulse'

!polarization unit vector(real part) for the incident pulse(x,y,z)
epdir_re1(1:3) = 0.00d0, 0.00d0, 1.00d0
!--- Caution ---------------------------------------------------------!
! Definition of the incident pulse is written in:                     !
! https://www.sciencedirect.com/science/article/pii/S0010465518303412 !
!---------------------------------------------------------------------!
/

ae_shape1 specifies the envelope of the field. For a linear response calculation, as_shape1='impulse' is used. It indicates that a weak impulsive perturbation is applied at $$t=0$$.
epdir_re1(i) specifies the i-th component of the real part of the polarization unit vector.
&analysis
!energy grid size and number of energy grids for output files
de      = 0.01d0
nenergy = 2000
/

de specifies the energy grid size for frequency-domain analysis.
nenergy specifies the number of energy grid points for frequency-domain analysis.
&atomic_red_coor
!cartesian atomic reduced coodinates
'Si'       .0      .0      .0      1
'Si'       .25     .25     .25     1
'Si'       .5      .0      .5      1
'Si'       .0      .5      .5      1
'Si'       .5      .5      .0      1
'Si'       .75     .25     .75     1
'Si'       .25     .75     .75     1
'Si'       .75     .75     .25     1
!--- Format ---------------------------------------------------!
! 'symbol' x y z index(correspond to that of pseudo potential) !
!--------------------------------------------------------------!
/

&atomic_red_coor specifies spatial coordinates of atoms in reduced coordinate system.

#### Execusion¶

In a multiprocess environment, calculation will be executed as:

$mpiexec -n NPROC salmon < Si_rt_response.inp > Si_rt_response.out  where NPROC is the number of MPI processes. A standard output will be stored in the file Si_rt_response.out. #### Output files¶ After the calculation, following output files are created in the directory that you run the code in addition to the standard output file,  file name description Si_response.data conductivity and dielectric function as functions of energy Si_rt.data vector potential, electric field, and matter current as functions of time Si_rt_energy total energy and electronic excitation energy as functions of time PS_Si_KY_n.dat information on pseodupotential file for silicon atom You may download the above files (zipped file) from: We first explain the standard output file. In the beginning of the file, input variables used in the calculation are shown. ############################################################################## # SALMON: Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience # # Version 2.0.1 ############################################################################## Libxc: [disabled] theory= tddft_response Total time step = 6000 Time step[fs] = 2.000000000000000E-003 Energy range = 2000 Energy resolution[eV]= 1.000000000000000E-002 Field strength[a.u.] = 1.000000000000000E-002 use of real value orbitals = F r-space parallelization: off ====== ........  After that, the time evolution loop starts. At every 10 iteration steps, electric current density in three Cartesian direction, the total number of electrons, and total energy are displayed.  time-step time[fs] Current(xyz)[a.u.] electrons Total energy[eV] #---------------------------------------------------------------------- 10 0.02000000 0.11911770E-11 -0.40018285E-13 0.24902126E-03 32.00000000 -850.72273308 20 0.04000000 0.17745321E-11 0.13712105E-12 0.21977876E-03 31.99999999 -850.72273319 30 0.06000000 0.31016197E-11 0.24481043E-12 0.20049151E-03 31.99999999 -850.72272966 40 0.08000000 0.36611565E-11 0.49184860E-12 0.17937042E-03 31.99999999 -850.72272925 50 0.10000000 0.36920991E-11 0.63805259E-12 0.15246564E-03 31.99999998 -850.72272922 60 0.12000000 0.32347636E-11 0.11280947E-11 0.12248647E-03 31.99999998 -850.72272655 70 0.14000000 0.25978450E-11 0.15550074E-11 0.91933957E-04 31.99999998 -850.72272293 80 0.16000000 0.20087959E-11 0.17983589E-11 0.62968342E-04 31.99999997 -850.72272036 90 0.18000000 0.90623268E-12 0.18067974E-11 0.38824129E-04 31.99999997 -850.72271918  Explanations of other output files are given below: Si_rt.data Results of time evolution calculation for vector potential, electric field, and matter current density are shown. In the first several lines, explanations of included data are given. # Real time calculation: # Ac_ext: External vector potential field # E_ext: External electric field # Ac_tot: Total vector potential field # E_tot: Total electric field # Jm: Matter current density (electrons) # 1:Time[fs] 2:Ac_ext_x[fs*V/Angstrom] 3:Ac_ext_y[fs*V/Angstrom] 4:Ac_ext_z[fs*V/Angstrom] # 5:E_ext_x[V/Angstrom] 6:E_ext_y[V/Angstrom] 7:E_ext_z[V/Angstrom] 8:Ac_tot_x[fs*V/Angstrom] # 9:Ac_tot_y[fs*V/Angstrom] 10:Ac_tot_z[fs*V/Angstrom] 11:E_tot_x[V/Angstrom] # 12:E_tot_y[V/Angstrom] 13:E_tot_z[V/Angstrom] 14:Jm_x[1/fs*Angstrom^2] # 15:Jm_y[1/fs*Angstrom^2] 16:Jm_z[1/fs*Angstrom^2]  Using first column (time in femtosecond) and 16th column (matter current density in z direction), the following graph can be drawn. Si_response.data Time-frequency Fourier transformation of the macroscopic current density gives the conductivity of the system. The dielectric function is then calculated from the conductivity. They are stored in this file. # Fourier-transform spectra: # sigma: Conductivity # eps: Dielectric constant # 1:Energy[eV] 2:Re(sigma_x)[1/fs*V*Angstrom] 3:Re(sigma_y)[1/fs*V*Angstrom] # 4:Re(sigma_z)[1/fs*V*Angstrom] 5:Im(sigma_x)[1/fs*V*Angstrom] # 6:Im(sigma_y)[1/fs*V*Angstrom] 7:Im(sigma_z)[1/fs*V*Angstrom] 8:Re(eps_x)[none] # 9:Re(eps_y)[none] 10:Re(eps_z)[none] 11:Im(eps_x)[none] 12:Im(eps_y)[none] # 13:Im(eps_z)[none]  Using first column (energy in eV) and 10th (real part of the dielectric function) and 13th (imaginary part), we obtain the following graph. The imaginary part appears above the direct bandgap energy that is about 2.4 eV in the present calculation using local density approximation. Dielectric function below 1 eV are not accurate and and are not shown. Si_rt_energy Eall and Eall-Eall0 are total energy and electronic excitation energy, respectively. # Real time calculation: # Eall: Total energy # Eall0: Initial energy # 1:Time[fs] 2:Eall[eV] 3:Eall-Eall0[eV]  ### Exercise-6: Electron dynamics in crystalline silicon under a pulsed electric field¶ In this exercise, we learn the calculation of electron dynamics in crystalline silicon. A cubic unit cell that contains eight silicon atoms is used in the calculation. This exercise should be carried out after finishing the ground state calculation that was explained in Exercise-4. In the calculation, a pulsed electric field specified by the following vector potential will be used, $$A(t) = - \frac{E_0}{\omega} \hat z \cos^2 \frac{\pi}{T} \left( t - \frac{T}{2} \right) \sin \omega \left( t - \frac{T}{2} \right), \hspace{5mm} (0 < t < T).$$ The electric field is given by $$E(t) = -(1/c)(dA(t)/dt)$$. The parameters that characterize the pulsed field such as the amplitude $$E_0$$, frequency $$\omega$$, pulse duration $$T$$, polarization direction $$\hat z$$, are specified in the input file. Time-dependent Kohn-Sham equation for Bloch orbitals are calculated in real time, $$i\hbar \frac{\partial}{\partial t} u_{n{\bf k}}({\bf r},t) = H_{{\bf k} + (e/\hbar c){\bf A}(t)} u_{n{\bf k}}({\bf r},t).$$ #### Input files¶ To run the code, following files in samples are necessary:  file name description Si_rt_pulse.inp input file that contain input keywords and their values Si_rps.dat pseodupotential file for Carbon restart directory created in the ground state calculation (rename the directory from data_for_restart to restart) First two files are prepared in the directory SALMON/samples/exercise_06_bulkSi_rt/. The file Si_rt_pulse.inp contains input keywords and their values. The pseudoopotential file should be the same as that used in the ground state calculation. In the directory restart, those files created in the ground state calculation and stored in the directory data_for_restart are included. Therefore, coopy the directory as cp -R data_for_restart restart if you calculate at the same directory as you did the ground state calculation. In the input file Si_rt_pulse.inp, input keywords are specified. Most of them are mandatory to execute the electron dynamics calculation. This will help you to prepare the input file for other systems that you want to calculate. A complete list of the input keywords that can be used in the input file can be found in List of input keywords. !########################################################################################! ! Excercise 06: Electron dynamics in crystalline silicon under a pulsed electric field ! !----------------------------------------------------------------------------------------! ! * The detail of this excercise is explained in our manual(see chapter: 'Exercises'). ! ! The manual can be obtained from: https://salmon-tddft.jp/documents.html ! ! * Input format consists of group of keywords like: ! ! &group ! ! input keyword = xxx ! ! / ! ! (see chapter: 'List of input keywords' in the manual) ! !----------------------------------------------------------------------------------------! ! * Conversion from unit_system = 'a.u.' to 'A_eV_fs': ! ! Length: 1 [a.u.] = 0.52917721067 [Angstrom] ! ! Energy: 1 [a.u.] = 27.21138505 [eV] ! ! Time : 1 [a.u.] = 0.02418884326505 [fs] ! !----------------------------------------------------------------------------------------! ! * Copy the ground state data directory('data_for_restart') (or make symbolic link) ! ! calculated in 'samples/exercise_04_bulkSi_gs/' and rename the directory to 'restart/'! ! in the current directory. ! !########################################################################################! &calculation !type of theory theory = 'tddft_pulse' /  theory specifies which theoretical method is used in the calculation. &control !common name of output files sysname = 'Si' /  sysname is a prefix for filenames of output files. &units !units used in input and output files unit_system = 'A_eV_fs' /  unit_system specifies which unit system is used in the input and output files. &system !periodic boundary condition yn_periodic = 'y' !grid box size(x,y,z) al(1:3) = 5.43d0, 5.43d0, 5.43d0 !number of elements, atoms, electrons and states(bands) nelem = 1 natom = 8 nelec = 32 nstate = 32 /  yn_periodic specifies whether or not periodic boundary condition is applied. al(i) specifies the side length of the unit cell. nelem is the number of elements in the system. natom is the number of atoms in the system. nelec is the number of electrons in the system. nstate is the number of orbitals that are used in the calculation. &pseudo !name of input pseudo potential file file_pseudo(1) = './Si_rps.dat' !atomic number of element izatom(1) = 14 !angular momentum of pseudopotential that will be treated as local lloc_ps(1) = 2 !--- Caution -------------------------------------------! ! Index must correspond to those in &atomic_red_coor. ! !-------------------------------------------------------! /  file_pseudo(n) specifies the filename of the pseudopotential file of the n-th element. izatom(n) is the atomic number of the n-th element. lloc_ps(n) specifies which angular momentum component is chosen as the local potential for the n-th element. &functional !functional('PZ' is Perdew-Zunger LDA: Phys. Rev. B 23, 5048 (1981).) xc = 'PZ' /  xc specifies the exchange-correlation potential to be used in the calculation. &rgrid !number of spatial grids(x,y,z) num_rgrid(1:3) = 12, 12, 12 /  num_rgrid(i) specifies the number of real-space grid point in i-th direction. &kgrid !number of k-points(x,y,z) num_kgrid(1:3) = 4, 4, 4 /  num_kgrid(i) specifies the number of k-points for i-th direction discretizing the Brillouin zone. &tgrid !time step size and number of time grids(steps) dt = 0.002d0 nt = 6000 /  dt specifies the time step. nt is the number of time steps for the time propagation. &emfield !envelope shape of the incident pulse('Acos2': cos^2 type envelope for vector potential) ae_shape1 = 'Acos2' !peak intensity(W/cm^2) of the incident pulse I_wcm2_1 = 1.0d12 !duration of the incident pulse tw1 = 10.672d0 !mean photon energy(average frequency multiplied by the Planck constant) of the incident pulse omega1 = 1.55d0 !polarization unit vector(real part) for the incident pulse(x,y,z) epdir_re1(1:3) = 0.0d0, 0.0d0, 1.0d0 !--- Caution ---------------------------------------------------------! ! Definition of the incident pulse is written in: ! ! https://www.sciencedirect.com/science/article/pii/S0010465518303412 ! !---------------------------------------------------------------------! /  ae_shape1 specifies the envelope of the field. I_wcm2_1 specify the intensity of the pulse in unit of W/cm2. tw1 specifies the duration of the pulse. omega1 specifies the mean photon energy of the pulse. epdir_re1(i) specifies the i-th component of the real part of the polarization unit vector. &analysis !energy grid size and number of energy grids for output files de = 0.01d0 nenergy = 3000 /  de specifies the energy grid size for frequency-domain analysis. nenergy specifies the number of energy grid points for frequency-domain analysis. &atomic_red_coor !cartesian atomic reduced coodinates 'Si' .0 .0 .0 1 'Si' .25 .25 .25 1 'Si' .5 .0 .5 1 'Si' .0 .5 .5 1 'Si' .5 .5 .0 1 'Si' .75 .25 .75 1 'Si' .25 .75 .75 1 'Si' .75 .75 .25 1 !--- Format ---------------------------------------------------! ! 'symbol' x y z index(correspond to that of pseudo potential) ! !--------------------------------------------------------------! /  &atomic_red_coor specifies spatial coordinates of atoms in reduced coordinate system. #### Execusion¶ In a multiprocess environment, calculation will be executed as: $ mpiexec -n NPROC salmon < Si_rt_pulse.inp > Si_rt_pulse.out


where NPROC is the number of MPI processes. A standard output will be stored in the file Si_rt_pulse.out.

#### Output files¶

After the calculation, following output files are created in the directory that you run the code in addition to the standard output file,

 file name description Si_pulse.data time-frequency Fourier transform of matter current and electric field Si_rt.data vector potential, electric field, and matter current as functions of time Si_rt_energy total energy and electronic excitation energy as functions of time PS_Si_KY_n.dat information on pseodupotential file for silicon atom

We first explain the standard output file. In the beginning of the file, input variables used in the calculation are shown.

##############################################################################
# SALMON: Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
#
#                             Version 2.0.1
##############################################################################
Libxc: [disabled]
theory= tddft_pulse

Total time step      =        6000
Time step[fs]        =  2.000000000000000E-003
Energy range         =        3000
Energy resolution[eV]=  1.000000000000000E-002
Laser frequency     = 1.55[eV]
Pulse width of laser=     10.67200000[fs]
Laser intensity     =  0.10000000E+13[W/cm^2]
use of real value orbitals =  F
r-space parallelization: off
======
........


After that, the time evolution loop starts. At every 10 iterations, the time, current in three Cartesian directions, the number of electrons, and the total energy are displayed.

  time-step  time[fs]                               Current(xyz)[a.u.]      electrons Total energy[eV]
#----------------------------------------------------------------------
10    0.02000000  0.11847131E-11 -0.47534543E-13 -0.43120486E-08    32.00000000     -850.76385276
20    0.04000000  0.17733186E-11  0.12820952E-12 -0.33012195E-07    32.00000000     -850.76385276
30    0.06000000  0.30965601E-11  0.23626542E-12 -0.10736819E-06    32.00000000     -850.76385275
40    0.08000000  0.36612711E-11  0.47687574E-12 -0.24607217E-06    32.00000000     -850.76385272
50    0.10000000  0.36958981E-11  0.62315158E-12 -0.46548014E-06    32.00000000     -850.76385263
60    0.12000000  0.32186097E-11  0.11429104E-11 -0.77911390E-06    32.00000000     -850.76385239
70    0.14000000  0.25712602E-11  0.15689467E-11 -0.11971541E-05    32.00000000     -850.76385186
80    0.16000000  0.19447699E-11  0.18250920E-11 -0.17261976E-05    32.00000000     -850.76385082
90    0.18000000  0.80514520E-12  0.18683828E-11 -0.23692381E-05    32.00000000     -850.76384896


Explanations of other output files are given below:

Si_rt.data

Results of time evolution calculation for vector potential, electric field, and matter current density.

# Real time calculation:
# Ac_ext: External vector potential field
# E_ext: External electric field
# Ac_tot: Total vector potential field
# E_tot: Total electric field
# Jm: Matter current density (electrons)
# 1:Time[fs] 2:Ac_ext_x[fs*V/Angstrom] 3:Ac_ext_y[fs*V/Angstrom] 4:Ac_ext_z[fs*V/Angstrom]
# 5:E_ext_x[V/Angstrom] 6:E_ext_y[V/Angstrom] 7:E_ext_z[V/Angstrom]
# 8:Ac_tot_x[fs*V/Angstrom] 9:Ac_tot_y[fs*V/Angstrom] 10:Ac_tot_z[fs*V/Angstrom]
# 11:E_tot_x[V/Angstrom] 12:E_tot_y[V/Angstrom] 13:E_tot_z[V/Angstrom]
# 14:Jm_x[1/fs*Angstrom^2] 15:Jm_y[1/fs*Angstrom^2] 16:Jm_z[1/fs*Angstrom^2]


The applied electric field is drawn using the first column (time in femtosecond) and the 7th column (electric field in z direction).

The matter current density is drawn using the first column (time in femtosecond) and 16th column (matter current density in z direction).

Si_pulse.data

Time-frequency Fourier transformation of the matter current and electric field.

# Fourier-transform spectra:
# energy: Frequency
# Jm: Matter current
# E_ext: External electric field
# E_tot: Total electric field
# 1:energy[eV] 2:Re(Jm_x)[1/Angstrom^2] 3:Re(Jm_y)[1/Angstrom^2] 4:Re(Jm_z)[1/Angstrom^2]
# 5:Im(Jm_x)[1/Angstrom^2] 6:Im(Jm_y)[1/Angstrom^2] 7:Im(Jm_z)[1/Angstrom^2]
# 8:|Jm_x|^2[1/Angstrom^4] 9:|Jm_y|^2[1/Angstrom^4] 10:|Jm_z|^2[1/Angstrom^4]
# 11:Re(E_ext_x)[fs*V/Angstrom] 12:Re(E_ext_y)[fs*V/Angstrom]
# 13:Re(E_ext_z)[fs*V/Angstrom] 14:Im(E_ext_x)[fs*V/Angstrom]
# 15:Im(E_ext_y)[fs*V/Angstrom] 16:Im(E_ext_z)[fs*V/Angstrom]
# 17:|E_ext_x|^2[fs^2*V^2/Angstrom^2] 18:|E_ext_y|^2[fs^2*V^2/Angstrom^2]
# 19:|E_ext_z|^2[fs^2*V^2/Angstrom^2] 20:Re(E_tot_x)[fs*V/Angstrom]
# 21:Re(E_tot_y)[fs*V/Angstrom] 22:Re(E_tot_z)[fs*V/Angstrom]
# 23:Im(E_tot_x)[fs*V/Angstrom] 24:Im(E_tot_y)[fs*V/Angstrom]
# 25:Im(E_tot_z)[fs*V/Angstrom] 26:|E_tot_x|^2[fs^2*V^2/Angstrom^2]
# 27:|E_tot_y|^2[fs^2*V^2/Angstrom^2] 28:|E_tot_z|^2[fs^2*V^2/Angstrom^2]


The power spectrum of the matter current density, $$|J(\omega)|^2$$ is shown in logarithmic scale as a function of the energy, $$\hbar\omega$$. High harmonic generations are visible in the spectrum.

Si_rt_energy

Energies are stored as functions of time.

# Real time calculation:
# Eall: Total energy
# Eall0: Initial energy
# 1:Time[a.u.] 2:Eall[a.u.] 3:Eall-Eall0[a.u.]


Eall and Eall-Eall0 are total energy and electronic excitation energy, respectively. The figure below shows the electronic excitation energy per unit cell volume as a function of time, using the first column (time in femtosecond) and the 3rd column (Eall-Eall0). Although the frequency is below the direct bandgap of silicon (2.4 eV in the LDA calculation), electronic excitations take place because of nonlinear absorption process.

## Maxwell + TDDFT multiscale simulation¶

### Exercise-7: Pulsed-light propagation through a silicon thin film¶

In this exercise, we learn the calculation of a propagation of pulsed light through a thin film of crystalline silicon. We consider an irradiation of a few-cycle, linearly polarized pulsed light normally on a thin film of 40 nm thickness. This exercise should be carried out after finishing the ground state calculation that was explained in Exercise-4.

In the calculation, macroscopic Maxwell equation that describes the light propagation and microscopic time-dependent Kohn-Sham equation that describes the electron dynamics are solved simultaneously. The light propagation is described by a one-dimensional light-propagation equation for the vector potential,

$$\frac{1}{c^2} \frac{\partial^2}{\partial X^2} A(X,t) - \frac{\partial^2}{\partial X^2} A(X,t) = \frac{4\pi}{c} I(X,t).$$

The direction of the propagation is set to x direction and the polarization of the pulse is set to z direction. The time profile of an incident pulse is given by

$$A(t) = - \frac{E_0}{\omega} \hat z \cos^2 \frac{\pi}{T} \left( t - \frac{T}{2} \right) \sin \omega \left( t - \frac{T}{2} \right), \hspace{5mm} (0 < t < T),$$

and is set in the vacuum region in front of the thin film. The parameters that characterize the pulsed field such as the amplitude $$E_0$$, frequency $$\omega$$, pulse duration $$T$$ are specified in the input file.

To discribe the light propagation, macroscopic coordinate $$X$$ is discretized as $$X_i$$. At each grid point inside the silicon thin film, for which we take eight points $$i=1 \cdots 8$$ in this exercise, time-dependent Kohn-Sham equation for Bloch orbitals are calculated in real time,

$$i\hbar \frac{\partial}{\partial t} u_{i n{\bf k}}({\bf r},t) = H_{{\bf k} + (e/\hbar c){\bf A}_i(t)} u_{i n{\bf k}}({\bf r},t).$$

From the Bloch orbital $$u_{in{\bf k}}({\bf r},t)$$, we calculate the electric current $$I(X_i,t)$$. We thus obtain a closed set of equations. Solving these equations simultaneously, we can describe macroscopic light propagation and microscopic electron dynamics at the same time.

#### Input files¶

To run the code, following files in samples are used:

 file name description Si_rt_multiscale.inp input file that contain input keywords and their values. Si_rps.dat pseodupotential file for silicon restart directory created in the ground state calculation (rename the directory from data_for_restart to restart)

First two files are prepared in the directory SALMON/samples/exercise_07_bulkSi_multiscale/. The file Si_rt_multiscale.inp contains input keywords and their values. The pseudoopotential file should be the same as that used in the ground state calculation. In the directory restart, those files created in the ground state calculation and stored in the directory data_for_restart are included. Therefore, coopy the directory as cp -R data_for_restart restart if you calculate at the same directory as you did the ground state calculation.

In the input file Si_rt_multiscale.inp, input keywords are specified. Most of them are mandatory to execute the electron dynamics calculation. This will help you to prepare the input file for other systems that you want to calculate. A complete list of the input keywords that can be used in the input file can be found in List of input keywords.

!########################################################################################!
! Excercise 07: Maxwell+TDDFT multiscale simulation                                      !
!               (Pulsed-light propagation through a silicon thin film)                   !
!----------------------------------------------------------------------------------------!
! * The detail of this excercise is explained in our manual(see chapter: 'Exercises').   !
!   The manual can be obtained from: https://salmon-tddft.jp/documents.html              !
! * Input format consists of group of keywords like:                                     !
!     &group                                                                             !
!       input keyword = xxx                                                              !
!     /                                                                                  !
!   (see chapter: 'List of input keywords' in the manual)                                !
!----------------------------------------------------------------------------------------!
! * Conversion from unit_system = 'a.u.' to 'A_eV_fs':                                   !
!   Length: 1 [a.u.] = 0.52917721067    [Angstrom]                                       !
!   Energy: 1 [a.u.] = 27.21138505      [eV]                                             !
!   Time  : 1 [a.u.] = 0.02418884326505 [fs]                                             !
!----------------------------------------------------------------------------------------!
! * Copy the ground state data directory('data_for_restart') (or make symbolic link)     !
!   calculated in 'samples/exercise_04_bulkSi_gs/' and rename the directory to 'restart/'!
!   in the current directory.                                                            !
!########################################################################################!

&calculation
!type of theory
theory = 'multi_scale_maxwell_tddft'
/

theory specifies which theoretical method is used in the calculation.
&control
!common name of output files
sysname = 'Si'
/

sysname is a prefix for filenames of output files.
&units
!units used in input and output files
unit_system = 'A_eV_fs'
/

unit_system specifies which unit system is used in the input and output files.
&system
!periodic boundary condition
yn_periodic = 'y'

!grid box size(x,y,z)
al(1:3) = 5.43d0, 5.43d0, 5.43d0

!number of elements, atoms, electrons and states(bands)
nelem  = 1
natom  = 8
nelec  = 32
nstate = 32
/

yn_periodic specifies whether or not periodic boundary condition is applied.
al(i) specifies the side length of the unit cell.
nelem is the number of elements in the system.
natom is the number of atoms in the system.
nelec is the number of electrons in the system.
nstate is the number of orbitals that are used in the calculation.
&pseudo
!name of input pseudo potential file
file_pseudo(1) = './Si_rps.dat'

!atomic number of element
izatom(1) = 14

!angular momentum of pseudopotential that will be treated as local
lloc_ps(1) = 2
!--- Caution -------------------------------------------!
! Index must correspond to those in &atomic_red_coor.   !
!-------------------------------------------------------!
/

file_pseudo(n) specifies the filename of the pseudopotential file of the n-th element.
izatom(n) is the atomic number of the n-th element.
lloc_ps(n) specifies which angular momentum component is chosen as the local potential for the n-th element.
&functional
!functional('PZ' is Perdew-Zunger LDA: Phys. Rev. B 23, 5048 (1981).)
xc = 'PZ'
/

xc specifies the exchange-correlation potential to be used in the calculation.
&rgrid
!number of spatial grids(x,y,z)
num_rgrid(1:3) = 12, 12, 12
/

num_rgrid(i) specifies the number of real-space grid point in i-th direction.
&kgrid
!number of k-points(x,y,z)
num_kgrid(1:3) = 4, 4, 4
/

num_kgrid(i) specifies the number of k-points for i-th direction discretizing the Brillouin zone.
&tgrid
!time step size and number of time grids(steps)
dt = 0.002d0
nt = 8000
/

dt specifies the time step.
nt is the number of time steps for the time propagation.
&emfield
!envelope shape of the incident pulse('Acos2': cos^2 type envelope for vector potential)
ae_shape1 = 'Acos2'

!peak intensity(W/cm^2) of the incident pulse
I_wcm2_1 = 1.0d12

!duration of the incident pulse
tw1 = 10.672d0

!mean photon energy(average frequency multiplied by the Planck constant) of the incident pulse
omega1 = 1.55d0

!polarization unit vector(real part) for the incident pulse(x,y,z)
epdir_re1(1:3) = 0.0d0, 0.0d0, 1.0d0
!--- Caution ---------------------------------------------------------!
! Defenition of the incident pulse is written in:                     !
! https://www.sciencedirect.com/science/article/pii/S0010465518303412 !
!---------------------------------------------------------------------!
/

ae_shape1 specifies the envelope of the field.
I_wcm2_1 specify the intensity of the pulse in unit of W/cm2.
tw1 specifies the duration of the pulse.
omega1 specifies the mean photon energy of the pulse.
epdir_re1(i) specifies the i-th component of the real part of the polarization unit vector.
&multiscale
!number of macro grids in electromagnetic analysis for x, y, and z directions
nx_m = 8
ny_m = 1
nz_m = 1

!macro grid spacing for x, y, and z directions
hx_m = 50.0d0
hy_m = 50.0d0
hz_m = 50.0d0

!number of macroscopic grids for vacumm region
!(nxvacl_m is for negative x-direction in front of material)
!(nxvacr_m is for positive x-direction behind material)
nxvacl_m = 1000
nxvacr_m = 1000
/

nx_m, ny_m, nz_m specify the number of macroscopic grid points inside the material.
hx_m, hy_m, hz_m specify the grid spacing of macroscopic coordinates.
nxvacl_m / nxvacr_m specifies the number of grid points in the vacuum region in the left / right side of the material.
&maxwell
!boundary condition of electromagnetic analysis
!first index(1-3 rows) corresponds to x, y, and z directions
!second index(1-2 columns) corresponds to bottom and top of the directions
!('abc' is absorbing boundary condition)
boundary_em(1,1) = 'abc'
boundary_em(1,2) = 'abc'
/

boundary_em(i,n) specifies the boundary condition for the electromagnetic analysis. The first index i corresponds to the x,y, and z direction. The second index n specifies left or right side of the material.
&atomic_red_coor
!cartesian atomic reduced coodinates
'Si'      .0      .0      .0      1
'Si'      .25     .25     .25     1
'Si'      .5      .0      .5      1
'Si'      .0      .5      .5      1
'Si'      .5      .5      .0      1
'Si'      .75     .25     .75     1
'Si'      .25     .75     .75     1
'Si'      .75     .75     .25     1
!--- Format ---------------------------------------------------!
! 'symbol' x y z index(correspond to that of pseudo potential) !
!--------------------------------------------------------------!
/

&atomic_red_coor specifies spatial coordinates of atoms in reduced coordinate system.

#### Execusion¶

In a multiprocess environment, calculation will be executed as:

\$ mpiexec -n NPROC salmon < Si_rt_multiscale.inp > Si_rt_multiscale.out


where NPROC is the number of MPI processes. A standard output will be stored in the file Si_rt_multiscale.out.

#### Output files¶

After the calculation, following output files and directories are created in the directory that you run the code in addition to the standard output file.

 file name description Si_m/mxxxxxx/Si_rt.data vector potential, electric field, and matter current at macroscopic position xxxxxx as functions of time Si_m/mxxxxxx/Si_rt_energy.data total energy and electronic excitation energy at macroscopic position xxxxxx as functions of time Si_m/mxxxxxx/PS_Si_KY_n.dat information on pseodupotential file for silicon atom at macroscopic position xxxxxx Si_RT_Ac/Si_Ac_yyyyyy.data vector potential, electric field, magnetic field, electromagnetic current density at time step yyyyyy as function of spatial position Si_wave.data waveform of incident, reflected, and transmitted waves

We first explain the standard output file. In the beginning of the file, input variables used in the calculation are shown.

##############################################################################
# SALMON: Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
#
#                             Version 2.0.1
##############################################################################
Libxc: [disabled]
theory= multi_scale_maxwell_tddft
Initializing macropoint:     1-     8

Total time step      =        8000
Time step[fs]        =  2.000000000000000E-003
Energy range         =        1000
Energy resolution[eV]=  1.000000000000000E-002
Laser frequency     = 1.55[eV]
Pulse width of laser=     10.67200000[fs]
Laser intensity     =  0.10000000E+13[W/cm^2]
use of real value orbitals =  F
r-space parallelization: off
======
.........


After that, the time evolution loop starts. At every 100 iterations, the step, grid point index, time, current in three Cartesian directions, the number of electrons, and the total energy are displayed.

   Step  Macro     Time                          Current      Electrons  Eabs/cell
fs                  1/fs*Angstrom^2                        eV
#------------------------------------------------------------------------------------------
100      1    0.200  5.45E-010 -4.60E-011  2.70E-004    32.00000000  2.36E-006
100      2    0.200  5.45E-010 -1.56E-011  1.83E-004    32.00000000  1.06E-006
100      3    0.200  5.45E-010  7.19E-012  1.23E-004    32.00000000  4.62E-007
100      4    0.200  5.45E-010  2.11E-011  8.14E-005    32.00000000  1.97E-007
100      5    0.200  5.45E-010  2.11E-011  5.28E-005    32.00000000  8.04E-008
100      6    0.200  5.45E-010  7.20E-012  3.34E-005    32.00000000  3.11E-008
100      7    0.200  5.45E-010 -1.56E-011  2.03E-005    32.00000000  1.10E-008
100      8    0.200  5.45E-010 -4.60E-011  1.13E-005    32.00000000  3.27E-009
200      1    0.400  1.77E-011 -2.93E-013  9.70E-004    32.00000000  5.80E-005
200      2    0.400  1.78E-011 -3.64E-011  7.50E-004    32.00000000  3.25E-005
200      3    0.400  1.78E-011 -5.58E-011  5.75E-004    32.00000000  1.80E-005
200      4    0.400  1.78E-011 -6.66E-011  4.38E-004    32.00000000  9.89E-006


Explanations of other output files are given below:

Si_wave.data

Waveforms of incident, reflected, and transmitted waves.

# 1D multiscale calculation:
# E_inc: E-field amplitude of incident wave
# E_ref: E-field amplitude of reflected wave
# E_tra: E-field amplitude of transmitted wave
# 1:Time[fs] 2:E_inc_x[V/Angstrom] 3:E_inc_y[V/Angstrom] 4:E_inc_z[V/Angstrom]
# 5:E_ref_x[V/Angstrom] 6:E_ref_y[V/Angstrom] 7:E_ref_z[V/Angstrom] 8:E_tra_x[V/Angstrom]
# 9:E_tra_y[V/Angstrom] 10:E_tra_z[V/Angstrom]


The figure below shows the incident, reflected, and transmitted electric fields that are drawn using the first column (time in femtosecond) and the 4th column (incident), 7th column (reflected), and 10th column (transmitted).

We find that the amplitude of the reflected pulse is comparable to the amplitude of the incudent pulse, while the phase is different by $$\pi$$. The amplitude of the transmitted pulse is smaller than the incident pulse.

Si_m/mxxxxxx/Si_rt.data

The number xxxxxx in the directory name mxxxxxx specifies the position of macroscopic grid point. Vector potential, electric field, and matter current density as functions of time are included in the file.

# Real time calculation:
# Ac_ext: External vector potential field
# E_ext: External electric field
# Ac_tot: Total vector potential field
# E_tot: Total electric field
# Jm: Matter current density (electrons)
# 1:Time[fs] 2:Ac_ext_x[fs*V/Angstrom] 3:Ac_ext_y[fs*V/Angstrom] 4:Ac_ext_z[fs*V/Angstrom]
# 5:E_ext_x[V/Angstrom] 6:E_ext_y[V/Angstrom] 7:E_ext_z[V/Angstrom] 8:Ac_tot_x[fs*V/Angstrom]
# 9:Ac_tot_y[fs*V/Angstrom] 10:Ac_tot_z[fs*V/Angstrom] 11:E_tot_x[V/Angstrom]
# 12:E_tot_y[V/Angstrom] 13:E_tot_z[V/Angstrom]  14:Jm_x[1/fs*Angstrom^2]
# 15:Jm_y[1/fs*Angstrom^2] 16:Jm_z[1/fs*Angstrom^2]


The figure below shows the electric field at front and back surfaces. Using 1st column (time in femtosecond) and 13th column (total electric field in z direction), electric field at a macroscopic poisition inside the thin film can be plotted. Using the file /m000001/Si_rt.data, electric field at the front surface is drawn by red curve. Using the file /m000008/Si_rt.data, electric field at the back surface is drawn by blue curve.

We find that the amplitude of the electric field at the front surface is small. It is consistent with the previous figure that showed incident and reflected pulses with a similar amplitude and opposite phase.

Si_m/mxxxxxx/Si_rt_energy.data

The number xxxxxx in the directory name mxxxxxx specifies the position of macroscopic grid point. Eall and Eall-Eall0 are total energy and electronic excitation energy, respectively.

# Real time calculation:
# Eall: Total energy
# Eall0: Initial energy
# 1:Time[fs] 2:Eall[eV] 3:Eall-Eall0[eV]


The figure below shows the electronic excitation energy per unit cell volume at front and back surfaces using 1st columnn (time in femtosecond) and 3rd column (Eall-Eall0). Using the file /m000001/Si_rt_energy.data, the excitation energy at the front surface is drawn by red curve. Using the file /m000008/Si_rt_energy.data, the excitation energy at the back surface is drawn by blue curve.

The excitation energy is much larger at the back surface compared with the energy at the front surface. This is because the amplitude of the electric field at the back surface is larger than that of the front surface, as seen in the previous figure, and the excitation is a nonlinear process.

Si_RT_Ac/Si_Ac_yyyyyy.data

The number yyyyyy in the file name Si_Ac_yyyyyy.data specifies the time step. Various quantities at the time step are included in the file as functions of macroscopic position index.

# Multiscale TDDFT calculation
# IX, IY, IZ: FDTD Grid index
# x, y, z: Coordinates
# Ac: Vector potential field
# E: Electric field
# J_em: Electromagnetic current density
# 1:IX[none] 2:IY[none] 3:IZ[none] 4:Ac_x[fs*V/Angstrom] 5:Ac_y[fs*V/Angstrom]
# 6:Ac_z[fs*V/Angstrom] 7:E_x[V/Angstrom] 8:E_y[V/Angstrom] 9:E_z[V/Angstrom] 10:B_x[a.u.]
# 11:B_y[a.u.] 12:B_z[a.u.] 13:Jem_x[1/fs*Angstrom^2] 14:Jem_y[1/fs*Angstrom^2]
# 15:Jem_z[1/fs*Angstrom^2] 16:E_em[eV/vol] 17:E_abs[eV/vol]


The figure below shows spatial dependence of the electric field at three times, $$t=0$$ fs (initial), $$t=8$$ fs (pulse goes through the film), and $$t=16$$ fs (final). It is drawn using the first column multiplied by the step size of $$X$$ and 9th column (electric field).

## Geometry optimization and Ehrenfest molecular dynamics¶

### Exercise-8: Geometry optimization of C2H2 molecule¶

In this exercise, we learn the calculation of geometry optimization of acetylene (C2H2) molecule, solving the static Kohn-Sham equation. This exercise will be useful to learn how to set up calculations in SALMON for any isolated systems such as molecules and nanoparticles.

#### Input files¶

To run the code, following files in samples are used:

 file name description C2H2_opt.inp input file that contains input keywords and their values C_rps.dat pseodupotential file for carbon atom H_rps.dat pseudopotential file for hydrogen atom

In the input file C2H2_opt.inp, input keywords are specified. Most of them are mandatory to execute the geometry optimization. This will help you to prepare an input file for other systems that you want to calculate. A complete list of the input keywords that can be used in the input file can be found in List of input keywords.

!########################################################################################!
! Excercise 08: Geometry optimization of C2H2 molecule                                   !
!----------------------------------------------------------------------------------------!
! * The detail of this excercise is expained in our manual(see chapter: 'Exercises').    !
!   The manual can be obtained from: https://salmon-tddft.jp/documents.html              !
! * Input format consists of group of keywords like:                                     !
!     &group                                                                             !
!       input keyword = xxx                                                              !
!     /                                                                                  !
!   (see chapter: 'List of input keywords' in the manual)                                !
!----------------------------------------------------------------------------------------!
! * Conversion from unit_system = 'a.u.' to 'A_eV_fs':                                   !
!   Length: 1 [a.u.] = 0.52917721067    [Angstrom]                                       !
!   Energy: 1 [a.u.] = 27.21138505      [eV]                                             !
!   Time  : 1 [a.u.] = 0.02418884326505 [fs]                                             !
!########################################################################################!

&calculation
!type of theory
theory = 'dft'

!geometry optimization option
yn_opt = 'y'
/

theory specifies which theoretical method is used in the calculation.
yn_opt is a switch to carry out the structure optimization.
&control
!common name of output files
sysname = 'C2H2'
/

sysname is a prefix for filenames of output files.
&units
!units used in input and output files
unit_system = 'A_eV_fs'
/

unit_system specifies which unit system is used in the input and output files.
&system
!periodic boundary condition
yn_periodic = 'n'

!grid box size(x,y,z)
al(1:3) = 12.0d0, 12.0d0, 16.0d0

!number of elements, atoms, electrons and states(orbitals)
nelem  = 2
natom  = 4
nelec  = 10
nstate = 6
/

yn_periodic specifies whether or not periodic boundary condition is applied.
al(i) specifies the spatial box size of the cubiod cell.
nelem is the number of elements in the system.
natom is the number of atoms in the system.
nelec is the number of electrons in the system.
nstate is the number of orbitals that are used in the calculation.
&pseudo
!name of input pseudo potential file
file_pseudo(1) = './C_rps.dat'
file_pseudo(2) = './H_rps.dat'

!atomic number of element
izatom(1) = 6
izatom(2) = 1

!angular momentum of pseudopotential that will be treated as local
lloc_ps(1) = 1
lloc_ps(2) = 0
!--- Caution ---------------------------------------!
! Indices must correspond to those in &atomic_coor. !
!---------------------------------------------------!
/

file_pseudo(n) specifies the filename of the pseudopotential file of the n-th element.
izatom(n) is the atomic number of the n-th element.
lloc_ps(n) specifies which angular momentum component is chosen as the local potential for the n-th element.
&functional
!functional('PZ' is Perdew-Zunger LDA: Phys. Rev. B 23, 5048 (1981).)
xc = 'PZ'
/

xc specifies the exchange-correlation potential to be used in the calculation.
&rgrid
!spatial grid spacing(x,y,z)
dl(1:3) = 0.20d0, 0.20d, 0.20d0
/

dl(i) specifies the spatial grid spacing in i-th direction.
&scf
!maximum number of scf iteration and threshold of convergence for ground state calculation
nscf      = 300
threshold = 1.0d-9
/

nscf specifies the maximum number of SCF iterations.
threshold specifies the threshold to judge the convergence.
&opt
!threshold(maximum force on atom) of convergence for geometry optimization
convrg_opt_fmax = 1.0d-3
/

&atomic_coor
!cartesian atomic coodinates
'C'    0.0    0.0    0.6  1  y
'H'    0.0    0.0    1.7  2  y
'C'    0.0    0.0   -0.6  1  y
'H'    0.0    0.0   -1.7  2  y
!--- Format -------------------------------------------------------!
! 'symbol' x y z index(correspond to that of pseudo potential) y/n !
!--- Caution ------------------------------------------------------!
! final index(y/n) determines free/fix for the atom coordinate.    !
!------------------------------------------------------------------!
/

&atomic_coor specifies spatial coordinates of atoms.

#### Output files¶

After the calculation, following output files and a directory are created in the directory that you run the code,

 name description C2H2_info.data information on ground state solution C2H2_eigen.data 1 particle energies C2H2_trj.xyz atomic coordinates during the geometry optimization C2H2_k.data k-point distribution (for isolated systems, only gamma point is described) data_for_restart directory where files used in the real-time calculation are contained PS_C_KY_n.dat information on pseodupotential file for carbon atom PS_H_KY_n.dat information on pseodupotential file for hydrogen atom
You may download the above files (zipped file, except for the directory data_for_restart) from:

Main results of the calculation such as orbital energies are included in C2H2_info.data. Explanations of the C2H2_info.data and other output files are below:

C2H2_info.data

Calculated orbital and total energies as well as parameters specified in the input file are shown in this file.

C2H2_eigen.data

1 particle energies.

#esp: single-particle energies (eigen energies)
#occ: occupation numbers, io: orbital index
# 1:io, 2:esp[eV], 3:occ


C2H2_trj.xyz

The atomic coordinates during the geometry optimization in xyz format.

C2H2_k.data

k-point distribution(for isolated systems, only gamma point is described).

# ik: k-point index
# kx,ky,kz: Reduced coordinate of k-points
# wk: Weight of k-point
# 1:ik[none] 2:kx[none] 3:ky[none] 4:kz[none] 5:wk[none]
# coefficients (2*pi/a [a.u.]) in kx, ky, kz


### Exercise-9: Ehrenfest molecular dynamics of C2H2 molecule¶

In this exercise, we learn the calculation of the molecular dynamics in the acetylene (C2H2) molecule under a pulsed electric field, solving the time-dependent Kohn-Sham equation and the Newtonian equation. As outputs of the calculation, time-evolution of the electron density as well as molecular structures and associated quantities such as the electron and ion kinetic energies, the electric dipole moment of the system and temperature as functions of time are calculated.. This tutorial should be carried out after finishing the geometry optimization that was explained in Exercise-8. In the calculation, a pulsed electric field that has $$\cos^2$$ envelope shape is applied. The parameters that characterize the pulsed field such as magnitude, frequency, polarization direction, and carrier envelope phase are specified in the input file.

#### Input files¶

To run the code, following files in samples are used. The directory restart is created in the ground state calculation as data_for_restart. Pseudopotential files are already used in the geometry optimization. Therefore, C2H2_md.inp that specifies input keywords and their values for the pulsed electric field and molecular dynamics calculations is the only file that the users need to prepare.

 file name description C2H2_md.inp input file that contain input keywords and their values. C_rps.dat pseodupotential file for carbon H_rps.dat pseudopotential file for hydrogen restart directory created in the geometry optimization (rename the directory from data_for_restart to restart)

In the input file C2H2_md.inp, input keywords are specified. Most of them are mandatory to execute the calculation of electron dynamics induced by a pulsed electric field. This will help you to prepare the input file for other systems and other pulsed electric fields with molecular dynamics calculation that you want to calculate. A complete list of the input keywords that can be used in the input file can be found in List of input keywords.

!########################################################################################!
! Excercise 09: Ehrenfest molecular dynamics of C2H2 molecule                            !
!----------------------------------------------------------------------------------------!
! * The detail of this excercise is expained in our manual(see chapter: 'Exercises').    !
!   The manual can be obtained from: https://salmon-tddft.jp/documents.html              !
! * Input format consists of group of keywords like:                                     !
!     &group                                                                             !
!       input keyword = xxx                                                              !
!     /                                                                                  !
!   (see chapter: 'List of input keywords' in the manual)                                !
!----------------------------------------------------------------------------------------!
! * Conversion from unit_system = 'a.u.' to 'A_eV_fs':                                   !
!   Length: 1 [a.u.] = 0.52917721067    [Angstrom]                                       !
!   Energy: 1 [a.u.] = 27.21138505      [eV]                                             !
!   Time  : 1 [a.u.] = 0.02418884326505 [fs]                                             !
!----------------------------------------------------------------------------------------!
! * Ehrenfest-MD option is still trial.                                                  !
! * Copy the ground state data directory ('data_for_restart') (or make symbolic link)    !
!   calculated in 'samples/exercise_08_C2H2_opt/' and rename the directory to 'restart/' !
!   in the current directory.                                                            !
!########################################################################################!

&calculation
!type of theory
theory = 'tddft_pulse'

!molecular dynamics option
yn_md  = 'y'
/

theory specifies which theoretical method is used in the calculation.
yn_md is a switch for Ehrenfest molecular dynamics.
&control
!common name of output files
sysname = 'C2H2'
/

sysname is a prefix for filenames of output files.
&units
!units used in input and output files
unit_system = 'A_eV_fs'
/

unit_system specifies which unit system is used in the input and output files.
&system
!periodic boundary condition
yn_periodic = 'n'

!grid box size(x,y,z)
al(1:3) = 12.0d0, 12.0d0, 16.0d0

!number of elements, atoms, electrons and states(orbitals)
nelem  = 2
natom  = 4
nelec  = 10
nstate = 6
/

yn_periodic specifies whether or not periodic boundary condition is applied.
al(i) specifies the spatial box size of the cubiod cell.
nelem is the number of elements in the system.
natom is the number of atoms in the system.
nelec is the number of electrons in the system.
nstate is the number of orbitals that are used in the calculation.
&pseudo
!name of input pseudo potential file
file_pseudo(1) = './C_rps.dat'
file_pseudo(2) = './H_rps.dat'

!atomic number of element
izatom(1) = 6
izatom(2) = 1

!angular momentum of pseudopotential that will be treated as local
lloc_ps(1) = 1
lloc_ps(2) = 0
!--- Caution ---------------------------------------!
! Indices must correspond to those in &atomic_coor. !
!---------------------------------------------------!
/

file_pseudo(n) specifies the filename of the pseudopotential file of the n-th element.
izatom(n) is the atomic number of the n-th element.
lloc_ps(n) specifies which angular momentum component is chosen as the local potential for the n-th element.
&functional
!functional('PZ' is Perdew-Zunger LDA: Phys. Rev. B 23, 5048 (1981).)
xc = 'PZ'
/

xc specifies the exchange-correlation potential to be used in the calculation.
&rgrid
!spatial grid spacing(x,y,z)
dl(1:3) = 0.20d0, 0.20d0, 0.20d0
/

dl(i) specifies the spatial grid spacing in i-th direction.
&tgrid
!time step size and number of time grids(steps)
dt = 1.00d-3
nt = 5000
/

dt specifies the time step.
nt is the number of time steps for the time propagation.
&emfield
!envelope shape of the incident pulse('Ecos2': cos^2 type envelope for scalar potential)
ae_shape1 = 'Ecos2'

!peak intensity(W/cm^2) of the incident pulse
I_wcm2_1 = 1.00d8

!duration of the incident pulse
tw1 = 6.00d0

!mean photon energy(average frequency multiplied by the Planck constant) of the incident pulse
omega1 = 9.28d0

!polarization unit vector(real part) for the incident pulse(x,y,z)
epdir_re1(1:3) = 0.00d0, 0.00d0, 1.00d0

!carrier emvelope phase of the incident pulse
!(phi_cep1 must be 0.25 + 0.5 * n(integer) when ae_shape1 = 'Ecos2')
phi_cep1 = 0.75d0
!--- Caution ---------------------------------------------------------!
! Defenition of the incident pulse is wrriten in:                     !
! https://www.sciencedirect.com/science/article/pii/S0010465518303412 !
!---------------------------------------------------------------------!
/

ae_shape1 specifies the envelope of the field.
I_wcm2_1 specify the intensity of the pulse in unit of W/cm2.
tw1 specifies the duration of the pulse.
omega1 specifies the mean photon energy of the pulse.
epdir_re1(i) specifies the i-th component of the real part of the polarization unit vector.
phi_cep1 specifies the carrier-envelope phase of the pulse.
&md
!ensemble
ensemble = 'NVE'

!set of initial velocities
yn_set_ini_velocity = 'y'

!setting temperature [K] for NVT ensemble, velocity scaling,
!and generating initial velocities
temperature0_ion_k = 300.0d0

!time step interval for updating pseudopotential
step_update_ps = 20
/

ensemble specifies the choice of the ensemble.
yn_set_ini_velocity is a switch to prepare initial velocity for atoms.
temperature0_ion_k specifies the temperature that is used to generate initial velocity of ions.
step_update_ps specifies the time step interval to update projector for the nonlocal pseudopotential.

#### Output files¶

After the calculation, following output files are created in the directory that you run the code,

 file name description C2H2_pulse.data dipole moment as functions of energy C2H2_rt.data components of change of dipole moment (electrons/plus definition) and total dipole moment (electrons/minus + ions/plus) as functions of time C2H2_rt_energy.data components of total energy and difference of total energy as functions of time C2H2_trj.xyz Trajectory of atoms(ions): Atomic coordinates, velocities, and forces are printed PS_C_KY_n.dat information on pseodupotential file for carbon atom PS_H_KY_n.dat information on pseodupotential file for hydrogen atom

Explanations of the files are described below:

C2H2_pulse.data

Time-frequency Fourier transformation of the dipole moment.

# Fourier-transform spectra:
# energy: Frequency
# dm: Dopile moment
# 1:energy[eV] 2:Re(dm_x)[fs*Angstrom] 3:Re(dm_y)[fs*Angstrom] 4:Re(dm_z)[fs*Angstrom] 5:Im(dm_x)[fs*Angstrom] 6:Im(dm_y)[fs*Angstrom] 7:Im(dm_z)[fs*Angstrom] 8:|dm_x|^2[fs^2*Angstrom^2] 9:|dm_y|^2[fs^2*Angstrom^2] 10:|dm_z|^2[fs^2*Angstrom^2]


C2H2_rt.data

Results of time evolution calculation for vector potential, electric field, and dipole moment.

# Real time calculation:
# Ac_ext: External vector potential field
# E_ext: External electric field
# Ac_tot: Total vector potential field
# E_tot: Total electric field
# ddm_e: Change of dipole moment (electrons/plus definition)
# dm: Total dipole moment (electrons/minus + ions/plus)
# 1:Time[fs] 2:Ac_ext_x[fs*V/Angstrom] 3:Ac_ext_y[fs*V/Angstrom] 4:Ac_ext_z[fs*V/Angstrom] 5:E_ext_x[V/Angstrom] 6:E_ext_y[V/Angstrom] 7:E_ext_z[V/Angstrom] 8:Ac_tot_x[fs*V/Angstrom] 9:Ac_tot_y[fs*V/Angstrom] 10:Ac_tot_z[fs*V/Angstrom] 11:E_tot_x[V/Angstrom] 12:E_tot_y[V/Angstrom] 13:E_tot_z[V/Angstrom] 14:ddm_e_x[Angstrom] 15:ddm_e_y[Angstrom] 16:ddm_e_z[Angstrom] 17:dm_x[Angstrom] 18:dm_y[Angstrom] 19:dm_z[Angstrom]


C2H2_rt_energy.data

Eall and Eall-Eall0 are total energy and electronic excitation energy, respectively.

# Real time calculation:
# Eall: Total energy
# Eall0: Initial energy
# Tion: Kinetic energy of ions
# Temperature_ion: Temperature of ions
# E_work: Work energy of ions(sum f*dr)
# 1:Time[fs] 2:Eall[eV] 3:Eall-Eall0[eV] # 4:Tion[eV] 5:Temperature_ion[K] 6:E_work[eV]


C2H2_trj.xyz

Atomic coordinates [Angstrom], velocities [a.u.] and forces [a.u.] are printed along the time evolution in xyz format.

## FDTD simulation(electromagnetic analysis)¶

### Exercise-10: Pulsed electric field response of a metallic nanosphere in classical electromagnetism(FDTD simulation)¶

In this exercise, we learn the pulsed electric field response in the metallic nanosphere, solving the time-dependent Maxwell equations. As outputs of the calculation, the time response of the electromagnetic field is calculated. A pulsed electric field that has $$\cos^2$$ envelope shape is applied. The parameters that characterize the pulsed field such as magnitude, frequency, polarization direction, and carrier envelope phase are specified in the input file.

#### Input files¶

To run the code, the input file classicEM_rt_pulse.inp that contains input keywords and their values for the pulsed electric field calculation is required. The shape file of the metallic nanosphere shape.cube is also required.

The shape file can be generated by program FDTD_make_shape in SALMON utilities: https://salmon-tddft.jp/utilities.html

shape.inp is an input file for FDTD_make_shape to generate shape.cube.

The input files are in samples

 file name description classicEM_rt_pulse.inp input file that contain input keywords and their values. shape.cube shape file for fdtd shape.inp input file for FDTD_make_shape

In the input file classicEM_rt_pulse.inp, input keywords are specified. Most of them are mandatory to execute the linear response calculation. This will help you to prepare the input file for other systems that you want to calculate. A complete list of the input keywords that can be used in the input file can be found in List of input keywords.

!########################################################################################!
! Excercise 10: Pulsed electric field response of a metallic nanosphere                  !
!               in classical electromagnetism(FDTD simulation)                           !
!----------------------------------------------------------------------------------------!
! * The detail of this excercise is expained in our manual(see chapter: 'Exercises').    !
!   The manual can be obtained from: https://salmon-tddft.jp/documents.html              !
! * Input format consists of group of keywords like:                                     !
!     &group                                                                             !
!       input keyword = xxx                                                              !
!     /                                                                                  !
!   (see chapter: 'List of input keywords' in the manual)                                !
!----------------------------------------------------------------------------------------!
! * Conversion from unit_system = 'a.u.' to 'A_eV_fs':                                   !
!   Length: 1 [a.u.] = 0.52917721067    [Angstrom]                                       !
!   Energy: 1 [a.u.] = 27.21138505      [eV]                                             !
!   Time  : 1 [a.u.] = 0.02418884326505 [fs]                                             !
!----------------------------------------------------------------------------------------!
! * The read-in file 'shape_file' in &maxwell category can be generated by program       !
!   'FDTD_make_shape' in SALMON utilities(https://salmon-tddft.jp/utilities.html).       !
!   'shape.inp' is an input file for 'FDTD_make_shape' to generate 'shape.cube'.         !
! * Results can be visualized by program 'FDTD_make_figani' in SALMON utilities.         !
!########################################################################################!

&calculation
!type of theory
theory = 'maxwell'
/

theory specifies which theoretical method is used in the calculation.
&control
!name of directory where output files are contained
base_directory = 'result'
/

base_directory specifies the directory name where output files are generated.
&units
!units used in input and output files
unit_system = 'A_eV_fs'
/

unit_system specifies which unit system is used in the input and output files.
&system
!periodic boundary condition
yn_periodic = 'n'
/

yn_periodic specifies whether or not periodic boundary condition is applied.
&emfield
!envelope shape of the incident pulse('Ecos2': cos^2 type envelope for scalar potential)
ae_shape1 = 'Ecos2'

!peak intensity(W/cm^2) of the incident pulse
I_wcm2_1 = 1.00d8

!duration of the incident pulse
tw1 = 4.60d0

!mean photon energy(average frequency multiplied by the Planck constant) of the incident pulse
omega1 = 5.49d0

!polarization unit vector(real part) for the incident pulse(x,y,z)
epdir_re1(1:3) = 0.00d0, 0.00d0, 1.00d0

!carrier emvelope phase of the incident pulse
!(phi_cep1 must be 0.25 + 0.5 * n(integer) when ae_shape1 = 'Ecos2')
phi_cep1 = 0.75d0
!--- Caution ---------------------------------------------------------!
! Defenition of the incident pulse is wrriten in:                     !
! https://www.sciencedirect.com/science/article/pii/S0010465518303412 !
!---------------------------------------------------------------------!
/

ae_shape1 specifies the envelope of the field.
I_wcm2_1 specify the intensity of the pulse in unit of W/cm2.
tw1 specifies the duration of the pulse.
omega1 specifies the mean photon energy of the pulse.
epdir_re1(i) specifies the i-th component of the real part of the polarization unit vector.
phi_cep1 specifies the carrier-envelope phase of the pulse.
&maxwell
!box size and spacing of spatial grid(x,y,z)
al_em(1:3) = 120d0, 120d0, 120d0
dl_em(1:3) = 1.2d0, 1.2d0, 1.2d0

!time step size and number of time grids(steps)
dt_em = 2.30d-4
nt_em = 20000

!name of input shape file and number of media in the file
shape_file = './shape.cube'
media_num  = 1

!*** MEDIA INFORMATION(START) **************************************!
!type of media(media ID)
media_type(1) = 'lorentz-drude'
!--- Au described by Lorentz-Drude model ---------------------------!
! The parameters are determined from:                               !
! (https://www.osapublishing.org/ao/abstract.cfm?uri=ao-37-22-5271) !
!-------------------------------------------------------------------!

!number of poles and plasma frequency of media(media ID)
pole_num_ld(1) = 6
omega_p_ld(1)  = 9.030d0

!oscillator strength, collision frequency,
!and oscillator frequency of media(media ID,pole ID)
f_ld(1,1:6)     = 0.760d0, 0.024d0, 0.010d0, 0.071d0, 0.601d0, 4.384d0
gamma_ld(1,1:6) = 0.053d0, 0.241d0, 0.345d0, 0.870d0, 2.494d0, 2.214d0
omega_ld(1,1:6) = 0.000d0, 0.415d0, 0.830d0, 2.969d0, 4.304d0, 13.32d0
!*** MEDIA INFORMATION(END) ****************************************!

!*** SOURCE INFORMATION(START) *************************************!
!type of method to generate the incident pulse
!('source': incident current source)
wave_input = 'source'

!location of source(x,y,z)
source_loc1(1:3) = -37.8d0, 0.0d0, 0.0d0

!propagation direction of the incident pulse(x,y,z)
ek_dir1(1:3) = 1.0d0, 0.0d0, 0.0d0
!*** SOURCE INFORMATION(END) ***************************************!

!*** OBSERVATION INFORMATION(START) ********************************!
!number of observation points
obs_num_em = 1

!time step interval for sampling
obs_samp_em = 20

!location of observation point(observation ID,x,y,z)
obs_loc_em(1,1:3) = 0.0d0, 0.0d0, 0.0d0

!output flag for electrmagnetic field distribution(observation ID)
yn_obs_plane_em(1) = 'n'
!--- Make of animation file ----------------------------------------!
! When yn_obs_plane_em(1) = 'y', animation file can be made         !
! by program 'FDTD_make_figani' in SALMON utilities.                !
! The animation file visualizes electromagnetic field distributions !
! on the cross-section including the observation point              !
! whose location is determined by obs_loc_em.                       !
!-------------------------------------------------------------------!
!*** OBSERVATION INFORMATION(END) **********************************!
/

al_em(i) specifies the lengths of three sides of the cuboid where the grid points are prepared.
dl_em(i) specifies the grid spacings in three Cartesian directions.
dt_em specifies the time step of the time evolution calculation.
nt_em specifies the number of time steps in the calculation.
shape_file specifies the filename of the shape file.
media_num specifies the number of the types of media that is provided in the shape file.
media_type(n) specifies the type of the n-th media.
pole_num_ld(n) and omega_p_ld(n) specify the number of poles and the plasmal frequency of the n-th media, respectively.
f_ld(n,m), omega_ld(n,m), gamma_ld(n,m) specify the oscillator strength, oscillator frequency, and collision frequency of the m-th pole of the n-th media, respectively.
wave_input specifies an electric current source that is used for the generation of the pulse.
source_loc1(i) specifies the coordinate of the current source.
ek_dir1(i) specifies the propagation direction of the pulse.
obs_num_em specifies the number of the observing point.
obs_samp_em specifies the sampling interval.
obs_loc_em(n,i) specifies the coordinate of n-th observing point.
yn_obs_plane_em(n) is a switch to output the electromagnetic fields on the xy, yz, and xz planes that include the n-th observation point.

#### Output files¶

After the calculation, following output files are created in the directory result,

 file name description obs0_info.data information to generate animation obs1_at_point_rt.data components of electric and magnetic fields as functions of time

Explanations of the files are described below:

obs0_info.data

This file is used to generate animation files by using SALMON utilities: https://salmon-tddft.jp/utilities.html

obs1_at_point_rt.data

Results of time evolution calculation for electric and magnetic fields at observation point 1.

# Real time calculation:
# E: Electric field
# H: Magnetic field
# 1:Time[fs] 2:E_x[V/Angstrom] 3:E_y[V/Angstrom] 4:E_z[V/Angstrom] 5:H_x[A/Angstrom] 6:H_y[A/Angstrom] 7:H_z[A/Angstrom]