Timepropagation TDDFT¶
Optical photoabsorption spectrum as well as nonlinear effects can be studied using time propagation TDDFT. This approach scales better than linear response, but the prefactor is so large that for small and moderate systems linear response is significantly faster.
Ground state¶
To obtain the ground state for TDDFT, one has to just do a standard ground state with a larger simulation box. A proper distance from any atom to edge of the simulation box is problem dependent, but a minimum reasonable value is around 6 Ångströms and recommended between 810 Ång. In TDDFT, one can use larger grid spacing than for geometry optimization. For example, if you use h=0.25 for geometry optimization, try h=0.3 for TDDFT. This saves a lot of time.
A good way to start is to use too small box (vacuum=6.0), too large grid spacing (h=0.35), and too large time step (dt=16.0). Then repeat the simulation with better parameter values and compare. Probably lowest peaks are already pretty good, and far beyond the ionization limit, in the continuum, the spectrum is not going to converge anyway. The first run takes only fraction of the time of the second run.
For a paralleloverstates TDDFT calculation, you must choose the number of states so, that these can be distributed equally to processors. For example, if you have 79 occupied states and you want to use 8 processes in parallelization over states, add one unoccupied state to get 80 states in total.
Ground state example:
# Standard magic
from ase import Atoms
from gpaw import GPAW
# Beryllium atom
atoms = Atoms(symbols='Be',
positions=[(0, 0, 0)],
pbc=False)
# Add 6.0 ang vacuum around the atom
atoms.center(vacuum=6.0)
# Create GPAW calculator
calc = GPAW(nbands=1, h=0.3)
# Attach calculator to atoms
atoms.set_calculator(calc)
# Calculate the ground state
energy = atoms.get_potential_energy()
# Save the ground state
calc.write('be_gs.gpw', 'all')
Timepropagation TDDFT is also available in LCAO mode.
Optical photoabsorption spectrum¶
Optical photoabsorption spectrum can be obtained by applying a weak delta pulse of dipole electric field, and then letting the system evolve freely while recording the dipole moment. A timestep around 4.08.0 attoseconds is reasonable. The total simulation time should be few tens of femtoseconds depending on the desired resolution.
Example:
from gpaw.tddft import *
time_step = 8.0 # 1 attoseconds = 0.041341 autime
iterations = 2500 # 2500 x 8 as => 20 fs
kick_strength = [0.0,0.0,1e3] # Kick to zdirection
# Read ground state
td_calc = TDDFT('be_gs.gpw')
# Kick with a delta pulse to zdirection
td_calc.absorption_kick(kick_strength=kick_strength)
# Propagate, save the timedependent dipole moment to 'be_dm.dat',
# and use 'be_td.gpw' as restart file
td_calc.propagate(time_step, iterations, 'be_dm.dat', 'be_td.gpw')
# Calculate photoabsorption spectrum and write it to 'be_spectrum_z.dat'
photoabsorption_spectrum('be_dm.dat', 'be_spectrum_z.dat')
Note
Make sure to number of iterations is divisible by the dump interval such that the last iteration will be stored in the restart file. Otherwise append td_calc.write(‘be_td.gpw’, mode=’all’) to the script.
When propagating after an absorption kick has been applied, it is a good idea to periodically write the timeevolution state to a restart file. This ensures that you can resume adding data to the dipole moment file if you experience artificial oscillations in the spectrum because the total simulation time was too short.
Example:
from gpaw.tddft import *
time_step = 8.0 # 1 attoseconds = 0.041341 autime
iterations = 2500 # 2500 x 8 as => 20 fs
# Read restart file with result of previous propagation
td_calc = TDDFT('be_td.gpw')
# Propagate more, appending the timedependent dipole moment to the
# already existing 'be_dm.dat' and use 'be_td2.gpw' as restart file
td_calc.propagate(time_step, iterations, 'be_dm.dat', 'be_td2.gpw')
# Recalculate photoabsorption spectrum and write it to 'be_spectrum_z2.dat'
photoabsorption_spectrum('be_dm.dat', 'be_spectrum_z2.dat')
Typically in experiments, the spherically averaged spectrum is measured. To obtain this, one must repeat the timepropagation to each Cartesian direction and average over the Fourier transformed dipole moments.
Fourier transformed density¶
If one merely wishes to record the timeevolution of the dipole moment and analyze the resulting spectrum, TDDFT offers little advantage over the much faster LrTDDFT. Further, one must bear in mind that only excitations induced by the absorption kick will show up in the spectrum.
However, propagating a slightly perturbed ground state density may offer much more structural information, starting with the ability to distinguish which spectral peaks correspond to which principal directions in a lattice.
Since the dipole moment is generated by displacements in the charge density, most strong peaks in the optical photoabsorption spectrum signify nearly harmonic oscillations herein. Therefore, taking Fourier transforms of the timeevolution of the density at the resonant frequencies is a great way of analyzing the spatial extent of the oscillating modes.
The discrete movingaverage Fourier transform of the pseudoelectron density \(\tilde{n}(\mathbf{r},t)\) is defined:
, where we allow for variable timestep \(\Delta t_j\) along the \(N\) propagation steps in the timeseries \(j=0,1,\ldots,N\). With a total propagation time of \(t_N\), the Fourier transforms are taken relative to the timeaverage \(\overline{n}_N(\mathbf{r})\) of the pseudo density:
Regrettably, having arrived at time \(t_N\) will not enable us to perform the above summations because recording \(N\sim 10^4\) sets of grid data is completely intractable. Instead, an iterative cumulation scheme is implemented, which only requires data from one timestep at a time.
XXX more on this later
The class gpaw.tddft.fourier.DensityFourierTransform
is used to
calculate and maintain Fourier transforms of the pseudo electron density. It
functions by attaching itself to a TDDFT instance, which in turn notifies
it after each timestep and allows it to update the density Fourier transforms.
Important
An incontestable restriction of the iterative approach is the requirement that the frequencies must be given upon initialization (i.e. time zero). To avoid wasted effort, getting the peak frequencies right is essential.
It is recommended to use either LrTDDFT or a somewhat cruder
timepropagation to estimate which frequencies could be of interest. In the
latter case, applying a weak kick [1e3, 1e3, 1e3]
will probably be
sufficient to excite and detect all the relevant modes in a short timespan.
For quick estimates, using the ECN
propagator and the CSCG
eigensolver
with a tolerance around 1e4
works reasonably well for timesteps of 510 as.
Tip
Using a finite width \(\sigma\) around 0.1 eV
will make any
ballpark figure a much safer bet. Be aware that peaks found using
LrTDDFT may shift slightly.
Example:
from gpaw.tddft import TDDFT
from gpaw.tddft.fourier import DensityFourierTransform
time_step = 4.0 # 1 attoseconds = 0.041341 autime
iterations = 5000 # 5000 x 4 as => 20 fs
kick_strength = [0.0,5e3,0.0] # Kick to ydirection
frequencies = [4.26,6.27,13.0, \
16.9,18.1,19.9] # Predetermined peak frequencies in eV
sigma = 0.05 # Width of Gaussian envelope in eV
# Read ground state
td_calc = TDDFT('bda_gs.gpw')
# Kick with a delta pulse to ydirection
td_calc.absorption_kick(kick_strength=kick_strength)
# Create and attach Fourier transform observer
obs = DensityFourierTransform(timestep, frequencies, sigma)
obs.initialize(td_calc)
# Propagate, save the timedependent dipole moment to 'bda_dm.dat',
# (just for comparison) and use 'bda_td.gpw' as restart file
td_calc.propagate(time_step, iterations, 'bda_dm.dat', 'bda_td.gpw')
# Save result of the Fourier transformations to a .ftd file
obs.write('bda_fourier.ftd')
You can now resume adding data to both the dipole moment file and the density fourier transform if the spectrum is not sufficiently evolved because the total simulation time was too short.
Example:
from gpaw.tddft import TDDFT
from gpaw.tddft.fourier import DensityFourierTransform
time_step = 4.0 # 1 attoseconds = 0.041341 autime
iterations = 5000 # 5000 x 4 as => 20 fs
frequencies = [4.26,6.27,13.0, \
16.9,18.1,19.9] # Predetermined peak frequencies in eV
sigma = 0.05 # Width of Gaussian envelope in
# Read restart file with result of previous propagation
td_calc = TDDFT('bda_td.gpw')
# Create and attach Fourier transform observer
obs = DensityFourierTransform(timestep, frequencies, sigma)
obs.initialize(td_calc)
# Read previous result of the corresponding Fourier transformations
obs.read('bda_fourier.ftd')
# Propagate more, appending the timedependent dipole moment to the
# already existing 'bda_dm.dat' and use 'bda_td2.gpw' as restart file
td_calc.propagate(time_step, iterations, 'bda_dm.dat', 'bda_td2.gpw')
# Save result of the improved Fourier transformations to an .ftd file
obs.write('bda_fourier2.ftd')
Time propagation¶
Since the total CPU time also depends on the number of iterations performed
by the linear solvers in each timestep, smaller timesteps around 2.04.0
attoseconds might prove to be faster with the ECN
and SICN
propagators because they have an embedded Euler step in each predictor step:
, where \(\hat{S}^{\;1}_\mathrm{approx.}\) is an inexpensive operation which approximates the inverse of the overlap operator \(\hat{S}\). See the Developers Guide for details.
Therefore, as a ruleofthumb, choose a timestep small enough to minimize the number of iterations performed by the linear solvers in each timestep, but large enough to minimize the number of timesteps required to arrive at the desired total simulation time.
TDDFT reference manual¶
The TDDFT
class and keywords:
Keyword  Type  Default  Description 

ground_state_file 
string 
Name of the ground state file  
td_potential 
TDPotential 
None 
Timedependent external potential 
propagator 
string 
'SICN' 
Timepropagator ('ECN' /'SICN' /'SITE' /'SIKE' ) 
solver 
string 
'CSCG' 
Linear equation solver ('CSCG' /'BiCGStab' ) 
tolerance 
float 
1e8 
Tolerance for linear solver 
Keywords for absorption_kick()
:
Keyword  Type  Default  Description 

kick_strength 
float[3] 
[0,0,1e3] 
Kick strength 
Keywords for propagate()
:
Keyword  Type  Default  Description 

time_step 
float 
Time step in attoseconds (1 autime = 24.188 as ) 

iterations 
integer 
Iterations  
dipole_moment_file 
string 
None 
Name of the dipole moment file 
restart_file 
string 
None 
Name of the restart file 
dump_interal 
integer 
500 
How often restart file is written 
Keywords for gpaw.tddft.photoabsorption_spectrum()
:
Keyword  Type  Default  Description 

dipole_moment_file 
string 
Name of the dipole moment file  
spectrum_file 
string 
Name of the spectrum file  
folding 
string 
Gauss 
Gaussian folding (or Lorentzian in future) 
width 
float 
0.2123 
Width of the Gaussian/Lorentzian (in eV) 
e_min 
float 
0.0 
Lowest energy shown in spectrum (in eV) 
e_max 
float 
30.0 
Highest energy shown in spectrum (in eV) 
delta_e 
float 
0.05 
Resolution of energy in spectrum (in eV) 

class
gpaw.tddft.
TDDFT
(filename, td_potential=None, propagator='SICN', calculate_energy=True, propagator_kwargs=None, solver='CSCG', tolerance=1e08, **kwargs)[source]¶ Timedependent density functional theory calculation based on GPAW.
This class is the core class of the timedependent density functional theory implementation and is the only class which a user has to use.
Create TDDFTobject.
Parameters:
 filename: string
 File containing ground state or timedependent state to propagate
 td_potential: class, optional
 Function class for the timedependent potential. Must have a method ‘strength(time)’ which returns the strength of the linear potential to each direction as a vector of three floats.
 propagator: {‘SICN’,’ETRSCN’,’ECN’,’SITE’,’SIKE4’,’SIKE5’,’SIKE6’}
 Name of the time propagator for the KohnSham wavefunctions
 solver: {‘CSCG’,’BiCGStab’}
 Name of the iterative linear equations solver for time propagation
 tolerance: float
 Tolerance for the linear solver
The following parameters can be used: \(txt\), \(parallel\), \(communicator\) \(mixer\) and \(dtype\). The internal parameters \(mixer\) and \(dtype\) are strictly used to specify a dummy mixer and complex type respectively.

absorption_kick
(kick_strength)[source]¶ Delta absorption kick for photoabsorption spectrum.
Parameters:
 kick_strength: [float, float, float]
 Strength of the kick, e.g., [0.0, 0.0, 1e3]

propagate
(time_step, iterations, dipole_moment_file=None, restart_file=None, dump_interval=100)[source]¶ Propagates wavefunctions.
Parameters:
 time_step: float
 Time step in attoseconds (10^18 s), e.g., 4.0 or 8.0
 iterations: integer
 Iterations, e.g., 20 000 as / 4.0 as = 5000
 dipole_moment_file: string, optional
 Name of the data file where to the timedependent dipole moment is saved
 restart_file: string, optional
 Name of the restart file
 dump_interval: integer
 After how many iterations restart data is dumped

read
(filename)[source]¶ Read atoms, parameters and calculated properties from output file.
Read result from self.label file. Raise ReadError if the file is not there. If the file is corrupted or contains an error message from the calculation, a ReadError should also be raised. In case of succes, these attributes must set:
 atoms: Atoms object
 The state of the atoms from last calculation.
 parameters: Parameters object
 The parameter dictionary.
 results: dict
 Calculated properties like energy and forces.
The FileIOCalculator.read() method will typically read atoms and parameters and get the results dict by calling the read_results() method.

gpaw.tddft.
photoabsorption_spectrum
(dipole_moment_file, spectrum_file, folding='Gauss', width=0.2123, e_min=0.0, e_max=30.0, delta_e=0.05)[source]¶ Calculates photoabsorption spectrum from the timedependent dipole moment.
Parameters:
 dipole_moment_file: string
 Name of the timedependent dipole moment file from which the spectrum is calculated
 spectrum_file: string
 Name of the spectrum file
 folding: ‘Gauss’ or ‘Lorentz’
 Whether to use Gaussian or Lorentzian folding
 width: float
 Width of the Gaussian (sigma) or Lorentzian (Gamma) Gaussian = 1/(sigma sqrt(2pi)) exp((1/2)(omega/sigma)^2) Lorentzian = (1/pi) (1/2) Gamma / [omega^2 + ((1/2) Gamma)^2]
 e_min: float
 Minimum energy shown in the spectrum (eV)
 e_max: float
 Maximum energy shown in the spectrum (eV)
 delta_e: float
 Energy resolution (eV)

class
gpaw.tddft.fourier.
DensityFourierTransform
(timestep, frequencies, width=None, interval=1)[source]¶ Parameters
 timestep: float
 Time step in attoseconds (10^18 s), e.g., 4.0 or 8.0
 frequencies: NumPy array or list of floats
 Frequencies in eV for Fourier transforms
 width: float or None
 Width of Gaussian envelope in eV, otherwise no envelope
 interval: int
 Number of timesteps between calls (used when attaching)