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# X-Ray Absorption from δ-Kick
This tutorial shows how to run a Real-Time Time-Dependent DFT calculation using the so-called δ-kick
approach to compute absorption electronic spectra.
We warmly recommend having a look first at the Linear-Response approach, for instance at
[the X-Ray frequencies](./tddft). See also this article and the supplementary information: LINK
This tutorial is structured as follows:
- Brief theory overview
- CP2K input file overview
Then, the link between the CP2K output (time-dependent dipole moment) and the quantity targetted
(the absorption spectra in the frequency domain) is performed in this jupyter notebook:
[download this file](https://www.cp2k.org/_media/howto:delta_kick_analyse.zip). Of course, one can
use its own script and method. Yet, as these analyses present some subtilities, we propose our which
shows quantitative results with the Linear Response one.
## Theory Overview
### Response theory reminders
In the perturbative regime and at the linear order, the time-dependent response of an electronic
cloud can be decomposed in the Fourier space: the response at a given frequency is proportional to
the perturbation acting on the system at the same frequency. In particular, the induced dipole
moment (or equivalently the induced current) at a given frequency, $\mu^\omega$, is given by the
product of the polarizability matrix, $\alpha(\omega, \omega)$, with the perturbative electric field
at the same frequency, $F^\omega$, within the dipolar approximation.
$$
\mu^\omega = \alpha(\omega, \omega) \cdot F^\omega
$$
This quantity involved a dot product between the field vector and the polarizability matrix. For the
next paragraphs, we will drop the vectorial behavior and discuss it later.
In Linear-Response-TDDFT approaches, one computes the excited state promoted by an electric field
oscillating at a specific frequency $\omega$ and then computes the transition dipole moment
associated with such electronic transition. This transition dipole moment is then used to compute
the polarizability for resonant or non-resonant frequency.
In the Real-Time approach presented here, we will excite **all** the possible electronic transitions
at the beginning of the simulation and then propagate this excited electronic cloud to observe the
induced dipole moment. We will deduce the polarizability tensor for any frequency from this induced
dipole moment.
### The δ-kick technic
In the δ-kick technic, the electronic structure is first obtained at the ground state, for instance
using DFT, and then perturbed with an instantaneous electric field named a δ-kick:
$$
F(t) = F^0 \delta(t)
$$
This δ-kick perturbes the ground state wave-function at $t=0^-$ using either the dipole moment
operator (length gauge) or using the momentum one (velocity gauge). Then, this excited wave-function
is propagated in real time by numerically integrating the DFT equivalent of the Schrodinger
equation. For more information, have a look for instance at the recent work in CP2K by
[](#Mattiat2022).
The electric δ-kick can be written in the frequency domain:
$$
F^\omega = \frac{F^0}{2 \pi} \int_{-\infty}^{+ \infty} \delta(t) e^{i \omega t} dt = \frac{F^0}{2 \pi} e^{i \omega \times 0} = \frac{F^0}{2 \pi}
$$
The field amplitude in the Fourier space is $F^0 / 2 \pi$ for all frequencies: this instantaneous
perturbation does indeed contain all the frequencies. The time-dependent wave-function can be thus
described to be the one at the ground state plus all the possible excited states. During the time
propagation, the dipole moment of the molecule will be given by the superposition of all possible
oscillations related to all the excited states. This complex behavior in the time domain became
simple in the frequency one since we know the amplitude of the perturbation applied at each
frequency:
$$
\mu^\omega = \frac{1}{2 \pi} \alpha(\omega, \omega) F^0
$$
Therefore, in order to get the polarizability $\alpha(\omega, \omega)$ one prepares the perturbed
state at $t=0^-$ for a given field amplitude, then propagates the wave-function in real time and
compute the time-dependent dipole moment. The dipole moment is Fourier transformed and the
polarizability is extracted using:
$$
\text{Re} \left[ \alpha(\omega, \omega) \right] = 2 \pi \frac{\text{Re} \left[ \mu^\omega \right] }{F^0} \\
\text{Im} \left[ \alpha(\omega, \omega) \right] = 2 \pi \frac{\text{Im} \left[ \mu^\omega \right] }{F^0}
$$
The amplitude of the dipole moment in the Fourier space may be very small if the frequencies are far
from resonances. But, in principle, one can extract the whole spectra from one Real-Time
calculation. Of course, numerically speaking this is not the case since the time scale involved for
UV frequencies is quite different from the one involved for X-Rays: a Real-Time propagation
calculation is thus run for a specific range of frequency.
### Absorption spectrum
Using the time-dependent dipole moment, we got the frequency-dependent polarizability $\alpha$
assuming a perturbative and linear response regime. The real and imaginary part of the
polarizability defines the nature of the response. If the frequency is at an electronic resonance,
then the polarizability has an imaginary part. The polarizability is pure real off resonances.
From one Real-Time propagation, one can obtain 3 components of the polarizability tensor. For
instance, applying a δ-kick along the $x$ direction will provides the components $\alpha_{xx}$,
$\alpha_{yx}$ and $\alpha_{zx}$ by Fourier Transform the dipole moment along the $x$, $y$ and $z$
direction respectively. Therefore, to get the full polarizability tensor, one should run 3 Real-Time
propagations using a perturbation along 3 orthogonal directions, for instance, $x$, $y$, and $z$.
To compare with experiments, one often assumes that the system is averaged over all possible
orientations in space. Then, the absorption spectrum $I(\omega)$ is proportional to:
$$
I(\omega) \propto \sum_{i=x,y,z} \text{Im} \left[ \alpha_{ii}(\omega, \omega) \right]
$$
## CP2K Input
In this tutorial, we will study the response of a carbon-monoxide in the gas phase upon a δ-kick
along its perpendicular direction. We will focus on the X-Ray range, but in principle, other ranges
of frequency can be sampled using this technic.
This has been done for isolated carbon-monoxide at the DFT/PBEh level using a PCSEG-2 basis set. One
can look at the input file contained in RTP.inp, in the following we will emphasize a few important
points related to the δ-kick technic.
```none
&GLOBAL
PROJECT RTP
RUN_TYPE RT_PROPAGATION
&END GLOBAL
&MOTION
&MD
ENSEMBLE NVE
STEPS 50000
TIMESTEP [fs] 0.00078
TEMPERATURE [K] 0.0
&END MD
&END MOTION
&FORCE_EVAL
METHOD QS
&DFT
&REAL_TIME_PROPAGATION
APPLY_DELTA_PULSE .TRUE.
DELTA_PULSE_DIRECTION 1 0 0
DELTA_PULSE_SCALE 0.001
MAX_ITER 100
MAT_EXP ARNOLDI
EPS_ITER 1.0E-11
INITIAL_WFN SCF_WFN
PERIODIC .FALSE.
&END REAL_TIME_PROPAGATION
BASIS_SET_FILE_NAME BASIS_PCSEG2
POTENTIAL_FILE_NAME POTENTIAL
&MGRID
CUTOFF 1000
NGRIDS 5
REL_CUTOFF 60
&END MGRID
&QS
METHOD GAPW
EPS_FIT 1.0E-6
&END QS
&SCF
MAX_SCF 500
SCF_GUESS RESTART
EPS_SCF 1.0E-8
&END SCF
&POISSON
POISSON_SOLVER WAVELET
PERIODIC NONE
&END POISSON
&XC
&XC_FUNCTIONAL PBE
&PBE
SCALE_X 0.55
&END
&END XC_FUNCTIONAL
&HF
FRACTION 0.45
&INTERACTION_POTENTIAL
POTENTIAL_TYPE TRUNCATED
CUTOFF_RADIUS 7.0
&END INTERACTION_POTENTIAL
&END HF
&END XC
&PRINT
&MULLIKEN OFF
&END MULLIKEN
&HIRSHFELD OFF
&END HIRSHFELD
&MOMENTS
PERIODIC .FALSE.
FILENAME =dipole
COMMON_ITERATION_LEVELS 100000
&EACH
MD 1
&END EACH
&END MOMENTS
&END PRINT
&END DFT
&SUBSYS
&CELL
ABC 10 10 10
ALPHA_BETA_GAMMA 90 90 90
PERIODIC NONE
&END CELL
&TOPOLOGY
COORD_FILE_NAME carbon-monoxide_opt.xyz
COORD_FILE_FORMAT XYZ
&END TOPOLOGY
&KIND C
BASIS_SET pcseg-2
POTENTIAL ALL
&END KIND
&KIND O
BASIS_SET pcseg-2
POTENTIAL ALL
&END KIND
&END SUBSYS
&END FORCE_EVAL
```
### Time step
The time step used to propagate the wave-function is related to the frequency of interest.
Typically, one should do a trade-off between the maximal frequency that can be sampled, this gives
the time step value, and the accuracy of the sampling itself, related to the total simulation time.
Typically, the time step should be about one order of magnitude lower than the frequency of
interest. Here, we will tackle the Oxygen K-edge: the electronic transition between the Oxygen 1s
and the first available excited state.
According to Linear Response TDDFT calculation (using the XAS_TDP module, see
[this tutorial](./tddft)), the first transition occurs at 529 eV with an oscillator strength of
0.044 a.u. In our δ-kick approach, it means that the induced dipole moment should have an
oscillatory component around the frequency corresponding to 529 eV. Hence, we should have a time
step smaller than this frequency. For this tutorial, we have set the
[TIMESTEP](#CP2K_INPUT.MOTION.MD.TIMESTEP) to `[fs] 0.00078`, which leads to about 10 time steps per
period for an oscillation at 529 eV.
The total time of the simulation is then given by the number of time steps required, here 500000,
and is related to the precision of the calculation: the more total time, the better the accuracy.
This value is less clear to determine: it depends on the strength of the perturbation, the error
thresholds used for the Real Time simulation, the amplitude of the polarizability value itself, and
the overall noise of the calculation. A good way to check that the total simulation time is enough
is to compute the polarizability for the targetted frequency using either 80% of the simulation time
or 100%.
### Field properties
The field perturbation (the δ-kick) is defined by its amplitude and polarization.
The amplitude depends on the system of interest, typically $10^{-3}$ would do, and a good way to
check it is to run 2 simulations: one with once the amplitude and another with twice the amplitude.
Within the linear regime, the response of the system should have doubled as well. If this is not the
case, lower the field amplitude. For instance, by one or two orders of magnitude. Note that applying
a too-low field may lead to numerical noise. For isolated carbon monoxide, we have checked that
$10^{-3}$ is within the linear regime.
**Please note that the actual perturbation applied in CP2K is not
[DELTA_PULSE_SCALE](#CP2K_INPUT.FORCE_EVAL.DFT.REAL_TIME_PROPAGATION.DELTA_PULSE_SCALE) and depends
on the cell size. Look at the output file just before the Real Time propagation part to know the
exact perturbation amplitude used**
The polarization of the field determines which excited state is triggered: the electric field and
the transition dipole moment should be non-perpendicular. One way to know how to select the
polarization is to run a Linear-Response TDDFT calculation and to look at the transition dipole
moment decomposition along the laboratory axis (or the oscillator strength decomposition). For
instance, have a look at the first part of [this tutorial](./resonant_excitation) on Real Time TDDFT
with a time-dependent field for isolated carbon monoxide. Relying on this calculation, the
electronic transition at 529 eV is perpendicular to the CO bond. For the geometry we are using, it
means that the field polarization should be along x (or y) by setting
[DELTA_PULSE_DIRECTION](#CP2K_INPUT.FORCE_EVAL.DFT.REAL_TIME_PROPAGATION.DELTA_PULSE_DIRECTION) to
`1 0 0` and [DELTA_PULSE_SCALE](#CP2K_INPUT.FORCE_EVAL.DFT.REAL_TIME_PROPAGATION.DELTA_PULSE_SCALE)
to `0.001`.
### Note about the choice of the Gauge
For an isolated system, use the length implementation because the perturbation is applied up to all
perturbative orders by setting [PERIODIC](#CP2K_INPUT.FORCE_EVAL.DFT.REAL_TIME_PROPAGATION.PERIODIC)
to `.FALSE.`.
For condensed phase systems, use the velocity gauge instead by setting
[PERIODIC](#CP2K_INPUT.FORCE_EVAL.DFT.REAL_TIME_PROPAGATION.PERIODIC) to `.TRUE.`. In this case, the
perturbation will be applied only within the first order. We have used the length one for this
tutorial.
## Analyzing the results
To analyze the Real Time propagation simulation, we propose to go through a jupyter notebook:
[download this file](https://www.cp2k.org/_media/howto:delta_kick_analyse.zip). Note that this zip
file also contains the data needed to perform the analyses (the output of the CP2K run).

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---
tddft
delta-kick
correction_scheme
```