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improved documentation RTP with delta kick for spectra calculations (#3529)

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docs/methods/properties/x-ray/delta-kick.md
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@ -1,7 +1,7 @@
# X-Ray Absorption from δ-Kick
# X-Ray Absorption from RTP and $\delta$-Kick perturbation
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.
This tutorial shows how to run a Real-Time Time-Dependent DFT calculation using the so-called
$\delta$-kick approach to compute absorption electronic spectra.
For the corresponding Linear-Response approach, read the tutorial [the X-Ray frequencies](./tddft).
@ -33,20 +33,21 @@ This quantity involves a dot product between the field vector and the polarizabi
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.
oscillating at a specific frequency $\omega$ and then the transition dipole moment associated with
such electronic transition. This transition dipole moment is then used to derive 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.
at the beginning of the simulation by providing an istantaneous pulse containing all the frequencies
The perturbed electronic wavefunction is then propagated and the fluctuations of the induced time
dependent dipole moment are recorded . The polarizability tensor at any frequency is finally derived
from the induced dipole moment.
### The δ-kic
### The $delta$-kick perturbation
To apply the δ-kck, i.e., an intense electric field in a very sort time, the electronic structure is
first obtained at the ground state, for instance using DFT, and then perturbed with an instantaneous
electric field:
To apply the $delta$-kick, i.e., an intense electric field in a very sort time, the electronic
structure is first obtained at the ground state, for instance using DFT, and then perturbed with an
instantaneous electric field:
$$
F(t) = F^0 \delta(t)
@ -54,9 +55,9 @@ $$
The same result is obtained by applying a constant field with a very narrow Gaussian envelope. This
field perturbes the ground state wave-function at $t=0^-$. Then, this excited wave-function is
propagated in real time by numerically integrating the time dependent Schrodinger equation.
propagated in real time by numerically integrating the time dependent Schroedinger equation.
The electric δ-kick can be written in the frequency domain:
Theinstantaneaous field can be written in the frequency domain as
$$
F^\omega = \frac{F^0}{2 \pi} \int_{-\infty}^{+ \infty} \delta(t) e^{i \omega t} dt = \frac{F^0}{2 \pi} e^{i \omega \times 0} = \frac{F^0}{2 \pi}
@ -64,11 +65,10 @@ $$
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
described to be the ground state one plus all possible excited states. The total dipole
moment\
results from the superposition of all possible oscillations related to all the excited
states. This complex behavior in the time domain becomes simple in the frequency domain, since we
know the amplitude of the perturbation applied at each frequency:
described to be the ground state one plus all possible excited states. The total dipole moment
results from the superposition of all possible oscillations related to all the excited states. This
complex behavior in the time domain becomes simple in the frequency domain, since we know the
amplitude of the perturbation applied at each frequency:
$$
\mu^\omega = \frac{1}{2 \pi} \alpha(\omega, \omega) F^0
@ -100,8 +100,7 @@ pure real off resonances.
From one Real-Time propagation, one can obtain three components of the polarizability tensor. For
instance, applying a $\delta$-kick along $x$ provides the components $\alpha_{xx}$, $\alpha_{yx}$
and $\alpha_{zx}$, corresponding to the Fourier transform in $x$, $y$ and $z$, respectively.
Therefore, to get the full polarizability tensor, one should run 3 Real-Time propagations each
perturbing along one of the three Cartesian axis.
Therefore, to get the full polarizability tensor, three RTP runs are needed, one per Cartesian axis.
To compare with experiments, one often assumes that the system is averaged over all possible
orientations in space. Then, the absorption spectrum $I(\omega)$ is proportional to:
@ -112,14 +111,13 @@ $$
## CP2K Input
In this tutorial, we will study the response of a carbon-monoxide in the gas phase obtained by
applying the $\delta$-kick along its perpendicular direction. We will focus on the X-Ray range, but
in principle, other ranges of frequency can be sampled using the same approach.
The following example is to simulate the response of carbon-monoxide in the gas phase when applying
the $\delta$-kick along the perpendicular to the CO bond. The analysis is done within the X-Ray
range, but in principle, any other range of frequency can be sampled using the same approach,
providing that fime step and length of the propagation are opportunely adjusted.
The input file RTP.inp presents an example to propagate the electronic density of an isolated
carbon-monoxide molecule, upon the application of a predefined $\delta$-kick. The simulation is
performed at the DFT/PBEh level of theory. For the expansion of the density the all-electron PCSEG-2
basis sets are used.
The ifollowing input file `RTP.inp` is for a simulation at the DFT/PBEh level of theory, it uses the
GAPW approah and the density is expandedn in the all-electron PCSEG-2 basis sets.
```none
&GLOBAL
@ -231,25 +229,24 @@ corresponding to the maximum frequency to be resolved. For core excitation from
K-edge is around 530 eV, corresponding to a period of 0.0078 fs.
According to Linear Response TDDFT calculations (using the XAS_TDP module, see
[this tutorial](./tddft)), the first transition occurs at 529 eV with an oscillator strength of
[this tutorial](./tddft)), the first transition occurs at 529 eV, with an oscillator strength of
0.044 a.u. For the $\delta$-kick approach, this means that the induced dipole moment should have an
oscillatory component around the frequency corresponding to 529 eV. Hence, the
[TIMESTEP](#CP2K_INPUT.MOTION.MD.TIMESTEP) is set ten times smaller than the maximum frequency,
i.e., to `[fs] 0.00078`, which allows the sampling of the fasted oscillation by means of at 10
propagation steps.
The total time of the simulation is then given by the number of time steps, here 500000, and it
affects the smoothness and sharpness of the resulting spectrum. Longer simulation times give a
better resolution of the spectral peaks, i.e., a more accurate determination of the resonances.
There is no always valid rule to decide how long the propagation needs to be extended, because the
fluctuations might be strongly system dependent. It is always a good idea to check whether the
obtained spectrum has converged with respect to further extension of the sampling.
The total time of the simulation is then determined by the maximum number of time steps, here
500000\. The longer is the sampling, the smoother and sharper is the resulting spectrum. Longer
simulation times give a better resolution of the spectral peaks, i.e., a more accurate determination
of the resonances. There is no always valid rule to decide how long the propagation needs to be
extended, because the fluctuations might be strongly system dependent. It is always a good idea to
check whether the obtained spectrum has converged with respect to further extension of the sampling.
### Field properties
The field perturbation (the $\delta$-kick) is defined by its amplitude and polarization.
The amplitude depends on the system of interest, a typical value is $10^{-3}$. Also in this case a
The field perturbation (the $\delta$-kick) is defined by its amplitude and polarization. The
amplitude depends on the system of interest, a typical value is $10^{-3}$. Also in this case a
convergence check by running more than one propagation at different amplitudes is the best way to
asses the parameter. Within the linear regime, the response of the system should double by twice the
field's amplitude. Note that when applying a too-low field, numerical noise might become dominant.
@ -257,29 +254,24 @@ For isolated CO, $10^{-3}$ is a good value.
**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. The corresponding amplitude value is written in the output file just before, when
RTP is initialized**
on the cell size. The corresponding amplitude value is written in the output file, as part of the
RTP initialization**
The field's polarization determines what excited states are most probably triggered since the
electric field and the transition dipole moment should be non-perpendicular. For the present
example, we know that 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 $y$) , i.e.,
The field's polarization determines what excited states are most probably triggered, according to
the selection rules. For the present example, we know that 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 $y$) , i.e.,
[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 gauge implementation because the perturbation is applied up
to all perturbative orders by setting
For an isolated system, the length gauge form coupling coupling the position and the electric field
operators can be employed 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
For condensed phase systems, the velocity gauge form, implying a gauge transformation involving the
vector potential is instead required, and in CP2K it is activated 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.
## Analyzing the results
The a jupyter notebook:
[download this file](https://www.cp2k.org/_media/howto:delta_kick_analyse.zip). provides some tools
to analyse and verify the results obtained by running this example.