Note: this tutorial is still under construction

In this tutorial, we will present a simulation of resonant X-Ray excitation of an isolated carbon monoxide in real-time using a time-dependent field. On this page, you will find an overview of the method, some equations, and the CP2K input file. A longer version is available in the form of a jupyter notebook file in this zip file along with the simulated data so that you do not need to run this calculation yourself to perform the analysis. This kind of calculation is not easy to grasp: do not hesitate to have a first look before diving into the equations and details! This tutorial is connected to this article REF where you can find complementary information.

Such real-time calculation aims to promote a small number of electrons from one specific state to another using an electromagnetic field. The electronic dynamics are propagated as the field is applied, leading to an electronic excitation increasing with time if the field frequency matches the targetted electronic transition energy.

The light is treated classically using either the length or velocity gauge to interact with the electrons. The simulation starts usually from the electronic ground state, but one can also construct another starting state in CP2K. The field is applied as the wave function is propagated in real-time. This time-dependent field $\textbf{F}(t)$ has typically an envelope, $E_{\text{env}}$, and is defined as:

$$ \textbf{F}(t) = \textbf{F} E_{\text{env}}(t) \cos \left( \omega_{0} t + \phi \right). $$

Where $\textbf{F}$ is the field polarization, $\omega_{0}$ is the carrying frequency at which the field oscillates, and $\phi$ its initial phase.

The DFT equivalent of the Schrodinger equation is solved numerically time step per time step:

$$ i \frac{\text{d}}{\text{d} t} |\psi_i(t)> = \hat{H}^{KS}(t) |\psi_i(t)>. $$

Where $|\psi_i>$ is the $i^{\text{th}}$ time-dependent Molecular Orbital (MO) describing the $i^{\text{th}}$ electrons and $\hat{H}^{KS}$ the time-dependent Kohn-Sham Hamiltonian. This Hamiltonian is time-dependent because of the evolution of the electronic degrees of freedom and the field evolution. In the case of Ehrenfest dynamics, the nuclei also move in real-time.

This equation is propagated for each $i$ electron with a time step $\Delta t$: the MOs evolved collectively over time in a continuous manner. Especially, the energy evolution is continuous: the electrons cannot absorb exactly one photon at a given frequency to instantaneously go from one electronic state to another. If one applies a field resonant with a given electronic transition, the MOs involved in this transition will gradually be transformed from their initial state to the corresponding excited state. This will happen in real-time without defining any specific electronic transition before starting the simulation but only the field parameters. Yet, it is better to characterize first what electronic transition we want to address before running such Real-Time Propagation (RTP) calculation. To do so, we will use the Linear Response TDDFT scheme.

For X-Rays, we use the XAS_TDP scheme implemented in CP2K, see for instance this tutorial for more information. For isolated carbon monoxide and with our calculation parameters (PBEh functional and PCSEG2 basis set), the first available excited state for the Oxygen 1s has a resonant frequency of $\omega_{res} = \omega = 529$ ev with an associated oscillator strength of 0.044 a.u. Note that this result is also confirmed using a real-time approach, the $\delta$-kick technic.

Hence, applying an electric field with a carrying frequency of 529 ev should promote electrons from the oxygen 1s toward this excited state, noted $|\omega>$. Within the RT-TDDFT approach, it means that the Molecular Orbital corresponding to the Oxygen 1s, $|1s(t)>$, will be transformed with time as:

$$ |1s(t)> = \sqrt{\rho_{GS}(t)} |1s> + \sqrt{\rho_{Exc}(t)} |\omega>. $$

$\rho_{GS}$ is the time-dependent ground state population and $\rho_{Exc}$ the excited state one. Note that this equation leaves aside the time-dependent phase factor between these two states.

If the field is resonant with only one electronic transition and if we assume that the rest of the electrons do not evolve we can use the generalized version of the Rabi oscillation formula to get a prediction of the excited state population with time:

$$ \rho_{\text{Exc}}(t) = \sin \left( \frac{ | \mathbf{\mu}^{res} \cdot \mathbf{F}| \times A(t)}{2} \right)^2; \ \ \ A(t) = \int_{- \infty}^t E_{env}(t') dt' $$

Where $\mathbf{\mu}^{res}$ is the transition dipole moment between the ground and excited state, $\mathbf{F}$ is the polarization of the field, and $A(t)$ is the integral of the field envelope. In today's case where the amount of electron promoted is small, one can use this approximate formula:

$$ \rho_{\text{Exc}}(t) \approx \tau \left( \frac{A(t)}{2} \right)^2. $$

Where $\tau = |\mathbf{\mu}^{res} \cdot \mathbf{F}|^2 $ can be seen as an instantaneous promoting rate for the core electron upon the resonant field.

In the perturbative regime, one can expect the above-mentioned excited population prediction to be correct as the oscillator strength is provided by Linear Response. Thus, one can track in real-time these ground and excited population using projection of the time-dependent molecular orbital:

$$ \rho_{GS}(t) = |<1s | 1s(t)>|^2 ; \ \ \ \rho_{Exc}(t) = |<\omega | 1s(t)>|^2 $$

Moreover, we should also have: $\rho_{Exc}(t) = 1 - \rho_{GS}(t)$. These projections are written for the Molecular Orbital originally corresponding to the Oxygen 1s, but we can also write them for the other electrons. Yet, in the case of core X-Ray excitation, we do not expect the other electrons to be affected within the perturbative regime: for any Molecular Orbital $i$ which is not the one corresponding to the Oxygen 1s we should have:

$$ |\psi_i(t)> \approx e^{ -i E_i t} |\psi_i(t=0)>, $$

with $E_i$ the energy associated to the $i^[\text{th}} molecular orbital. Let us define the projection for all the molecular orbital either toward their ground state or the targetted excited state:

$$ \rho^i_{GS}(t) = |<\psi_i(t=0) | \psi_i(t)>|^2 ; \ \ \ \rho^i_{Exc}(t) = |<\omega | \psi_i(t)>|^2 $$

For all MOs except the ones corresponding to the Oxygen 1s we should have $\rho^i_{GS}(t) = 1$ and $\rho^i_{Exc}(t) \approx 0$. Therefore, here is the generalized formula for the ground state population and the excited state one for the whole wave-function:

$$ \rho_{GS}(t) = \sum_{i=1}^{N_e} |<\psi_i(t=0) | \psi_i(t)>|^2 ; \ \ \ \rho_{Exc}(t) = \sum_{i=1}^{N_e} |<\omega| \psi_i(t)>|^2. $$

We should have in the perturbative regime $\rho_{GS}(t)$ close to the number of electron $N_e$: $\rho_{GS}(t) = N_e - \rho_{Exc}(t)$.

Finally, if we neglect the off-resonance interaction with the field, the evolution of the energy is:

$$ \Delta E(t) = E(t) - E_{GS} = \hbar \omega \rho_{Exc}(t) $$

RTP.inp is the input file used for such simulation:

RTP.inp

@set EXC_STATE_1     LR-xasat2_1s_singlet_idx1-1.wfn
@set EXC_STATE_2     LR-xasat2_1s_singlet_idx2-1.wfn

&GLOBAL
  PROJECT RTP
  RUN_TYPE RT_PROPAGATION
  PRINT_LEVEL MEDIUM
&END GLOBAL

&MOTION
  &MD
    ENSEMBLE NVE
    STEPS 5000
    TIMESTEP [fs] 0.00078
    TEMPERATURE [K] 0.0
  &END MD
&END MOTION

&FORCE_EVAL
  METHOD QS
  &DFT
    &REAL_TIME_PROPAGATION
      MAX_ITER 100
      MAT_EXP ARNOLDI
      EPS_ITER 1.0E-11
      INITIAL_WFN SCF_WFN
      &PRINT
        &FIELD
          FILENAME =applied_field
        &END FIELD
        &PROJECTION_MO
	  REFERENCE_TYPE SCF
          REF_MO_FILE_NAME RTP-RESTART.wfn
          REF_MO_INDEX -1 
          SUM_ON_ALL_REF .FALSE.
          TD_MO_INDEX -1 
          SUM_ON_ALL_TD .FALSE.
          &PRINT
            &EACH
              MD 1
            &END EACH 
          &END PRINT
	&END PROJECTION_MO
	&PROJECTION_MO
          REFERENCE_TYPE XAS_TDP
          REF_MO_FILE_NAME ${EXC_STATE_1}
          TD_MO_INDEX -1
          SUM_ON_ALL_TD .FALSE.
          &PRINT
            &EACH
              MD 1
            &END EACH
          &END PRINT
        &END PROJECTION_MO
	&PROJECTION_MO
          REFERENCE_TYPE XAS_TDP
          REF_MO_FILE_NAME ${EXC_STATE_2}
          TD_MO_INDEX -1
          SUM_ON_ALL_TD .FALSE.
          &PRINT
            &EACH
              MD 1
            &END EACH
          &END PRINT
        &END PROJECTION_MO
      &END PRINT
    &END REAL_TIME_PROPAGATION
    &EFIELD
      ENVELOP GAUSSIAN
      ! gaussian env in fs units
      &GAUSSIAN_ENV
        SIGMA [fs] 0.3073
        T0 [fs] 1.3190
      &END GAUSSIAN_ENV
      ! in W.cm-2
      INTENSITY 4.08E+13
      PHASE 0.0
      POLARISATION 1 0 0
      ! this is 529 ev:
      WAVELENGTH 2.34374655955
    &END EFIELD
    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
      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

We will not detail all the parameters and instead focus on the one related to the real-time propagation, the field, and the time-dependent projection part.

In this simulation, we start from an SCF-optimized state which corresponds to the ground state:

INITIAL_WFN SCF_WFN

Before the Real-Time Propagation starts, a ground state calculation will thus be performed. We use a Molecular Orbital-based description of the wave function for the propagation along with the Arnoldi approach to compute the exponential:

      MAT_EXP ARNOLDI

Note that for extensive systems, the density-based method can be used for linear scaling (see the DENSITY_PROPAGATION keyword). For each time step, the AERTS algorithm is used with a convergence threshold defined by

      EPS_ITER 1.0E-11

Note that the smaller this threshold, the more iterations per time step will be needed to converge. Hence, we have set the maximal iteration number at quite high value already :

      MAX_ITER 100

The time step used is rather small since we have to describe a core-hole excitation process that takes place with a typical frequency of 529 ev. As a rule of thumb, the time step should be about tens of the field frequency used:

      TIMESTEP [fs] 0.00078

The interaction between the field and the electron is done through the length gauge by default. For periodic systems (or isolated ones), use the velocity gauge by setting the VELOCITY_GAUGE keyword to True. The time-dependent electric field is defined for both gauges in the EFIELD section. It should be noted that one can define several EFIELD sections to apply several fields within the same simulation.

The field is defined by its envelope, its intensity, its polarization along the laboratory x, y, and z-axis, its wavelength, and the original phase. Several types of field envelopes can be used. Here we use a Gaussian one with a width of $\sigma=0.3073$ fs and centered at $T0=1.3190$ fs. The intensity used is 4.08E+13 W.cm$^{-2}$. Along with a carrying frequency of 529 ev (approximately 2.34374655955 nm), it should promote about $10^{-3}$ from the Oxygen 1s to the first available excited state. See the jupyter notebook tutorial for more information about how to choose these parameters depending on your system.

    &EFIELD
      ENVELOP GAUSSIAN
      &GAUSSIAN_ENV
        SIGMA [fs] 0.3073
        T0 [fs] 1.3190
      &END GAUSSIAN_ENV
      INTENSITY 4.08E+13
      PHASE 0.0
      POLARISATION 1 0 0
      WAVELENGTH 2.34374655955
    &END EFIELD

The total number of steps, STEPS 5000, is determined so that the field envelope fits in the period of time simulated.

Time-dependent projections can be defined in the PRINT section of REAL_TIME_PROPAGATION. One can define several PROJECTION_MO which will be done consequitively and the results stored in different files.

First, let us have a look at the projection toward the ground state:

&PROJECTION_MO
   REFERENCE_TYPE SCF
   REF_MO_FILE_NAME RTP-RESTART.wfn
   REF_MO_INDEX -1 
   SUM_ON_ALL_REF .FALSE.
   TD_MO_INDEX -1 
   SUM_ON_ALL_TD .FALSE.
   &PRINT
     &EACH
       MD 1
     &END EACH 
   &END PRINT
&END PROJECTION_MO

All the time-dependent Molecular Orbitals are projected (TD_MO_INDEX = -1) and these projections are stored separately (SUM_ON_ALL_TD .FALSE.). This calculation is spin-independent so one does not have to define the spin of the MO to project. The reference to projected to is loaded from the file RTP-RESTART.wfn, which is the ground state obtained after the SCF cycles. Note that you can define a wave-function that is not the ground state as long as the wave-function description (basis set, number of electrons…) is the same as the one used for the real-time propagation. The 'type' of the reference file is the default one, REFERENCE_TYPE SCF. All the molecular orbitals available in this reference wave-function will be used as a reference for the projection (REF_MO_INDEX -1 ), and stored in separate files (SUM_ON_ALL_REF .FALSE.). The projection will be computed at each time step.

There are $N_e/2 = 7$ molecular orbitals for carbon monoxide. There are thus 7 time-dependent MOs (the $i$s) and 7 reference MO (the $j$s), so that there will be $7x7=49$ projection per time step:

$$ \rho_j^i(t) = |<\psi_j(t=0) | \psi_i(t)>|^2> = |\sum_{ab} c_j^a(t=0)^* c_i^b(t) S_{ab} |^2 $$

Where $c_j^a(t=0)$ is the $a^{\text{th}}$ atomic coefficient of the $j^{\text{th}}$ reference MO, $c_i^b(t)$ the $b^{\text{th}}$ atomic coefficient of the $i^{\text{th}}$ time-dependent MO and $S_{ab}$ the overlap matrix between atomic orbital $a$ and $b$. Using this definition, the time-dependent ground state population is:

$$ \rho_{GS} = \sum_{i,j} \rho_j^i(t) $$

Now, let us focus on the second and third projections which can be viewed as projections toward excited-states:

&PROJECTION_MO
   REFERENCE_TYPE XAS_TDP
   REF_MO_FILE_NAME ${EXC_STATE_1}
   TD_MO_INDEX -1
   SUM_ON_ALL_TD .FALSE.
   &PRINT
      &EACH
         MD 1
      &END EACH
   &END PRINT
&END PROJECTION_MO

In this case, all the time-dependent MO are involved in the projection and stored separately. This time, the reference wave function is supposed to be from an XAS_TDP calculation, see XAS_TDP%PRINT%RESTART_WFN section or the Linear Response input file in the tutorial archive. Running with the proper parameters, this XAS_TDP run saves one .wfn file per excited state: it is the $|\omega>$ presented before. Using REFERENCE_TYPE XAS_TDP, the projection uses automatically the $|\omega>$ saved in the .wfn file as the reference file:

$$ \rho_{\omega}^i(t) = |<\omega | \psi_i(t)>|^2 = |\sum_{ab} \left( c_\omega^a \right)^* c_i^b(t) S_{ab} |^2 $$

Where $c_\omega^a$ is the $a^{\text{th}}$ atomic coefficient of the excited state found in the XAS_TDP module. There will be 7 projections per time step in this case: one for each time-dependent MO. The excited state population associated with this excited state is:

$$ \rho_{\omega}(t) = \sum_i \rho_{\omega}^i(t) $$

The carbon monoxide molecule has a rotational symmetry along its CO bond. If one notes this axis $z$, then the $x$ and $y$-axis are equivalent by symmetry. It happens that the first available excited state for the Oxygen 1s is degenerate: there are two available excited states orthogonal in the $xy$-plane. That is why a third projection is requested which uses the other excited state proposed by the XAD_TDP module.

Therefore, the excited state should be understood as the sum over the two equivalent excited states:

$$ \rho_{Exc}(t) = \rho_{\omega}(t) + \rho_{\omega'}(t) $$

Where $\omega'$ stands for the other equivalent excited state, with $\omega=\omega'=529$ ev.

In this tutorial, we have presented the ingredients needed to perform a Real-Time simulation with an explicit field in order to trigger a specific electronic transition. Such approaches can be used for other frequencies without much changes in the parameters and also extended to Ehrenfest dynamics. Note that in this case, the projection toward the reference state is less trivial to define, see the PROJECTION_MO%PROPAGATE_REF keyword.

This has been done within the perturbative regime (the amount of excited electron is small compared to 1) and the jupyter notebook presents an extensive connection to the Linear Response prediction and also discusses the obtained results. Please note that applying a field with the intention to promote about one electron can bring more challenges as well as applying a field with a large amplitude.