Real-Time Bethe-Salpeter Propagation

Instead of solving the Casida equation in the linear response regime, an explicit time-integration of the equation of motion of electrons can be carried out to determine the excitation frequencies. In the real-time Bethe-Salpeter propagation (RTBSE) method, the equation of motion is the von Neumann equation for the single particle density matrix \(\hat{\rho}\) with an effective Hamiltonian \(\hat{H}\)

\[ \frac{\mathrm{d} \hat{\rho}}{\mathrm{d} t} = -\mathrm{i} [\hat{H}(t), \hat{\rho}(t)] \]

The accuracy of such method is mainly determined by the choice of interaction model in the effective Hamiltonian. Instead of using TDDFT functionals, the COHSEX approximation to the self-energy is employed to calculate the time dependent behaviour of the density matrix [Attaccalite2011]. This requires a previous determination of the screened Coulomb potential, done via the bandstructure GW calculation.

The equation of motion is solved in steps

\[ \hat{\rho} (t + \Delta t) = \mathrm{e} ^ {- i \hat{H} (t+\Delta t) \Delta t/2} \mathrm{e} ^ {-i \hat{H}(t) \Delta t/2} \hat{\rho} (t) \mathrm{e} ^ {i \hat{H}(t) \Delta t/2} \mathrm{e} ^ {i \hat{H} (t + \Delta t) \Delta t/2}\]

which is called the enforced time reversal scheme[Castro2004]. The effective Hamiltonian is given as

\[ \hat{H}(t) = \hat{h}^{G0W0} + \hat{U} (t) + \hat{V}^{\mathrm{Hartree}} [\hat{\rho}(t)] - \hat{V}^{\mathrm{Hartree}} [\hat{\rho}_0] + \hat{\Sigma}^{\mathrm{COHSEX}}[\hat{\rho}(t)] - \hat{\Sigma}^{\mathrm{COHSEX}}[\hat{\rho}_0] \]

where \(\hat{\rho}_0\) is the density matrix determined from the molecular orbitals used in GW and \(\hat{U}(t)\) is the external applied field.

Excitation scheme

Without the external field \(\hat{U}(t)\), the density matrix only rotates in phase but does not produce any measurable dynamics. The excitation of the dynamics can be done either by a real time pulse (i.e. at each point, \(\hat{U}(t)\) follows form due to some finite time dependent field \(\vec{E}(t)\)) or by an infinitely sharp delta pulse, which we can understand as the limit of \(\vec{E}(t) \to I \vec{e} \delta(t)\), where \(I\) is the delta pulse intensity and \(\vec{e}\) its direction.

Observables

The dynamics can be traced through time with electric dipole moment associated with the density matrix

\[ \mu_i(t) = \mathrm{Tr} (\hat{\rho}(t) (\hat{x}_i - x_{i,\mathrm{CC}})) \: , \]

where \(x_{i,\mathrm{CC}}\) are the coordinates of center of molecular charge and \(\hat{x}_i\) is the position operator.

The electric polarizability (which is related to the photon absorption spectrum) is then determined as

\[ \alpha_{ij} (\omega) = \frac{\mu_i(\omega)}{E_j(\omega)} \]

where we Fourier transformed to the frequency domain. In order to stabilise the Fourier transform of infinitely oscillating dipole moments, we introduce a damping factor \(\gamma\)[Müller2020]

\[ \mu_i(\omega) = \int _ 0 ^ T \mathrm{d}t \mathrm{e}^{-\gamma t} \mathrm{e} ^ {i \omega t} \mu_i(t) = \int _ 0 ^ T \mathrm{d}t \mathrm{e}^{i(\omega + i \gamma) t} \mu_i (t) \]

One can easily verify that for real FT of the applied field, this leads Lorentzian peaks at the frequencies of the oscillations of the moments present in the imaginary part of the corresponding polarizability element.

Running the Propagation

To run the RTBSE propagation, include the REAL_TIME_PROPAGATION section in the input file and set the RTP_METHOD to RTBSE - otherwise, the standard TDDFT propagation is employed.

Furthermore, the TIMESTEP and STEPS influence the size of each timestep and the total time of propagation. From the properties of the Fourier transform, one can determine that smaller TIMESTEP increases the maximum energy \(\omega\) that is captured by the transform, while larger total propagation time (influenced by STEPS) leads to a better energy resolution (smaller \(\Delta \omega\)).

For gas phase/isotropic calculation of polarizability, one needs to run 3 calculations to determine the trace of the polarizability tensor.

ETRS Precision

The precision of the self-consistency in the ETRS loop is controlled by the EPS_ITER keyword. Smaller threshold (larger precision) lead to more stable propagation, but might require smaller timestep/more self-consistent iterations.

MAX_ITER keyword is used to determine the maximum number of self-consistent iterations for a single time step before the cycle is broken and non-convergence is reported.

If the propagation is converging poorly (>50 ETRS iterations), smaller TIMESTEP may stabilize the propagation. A typical setup prints a status after each successful ETRS iteration, similar to the following

 RTBSE| Simulation step         Convergence     Electron number  ETRS Iterations
 RTBSE|               0     0.55891101E-008     0.16000000E+002                5
 RTBSE| Simulation step         Convergence     Electron number  ETRS Iterations
 RTBSE|               1     0.31847656E-008     0.16000000E+002                5
 RTBSE| Simulation step         Convergence     Electron number  ETRS Iterations
 RTBSE|               2     0.38793291E-008     0.16000000E+002                5

Exponential Method

The method used for the exponentiation of the Hamiltonian is set in the MAT_EXP keyword, with the following methods implemented for both TDDFT and RTBSE

BCH - calculates the effect of matrix exponential by series of commutators using Baker-Campbell-Hausdorff expansion

and the following methods implemented only for RTBSE

EXACT - Diagonalizes the instantaneous Hamiltonian to determine the exponential exactly

For inexact methods, a threshold for the cutoff of exponential series is provided by the EXP_ACCURACY keyword. For these, the MAX_ITER keyword also sets the maximum number of iterations before the program is stopped and non-convergence is reported.

Use RTP_METHOD to start the calculation by setting it to TDAGW.

Excitation Method

The real time pulse can be specified in the EFIELD section.

If delta pulse is required instead, use APPLY_DELTA_PULSE with DELTA_PULSE_DIRECTION used for defining the \(\vec{e}\) vector and DELTA_PULSE_SCALE setting the \(I\) scale of the delta pulse (in atomic units). Note that the definition of the vector is different from the definition used in the TDDFT method.

The actual value of \(I \vec{e}\) is printed out in atomic units, as well as the absolute value of the maximum element difference between the density matrix before and after the application of the delta pulse - so called metric difference after delta kick.

 RTBSE| Applying delta puls
 RTBSE| Delta pulse elements (a.u.) :   -0.1000E-003  -0.0000E+000  -0.0000E+000
 RTBSE| Metric difference after delta kick                       0.61399576E-004

If this metric difference is approaching 1.0, the ETRS cycle might have trouble converging - we recommend reducing the DELTA_PULSE_SCALE.

Printing observables

The code is so far optimised for printing the polarizability elements, which are linked to the absorption spectrum. The printing of all available properties is controlled in the PRINT section - the ones relevant for RTBSE propagation are listed here

DENSITY_MATRIX - Prints the elements of the density matrix in the MO basis into a file at every timestep
FIELD - Prints the elements of the electric field applied at every time step
MOMENTS - Prints the electric dipole moment elements reported at every time step
MOMENTS_FT - Prints the Fourier transform of the dipole moment elements time series
POLARIZABILITY - Prints an element of the Fourier transform of polarizability.
RESTART - Controls the name of the restart file

When RESTART, MOMENTS and FIELD are saved into files, one can continue running the calculation in the same directory for longer time without rerunning the already calculated time steps. Note that total length of the propagation time controls the energy/frequency precision, while timestep size controls the energy/frequency range.

Example Input

A typical input file which runs the RTBSE propagation will have the REAL_TIME_PROPAGATION section similar to this one

&REAL_TIME_PROPAGATION
    RTP_METHOD RTBSE ! Start the RTBSE method
    EPS_ITER 1.0E-8 ! Check convergence
    MAT_EXP BCH
    EXP_ACCURACY 1.0E-14 ! Less than EPS_ITER
    INITIAL_WFN RT_RESTART
    APPLY_DELTA_PULSE
    DELTA_PULSE_DIRECTION 1 0 0
    DELTA_PULSE_SCALE 0.0001 ! Small
    &PRINT
        &MOMENTS
            FILENAME MOMENTS
        &END MOMENTS
        &MOMENTS_FT
            FILENAME MOMENTS-FT
            DAMPING 0.1 ! Exponential damping
            START_TIME ! Fourier transform offset
        &END MOMENTS_FT
        &FIELD
            FILENAME FIELD
        &END FIELD
        &POLARIZABILITY
            FILENAME POLARIZABILITY
            ELEMENT 1 1
        &END POLARIZABILITY
    &END PRINT
&END REAL_TIME_PROPAGATION

A complete example input file is available in the cp2k-examples repository.