PROMDENS: Promoted Density Approach code

PROMDENS is a Python code implementing the Promoted Density Approach (PDA) and its version for windowing (PDAW), freely available to the scientific community under the MIT license. PDA and PDAW are techniques for implicit inclusion of laser pulses in trajectory-based nonadiabatic dynamics, such as trajectory surface hopping or ab initio multiple spawning. PDA generates initial conditions (positions, momenta and excitation times) for the excited-state dynamics, while PDAW provides weights and convolution functions for the trajectories created using the vertical excitation approach. Both methods take as input ground-state positions and momenta with corresponding excitation energies and transition dipole moments (quantities readily available from absorption spectra calculation with the Nuclear Ensemble Approach). Description of PDA and PDAW, their derivation and benchmark against quantum dynamics can be found in our recent paper in JPCL.

Installation

The code is published on PyPI and can be installed with pip

pip install promdens

After installation, the code is available as a script via the promdens command. To print help, run:

promdens --help

The minimum supported Python version is 3.7. The code depends on numpy and matplotlib libraries that are automatically installed by pip. However, since pip by default installs packages into a global Python environment, it can break previously installed packages e.g. by installing an incompatible version of numpy.

Therefore, we recommend using tools like pipx or uv which install the dependencies into an isolated Python environment but still make the promdens command globally available. See the example for pipx

pip install pipx
pipx install promdens

and uv

pip install uv
uv tool install promdens

Brief outline of the code

The code makes several steps and analysis procedures before it calculates the promoted density quantities. The PROMDENS workflow is briefly summarized as:

1. The absorption spectrum is calculated using the Nuclear Ensemble Approach and decomposed into different excited states.
2. The laser field and its characteristics such as spectral intensity are calculated.
3. The code checks if the pulse fulfils Maxwell equations, i.e., is not too short.
4. The excitation times for PDA or weights for PDAW are calculated.

All the steps are commented in the verbose code output.

Input

Both PDA and PDAW require as an input excitation energies and magnitudes of transition dipole moments for each position-momentum pair in the sample of initial (ground-state) density. These quantites are usually readily available from calculations of absorption spectra using the Nuclear Ensemble Approach. The code does not need the positions and momenta itself but only their excitation energies and magnitudes of transition dipole moments, from which it selects the right position-momentum pairs and assigns them excitation times.

Structure of the input file

The input file contains excitation energies to all excited states considered and their corresponding magnitudes of the transition dipole moments for each position-momentum pair (labelled by an index number). Acceptable units are specified in the code description available through promdens -h. In the following, we provide an example of the input file for the first two excited states of protonated formaldimine:

#index    dE12 (a.u.)  |mu_12| (Debye)  dE13 (a.u.)  |mu_13| (Debye)
1         0.32479719       0.1251       0.40293672       1.351
2         0.32070472       0.2434       0.40915241       1.289
3         0.34574925       0.7532       0.38595754       1.209
4         0.33093699       0.1574       0.36679075       1.403
5         0.31860215       0.1414       0.36973886       1.377
6         0.31057768       0.0963       0.40031651       1.390
7         0.33431888       0.1511       0.40055704       1.358
8         0.31621589       0.0741       0.36644659       1.425
9         0.32905912       0.5865       0.36662982       1.277
10        0.31505412       0.2268       0.35529522       1.411

The first column always considers the index (label) of the given position-momentum pair. This index is used in the output to specify the selected position-momentum pairs. The following columns come in pairs for each excited state: the first column contains the excitation energy while the second column bears the magnitudes of the transition dipole moment. Note that the header is not mandatory in the input file. We will use this input file in the following examples and refer to it as to input_file.dat.

Usage

The code requires information about the method (PDA or PDAW, --method), the number of excited states to consider (--nstates), the number of initials conditions to be generated (--npsamples), and the characteristics of the laser pulse, such as the envelope type (Gaussian, Lorentzian, sech, etc., --envelope_type), the pulse frequency (--omega), the linear chirp parameter (--linear_chirp), and the full width at half maximum parameter (--fwhm). The units of excitation energies (--energy_unit) and transition dipole moments can be specified (--tdm_unit debye). An optional flag is --plot, which produces and saves plots of the absorption spectrum, laser pulse characteristics and the final analysis of the results.

PDA

The PDA method can be launched from a terminal with a series of flags as follows

promdens --method pda --energy_unit a.u. --tdm_unit debye --nstates 2 --fwhm 3 --omega 0.355 --npsamples 10 --envelope_type gauss input_file.dat

Using this input file input_file.dat shown above produces the following output file called pda.dat containing information about excitation times and initial excited states:

# Sampling: number of ICs = 10, number of unique ICs = 5
# Field parameters: omega = 3.55000e-01 a.u., linear_chirp = 0.00000e+00 a.u., fwhm = 3.000 fs, t0 = 0.000 fs, envelope type = 'gauss'
# index        exc. time (a.u.)   el. state     dE (a.u.)       |tdm| (a.u.)
       3        15.09731061            1       0.34574925       0.29635106
       3        25.94554064            1       0.34574925       0.29635106
       3        61.98106992            1       0.34574925       0.29635106
       4         7.38522206            2       0.36679075       0.55201877
       8       -14.27561557            2       0.36644659       0.56067480
       9       155.72500917            2       0.36662982       0.50244331
       9       -44.31379959            2       0.36662982       0.50244331
      10        94.19109952            2       0.35529522       0.55516642
      10        -9.13220842            2       0.35529522       0.55516642
      10        31.75086044            2       0.35529522       0.55516642

Inspecting this output file shows that the code generated 10 initial conditions accounting for the effect of the laser pulse, yet only 5 unique ground-state samples (pairs of nuclear positions and momenta) were used: indexes 3, 9, and 10 were selected more than once. The initial conditions are also spread over both excited states. The user should then run only 5 nonadiabatic simulations: initiating the nuclear position-momentum pair with index 3 in the first excited state and the nuclear position-momentum pairs with indexes 4, 8, 9, and 10 in the second excited state. The trajectories 3, 9 and 10 are then accounted for in the trajectory ensemble multiple times, but always with different excitation times. This multiple accounting of the same trajectory mimics the temporal spread of the laser pulse. A larger number of samples is recommended to get smooth curves fully accounting for the pulse duration.

Let us now briefly comment on the generation of initial conditions with PDA and the processing of the data. The code selects npsamples initial conditions, yet some of them will be generally taken multiple times. Thus, the number of samples used for nonadiabatic dynamics will be smaller than npsamples. However, the restriction usually lies in the maximum number of trajectories calculated, i.e., the number of unique indexes (position-momentum pairs) selected. Imagine we want to run 100 trajectories. We run the code with --npsamples 1000 generating 1000 initial conditions, which will contain for example 250 unique indexes. Since we cannot run 250 trajectories, we need to decrease npsamples and optimize the value until we obtain the required 100 unique position-momentum pairs.

PDAW

If the same command would be used with PDAW instead of PDA (--method pdaw), the output file pdaw.dat would look like

# Convolution: I(t) = exp(-4*ln(2)*(t-t0)^2/fwhm^2)
# Parameters:  fwhm = 3.000 fs, t0 = 0.000 fs
# index        weight S1        weight S2
       1      1.78475e-05      9.66345e-07
       2      1.56842e-05      2.59858e-08
       3      6.31027e-02      1.29205e-03
       4      1.79107e-04      1.62817e-01
       5      2.31817e-06      1.01665e-01
       6      2.96548e-08      3.90152e-06
       7      3.81650e-04      3.33694e-06
       8      2.36147e-07      1.75628e-01
       9      1.47188e-03      1.37747e-01
      10      1.33347e-06      3.55670e-01

The code provides the pulse intensity $I$ and weights $w_i$ necessary for calculating any time-dependent observable $\mathcal{O}(t)$ based on equation

$$\mathcal{O}(t) = \int_{-\infty}^{\infty} \bar{I}(t-t^{\prime}) \frac{\sum_i w_i \mathcal{O}_i(t^\prime)}{\sum_i w_i} \mathrm{d} t^{\prime} , $$

where $\bar{I}$ is normalized pulse intensity and $\mathcal{O}_i$ is the observable calculated for trajectory $i$. Note that the intensity should be normalized before being used in convolution. If only a restricted amount of trajectories can be calculated, the user should choose the indexes and initial excited states corresponding to the largest weights in the file. For example, if we could run only 10 trajectories of protonated formaldimin, we would run ground-state position-momentum pairs with indexes 3, 4, 7, and 9 starting in $S_1$ and indexes 3, 4, 5, 8, 9, and 10 starting in $S_2$.

If the user selects option --plot, the code will produce a series of plots analyzing the provided data and calculated results, e.g. the absorption spectrum calculated with the nuclear ensemble method, the pulse spectrum or the Wigner pulse transform.

Laser field representation

The laser field in the code is represented as an envelope $\varepsilon(t)$ multiplied by $\cos\left[(\omega_0+\beta t) t\right]$ where $\omega_0$ is the central pulse frequency (omega) and $\gamma$ is the linear chirp parameter linear_chirp. The code allow to select several pulse envelopes which are defined through $t_0$ (t0) and the full width at half maximum (FWHM) parameter $\tau$ (fwhm). Note that the FWHM parameter is defined for the intensity of the pulse, i.e., the square of the pulse envelope.

Guassian envelope (gauss):

$$\varepsilon(t)=\exp\left[-2\ln2\left( \frac{t - t_0}{\tau} \right)^2\right]$$

Lorentzian envelope (lorentz):

$$\varepsilon(t)=\left[1 + \frac{4}{1 + \sqrt{2}}\left(\frac{t - t_0}{\tau}\right)^2\right]^{-1}$$

Hyperbolic secant envelope (sech):

$$\varepsilon(t)=\mathrm{sech}\left[2\ln(1 + \sqrt{2})\frac{t - t_0}{\tau}\right]$$

Sinusoidal envelope (sin):

$$\varepsilon(t)=\sin\left[\pi\frac{t - t_0 + \tau}{2\tau}\right] \quad \quad \text{ if } \quad t \in [t_0 - \tau, t_0 + \tau]$$ $$0 \quad \text{elsewhere}$$

Sinusoidal squared envelope (sin2):

$$\varepsilon(t)=\sin^2\left[\pi\frac{t - t_0 + T}{2T}\right] \quad \quad \text{ if } \quad t \in [t_0 - T, t_0 + T]$$ $$0 \quad \quad \text{elsewhere}$$

where

$$T = \frac{1}{2-4/\pi\arcsin(2^{-1/4})} \tau = 1.373412575\tau$$

Characteristics of the envelopes

The following figure shows the envelope $\varepsilon(t)$, the pulse intensity $I(t)\propto\varepsilon^2(t)$, the pulse spectrum $|\varepsilon(\omega)|$ (Fourier transform of $\varepsilon(t)$) and the pulse spectral intensity $S(\omega)\propto\varepsilon^2(\omega)$.

The pulse envelope Wigner transforms for the presented pulses are presented in the next figure in standard and logarithmic scales. The red colour represents the negative regions of the Wigner transform. The only envelope free of negative parts is the Gaussian envelope. However, the negative regions in the remaining envelopes are located at edges of the distribution with negligible intensities. The code allows to treat the negative regions by either ignoring them (--neg_handling ignore) or taking the absolute values (--neg_handling abs). As default, the code issues error to inform the user that the negative probabilities must be handled.

Notes on some keywords

While most of the keywords are explained in the code description available by promdens -h, some of them require some further information. Here, we summarize these keywords and provide recommendations for their usage.

• --preselect: This keyword preselects provided samples using the pulse spectral intensity and discards samples that far from resonance. We recommend this keyword in case the pulse spectrum targets only a part of the absorption spectrum, or when a lot of states are calculated. It saves a significant amount of computational time in such cases as it prevents the code from testing samples with negligible excitation probability. However, final production calculations should be always done without preselection
• --random_seed: The PDA algorithm generates random numbers and compares them with the excitation probability in order to generate excitation times, which makes it stochastic (note that PDAW is fully deterministic and unaffected by this choice). We recommend using the default option and generating a new random seed for each calculation. However, if reproducibility is required (as in our test suite), the random seed can be set.
• --file_type: Currently the only input file type available is described in section Input, no flexibility is available. In the future, we intend to implement other input file types based on standard output files from widely used code such as SHARC or NEWTON-X. If the users would find a use of any specific input file type which is not currently implemented, feel free to contact the developers for implementation.

Technical details

The following section contains information about some technical aspects of the PROMDENS implementation.

Evaluation of the Wigner pulse transform

The PDA and PDAW are based on the Wigner pulse transform which requires evaluating the Wigner integral

$$\mathcal{W}_E(t^\prime,\omega)=\int _{-\infty}^{\infty} E\left(t^\prime+\frac{s}{2}\right) E^*\left(t^\prime-\frac{s}{2}\right) \mathrm{e}^{-i\omega s} \mathrm{d} s$$

where the frequency is given by the excitation energy ($\omega=\Delta E/\hbar$) and $t^\prime$ is the excitation time. To simplify the integral evaluation, we implemented a simpler Wigner pulse envelope transform (see the SI of our article for more details):

$$\mathcal{W}_\varepsilon(t^\prime,\omega) = \int _{-\infty}^{\infty} \varepsilon\left(t^\prime+\frac{s}{2}\right) \varepsilon\left(t^\prime-\frac{s}{2}\right) \mathrm{e}^{-i\omega s} \mathrm{d}s$$

where $\varepsilon$ is the pulse envelope. The relation between the two Wigner transforms reads:

$$\mathcal{W}_ {E}(t^\prime,\omega) = \mathcal{W}_\varepsilon(t^\prime,\omega-(\omega_0+2\beta t^\prime))$$

where $\omega_0+2\beta t$ is the immediate frequency for pulse described as $E(t) = E_0 \varepsilon(t)\cos[(\omega_0+\beta t)t]$.

While the gauss and sin envelope Wigner transforms are calculated analytically, lorentz, sin2 and sech envelopes are calculated numerically with the trapezoid rule since we so far have not derived analytic formulas.

Analytic formulas for the Wigner pulse envelope transform

The analytic formulas for the Wigner pulse envelope transform, $\mathcal{W}_\varepsilon$, are available only for the Gaussian and Sinusoidal envelopes:

Gaussian envelope (gauss):

$$\mathcal{W}_\varepsilon(t^\prime,\omega)=\tau\sqrt{\frac{\pi}{\ln2}}16^{-\frac{(t^\prime - t_0)^2}{\tau^2}}\exp\left(-\frac{\tau^2\omega^2}{\ln16}\right)$$

Sinusoidal envelope (sin):

$\mathcal{W}_\varepsilon(t^\prime,\omega)=$

$$\pi\frac{-2\tau\omega\cos(2(t^\prime - t_0 + \tau)\omega)\sin(\pi(t^\prime - t_0)/\tau) +\pi\cos(\pi(t^\prime - t_0)/\tau)\sin(2(t^\prime - t_0 + \tau)\omega))}{\omega(\pi^2 - 4\tau^2\omega^2)} \quad \quad \text{if} \quad t^\prime < t_0 \quad \text{and} \quad t^\prime > t_0 - \tau $$

$$\pi\frac{2\tau\omega\cos(2(-t^\prime + t_0 + \tau)\omega)\sin(\pi(t^\prime - t_0)/\tau) +\pi\cos(\pi(t^\prime - t_0)/\tau)\sin(2(-t^\prime + t_0 + \tau)\omega))}{\omega(\pi^2 - 4\tau^2\omega^2)} \quad \quad \text{if} \quad t^\prime \ge t_0 \quad \text{and} \quad t^\prime < t_0 - \tau $$

$$0 \quad \quad \text{elsewhere}$$

Numerical evaluation of the Wigner pulse envelope transform

For lorentz, sin2 and sech, the analytic formulas are currently not available. Instead, we solve the integral numerically. Using the symmetry of the integrand, it can be shown that

$$\mathcal{W}_ \varepsilon(t^\prime,\omega) = 2\int_{0}^{\infty} \varepsilon\left(t^\prime+\frac{s}{2}\right) \varepsilon\left(t^\prime-\frac{s}{2}\right) \cos(\omega s)\mathrm{d} s$$

which simplifies the integral evaluation since the lower integral limit is 0 instead of $-\infty$ and also allows to use only real numbers instead of complex numbers due to the complex exponential. The speed up si rougly four times. In practice, the integral is evaluated using the trapezoidal rule with adaptive upper limit and integration step set according to the frequency and envelope.