TEPID: Time-Dependent Photoionization Code

• TEPID is a time-dependent photoionization code that models the evolution of ionized gases and molecular systems under variable radiation.
• It employs advanced numerical methods—including implicit solvers, domain decomposition, and surface flux techniques—to solve coupled quantum and kinetic equations.
• The framework is applied to diverse scenarios from AGN outflows and GRB environments to ultrafast pump-probe experiments, despite challenges in computational cost and atomic data completeness.

The Time-Dependent Photoionization Code (TEPID) is a computational framework for modeling the evolution of ionization, thermal, and radiative properties of gas exposed to time-variable ionizing radiation. Such codes are indispensable for interpreting observations in astrophysics and ultrafast atomic/molecular physics where non-equilibrium effects, finite equilibration timescales, and strong field-induced dynamics invalidate steady-state assumptions. TEPID employs rigorous, time-dependent approaches—spanning from the full quantum mechanical time-dependent Schrödinger equation for atomic/molecular systems to coupled kinetic and radiative transfer equations for astrophysical plasmas—to accurately track the interplay of photoionization, recombination, heating/cooling, and radiative transport.

1. Fundamental Principles and Key Equations

The core of TEPID lies in the numerical solution of coupled, time-dependent equations that govern both the microscopic (quantum mechanical) and macroscopic (plasma or gas) states. At the atomic/molecular level, TEPID implements the time-dependent Schrödinger equation (TDSE):

$$i\hbar \frac{\partial \Psi(\mathbf{r},t)}{\partial t} = \hat{H}(t)\Psi(\mathbf{r},t)$$

where the Hamiltonian includes the electron kinetic energy, electron-ion or electron-nuclear interaction, and coupling to external (e.g., laser or astrophysical) fields.

For astrophysical/kinetic applications, TEPID solves a hierarchical system comprising:

• Ionization rate equations:

$$\frac{dn^i_X}{dt} = - [F^i_X + C^i_X n_e + \alpha_{rec}(X^i,T) n_e] n^i_X + (\text{ionization/recombination from adjacent stages})$$

• Energy (thermal) equation:

$$\frac{dT}{dt} = \sum_{X,i} [H(X^i) - \Lambda(X^i)\, n_e] Z_X n^i_X + \Theta(\mathcal{F}_{ion}, T)$$

where $H(X^i)$ and $\Lambda(X^i)$ are heating and cooling rates for ion $i$ of element X, $Z_X$ is abundance, $n^i_X$ is ion number fraction, and $\Theta$ includes external energy exchange.

• Radiative transfer equation (for non-local thermodynamic equilibrium systems):

$\frac{dI_\nu}{ds} = -\kappa_\nu I_\nu + j_\nu$

for specific intensity $I_\nu$, opacity $\kappa_\nu$, and emissivity $j_\nu$.

An essential parameter in time-dependent photoionization is the ionization parameter (or $\xi$), defined as: $\xi = \frac{L}{n_H r^2}$ or

$U = \frac{Q_{ion}}{4\pi R^2 n_H}$

for number or energy-based formulations, with $L$ the ionizing luminosity, $Q_{ion}$ the ionizing photon rate, $n_H$ the hydrogen number density, and $R$ the distance from the ionizing source.

2. Quantum Mechanical Time-Dependent Photoionization

For single atoms and small molecules under strong, rapid fields (e.g., laser pulses), TEPID uses several specialized quantum algorithms:

• Time-dependent surface flux (t-SURFF): Computes photoelectron spectra by recording the flux through a surface at $r = R_c$, employing minimal simulation volumes and absorbing outgoing waves via infinite-range exterior complex scaling (irECS). The extracted spectral amplitude $b(\mathbf{k})$ connects surface flux to measurable observables:

$$e^{iT^2/2} b(\mathbf{k}) = i \int_0^T dt\, \langle \chi_{\mathbf{k}}(t) | [H_c(t), \theta(R_c)] | \Psi_s(t) \rangle$$

• Time-dependent Configuration Interaction (TD-CI): For multi-electron systems, TEPID can invoke time-dependent restricted active space CI (TD-RASCI) or TD-CIS (singles) methods—partitioning the orbital basis, restricting electron excitations, and solving the TDSE on this reduced manifold.
• Hybrid Coupled Channels (haCC): Combines quantum chemical electronic structure for core/ionic states with explicit time propagation for “active” electrons, with anti-symmetrized bases and tSURFF for photoelectron extraction.
• Feshbach Partitioning: Expands the wavefunction in terms of bound/resonant ($\mathcal{Q}$) and continuum ($\mathcal{P}$) subspaces, capturing time-resolved resonant processes including autoionization and Fano profiles.
• Photoionization Time Delay: Through attosecond pump-probe or streaking (e.g., RABBITT), TEPID can extract photoemission delays, including Eisenbud-Wigner-Smith (EWS) delays and Coulomb-laser coupling (CLC).

3. Non-Equilibrium Astrophysical and Plasma Modeling

TEPID’s macroscopic modules address ionized gases and plasmas responding to variable radiation:

• Time-dependent ionization/energy balance: Coupled ODEs for ion populations and temperature, including photoionization, collisional ionization, radiative and dielectronic recombination, and energy exchange processes (e.g., Compton, photoelectric, collisional de-excitation).
• Adaptive time-stepping: Adaptive binning and implicit solvers address wide dynamical timescales and stiffness due to rapid fronts (e.g., for AGN or GRB environments).
• Radiative transfer: Implements one-stream or multi-stream (ray tracing) RT equations, self-consistently coupling local ionization and temperature evolution with the propagation of the flux.
• Astrophysical scenarios: Applications span variable AGN outflows, GRB circumburst media, H II regions, expanding superwinds, and cosmological UV backgrounds.

In such non-equilibrium settings, time-averaged states differ markedly from steady-state solutions, with phenomena such as over-ionization, broadened ionization fronts, and asynchronous behavior across the medium.

4. Algorithms, Numerical Strategies, and Computational Aspects

TEPID uses advanced algorithms to ensure accurate and efficient evolution:

• Implicit methods: Given the stiffness of the coupled ODEs, implicit or semi-implicit integration schemes are standard, with nonlinear equations (e.g., thermal) solved via root-finding (secant method).
• Domain decomposition: For higher efficiency, regions can evolve asynchronously—regions near ionization fronts use small timesteps, equilibrium regions advance with longer steps.
• Minimal simulation volumes and absorbing boundaries: Surface flux approaches and complex scaling (e.g., irECS) enable compact grids without introducing reflectivity artifacts or loss of accuracy.
• Interpolation of atomic/microphysics: Gridding of heating/cooling, force multiplier, and rate coefficients (e.g., from XSTAR or MAIHEM/Cloudy calculations) supports rapid lookup and consistency during hydrodynamic and radiative simulations.

5. Applications and Phenomenological Insights

TEPID enables exploration and quantitative prediction in:

• Photoelectron momentum and energy spectra: Full angular and energy-resolved distributions for atomic photoionization (including strong-field, multiphoton, above-threshold ionization, and nonlinear channels).
• Time-resolved spectroscopy: Modeling transient absorption/emission and diagnostic lines (e.g., Fe XIX in AGN, O VI/C IV in starburst outflows) in non-equilibrium regimes, providing constraints on density and location.
• Ionization/thermal front dynamics: Propagation of supersonic ionization/thermal fronts in nebulae or AGN/GRB-irradiated regions, including pressure imbalances and implications for shocks, fragmentation, and feedback.
• Cosmic/galactic scale ionization fluctuations: Time-dependent cosmological UV backgrounds (e.g., HI Lyα forest fluctuations), including finite source lifetimes and their imprint on large-scale power spectra and BAO feature positions.
• Feedback and wind modeling in AGN: Hydrodynamic simulations incorporating tabulated or coupled time-dependent photoionization microphysics to self-consistently model wind launching, acceleration, and impact on disc accretion.

6. Significance, Limitations, and Outlook

The adoption of time-dependent photoionization codes, such as TEPID, is critical when the timescale of ionizing source variability approaches or undercuts the local equilibration time of the gas. These tools produce time-resolved ionization and temperature diagnostics that cannot be captured by steady-state models, thereby enabling unbiased density and distance determinations, breaking degeneracies intrinsic to equilibrium modeling, and revealing the true response of ionized matter in both laboratory and astrophysical settings.

TEPID’s modularity allows straightforward extensions—for example, inclusion of more elaborate atomic/molecular/multielectron physics, spatially resolved radiation transfer, and integration with hydrodynamics/feedback loops. It paves the way for next-generation spectroscopic inference from high-resolution missions (e.g., XRISM, Athena), for tomographic mapping of the ISM/IGM, and for precise interpretation of ultrafast pump-probe photoionization experiments.

Limitations include the computational cost for full multidimensional, multi-electron, or radiation-coupled flows, and the level of atomic data completeness needed for high-accuracy predictions. Continued developments in algorithmic efficiency, atomic databases, and code–experiment synergy are expected to further expand the capabilities and scientific reach of TEPID-class models.