TACAW: Auxiliary Wavefunction Autocorrelation

• TACAW is a simulation framework that extracts EELS spectra via the time autocorrelation of auxiliary electron wavefunctions, effectively capturing dynamical diffraction and multiple scattering.
• It employs molecular dynamics and Fourier transforms, with extensions like TRPMD for cryogenic conditions and FTE for non-equilibrium pump–probe analyses.
• Software tools such as PySlice and torched-TACAW enable routine, large-scale EELS simulations that incorporate coupled phonon–magnon dynamics and realistic experimental comparisons.

Searching arXiv for TACAW and closely related papers to ground the article in the cited literature. Calling arXiv search… Time Autocorrelation of Auxiliary Wavefunctions (TACAW) is a dynamical simulation framework for electron energy loss spectroscopy (EELS) in which momentum- and energy-resolved scattering intensities are obtained from the time autocorrelation of an auxiliary electron wavefunction propagated through time-dependent atomistic or spin-lattice configurations. In the recent TACAW literature, the auxiliary object is variously termed an “auxiliary wave,” “auxiliary wavefunction,” or “auxiliary wavefunctions,” but the operational idea is consistent: the exit wave encodes the instantaneous scattering response of the specimen, and its temporal autocorrelation or, equivalently in practical implementations, its time-to-energy Fourier transform yields spectra that naturally include dynamical diffraction and multiple scattering (He et al., 21 Mar 2026). The method has been developed for vibrational EELS, extended to cryogenic regimes through thermostatted ring polymer molecular dynamics (TRPMD), generalized to non-equilibrium pump–probe settings through frozen trajectory excitation (FTE), and coupled to atomistic spin-lattice dynamics (ASLD) to describe phonon–magnon spectra (Walker et al., 10 Feb 2026).

1. Definition and formal structure

TACAW formulates the angle- and energy-resolved EELS intensity $I(\mathbf{q},E)$ as the Fourier transform of a time autocorrelation function of an auxiliary beam operator or auxiliary exit wavefunction. In the equilibrium formulation summarized for low-temperature vibrational EELS, the intensity is written as

$$I(\mathbf{q}, E) = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{-\frac{i}{\hbar} E t} c_{\phi\phi}(t)\, dt = c_{\phi\phi}(E),$$

with

$$c_{\phi\phi}(t) = \langle \hat{\phi}(t) \hat{\phi}(0) \rangle = \frac{1}{Z} \mathrm{Tr}\!\left[e^{-\beta \hat{H}_c}\hat{\phi}^{\dagger}(\mathbf{q},0)\hat{\phi}(\mathbf{q},t)\right].$$

Here $\hat{\phi}(\mathbf{q},t)$ is the auxiliary beam operator in the Heisenberg picture, propagated with the crystal Hamiltonian $\hat{H}_c$, $Z$ is the partition function, and $\beta=(k_BT)^{-1}$ (He et al., 21 Mar 2026).

A complementary formulation for equilibrium and non-equilibrium vibrational spectroscopy begins from state-to-state scattering amplitudes,

$$\psi_{n \rightarrow m}(\vec{q}) = \langle m|\hat{\phi}(\vec{q})|n\rangle,$$

and expresses the energy-resolved scattering intensity as

$$I(\vec{q},E)=\sum_{n,m}\frac{e^{-\beta E_n}}{Z}\left|\langle m|\hat{\phi}(\vec{q})|n\rangle\right|^2 \delta\!\left(E-(E_m-E_n)\right),$$

which is then recast through the wavefunction correlation function

$$I(\vec{q},E)=\int_{-\infty}^{\infty}\frac{dt}{2\pi\hbar}e^{-iEt/\hbar}c_{\phi\phi}(\vec{q},t).$$

The practical approximation is

$$I(\mathbf{q}, E) = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{-\frac{i}{\hbar} E t} c_{\phi\phi}(t)\, dt = c_{\phi\phi}(E),$$0

so the spectrum reduces to the power spectrum of the time-dependent exit wavefunction computed along an atomistic trajectory (Marciniak et al., 11 Jun 2026).

The same core idea appears in the coupled phonon–magnon formulation, where

$$I(\mathbf{q}, E) = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{-\frac{i}{\hbar} E t} c_{\phi\phi}(t)\, dt = c_{\phi\phi}(E),$$1

In that representation, $$I(\mathbf{q}, E) = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{-\frac{i}{\hbar} E t} c_{\phi\phi}(t)\, dt = c_{\phi\phi}(E),$$2 is the Fourier transform of the simulated auxiliary exit wavefunction at time $$I(\mathbf{q}, E) = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{-\frac{i}{\hbar} E t} c_{\phi\phi}(t)\, dt = c_{\phi\phi}(E),$$3, and the average is taken over time origins along the underlying trajectory (Castellanos-Reyes et al., 9 Aug 2025).

A widely used practical estimator replaces explicit autocorrelation evaluation by direct time-domain Fourier analysis of the wavefunction. In the PySlice implementation, the mean-subtracted exit wavefunction

$$I(\mathbf{q}, E) = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{-\frac{i}{\hbar} E t} c_{\phi\phi}(t)\, dt = c_{\phi\phi}(E),$$4

is Fourier transformed,

$$I(\mathbf{q}, E) = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{-\frac{i}{\hbar} E t} c_{\phi\phi}(t)\, dt = c_{\phi\phi}(E),$$5

and the spectrum is then

$$I(\mathbf{q}, E) = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{-\frac{i}{\hbar} E t} c_{\phi\phi}(t)\, dt = c_{\phi\phi}(E),$$6

That treatment is described as an application of the Wiener–Khinchin theorem relating power spectral density to autocorrelation (Walker et al., 10 Feb 2026).

2. Physical content and equilibrium vibrational EELS

In equilibrium TACAW, molecular dynamics generates a time series of atomic configurations, multislice electron scattering propagates the probe through each snapshot, and frequency-domain analysis extracts the vibrational signal from the resulting time-dependent exit wavefunction. This workflow is explicitly described as combining molecular dynamics, multislice electron scattering, and frequency-domain analysis to predict vibrational EELS from atomistic structures (Walker et al., 10 Feb 2026).

A central feature of TACAW is that it naturally includes dynamical diffraction and multiple scattering. The cryogenic vibrational EELS formulation states that the method “enables efficient computation of scattering intensities while naturally accounting for dynamical diffraction and multiple-scattering effects” (He et al., 21 Mar 2026). The same point is repeated in the large-scale STEM-EELS implementation, where TACAW is presented as retaining dynamical diffraction and naturally including multiple elastic and inelastic events (Osmera et al., 2 Jul 2026).

In practical equilibrium simulations, the exit wavefunction depends parametrically on the instantaneous atomic positions. PySlice describes the auxiliary wavefunction as the electron exit wavefunction $$I(\mathbf{q}, E) = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{-\frac{i}{\hbar} E t} c_{\phi\phi}(t)\, dt = c_{\phi\phi}(E),$$7 obtained by propagating the electron probe through the dynamic atomic potential at each molecular-dynamics snapshot using the multislice algorithm. The specimen is divided into thin slices; for each time step, atomic positions define the projected potential; and the transmission and free-space propagation operators are applied slice by slice to produce the exit wavefunction (Walker et al., 10 Feb 2026).

The equilibrium TACAW spectrum is directly comparable to experimental vibrational EELS. According to the non-equilibrium FTE/TACAW paper, even in equilibrium the method yields momentum- and energy-resolved scattering spectra by evaluating the time-correlation function of the auxiliary exit wavefunction and Fourier transforming it with respect to time, producing a spectrum comparable to vibrationally resolved $$I(\mathbf{q}, E) = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{-\frac{i}{\hbar} E t} c_{\phi\phi}(t)\, dt = c_{\phi\phi}(E),$$8TEM-EELS or EEGS (Marciniak et al., 11 Jun 2026).

For classical molecular-dynamics trajectories, the low-temperature vibrational EELS paper states that the quantum-corrected spectrum is obtained through a Kubo relation,

$$I(\mathbf{q}, E) = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{-\frac{i}{\hbar} E t} c_{\phi\phi}(t)\, dt = c_{\phi\phi}(E),$$9

with

$$c_{\phi\phi}(t) = \langle \hat{\phi}(t) \hat{\phi}(0) \rangle = \frac{1}{Z} \mathrm{Tr}\!\left[e^{-\beta \hat{H}_c}\hat{\phi}^{\dagger}(\mathbf{q},0)\hat{\phi}(\mathbf{q},t)\right].$$0

This factor is reused in later TACAW implementations as the standard classical-to-quantum correction for equilibrium spectra (He et al., 21 Mar 2026).

3. Cryogenic extension through TRPMD

At low temperatures, the vibrational EELS literature emphasizes that nuclear quantum effects, notably zero-point motion, dominate atomic dynamics and invalidate purely classical molecular dynamics. To address this, TRPMD has been incorporated into TACAW so that the nuclear ensemble is sampled from ring-polymer trajectories while the scattering signal is still computed from auxiliary-wave time correlations (He et al., 21 Mar 2026).

In the TRPMD-TACAW construction, the auxiliary wave is replaced by a bead-averaged estimator,

$$c_{\phi\phi}(t) = \langle \hat{\phi}(t) \hat{\phi}(0) \rangle = \frac{1}{Z} \mathrm{Tr}\!\left[e^{-\beta \hat{H}_c}\hat{\phi}^{\dagger}(\mathbf{q},0)\hat{\phi}(\mathbf{q},t)\right].$$1

and the intensity becomes

$$c_{\phi\phi}(t) = \langle \hat{\phi}(t) \hat{\phi}(0) \rangle = \frac{1}{Z} \mathrm{Tr}\!\left[e^{-\beta \hat{H}_c}\hat{\phi}^{\dagger}(\mathbf{q},0)\hat{\phi}(\mathbf{q},t)\right].$$2

Each atom is mapped onto a ring polymer of $$c_{\phi\phi}(t) = \langle \hat{\phi}(t) \hat{\phi}(0) \rangle = \frac{1}{Z} \mathrm{Tr}\!\left[e^{-\beta \hat{H}_c}\hat{\phi}^{\dagger}(\mathbf{q},0)\hat{\phi}(\mathbf{q},t)\right].$$3 beads evolving under the standard ring-polymer Hamiltonian, and the bead count is chosen to resolve quantum fluctuations for modes up to the highest relevant energy, summarized in the paper as $$c_{\phi\phi}(t) = \langle \hat{\phi}(t) \hat{\phi}(0) \rangle = \frac{1}{Z} \mathrm{Tr}\!\left[e^{-\beta \hat{H}_c}\hat{\phi}^{\dagger}(\mathbf{q},0)\hat{\phi}(\mathbf{q},t)\right].$$4 (He et al., 21 Mar 2026).

The reported physical consequences are specific. At high temperature, TRPMD-TACAW results reduce to classical MD–TACAW results. At low temperature, TRPMD-TACAW yields nonzero mean squared displacements as $$c_{\phi\phi}(t) = \langle \hat{\phi}(t) \hat{\phi}(0) \rangle = \frac{1}{Z} \mathrm{Tr}\!\left[e^{-\beta \hat{H}_c}\hat{\phi}^{\dagger}(\mathbf{q},0)\hat{\phi}(\mathbf{q},t)\right].$$5 because the radius of gyration of the ring polymer remains finite; the vibrational EELS spectrum remains finite as $$c_{\phi\phi}(t) = \langle \hat{\phi}(t) \hat{\phi}(0) \rangle = \frac{1}{Z} \mathrm{Tr}\!\left[e^{-\beta \hat{H}_c}\hat{\phi}^{\dagger}(\mathbf{q},0)\hat{\phi}(\mathbf{q},t)\right].$$6; and the method correctly predicts the nearly temperature-independent optical phonon peak intensities in silicon, consistent with the first Born approximation $$c_{\phi\phi}(t) = \langle \hat{\phi}(t) \hat{\phi}(0) \rangle = \frac{1}{Z} \mathrm{Tr}\!\left[e^{-\beta \hat{H}_c}\hat{\phi}^{\dagger}(\mathbf{q},0)\hat{\phi}(\mathbf{q},t)\right].$$7 for relatively high-energy phonons when $$c_{\phi\phi}(t) = \langle \hat{\phi}(t) \hat{\phi}(0) \rangle = \frac{1}{Z} \mathrm{Tr}\!\left[e^{-\beta \hat{H}_c}\hat{\phi}^{\dagger}(\mathbf{q},0)\hat{\phi}(\mathbf{q},t)\right].$$8 (He et al., 21 Mar 2026).

The same study also reports that TRPMD-TACAW correctly models the suppression of energy-gain features in EELS at low temperature, as dictated by detailed balance, and that blue-shift and decreased intensity of the phonon peaks with cooling are absent or suppressed in TRPMD relative to classical MD. The computational setup is described as combining large supercells, machine-learned SNAP force fields, multislice electron propagation for each nuclear bead configuration, and thermostat choices such as PILE-L and PILE-G to avoid spurious resonances (He et al., 21 Mar 2026).

A plausible implication is that TACAW is not tied to a single description of nuclear motion; rather, it functions as a spectroscopic post-processing layer over whatever trajectory ensemble is used, provided the trajectory estimator is consistent with the desired correlation function.

4. Non-equilibrium and time-resolved TACAW

TACAW has also been generalized to non-equilibrium lattice dynamics by combining it with frozen trajectory excitation. In this setting, a selected phonon excitation is first introduced into an equilibrium molecular-dynamics trajectory using FTE, and the subsequent relaxation is analyzed by short-time TACAW windows at different pump–probe delays (Marciniak et al., 11 Jun 2026).

The working non-equilibrium observable is

$$c_{\phi\phi}(t) = \langle \hat{\phi}(t) \hat{\phi}(0) \rangle = \frac{1}{Z} \mathrm{Tr}\!\left[e^{-\beta \hat{H}_c}\hat{\phi}^{\dagger}(\mathbf{q},0)\hat{\phi}(\mathbf{q},t)\right].$$9

where $\hat{\phi}(\mathbf{q},t)$0 is the time since excitation and $\hat{\phi}(\mathbf{q},t)$1 is the probe duration or analysis window. The average is taken over multiple independently relaxed trajectories with the same excitation (Marciniak et al., 11 Jun 2026). The paper states that this construction “inherently respects time–energy uncertainty,” because $\hat{\phi}(\mathbf{q},t)$2 sets the attainable energy resolution.

The combined FTE/TACAW workflow is described in four stages: generation of a long equilibrium trajectory; reciprocal-space excitation by overpopulating selected phonons or modes via a multiplicative filter; inverse transformation back to real space to generate an excited non-equilibrium trajectory; and multiple NVE relaxation simulations from random snapshots of that excited trajectory. At selected delays, short windows are extracted, the configuration-dependent auxiliary exit wavefunction is computed for all frames by multislice, a window function such as a Hann window is applied, the wavefunction is Fourier transformed, and the intensity is obtained by $\hat{\phi}(\mathbf{q},t)$3 followed by averaging over runs (Marciniak et al., 11 Jun 2026).

Because detailed balance is violated out of equilibrium when classical MD is used, the paper introduces an approximate mode- and time-dependent quantum correction. It defines

$\hat{\phi}(\mathbf{q},t)$4

and replaces the equilibrium Kubo factor by a corrected expression involving

$\hat{\phi}(\mathbf{q},t)$5

The paper presents this as a post-processing correction for non-equilibrium FTE/TACAW spectra (Marciniak et al., 11 Jun 2026).

The demonstrative systems are fcc-Ni and 3C-SiC. For fcc-Ni, the reported result is efficient energy transfer from an initially excited transverse acoustic mode on the $\hat{\phi}(\mathbf{q},t)$6 line to a longitudinal acoustic mode at $\hat{\phi}(\mathbf{q},t)$7, seen as a transient spectral peak that dissipates during relaxation. For 3C-SiC, the $\hat{\phi}(\mathbf{q},t)$8 excitation case shows slow redistribution primarily along the same transverse acoustic branch, while the $\hat{\phi}(\mathbf{q},t)$9 excitation case shows transfer from an initially excited longitudinal acoustic mode to a transverse optical mode, described as direct evidence of phonon–phonon mode coupling and sublattice-resolved energy transfer (Marciniak et al., 11 Jun 2026).

Across these examples, the paper emphasizes that equilibrium phonon branch structure is mostly preserved and that relaxation appears mainly as transient redistribution of spectral weight among existing features. This suggests that TACAW is particularly suited to observing mode coupling and energy flow without reducing the signal to energy-integrated diffuse scattering alone.

5. Spin, magnons, and coupled phonon–magnon spectra

TACAW is not restricted to lattice vibrations. The 2026 coupled-excitation study extends the method to atomistic spin-lattice dynamics in order to simulate the full EELS signal from body-centered cubic iron at $\hat{H}_c$0, including phonons, magnons, and their coupling within a unified dynamical formalism (Castellanos-Reyes et al., 9 Aug 2025).

In that framework, large supercells are evolved with ASLD using ab initio force constants and exchange or interaction parameters so that atomic positions and local spins evolve jointly. For each snapshot, a supercell potential including atomic and magnetic contributions is constructed, the electron beam is propagated through the sample using a Pauli multislice method, and the resulting exit wavefunction is projected to $\hat{H}_c$1. Time autocorrelations and Fourier transforms then deliver momentum- and energy-resolved spectra (Castellanos-Reyes et al., 9 Aug 2025).

The stated advantage over earlier phonon-only or magnon-only TACAW variants is that coupled dynamics directly encode interaction effects, including phonon–magnon coupling, mutual interference, and redistribution of spectral weight. The paper explicitly reports “non-additive spectral features arising from phonon-magnon coupling, including interference and energy redistribution effects,” and predicts experimental detectability of magnon signals under optimized detector conditions (Castellanos-Reyes et al., 9 Aug 2025).

Channel isolation remains possible. By selectively turning off atomic or spin degrees of freedom in the ASLD simulation, the study constructs “phonons only” and “magnons only” reference spectra against which the full coupled signal can be compared (Castellanos-Reyes et al., 9 Aug 2025). This permits TACAW to function not only as a forward simulator of total EELS but also as a diagnostic tool for disentangling coupled contributions within a common numerical framework.

The same paper presents TACAW-ASLD as preserving momentum and energy resolution, finite-temperature effects, dynamical diffraction, and multiple scattering while moving beyond additive-channel treatments. Within the scope of the reported results, TACAW thereby becomes a general dynamical formalism for low-energy EELS rather than a method specialized to phonons alone (Castellanos-Reyes et al., 9 Aug 2025).

6. Software implementations, numerical scaling, and related theoretical context

The TACAW literature now includes public software intended to make the method routine rather than bespoke. “PySlice” is described as “the first publicly available implementation of the Time Autocorrelation of Auxiliary Wavefunction (TACAW) method,” providing an automated workflow from atomic structures to momentum- and energy-resolved vibrational EELS spectra, phonon dispersions, spectral diffraction patterns, and spectrum images (Walker et al., 10 Feb 2026). Its workflow is organized around structure input, molecular dynamics using universal machine learning interatomic potentials, GPU-accelerated multislice propagation, and TACAW analysis. The code is further described as modular, Python-based, and also capable of conventional electron microscopy simulations (Walker et al., 10 Feb 2026).

PySlice emphasizes routine deployment with universal machine learning interatomic potentials such as ORB, MACE, CHGNet, Allegro-FM, and UMA, so that no per-system force-field development is required. The article reports applications to canonical transition-metal dichalcogenides, a bulk Si/Ge heterostructure interface, and a Si defect in graphene, with outputs including material-specific phonon dispersions, spectral diffraction patterns, confined optical phonon modes, and defect-localized low-frequency vibrational modes (Walker et al., 10 Feb 2026).

For larger and thicker specimens, “torched-TACAW” addresses scaling bottlenecks associated with molecular dynamics, multislice propagation, data throughput, and storage. The method combines foundational machine-learned interatomic potentials, partitioning of elongated supercells along the beam direction, and on-the-fly processing of multislice outputs. The implementation is described as a freely available TACAW component for efficient STEM-EELS simulations with tractable memory use and data flow (Osmera et al., 2 Jul 2026).

Three numerical issues receive particular emphasis in the torched-TACAW study. First, spectral leakage from finite trajectory chunks is handled with windowing and Welch’s method; the paper reports that the Hann window suppresses artifacts with minimal spectral broadening. Second, supercell partitioning along $\hat{H}_c$2 permits thick-sample simulations by stacking independently simulated sub-supercells, with only minor, noise-like differences in the resulting spectra. Third, out-of-core batch processing and region-of-interest restriction reduce the otherwise multi-terabyte storage burden of auxiliary wavefunctions (Osmera et al., 2 Jul 2026). The demonstration system is rutile TiO$\hat{H}_c$3, including atomic-resolution STEM-EELS simulations for thick samples.

Beyond direct EELS applications, several related arXiv works situate TACAW within a broader theory of time-correlation functions. A study of general Hamiltonian systems derives a formal series expansion for autocorrelation functions,

$\hat{H}_c$4

and analyzes exponential versus sub-exponential decay through the spectral measure and Laplace transform (Maiocchi et al., 2011). That paper explicitly presents the method as a classical analog related to TACAW. In another direction, a non-Hermitian quantum mechanics study defines two distinct time-correlation functions,

$\hat{H}_c$5

and discusses their equivalence conditions and a relative-difference diagnostic for positivity of the density operator (Sergi et al., 2014). For wavepacket-based spectroscopy, overlap relations between Hagedorn bases associated with different Gaussians are derived specifically to evaluate nonlocal-in-time quantities such as time correlation functions needed for computing spectra, avoiding numerical quadrature through exact algebraic recurrences (Vaníček et al., 2024).

These neighboring developments do not redefine TACAW itself, but they clarify its mathematical setting. TACAW can be viewed as one member of a broader class of autocorrelation-based spectral methods in which the central object is not a conventional density response function but a time-correlation built from an auxiliary dynamical representation of the evolving system.

7. Scope, terminology, and present trajectory

Across the literature summarized here, TACAW consistently denotes a framework in which a dynamically evolving auxiliary electron wavefunction is sampled along trajectories, processed in time, and converted into momentum- and energy-resolved electron scattering observables. Its documented domains now include equilibrium vibrational EELS, cryogenic vibrational EELS with nuclear quantum effects, non-equilibrium time-resolved vibrational EELS or EEGS after selective phonon excitation, and coupled phonon–magnon spectroscopy (He et al., 21 Mar 2026).

The terminology is not completely uniform. One paper refers to the “Time Autocorrelation of Auxiliary Wave” method (He et al., 21 Mar 2026), others to “Time Autocorrelation of Auxiliary Wavefunction” (Walker et al., 10 Feb 2026), and still others to “Time Autocorrelation of Auxiliary Wavefunctions” (Osmera et al., 2 Jul 2026). This suggests a naming variation rather than a substantive methodological split.

The present trajectory of the field, as stated in the software and scaling papers, is toward routine, large-scale, and experimentally comparable simulations. PySlice frames TACAW as an automated framework spanning molecular dynamics, GPU-accelerated electron scattering, and frequency-domain analysis (Walker et al., 10 Feb 2026), while torched-TACAW focuses on practical large-scale STEM-EELS for thick samples, defects, interfaces, impurities, and grain boundaries (Osmera et al., 2 Jul 2026). The TRPMD extension identifies low-temperature quantum nuclei as a necessary ingredient for cryogenic vibrational EELS (He et al., 21 Mar 2026), and the ASLD extension points toward unified treatment of lattice and spin excitations in magnetic materials (Castellanos-Reyes et al., 9 Aug 2025).

Taken together, these works establish TACAW as a correlation-based, trajectory-driven, multislice-compatible formalism for low-energy EELS whose distinctive feature is that the spectroscopy is extracted from the temporal statistics of an auxiliary exit wave rather than from explicit phonon or magnon mode summations.