# Adaptive Exponential Integrators for MCTDHF

## TL;DR

This paper evaluates various exponential integrators for the MCTDHF method, finding that exponential Lawson multistep methods with predictor/corrector steps offer optimal stability, accuracy, and efficiency, especially with adaptive time-stepping.

## Contribution

It introduces and assesses exponential Lawson multistep integrators with predictor/corrector steps for MCTDHF, demonstrating their superior stability and efficiency over traditional methods.

## Key findings

- Exponential Lawson multistep methods outperform other integrators in stability and accuracy.
- Predictor step enables reliable adaptive time-stepping without extra cost.
- Optimal balance of computational cost and accuracy achieved with these methods.

## Abstract

We compare exponential-type integrators for the numerical time-propagation of the equations of motion arising in the multi-configuration time-dependent Hartree-Fock method for the approximation of the high-dimensional multi-particle Schr{\"o}dinger equation. We find that among the most widely used integrators like Runge-Kutta, exponential splitting, exponential Runge-Kutta, exponential multistep and Lawson methods, exponential Lawson multistep methods with one predictor/corrector step provide optimal stability and accuracy at the least computational cost, taking into account that the evaluation of the nonlocal potential terms is by far the computationally most expensive part of such a calculation. Moreover, the predictor step provides an estimator for the time-stepping error at no additional cost, which enables adaptive time-stepping to reliably control the accuracy of a computation.