Abstract
In ultrafast science, the strong-field approximation (SFA) provides a powerful framework to describe high-order harmonic generation (HHG) and related phenomena. Meanwhile, within the current ab initio theoretical framework, the use of nonlocal potentials in calculating multi-electron molecular wave functions is almost unavoidable. We find that when such wave functions are directly applied to compute transition dipole moments for correcting SFA, it introduces a fundamental gauge transformation problem. Specifically, the nonlocal potential contributes an additional gauge-dependent phase function to the dipole operator, which directly modifies the phase of the transition dipole. As a consequence, the saddle-point equations acquire an entirely different structure compared to the standard SFA, leading to a splitting of the conventional short and long classical trajectories in HHG into multiple distinct quantum trajectories. Here, ‘‘complex molecules’’ refers to multi-center molecular systems whose nonlocal electronic structure leads to gauge-dependent strong-field responses. Our analysis highlights that the validity of gauge in-variation cannot be assumed universally in SFA framework. Our approach combines the molecular strong-field approximation with gauge transformation analysis, incorporating nonlocal pseudopotentials, saddle-point equations, and multi-center recombination effects.