This paper addresses the issue of angular momentum conservation in surface hopping (FSSH) dynamics, a popular method for simulating nonadiabatic molecular dynamics. The authors propose a one-electron operator, denoted as $\hat{O}$, whose matrix element $\langle J | \hat{O} | K \rangle$ represents the angular component of the derivative coupling between electronic states $|J\rangle$ and $|K\rangle$. They demonstrate that this operator can be constructed efficiently in a semi-local manner, providing physical insights into designing new surface hopping algorithms and practical utility for FSSH calculations. The introduction reviews the basics of surface hopping, emphasizing the importance of momentum conservation in both linear and angular directions. The authors explain that linear momentum conservation is typically addressed in FSSH, but angular momentum conservation is often overlooked. They derive the constraints for the total momentum and angular momentum operators $\boldsymbol{\Gamma}$, which must satisfy $\sum_{A} \boldsymbol{\Gamma}_{\mu \nu}^{A} = \frac{\boldsymbol{p}_{\mu \nu}}{i \hbar}$ and $\sum_{A} \boldsymbol{X}_{A} \times \boldsymbol{\Gamma}_{\mu \nu}^{A} = \frac{\boldsymbol{l}_{\mu \nu}}{i \hbar}$. The paper then details the construction of electron translation factors (ETFs) and electron rotation factors (ERFs). ETFs are well-established, while ERFs are less explored. The authors propose a new approach to construct ERFs by relaxing strict locality in favor of semi-locality, leading to the final expressions for $\mathbf{\Gamma}''$ in Equations 45-47. These expressions are shown to satisfy the required constraints and are localized around the atoms where the orbitals are centered. Numerical results for [5]helicene and methanol molecules demonstrate the stability and locality of the proposed ERFs. The choice of the weighting factor $w$ is critical for balancing locality and numerical stability, with $w = 0.3$ Bohr$^{-2}$ being recommended. The authors also discuss the translational and rotational invariance of the proposed operators, proving that the momentum-rescaling direction does not depend on the orientation or origin of the molecule. Finally, the paper highlights the broader implications of the derived ERFs, suggesting their potential applications in adiabatic propagation, non-adiabatic propagation, and modeling nuclear-electronic-spin dynamics.This paper addresses the issue of angular momentum conservation in surface hopping (FSSH) dynamics, a popular method for simulating nonadiabatic molecular dynamics. The authors propose a one-electron operator, denoted as $\hat{O}$, whose matrix element $\langle J | \hat{O} | K \rangle$ represents the angular component of the derivative coupling between electronic states $|J\rangle$ and $|K\rangle$. They demonstrate that this operator can be constructed efficiently in a semi-local manner, providing physical insights into designing new surface hopping algorithms and practical utility for FSSH calculations. The introduction reviews the basics of surface hopping, emphasizing the importance of momentum conservation in both linear and angular directions. The authors explain that linear momentum conservation is typically addressed in FSSH, but angular momentum conservation is often overlooked. They derive the constraints for the total momentum and angular momentum operators $\boldsymbol{\Gamma}$, which must satisfy $\sum_{A} \boldsymbol{\Gamma}_{\mu \nu}^{A} = \frac{\boldsymbol{p}_{\mu \nu}}{i \hbar}$ and $\sum_{A} \boldsymbol{X}_{A} \times \boldsymbol{\Gamma}_{\mu \nu}^{A} = \frac{\boldsymbol{l}_{\mu \nu}}{i \hbar}$. The paper then details the construction of electron translation factors (ETFs) and electron rotation factors (ERFs). ETFs are well-established, while ERFs are less explored. The authors propose a new approach to construct ERFs by relaxing strict locality in favor of semi-locality, leading to the final expressions for $\mathbf{\Gamma}''$ in Equations 45-47. These expressions are shown to satisfy the required constraints and are localized around the atoms where the orbitals are centered. Numerical results for [5]helicene and methanol molecules demonstrate the stability and locality of the proposed ERFs. The choice of the weighting factor $w$ is critical for balancing locality and numerical stability, with $w = 0.3$ Bohr$^{-2}$ being recommended. The authors also discuss the translational and rotational invariance of the proposed operators, proving that the momentum-rescaling direction does not depend on the orientation or origin of the molecule. Finally, the paper highlights the broader implications of the derived ERFs, suggesting their potential applications in adiabatic propagation, non-adiabatic propagation, and modeling nuclear-electronic-spin dynamics.