Theoretical investigation of noncentrosymmetric Ruddlesden-Popper germanium perovskites <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Cs</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mi>Ge

Inorganic halide perovskites possessing noncentrosymmetric (NCS) structures are highly desirable for anomalous photovoltaic, ferroelectric, and nonlinear optical (NLO) applications. In this work, we investigate the Ruddlesden-Popper (RP) germanium (Ge) halide perovskites ${\mathrm{Cs}}_{2}\mathrm{Ge}{X}_{4}$ ($X=\mathrm{Cl}$, Br, I) to identify low-energy NCS phases and evaluate their NLO properties via first-principles calculations. Starting from the centrosymmetric $I4$/mmm parent phase, we systematically introduce 40 symmetry-allowed structural distortion modes, including 37 octahedral rotation/tilt modes and 3 polar displacement modes, to generate a set of low-symmetry configurations. Our results reveal that NCS configurations are energetically favored in ${\mathrm{Cs}}_{2}\mathrm{Ge}{X}_{4}$ relative to their centrosymmetric counterparts, in contrast to analogous lead-based RP perovskites, which tend to stabilize centrosymmetric structures. The lowest energy NCS structure is identified as the polar $\mathit{Pc}$ phase. Symmetry-mode analysis indicates that the polar ${\mathrm{\ensuremath{\Gamma}}}_{5}^{\ensuremath{-}}$ distortion in these Ge-based systems originates from the stereochemically active lone-pair electrons of $\mathrm{G}{\mathrm{e}}^{2+}$, which drives inversion-symmetry breaking and stabilizes the NCS structure. Furthermore, we compute the second-order nonlinear susceptibility and second-harmonic-generation response of $\mathit{Pc}$-phase ${\mathrm{Cs}}_{2}\mathrm{Ge}{X}_{4}$, confirming its promising NLO performance. This study provides mechanistic insight into lone-pair-induced noncentrosymmetry in layered germanium halide perovskites and underscores their potential for advanced nonlinear optical applications.