validate - 赝势可转移性验证 ============================== 本模块提供赝势质量评估工具，包括范数守恒、对数导数匹配和幽灵态检测。 .. automodule:: atomppgen.validate :members: :undoc-members: :show-inheritance: 主要函数 -------- 范数守恒检验 ~~~~~~~~~~~~ .. autofunction:: atomppgen.validate.check_norm_conservation 对数导数匹配 ~~~~~~~~~~~~ .. autofunction:: atomppgen.validate.check_log_derivative 幽灵态检测 ~~~~~~~~~~ .. autofunction:: atomppgen.validate.check_ghost_states 完整验证流程 ~~~~~~~~~~~~ .. autofunction:: atomppgen.validate.run_full_validation 数据类 ------ .. autoclass:: atomppgen.validate.NormConservationResult :members: :undoc-members: .. autoclass:: atomppgen.validate.LogDerivativeResult :members: :undoc-members: .. autoclass:: atomppgen.validate.GhostStateResult :members: :undoc-members: .. autoclass:: atomppgen.validate.ValidationReport :members: :undoc-members: 使用示例 -------- 基础验证流程 ~~~~~~~~~~~~ .. code-block:: python from atomppgen import solve_ae_atom, tm_pseudize, invert_semilocal_potential from atomppgen.validate import run_full_validation # 1. 生成 Al 的全电子解和伪化 ae = solve_ae_atom(Z=13, spin_mode='LDA', lmax=0, grid_params={'n': 600}) tm_s = tm_pseudize(ae.r, ae.w, ae.u_by_l[0][-1], ae.eps_by_l[0][-1], l=0, rc=2.0) inv_s = invert_semilocal_potential(tm_s, ae.r) # 2. 构建通道字典 tm_dict = {0: tm_s} inv_dict = {0: inv_s} # 3. 完整验证 report = run_full_validation( ae, tm_dict, inv_dict, r_test=3.0, # 测试半径 (Bohr) E_range_Ry=(-0.5, 0.5), # 能量窗口 (Rydberg) E_step_Ry=0.05 # 能量步长 (Rydberg) ) # 4. 查看结果 print(f"整体通过: {report.overall_passed}") print(f"范数守恒 (s): {report.norm_results[0].passed}") print(f"对数导数 (s): {report.log_deriv_results[0].passed}") if report.ghost_result: print(f"幽灵态数量: {report.ghost_result.n_ghosts}") # 5. 导出 JSON 报告 import json with open('validation_report.json', 'w') as f: json.dump(report.to_dict(), f, indent=2) 范数守恒单项检验 ~~~~~~~~~~~~~~~~ .. code-block:: python from atomppgen import solve_ae_atom, tm_pseudize from atomppgen.validate import check_norm_conservation ae = solve_ae_atom(Z=13, spin_mode='LDA', lmax=0) tm = tm_pseudize(ae.r, ae.w, ae.u_by_l[0][-1], ae.eps_by_l[0][-1], l=0, rc=2.0) # 范数守恒检验 result = check_norm_conservation(tm, tolerance=1e-6) print(f"范数误差: {result.norm_error:.3e}") print(f"通过检验: {result.passed}") print(f"截断半径: {result.rc} Bohr") print(f"求解器收敛: {result.diagnostics['solver_converged']}") 对数导数匹配单项检验 ~~~~~~~~~~~~~~~~~~~~ .. code-block:: python from atomppgen.validate import check_log_derivative, _extract_ks_potential # 提取 KS 有效势和伪势 V_AE = _extract_ks_potential(ae) V_PS = inv.V_l # 对数导数匹配（缩小参数加速示例） result = check_log_derivative( V_AE, V_PS, ae.r, l=0, # s 轨道 r_test=3.0, # 测试半径 (Bohr) E_range_Ha=(-0.2, 0.2), # 能量窗口 (Hartree) E_step_Ha=0.05 # 能量步长 (Hartree) ) print(f"零点 RMS: {result.zero_crossing_rms:.6f} Ha") print(f"曲线 RMS: {result.curve_rms:.6f}") print(f"通过检验: {result.passed}") # 绘制对数导数曲线 import matplotlib.pyplot as plt plt.plot(result.energies, result.L_AE, label='AE') plt.plot(result.energies, result.L_PS, label='PS', linestyle='--') plt.axhline(0, color='k', linewidth=0.5) plt.xlabel('Energy (Ha)') plt.ylabel('L(E, r_test)') plt.legend() plt.savefig('log_derivative.png') 幽灵态检测单项检验 ~~~~~~~~~~~~~~~~~~ .. code-block:: python from atomppgen.validate import check_ghost_states # 幽灵态检测 (径向方法) result = check_ghost_states( inv, ae.r, ae.w, valence_energy=tm.eps, E_window_Ha=(-0.25, 0.25), method='radial' ) print(f"检测到幽灵态数量: {result.n_ghosts}") print(f"能量窗口内束缚态数: {result.diagnostics['n_bound_states_in_window']}") print(f"通过检验: {result.passed}") # 查看所有束缚态 if len(result.eigenvalues) > 0: print("窗口内本征值 (Ha):") for E in result.eigenvalues: print(f" {E:.6f}") 多通道验证 ~~~~~~~~~~ .. code-block:: python # 生成 s, p, d 三通道 ae = solve_ae_atom(Z=13, spin_mode='LDA', lmax=2) # s 通道 tm_s = tm_pseudize(ae.r, ae.w, ae.u_by_l[0][-1], ae.eps_by_l[0][-1], l=0, rc=2.0) inv_s = invert_semilocal_potential(tm_s, ae.r) # p 通道 (需调参) tm_p = tm_pseudize(ae.r, ae.w, ae.u_by_l[1][-1], ae.eps_by_l[1][-1], l=1, rc=1.9, continuity_orders=2) inv_p = invert_semilocal_potential(tm_p, ae.r) # d 通道 tm_d = tm_pseudize(ae.r, ae.w, ae.u_by_l[2][-1], ae.eps_by_l[2][-1], l=2, rc=2.4) inv_d = invert_semilocal_potential(tm_d, ae.r) # 完整验证 tm_dict = {0: tm_s, 1: tm_p, 2: tm_d} inv_dict = {0: inv_s, 1: inv_p, 2: inv_d} report = run_full_validation(ae, tm_dict, inv_dict) # 逐通道查看结果 for l in [0, 1, 2]: label = ['s', 'p', 'd'][l] norm_pass = report.norm_results[l].passed ld_pass = report.log_deriv_results[l].passed print(f"{label} 通道: 范数={norm_pass}, 对数导数={ld_pass}") 技术细节 -------- 能量单位转换 ~~~~~~~~~~~~ 所有内部计算使用 **Hartree (Ha)** 原子单位。外部 API 支持 Rydberg (Ry) 输入， 自动转换： .. code-block:: python # 用户指定 Rydberg E_range_Ry = (-0.5, 0.5) E_step_Ry = 0.05 # 内部转换为 Hartree E_range_Ha = (E_range_Ry[0] / 2, E_range_Ry[1] / 2) # (-0.25, 0.25) Ha E_step_Ha = E_step_Ry / 2 # 0.025 Ha Numerov 求解器 ~~~~~~~~~~~~~~ 对于非均匀网格（如 ``exp_transformed``），自动重采样到等距网格： .. code-block:: python # 检查网格均匀性 dr = np.diff(r) is_uniform = np.allclose(dr, dr[0], rtol=1e-6) if not is_uniform: # 重采样到均匀网格 n_uniform = len(r) r_uniform = np.linspace(r[0], r[-1], n_uniform) V_uniform = interp1d(r, V, kind='cubic')(r_uniform) # 在均匀网格上求解... # 结果插值回原网格 对数导数零点匹配 ~~~~~~~~~~~~~~~~ 零点通过符号变化检测，线性插值精确化： .. code-block:: python def find_zero_crossings(E, L): zeros = [] for i in range(len(L) - 1): if L[i] * L[i+1] < 0: # 符号变化 # 线性插值 E_zero = E[i] - L[i] * (E[i+1] - E[i]) / (L[i+1] - L[i]) zeros.append(E_zero) return zeros 幽灵态判定准则 ~~~~~~~~~~~~~~ 束缚态筛选条件： 1. 能量 :math:`E < E_{\\text{max}}` (默认 0 Ha) 2. 边界衰减：:math:`|\\psi(r_{\\text{max}})| < 0.1 \\cdot \\max|\\psi|` 幽灵态判定： - 在能量窗口 :math:`[-0.25, +0.25]` Ha 内 - 距已知价态 :math:`>0.1` Ha 的额外束缚态 性能优化建议 ~~~~~~~~~~~~ 1. **对数导数扫描**：粗扫 (0.1 Ry) 定位零点，细扫 (0.025 Ry) 精确化 2. **幽灵态检测**：限制网格点数 (≤300) 加速对角化 3. **多通道并行**：各通道独立，可并行计算（当前串行） 参考算法文档 ------------ 详细数学推导和原理见 :doc:`../algorithm/validation_methods`。