Formula: avg = 0.5 * (a + b) Example input: a=2.0, b=6.0 Example output: avg=4.0

Union-find for equivalent groups

Formula: rho_total(r) = rho_up_tilde(r) + rho_dn_tilde(r) + rho_aug(r) rho_up_eff(r) = rho_up_tilde(r) + 0.5 * rho_aug(r) rho_dn_eff(r) = rho_dn_tilde

Formula: v_eff_up(r) = v_ext(r) + v_h(r) + v_xc_up(r) v_eff_dn(r) = v_ext(r) + v_h(r) + v_xc_dn(r) v_eff_avg(r) = 0.5 * (v_eff_up(r) + v_eff_dn(r)) Ex

Formula: v_eff(r) = v_ext(r) + v_h(r) + v_xc(r) v_smooth(r) = v_ext(r) + v_h(r) Example input: v_ext=1.0 eV, v_h=0.2 eV, v_xc=-2.7211652 eV Example ou

Calculate and return erfc based valence valence exchange interactions.

If f_j is specified, custom occupation numbers will be used. Hund rules disabled if so.

Compute projector function overlaps B_ii = <pt_i | pt_i>.

Calculate and return the Yukawa based interaction.

Generate all 3x3 integer matrices with entries in {-1,0,1} and |det|=1. Returns column-major Mat3i (exactly 386 candidates). CONVENTION: We enumerate

Formula: y_clipped = max(0, y) Example input: y=-0.7 Example output: y_clipped=0.0

Formula: residual_up(r) = [rho_up_tilde_new(r) + 0.5*rho_aug_new(r)] - [rho_up_tilde_old(r) + 0.5*rho_aug_old(r)] residual_dn(r) = [rho_dn_tilde_new(r

Formula: residual(r) = [rho_tilde_new(r) + rho_aug_new(r)] - rho_total_old(r) residual_inf = max_r |residual(r)| Example input: rho_tilde_new=1.1, rho