Figure 6.

Approximations for N-site kinetic schemes for the R_ex contribution in R_1ρ experiments. (Solid) Numerical calculation of R_ex=−λ/sin2θ from the least negative real eigenvalue of the 12 × 12 evolution matrix, (dashed) calculation from the first order approximation from Eq. 11 in Koss et al., 2017, (dotted) calculation from the Woodbury approximation from Eq. 50 in Koss et al., 2017, (dashed-dotted) calculation from the new second-order eigenvalue (Halley) approximation from Eq. 27 in this paper and Eq. 11 in Koss et al, 2017. This (dashed-dotted) approximation overlaps almost perfectly with the numerical solution. Left: Approximations for the triangular scheme (see Fig. 3 for the respective CPMG figure). k₁₂ + k₂₁ = 200 s⁻¹; k₁₃ + k₃₁ = 300 s⁻¹; k₂₃ + k₃₂ = 600 s⁻¹; Δω_AB = 1800 s⁻¹; Δω_AC = −2600 s⁻¹; p_A = 0.85; p_B = 0.08; p_C = 0.07. Right: Approximations for the 4-site kite scheme. The insets exemplify regions in which the results of the calculations differ. Parameters used for all calculations were used for all calculations were ω₁ = 1250 s⁻¹; k₁₂ + k₂₁ = 140 s⁻¹, k₁₃ + k₃₁ = 350 s⁻¹, k₃₄ + k₄₃ = 700 s⁻¹ and k₁₄ + k₄₁ = 350 s⁻¹, Ω_B − Ω_A = −850 s⁻¹, Ω_C − Ω_A = 2550 s⁻¹, and Ω_D − Ω_A = −4250 s⁻¹. p_A = 0.79, p_B = 0.08, p_C = 0.06, p_D = 0.07.