The spectral density functions - J(ω)

The relaxation equations are themselves dependent on the calculation of the spectral density values J(ω). Within model-free analysis these are modelled by the original model-free formula (Lipari and Szabo, 1982b,a)

$${\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} \Bigg( {\frac{{S^2}}{{1 + (\omega \tau_i)^2}}} {\frac{{(1 - S^2)(\tau_e + \tau_i)\tau_e}}{{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}}} \Bigg)$$ (7.7)

where S² is the square of the Lipari and Szabo generalised order parameter and τ_e is the effective correlation time. The order parameter reflects the amplitude of the motion and the correlation time in an indication of the time scale of that motion. The theory was extended by Clore et al. (1990) by the modelling of two independent internal motions using the equation

$$\begin{multline} J(\omega) = \frac{2}{5} \sum_{i=-k}^k c_i \cdot \tau_i \Bigg( ... ...)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2} \Bigg). \end{multline}$$

where S²_f and τ_f are the amplitude and timescale of the faster of the two motions whereas S²_s and τ_s are those of the slower motion. S²_f and S²_s are related by the formula S² = S²_f⋅S²_s.

If these forms of the model-free spectral density functions are unfamiliar, that is because these are the numerically stabilised forms presented in d'Auvergne and Gooley (2008b). The original model-free spectral density functions presented in Lipari and Szabo (1982a) and Clore et al. (1990) are not the most numerically stable form of these equations. An important problem encountered in optimisation is round-off error in which machine precision influences the result of mathematical operations. The double reciprocal τ^-1 = τ_m^-1 + τ_e^-1 used in the equations are operations which are particularly susceptible to round-off error, especially when τ_e $\ll$ τ_m. By incorporating these reciprocals into the model-free spectral density functions and then simplifying the equations this source of round-off error can be eliminated, giving relax an edge over other model-free optimisation software.