In the original model-free analysis of Lipari and Szabo (1982a) the correlation function *C*(*τ*) of the XH bond vector is approximated by decoupling the internal fluctuations of the bond vector
*C*_{I}(*τ*) from the correlation function of the overall Brownian rotational diffusion
*C*_{O}(*τ*) by the equation

C(τ) = C_{O}(τ)⋅C_{I}(τ). |
(15.57) |

The overall correlation functions of the diffusion of a sphere, spheroid, and ellipsoid are presented respectively in section 15.9.1 on page , section 15.10.1 on page , and section 15.11.1 on page . These three different equations can be combined into one generic correlation function which is independent of the type of diffusion. This generic correlation function is

C_{O}(τ) = c_{i}⋅e^{-τ/τi}, |
(15.58) |

where *c*_{i} are the weights and *τ*_{i} are correlation times of the exponential terms.
In the original model-free analysis of Lipari and Szabo (1982a,b) the internal motions are modelled by the correlation function

C_{I}(τ) = S^{2} + (1 - S^{2})e^{-τ/τe}, |
(15.59) |

where *S*^{2} is the generalised Lipari and Szabo order parameter which is related to the amplitude of the motion and *τ*_{e} is the effective correlation time which is an indicator of the timescale of the motion, albeit being dependent on the value of the order parameter.
The order parameter ranges from one for complete rigidity to zero for unrestricted motions.
Model-free theory was extended by Clore et al. (1990) to include motions on two timescales by the correlation function

C_{I}(τ) = S^{2} + (1 - S^{2}_{f})e^{-τ/τf} + (S^{2}_{f} - S^{2})e^{-τ/τs}, |
(15.60) |

where the faster of the motions is defined by the order parameter *S*^{2}_{f} and the correlation time *τ*_{f}, the slower by the parameters *S*^{2}_{s} and *τ*_{s}, and the two order parameter are related by the equation
*S*^{2} = *S*^{2}_{f}⋅*S*^{2}_{s}.

The relaxation equations of Abragam (1961) are composed of a sum of power spectral density functions *J*(*ω*) at five frequencies.
The spectral density function is related to the correlation function as the two are a Fourier pair.
Applying the Fourier transform to the correlation function composed of the generic diffusion equation and the original model-free correlation function results in the equation

The Fourier transform using the extended model-free correlation function is

The relax user manual (PDF), created 2020-08-26.