To simplify the calculations of the gradients and Hessians the Ri'(θ) equations have been broken down into a number of components. These include the dipolar and CSA constants as well as the dipolar and CSA spectral density terms for each of the three transformed relaxation data types { R1, R2, σNOE}. The segregation of these components simplifies the maths as many partial derivatives of the components are zero.
The dipolar constant is defined as
This component of the relaxation equations is independent of the parameter of the spectral density function θj, the chemical exchange parameter ρex, and the CSA parameter Δσ. Therefore the partial and second partial derivatives with respect to these parameters is zero. Only the derivative with respect to the bond length r is non-zero being
The second derivative with respect to the bond length is
The CSA constant is defined as
The partial derivative of this component with respect to all parameters but the CSA parameter Δσ is zero. This derivative is
The CSA constant second derivative with respect to Δσ is
The Rex constant is defined as
The partial derivative of this component with respect to all parameters but the chemical exchange parameter ρex is zero. This derivative is
The Rex constant second derivative with respect to ρex is
For the dipolar component of the R1 equation (7.3a) on page the spectral density terms are
The partial derivative of these terms with respect to the spectral density function parameter θj is
The second partial derivative with respect to the spectral density function parameters θj and θk is
For the CSA component of the R1 equation (7.3a) on page the spectral density terms are
The partial derivative of these terms with respect to the spectral density function parameter θj is
The second partial derivative with respect to the spectral density function parameters θj and θk is
For the dipolar component of the R2 equation (7.3b) on page the spectral density terms are
The partial derivative of these terms with respect to the spectral density function parameter θj is
The second partial derivative with respect to the spectral density function parameters θj and θk is
For the CSA component of the R2 equation (7.3b) on page the spectral density terms are
The partial derivative of these terms with respect to the spectral density function parameter θj is
The second partial derivative with respect to the spectral density function parameters θj and θk is
For the dipolar component of the σNOE equation (7.3c) on page the spectral density terms are
The partial derivative of these terms with respect to the spectral density function parameter θj is
The second partial derivative with respect to the spectral density function parameters θj and θk is
The relax user manual (PDF), created 2020-08-26.