- Dipolar constant
- CSA constant
*R*_{ex}constant- Spectral density terms of the
R
_{1}dipolar component - Spectral density terms of the
R
_{1}CSA component - Spectral density terms of the
R
_{2}dipolar component - Spectral density terms of the
R
_{2}CSA component - Spectral density terms of the
*σ*_{NOE}dipolar component

To simplify the calculations of the gradients and Hessians the
R_{i}'(*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 {
R_{1},
R_{2},
*σ*_{NOE}}.
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 *θ*_{j}, the chemical exchange parameter *ρ*_{ex}, 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 *R*_{ex} constant is defined as

The partial derivative of this component with respect to all parameters but the chemical exchange parameter *ρ*_{ex} is zero.
This derivative is

The *R*_{ex} constant second derivative with respect to *ρ*_{ex} is

For the dipolar component of the
R_{1} equation (7.3a) on page the spectral density terms are

The partial derivative of these terms with respect to the spectral density function parameter *θ*_{j} is

The second partial derivative with respect to the spectral density function parameters *θ*_{j} and *θ*_{k} is

For the CSA component of the
R_{1} equation (7.3a) on page the spectral density terms are

The partial derivative of these terms with respect to the spectral density function parameter *θ*_{j} is

The second partial derivative with respect to the spectral density function parameters *θ*_{j} and *θ*_{k} is

For the dipolar component of the
R_{2} equation (7.3b) on page the spectral density terms are

The partial derivative of these terms with respect to the spectral density function parameter *θ*_{j} is

The second partial derivative with respect to the spectral density function parameters *θ*_{j} and *θ*_{k} is

For the CSA component of the
R_{2} equation (7.3b) on page the spectral density terms are

*θ*_{j} is

*θ*_{j} and *θ*_{k} is

For the dipolar component of the
*σ*_{NOE} equation (7.3c) on page the spectral density terms are

*θ*_{j} is

*θ*_{j} and *θ*_{k} is

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