Subsections

The model-free gradient of the extended spectral density function (15.63) is the vector of partial derivatives of the function with respect to the geometric parameter , the orientational parameter , the order parameters S2 and S2f, and the internal correlation times τf and τs. The positions in the vector correspond to the model parameters which are being optimised.

### partial derivative

The partial derivative of (15.63) with respect to the geometric parameter is

### partial derivative

The partial derivative of (15.63) with respect to the orientational parameter is

 = τi + + . (15.72)

### S2 partial derivative

The partial derivative of (15.63) with respect to the order parameter S2 is

 = ciτi - . (15.73)

### S2f partial derivative

The partial derivative of (15.63) with respect to the order parameter S2f is

 = - ciτi - . (15.74)

### τf partial derivative

The partial derivative of (15.63) with respect to the correlation time τf is

 = (1 - S2f)ciτi2. (15.75)

### τs partial derivative

The partial derivative of (15.63) with respect to the correlation time τs is

 = (S2f - S2)ciτi2. (15.76)

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