Subsections

## The extended model-free Hessian

The model-free Hessian of the extended spectral density function (15.63) is the matrix of second partial derivatives. The matrix coordinates correspond to the model parameters which are being optimised.

### - partial derivative

The second partial derivative of (15.63) with respect to the geometric parameters and is ### - partial derivative

The second partial derivative of (15.63) with respect to the geometric parameter and the orientational parameter is ### - S2 partial derivative

The second partial derivative of (15.63) with respect to the geometric parameter and the order parameter S2 is ### - S2f partial derivative

The second partial derivative of (15.63) with respect to the geometric parameter and the order parameter S2f is ### - τf partial derivative

The second partial derivative of (15.63) with respect to the geometric parameter and the correlation time τf is ### - τs partial derivative

The second partial derivative of (15.63) with respect to the geometric parameter and the correlation time τs is ### - partial derivative

The second partial derivative of (15.63) with respect to the orientational parameters and is ### - S2 partial derivative

The second partial derivative of (15.63) with respect to the orientational parameter and the order parameter S2 is =   τi  -  . (15.77)

### - S2f partial derivative

The second partial derivative of (15.63) with respect to the orientational parameter and the order parameter S2f is = -   τi  -  . (15.78)

### - τf partial derivative

The second partial derivative of (15.63) with respect to the orientational parameter and the correlation time τf is = (1 - S2f)  τi2 . (15.79)

### - τs partial derivative

The second partial derivative of (15.63) with respect to the orientational parameter and the correlation time τs is = (S2f - S2)  τi2 . (15.80)

### S2 - S2 partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2 twice is = 0. (15.81)

### S2 - S2f partial derivative

The second partial derivative of (15.63) with respect to the order parameters S2 and S2f is = 0. (15.82)

### S2 - τf partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2 and correlation time τf is = 0. (15.83)

### S2 - τs partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2 and correlation time τs is = -  ciτi2 . (15.84)

### S2f - S2f partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2f twice is = 0. (15.85)

### S2f - τf partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2f and correlation time τf is = -  ciτi2 . (15.86)

### S2f - τs partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2f and correlation time τs is =  ciτi2 . (15.87)

### τf - τf partial derivative

The second partial derivative of (15.62) with respect to the correlation time τf twice is = - (1 - S2f) ciτi2 (15.88)

### τf - τs partial derivative

The second partial derivative of (15.62) with respect to the correlation times τf and τs is = 0. (15.89)

### τs - τs partial derivative

The second partial derivative of (15.62) with respect to the correlation time τs twice is = - (S2f - S2) ciτi2 (15.90)

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