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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