Subsections

## The alternative extended model-free Hessian

The model-free Hessian of the extended spectral density function (15.63) is also complicated by the convolution resulting from the use of the parameters {S2f, S2s, τf, τs}. The second partial derivatives with respect to these parameters are presented below.

### - 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 ### - S2f partial derivative

The second partial derivative of (15.63) with respect to the geometric parameter and the order parameter S2f is ### - S2s partial derivative

The second partial derivative of (15.63) with respect to the geometric parameter and the order parameter S2s 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 ### - S2f partial derivative

The second partial derivative of (15.63) with respect to the orientational parameter and the order parameter S2f is =   τi  - +  . (15.96)

### - S2s partial derivative

The second partial derivative of (15.63) with respect to the orientational parameter and the order parameter S2s is = S2f  τi  -  . (15.97)

### - 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 . (15.98)

### - τs partial derivative

The second partial derivative of (15.63) with respect to the orientational parameter and the correlation time τs is = S2f(1 - S2s)  τi2 . (15.99)

### S2f - S2f partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2f twice is = 0. (15.100)

### S2f - S2s partial derivative

The second partial derivative of (15.63) with respect to the order parameters S2f and S2s is =  ciτi  -  . (15.101)

### S2f - τf partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2f and correlation time τf is = -  ciτi2 . (15.102)

### S2f - τs partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2f and correlation time τs is = (1 - S2s) ciτi2 . (15.103)

### S2s - S2s partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2s twice is = 0. (15.104)

### S2s - τf partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2s and correlation time τf is = 0. (15.105)

### S2s - τs partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2s and correlation time τs is = - S2f ciτi2 . (15.106)

### τf - τf partial derivative

The second partial derivative of (15.62) with respect to the correlation time τf twice is = - (1 - S2f) ciτi2 (15.107)

### τf - τs partial derivative

The second partial derivative of (15.62) with respect to the correlation times τf and τs is = 0. (15.108)

### τs - τs partial derivative

The second partial derivative of (15.62) with respect to the correlation time τs twice is = - S2f(1 - S2s) ciτi2 (15.109)

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