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

 = S2fτi - . (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

 = - S2fciτi2. (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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