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 {S^{2}_{f}, S^{2}_{s}, τ_{f}, τ_{s}}. The second partial derivatives with respect to these parameters are presented below.
The second partial derivative of (15.63) with respect to the geometric parameters and is
The second partial derivative of (15.63) with respect to the geometric parameter and the orientational parameter is
The second partial derivative of (15.63) with respect to the geometric parameter and the order parameter S^{2}_{f} is
The second partial derivative of (15.63) with respect to the geometric parameter and the order parameter S^{2}_{s} is
The second partial derivative of (15.63) with respect to the geometric parameter and the correlation time τ_{f} is
The second partial derivative of (15.63) with respect to the geometric parameter and the correlation time τ_{s} is
The second partial derivative of (15.63) with respect to the orientational parameters and is
The second partial derivative of (15.63) with respect to the orientational parameter and the order parameter S^{2}_{f} is
= τ_{i} - + . | (15.96) |
The second partial derivative of (15.63) with respect to the orientational parameter and the order parameter S^{2}_{s} is
= S^{2}_{f}τ_{i} - . | (15.97) |
The second partial derivative of (15.63) with respect to the orientational parameter and the correlation time τ_{f} is
= (1 - S^{2}_{f})τ_{i}^{2}. | (15.98) |
The second partial derivative of (15.63) with respect to the orientational parameter and the correlation time τ_{s} is
= S^{2}_{f}(1 - S^{2}_{s})τ_{i}^{2}. | (15.99) |
The second partial derivative of (15.63) with respect to the order parameter S^{2}_{f} twice is
= 0. | (15.100) |
The second partial derivative of (15.63) with respect to the order parameters S^{2}_{f} and S^{2}_{s} is
= c_{i}τ_{i} - . | (15.101) |
The second partial derivative of (15.63) with respect to the order parameter S^{2}_{f} and correlation time τ_{f} is
= - c_{i}τ_{i}^{2}. | (15.102) |
The second partial derivative of (15.63) with respect to the order parameter S^{2}_{f} and correlation time τ_{s} is
= (1 - S^{2}_{s})c_{i}τ_{i}^{2}. | (15.103) |
The second partial derivative of (15.63) with respect to the order parameter S^{2}_{s} twice is
= 0. | (15.104) |
The second partial derivative of (15.63) with respect to the order parameter S^{2}_{s} and correlation time τ_{f} is
= 0. | (15.105) |
The second partial derivative of (15.63) with respect to the order parameter S^{2}_{s} and correlation time τ_{s} is
= - S^{2}_{f}c_{i}τ_{i}^{2}. | (15.106) |
The second partial derivative of (15.62) with respect to the correlation time τ_{f} twice is
= - (1 - S^{2}_{f})c_{i}τ_{i}^{2} | (15.107) |
The second partial derivative of (15.62) with respect to the correlation times τ_{f} and τ_{s} is
= 0. | (15.108) |
The second partial derivative of (15.62) with respect to the correlation time τ_{s} twice is
= - S^{2}_{f}(1 - S^{2}_{s})c_{i}τ_{i}^{2} | (15.109) |
The relax user manual (PDF), created 2024-06-08.