Jacobian factor

To obtain thermodynamic perturbation or integration formulas for changing q, one must go back and forth between expressions of the configuration integral in Cartesian coordinates and in suitably chosen generalized coordinates [51]. This introduces Jacobian factors... 

Boresch, S. Karplus, M., The Jacobian factor in free energy simulations, J. Comp. Chem. 1996,105, 5145-5154... 

Calculations of relative free energies of binding often involve the alteration of bond lengths in the course of an alchemical simulation. When the bond lengths are subject to constraints, a correction is needed for variation of the Jacobian factor in the expression for the free energy. Although a number of expressions for the correction formula have been described in the literature, the correct expressions are those presented by Boresch and Karplus.21... 

The products of the endothermic as well as the exothermic reactions are widely scattered, and their distribution is clearly consistent with the formation of a complex capable of surviving several rotations before dissociating to produce a symmetric (about 6 = 90°) c.m. distribution that is distorted in the lab system by the (v ju Y Jacobian factor. This conclusion was confirmed by the presence at wider lab angles of a strong, sticky-collision peak in the distribution of M, resulting from the break-up of the complex to reform the original reagents rather than new products. 

Cowley [Cowley 2003] also criticized the data analysis on VESUVIO and argued that the angular dependent anomaly of an/am of NbH0.78 might be caused by the lack of proper corrections due to the incident neutron flux I (Eo) (see Eq. (4)) or due to a large Jacobian factor... 

Figure 7 A - Dependence of the involved incident energy on the scattering angle for different masses H (M = 1), D (M = 2), Zr (M = 91), and Nb (M = 93). The same incident energy is involved for Zr and Nb. B - Dependence of the involved Jacobians (see Eq. (8)) for H, D, Zr, and Nb. The same Jacobian factor is involved for both, Zr and Nb.

The SCT method extends the Marcus-Coltrin idea in a way that eliminates problems with the Jacobian becoming unphysical. Rather than including the Jacobian factor, the reduced mass for motion along the reaction coordinate /i is replaced by... 

Usually, the Jacobian matrix of the system (7.38) is nonsymmetric. Thus, it is neither possible to solve the linear system by means of the Cholesky algorithm nor to halve memory allocation. The most efficient methods (Gauss or PLR variant) adopted for Jacobian factorization require twice as much time and memory allocation as the Cholesky algorithm. 

The scalar Jacobian factor that more typically arises in a coordinate transformation equals det J. For a nondegenerate coordinate system, det 0 thus the potential energy gradient with respect to the generalized coordinates equals the zero vector at the transition state. 

In this appendix we wish to obtain the Jacobian factor for Eq. [63] and thence to evaluate the free energy density given by Eq. [70]. 

The Jacobian factor d /ds for the path defined by these turning points is expressed in terms of the curvature and turning points by... 

---