Hessian Matrix of Gibbs Energy

Pandat™ can calculate the determinant of Hessian matrix of Gibbs energy of a phase and the eigenvalues and eigenvectors of the Hessian matrix.

Since there is one dependent molar fraction for the molar fraction variables (x₁,x₂,⋯,x_n), one of the components is selected as the dependent one. Without loss of generality, x_n is selected as the one, i.e., the last component is considered as the solvent. Then, the second derivatives of Gibbs free energy of a phase form the Hessian matrix, which is an (n-1)×(n-1) symmetrical matrix.

Its determinant is given by

The determinant of Hessian matrix for phase f is available from “HSN(@f)”. The value of “HSN(@f)” is independent of the selection of the solvent component.

A Hessian matrix has real eigenvectors and each eigenvalue has a corresponding eigenvector. The eigenvalues and their eigenvectors are available from “eVal(#*@f)” and “eVec(*#*@f)”. Above Hessian matrix has eigenvalues of “eVal(#1@f)”, “eVal(#2@f)”, … ,“eVal(#n-1@f)”. Each eigenvalue has an eigenvector. For example, eVal(#1@f) has an eigenvector of (eVec(C1#1@f), eVec(C2#1@f),…, eVec(Cn-1#1@f)), where C_k is the name of the k^th component.The keywords HSN, eVal and eVec are listed in Table 1 in Section Pandat Table Format Syntax.

The “solvent” component can be defined while using “Thermodynamic Property” calculation, or defined in Pandat™ Batch file (.pbfx). If “solvent” component is not defined, Pandat™ will choose the first component (the smallest atomic number) as the default “solvent” component.