Energy minimisation methods applications

Most of this book concerns the development and application of theoretical thermodynamic models, as these are the basis of the CALPHAD method. However, none of this would be possible without the existence of the computational methods and software which allow these models to be applied in practice. In essence, the issues involved in computational methods are less diverse and mainly revolve around Gibbs energy minimisation. In addition, there are optimiser codes which are used for the thermodynamic assessment of phase equilibria. The essential aim of these codes is to reduce the statistical error between calculated phase equilibria, thermodynamic properties and the equivalent experimentally measured quantities. 

In summary, then, the complexity of zeolitic structures is too great to allow a complete enumeration, so that this approach could not be used to identify all new zeolite structure types from a comparison of observed diffraction profiles and those predicted from energy minimisation. Nevertheless, for structures where the symmetry is quite high the method may find application in structure solution. Furthermore, novel hypothetical structures which are chemically feasible and which would possess important new properties if made would be attractive targets for synthesis via the methods of templating described elsewhere in this book. 

Eqs. (l)-(3), (13), and (19) define the spin-free CGWB-AIMP relativistic Hamiltonian of a molecule. It can be utilised in any standard wavefunction based or Density Functional Theory based method of nonrelativistic Quantum Chemistry. It would work with all-electron basis sets, but it is expected to be used with valence-only basis sets, which are the last ingredient of practical CGWB-AIMP calculations. The valence basis sets are obtained in atomic CGWB-AIMP calculations, via variational principle, by minimisation of the total valence energy, usually in open-shell restricted Hartree-Fock calculations. In this way, optimisation of valence basis sets is the same problem as optimisation of all-electron basis sets, it faces the same difficulties and all the experience already gathered in the latter is applicable to the former. 

I is the identity matrix. The six first derivatives of the energy with respect to the strain components e, measure the forces acting on the unit cell. When combined with the atomic coordinates we get a matrix with 3N - - 6 dimensions. At a minimum not only should there be no force on any of the atoms but the forces on the unit cell should also be zero. Application of a standard iterative minimisation procedure such as the Davidon-Fletcher-Powell method will optimise all these degrees of freedom to give a strain-free final structure. In such procedures a reasonably accurate estimate of the initial inverse Hessian matrix is usually required to ensure that the changes in the atomic positions and in the cell dimensions are matched. 

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