Free energy minimisation method

Determine thermodynamic relationships between these species by using the free energy minimisation method, i.e., the fact that the free energy is minimum at equilibrium. 

Sutton AP (1992) Direct free energy minimisation methods apphcation to grain boimdaries. Phil Trans Roy Soc London A 341 233-245... 

This method has been applied to some CVD systems. In most cases, however, the chemical reactions are very complex there exist many gaseous species, such as the reaction intermediates. Consequently, solving the non-linear equations for so many gaseous species becomes very difficult or impossible because there are many unknowns for gaseous species and reaction paths. To deal with such complex situations, free-energy minimisation is generally more suitable, especially for complex chemical systems. 

A second possibility to calculate the equilibrium composition is by Gibbs free energy minimisation. The starting point is the system of equations generated by the relation (8.36). Phase equilibrium may be included in analysis. This method is particularly powerful, because it does not imply necessarily the knowledge of the stoichiometry. However, the user should consider only species representative for equilibrium. As in any optimisation technique, this approach might find local optimum. Specifying explicitly the equilibrium reactions is safer. 

Gale JD (1998) Analytical free energy minimisation of silica polymorphs. J Phys Chem B 102 5423-5431 Gale JD (1999) Modelhng the thermd expansion of zeolites, in Neutrons and Numerical Methods - N2M. 

The main advantages of the method are that it gives a built-in estimate of the sampling error, and it gives an estimate of the free energy landscape that minimises the statistical errors of the overlapping probability distributions for an ensemble of MD simulations. 

A CVD phase diagram provides important information about the equilibrium compositions of the solid phases present under given conditions of pressure, temperature and input concentration. It is usually constructed either by the equilibrium constant method or the minimisation of Gibbs free energy method. The detailed principles are discussed below. 

A simplified method to find the conversions in the two reactions is available as will be shown below, but a general method which can solve any ehemieal equilibrium problem is preferred. For this purpose two methods may be used. The first is minimisation of the Gibbs free energy [316], whereas the other one is the solution for conversions [468]. The first one may be attractive from a theoretical point of view and it is readily combined with phase equilibrimn, but the last one is preferred in catalysis, since no combination of reactions may proceed in all cases. The set of equations in Table 1.2 may be solved using the Newton-Raphson method with the conversions as independent variables. Some of the components (higher hydrocabons or oxygen) may almost disappear in the final mixture so it is necessary to handle elimination of reactions with almost complete conversion. 

The two possible initiations for the free-radical reaction are step lb or the combination of steps la and 2a from Table 1. The role of the initiation step lb in the reaction scheme is an important consideration in minimising the concentration of atomic fluorine (27). As indicated in Table 1, this process is spontaneous at room temperature [AG25 = —24.4 kJ/mol (—5.84 kcal/mol) ] although the enthalpy is slightly positive. The validity of this step has not yet been conclusively estabUshed by spectroscopic methods which makes it an unsolved problem of prime importance. Furthermore, the fact that fluorine reacts at a significant rate with some hydrocarbons in the dark at temperatures below —78° C indicates that step lb is important and may have Httie or no activation energy at RT. At extremely low temperatures (ca 10 K) there is no reaction between gaseous fluorine and CH or 2 6... 

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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