Newton-Raphson energy minimisation

We shall not discuss all the numerous energy minimisation procedures which have been worked out and described in the literature but choose only the two most important techniques for detailed discussion the steepest descent process and the Newton-Raphson procedure. A combination of these two techniques gives satisfactory results in almost all cases of practical interest. Other procedures are described elsewhere (1, 2). For energy minimisation the use of Cartesian atomic coordinates is more favourable than that of internal coordinates, since for an arbitrary molecule it is much more convenient to derive all independent and dependent internal coordinates (on which the potential energy depends) from an easily obtainable set of independent Cartesian coordinates, than to evaluate the dependent internal coordinates from a set of independent ones. Furthermore for our purposes the use of Cartesian coordinates is also advantageous for the calculation of vibrational frequencies (Section 3.3.). The disadvantage, that the potential energy is related to Cartesian coordinates in a more complex fashion than to internals, is less serious. 

In case the Newton-Raphson process is used for energy minimisation, the subsequent evaluation of the molecular vibrational frequencies is a fairly simple matter since the F-matrix is available (17). 

One method would be to use Eq. (3.62) and utilise a Newton-Raphson technique to perform a Gibbs energy minimisation with respect to the composition of either A or B. This has an advantage in that only the integral function need be calculated and it is therefore mathematically simpler. The other is to minimise the difference in potential of A and B in the two phases using the relationships... 

To determine the virtual orbitals which minimise the variational energy (15), the derivatives with respect to the basis set expansion, see Eq. (13), and the configuration coefficients, Eq. (11) are computed. Analytic gradients and second derivatives, including mixed terms, are computed and inserted into the Newton-Raphson stabilised algorithm [19]. The detailed expression of the derivatives... 

The energies of most of the defects were minimised using the Newton-Raphson method with BFGS [14] updating of the Hessian. However, it was sometimes necessary to use the more demanding Rational Function Optimiser (RFO) [15] which enforces the required number of imaginary eigenvalues of the Hessian, to be zero at the minimum and one when used to locate a transition state as discussed later. 

In practice, of course, the surface is only quadratic to a first approximation and so a number of steps will be required, at each of which the Hessian matrix must be calculated and inverted. The Hessian matrix of second derivatives must be positive definite in a Newton-Raphson minimisation. A positive definite matrix is one for which all the eigenvalues are positive. When the Hessian matrix is not positive definite then the Newton-Raphson method moves to points (e.g. saddle points) where the energy increases. In addition, far from a mimmum the harmonic approximation is not appropriate and the minimisation can become unstable. One solution to this problem is to use a more robust method to get near to the minimum (i.e. where the Hessian is positive definite) before applying the Newton-Raphson method. 

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. 

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