Energy minimisation

While the energy o f any structure can be calculated by these methods, it is unlikely to be observed experimentally if it does not correspond to an energy minimum where all atoms experience no net force. To determine the equilibrium configuration, the total energy o f the system is minimised with respect to the atomic co-ordinates and the cell dimensions. The energy is first calculated and differentiated with respect to some atomic perturbation, r, to obtain the gradient o f the potential energy surface for the initial structure. Atoms are then systematically moved so as to decrease the energy o f the system for a number o f cycles until the energy gradient calculated is less than a specified cut-off, e.g. 0.001 eV Â * used in the calculations presented here. Different minimisation methods implementing different algorithms were used to perturb the system between cycles. The default Newton-Raphson method was used initially for all calculations. By this method, the potential energy surface is approximated to a harmonic curve, and by employing a Taylor expansion, displacement towards the energy minimum can be estimated by:

Chapter 2 M ethodology o f interatomic potential computer models

dr = - d^U

dr^

This was used together with the Broyden, Fletcher, Goldfarb and Shanno (BFGS) adaptation [13] that only calculates the second derivative o f the energy surface at certain values o f r and dU/dr. At iterations between these points, the inverse derivative o f the matrix can be estimated from the dU/dr and r values and those o f the previous iteration, together with its inverse Hessian. This method o f calculating the inverse Hessian indirectly reduces the computational demand o f energy minimisation compared to the direct method. While this Newton-Raphson/BFGS method is effective in minimising initial structures that are further away from their minimum energy, it is less efficient to find the exact minimum by making small perturbations near equilibrium. For example it merely looks for a point where the energy gradient is zero, without ensuring that the second derivative o f the energy is positive. Since an energy maximum complies equally with this criterion as an energy minimum, sometimes saddle-points can be reached rather than energy minima. For this reason, when the gradient norm was reduced to 0.01 eV A '\ the minimisation method is switched to the alternative Rational Functional Optimisation (RFC) method [14]. This method is more computationally demanding, because the Hessian is calculated after every iteration in order to determine and eliminate any negative eigenvalues. However, its use ensures that a true minimum is reached.

This sequence o f energy minimisers was not always able to find an energy minimum, particularly when the initial geometry was far from the minimum energy geometry. In these cases, the conjugate gradient method was used, as an alternative initial minimisation method [15]. This avoids searching in the same direction as previous steps by finding a search direction that is independent to that o f previous iterations. The Hessian does not need to be calculated, which reduces the cost o f each iteration, but in general a greater number o f iterations are required. Alternative methods can be applied when the gradient norm is reduced to a sufficient level, e.g. the RFO method at gradient norms below 0.01 eV Â'^ in order to find the true minimum.

Cliapter 2 Metliodology o f interatomic potential computer models

However, a major limitation o f all o f these algorithms are that they are limited to finding the local minimum - the closest structure to the initial one that has zero gradient. In fact, it is the global minimum that is often sought and this may not always be found, as illustrated in Figure 2.2.

Figure 2.2 Determination o f local and global minima. X represents the initial structure fo r the calculation, Y represents the local minima that most algorithms will find, and Z

represents the global minima that most o f them would not

In order to avoid such a situation arising it is necessary to determine an initial structure with an energy as close to the global minimum as possible. The structures (unit cell parameters and motif) o f the siliceous frameworks comprising the zeolite lattices modelled were taken from a data base compiled from X-ray crystallographic data, which is part o f the CERJUS^ visualisation package [16]. These siliceous structures were optimised before the substitution o f acid sites into them, as well as afterwards, so as to reduce the minimisation required for the acid site calculations.

Our main subjects o f interest in the zeolite structures are the Bronsted acid sites substituted into the siliceous structures. Once a siliceous bridging site is selected for this acid site substitution, the positions o f the A1 and 01 ions are relatively straightforward to determine by simple substitution o f an adjacent Si and O, respectively. However, the position of the H ion requires more careful consideration. An appropriate starting position o f the H ion was determined to be at 1.0 Â from the 01 ion, in the same plane as the Si, 01 and A1 ions, such that the Si-Ol-H angle was equal to the Al-Ol-H angle, as shown in Figure 2.3.

Chapter 2 M ethodology o f interatomic potential computer models

01

Figure 2.3 Initial geometry o f Bronsted acid sites

At first these were set up by hand using the CERIUS^ visualisation package, which gave rise to a certain amount o f variation from this ideal structure, hut later C and FORTRAN computer programs were written implementing simple vector mathematics to calculate the H position from the co-ordinates o f the Al, 0 1 , and the adjacent Si in order to standardise this procedure.