POBLACION ASENTADA EN EL ÁREA DE LOCALIZACION DEL PROYECTO

The nudged elastic band (NEB) method is a method for finding the minimum energy path (MEP) across a potential energy surface (PES) between two metastable states. The MEP is the path for which the force, the negative potential energy gradient, is always directed along the path. It is also the path with the greatest statistical weight in transition state theory (TST). For an activated process, the MEP will always exhibit a saddle point, which is a local maximum on the PES in one direction and a local minimum in all others. The state of the system at the saddle point will be called the transition state.

The PES can be calculated as a function of ionic positions using any method, for example, force-field methods, but in this thesis DFT will be used to map out the PES. The determination of the MEP is useful, for the energy of the saddle point will give an approximation of the activation energy of a transition in harmonic TST (see Section 2.5) [103, 104, 105], and allow the calculation of the rate constant of such a transition.

The activation energy,Ea_{, is given through:}

Ea =Etrans−Einit. (2.65) where Etrans _{is the system energy at the transition state and E}init _{is the system}

energy in the reactant state.

The activation energy of a reaction can be determined experimentally, for example, by measuring the temperature dependence of a reaction. The results of such an exper- iment are given in Chapter 6, for porphycene tautomerisation, where the NEB method is used to determine the reaction path for and energy barrier to tautomerisation.

In the NEB method [106, 107], a series of images between the initial and final states are created, with coordinates R~i for the ionic positions of theith image. These

neighbouring image vectors, R~i and R~i+1 from one another. The MEP is then found

by minimising the total force of all the images.

What distinguishes the NEB method from other elastic band methods is that only the component of the spring force parallel to the tangent of the MEP at an image point is included in the total force of an image, along with the component of the true force perpendicular to this tangent. This prevents the true force from moving the images along the MEP and causing the images to slide away from the saddle point and towards the local energy minima.

In this scheme, therefore, the force on an image,i, is:

~

Fi=F~_isk− ∇E(R~i)⊥, (2.66)

whereF~s

ik is the spring force component parallel to the tangent of the MEP at image,

iand _−∇E(R~i)⊥ is the true force perpendicular to this tangent.

For this equation to be applied, the tangent unit vector to the MEP at i, ˆτi, must

be approximated. In the simplest approximation, ˆτi would be defined as:

ˆ τi= ~ Ri+1−R~i−1 |R~i+1−R~i−1| , (2.67)

but this can lead to convergence problems in the method due to ‘kinks’ in the elastic band developing when the true force perpendicular to the MEP is much bigger than the spring force. To deal with this problem Henkelman and coworkers [108] suggest a different approximation to the definition of the local tangent of the energy path. This approximation doesn’t calculate ˆτi as the displacement vector between images i+ 1

and i₋1 but as the displacement vector between image i and image i+ 1 or i₋1, whichever one has larger magnitude. When the image is a maximum or minimum, ˆτi

is given as a weighted average of the two displacement vectors. This definition ensures that ˆτi always points ‘downhill’ on the PES from a saddle point, and never ‘uphill’

from a minimum point, since the latter does not guarantee that a saddle point is being approached.

~

F_is_k =k(_|R~i+1−R~i| − |R~i−R~i−1|)ˆτi. (2.68)

This has an advantage over a conventional definition of a spring force (i.e. F~s

ik =

k(R~i+1−R~i−(R~i−R~i−1))ˆτi) because it ensures the images are kept equally spaced

along the energy path.

One further alteration of the NEB method that is used in Chapter 6 is in the treatment of the forces acting on the image that is highest in energy. This thesis uses an implementation of climbing image NEB (CI-NEB), where the image highest in energy is subject to forces that ensure it converges (or ‘climbs’) to the saddle point. The forces on the image highest in energy do not include contributions from the spring forces, rather they are defined to be:

~ Fs

i,max=−∇E(R~i,max) + 2(∇E(R~i,max)·τˆi)ˆτi. (2.69)

The effect of Eq. (2.69) is to invert the component of the true force along the path and so push the image up the path.

Once the activation energy,Ea_{, has been determined through the CI-NEB method}

it can be compared with activation energies obtained experimentally. However further corrections to account for the quantum nature of the reactants, for example through the inclusion of ZPE effects (as in Henkelman et al. [109]), may be required. In this case the calculation of the conformation at the saddle point can additionally be used as input in a determination of the rate constant of a process using transition state theory (described in Section 2.5). Such an approximation for the quantum nature of nuclei lowering the barrier is applied in Chapter 6.

2.4

The Harmonic Approximation of Molecular Vibra-