Ab Initio Approaches In Large Systems

One of the major headaches in computational material science is how accurate theoretical techniques can be applied in large systems. By "accurate theoretical techniques" we mean quantum chemistry methods, called ab initio, due to the fact that they do not enforce any parameters to the system but solve the Schrodinger equations from first principles. The grater advantage of the ab initio methods is that they can provide structural, electronic, and dynamic properties of the calculating system in high accuracy. On the other hand the computational cost increases dramatically with the number of the electrons in the system.

The problem that arises in polyatomic systems is how to compromise the relatively large size of the system and an accurate ab initio method without ending up with a prohibitively large calculation. There are three possible solutions to this dilemma of treating large systems with ab initio methods: (a) the periodic density functional theory model, (b) the mixed quantum mechanics/molecular mechanics model, and (c) the cluster model.

4.1. Periodic Density Functional Theory Model

The periodic density functional theory (DFT) model is schematically presented in Figure 1a for a (4,4) SWNT. A part of the system (central part of the tube presented

Figure 1. Three possible models for treating large systems with ab initio methods. (A) Periodic DFT model simulating a (4,4) SWNT A part of the system (blue colored carbon atoms) is separated and treated as unit cell for a periodic building of the system. (B) The QM/MM model simulating a (4,4) SWNT The total 200-atom tube was separated in three cylindrical parts. The inner one was treated with DFT (40 blue color carbon atoms) while the two outer parts with molecular mechanics (brown color carbon atoms). The dangling bonds at the ends of the tube were saturated with hydrogen atoms. (C) The cluster model. A part of the system is separated and treated as individual cluster.

with blue color carbon atoms) is separated and treated as a unit cell. This unit cell is periodically repeated in the space (for the tube in one dimension only) building an infinite system. Implementation of periodic boundaries conditions to the equations solves the mathematical part of the problem. The advantage of this approach is that the total system is treated with ab initio techniques. The disadvantage is that local interactions and defects studied in the unit cell are automatically repeated periodically. In this way local properties become periodic. An example of the use of this approach is presented in Section 5.3.1.

4.2. Quantum Mechanics/Molecular Mechanics Mixed Model

The mixed quantum mechanics/molecular mechanics (QM/MM) model is schematically presented in Figure 1b. The basic idea of this method is that the system is divided in two parts: one is treated by ab initio methods and the other by molecular mechanics. The bordering of the two parts is arranged by introducing link atoms. More details for this model can be found in [21].

For the (4,4) SWNT presented in Figure 1b, the total 200-atom tube was separated in three cylindrical parts. The inner one was treated with DFT (40 blue color carbon atoms) and the two outer parts with molecular mechanics (brown color carbon atoms). The dangling bonds at the ends of the tube were saturated with hydrogen atoms.

The advantage of the QM/MM model is that no periodic constrains are introduced to the systems. The disadvantage is that only a relatively small part of the system is treated quantum mechanically while the rest is used for constraining the boundaries. This method is suitable for studding local properties. An example of the implementation of the QM/MM approach in SWNTs is presented in Section 5.3.2.

4.3. The Cluster Model

In the cluster model (Fig. 1c) a part of the system is separated and treated as an individual cluster. Extra attention has to be given to the dangling bonds that are generated at the braking areas. These dangling bonds have to be saturated properly with the addition of extra atoms (for example hydrogens). In this way the boundary instabilities that arise from QM/MM and the periodicity problems introduced by periodic DFT can be eliminated. The drawback of this method is that only a relatively small part of the system can be used. An example using the cluster model approximation is presented in Section 6.1.

It is clearly shown that no one of all these three techniques gives a perfect and unique solution to the problem. All show advantages and disadvantages and the decision as to which one we should choose have to be made according to the problem we face each time.

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