Optimizing Supercell Volume and Shape

The calculations above allowed the positions of atoms to change within a supercell while holding the size and shape of the supercell constant. But in the calculations we introduced in Chapter 2, we varied the size of the supercell to determine the lattice constant of several bulk solids. Hopefully you can see that the numerical optimization methods that allow us to optimize atomic positions can also be extended to optimize the size of a supercell. We will not delve into the details of these calculations—you should read the documentation of the DFT package you are using to find out how to use your package to do these types of calculations accurately. Instead, we will give an example. In Chapter 2 we attempted to find the lattice constant of Cu in the hep crystal structure by doing individual calculations for many different values of the lattice parameters a and c (you should look back at Fig. 2.4). A much easier way to tackle this task is to create an initial supercell of hep Cu with plausible values of a and c and to optimize the supercell volume and shape to minimize 

In the exercises for Chapter 2 we suggested calculations for several materials, including Pt in the cubic and fee crystal structures and ScAl in the CsCl structure. Repeat these calculations, this time developing numerical evidence that your results are well converged in terms of sampling k space and energy cutoff. 

We showed how to find a minimum off(x) = e x cos x using the bisection method and Newton s method. Apply both of these methods to find the same minimum as was discussed above but using different initial estimates for the solution. How does this change the convergence properties illustrated in Fig. 3.6 This function has multiple minima. Use Newton s method to find at least two more of them. 

Newton s method is only guaranteed to converge for initial estimates that are sufficiently close to a solution. To see this for yourself, try applying Newton s method to find values of x for which g(x) = tan 1 x = 0. In this case, Newton s method is xl+ = x, (1 + xf) tan 1 x,. Explore how well this method converges for initial estimates including xo = 0.1,xo = 1, and 

We did not define a stopping criterion for the multidimensional version of Newton s method. How would you define such a criterion  

---