DFT Total Energies—An Iterative Optimization Problem

We cannot continue this process indefinitely because in an fV-dimensional space we can only make a vector orthogonal to at most (N 1) other vectors. So to make this a well-defined algorithm, we have to restart the process of defining search directions after some number of iterations less than N. 

3 What Do I Really Need to Know about Optimization  

The most basic type of DFT calculation is to compute the total energy of a set of atoms at prescribed positions in space. We showed results from many calculations of this type in Chapter 2 but have not said anything about how they actually are performed. The aim of this section is to show that this kind of calculation is in many respects just like the optimization problems we discussed above. 

As we discussed in Chapter 1, the main aim of a DFT calculation is to find the electron density that corresponds to the ground-state configuration of the system, p(r). The electron density is defined in terms of the solutions to the Kohn-Sham equations, i /-(r), by 

We also need to think about how to stop our iterative calculations. It is not necessarily convenient to directly compare two solutions for the electron density and determine how similar they are, even though this is the most direct test for whether we have found a self-consistent solution. A method that is easier to interpret is to calculate the energy corresponding to the electron density after each iteration. This is, after all, the quantity we are ultimately interested in finding. If our iterations are converging, then the difference in energy between consecutive iterates will approach zero. This suggests that the iterations can 

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