Why Real Space

A chapter in an earlier volume in this series devoted to quantum Monte Carlo (QMC) methods noted that there are many ways to skin a cat this chapter discusses yet another In quantum chemistry, the dominant theme over many decades has been basis set calculations. The basis sets consist of localized Slater-type orbitals or Gaussian functions that are adapted to provide accurate representations of the electron states in atoms. The main advantage of this approach is that the basis sets typically do not have to be terribly large in size since they already contain a lot of the detailed atomic information. A disadvantage is that it can be difficult to obtain an unambiguously converged result due to factors such as basis set superposition errors.  

Another real-space approach is to use a finite-element (FE) basis. The equations that result are quite similar to the FD method, but because localized basis functions are used to represent the solution, the method is variational. In addition, the FE method tends to be more easily adaptable than the FD method a great deal of effort has been devoted to FE grid (or mesh) generation techniques in science and engineering applications. Other real-space-related methods include discrete variable representations, Lagrange meshes,and wavelets.  

As we noted above, in an FD or FE solution, the exact result emerges as the grid spacing is reduced in that respect we can call these methods fully numerical in the same spirit as the plane-wave approach from physics. An added advantage 

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