Plane wave basis functions

Note that the energy depends quadratically on the k factor. For infinite systems, the molecular orbitals coalesce into bands, since the energy spacing between distinct levels vanishes. The electrons in a band can be described by orbitals expanded in a basis set of plane waves, which in three dimensions can be written as a complex function. 

While plane wave basis sets have primarily been used for periodic systems, they can also be used for molecular species by using a supercell approach, where the molecule is placed in a sufficiently large unit cell such that it does not interact with its own image in the neighbouring cells. Placing a relatively small molecule in a large supercell to avoid self-interaction consequently requires many plane wave functions, and such cases are handled more efficiently by localized Gaussian functions. A three-dimensional periodic system, on the other hand, may be better described by a plane wave basis than with nuclear-centred basis functions. 

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Formally, each orthogonalized-plane-wave basis function may be written as (1 - P), where ijjk is a plane wave and P is the projection operator such that Pif/k gives the core-state component of Il>k ... 

In order to tackle large and complex structures, new methods have recently been developed for solving the eleetronie part of the problem. These are mostly applied to the pseudopotential plane wave method, because of the simplicity of the Hamiltonian matrix elements with plane wave basis functions and the ease with which the Hellmann-Feynman forces can he found. Conventional methods of matrix diagonalization for finding the energy eigenvalues and eigenfunctions of the Kohn-Sham Hamiltonian in (9) can tackle matrices only up to about 1000 x 1000. As a basis set of about 100 plane waves per atom is needed, this restricts the size of problem to... 

Figure B3.2.4. A schematic illustration of an energy-independent augmented plane wave basis function used in the LAPW method. The black sine function represents the plane wave, the localized oscillations represent the augmentation of the function inside the atomic spheres used for the solution of the Schrodinger equation. The nuclei are represented by filled black circles. In the lower part of the picture, the crystal potential is sketched.

Recent developments have attempted to combine localized and plane wave basis functions, i.e. describing the core region by radial polynomials or Gaussian functions, and the valence region by plane waves. This price of this approach is increased computational complexity, since new integrals involving different types of basis function are required. 

This corresponds to determining a set of LMOs that minimizes the spatial extent, i.e. they are as compact as possible. For extended (periodic) systems described by plane wave basis functions, the equivalent of the Boys LMOs is called Wannier orbitals. Feng el al have shown that the Boys LMOs can be made even more compact by 10-25% by allowing the localized orbitals to be non-orthogonal, but this requires a general optimization procedure, rather than a simple unitary transformation. 

The valence electrons oscillate in the core region as is shown in Fig. A5, which is difficult to treat using plane wave basis functions. Since the core electrons are typically insensitive to the environment, they are replaced by a simpler smooth analytical function inside the core region. This core can also now include possible scalar relativistic effects. Both the frozen core and pseudopotential approximations can lead to significant reductions in the CPU requirements but one should always test the accuracy of such approximations. 

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