Planewave discretization

KSSOLV uses a planewave expansion to represent the Kohn-Sham eigenfunctions. For a \(R\)-periodic Kohn-Sham Hamiltonian, it follows from the Bloch theorem that an eigenfunction \(\psi_k(r)\) can be expressed as

(1)\[\psi_k(r) = e^{i k \cdot R} u_k(r),\]

where \(k\) is a reciprocal space vector in the Brillouin zone of the unit cell on which \(H\) is defined, and \(u_k(r) = u_k(r+R)\) is periodic, and can be further approximated as (a Fourier series):

\[u_k(r) = \sum_{j=1}^{n_w} c_{k,j} e^{i g_j \cdot r},\]

where \(c_{k,j}\) are the Fourier expansion coefficients, \(g_j\) is a reciprocal space vector, and \(n_w\) is the total number of expansion coefficients what determines the accuracy of the approximation.

As a result, \(\psi_k(r)\) can be written (approximately) as

\[\psi_k(r) = \sum_{j=1}^{n_w} c_{k,j} e^{i (k + g_j) \cdot r}.\]

The choice of \(k\) in (1) is determined by an additional boundary condition on multiple unit cells, and is discussed in BlochHam.

When the unit cell is sufficiently large, i.e., \(|R|\) is large, the Brillouin zone is sufficiently small that it can be approximated by a single point at the center of the zone (often referred to as the gamma point.) As a result, we can leave out the \(k\) index in the eigenfunction.

For an isolated system such as a molecule or nanocluster, which is not periodic, we can place the system in a ficticious supercell that is periodically replicated in space. When the supercell is sufficiently large, planewave expansion can be used to approximate eigenfunctions of a Kohn-Sham Hamitonian.