Chemistry Reference
In-Depth Information
Enforcing some localization constraint on the solved-for orbitals leads to
a different kind of problem than the eigenvalue problem. The total energy can
be expressed in terms of the localized orbitals, and then solving the electronic
structure problem corresponds to minimizing that total energy with respect to
variations of the values of the functions on the grid. Often the localized orbi-
tals are allowed to be nonorthogonal. So for the present discussion, the rele-
vant definitions are: (1)
wave functions
or
eigenfunctions
are solutions to the
Kohn-Sham system of equations; they possess the properties of normalization
and orthogonality, and they span the whole physical domain, and (2)
orbitals
may be localized functions on the grid that are used to represent the total elec-
tronic energy. They are often used to construct the density matrix. For case 2
we do not solve the eigenvalue problem in its standard form. Another common
term from quantum chemistry is
basis function
; basis functions are used to
''build up'' the eigenfunctions or orbitals from simpler functions. The basis
functions could be finite elements, Gaussian functions, or atomic-like func-
tions. The FD method is
not
a basis function method since it results from a
Taylor expansion of the function about a grid point.
The FD representation is the simplest approach and is thus the easiest to
implement in MG solvers. It leads to a highly structured and banded Hamil-
tonian matrix, as can be inferred from Eq. [13] for the Poisson problem. That
highly structured form is helpful when passing functions between grids of dif-
ferent resolution, which makes MG code development more straightforward.
Several early works in real-space electronic structure employed FD representa-
tions, usually using high-order forms.
59-61,159,160,163
The high-order FD repre-
sentation can yield accurate DFT results on grids with reasonable spacings
(roughly 0.2-0.3 au).
After initial testing on small systems, Chelikowsky's group extended
their real-space code (now called PARSEC) for a wide range of challenging
applications.
164
The applications include quantum dots, semiconductors,
nanowires, spin polarization, and molecular dynamics to determine photoelec-
tron spectra, metal clusters, and time-dependent DFT (TDDFT) calculations
for excited-state properties. PARSEC calculations have been performed on sys-
tems with more than 10,000 atoms. The PARSEC code does not utilize MG
methods but rather employs Chebyshev-filtered subspace acceleration
165,166
and other efficient techniques
167
during the iterative solution process. When
possible, symmetries may be exploited to reduce the numbers of atoms treated
explicitly.
Two of the first efforts to adapt MG methods for DFT calculations were
made by the groups of Bernholc
60,61,163
and Beck.
58,85,115,159-162
The method
developed by Briggs, Sullivan, and Bernholc
60
employed an alternative FD
representation called the Mehrstellen discretization. In this discretization,
both the kinetic and potential portions contain off-diagonal contributions in
the real-space Hamiltonian, as opposed to the standard high-order FD repre-
sentation, where the kinetic energy piece contains off-diagonal terms, but the
