Chemistry Reference
In-Depth Information
The predominant computational motif for solid-state physicists, in
contrast, has centered on expansions of wave functions in plane-wave basis
sets.
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Advantages of this approach include: (1) simplicity of the basis, (2) abil-
ity to use efficient fast Fourier transform (FFT) operations during minimization,
(3) systematic convergence to the exact result with increasing basis set size (for a
given model), and (4) lack of
Pulay forces
during molecular dynamics simula-
tions.
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While this basis is well suited for periodic systems, a disadvantage is
that it is entirely delocalized in coordinate space—the basis functions have uni-
form magnitudes over all space. Thus, if we are interested in a localized system,
a lot of effort must be expended to add up many functions so as to get a result of
zero for regions that are far away from the molecule or cluster.
Real-space methods
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are conceptually situated somewhere between
the two limits of traditional basis functions and plane waves. The desired func-
tions (electrostatic potential, wave functions, electron density, etc.) are repre-
sented directly on a grid in coordinate space, and several techniques exist for
real-space representations of the partial differential equations to be solved on
those grids. The simplest is finite differences (FD);
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here the functions are
Taylor expanded about a grid point, and those expansions are manipulated to
obtain approximate representations of the derivatives. An important technical
point is that this approach is
not necessarily variational
in the sense of yielding
a result above the ground-state energy. The computed energy can lie above or
below the ground-state energy depending on the sign of terms omitted in the
expansion. The calculations do converge to the exact result as the grid spacing
is reduced, however.
Another real-space approach is to use a finite-element (FE) basis.
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The equations that result are quite similar to the FD method, but because loca-
lized basis functions are used to represent the solution, the method is varia-
tional. 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.
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Other real-
space-related methods include discrete variable representations,
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Lagrange
meshes,
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and wavelets.
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The main feature of real-space methods is that, if we represent the Lapla-
cian operator (Poisson equation) or Hamiltonian (eigenvalue problem) on a
grid, the application of the operator to the function requires information
only from a local region in space. We will develop these concepts below,
but here we note that only several neighboring grid points are required in gen-
eral to solve such problems (depending on the approximation order). The
matrix representation of these operators is thus highly banded, and application
of the operator to one function requires a number of operations that is the
width of the band times the number of grid points.
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
