Chemistry Reference
In-Depth Information
of real-space methods is that it is not terribly difficult to place different resolu-
tion domains in different parts of space.
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That is, the methods can uti-
lize adaptivity, and this can be done without sacrificing the numerical efficiency
of the real-space solver. In contrast, plane-wave methods utilize FFT operations
at their core, and it is difficult (although not impossible
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) to develop adaptive
meshes for these processes. Finally, real-space methods, due to their near local-
ity, are ideal for parallel computing. We can partition the problem into domains
in space—most of the operations are contained within each of the domains, with
relatively low communication overhead across the boundaries.
The real strength of real-space methods comes into play when we combine
the localized computational representation together with the physical localiza-
tion that exists in most chemical systems. For example, it is well known in
chemistry that relatively localized orbitals can represent accurately directional
chemical bonds in large molecules, thus allowing us to employ localized (often
nonorthogonal) orbitals as we minimize the total electronic energy.
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Those
localized orbitals are set to zero outside a specified domain in space. So, by
operating solely in real space, we can exploit this physical locality directly.
Despite the claims in chaos theory that a butterfly flapping its wings on the
other side of the globe might affect the weather here, small chemical changes
at a point in space typically do not have a great influence far away due to
screening. Kohn calls this the
near-sightedness
principal of matter.
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The com-
bination of a localized computational representation and physical locality can
lead to linear-scaling algorithms for electronic structure,
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a major aim in
computational chemistry.
This chapter discusses numerical methods for solving several important
differential equations in computational chemistry. It does not cover the con-
ceptually related problem of coarse-grained modeling of large amplitude
motions in polymers and biological macromolecules.
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REAL-SPACE BASICS
As outlined above, several means of representing partial differential
equations in real space exist. Here, for the most part, we choose the simplest
(the FD representation), restrict ourselves to second-order-accurate representa-
tions, and operate in one dimension so as to bring out all the important con-
cepts and avoid getting wrapped up in details. A short introduction to FE
representations is also included.
Equations to Be Solved
We consider here two of the most basic equations in computational
chemistry: the Poisson equation and the Schr¨ dinger equation. The Poisson
equation yields the electrostatic potential due to a fixed distribution of
