Chemistry Reference
In-Depth Information
approach for studying nanoscale phenomena, where atomic behavior becomes
important but including quantum-level effects is less vital than modeling rela-
tively large systems (up to many millions of atoms). Lastly, quantummechanical
simulations are often the most appropriate way to investigate atomic-scale pro-
cesses where the detailed electronic behavior can be crucial.
We begin by briefly reviewing some of the computational methodologies
most commonly used in studies of the solid state, starting with the finite-
element method (FEM). The FEM (see Refs. 3-6 among many) is the compu-
tational technique most commonly used to investigate macroscale processes
such as stress-strain and thermal behaviors; it is based upon the idea of discre-
tizing the continuum into a set (mesh) of discrete subdomains (elements). The
partial differential equations that determine the material behavior are solved
at each mesh point (node). The behavior away from the nodes is then
determined using an interpolation scheme.
Atomistic modeling using semiempirical classical potentials (e.g., see
Refs. 7-13) is widely used to model phenomena at the nanoscale. In this
approach, the interatomic potential energy function is computed either using
a relatively simple analytic functional form or an interpolation of empirical
data points. In both cases, the information describing the specific material is
entered into the model through empirical parameters that are determined by
fitting experimental or ab-initio data. The equilibrium atomic configuration
of the system is then found using minimization procedures such as conjugate
gradient
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or Monte Carlo techniques (see Refs. 15-17 among many). Using
semiempirical classical potentials, the dynamical evolution of the system can
be determined as well, by deriving the forces acting on the atoms from the
expression for the energy, and applying a molecular dynamics (MD)
approach.
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In this level of theory, no quantum mechanical behavior is
explicitely included, so results are only as good as the parametrization used,
which also means that transferability can be a problem.
Finally, the atomic scale can be investigated by using tight-binding (TB),
density functional theory (DFT), or even lower level ab initio calculations. All
of these methodologies are within the quantum mechanical framework, with
TB
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being the least accurate (because of the many approximations) but
also the least computationally demanding. TB describes the electronic states
starting from the limit of isolated-atom orbitals, and is well suited for the inves-
tigation of materials characterized by fairly localized electrons, such as transition
metals and their alloys, or by covalent bonding, such as semiconductors and insu-
lators. A significant advantage of TB, beyond its computational speed, is that it
can be easily coupled with molecular dynamics (TBMD), providing a computa-
tionally efficient way to investigate the dynamical evolution of a system, while
retaining the most important aspects of its quantum behavior. In most applica-
tions, TB is used in its semiempirical form, i.e., the energy is approximated as the
sum of a ''band structure'' energy and a ''repulsive'' energy, where only the band
structure energy is actually found through a quantum mechanical calculation.
