Methods Using Quantum-based Ideas
From simple QM arguments, basically from tight-binding (or Debye-Huckl)
some things can be learned about what a more accurate potential function
should have in it. To form
a solid starting from a separated collection of atoms or molecules
(which have there own inherent distribution of orbital states vs. Energy),
you bring them closer together and, due to bonding, this distribution
of states (usually called Density of States) broadens in energy so
that it is now quasi-continuous rather than discrete set. The new
distribution allows electrons to occupy the lower-energy states forming
stabilizing, bonding-type states. This is the standard
picture
of bonding (in solids or molecules).
As any distribution may be characterized by its moments, the
moments of the density of states, say for a d-band metal, gives:
- 1st moment: mean (or energy centroid of d-band DOS).
- 2nd moment: variance (energy width of the d-band DOS).
- 3rd moment: skew (shape relative to mean).
- 4th moment: kurtosis (amount of DOS in middle compared to ends).
Using this we find that:
- The binding energy should be proportional to the width (or 2nd
moment) because the binding results from the occupation of the lower
energy states and so must be related to width.
Thus, Potential Energy U ~ (2nd-mom)1/2.
- A projection of the DOS site-by-site yields the "Local DOS".
The moments of the LDOS may be related to the local bonding topology
via the so-called Moments Theorem:
The Nth moment of the LDOS on an atom is determined by the
sum over all paths comprised of N steps between neighboring atoms
that begin and end on the same atom.
For example, the 2nd moment then arises from all the "steps"
from the atom to its nearest-neighbor and back, which is just the
number of nearest neighbors, Z.
- Thus, Uatom~ (Z)1/2.
- For this Atomic Potential Energy, Finnis and Sinclair
(Phil. Mag. A50, 45 (1985) then argued to replace Z with sum of
exponentials to represent the valence electrons and balance this
attractive contribution with hard-core repulsion also given by
sum of exponetials, i.e,
this form. Bond-order potentials are slightly different.
A nice pictorial of this discussion may be found
from D.W. Brenner at NCSU.
This is the basic form for many "many-body" effective potentials
for metals, such as Finnis-Sinclair, Embedded Atom Method, Effective
Medium Theory. Because, however, there are no angular terms, these
potentials really only work reasonably well for close-packed FCC
methods. BCC metals have more interstitial region than FCC, hence,
more open and these forms break down. Note that
EAM does not
use exponential forms; rather it is an embedding function that
replaces the square root function above.
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