My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
value. However, because these points (which constitute a region of zero volume) are
never sampled in a QMC simulation, their contribution is missed, and any expectation
values involving the Laplacian will not be accurate. This is not the case for the
integrable point singularities in the Laplacian of the electron-electron cusps, which
do not generate errors in QMC.
Therefore, in order to achieve proper sampling in QMC, and also to implement
the electron-electron cusp conditions correctly, it is necessary to smooth out the
plasmon wave function: both u pl and χ bulk must be modified to have continuous
first and second derivatives.
8.3.1 Removing undesirable cusps
The cusps in u bulk , u surf and χ bulk are a consequence of the piecewise way in which
these functions are constructed. A simple and appealing method of removing the
cusps is to blur the boundaries between the regions of definition.
For the purpose of illustration, consider an arbitrary function f(x) with the
following form near the point x = x 0 :
f(x) = f 1 (x)Θ(x − x 0 ) + f 2 (x)Θ(x 0 − x). (8.106)
A smooth approximation to f is
f s (x) = f 1 (x)T (x − x 0 ) + f 2 (x)T (x 0 − x) (8.107)
where the smoothing function T has the following properties:
• the value and first and second derivatives are continuous;
• lim
x→∞
T (x) = 1;
• lim T (x) = 0;
x→−∞
• the transition between these limiting values takes place over some quantifiable
distance ∆x.
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