My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 5.
THE JELLIUM SLAB
At the minimum, 5 this quantity must be stationary, giving
[
T i (α) − ¯T (α) + V i − ¯V
∂O
∂α = 2 ∑ i
][ ∂Ti
∂α − 1 ∑
N
j
]
∂T j
= 0. (5.10)
∂α
As well as the desirable minimum, when E L is constant, there is at least one other
stationary point, when ∂T i /∂α is constant; it is not trivial to determine whether or
not this is a (local) minimum. It may be the case that there is a false minimum for
the jellium slab in which the variance minimisation procedure becomes stuck.
The question of the correct form for the Jastrow factor is also a difficult one.
If the form is wrong, giving too much or too little flexibility, then the variance
minimisation procedure cannot be expected to work.
Several different forms for
both the one- and two-body terms were tested and found to be equally unsuccessful
under variance minimisation.
The result is that in order to perform realistic VMC or efficient DMC simulations,
manual optimisation is required. One parameter is adjusted at a time: VMC
simulations for different values of the parameter are performed, and the results are
fitted to a quadratic form.
The traditional u function described by equation (3.79), which has been successful
in simulations of the homogeneous electron gas, was used in the calculations of
chapter 6. Including this two-body term causes electrons to spill out of the slab;
the one-body term must restore the correct electron density, as discussed in section
3.4.1.
Chapters 7 and 8 are devoted to improving the Jastrow factor; the new version
is used in the calculations of chapter 9.
5 For an exact eigenstate, the true minimum has O = 0; however, this point is not generally
accessible.
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