CASINO manual - Theory of Condensed Matter

When plotting a binned estimator, it is convenient to draw a single point per bin rather than a
horizontal line, and this point should be located at the geometrical centre of the bin,
r p = 1 Ω p
∫Ω p
rdr . (315)
If any symmetry constraints are imposed, e.g., the function is spherically symmetric and it is accumulated
in spherical bins along a radial grid, the above expression should be modified so that the
integrand remains the coordinate on the left-hand side. For example, to calculate the radius r p at
which we should plot ρ p ≈ ¯ρ(r p ) in the spherical annulus Ω p , we would use the expression
r p = 1 Ω p
∫Ω p
rdr =
4π
4π (b 3 − a 3 ) /3
∫ b
a
r 3 dr =
where a and b are the inner and outer radii of Ω p , respectively.
(
b 4 − a 4) /4
(b 3 − a 3 ) /3 = 3 b 3 + b 2 a + ba 2 + a 3
4 b 2 + ba + a 2 , (316)
Using similar analysis, we can calculate the standard error in ¯ρ, σ¯ρ (r), given by
∑
σ 2¯ρ(r) M
m=1
=
w m (ρ(R m ) − ¯ρ(R m )) 2
( ∑ M
) . (317)
∑M
M
m=1 w m=1
m −
w2 m
∑ M
m=1 wm
Note that contribution to the error from serial correlation has been neglected; the configurations are
assumed to be independent.
We are also interested in expectation values that represent a quantity associated with the relative
vectorial position of two particles (the ith and the jth particles, for instance) being fixed at r, such
as the intracule density, h(r).
The spherical average of the intracule density, h sph (r) is
h sph (r) = 1 ∫
1 ∑ f(R)δ(|rij | − r)dR
∫
2 4πr 2 , (318)
f(R)dR
such that ∫ 4πr 2 h sph (r)dr = N(N − 1)/2.
i≠j
This accumulation process in this case is analogous to that in the single-particle case, and one only
needs to include a factor of 1/2 and replace r i with r ij in the formulae in this section to adapt them
to the two-particle case.
The n th radial moment µ n of a function g(r) is defined as
∫
∫ ∞
µ n = |r| n g(r)dr = 4π r n+2 g sph (r)dr . (319)
The moments of the charge density are given by g(r) = ρ(r) and the moments of the intracule density
are given by g(r) = h(r). They are calculated for n = −2, −1, 1, 2, 3 and written to rad mom.dat.
The systems we deal with usually contain several indistinguishable particles for which the expectation
value of any r i -dependent observable should give the same result. It is possible to take statistical
advantage of the presence of indistinguishable particles by averaging any QMC-accumulated quantities
over all particles of the same type.
Moreover, in cases where different particle classes are equivalent (e.g., when there are the same number
of up- and down-spin electrons in a system without magnetic fields, provided g(r i ) does not itself
depend on the spin of the particles) one can average over classes as well.
For r ij -dependent expectation values it is also possible to make use of the presence of indistinguishable
particles to achieve better statistics. In this case one has to average any QMC-accumulated quantities
over all particle pairs of the same relative type (e.g., all parallel-spin electron pairs).
This average has been omitted in this document for simplicity. However, it is important to do this in
practice for improved statistics.
0
33.2.3 Spin-density matrix
K eywords: density or spin density in a non-collinear spin system (both produce the same thing).
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