CASINO manual - Theory of Condensed Matter

33.2.2 Atomic densities
K eywords: density or spin density (both produce the same thing)
Density accumulation for finite systems is only available for single atoms; the molecular case has not
been coded in casino.
In the single-atom case, the charge density is obviously spherically symmetric, ρ(r) = ρ(r).
spherical average of the charge density is
ρ sph (r) = 1
4πr 2
N ∑
i=1
The
∫
f(R)δ(|ri | − r)dR
∫
f(R)dR
, (307)
where ∫ f(R) is the distribution of configurations.
4πr 2 ρ sph (r)dr = N.
The charge density is normalized such that
The analytical behaviour of ρ(r), ρ sph (r) at r → 0 and r → ∞ is known,
ρ ′ sph(0) = −2Zρ sph (0) , (308)
where Z is the atomic number, and
(
ρ sph (r) → Ar 2B exp −2 √ )
2Ir , (309)
r→∞
where A is a constant, B = (Z − N + 1)/ √ 2I − 1, and I is the ionization energy.
The method used to calculate the charge density in QMC is presented below for the general case of
the charge density, but it is easily applied to quantities such as the spherical average of the charge
density.
Suppose we are performing a QMC calculation, sampling a distribution f(R). In the case of VMC,
f(R) = |Ψ(R)| 2 , while in DMC the distribution is f(R) = Ψ(R)Φ(R). The charge density can be
evaluated in QMC as
∑ M
m=1
ρ(r) ≈ ¯ρ(r) =
w mρ(R m )
∑ M
m=1 w , (310)
m
where {R m } M m=1 are the M configurations visited during the simulation and w m is the weight of each
of them, which is constant in VMC and varies from configuration to configuration in DMC. The
‘approximately equal’ sign becomes an ‘equal’ sign in the limit of perfect sampling, that is, when
M → ∞.
We can identify ∑ i δ(r i − r) with ρ(R), resulting in
∑ M ∑
m=1 i
ρ(r) ≈ ¯ρ(r) = w mδ(r i,m − r)
∑ M
m=1 w . (311)
m
However, Eq. (311) is not useful for accumulating ¯ρ(r) since Dirac delta functions cannot be evaluated
outside an integral.
We overcome this by accumulating data in bins of finite width. Let us partition three-dimensional
space into N exp disjoint regions or bins {Ω p }, and let us define the step function
{ 1 if r ∈ Ωp
Θ p (r) =
. (312)
0 if r /∈ Ω p
Denoting the volume of region Ω p by the same symbol, Ω p , we can construct an orthogonal functional
basis h p (r) = Ω −1/2
p Θ p (r). Then, Eq. (311) becomes
¯ρ(r) ≈
N exp
∑
p=1
that is, the function inside Ω p is approximated by a single value
∑ M ∑
m=1 i w mΘ p (r i,m )
∑ M
Ω p m=1 w Θ p (r) , (313)
m
∑ M ∑
m=1 i
ρ p = w mΘ p (r i,m )
∑ M
Ω p m=1 w . (314)
m
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