CASINO manual - Theory of Condensed Matter

19.4.3 1D Coulomb interaction
Coulomb sums in systems that are periodic in one dimension (the x-direction, according to casino)
are performed using an algorithm based on the Euler–Maclaurin summation formula.
Such systems are done infrequently enough that it probably isn’t worth writing out all the theory
here. Please refer to Ref. [40], particularly noting Eq. (4.8).
Note that the first derivatives of the 1D Coulomb interaction have never been implemented in casino,
so core-polarization potentials may not be used in one-dimensionally periodic systems with atoms.
19.4.4 MPC interaction
The model periodic Coulomb (MPC) interaction [41, 42, 30] is used to reduce finite size effects in
periodic calculations. The exact MPC interaction operator is
Ĥe−e exact = ∑ f(r i − r j ) + ∑ ∫
ρ(r) [v E (r i − r) − f(r i − r)] dr
i>j
i WS
− 1 ∫
ρ(r)ρ(r ′ ) [v E (r − r ′ ) − f(r − r ′ )] dr dr ′ , (126)
2
WS
where f(r) is the 1/r Coulomb interaction treated within the ‘minimum-image’ convention, which
corresponds to reducing the vector r into the Wigner-Seitz (WS) cell of the simulation cell, v E is the
Ewald potential, and ρ is the electronic charge density from the many-electron wave function Ψ. The
electron–electron interaction energy is
which gives
Ee−e exact = 1 ∫
ρ(r)ρ(r ′ )v E (r − r ′ ) dr dr ′
2 WS
⎛
∫
+ ⎝ |Ψ| ∑ 2 f(r i − r j ) Π k dr k − 1 ∫
WS
2
i>j
E exact
e−e = 〈Ψ|Ĥexact e−e |Ψ〉 , (127)
WS
ρ(r)ρ(r ′ )f(r − r ′ ) dr dr ′ ⎞
⎠ , (128)
where the first term on the right-hand side is the Hartree energy and the term in brackets is the XC
energy. We can see that the Hartree energy is calculated with the Ewald interaction while the XC
energy (expressed as the difference between a full Coulomb term and a Hartree term) is calculated
with the cutoff interaction f.
In a DMC calculation we require the local energy at every step, but we only know the DMC charge
density, ρ, at the end of the run. Normally we have a good approximation to the charge density, ρ A ,
either from an independent particle calculation or a VMC calculation. We can avoid the need to know
ρ exactly by constructing a new interaction operator which involves only ρ A ,
Ĥ e−e = ∑ f(r i − r j ) + ∑ ∫
ρ A (r) [v E (r i − r) − f(r i − r)] dr
i>j
i WS
− 1 ∫
ρ A (r)ρ A (r ′ ) [v E (r − r ′ ) − f(r − r ′ )] dr dr ′ . (129)
2
The interaction energy becomes
Noting that
WS
E e−e = 〈Ψ|Ĥe−e|Ψ〉
∫
= ρ(r)ρ A (r ′ )v E (r − r ′ ) dr dr ′ − 1 ∫
ρ A (r)ρ A (r ′ )v E (r − r ′ ) dr dr ′
WS 2 WS
∫
+ |Ψ| ∑ ∫
2 f(r i − r j ) Π k dr k − ρ(r)ρ A (r ′ )f(r − r ′ ) dr dr ′
WS i>j
WS
+ 1 ∫
ρ A (r)ρ A (r ′ )f(r − r ′ ) dr dr ′ . (130)
2
E e−e = E exact
e−e − 1 2
∫
WS
WS
[ρ(r) − ρ A (r)] [ρ(r ′ ) − ρ A (r ′ )] [v E (r − r ′ ) − f(r − r ′ )] dr dr ′ , (131)
143