CASINO manual - Theory of Condensed Matter

with
F HFT
mix = −
F P mix = −
F V mix = −
F N mix = −
∫
ΦΨT
(
Ŵ ′ Ψ T
Ψ T
)
∫
ΦΨT dV
dV
−
∫ [ (
Ŵ Ψ
′
ΦΨT
T
Ψ T
− Ŵ ΨT
∫
ΦΨT dV
∫
ΦΨT
[
Φ ′
]
(Ĥ−ED)ΨT
Φ Ψ T
∫
ΦΨT dV
∫
ΦΨT
[
( Ĥ−E D)Ψ ′ T
Ψ T
]
∫
ΦΨT dV
∫
ΦΨT V ′
loc dV
∫
ΦΨT dV
) ]
Ψ
′
T
Ψ T Ψ T
dV
dV
+ Z α
∑
β (β≠α)
Z β
R α − R β
|R α − R β | 3 (422)
(423)
(424)
dV
. (425)
is the mixed DMC HFT force and the other expressions are Pulay terms. The HFT force in Eq.
(422) contains two contributions from the pseudopotential, one from its local part V loc and one from
its nonlocal part Ŵ , and a third contribution from the nucleus–nucleus interaction. In this nucleus–
nucleus interaction term, R α represents the 3-dimensional position vector of the αth nucleus, and Z α
is the associated charge. The three Pulay terms in Eqs. (423)–(425) are identified as follows: Fmix
P
results from the PLA and is therefore called the pseudopotential Pulay term, Fmix V is the volume term,
and Fmix N is called the mixed DMC nodal term since it can be written as an integral over the nodal
surface [102]. Note that all terms in Eqs. (422)–(425) take the same form under both localization
schemes; the only difference is the distribution (Ψ T Φ A or Ψ T Φ B ) used to evaluate the expectation
values. A simple way to understand this is to note that Ĥ always acts on the trial wave function Ψ T
and ĤAΨ T = ĤBΨ T .
F HFT
mix
In mixed DMC simulations, it is straightforward to evaluate the contributions to the force, except for
the volume term Fmix V , because it depends on the derivative of the DMC wave function, Φ′ . Since
it is unclear how to evaluate Φ ′ in mixed DMC calculations, we use the Reynolds’ approximation
[103, 104],
Φ ′
Φ ≃ Ψ′ T
, (426)
Ψ T
which is exact on the nodal surface [see Eqs. (4) and (16) of Ref. [102]] but introduces an error of first
order in (Ψ T − Φ) away from the nodal surface.
34.4 The pure DMC forces
The total force in the pure DMC method, F tot
pure, is obtained by setting Ψ = Φ in Eq. (420). After
some manipulations, we obtain
F tot
pure = F HFT
pure + F P pure + F N pure, (427)
with
F HFT
pure =
F P pure =
⎧
⎪⎨
⎪⎩
−
⎧
⎪⎨
⎪⎩
F N pure = − 1 2
−
∫
Φ A Φ A
(
Ŵ ′ Ψ T
Ψ T
)
dV
∫
∫
Φ AΦ A dV
Φ B Φ B
(
(Ŵ + ) ′ Ψ T
Ψ T
− ∫
∫
ΦΦV
′
loc
dV
∫
ΦΦ dV
+ Z α
∑
∫
−
∫
−
∫
Φ A Φ A
[
Ŵ Ψ
′
T
Ψ T
∫
Φ B Φ B
[
Ŵ
+ Ψ
′
T
Ψ T
Φ B Φ B dV
+ (Ŵ − ) ′ Φ B
Φ B
)
dV
Z β
β (α≠β)
( ) ]
Ŵ Ψ Ψ ′ − T T
Ψ T Ψ dV T
Φ A Φ A dV
( ) ]
Ŵ
−
+ Ψ Ψ ′ T T
Ψ T Ψ dV T
∫
Φ B Φ B dV
⎫
⎪⎬
⎪⎭
R α − R β
|R α − R β | 3
(428)
(429)
Γ |∇ rΦ|Φ ′ dS
∫ . (430)
ΦΦ dV
199