CASINO manual - Theory of Condensed Matter

ather than dynamical variables. Section 34.1 states the force expression in VMC. Section 34.2 reports
the exact and approximate expressions for the forces in DMC under two different localization schemes.
Section 34.5 describes the implementation in casino and gives some practical advice.
34.1 Forces in the VMC method
We write the valence Hamiltonian for a many-electron system as
Ĥ = Ĥloc + Ŵ , (416)
where Ĥloc consists of the kinetic energy, the Coulomb interaction between the electrons and the
local pseudopotential, and Ŵ is the nonlocal pseudopotential operator. We now consider a general
parameter λ, e.g., a nuclear coordinate, which is used to vary the Hamiltonian, and upon which the
trial wave function Ψ T depends. Taking the derivative of the VMC energy E VMC with respect to λ
gives
∫ ( )
ΨT Ψ Ĥ′ Ψ T
∫
T Ψ T
dV ΨT Ψ T (E L − E VMC ) Ψ′ T
Ψ
F VMC = − ∫ − 2
T
dV
∫ . (417)
ΨT Ψ T dV
ΨT Ψ T dV
We use the notation α ′ = dα/dλ where α can be a function or an operator. The first term in Eq.
(417) is the Hellmann–Feynman theorem (HFT) force [97, 98] and the others are the Pulay terms [99].
34.2 Forces in the DMC method
In the DMC method, we may use two different pseudopotential localization approximation (PLA)
schemes to evaluate the nonlocal action of Ŵ on the DMC wave function Φ. In these schemes Ĥ is
replaced by an effective Hamiltonian [100, 20],
Ĥ A = Ĥloc + Ŵ Ψ T
Ψ T
, Ĥ B = Ĥloc + Ŵ + Ψ T
Ψ T
+ Ŵ − . (418)
The nonlocal pseudopotential operator Ŵ + corresponds to all positive matrix elements 〈r ′ | Ŵ + |r ′ 〉,
and Ŵ − corresponds to all negative matrix elements [20], where r is the 3N-dimensional position
vector for the N-electron system and N is the total number of electrons. Following Ref. [101], these
two approximations are referred to as the full-PLA (FPLA) and semi-PLA (SPLA) when using ĤA
and ĤB, respectively. The corresponding fixed-node DMC ground-state wave functions are denoted
by Φ A and Φ B .
The DMC energy can be written in the form
E D =
∫
Φ ĤΨ dV
∫
ΦΨ dV
, (419)
which includes the mixed DMC (Ψ = Ψ T ) and pure DMC (Ψ = Φ) estimates of the energy. In all
later expressions, Φ stands for either Φ A or Φ B , and Ĥ for either ĤA or ĤB. Taking the derivative
of the DMC energy with respect to λ gives
dE D
dλ
∫ ∫ [ )
) ]
Φ = Ĥ ′ Ψ dV Φ
(Ĥ ′ − ED Ψ + Φ
(Ĥ − ED Ψ ′ dV
∫ +
∫ , (420)
ΦΨ dV ΦΨ dV
for both the mixed and pure DMC methods. The first term in Eq. (420) is the HFT force [97, 98] and
the other terms are Pulay terms [99].
34.3 The mixed DMC forces
The total force in the mixed DMC method, F tot
mix , is obtained by setting Ψ = Ψ T in Eq. (420). After
some rearrangements, we arrive at
F tot
mix = F HFT
mix + F P mix + F V mix + F N mix, (421)
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