CASINO manual - Theory of Condensed Matter

The energy minimization method used in casino has two additional benefits. Firstly, it does not re-use
the same set of configurations with different parameters. This means that, when optimizing parameters
which change the nodal surface, it will not encounter difficulties caused by old configurations lying
very close to the new nodal surface. Secondly, it is especially accurate when optimizing parameters
which appear linearly in the wave function (i.e., determinant coefficients). When only optimizing such
parameters, the global energy minimum is found, usually in one cycle.
Sections 25.3.2 and 25.3.3 briefly summarize the theory of the method, and Sec. 25.3.6 explains its use.
The theoretical background given here is far from exhaustive. For a much more thorough discussion,
see Refs. [60, 61, 62, 63].
25.3.2 Basic theory
Consider a (generally complex) trial wave function Ψ which depends upon a set of p real, variable
parameters α. If the cycle Ψ (n) → Ψ (n+1) involves parameter changes α (n) → α (n) + δα (n) , then we
can Taylor-expand Ψ (n+1) = Ψ(α (n+1) ) as:
where Ψ (n+1)
lin
Ψ(α (n+1) ) = Ψ(α (n) ) +
is the linear sum
= Ψ (n+1)
lin
+ O
p∑
i=1
Ψ (n+1)
lin
=
δα (n)
i
with the coefficients {a i } and the basis functions {φ i } defined as:
∣
∂Ψ ∣∣∣α (
+ O [δα (n) ] 2) (203)
∂α i (n)
(
[δα (n) ] 2) , (204)
p∑
a i φ i , (205)
i=0
a i =
φ i =
{ 1 i = 0
δα (n)
i i ≠ 0
{ Ψ(α (n) ) i = 0
∂Ψ
∂α i
∣ ∣∣α
(n)
i ≠ 0.
(206)
(207)
The form of Ψ (n+1)
lin
allows {a i } to be optimized using diagonalization (the freedom of normalization
in the resulting eigenvectors of coefficients can be exploited to demand that a 0 = 1). The energy
minimization method in casino makes the approximation that
Ψ (n+1) ≃ Ψ (n+1)
lin
, (208)
and determines the parameter changes {δα i } at each cycle by optimizing {a i } using diagonalization,
and taking the eigenvector of coefficients corresponding to the lowest eigenvalue. This approach is
motivated by the existence of a formulation of diagonalization for VMC which incorporates a zerovariance
principle similar to that enjoyed by variance minimization [63].
In the standard derivation of diagonalization, Ψ (n+1)
lin
would be optimized by demanding that the
variational energy be stationary with respect to the variable parameters:
∂ 〈Ψ (n+1)
lin
∂α i
which leads to the generalized eigenproblem
|Ĥ|Ψ(n+1) lin
〉
〈Ψ (n+1)
lin
|Ψ (n+1)
lin
〉
= 0, (209)
(H + H T )a = E(S + S T )a, (210)
where:
S ij =
H ij =
〈φ i |φ j 〉
〈Ψ (n+1)
lin
|Ψ (n+1)
lin
〉 ; (211)
〈φ i |Ĥ|φ j〉
〈Ψ (n+1)
lin
|Ψ (n+1)
lin
〉 . (212)
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