CASINO manual - Theory of Condensed Matter

valid. The first problem, which arises from redundancies in the parameters, leads to the algorithm
attempting to invert a singular matrix. Exact redundancies in the parameters should be avoided
when setting up the wave function, but parameters which are approximately redundant (to within
the numerical precision of the computer) are harder to detect. The solution, as described in Ref.
[63], is to use Singular Value Decomposition [64] for matrix inversions. The practical problems of
redundant parameters are discussed in more detail in Sec. 25.3.6. The second problem can cause
the optimization to diverge. Two techniques are used in casino to prevent divergence and stabilize
the algorithm: semi-orthogonalization of the basis functions, and ‘level-shifting’ [65, 66] in either the
H + H T matrix or the (S + S T ) −1 (H + H T ) matrix. Both ideas were first introduced to the method
by Umrigar et al. [60, 61].
25.3.4 Semi-orthogonalization
Consider multiplying Ψ by a (complex) renormalizing factor A(α) which does not depend on R:
˜Ψ(R, α) = A(α)Ψ(R, α). (219)
The same parameter set minimizes the energy expectation value taken using both Ψ and ˜Ψ, so we
are free to choose any A(α). However, the energy expectation values taken using Ψ lin and ˜Ψ lin are
minimized by different parameter sets. We can choose A(α) to improve the parameter changes. The
new basis { ˜φ i } is given by:
˜φ 0 = A(α (n) )φ 0 ; (220)
˜φ i = ∂(AΨ) ∣ ∣∣∣α
∂α i (n)
∣
= A(α (n) ∂A ∣∣∣α
)φ i + φ 0 (i ≠ 0). (221)
∂α i (n)
Following Ref. [60, 61], we choose A(α (n) ) = 1 (such that ˜φ 0 = φ 0 ), and
∣
∂A ∣∣∣α
= A i (α (n) ) = − 〈Λ|φ i〉
∂α i (n)
〈Λ|Ψ (n) 〉
∀ i. (222)
The basis { ˜φ i } is ‘semi-orthogonalized’, in the sense that, while the basis functions are not orthogonal
to one another, they are all orthogonal to a chosen wave function Λ. Umrigar et al. suggest choosing
Λ = ξ Ψ(n)
||Ψ (n) ||
Ψ(n+1)
lin
+ σ(1 − ξ)
||Ψ (n+1)
lin
|| , (223)
where 0 ≤ ξ ≤ 1 is a parameter which can be chosen freely, and σ takes the value 1 when the angle
between Ψ (n) and Ψ (n+1)
lin
is acute, and −1 when it is obtuse, i.e.,
[
]
σ = 1 × sgn Re(〈Ψ (n) |Ψ (n+1)
lin
〉) . (224)
The primary purpose of this choice of Λ is simply to reduce the size of the parameter changes, although
it also achieves certain other conditions for particular choices of ξ. This can be seen using geometrical
arguments, presented and explained 26 in Ref. [62]. Although it is possible to vary ξ in casino (see
Sec. 25.3.6), the default value of 0.5 is believed to be effective in all circumstances.
To use the semi-orthogonalized basis { ˜φ i }, an expression for A i (α (n) ) is needed. From expanding Eq.
(222):
A i = −ξ ¯S 0i D − σ(1 − ξ)( ¯S 0i + ∑ p
j=1 δα ¯S j ji )
ξD + σ(1 − ξ)(1 + ∑ p
j=1 δα ¯S , (225)
j j0 )
where
√
p∑
p∑
D = √1 + δα j ( ¯S j0 + ¯S 0j ) + δα j δα k ¯Sjk , (226)
j=1
and ¯S ij is the VMC estimate of S ij (note that, unlike for H ij , there is no ambiguity over how to do
these VMC estimates). These expressions differ very slightly from those appearing in Refs. [60, 61, 62],
because here the wave function is allowed to be complex.
26 The treatment in Ref. [62] does not include σ, but its necessity (noted by Umrigar et al.) is clear when one considers
the case where the angle between Ψ (n) and Ψ (n+1) is obtuse. Also, in Ref. [62], Ψ is assumed to be real, although the
lin
extension to the complex case is simple, and the correct expressions are given here.
j,k=1
161