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