CASINO manual - Theory of Condensed Matter

Since H T = H ∗ , there is more than one possible way to VMC-estimate H + H T . For example, the
choice
〈 ( ) )
∗ ( ) )〉
∗ φi
(Ĥφj φj
(Ĥφi
+
(213)
φ 0 φ 0 φ 0 φ 0
|φ 0| 2
ensures that the VMC estimate of the H matrix is symmetric, but does not ensure that it is real. The
different choice
〈 ( ) )
∗ ( )
φi
(Ĥφj
( ) ∗ 〉
φi Ĥφ j
+
(214)
φ 0 φ 0 φ 0 φ 0
|φ 0| 2
ensures that it is real, but not necessarily symmetric. (The exact H+H T is both real and symmetric.)
In Ref. [63], Nightingale and Melik-Alaverdian show that it is possible to rederive diagonalization as a
least-squares fit, over a finite number of VMC configurations, to the ideal situation in which the basis
functions {φ i } span an invariant subspace of the Hamiltonian. Because this approach incorporates
finite-sampling from the beginning, it removes the ambiguity over how to VMC-estimate H + H T .
The analysis introduced in Ref. [63] (and used in Ref. [60, 61, 62]) assumes the wave function is real,
but it is not difficult to extend it to the complex case: what follows is a very concise outline.
Assume that the basis functions span an invariant subspace of the Hamiltonian, i.e.,
Ĥ|φ i 〉 =
p∑
E ji |φ j 〉 ∀ i, (215)
j=0
meaning that Ψ (n+1)
lin
will be an eigenstate of Ĥ if a is an eigenvector of E. In general, {φ i } are
complex, but we choose to restrict {E ji } to be real. Equation (215) cannot be exactly satisfied, so
instead we seek an approximate solution for E by minimizing:
2
M∑ p∑
p∑
χ 2 =
Ĥφ i (R σ ) − E ji φ j (R σ )
. (216)
∣
∣
Demanding that
σ=1 i=0
j=0
∂χ 2
∂E pq
= 0 ∀ p, q, (217)
and seeking the eigenvalues and eigenvectors of the resulting approximation to E, leads to exactly
the same eigenproblem as in Eq. (210). The difference is that the VMC estimate of H + H T is now
specified, as:
〈 ( ) )
∗ ( )
φi
(Ĥφj
( ) ∗ 〉
φi Ĥφ j
+
. (218)
φ 0 φ 0 φ 0 φ 0
|φ 0| 2
Using this expression to estimate H + H T gives the method a zero-variance principle: if the basis
functions {φ i } genuinely do span an invariant subspace of the Hamiltonian, exact diagonalization will
be achieved for any number of configurations greater than p. In practice, this condition never holds
true. Nevertheless, using this expression to VMC-estimate H + H T massively reduces the statistical
noise in the diagonalization process. As noted above, this expression does not guarantee that the
estimate of H + H T is symmetric. This means that the eigenvalues {E} and {a} may not be real,
but, since their imaginary parts only arise from statistical noise, we can simply discard them.
Using this method to optimize the trial wave function relies on the accuracy of the approximation
made in Eq. (208). If all the parameters being optimized appear linearly in the wave function, the
approximation is exact, and the global minimum of the energy will be found in one cycle. 25 If other
parameters are included in the optimization, the energy will converge to a local minimum over several
cycles, provided that the central approximation holds true.
25.3.3 Stabilization
The basic method described above can encounter two main problems: (i) linear dependencies in
the basis functions {φ i }; (ii) parameter changes too large for the approximation of Eq. (208) to be
25 In fact, a second cycle may sometimes improve the accuracy of the optimized parameters. The parameters resulting
from the first cycle will be inexact because of statistical noise. The second cycle uses an improved wave function, which
will result in lower noise when determining the second cycle’s parameters.
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