My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
3.4 Trial wave functions
In VMC, the quality of the trial wave function sets a limit on the accuracy of the
calculation. In fixed-node DMC, this is also true to some extent: the nodes are
determined by the trial wave function, which therefore also determines the fixednode
error. However, the statistical efficiency of a DMC calculation strongly depends
on the quality of the trial wave function, as this determines the importance sampling.
In addition, from the computational point of view, evaluating the trial wave function
normally constitutes the major part of the calculation. The form of the trial wave
function is therefore very important; it must be both accurate and easy to evaluate.
The wave functions normally used in QMC simulations are of the Slater-Jastrow
type, consisting of a Slater determinant multiplied by an exponential Jastrow factor:
Ψ(X) = e J(X) D(X), (3.76)
where D(X) is a determinant of one-electron orbitals, exactly as in equation (2.13).
In fact, the computation is made significantly faster by the use of wave functions
of the form
Ψ(X) = e J(X) D ↑ (R ↑ )D ↓ (R ↓ ). (3.77)
The spin-dependence of the one-electron orbitals in the determinants has been removed,
and the evaluation of the two smaller determinants D ↑ and D ↓ is more
efficient than that of the large determinant D. This function is no longer antisymmetric
on exchange of electrons with opposite spins; however, the expectation value
of any spin-independent operator is unaffected by this alteration.
The single-electron orbitals may be obtained from density-functional theory or
Hartree-Fock calculations. The optimal orbitals in these two mean-field schemes
are usually very similar. The nodal surface of the resulting trial wave function
is completely defined by these orbitals, since the Jastrow factor is never zero; for
DMC calculations, this means the expectation value of the energy is not affected by
including the Jastrow factor. However, the variance of the energy is affected, and
the Jastrow factor is an important part of the trial wave function.
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