Computational Physics

12.3 Diffusion Monte Carlo 395

we assume the spins of theNfermions to be given, then the nodes form (3N−
1)-dimensional hypersurfaces in the 3N-dimensional configurational space. The
obvious zeroes ofψwheneverxi =xjfor any pairi =jdefine a (3N−3)-
dimensional scaffolding for the nodal surface structure. This scaffolding does not
depend on the particular form of the trial function. A node of a one-electron orbital
in the Slater determinant occurring in the wave function should not be confused
with a ‘fermionic zero’, as such an orbital node does not force the many-electron
wave function to vanish: one of the electrons, sayi, might be at a zero of some
orbital, but the wave function also contains contributions with the coordinates of
the electrons permuted, and in general the coordinates of the other electrons are
different from those of electroni.
Changing the diffusion Monte Carlo method to a fixed-node simulation is easy.
Simply add the following step just after having generated a new trial position
of a particle, sayi. Check whether the trial wave function changes sign for this
displacement. If this is the case, the move is not accepted, otherwise proceed as in
the boson case. The interested reader can implement the fixed-node extension and
test it, for example, for the lithium atom, taking an appropriate Slater determinant
fortheguidefunction.MoredetailscanbefoundinRef. [9].

∗The transient estimator method

In view of the variational error present in the fixed-node method it is worthwhile to
devise other methods. A method which does not depend on fixed nodal surfaces is
thetransient estimatormethod. To understand how and why this method works, it is
important to realise that the Hamiltonian and hence the time-evolution operator are
the same for fermions and for bosons. However, because the time-evolution operator
is symmetric with respect to particle permutations, an antisymmetric (fermionic)
initial state will remain antisymmetric and a symmetric (bosonic) state remains
symmetric.
Let us split an arbitrary fermion wave functionφinto two parts,φ−andφ+,
which contain the negative and positive parts ofφrespectively (all wave functions
depend on all the spin-orbit coordinatesX=(x 1 ,x 2 ,...,xN), and on imaginary
timeτ):

φ+=^12 (|φ|+φ) (12.63a)

φ−=^12 (|φ|−φ), (12.63b)

so that

φ=φ+−φ−. (12.64)