The FN-DMC method [RCAJ82] modifies the DMC method to allow an approximate
treatment of excited states of many-body systems. The idea of the FN-DMC method is
to treat excited states by ``enforcing'' the positive definiteness of the
probability distribution
. The function
is positive definite everywhere in configuration space, and can hence
be interpreted as a probability distribution, if
and
change sign together, and thus share the same (high-dimensional) nodal surface. To
ensure positive definiteness of
, the trial wave function
imposes a nodal constraint, which is fixed during the calculation. Within this
constraint, the function
is propagated (following a scheme very similar
to that outlined in Sec. 2.3), and reaches an asymptotic distribution for
large
,
. In the FN-DMC method,
is an approximation to the exact excited eigenfunction of the many-body
Schrödinger equation (and not the groundstate wave function as in the DMC method).
It can be proven that, due to the nodal constraint, the fixed-node energy is a
variational upper bound to the exact eigenenergy for a given
symmetry [RCAJ82]. In particular, if the nodal surface of
were
exact, then
would be exact. Thus, the FN-DMC energy depends
crucially on a good parameterization of the many-body nodal surface.
Thus, in a FN-DMC simulation the function
, where
denotes the wave function of the system and
is a trial function used for importance sampling, is evolved in
imaginary time according to the Schrödinger equation
In the above equation
,
denotes the local energy,
is the quantum drift force,
plays the role of an effective diffusion constant, and
is a reference energy introduced to stabilize the numerics. The energy and other
observables of the state of the system are calculated from averages over the
asymtpotic distribution function
. To ensure positive
definiteness of the probability distribution
for fermions, the nodal structure
of
is imposed as a constraint during the calculation. It can be proved
that, due to this nodal constraint, the calculated energy is an upper bound to the
eigenenergy for a given symmetry [RCAJ82]. In particular, if the nodal
surface of
were exact, the fixed-node energy would also be exact.
Construction of the trial wave function, as well as evaluation of the energy, is described in Sec. 2.6.