The Monte Carlo solution to the diffusion equation can be written as a
function of position, 
, and imaginary time, 
, as
follows
where the coefficients, 
, are the overlap integrals of
with the eigenfunctions of the many-body Hamiltonian,
. The DMC method relies on the fact that in the limit of
large imaginary time 
is dominated by the lowest
energy solution, 
.
However, in the initial short imaginary time regime, it is clear that
the above equation contains information about the energy differences,
. For example the time-dependence of the energy
estimate is given by
where the 
are the overlap integrals of the guiding wavefunction,
, with the eigenfunctions of the many-body Hamiltonian,
. Therefore, if one was to compute the energy estimate as a
function of time, then standard curve fitting methods could be used to
extract the excited state energies, 
. In practice
however, obtaining energies from Eq.(
) would be
extremely difficult due to the statistical noise of the Monte Carlo
simulation.
More sophisticated methods have been
devised[80, 81, 82] to
specifically measure the time dependence. Instead of the energy in
Eq.(), consider the expectation value of the
Green's function,
which can be sampled from the random walk by evaluating
where W is the cumulative branching weight,
which is essentially the total population. If we insert a complete
set of eigenstates into Eq.(), the time behaviour of
is
This is a simpler expression to attempt to fit than
Eq.(). However, in practice it is still dominated
by the statistical noise[23].
