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.