The energy is a direct output of the DMC algorithm. In fact, the
population of walkers is stable only if the energy shift is
equal to the value of ground state energy.
The ground state energy can be expressed as an integral ratio
(3.44)
where
is the ground state
eigenfunction of the Hamiltonian
.
By multiplying and dividing the integrand in the numerator
by
, the formula (3.44) can be rewritten as
(3.45)
The average of the Hamiltonian over
the trial wavefunction is the local energy (see definition (3.10)).
Since in the large time limit the distribution function
is proportional to the product of the
trial and the ground-state wavefunctions (see eq. (3.5))
(3.46)
The calculation of the mean local energy of the walkers provides
the value of the ground state energy3.1