Energy

The energy is a direct output of the DMC algorithm. In fact, the population of walkers is stable only if the energy shift $E$ is equal to the value of ground state energy. The ground state energy can be expressed as an integral ratio

$$E_0 = \frac{\int\psi_T({\bf R})\hat H \phi_0({\bf R}){\bf dR}} {\int\psi_T({\bf R})\phi_0({\bf R})\,{\bf dR}},$$ (3.44)

where $\phi_0({\bf R})$ is the ground state eigenfunction of the Hamiltonian $\hat H\phi_0({\bf R})~=~E_0\phi_0({\bf R})$. By multiplying and dividing the integrand in the numerator by $\psi_{T}({\bf R})$, the formula (3.44) can be rewritten as

$$E_0 = \frac{\int\psi_T^{-1}({\bf R})\hat H\psi_T({\bf R})\psi_T({\bf R})\phi_0({\bf R})\,{\bf dR}} {\int\psi_T({\bf R})\phi_0({\bf R})\,{\bf dR}}$$ (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 $f$ is proportional to the product of the trial and the ground-state wavefunctions (see eq. (3.5))

$$\lim\limits_{t \to \infty} f({\bf R},t) = c_0 \psi_T({\bf R}) \phi_0({\bf R},t)$$ (3.46)

The calculation of the mean local energy of the walkers provides the value of the ground state energy^3.1

$$E_0 = \frac{\int E_L({\bf R})f({\bf R}){\bf dR}}{\int f({\bf R}){\bf dR}} = \frac{1}{N_W} \sum\limits_{i=1}^{N_W} E_L({\bf R_i})$$ (3.47)