DMC Theory¶

Imaginary Time Evolution¶

The imaginary time Schrödinger equation is a diffusion equation with a source/sink term.

\[ -\frac{\partial \Psi}{\partial \tau} = -\frac{1}{2} \nabla^2 \Psi + (V - E_T) \Psi \]

By simulating a population of walkers that diffuse (kinetic energy) and branch (potential energy), DMC samples the ground state distribution.

Importance Sampling¶

To improve efficiency, we sample the distribution \(f(\mathbf{R}, \tau) = \Psi_T(\mathbf{R}) \Psi(\mathbf{R}, \tau)\). The evolution equation for \(f\) includes a drift term guided by the trial wavefunction.

Fixed-Node Approximation¶

The fixed-node approximation enforces the nodes (zeros) of the wavefunction to be the same as those of the trial wavefunction \(\Psi_T\). This prevents the "fermion sign problem" where the wavefunction would otherwise collapse to the bosonic ground state.

The fixed-node energy is variational: \(E_{FN} \ge E_0\). The error depends on the quality of the nodal surface of \(\Psi_T\).

Non-Local Pseudopotentials¶

When using non-local pseudopotentials, the locality approximation or T-moves (Casula moves) are used to handle the non-local integration. In CHAMP, icasula controls this behavior.