Time Step Control¶

The time step \(\tau\) is a crucial parameter in DMC.

Time Step Error¶

The DMC algorithm involves a short-time approximation for the Green's function, which introduces a bias proportional to \(\tau\) (or \(\tau^2\) depending on the algorithm).

To obtain accurate results, one should: 1. Perform calculations at multiple time steps (e.g., \(\tau = 0.01, 0.005, 0.002\)). 2. Extrapolate the energy to \(\tau \to 0\).

Choosing \(\tau\)¶

• Large \(\tau\): Faster equilibration, smaller statistical error for same CPU time, but larger bias.
• Small \(\tau\): Smaller bias, but requires more steps to reduce statistical error.

A good starting point is often \(\tau \approx 0.01\) - \(0.05\) a.u. for all-electron calculations of light atoms, or larger for pseudopotential calculations.

Acceptance Ratio¶

The acceptance ratio of the Metropolis step in DMC is a good indicator. It should be high (> 99%) for the time step error to be small. If it drops significantly, \(\tau\) is likely too large.