8. Mixed estimators, timestep, and population bias
Position in the course: Lesson 8 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: DMC; covariance; correlated errors
Learning goal: explain the mathematical steps, reproduce the analytical examples, and state the conditions under which the conclusions hold.
1. Know which distribution produced the estimate
For real states under consistent boundaries, the DMC mixed expectation is \(A_M=\langle\Psi_T|A|\Phi\rangle/\langle\Psi_T|\Phi\rangle\). A pure estimate instead uses \(A_P=\langle\Phi|A|\Phi\rangle/\langle\Phi|\Phi\rangle\). For projected energy, \(H\Phi=E\Phi\) gives \(H_M=E\), a special eigenstate identity. A position-dependent operator generally does not share this property. The trial density still influences the mixed estimate of density, dipole, or pair correlation.
2. Derive the extrapolated estimator's order
Let normalized \(\Psi_T=\Phi+\delta\) to first order, choosing \(\langle\Phi|\delta\rangle=0\) at that order. For real Hermitian \(A\), define \(a=\langle\delta|A|\Phi\rangle\). Then \(A_M=A_P+a+O(\delta^2)\), while \(A_V=A_P+2a+O(\delta^2)\). Therefore
This removes a first-order trial-state contribution, not all bias. Pure-estimator methods such as forward walking or reptation can avoid this particular approximation, but introduce their own projection-length and correlation issues. Error propagation for \(2A_M-A_V\) must account for covariance if shared data are used.
3. Timestep extrapolation is a model to test
A finite-step propagator produces \(E(h)=E(0)+ch+\cdots\) or another leading power depending on the algorithm and Hamiltonian. Do not assume universal quadratic convergence. With a justified linear regime and values at \(h\) and \(h/2\), \(E(0)\approx2E(h/2)-E(h)\). Two points always define a line and cannot validate linearity: use more steps, compare fit ranges, and propagate fit uncertainty. The imaginary-time duration must remain adequate when shrinking steps; holding step count fixed changes projection time.
4. Population control creates a different bias
Branching and feedback correlate descendants and normalization. Finite populations can bias estimates even at zero timestep. Increasing population, varying feedback/resampling choices, and checking stability are separate from increasing recorded blocks. A heuristic inverse-population fit may be useful in a demonstrated regime, but should not be assumed valid universally. Walker number is not an independent-sample count; analyze block time series and repeat independent runs.
A fit residual comparable to sampling noise does not prove the assumed functional form, especially with few points. Compare algorithmically justified forms and sensitivity to omitting coarse steps. Retain covariances for shared random streams or population histories. Report raw convergence points alongside extrapolated values so assumptions can be reviewed. A single precise extrapolated number can hide both an uncertain model and unresolved systematic limitations.
5. Worked extrapolation and exercises
For a constructed linear model \(E(h)=-1+0.2h\) hartree, \(E(0.1)=-0.98\) and \(E(0.05)=-0.99\). Combining them gives \(-1\). These are algebraic inputs, not executed DMC outputs. If each independent estimate has standard error \(0.001\), the extrapolated error is \(\sqrt{4+1}\,0.001\approx0.00224\), larger than either individual error.
Exercise: Does a mixed local-energy estimate's exact eigenstate identity prove that a mixed radius is pure?
Solution
No. The identity used \(H\Phi=E\Phi\). Multiplication by radius has no such eigenstate relation and retains trial-state bias.
Exercise: What systematic errors remain after timestep extrapolation?
Solution
Population, nodal/phase, finite-size, pseudopotential and observable-estimator errors can remain, along with statistical and extrapolation uncertainty. Each needs evidence or an explicit limitation.
Original teaching schematic of the mathematics or algorithm; it is not simulation or experimental data.
6. Sources and connections
Related theory: Classical Monte Carlo · Molecular methods · Density functional theory
Software connection: CP2K · Quantum ESPRESSO · Gaussian
These software courses provide related background on energies, orbitals, or convergence management; they do not imply that the Monte Carlo or QMC examples on this page were executed there.
7. Related theory and practice
Quantum mechanics · Molecular methods · Monte Carlo · Molecular dynamics · QMCPACK