6. Importance-sampled diffusion Monte Carlo
Position in the course: Lesson 6 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Imaginary-time projection; logarithmic derivatives
Learning goal: explain the mathematical steps, reproduce the analytical examples, and state the conditions under which the conclusions hold.
1. Transform the evolving amplitude
Take a real time-independent guide \(g=\Psi_T\) positive inside a chosen nodal pocket and set \(f=g\Phi\). Define drift \(v_D=\nabla\ln g\) and local energy \(E_L=(Hg)/g\). Begin with \(\partial_\tau f=g\nabla^2(f/g)/2-(V-E_{\mathrm{ref}})f\). The product rule gives
Since \(\nabla^2g/g=\nabla\cdot v_D+|v_D|^2\), regrouping produces
Diffusion and drift guide configurations toward important regions; local-energy differences control branching. Some literature defines the quantum force as \(F=2v_D\), so its drift term is \(F/2\). Check the convention before transferring equations to an implementation.
2. A short-step algorithm and its limitations
A simple Euler proposal is \(R'=R+v_D(R)\Delta\tau+\sqrt{\Delta\tau}\,\xi\), where every component of \(\xi\) is standard normal. A symmetric endpoint branching approximation uses \(w=\exp[-\Delta\tau(E_L(R)+E_L(R')-2E_{\mathrm{ref}})/2]\). These expressions illustrate the generator; production methods may use acceptance corrections, bounded drifts, improved Green functions, and specific nodal-boundary handling. Near a node \(\nabla\ln|\Psi_T|\) can diverge. Calling the Euler schematic an exact DMC solver would be misleading.
3. Worked oscillator drift and a finite-step warning
For exact guide \(g=e^{-x^2/2}\), \(v_D=-x\) and \(E_L=1/2\). Taking reference \(1/2\) makes branching constant. The continuous stochastic differential equation \(dx=-x\,d\tau+dW\) has stationary variance \(1/2\). Euler instead gives \(x'=(1-h)x+\sqrt h\xi\). If its stationary variance is \(s^2\), then \(s^2=(1-h)^2s^2+h\), so
At \(h=0.1\), \(s^2\approx0.5263\), already differing from \(0.5\). This exact recurrence isolates discretization bias despite perfect guiding and no branching noise. It does not characterize all corrected DMC implementations.
4. The asymptotic distribution and energy
The stationary importance-sampled product is \(\Psi_T\Phi_{\mathrm{FN}}\), rather than \(|\Phi_{\mathrm{FN}}|^2\). Its local-energy average is the mixed energy. For a local Hamiltonian and consistent fixed-node boundaries, self-adjointness connects this average to the projected eigenenergy. General observables require separate estimators. The reference-energy feedback used to control population is not automatically the preferred energy estimate.
In an ideal unconstrained positive local problem, different admissible guides should produce the same projected energy after limits are controlled. Constrained fermionic nodes or phase can instead change the solution itself. At finite timestep, finite population, or with localization approximations, additional guide dependence can appear. State which ideal invariance is expected and which approximations remain before interpreting guide-dependent results as nodal differences.
5. Exercises and answers
Exercise: What are the noise variance and standard deviation in one coordinate of the Euler proposal?
Solution
Variance is \(h\) and standard deviation is \(\sqrt h\), not \(h\). Confusing them changes the diffusion coefficient and therefore the represented operator.
Exercise: Does more sampling remove the Euler stationary-variance error?
Solution
No. It converges more accurately to the finite-step process's biased variance. Reduce timestep or use an appropriate corrected transition.
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