5. Imaginary-time projection and branching
Position in the course: Lesson 5 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Operator exponentials; eigenstate expansions
Learning goal: explain the mathematical steps, reproduce the analytical examples, and state the conditions under which the conclusions hold.
1. Replace oscillation by exponential filtering
Real-time evolution multiplies eigenstate \(n\) by \(e^{-iE_nt}\), retaining its magnitude. Set \(t=-i\tau\) to obtain imaginary-time evolution \(e^{-\tau(H-E_{\mathrm{ref}})}\). For an initial state \(\Phi_0=\sum_nc_n\Psi_n\),
Relative excited amplitudes decay as \(e^{-\tau(E_n-E_0)}\) when \(c_0\ne0\) and the relevant spectrum permits projection. An exact orthogonality to the desired state prevents its appearance. The reference energy adjusts normalization, not the relative filtering. Degenerate ground states, small gaps, and a restricted symmetry sector need separate interpretation. Imaginary time is not a physical clock for electronic trajectories.
2. Diffusion and reaction from the Schrödinger operator
With \(H=-\nabla_R^2/2+V(R)\),
The Laplacian is the generator of diffusion; the second term grows or removes amplitude. A small-step free diffusion kernel is Gaussian with coordinate variance \(\Delta\tau\). A simple splitting approximates potential weighting by \(e^{-\Delta\tau(V-E_{\mathrm{ref}})}\). Singular Coulomb potentials make this unguided formulation inefficient and unstable, motivating importance sampling in the next lesson. Operator splitting is approximate because kinetic and potential operators do not commute.
3. What branching represents
Positive weights can be represented by a population of walkers: diffuse configurations, weight them, then replicate or remove representatives according to weight. Alternatively use weighted walkers with resampling. The population represents amplitude, not a conserved number of electrons. Each walker still contains all \(N\) electronic coordinates. Stabilizing the number of walkers with feedback introduces finite-population effects; different resampling strategies produce correlations between descendants.
4. Worked spectral filter
Take a two-level initial amplitude with \(c_1/c_0=1\) and gap \(\Delta=0.5\) hartree. At \(\tau=4\) inverse hartree, the excited-to-ground amplitude ratio is \(e^{-2}\approx0.1353\). Its squared weight relative to the ground is \(e^{-4}\approx0.0183\). The difference between amplitude and probability exponents matters. This is an exact algebraic model, not a recorded walker calculation. To suppress the amplitude ratio below \(10^{-3}\) requires \(\tau>\ln(1000)/\Delta\approx13.82\) inverse hartree.
Projection duration and population size are separate controls. Duration determines whether transient excited components decay; population determines the representation and feedback correlations. A large population followed briefly may retain initial-state bias. Long projection with a small population can retain control bias. Report equilibration in imaginary-time units as well as step count when comparing timesteps, and vary these controls independently.
5. Exercises and explanations
Exercise: If \(E_{\mathrm{ref}}\) is increased by a constant \(c\), how does \(\Phi(\tau)\) change?
Solution
All components gain a common factor \(e^{c\tau}\). Relative amplitudes remain unchanged, so ideal projection physics is unaffected while normalization and walker growth change.
Exercise: Why can a positive walker population not directly project an arbitrary antisymmetric state?
Solution
An antisymmetric state has signs and cancellation. Positive amplitudes alone cannot encode that structure; unconstrained positive projection instead favors a lower bosonic-like state. A sign or nodal treatment is required.
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