The Quest for Better Wavefunctions

One of the nice features of QMC is the freedom in the choice of wavefunction. The downside is it can be time consuming and unwieldy to optimize them in VMC.

In a recent article - Backflow Correlations for the Electron Gas and Metallic Hydrogen - Holzmann, et al. look at improving wavefunctions through adding analytic information.

The section with the Feynman-Kacs approach uses a cumulant expansion. In case you (read: me) need a reminder of the cumulant expansion, look at this online statistics text (chapter 8.4).

Followup to "Testing Monte Carlo"

I wrote a program that uses Simpson's rule to compute the energy of two particles in a box interacting via a Lennard-Jones potential


The web page and program are here

It runs in a second or so on my 1 GHz P3. I expect 3 particles should be doable.

Edit 9/4/2004, fixed the link.

Testing Monte Carlo

You've just written a brand new Monte Carlo code. It's producing vast amounts of data on your fast new computer. How do know if that random looking time series will average to a beautiful result and lead to deep insight, or is it simply random noise from the physics of some other universe?

There are several techniques we can use to verify that the codes we write are working correctly.

1. Unit testing
   This is standard software engineering (or *should* be standard). This involves small drivers to test individual classes or routines. It's good for verifying small, easily understood pieces before integrating into a larger system and for regression testing (making sure the latest change didn't break something).
   
2. Solve simple systems with analytic or well-known answers.
   The harmonic oscillator is popular. For QMC, H₂ is very well known, and very accurate answers are known from other techniques.
   
3. Compute internal quantities in multiple ways.
   The best example in QMC is the local energy - compute the derivatives analytically and numerically. In PIMC there are multiple energy estimators.
   
4. Compute system values with another integration method.
   It seems that computers are fast enough that we should be able to use grid-based methods to compute answers for small numbers (1-3) of particles.
   For QMC, shutting off the electron-electron interaction simplifies the problem, and Mathematica can handle the resulting integrals (numerically, not symbolically)
   
5. Comparison with literature
   Compare with experiment and answers obtained by other methods.
   

Do you believe your error bars?

Computing uncertainty estimates from Monte Carlo data usually assumes the data is normal (or can be sufficiently reblocked to be normal, according to the central limit theorem). But what if we don't have quite that much data - what effect would the underlying distribution have on the uncertainty estimates (error bars)?

Mervlyn Moodley looks at this question in the paper The Lognormal Distribution and Quantum Monte Carlo Data.

The interesting figures seem to be figures 3 and 9. In Figure 3, we see the confidence intervals change as the data is reblocked (and hence comes closer to normal). Figure 9 is generated from a real simulation, and it seems there is very little difference between the error bars computed assuming normality and those taking the underlying lognormal distribution into account.

Bilinear QMC

I'm reading a recent paper by Arias de Saavedra and Kalos titled "Bilinear diffusion quantum Monte Carlo methods" (PRE 67, 026708). They present an algorithm that uses a pair of walkers (the bilinear part) to sample the square of the ground state wavefunction (rather than first power of the wavefunction like DMC).

They claim it's useful for computing unbiased expectation values (see my post from Monday, Feb. 24 - although I'm not sure it would help with derivative operators) and energy differences.

The part I find most intriguing is the way they use importance sampling - they get reasonable results for hydrogen without it. And when they do use importance sampling, it's only to remove singularities due to cusps. I would guess that any successful scheme (ie, able to scale reasonably with system size) needs to use as an accurate a trial wavefunction as possible to guide the sampling.

The more I study the paper, the less I understand it. I guess the first step is to try reproduce their results for the 1-D harmonic oscillator.

Steal this research idea

Automatic control of Monte Carlo simulations - I want to know how to do it, but I don't necessarily want to figure it out myself. :)

• Determining equilibrium and phase transitions.
  Typically the particles are started in a lattice or random configuration and let run a whlie to equilibrate. I want automatic detection of equilibrium, which is especially tricky near phase boundaries. Also near phase transitions, the system can suddenly change phases.
  
• Adjustment of parameters for optimum efficiency
  The key quantity affecting the efficiency is the correlation time, but it's very noisy. Is there some other quantity that we could use that would behave similarly, but with less noise? Since this will only affect the efficiency, approximations are quite okay. Like assuming some analytic form for the autocorrelation function with a small number of parameters and fitting to it?
  

  
  Also note that SGA-like algorithms can be used for control as well as optimization. The decay parameter reaches a constant to follow the system, rather than continuing to decrease, as in optimization.
  

• Better control of DMC population
  
  Occasionally, I have trouble with the number of walkers increasing rapidly. In my codes, this means it exceeds some maximum value and stops the program. Sometimes the number of walkers jumps to a large but stable value, and the energy becomes far too low and clearly wrong. The hard part is that population control introduces a bias, and can't be too intrusive.
  

  
  For these items, are there any concepts from control theory or signal processing that would be useful?

Dynamic Monte Carlo

Checking the arxiv.org preprint server, there was an article titled An Introduction To Monte Carlo Simulations Of Surface Reactions by A.P.J. Jansen, which describes Dynamic Monte Carlo methods (Kinetic Monte Carlo is similar). One important input is transition probabilities. On page 21, "Estimates of the error made using DFT for such systems are at least about 10 kJ/mol."

Can QMC do better?