The Way of Antisymmetry

The traditional QMC wavefunction is the product of a determinant of single particle orbitals and a Jastrow factor (which contains the electron-electron correlation). The determinant ensures the required spatial antisymmetry of the wavefunction, and is computable in O(N³) time (where N is the number of electrons). But it is not the most general form possible.

The most general form is obtained by explicitly antisymmetrizing: summing the values of a function with all possible permutations of the electron coordinates (and multiplying by the sign appropriate for the permutation). For example, with 2 particles and a function f(r₁,r₂), the antisymmetrized form is f(r₁,r₂) - f(r₂,r₁).

Unfortunately, the number of terms is N!. (The determinant form is obtained by applying this procedure when the function has the special form of the product of functions that each depend only on a single electron coordinate - these functions are the single particle orbitals. In the two particle example, f = f₁(r₁)f₂(r₂). The antisymmetrized form is f₁(r₁)f₂(r₂) - f₁(r₂)f₂(r₁), which is the determinant of a matrix.)

Dario Bressanini, Massimo Mella, Gabriele Morosi, and Luca Bertini have several papers using this form of the wavefunction for small molecules, and they get very good results. The papers can be found on Massimo Mella's publication list (edit 12/3/2005 - removed nonworking link). They are (this may not be a complete list):

• Many-electron correlated exponential wavefunctions. A Quantum Monte Carlo application to H2 and He2+
  
  Chemical Physics Letters 240, 566 (1995)
  
  Paper available at this Citeseer page
  
  
• Nonadiabatic wavefunctions as linear expansions of correlated exponentials. A Quantum Monte Carlo application to H2+ and Ps2
  
  Chem. Phys. Lett. 272, 370 (1997)
  
  Paper available at this Citeseer page
  
  
• Linear expansions of correlated functions: a variational Monte Carlo case study
  
  Int. J. Quantum. Chem. 74, 23 (1999)
  
  Paper available at this Citeseer page
  
  
• Explicitly correlated trial wave functions in quantum Monte Carlo calculations of excited state of Be and Be-
  
  J. Phys. B, (2001)
  

Not all N! terms will give a significant contribution to the result. Is it possible to use techniques similar to O(N) approaches to reduce the number of terms? Is it possible to sample terms from the sum? These might have trouble near the nodes, since the wavefunction vanishes due to the cancelation of all the terms.

DMC of porphyrin

An all electron calculation of a large molecule (large by QMC standards, anyway):
Quantum Monte Carlo for electronic excitations of free-base porphyrin by Alan Aspuru-Guzik, Ouafae El Akramine, Jeffrey Grossman, and William Lester. The porphyrin they used has the formula C₂₀ N₄ H₁₄, so 168 electrons.

Solid vs. Liquid, The Free Energy Battle

Determining the free energy difference between the solid and liquid states is essential for accurately mapping the phase diagram of a substance. Typically, one determines the free energies of the solid and liquid states through separate thermodynamic integrations, where each integration has one end point connected to a state with an analytically known free energy. In the solid state, the Einstein crystal (springs attaching particles to lattice sites) is often used as a starting point. In the liquid state, the low density gas can be used as an end point. Other more complicated scalings of the potential are also possible for the integration path.

A direct integration path between the solid and liquid states would be useful.
Gregory Grochola has published one such path in "Constrained fluid lambda-integration: Constructing a reversible thermodynamic path between the solid and liquid state"(JCP 120, 2122).

The basic path is to start with the liquid state and turn off the intermolecular potential. Then turn on a potential constraining particles to lattice sites (He used a Gaussian potential). The last step is to turn off the lattice potential while turning the intermolecular potential back on.

He mentions at the end that this method is not very computationally efficient. I would expect using Bennett's method for free energy differences would cut down on the number of intermediate states needed.

Questions

1. Compare and contrast this path with the phase switching Monte Carlo technique (there's a reference in the article)
   
2. Thermodynamic integration requires there be no phase transitions on the path. Is it immediately obvious that this path has no transitions? Under what conditions, if any, would a phase transition occur on the path?