Quantum Monte Carlo

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Electronic structure methods
Tight binding
Hartree-Fock
Møller-Plesset perturbation theory
Configuration interaction
Coupled cluster
Multi-configurational self-consistent field
Density functional theory
Quantum Monte Carlo
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Quantum Monte Carlo is a large class of computer algorithms that simulate quantum systems with the idea of solving the many-body problem. They use, in one way or another, the Monte Carlo method to handle the many dimensional integrals that arise. Quantum Monte Carlo allows a direct representation of many-body effects in the wavefunction, at the cost of statistical uncertainty that can be reduced with more simulation time. For bosons, there exist numerically exact and polynomial-scaling algorithms. For fermions, there exist very good approximations and numerically exact exponentially scaling quantum Monte Carlo algorithms, but none that are both.

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[edit] Background

In principle, any physical system can be described by the many-body Schrödinger equation, as long as the constituent particles are not moving 'too' fast; that is, they are not moving near the speed of light. This includes the electrons in almost every material in the world, so if we could solve the Schrödinger equation, we could predict the behavior of any electronic system, which has important applications in fields from computers to biology. This also includes the nuclei in Bose-Einstein condensates and superfluids like liquid helium. The difficulty is that the Schrödinger equation involves a function of three times the number of particles(in 3 dimensions), and is difficult(and impossible in the case of fermions) to solve in a reasonable amount of time. Traditionally, theorists have approximated the many-body wave function as an antisymmetric function of one-body orbitals: \Psi(x_1,x_2,\dots,x_n)=f(\Phi_1(x_1),\Phi_2(x_1),\dots, \Phi_n(x_1);\Phi_1(x_2) \Phi_2(x_2),\dots), for an example, see Hartree-Fock theory. This kind of formulation either limits the possible wave functions, as in the case of Hartree-Fock, or converges very slowly, as in configuration interaction. One of the reasons for the difficulty with a Hartree-Fock ansatz is that it is very difficult to model the electronic and nuclear cusps in the wavefunction. As two particles approach each other, the wavefunction has exactly known derivatives.

Quantum Monte Carlo is a way around these problems because it allows us to model a many-body wave function of our choice directly. Specifically, we can use a Hartree-Fock wavefunction as our starting point, but then multiply it by any symmetric function, of which Jastrow functions are typical, designed to enforce the cusp conditions. Most methods aim at computing the ground state wave function of the system, with the exception of path integral Monte Carlo and finite-temperature auxiliary field Monte Carlo, which calculate the density matrix.

Although Quantum Monte Carlo theoretically only scales as O(N3), better than any other ab initio quantum chemistry method, it is still very expensive because the prefactor in the scaling is large. It is currently more of a theoretical tool than something one might use every day like the better known DFT or Coupled-Cluster methods. Calculations on molecules probably requires psuedopotentials for any nuclei larger than Neon and will require a CPU cluster for anything larger than a couple atoms. Fortunately, as a Monte Carlo method, it is very easy to parallelize. In fact, it nearly qualifies as embarrassingly parallel, however it isn't because the non-statistics producing equilibration phase must be performed for each processor.

To start a calculation, one might obtain a Hartree-Fock wavefunction for the system of choice and then choose a set of Jastrow functions. It is known that Hartree-Fock wavefunctions have remarkably well placed wavefunction nodes. A typical calculation might then involve performing a Variational Monte Carlo run. Because a lower statistical variance corresponds to a wavefunction closer to the exact wavefunction, a VMC calculation could be used to optimize some or all of the parameters in the wavefunction. Once a suitable wavefunction is chosen, (the VMC step could even be skipped) one would proceed to the exact Diffusion Monte Carlo procedure which only uses the wavefunction for importance sampling, and is able to project the exact ground state out of the wavefunction given. This is far from a black box method, since a DMC calculation is only exact in the limit of an infinite number of walkers, and a time step of 0.

There are several quantum Monte Carlo flavors, each of which uses Monte Carlo in different ways to solve the many-body problem:

[edit] Flavors of quantum Monte Carlo

[edit] See also

[edit] References

[edit] External links

[edit] Lecture notes

[edit] Computer programs

[edit] Conferences, workshops, and schools

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