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 simultation time. For bosons, there exist numerically exact and polynomial-scaling algorithms. For fermions, there exist very good approximations and exponentially scaling Quantum Monte Carlo algorithms, but none that are both.

Background

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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 a separable function of one-body wave functions: $\backslash Psi(x\_1,x\_2,...,x\_n)=f(\backslash Phi\_1(x\_1),\backslash Phi\_2(x\_1),...\; \backslash Phi\_n(x\_1);\backslash Phi\_1(x\_2)\; \backslash Phi\_2(x\_2),...)$, 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.

Quantum Monte Carlo is a way around these problems, which allows us to work with the full many-body wave function directly. 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. What follows is a sampling of some of the Quantum Monte Carlo flavors, which approach the problem in different ways.

Variational Monte Carlo

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Variational Monte Carlo takes advantage of the Variational method in quantum mechanics to approximate the ground state. The expectation value necessary can be written in the $x$ representation as $\backslash frac\{\backslash langle\; \backslash Psi(a)\; |\; H\; |\; \backslash Psi(a)\; \backslash rangle\}\; \{\backslash langle\; \backslash Psi(a)\; |\; \backslash Psi(a)\; \backslash rangle\; \}\; =\; \backslash frac\{\backslash int\; \backslash Psi(X,a)^2\; \backslash frac\{H\backslash Psi(X,a)\}\{\backslash Psi(X,a)\}\; dX\}\; \{\; \backslash int\; \backslash Psi(X,a)^2\; dX\}$. Following the Monte Carlo method for evaluating integrals, we can interpret $\backslash frac\{\; \backslash Psi(X,a)^2\; \}\; \{\; \backslash int\; \backslash Psi(X,a)\; dX\; \}$ as a probability distribution function, sample it, and evaluate the energy expectation value $E(a)$ as the average of the local function $\backslash frac\{H\backslash Psi(X,a)\}\{\backslash Psi(X,a)\}$, and minimize $E(a)$.

This is no different from any other variational method, except that since the many-dimensional integrals are evaluated numerically, we only need to calculate the value of the possibly very complicated wave function, which gives a large amount of flexibility to the method. One of the largest gains in accuracy over writing the wave function separably comes from the introduction of the so-called Jastrow factor, where the wave function is written as $exp(\backslash sum\{u(r\_\{ij\})\})$, where $r\_\{ij\}$ is the distance between a pair of quantum particles. With this factor, we can explicitly account for particle-particle correlation, but the many-body integral becomes unseparable, so Monte Carlo is the only way to evaluate it efficiently. In chemical systems, slightly more sophisticated versions of this factor can obtain 80-90% of the correlation energy(see Electronic correlation) with less than 30 parameters. In comparison, a Configuration Interaction calculation may require around 50,000 parameters to reach that accuracy, although it depends greatly on the particular case being considered. In addition, VMC usually scales as a small power of the number of particles in the simulation, usually something like N^{2-4 for calculation of the energy expectation value, depending on the form of the wave function.}

Diffusion Monte Carlo

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Diffusion Monte Carlo(DMC) is potentially numerically exact, meaning that it can find the exact ground state energy within a given error for any quantum system. When actually attempting the calculation, one finds that for bosons, the algorithm is scales as a polynomial with the system size, but for fermions, DMC is exponentially scaling with the system size. This makes exact large-scale DMC simulations for fermions impossible; however, with a clever approximation known as fixed-node, very accurate results can be obtained. What follows is an explanation of the basic algorithm, how it works, why fermions cause a problem, and how the fixed-node approximation resolves this problem.

Flavors of Quantum Monte Carlo

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• Reptation Monte Carlo
• Path Integral Monte Carlo
• Auxiliary Field Monte Carlo

See also

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• Monte Carlo method
• Quantum chemistry

References

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• Auxiliary Field Monte Carlo
  • ^* R. Blankenbecler, D.J. Scalapino and R.L. Sugar, Phys. Rev. D 24,2278(1981)
  • ^* D. Ceperley, G.V. Chester and M.H. Kalos, Phys. Rev. B 16,3081(1977)
• Diffusion Monte Carlo
  • R.C. Grimm and R.G. Storer, J. Comput. Phys. 7,134(1971)
  • ^* J. Anderson, J. Chem. Phys. 63, 1499(1975)
  • B.L. Hammond, W.A Lester, Jr. & P.J. Reynolds "Monte Carlo Methods in Ab Initio Quantum Chemistry" (World Scientific, 1994)
• Path Integral Monte Carlo
  • D.M. Ceperley, Rev. Mod. Phys. 67,279(1995)
• Reptation Monte Carlo
  • ^* S. Baroni and S. Moroni, Phys. Rev. Lett. 82,4745(1999)
• Variational Monte Carlo
  • ^* W.L. McMillan, Phys. Rev. 138,A442(1964)
  • ^* D. Ceperley, G.V. Chester and M.H. Kalos, Phys. Rev. B 16,3081(1977)
• Valence-bond Monte Carlo
  • ^* Anders. W. Sandvik, Phys. Rev. Lett 95,207203(2005)

External links

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Lecture notes

• Joint DEMOCRITOS-ICTP School on Continuum Quantum Monte Carlo Methods

Computer programs

• CASINÒ
• CHAMP
• QMcBeaver
• QMCMOL

Quantum chemistry | Monte Carlo methods