Electronic structure methods Tight binding Hartree-Fock Møller-Plesset perturbation theory Configuration interaction Coupled cluster Multi-configurational self-consistent field Density functional theory Quantum chemistry composite methods Quantum Monte Carlo This box: view • talk • edit

Quantum Monte Carlo is a large class of computer algorithms that simulate quantum systems with the idea of solving the many-body problem. In the tight binding model for a solid-state lattice of atoms it is assumedthat the full Hamiltonian H of the system may be approximated by theHamiltonian of an isolated In Computational physics and Computational chemistry, the Hartree-Fock ( HF) method is an approximate method for the determination of the ground-state Møller-Plesset perturbation theory (MP is one of several Quantum chemistry Post-Hartree-Fock ab initio methods in the field of Computational chemistry Configuration interaction ( CI) is a Post Hartree-Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Coupled cluster ( CC) is a numerical technique used for describing Many-body systems Its most common use is as one of several quantum chemical Multi-configurational self-consistent field (MCSCF is a method in Quantum chemistry used to generate qualitatively correct reference states of molecules in cases where Density functional theory (DFT is a quantum mechanical theory used in Physics and Chemistry to investigate the Electronic structure (principally Quantum chemistry composite methods are ab initio post-Hartree-Fock methods in Computational chemistry that aim for high accuracy by combining the results Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The many-body problem may be defined as the study of the effects of interaction between bodies on the behaviour of a many-body system i They use, in one way or another, the Monte Carlo method to handle the many dimensional integrals that arise. Monte Carlo methods are a class of Computational Algorithms that rely on repeated Random sampling to compute their results 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. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system For bosons, there exist numerically exact and polynomial-scaling algorithms. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein For fermions, there exist very good approximations and numerically exact exponentially scaling quantum Monte Carlo algorithms, but none that are both. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi.

## 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. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system 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. In Computational physics and Computational chemistry, the Hartree-Fock ( HF) method is an approximate method for the determination of the ground-state 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. In Computational physics and Computational chemistry, the Hartree-Fock ( HF) method is an approximate method for the determination of the ground-state Configuration interaction ( CI) is a Post Hartree-Fock linear variational method for solving the nonrelativistic Schrödinger equation within the 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. In Computational physics and Computational chemistry, the Hartree-Fock ( HF) method is an approximate method for the determination of the ground-state 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. Path integral Monte Carlo is a Quantum Monte Carlo method in the Path integral formulation of quantum mechanics. Auxiliary field Monte Carlo is a method that allows the calculation by use of Monte Carlo techniques of averages of operators in many-body quantum mechanical (Blankenbecler

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

## Flavors of quantum Monte Carlo

• Variational Monte Carlo : A good place to start; it is commonly used in many sorts of quantum problems. Variational Monte Carlo (VMC is a Quantum Monte Carlo method that applies the variational method to approximate the ground state of the system
• Diffusion Monte Carlo : The most common high-accuracy method for electrons (that is, chemical problems), since it comes quite close to the exact ground state energy fairly efficiently. Diffusion Monte Carlo (DMC is a quantum Monte Carlo method that utilizes a Green function to solve the Schrödinger equation. Also used for simulating the quantum behavior of atoms, etc.
• Path integral Monte Carlo : Finite temperature technique mostly applied to bosons where temperature is very important, especially superfluid helium. Path integral Monte Carlo is a Quantum Monte Carlo method in the Path integral formulation of quantum mechanics.
• Auxiliary field Monte Carlo : Usually applied to lattice problems, although there has been recent work on applying it to electrons in chemical systems. Auxiliary field Monte Carlo is a method that allows the calculation by use of Monte Carlo techniques of averages of operators in many-body quantum mechanical (Blankenbecler
• Reptation Monte Carlo : Recent zero-temperature method related to Path integral Monte Carlo, with applications similar to Diffusion Monte Carlo, but with some different tradeoffs. Reptation Monte Carlo is a quantum Monte Carlo method
• Gaussian quantum Monte Carlo