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EB

EB

ONLINE

1st ed.

Bristol : Institute of Physics Publishing, 2025

1 online zdroj (290 stran)

ISBN 9780750363105 (e-knihy)

ISBN 9780750363112

IOP Series in Quantum Technology Series

Print version: Apaja, Vesa A Practical Course on Quantum Monte Carlo Bristol : Institute of Physics Publishing,c2025 ISBN 9780750363112

This book showcases that Quantum Monte Carlo isn’t just a numerical tool--it’s an expedition that reveals the hidden facets of the quantum world. It’s an ideal text for researchers and readers who want to see how statistical and quantum physics work in numerical applications..

002001789

Intro // Author biography // Vesa Apaja // Chapter Introduction // 1.1 The Schrödinger equation // 1.2 Indistinguishable particles and the ground state // 1.2.1 Introducing orbitals // 1.2.2 Exact multielectron wave function from spin orbitals // 1.2.3 Solving single-particle orbitals // 1.2.4 Boson ground state // 1.2.5 Fermion ground state // 1.3 Solving quantum problems with Monte Carlo // 1.3.1 One-dimensional integrals // 1.3.2 Why use Monte Carlo? // 1.3.3 Integrals in quantum mechanics // 1.3.4 Zero variance principle // 1.3.5 Sampling from a distribution // References // Chapter Variational Monte Carlo (VMC) // 2.0.1 Metropolis algorithm // 2.0.2 Application to He atom ground state // 2.1 Spin in expectation values // 2.2 The multielectron wave function // 2.2.1 The Slater-Jastrow wave function // 2.2.2 The multideterminant and the configuration interaction wave function // 2.2.3 Configuration state functions (CSFs) // 2.2.4 Cusp corrections to Gaussian orbitals // 2.2.5 Nodal surface and the backflow transformation // 2.3 The Jastrow factor // 2.3.1 Optimization of the Jastrow factor // References // Chapter Principles of wave function optimization // 3.1 Energy and variance optimization // 3.1.1 H atom: analytical variance optimization // 3.2 Direct and correlated sampling // 3.2.1 He atom: comparing direct and correlated sampling // 3.3 He atom: energy and variance optimization // 3.3.1 About cusp conditions and ground-state optimization // 3.3.2 Gradients in variance and energy optimization // 3.3.3 Multielectron optimizations with paper and pen // 3.3.4 Stochastic gradient descent // 3.3.5 Linearized optimization method // References // Chapter Diffusion Monte Carlo // 4.1 Projection on the ground state // 4.2 Imaginary-time Schrödinger equation // 4.3 Stochastic representation of a wave function.

4.3.1 Stochastic representation of a distribution // 4.3.2 Diffusion // 4.3.3 The source term // 4.3.4 A crude DMC algorithm // 4.4 Importance sampling DMC // 4.4.1 Importance sampling in the evolution integral equation // 4.4.2 Importance sampling in the evolution differential equation // 4.4.3 Combined drift, diffusion, and source // 4.4.4 Correcting sampling using detailed balance // 4.4.5 A first-order DMC algorithm // 4.4.6 Measurements in importance sampling DMC // 4.4.7 A second-order DMC algorithm // 4.4.8 About fourth-order DMC algorithms // 4.5 Case study: importance sampling DMC for a particle in a box // 4.5.1 Drift trajectories // 4.5.2 Persistent configurations (aka trapped walkers) // 4.5.3 Dealing with drift and and local energy divergence near boundaries // 4.6 Case study: DMC for a particle in a box without importance sampling // 4.7 DMC for bosons and fermions // 4.7.1 Two identical particles in a box // 4.7.2 More diffusion near walls or nodes // 4.7.3 Tiling property // 4.8 Fixed-node DMC of atoms and small molecules // 4.8.1 Beryllium atom ground state // 4.8.2 Molecules // 4.8.3 Antisymmetrized geminal power wave functions // References // Chapter Path integral Monte Carlo // 5.0.1 Boson and fermion path integrals // 5.1 From real-time path integrals to imaginary-time path integrals // 5.2 High-temperature density matrix // 5.2.1 Isomorphism of PIMC paths and classical polymers // 5.3 Path generation // 5.3.1 Collective slice coordinates x // 5.3.2 Path generation for free particles // 5.3.3 Staging algorithm // 5.3.4 Bisection algorithm // 5.3.5 Bisection with interaction in the primitive approximation // 5.3.6 Harmonic oscillator // 5.4 Worm algorithm and permutation sampling // 5.4.1 Worm open update // 5.4.2 Worm close update // 5.4.3 Worm insert and remove updates // 5.4.4 Worm advance and recede updates.

5.4.5 Worm wiggle tail update // 5.4.6 Worm swap update // 5.4.7 Periodic boundary conditions // 5.5 Approximate action // 5.5.1 Chin action // 5.6 PIMC measurements // 5.6.1 Superfluidity and winding paths // 5.6.2 Bose-Einstein condensation // 5.6.3 Energy estimators in PIMC // 5.6.4 Fermion PIMC and the sign problem // 5.7 Practical suggestions for testing a PIMC code // 5.7.1 First test system // 5.7.2 Second test system // 5.7.3 Third test system // 5.7.4 About the sample PIMC code // 5.8 Stochastic series expansion // References // Chapter Path integral ground-state Monte Carlo // References // A.1 Cauchy distribution // D.0.1 Drift and local energy of the Slater-Jastrow trial wave function // D.0.2 Gradients of a Slater determinant // D.0.3 Updating the inverse Slater matrix // F.1 Definitions // F.2 Biased and unbiased estimators of variance // F.3 Integrated autocorrelation time // F.4 Block averaging // F.5 Resampling methods // F.5.1 Jackknife // F.5.2 Bootstrap // Chapter.

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