# Quantum monte carlo

## Top PDF Quantum monte carlo: ### Generalized Pairing Wave Functions and Nodal Properties for Electronic Structure Quantum Monte Carlo

There are several Quantum Monte Carlo methods. They all apply a stochastic approach to find a solutions of a stationary Schr¨ odinger equation of quantum systems. However, in this dissertation we will be restricted to description of only the variational and diffusion Monte Carlo methods. In the variational Monte Carlo the expectation values are evaluated via stochastic integration over 3N dimensional space. A variational theorem ensures that the expectation value of Hamiltonian with respect to given trial wave function is a true upper bound to the exact ground-state energy and sorely depends on the accuracy of the trial wave function. The second method, the diffusion Monte Carlo, removes some deficiency in the accuracy of trial wave function by employing imaginary-time projection of trial wave function onto the ground state. The nodal structure of trial wave function is enforced on the projected wave function by fixed node approximation. Hence, if the nodes of trial wave function were identical to the nodes of an exact one, we would know the ground state energy of a quantum system in polynomial time. ### Quantum Monte Carlo Studies of Ultra-cold Molecules and Rashba Interactions.

Quantum mechanics have been successful in interpreting the properties of condensed mat- ter systems at microscale. It describes the nature of fundamental particles in our daily world: atoms, molecules, electrons, etc. The Schr¨ odinger equation, proposed in 1925, pro- vide us an exact mathematical formulation of non-relativistic quantum physics . With the solution to the Schr¨ odinger equation, we are able to know and predict many properties of quantum systems. However, while physicists are often proud of being able to precisely describe how a system behaves quantitatively, solving the equations underlying the the- ory is a great challenge, especially when the size of the system becomes large. Thanks to the tremendous development in computer technology, many calculations which used to take immense time to perform, can be quickly done on modern computers. But there still exist many unsolved problems because they suffer difficulties beyond the computational efficiency. Theoretical physicists aim to exactly solve problems relying on the equations from first principles and many people have been devoting their lives to developing various kinds of methods. Quantum Monte Carlo, presented in this thesis, is one of the meth- ods to simulate quantum systems. Generally, the term “Monte Carlo” refers to a class of computer algorithms to compute the results based on random sampling. The word “quantum” indicates we are using the Monte Carlo technique to solve problems of quan- tum systems based on the knowledge of quantum mechanics. In this thesis, we will focus on electronic structure problems, a particular area of the quantum system problems. ### The quantum Monte Carlo method : application to problems in statistical physics

Section 3.) of Chapter 2 and the curves presented in Figure 3.5 are the results of superimposing many segments of the random walk trajectory. As discussed in Chapter 2, estimates of the expectation values of the potential and kinetic energy can be obtained from the asymptotic behaviour of the curves presented in Figures 3.5. The values of these quantities are <V> = 2370 ± 50 K/molecule and <T> = 3400 ± 100 K/molecule giving the total ground state energy of the water dimer as <E0 > = 5770 ± 100 K/molecule. To within the statistical fluctuations inherent in these calculations this agrees with the average value of Vref <vref> = 5730 ± 20 K/molecule obtained during the run. As discussed earlier, this agreement provides a self-consistent check of the quantum Monte Carlo method and indicates that the Virial Theorem is satisfied. The eigenvalue estimate calculated from the average energy reference has less statistical uncertainty than the value obtained from the sum of the potential and kinetic energies since the evaluation does not rely on the ( j ) 02 generation procedure. ### Towards Heavy Element Materials with Electronic Structure Quantum Monte Carlo Methods.

When using the conventional Ne-core and 3s3p valence, the transferability of the ECPs in particular environments was observed to be less than satisfying. This is particularly true for compressed polar bonds (as will be shown). In order to reach high many-body and single-body accuracies in these regimes as well, we have also generated He-core ECPs for all of the second row atoms. Though more expensive than their Ne-core counterparts, the He-core ECPs still provide a non- negligible computational savings when considering their use in quantum Monte Carlo (QMC) which scales with the atomic number as O (Z 5.5−6.5 ) and within planewave codes where cal- culations with the full Coulomb potential are not feasible. For these small-core ECPs, the optimization procedure was nearly identical, except that we took the neutral ground states for all references in the norm-conserving component and we gener- ated the cutoff radii from the semi-core orbitals. In this case, for each angular momentum channel, the cutoff radius was taken as the inner most finite extremum of rφ AE ### Finite size errors in continuum quantum Monte Carlo calculations

We analyze the problem of eliminating finite-size errors from quantum Monte Carlo 共 QMC 兲 energy data. We demonstrate that both 共 i 兲 adding a recently proposed 关 S. Chiesa et al., Phys. Rev. Lett. 97, 076404 共 2006 兲兴 finite-size correction to the Ewald energy and 共 ii 兲 using the model periodic Coulomb 共 MPC 兲 interaction 关 L. M. Fraser et al., Phys. Rev. B 53, 1814 共 1996 兲 ; P. R. C. Kent et al., Phys. Rev. B 59, 1917 共 1999 兲 ; A. J. Williamson et al., Phys. Rev. B 55, R4851 共 1997 兲兴 are good solutions to the problem of removing finite-size effects from the interaction energy in cubic systems provided the exchange-correlation 共 XC 兲 hole has con- verged with respect to system size. However, we find that the MPC interaction distorts the XC hole in finite systems, implying that the Ewald interaction should be used to generate the configuration distribution. The finite-size correction of Chiesa et al. 关 Phys. Rev. Lett. 97, 076404 共 2006 兲兴 is shown to be incomplete in systems of low symmetry. Beyond-leading-order corrections to the kinetic energy are found to be necessary at intermediate and high densities; we investigate the effect of adding such corrections to QMC data for the homogeneous electron gas. We analyze finite-size errors in two-dimensional systems and show that the leading-order behavior differs from that which has hitherto been supposed. We compare the efficiencies of different twist-averaging methods for reducing single-particle finite-size errors and we examine the perfor- mance of various finite-size extrapolation formulas. Finally, we investigate the system-size scaling of biases in diffusion QMC. ### Quantum Monte Carlo calculations of energy gaps from first principles

We review the use of continuum quantum Monte Carlo (QMC) methods for the calculation of energy gaps from first principles, and present a broad set of excited-state calculations carried out with the variational and fixed-node diffusion QMC methods on atoms, molecules, and solids. We propose a finite-size-error correction scheme for bulk energy gaps calculated in finite cells subject to periodic boundary conditions. We show that finite-size effects are qualitatively different in two- dimensional materials, demonstrating the effect in a QMC calculation of the band gap and exciton binding energy of monolayer phosphorene. We investigate the fixed-node errors in diffusion Monte Carlo gaps evaluated with Slater-Jastrow trial wave functions by examining the effects of backflow transformations, and also by considering the formation of restricted multideterminant expansions for excited-state wave functions. For several molecules, we examine the importance of structural relaxation in the excited state in determining excited-state energies. We study the feasibility of using variational Monte Carlo with backflow correlations to obtain accurate excited-state energies at reduced computational cost, finding that this approach can be valid. We find that diffusion Monte Carlo gap calculations can be performed with much larger time steps than are typically required to converge the total energy, at significantly diminished computational expense, but that in order to alleviate fixed-node errors in calculations on solids the inclusion of backflow correlations is sometimes necessary. ### Mixtures of ultracold Bose gases in one dimension: A Quantum Monte Carlo study

and inter-species interactions. Mean-field theories are valid for small values of γ and η, which can be obtained either at weak couplings or high densities, while at large values of the coupling strength g tunneling between particles is suppressed and in the limit of infinite repulsion the atoms become impen- etrable. While no exact analytical solutions are known in higher dimensions, exact solutions can be found in one-dimensional systems in some limiting regimes. In the case of a single component the model is known as the Lieb- Liniger model which can be solved exactly using Bethe Ansatz techniques (BA). In the limit of infinite repulsion one realizes the Tonks- Girardeau gas (TG) where one can prove that all static properties are equal to those of a free spinless Fermi gas, as the particle impenetrability plays the role of an effective Pauli principle. An analytical solution is also available in mixtures in the limit of very strong intra-species repulsion and arbitrary inter-species interaction , as the system can be mapped to the Yang-Gaudin model[6, 8, 9] , describing a gas of spin 1/2 fermions with contact interactions. For finite repulsion strength, however, no analytical solutions are known and one must resort to numerical techniques. In particular we use essentially exact zero- temperature Quantum Monte-Carlo techniques, which are able to tackle the strong correlations present in these systems. ### New quantum Monte Carlo algorithms to efficiently utilize massively parallel computers

A Jastrow function with parameters which are transferable to large classes of hy- drocarbons is demonstrated . Such a Generic Jastrow function can lead to im- proved initial guesses for variational Quantum Monte Carlo (VMC) wave function optimizations. Furthermore, with more development, it may be possible to construct wave functions for diffusion Quantum Monte Carlo (DMC) calculations without first performing a VMC wave function optimization. Both possibilities will significantly reduce the time necessary to perform a Quantum Monte Carlo (QMC) calculation of a molecular system. ### Energetics and dipole moment of transition metal monoxides by quantum Monte Carlo

Many authors 共most recently, Refs. 1 and 2兲 have studied the transition metal monoxides using density functional theory and post-Hartree-Fock methods. The performance of these methods is less reliable whenever transition metals are included in a system. In particular, the calculation of dipole moments is challenging because it is rather sensitive to the details of calculations and sizes of employed basis sets. The results can sometimes be far from experiment. The transition metal 共TM兲–O bond is the driving force behind many perov- skite and earth materials, which have been noted to have significant errors in the unit cell volume within density func- tional theory 共 DFT 兲 , 3,4 while being too large for post- Hartree-Fock methods to be applied. We have had some success 5 applying ground-state quantum Monte Carlo 共 QMC 兲 to the binding energies of the TiO and MnO mol- ecules, which hinted that QMC may be able to treat these systems more accurately. QMC also has the property of scal- ing well with system size, although with a large prefactor, and has seen limited application to TM solids. As more com- puting power becomes available, QMC calculations of solids will become routine, and this study offers some insight as to the accuracy that will be achieved. ### Extracting the Single-Particle Gap in Carbon Nanotubes with Lattice Quantum Monte Carlo

Abstract. We show how lattice Quantum Monte Carlo simulations can be used to calculate electronic properties of carbon nanotubes in the presence of strong electron-electron correlations. We employ the path integral formalism and use methods developed within the lattice QCD community for our numerical work and compare our results to empirical data of the Anti-Ferromagnetic Mott Insulating gap in large diameter tubes. ### All electron quantum Monte Carlo calculations for the noble gas atoms He to Xe

We report all-electron variational and diffusion quantum Monte Carlo 共 VMC and DMC 兲 calculations for the noble gas atoms He, Ne, Ar, Kr, and Xe. The calculations were performed using Slater-Jastrow wave functions with Hartree-Fock single-particle orbitals. The quality of both the optimized Jastrow factors and the nodal surfaces of the wave functions declines with increasing atomic number Z, but the DMC calculations are tractable and well behaved in all cases. We discuss the scaling of the computational cost of DMC calculations with Z. ### Assessment of multireference approaches to explicitly correlated full configuration interaction quantum Monte Carlo.

The Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method has proved able to provide near- exact solutions to the electronic Schr¨odinger equation within a finite orbital basis set, without relying on an expansion about a reference state. However, a drawback to the approach is that being based on an expansion of Slater determinants, the FCIQMC method suffers from a basis set incompleteness error that decays very slowly with the size of the employed single particle basis. The FCIQMC results obtained in a small basis set can be improved significantly with explicitly correlated techniques. Here, we present a study that assesses and compares two contrasting ‘universal’ explicitly correlated approaches that fit into the FCIQMC framework; the  R12 method of Valeev et al., and the explicitly correlated canonical transcorrelation approach of Yanai ### Applications of the Variational Quantum Monte Carlo Method to the Two-Electron Atoms

For many-electron atoms, many researchers studied the effect of confinement by impenetrable as well as non- impenetrable spherical boxes. Most of the studies have considered the case of the helium atom as the simplest few- body system to study electron correlation effects as a function of the cavity dimension into which they embedded. The effect of confinement on the electron correlation arises due to the Coulomb interaction between the two electrons. Methods such as the variational method, self-consistent, configuration interaction (CI) and quantum Monte Carlo (QMC) methods have also been used to study the properties of the helium atom and its ions confined in an impenetrable spherical box. ### Quantum Monte Carlo, density functional theory, and pair potential studies of solid neon

One of the most important goals of ab initio computa- tional electronic-structure theory is the development of accu- rate methods for describing interatomic bonding. Quantum Monte Carlo 共QMC兲 techniques are useful in this regard, as they can provide a highly accurate description of electron correlation effects. Although QMC methods are computa- tionally expensive, they can be applied to systems that are large enough to model condensed matter. ### A Quantum Monte Carlo Study of Atomic Fermi Gas.

In this Chapter, comprehensive quantum Monte Carlo results of various quantities of interest for the unpolarized Fermi gas in a periodic box are provided. The first Green’s function and diffusion Monte Carlo calculations were carried out in 2003 and 2004[22, 4]. Since then, various QMC calculations have been performed. However, two fundamental questions regarding the ground state energies are still not clear. First, the unitary limit is defined using point-like interactions with the delta-function. However, in real simulations, the interaction with finite effective range are always used. It is therefore important to figure out whether the system really reaches the unitary limit. Second, the DMC provides only the upper-bound for the energy which is determined by the nodal surface of the wave function. Therefore, the question of whether the pair orbitals can be further improved to provide better nodal surface should be addressed. These questions motivate the further study of the unpolarized Fermi gas performed in this Chapter. ### Applications of the Fixed-node Quantum Monte Carlo Method.

Diffusion quantum Monte Carlo (DMC) is a projector quantum Monte Carlo (QMC) method for solving quantum many-body prob- lems. Its name originates from the formal similarity between the Schrödinger’s equation in imaginary time and a classical diffusion equation. Based on stochastic sampling and evolution of particle conﬁgurations, DMC projects out the ground state of a given sym- metry from any trial wave function with nonzero overlap. In deal- ing with fermions, DMC has to adopt some strategy for overcoming the infamous fermion sign problem which would make it dramat- ically inefﬁcient for many-particle systems. It is common to cir- cumvent the sign problem by forcing the ground state nodes (zero-locus) to be the same as the nodes of the best available trial function. This nodal constraint ensures that the product of the ground state and the trial wave function becomes nonnegative throughout the conﬁguration space. This is known as the ﬁxed- node (FN) approximation [1–3]. ### Determinant quantum Monte Carlo study of exciton condensation in the bilayer Hubbard model

We studied the possibility of exciton condensation in a strongly correlated bilayer extended Hubbard model using determinant quantum Monte Carlo. To model both the on-site repulsion U and the interlayer interaction V we introduced an update scheme extending the standard Sherman-Morrison update. We observe that the sign problem increases dramatically with the inclusion of the interlayer interaction V , which prohibits at this stage an unequivocal conclusion regarding the presence of exciton condensation. However, enhancement of the interlayer tunneling results suggest that the strongest exciton condensation tendency lies around 10–20% p/n doping. Magnetic properties and conductivity turn out to be relatively independent of the interlayer interaction. DOI: 10.1103/PhysRevB.88.235115 PACS number(s): 71.10.Fd, 02.70.Ss, 71.27.+a, 73.21.−b ### Chiral 2N and 3N interactions and quantum Monte Carlo applications

Abstract. Chiral Eﬀective Field Theory (EFT) two- and three-nucleon forces are now widely employed. Since they were originally formulated in momentum space, these in- teractions were non-local, making them inaccessible to Quantum Monte Carlo (QMC) methods. We have recently derived a local version of chiral EFT nucleon-nucleon and three-nucleon interactions, which we also used in QMC calculations for neutron matter and light nuclei. In this contribution I go over the basics of local chiral EFT and then summarize recent results. ### Binding energies of excitonic complexes in type II quantum rings from diffusion quantum Monte Carlo calculations

Excitonic complexes in type-II quantum-ring heterostructures may be considered as artificial atoms due to the confinement of only one charge-carrier type in an artificial nucleus. Binding energies of excitons, trions, and biexcitons in these nanostructures are then effectively ionization energies of these artificial atoms. The binding energies reported here are calculated within the effective-mass approximation using the diffusion quantum Monte Carlo method and realistic geometries for gallium antimonide rings in gallium arsenide. The electrons form a halo outside the ring, with very little charge density inside the central cavity of the ring. The de-excitonization and binding energies of the complexes are relatively independent of the precise shape of the ring. 