A Green's function quantum Monte Carlo (GF QMC) method is described which renders possible the determination of the exact ground state energy of any π system in the computational framework of effective π Hamiltonians. An instruction to evaluate the growth estimator in the simultaneous presence of positive and negative phases is suggested. Numerical results for five annulene molecules with 4–12 atomic centres as well as 10 further π molecules with different topology (i.e., monocycles with methylene groups, bi- and polycycles) are given. Negative phases occur in 13 of the 15 π networks considered. The statistical QMC results are compared with data derived by conventional (many-body) diagonalization techniques. The interaction-free Hückel Hamiltonian has been adopted as most simple setup to test the applicability of the Monte Carlo approach. Many-body results have been derived for the Hubbard and Pariser-Parr-Pople variants of the π-electron Hamiltonian. It is demonstrated that GF QMC results can be derived whose accuracy does not depend on the presence of negative signs in the numerical simulation.
A Green's function quantum Monte Carlo (GF QMC) method is described which renders possible the determination of the exact ground state energy of any π system in the computational framework of effective π Hamiltonians. An instruction to evaluate the growth estimator in the simultaneous presence of positive and negative phases is suggested. Numerical results for five annulene molecules with 4–12 atomic centres as well as 10 further π molecules with different topology (i.e., monocycles with methylene groups, bi- and polycycles) are given. Negative phases occur in 13 of the 15 π networks considered. The statistical QMC results are compared with data derived by conventional (many-body) diagonalization techniques. The interaction-free Hückel Hamiltonian has been adopted as most simple setup to test the applicability of the Monte Carlo approach. Many-body results have been derived for the Hubbard and Pariser-Parr-Pople variants of the π-electron Hamiltonian. It is demonstrated that GF QMC results can be derived whose accuracy does not depend on the presence of negative signs in the numerical simulation.
Green's function quantum Monte Carlo (GF QMC) simulations of fermionic ensembles require the definition of a so-called spectral parameter W to yield trustworthy estimates of the fully correlated ground state energy E. In the present work we discuss the influence of the shift parameter, W, on the convergence behaviour of GF QMC simulations. As model systems we have considered some π molecules which are studied in the framework of the Pariser–Parr–Pople (PPP) Hamiltonian. The influence of the W parameter on the convergence of the GF QMC simulations in many-electron systems with an odd number of electronic permutations within one spin direction exceeds the influence observed in systems without such odd permutations. They do not occur in polyenes and Huckel annulenes with an electron count of (4n + 2) (n = 0, 1, 2…). The GF QMC technique adopted as a computational tool has been developed to study π molecules with fermionic sign problems owing to odd electronic interchanges within one spin direction.
Electronic structure quantum Monte Carlo Quantum Monte Carlo (QMC) is an advanced simulation methodology for studies of manybody quantum systems. The QMC approaches combine analytical insights with stochastic computational techniques for efficient solution of several classes of important many-body problems such as the stationary Schrödinger equation. QMC methods of various flavors have been applied to a great variety of systems spanning continuous and lattice quantum models, molecular and condensed systems, BEC-BCS ultracold condensates, nuclei, etc. In this review, we focus on the electronic structure QMC, i.e., methods relevant for systems described by the electron-ion Hamiltonians. Some of the key QMC achievements include direct treatment of electron correlation, accuracy in predicting energy differences and favorable scaling in the system size. Calculations of atoms, molecules, clusters and solids have demonstrated QMC applicability to real systems with hundreds of electrons while providing 90-95% of the correlation energy and energy differences typically within a few percent of experiments. Advances in accuracy beyond these limits are hampered by the so-called fixed-node approximation which is used to circumvent the notorious fermion sign problem. Many-body nodes of fermion states and their properties have therefore become one of the important topics for further progress in predictive power and efficiency of QMC calculations. Some of our recent results on the wave function nodes and related nodal domain topologies will be briefly reviewed. This includes analysis of few-electron systems and descriptions of exact and approximate nodes using transformations and projections of the highly-dimensional nodal hypersurfaces into the 3D space. Studies of fermion nodes offer new insights into topological properties of eigenstates such as explicit demonstrations that generic fermionic ground states exhibit the minimal number of two nodal domains. Recently proposed trial wave functions based on Pfaffians with pairing orbitals are presented and their nodal properties are tested in calculations of first row atoms and molecules. Finally, backflow "dressed" coordinates are introduced as another possibility for capturing correlation effects and for decreasing the fixed-node bias.
Although stoquastic Hamiltonians are known to be simulable via sign-problem-free quantum Monte Carlo (QMC) techniques, the nonstoquasticity of a Hamiltonian does not necessarily imply the existence of a QMC sign problem.We give a sufficient and necessary condition for the QMC-simulability of Hamiltonians in a given basis: We prove that a QMC simulation will be sign-problem-free if and only if all the overall total phases along the chordless cycles of the weighted graph whose adjacency matrix is the Hamiltonian are zero (modulo 2π ).We use our findings to provide a construction for nonstoquastic, yet sign-problem-free and hence QMC-simulable, quantum many-body models.We also demonstrate why the simulation of truly sign-problematic models using the QMC weights of the stoquasticized Hamiltonian is generally suboptimal.We offer a superior alternative.
Fermion quantum Monte Carlo (QMC) methods that work in a general basis space may be more effective for some problems than the traditional diffusion QMC method. Projection of the ground state energy is achieved by random walks in the space of Slater determinants whose one‐particle orbitals are expressed in terms of the chosen basis set. A complication is that the determinants will in general acquire complex phases. This a consequence of ground state Monte Carlo projection using the Hubbard‐Stratonovich transformation of the two‐body interaction. To control the resulting “sign” decay, we describe a method we have recently introduced for the propagation of phaseless determinants. The approximation relies on importance sampling with a trial wave function. The approximation has features in common with diffusion MC fixed node and lattice‐model constrained path methods. Using a planewave basis and non‐local pseudopotentials, we apply the method to Si atom, dimer, and 2, 16, 54 atom (216 electrons) bulk supercells. Single Slater determinant wave functions from density functional theory calculations were used as |ΨT〉 with no additional optimization. The calculated binding energy of Si2 and cohesive energy of bulk Si are in excellent agreement with experiments and are comparable to the best existing theoretical results.
The shell model provides a powerful framework for nuclear structure calculations. The nucleons beyond an inert magic core are confined in a valence shell and interact through an effective two-body potential generally determined from the G-matrix method. However, the applicability of the shell model remains limited by the exponential growth of the many-body space with the number of valence nucleons and with the size of the one-body space. The quantum Monte Carlo (QMC) methods may thus be considered as a potentially attractive alternative to the direct diagonalization of the Hamiltonian matrix. The many-body problem is indeed reduced to a set of stochastic one-body problems describing independent particles in a fluctuating external field. Up to date, the QMC approaches of the shell model lead to the ground-state and the thermodynamic properties, but a detailed spectroscopy of nuclei is not reachable. Furthermore, they suffer from a common pathology of fermionic QMC methods, namely the sign/phase problem, which strongly contaminates the convergence of the calculation. In this context, we will present a new QMC method, providing the "yrast spectroscopy" of nuclei, and in which the sign/phase problem is managed through the constrained path approximation that is widely used in the ab initio nuclear calculations and in condensed matter physics.
A recent technique, proposed to alleviate the ``sign problem disease,'' is discussed in detail. As is well known, the ground state of a given Hamiltonian H can be obtained by applying the propagator ${e}^{\ensuremath{-}H\ensuremath{\tau}}$ to a trial wave function ${\ensuremath{\psi}}_{T}$ and sampling statistically the state ${\ensuremath{\psi}}_{\ensuremath{\tau}}{=e}^{\ensuremath{-}H\ensuremath{\tau}}{\ensuremath{\psi}}_{T}$ for large imaginary time $\ensuremath{\tau}.$ However, the sign problem may appear in the simulation and such statistical propagation would be practically impossible without employing some approximation such as the ``fixed node'' (FN) one. The present method allows the improvement of the FN dynamics with a systematic correction scheme. This is possible by the simple requirement that, after a short imaginary time propagation via the FN Hamiltonian, a number p of correlation functions can be further constrained to be exact by small perturbations of the FN state, which is free from the sign problem. By iterating this procedure, the Monte Carlo average sign, which is almost zero when there is a sign problem, remains stable and finite even for large $\ensuremath{\tau}.$ The proposed algorithm is tested against exact diagonalization data available on finite lattices. It is also shown, in some test cases, that the dependence of the results upon the few parameters entering the stochastic technique can be very easily controlled, unless for exceptional cases.