ABSTRACT
With the development of peta-scale computers and exa-scale only a few years away, the **quantum** **Monte** **Carlo** (QMC) method, with favorable scaling and inherent parrallelizability, is poised to increase its impact on the electronic structure community. The most widely used variation of QMC is the diffusion **Monte** **Carlo** (DMC) method. The accuracy of the DMC method is only limited by the trial wave function that it employs. The effect of the trial wave function is studied here by initially developing correlation-consistent Gaussian basis sets for use in DMC calculations. These basis sets give a low variance in variance **Monte** **Carlo** calculations and improved convergence in DMC. The orbital type used in the trial wave function is then investigated, and it is shown that Brueckner orbitals result in a DMC energy comparable to a DMC energy with orbitals from density functional **theory** and significantly lower than orbitals from Hartree-Fock **theory**. Three large weakly interacting systems are then studied; a water-16 isomer, a methane clathrate, and a carbon dioxide clathrate. The DMC method is seen to be in good agreement with MP2 calculations and provides reliable benchmarks. Several strongly correlated systems are then studied. An H4 model system that allows for a fine tuning of the multi-configurational character of the wave function shows when the accuracy of the DMC method with a single Slater-determinant trial function begins to deviate from multi-reference benchmarks. The weakly interacting face-to-face ethylene dimer is studied with and without a rotation around the π bond, which is used to increase the multi-configurational nature of the

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We report **quantum** **Monte** **Carlo** 共 QMC 兲 , plane-wave density-functional **theory** 共 DFT 兲 , and interatomic pair-potential calculations of the zero-temperature equation of state 共 EOS 兲 of solid neon. We find that the DFT EOS depends strongly on the choice of exchange-correlation functional, whereas the QMC EOS is extremely close to both the experimental EOS and the EOS obtained using the best semiempirical pair potential in the literature. This suggests that QMC is able to give an accurate treatment of van der Waals forces in real materials, unlike DFT. We calculate the QMC EOS up to very high densities, beyond the range of values for which experimental data are currently available. At high densities the QMC EOS is more accurate than the pair-potential EOS. We generate a different pair potential for neon by a direct evaluation of the QMC energy as a function of the separation of an isolated pair of neon atoms. The resulting pair potential reproduces the EOS more accurately than the equivalent potential generated using the coupled-cluster CCSD 共 T 兲 method. DOI: 10.1103/PhysRevB.73.024107 PACS number 共 s 兲 : 64.30. ⫹ t, 71.10. ⫺ w

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Abstract
The neutron-matter equation of state connects several physical systems over a wide density range, from cold atomic gases in the unitary limit at low densities, to neutron-rich nuclei at intermediate densities, up to neutron stars which reach supranuclear densities in their core. An accurate description of the neutron-matter equation of state is therefore crucial to describe these systems. To calculate the neutron-matter equation of state reliably, precise many-body methods in combination with a systematic **theory** for nuclear forces are needed. Chiral effective field the- ory (EFT) is such a **theory**. It provides a systematic framework for the description of low-energy hadronic interactions and enables calculations with controlled theoretical uncertainties. Chiral EFT makes use of a momentum-space expansion of nuclear forces based on the symmetries of **Quantum** Chromodynamics, which is the fundamental **theory** of strong interactions. In chiral EFT, the description of nuclear forces can be systematically improved by going to higher orders in the chiral expansion. On the other hand, continuum **Quantum** **Monte** **Carlo** (QMC) methods are among the most precise many-body methods available to study strongly interacting systems at finite densities. They treat the Schrödinger equation as a diffusion equation in imaginary time and project out the ground-state wave function of the system starting from a trial wave function by propagating the system in imaginary time. To perform this propagation, continuum QMC methods require as input local interactions. However, chiral EFT, which is naturally formulated in momentum space, contains several sources of nonlocality.

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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.

The objective of pure-sampling **quantum** **Monte** **Carlo** is to calculate physical prop- erties that are independent of the importance sampling function being employed in the calculation, save for the mismatch of its nodal hypersurface with that of the ex- act wave function. To achieve this objective, we describe a pure-sampling algorithm that combines features of forward-walking methods of pure-sampling and reptation **quantum** **Monte** **Carlo**. The importance sampling is performed by using a single- determinant basis set composed of Slater-type orbitals. We implement our algorithm by systematically increasing an algorithmic parameter until the properties sampled from the electron distributions converge to statistically equivalent values, extrap- olated in the limit of zero time-step. In doing so, we are able to unambiguously determine the values for the ground-state fixed-node energies and one-electron prop- erties of various molecules. These quantities are free from importance sampling bias, population control bias, time-step bias, extrapolation-model bias, and the finite-field approximation. We applied our algorithm to the ground-states of lithium hydride, water and ethylene molecules, and found excellent agreement with the accepted liter- ature values for the energy and a variety of other properties for those systems. Some of our one-electron properties of ethylene had not been calculated before at any level of **theory**. In a detailed comparison, we found reptation **quantum** **Monte** **Carlo**, our closest competitor, to be less efficient by at least a factor of two. It requires different sets of time-steps to accurately determine the ground-state energy and one-electron properties, whereas our algorithm can achieve the same objective by using a single set of time-step values.

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Single point calculations of the ground state electronic structure of peroxynitrite anion have been performed at the optimized cis geometry using the restricted Hartree–Fock 共 RHF 兲 , Møller Plesset second order perturbation **theory** 共 MP2 兲 , generalized gradient approximation density functional **theory** 共 GGA DFT 兲 in the B3LYP form and two **quantum** **Monte** **Carlo** 共 QMC 兲 methods, variational **Monte** **Carlo** 共 VMC 兲 and diffusion **Monte** **Carlo** 共 DMC 兲 . These calculations reveal differences in atomization energies estimated by B3LYP 共 287.03 kcal/mol 兲 , MP2 共 290.84 kcal/mol 兲 , and DMC, 307.4 共 1.9 兲 kcal/mol, as compared to experiment, 313 共 1 兲 kcal/mol. The error associated with MP2 and B3LYP methods is attributed largely to differential recovery of correlation energies for neutral nitrogen and oxygen atoms relative to the oxygen and peroxynitrite anions. In addition, basis set studies were carried out to determine potential sources of error in MP2 and B3LYP valence energy values. Our studies indicate that MP2 and B3LYP valence energies are strongly dependent on the presence of diffuse functions for the negative ions O ⫺ and ONOO ⫺ . © 2003 American Institute of Physics. 关 DOI: 10.1063/1.1544732 兴

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By this point, QMC has established itself as a generator of excellent energetic properties, usually producing the most accurate cohesive energies available in a reasonably scaling method in materials ranging from organic molecules to transition metal systems[6, 7, 8, 9]. Band gaps and excitation energies have been calculated as well[10, 8, 11] with excellent agreement with experiment. There has been much work on forces[12, 13, 14, 15, 16], but there has not been a definitive method for calculating the energy derivative in all systems. There has also been some work on non-energetic quantities, such as the dipole moment[17], but most of these studies used the so-called pure Diffusion **Monte** **Carlo** method[18, 19], which does not work for larger systems. Recently, the Reptation **Monte** **Carlo**[20] method has been developed, which shows promise for larger systems, but has not been applied to chemical problems.

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Statistical sampling had been known for centuries, but it was really the advent of com- puters that made this approach feasible for attacking many problems of physics. The **Monte** **Carlo** method was part of the picture from the very beginning, thanks to Nicholas Metropolis, who was the leader of the team that designed and built one of the very first electric computing machines ENIAC and MANIAC in the Manhattan project in Los Alamos. Metropolis was also involved in improving the method by introducing a tech- nique, which is today known by the name “importance sampling”.

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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.

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Although this is the seventh excited state for three electrons in the Coulomb potential, it is the ground state of its symmetry. The chosen state and system exhibit a near-degeneracy effect anal- ogous to the one in the Be atom, so that we can make a direct com- parison between the two cases. We have therefore carried out DMC calculations with trial wave functions constructed at the HF and multi-reference levels of **theory**, and we analyzed the correspond- ing ﬁxed-node errors as well as associated nodal structures. We have estimated the exact energies using an alternative method, full Conﬁguration Interaction extrapolated to the complete basis set limit (FCI/CBS). The density dependence was varied by changing the nuclear charge of the system. The results show that the ﬁxed-node error in the HF wave function is quite signiﬁcant, while the two-conﬁguration trial wave function enabled DMC to recover almost all of the correlation energy. In addition, the results clearly

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Variational and di usion Monte Carlo, the most ommonly used ele troni stru ture QMC variants, use sto hasti methods to optimize wavefun tions and al ulate expe tation values [56℄ and an [r]

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We report an accurate study of interactions between benzene molecules using variational **quantum** **Monte** **Carlo** (VMC) and di ffusion **quantum** **Monte** **Carlo** (DMC) methods. We compare these results with density functional **theory** using di fferent van der Waals functionals. In our **quantum** **Monte** **Carlo** (QMC) calculations, we use accurate correlated trial wave functions including three-body Jastrow factors and backflow transformations. We consider two benzene molecules in the parallel displaced geometry, and find that by highly optimizing the wave function and introducing more dynamical correlation into the wave function, we compute the weak chemical binding energy between aromatic rings accurately. We find optimal VMC and DMC binding energies of −2.3(4) and −2.7(3) kcal /mol, respectively. The best estimate of the coupled-cluster **theory** through perturbative triplets/complete basis set limit is −2.65(2) kcal/mol [Miliordos et al., J. Phys. Chem. A 118, 7568 (2014)]. Our results indicate that QMC methods give chemical accuracy for weakly bound van der Waals molecular interactions, comparable to results from the best **quantum** chemistry methods. C 2015 AIP Publishing

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After the introduction of basic concept and **theory** of **quantum** **Monte** **Carlo** and atomic Fermi gas system, we study the two component atomic Fermi gas at unitary limit in the periodic box using variational and diffusion **Monte** **Carlo** methods. By tuning the scattering length to infinity and extrapolating the effective range of the interacting potential close to zero, we find the Bertsch parameter drops approximately 5% compared to the previous **Monte** **Carlo** study without the effective range extrapolation. To examine the size of the fixed-node error, we also carried out released-node diffusion **Monte** **Carlo** calculations. We find the fixed-node error for the smallest system of four atoms is marginal. We calculate the ground state properties for a larger system of sixty-six atoms. We calculate pair correlation function and one-body and two-body density matrices, from which we extract the condensate fraction. We find that more than half of the atoms are involved in pairing. We also analyse the nodal surface properties for the system in BCS, unitary and BEC regime and extract Tan’s constant from calculated results.

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Replacing ω by i|ω n |, the Green’s function satisfies
[(r, i|ω n |)ω n 2 δ il + ∇ im × ∇ ml ×] D lk (r, r 0 ; i|ω n |) = −4πδ ik δ(r − r 0 ). (A.8)
It is possible to find a general form for the vdW part of the thermodynamic quantities for an arbitrary inhomogeneous medium based on the **quantum** field **theory**. In **quantum** field **theory**, physical quantities are described in perturbation series whose terms can be described by an appropriate Feynman diagram and computed based on Feynman technique. The advantage of the diagram technique is that the terms in the perturbation series can be infinite and the problem can be solved easily by taking a summation over all the infinite sequences which is called “principal diagrams”. Every interline of the diagram are associated with a temperature Green’s function for the free particle or a free photon Green’s function D and each intersection of lines (vertex) is related to an interaction operator. Finally, an integration is carried out over the four dimensional coordinates of each vertex in the diagram. The average value of any quantity in the field **theory** is computed by the equations of motion for the field operators. To preserve the formal similarity with the usual equations of motion, the time t from the real value shifts to the imaginary value of τ which varies from [−1/T, 1/T ]. The natural unit system ~ = c = K B = 1 is used here. In a perturbed system, the Hamiltonian is

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6.1 Summary of highlights
Inspired by the interesting deconfined **quantum** criticality (DQC) **theory** that predicts the deconfinement of spinons at certain critical points, I developed a technique to quantitatively define the S = 1/2 emergent excitation spinons at **quantum** phase transitions (QPTs) in **quantum** spin models. Though the method only works with the valence-bond basis, I have illustrated that the measured spinon size λ and confinement length Λ are inherited in the spin-spin correlation functions as well. Therefore, these two lengths appear to be basis- independent physical observables. My test results in 1D have demonstrated that, spinons are not well-defined in the N´ eel states, marginally-defined at the critical point and well- defined in the valence-bond solid state, which agree well with theoretical expectations. The main purpose of this work is to test the spinon deconfinement in the 2D AFM—VBS phase transition, which violates the transitional Landau-Ginzburg-Wilson **theory**. I have success- fully demonstrated that spinons are deconfined in 1D spin systems. I have also observed that spinons become confined in the dimerized 1D spin chain or ladder systems. In 2D system, by simulating the J − Q 3 model, which holds the AFM to VBS QPT with the spontaneous

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Dynamic correlation accounts for the correlated motion of particles approaching one another. Concerning electrons of the same spin one speaks of Fermi correlation in this context and defines an area of reduced probability around one electron for finding a second electron with the same spin. A description of this non-classical effect is guaranteed by the anti- symmetry of the SD. The analogue electrostatic phenomenon is called Coulomb correlation. They determine the correlation between all particles included, i.e. electrons of both spins and nuclei. For a description of Coulomb correlation the wave function must contain the proper electron electron and electron nucleus cusps not all present in the HF determinant. In **quantum** **Monte** **Carlo** for example, it is included by multiplying the Slater determinant(s) by a symmetric Jastrow (cusp) function. DFT methods have correlation functionals focussed on the problem. Perturbation **theory**, configuration interaction, and coupled cluster account for dynamic correlation by including several excited state Slater determinants in the wave functions.

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We now need to match this scalar Lagrangian to a DM-nucleon effec- tive interaction, non-perturbatively, at an energy scale corresponding to the chiral-symmetry-breaking scale of ∼ 1 GeV. In order to do that we use chiral perturbation **theory** [33–35], the effective **theory** that allows us to study the low-energy dynamics of QCD. χPT is a powerful tool that encapsulate the non-perturbative nature of QCD into a set of low energy constants (LECs), that can be determined from experimental data or calculated using lattice QCD.

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Diffusion **Monte** **Carlo** methods have been used mainly in studying electronic systems where the particle interactions vary relatively slowly with distance. Thus the finite time step approximation is expected to be valid. The most significant problem in using **quantum** **Monte** **Carlo** methods to study electronic systems is the treatment of identical particle statistics (Kalos (1984)). In most of the electronic **applications** of the zero temperature **quantum** **Monte** **Carlo** methods, approximate information about nodal surfaces in the Fermion wave function is used to provide boundary conditions for the random walks. Wave functions obtained from variational calculations are often used for this purpose. With the fixed node approximate methods (Reynolds et al. (1982)) the random walk results are dependent on the nodal surfaces used in the calculation. Methods for "relaxing" the nodes have been developed (Ceperley and Alder (1984)) and essentially exact solutions which are antisymmetric with respect to particle inerchange may be obtained.

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aimed to resolve the controversy, pointing out an error in one set of the QMC calculations; we will show that there are other more important reasons for the discrepancy. This is very significant because QMC, and, in particular, diffusion **Monte** **Carlo** 共 DMC 兲 , is often considered to be the most ac- curate method available and is used as a benchmark. If QMC calculations were shown to be wrong for this model system, serious doubt would be cast on this status. On the other hand, if the QMC results were proven correct, there would be even more serious consequences: a good deal of surface-science **theory** would have to be reexamined.

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temperature, a dimerized state that breaks no symmetries of the microscopic interactions can be stabilized for a lattice with two (or, a larger even number of) S = 1/2 spins per unit cell by introducing a pattern of different Heisenberg couplings in such a manner that favours singlet formation on dimers (or, larger units of an even number of spins). Such dimerized states preserves all the symmetries of the interactions and are dubbed **quantum**-disordered states to distinguish them from VBC states (that spontaneously break lattice symmetries) or spin liquid states (that have fractional excitations and topological order). Both the Shastry Sutherland model [11] and the bilayer Heisenberg model [33] (the latter will be discussed in more detail later) are Hamiltonians where such **quantum**-disordered ground states are realized. **Quantum** field **theory** techniques provide a very general framework for addressing low-energy properties of such interacting spin systems [13]. However, in many cases, while such an approach is really useful for the classification of possible phases on general grounds, several other details at the lattice level are important in deciding which phase is ultimately realized by a microscopic lattice Hamiltonian. Furthermore, any field **theory** has certain free parameters that have to be determined from a microscopic calculation to match its results (that apply at low-energy or equivalently at long distances) to an actual lattice model. **Quantum** **Monte** **Carlo** (QMC) methods provide an unbiased route for calculating properties of spin models that do not suffer from the notorious “sign problem” [14] for system sizes that typically far exceed other numerical methods like exact diagonalization. However, unlike exact diagonalization which gives access to all possible observables, QMC simulations typically allow for calculations of quantities like energy, uniform magnetic susceptibility, spin stiffness, and certain kinds of imaginary-time- dependent correlation functions. A general introduction to computational methods in **quantum** spin systems can be found in Ref. 15.

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