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.

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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.

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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

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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.

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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.

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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[5] which can be solved exactly using Bethe Ansatz techniques (BA)[6]. In the limit of infinite repulsion one realizes the Tonks- Girardeau gas (TG)[7] 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.

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A Jastrow function with parameters which are transferable to large classes of hy- drocarbons is demonstrated [77]. 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.

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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.

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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.

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.

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 [2] R12 method of Valeev et al., and the explicitly correlated canonical transcorrelation approach of Yanai

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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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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.

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.

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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].

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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

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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.

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.

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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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