When applied to many-**electron** systems, the well-known fer- mion sign problem motivates the use of the fixed-node ap- proximation. Instead of the true ground state, the method then projects out the lowest-**energy** state with the same nodal **surface** as the trial wave function. 共The nodal **surface** is the set of points where the wave function is zero.兲 The fixed- node approximation introduces a small error in the total en- ergy, which we believe is different for bulk and slab simula- tions; the nodal **surface** of the trial wave function is closer to that of the true ground state in the bulk system than it is in the slab. Although the difference in errors is very small, the sensitive nature of the problem means that it 共along with the very slow convergence in L 兲 is the major source of inaccu- racy in previous QMC **calculations**. We will return to this point later and justify this assessment with reference to our results.

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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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An unexpectedly high sensitivity of the nodal quality to the details of the pair orbital at large distances are also observed. Although the suboptimal orbitals used in the 4-atom RN-DMC simulations are modified only in their long-range tails (see the upper row of Fig. 4.6), the fixed- node energies raise by sizeable amounts. This suggests an explanation for the relatively slow convergence of the released-node **energy**: the long-range tails of the pair orbital affect the nodal hypersurfaces, although the **energy** cost of nodal hypersurfaces displacement is surprisingly low. One can further deduce that this makes the released-node method quite challenging to apply since it requires correcting the nodal **surface** change by sampling low-density regions with walkers travelling large distances. This is, however, difficult to achieve since the diffusive motion of walkers is slow, proportional to t 1/2 , while the growth of the noise is fast, proportional to exp(∆ BF t), where ∆ BF is the difference between the bosonic and fermionic ground-state

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In this work we solve an effective-mass model of exci- tons (X), positive and negative trions (X + and X − ), and biexcitons (XX) in type-II **quantum**-ring heterostruc- tures, focusing on GaSb rings in GaAs. The holes are confined to the ring, which is modeled as an infinite po- tential well, while the electrons are excluded from the ring but bound to the holes by an isotropically screened Coulomb interaction. The kinetic **energy** of the tightly confined holes is the dominant contribution to the total **energy** of each complex; however, the **electron**-hole at- traction is non-negligible, as is the hole-hole repulsion. The ring was chosen to have a rectangular cross-section for computational convenience. The ring is centered on the origin, orientated so that the axis of rotation is the z- axis and the midpoint in the z direction is the x-y plane. The three parameters defining the ring geometry are the half height of the ring R z , the inner radius of the ring

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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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In this work, we performed **Monte** **Carlo** simulations, based on ab-initio input data of the **energy**-dependent dielectric function, of REEL and SE spectra of graphite, taking into ac- count features related to the anisotropic structure of the target material. Graphite has indeed a layered structure and this must be considered in the treatment of the **electron** transport properties. In particular, the determination of the inelastic mean free path and of the **energy** loss was carried out by considering a linear combination of the dielectric properties along the two main orthogonal crystal directions (in-plane and out of plane). In our model, coef- ficients of this linear combinations depend on an anisotropy parameter f and on the angle between the transferred momentum ~ q and the **surface** normal vector ~c . Our approach for including a dependence of the dielectric properties on the target anisotropy clearly improves the agreement between simulated and experimental REEL spectra. Indeed, spectral features are well reproduced by MC **calculations** for a value of the anisotropy parameter f = 0.6. This means that the **energy** loss along the ~ q || ~c (inter-planar) direction contributes for 60% of inelastic collisions, while 40% of the spectral features are contributed by collisions along the ~

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that localize the protons around the lattice sites. Ceper- ley and Alder (1987) explored both molecular and atomic phases, predicting a transition from the freely rotating molecular **quantum** crystal to a rotationally or- dered molecular phase at about 100 GPa, which was subsequently identified in experiments (Lorenzana, Sil- vera, and Goettel, 1990), and a further transition to atomic phases at about 300 GPa. The DMC **calculations** of Natoli, Martin, and Ceperley (1993) showed that the zero-point motion of the protons is a very important factor in determining the relative stabilities of the high- pressure monatomic phases of hydrogen. The electronic **energy** favors low coordination numbers at these densi- ties, but this is counteracted by the effect of the zero- point **energy**, which favors high coordination and sym- metry. At still higher pressures the Madelung **energy** of the proton lattice should result in a transition to close- packed structures. Natoli, Martin, and Ceperley (1993) found the diamond structure to be the most stable of the monatomic phases up to pressures of about 400 GPa. They concluded that the usual estimates of zero-point energies from mean-field theories assuming the har- monic approximation were inaccurate, in some cases by a factor of two. Natoli, Martin, and Ceperley (1995) found that insulating molecular phases with canted ori- entations of molecules on an hcp lattice were favored at pressures above about 123 GPa. Combining these data with their earlier DMC results on monatomic phases of hydrogen, they predicted that the transition to a mon- atomic diamond-structure phase would occur at around FIG. 12. Valence Compton profiles for Li in the [100], [110],

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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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DMC (Diffusion **Monte** **Carlo**) benchmark **calculations** in history served as Rosetta stone for modern electronic structure **calculations**. Heavy computational costs have been paid to evaluate and improve (projecting out ground state) explicit many-body wave functions in DMC in order to gain highly accurate results. One of the hallmark features of this approach is that **energy** benchmarks in DMC in theory are strict upper bounds for true ground-state energies even if they are not the exact. This means the energies we calculate in DMC are expectation values satisfying variational principle of **quantum** mechanics. This can be explained in simple terms as following

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Efforts to construct wave functions with accurate nodal surfaces have continued since the introduction of the fixed- node approximation. Single-determinant wave functions of- ten provide good nodal surfaces for closed-shell systems, and multideterminant wave functions can do so for small open- shell systems, although the required number of determinants becomes excessive for large systems. Compact pairing wave functions consisting of an antisymmetrized product of two- **electron** “geminals” 关 4 兴 were introduced long ago 关 5,6 兴 and have recently been used in QMC **calculations** for atoms and molecules 关7,8兴. Triplet-pairing Pfaffian wave functions were first used in QMC **calculations** for liquid 3 He by Bouchaud and Lhuillier 关 9 兴 , and recently this approach has been ex- tended by Bajdich et al. 关10兴, who considered atomic and molecular systems in which both parallel- and antiparallel- spin electrons are paired.

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We report a systematic analysis of the performance of a widely used set of Dirac-Fock pseudopo- tentials for **quantum** **Monte** **Carlo** (QMC) **calculations**. We study each atom in the periodic table from hydrogen (Z = 1) to mercury (Z = 80), with the exception of the 4f elements (57 ≤ Z ≤ 70). We demonstrate that ghost states are a potentially serious problem when plane-wave basis sets are used in density functional theory (DFT) orbital-generation **calculations**, but that this problem can be almost entirely eliminated by choosing the s channel to be local in the DFT calculation; the d channel can then be chosen to be local in subsequent QMC **calculations**, which generally leads to more accurate results. We investigate the achievable **energy** variance per **electron** with different levels of trial wave function and we determine appropriate plane-wave cutoff energies for DFT calcu- lations for each pseudopotential. We demonstrate that the so-called “T-move” scheme in diffusion **Monte** **Carlo** is essential for many elements. We investigate the optimal choice of spherical integra- tion rule for pseudopotential projectors in QMC **calculations**. The information reported here will prove crucial in the planning and execution of QMC projects involving beyond-first-row elements. PACS numbers: 02.70.Ss, 71.15.Dx

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Numerous all-**electron** DMC studies have been reported 关 5–17 兴 for atoms up to Z = 10, but very few have included heavier atoms. DMC studies of heavier atoms have normally used pseudopotentials to remove the chemically inert core electrons from the problem. However, pseudopotentials in- evitably introduce some errors and it may be useful to con- sider how much progress can be made with all-**electron** DMC **calculations**. Accurate all-**electron** **calculations** for at- oms may also be useful in constructing pseudopotentials which incorporate many-body effects. In this paper we report VMC and DMC **calculations** for the noble **gas** atoms He, Ne, Ar, Kr, and Xe, which extends the range of atoms studied within VMC and DMC up to Z = 54. The main aims of this paper are to investigate how well current all-**electron** DMC methods perform for heavy atoms and to study the scaling of the computational cost with Z.

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Methods developed to improve the scaling of QMC **calculations** 90,91 may find use in excitation **calculations**. By localizing low-lying states which are not directly in- volved in excitations, the number of nonzero orbitals to evaluate at a given point r is reduced, and the Slater matrix is made sparse, improving the cost scaling of the Slater part of the wave function by a factor of N . An additional side effect of this is to reduce the computa- tional expense of the inclusion of backflow correlations (whose dominant cost arises at the orbital-evaluation stage of a calculation). However, a major problem with the use of localized orbitals is that, in order to obtain efficiency increases, one sacrifices accuracy in individual total energies by truncating localized orbitals to zero at finite range. The extent to which this loss of accuracy will affect total-**energy** differences in solids is unclear, al- though early studies on molecules have provided positive results. 42 Given that other biases (single-particle finite- size effects, time-step bias, etc.) cancel so well in gap **calculations** in solids (see Sec. IV C 1) we expect the loss in accuracy in **energy** gaps due to the truncation of local- ized low-lying electronic states to be very small. On the other hand, computational expense is often dominated by other factors such as the evaluations of two-body terms in the Jastrow factor and updates to the Slater matrix, limiting the scope for speedup.

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Excitonic effects play a particularly important role in the optoelectronic behavior of two- dimensional (2D) semiconductors. To facilitate the interpretation of experimental photoabsorp- tion and photoluminescence spectra we provide statistically exact diffusion **quantum** **Monte** **Carlo** binding-**energy** data for Mott-Wannier models of excitons, trions, and biexcitons in 2D semiconduc- tors. We also provide contact pair densities to allow a description of contact (exchange) interactions between charge carriers using first-order perturbation theory. Our data indicate that the binding **energy** of a trion is generally larger than that of a biexciton in 2D semiconductors. We provide inter- polation formulas giving the binding **energy** and contact density of 2D semiconductors as functions of the **electron** and hole effective masses and the in-plane polarizability.

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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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structure **calculations** are, however, good even in the most scattered one. So, the reason for the small scattering may not be attributable to the poor calculation, but to the high sensitivity of the bond order value to the bond lengths between ions which are determined from diﬀraction experi- ments in the wide temperature range and used for the present calculation. Also, as shown in Fig. 10, there is a recovery of the Ru-O bond order by the tilting of RuO 6 octahedra in

estimation of the flow at a point isotropic source. The absorbed dose rates in the air, dose accumulation factors, and **energy** distributions of the photon flux density at different distances from the source were taken as characteristics of the photon radiation fields. At the same time, the average of these values on the cylinder axis were estimated for a unidirectional source. The FLUKA program calculates the **energy** distributions of the photon flux density in the material under consideration. These distributions were calculated at distances from the source from 0.25 to 30 mean free paths. In this case, the photon mass attenuation coefficients [2, 6] given in table 1 were used for the transition from the mean free path to the true distance in linear

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In this paper, we investigate three existing methods for outperforming MC, namely, multilevel **Monte** **Carlo** (MLMC) [3], quasi-**Monte** **Carlo** (QMC) [4] and multilevel quasi-**Monte** **Carlo** (MLQMC) [5]. We apply these methodologies to the problem of travel time estimation in heterogeneous porous media. This is of central importance in a series of engineering applications ranging from groundwater management to groundwater remediation. It also involves the development of mathematical models for reactive transport in porous media. These models are used to assess, for instance, groundwater contamination, CO 2 sequestration, residence time distributions, etc. The QoI considered in this study will be the result

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In the second part of this study, cutout correction factors for both regular [square] and irregular shaped cutouts are measured and compared with the corresponding eMC calculated data. For this, a simple and direct relative dosimetric method is developed. A cutout is an insert made up of an alloy cerrobend which is placed on the applicator to produce customized shape for radiation beam in different clinical use (figure 2). The cerrobend is a low-temperature melting alloy containing bismuth, lead, tin, and cadmium in 50.0%, 26.7%, 13.3% and 10.0% by weight, respectively. The shielding thickness of the cutouts should he approximately equal to the maximum range of the highest **electron** **energy** beam passing through it [15]. Different square cutouts of sizes from 10X10 cm 2 to 3X3 cm 2 are prepared in the mould room of our therapy center. The irregular cutouts were obtained from 10 patients with different sites of cancer who had already completed their treatment recently. QA plans for all these cutouts are prepared in Eclipse TM using eMC algorithm and the corresponding MUs are noted. The required MU can also be calculated manually by using initially measured O.F. The effect of applicator correction, not the cutout correction factor has been considered during this calculation. Now measurements are carried out by delivering above MU on the phantom using corresponding cutouts (both regular and irregular). From the measured data, the cutout correction factors can be calculated. This correction will be applied to the manually calculated MU to get the corrected MU (MU corr ), which is shown below in

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