bluuebunny/crossref_metadata_2025 huggingface dataset:
Stochastic wave function for atomic helium
Authors: Kalos M H
Published: November 2023

This paper reports a first application of the Green’s function approach (GFQMC) introduced by Kalosa to the calculation of the wavefunction for an atomic system. Although the precision of the results obtained was (far) “less than that already achieved for helium in a number of existing calculations,” the study provided a useful exploration of the method. It revealed some of the problems associated with the sign change for particle weights introduced by the potential energy term (V/ E) occurring in the expression for sampling with a Green’s function. Kalos investigated several methods for avoiding or minimizing the problem. Since the ground-state wavefunction for helium is nodeless, the sign problem associated with fermion nodes does not present itself in this case.
A soft Hartree-Fock pseudopotential for carbon with application to quantum Monte Carlo
Authors: Greeff C W , Lester W A
Published: November 2023

The popular norm-conserving Hartree-Fock pseudopotentials of Stevens, Basch, and Krauss0 have been highly successful in reducing the computational effort required in several types of ab initio electronic structure calculations, including both variational and diffusion QMC calculations. These pseudopotentials create a problem specific to diffusion calculations that leads to occasional large fluctuations in local energies and correspondingly higher uncertainties in the calculated energies. In this paper Greeff and Lester propose and describe a new pseudopotential expression for carbon atoms that avoids this problem. The new pseudopotential is “soft.” It is a smooth even function of the electron-nucleus distance r, which is nonzero at r = 0 and does not diverge at r = 0. It is written in a standard form for use in quantum chemistry programs. Tests with analytic Hartree-Fock calculations showed excellent agreement with HF energies from all-electron calculations and others using standard pseudopotential functions for the small hydrocarbons CH, CH4, and CH2CH2, as well as for the triplet-singlet splitting of CH2. Tests with the new pseudopotential in VQMC and DQMC calculations for the species C and CH showed a much greater stability, allowed larger time steps without stability problems, and exhibited a much lower time-step dependence than those for earlier pseudopotentials. The C-H bond energy determined from these calculations was in good agreement with the experimental value and with values from other high-level calculations. By making possible the use of larger time steps, these soft pseudopotentials significantly increase the range of practical applications of QMC.
Quantum Monte Carlo study of Be2 and group 12 dimers :tvh (M = Zn, Cd, Hg)
Authors: Schautz F , Flad H J , Dolg M
Published: November 2023

This study was undertaken as a first step toward QMC calculations for clusters of Zn and Cd atoms, following earlier studies of Hg atoms. These dimers are weakly bound in their ground states, but they have some strongly bound excited states which make them interesting candidates for use in eximer lasers, and this has stimulated several successful coupled cluster calculations and spectroscopic studies for these species. As a result, there are sufficient data for comparisons of QMC results. The QMC calculations were fixed-node diffusion calculations carried out with relativistic energy-consistent large-core pseudopod tentials. These ab initio pseuodopotentials eliminated 28, 46, and 78 electrons in Zn, Cd, and Hg, leaving only 2 electrons per atom and 4 for each dimer - like the helium dimer He2. Semi-empirical core-polarization terms were added to the potential energy expression. Relative energies, as well as static dipole polarizabilities, were determined for both atoms and dimers. The polarizabilities required an added finite-difference approach with applied fields. From the complete set of results, the spectroscopic constants Re, De, and We for Zn2 and Cd2 could be derived. As for earlier calculations for Hg2 and for test calculations, these were found to give good agreement with values derived from spectroscopic measurements and from coupled cluster calculations. Similar agreement was found for the polarizabilities.
Adiabatic diffusion Monte Carlo approaches for studies of ground and excited state properties of van der Waals complexes
Authors: Lee H S , Herbert J M , Mccoy A B
Published: November 2023

This paper introduced an “adiabatic” extension to simple diffusion QMC in order to provide (1) a means for generating ‘¢ 2 distributions for evaluating expectation values, and (2) a means for optimizing single-parameter node locations for excited states. The procedures were illustrated with applications to the low-energy vibrational states of the loosely bound van der waals complexes NeSH and Ar2HCl. The adiabatic scheme is based on first-order perturbation theory with a Hamiltonian expressed as fI = fI(o) +,\W and an energy approximated by En = E o) +>.E 1). If a typical diffusion simulation in imaginary time T is modified by adding to the Hamiltonian a perturbation, >.W, which changes slowly with time as d>./dr (the adiabatic approximation), the average energy E( r) evolves linearly in >.. This provides the required ‘¢ 2 sampling for expectation values of W. In the case of a nodal surface which is a function of a single parameter T/, the correct node location may be determined by varying the position T/ with time in a similar fashion. For two simulations, one on each side of a nodal surface, the value of T/ which produces the same energy for both is the optimized node location. This replaces the use of repeated simulations with T/ fixed in each. The calculations were successful in reproducing ground-state expectation values for several interatomic distances and angles determined in analytic variational calculations for each of the species NeSH and Ar2HCl. For the lowest-energy vibrational states of Ar2HCl, the nodes in some cases could be defined by symmetry considerations. In others, the node locations were successfully approximated using the adiabatic diffusion method as described, and they produced energies in good agreement with alternate predictions. The advantages of this method arise in the simplification of the search for correct values of the ,\ and T/ parameters for expectation values and node location.
Energetics of carbon clusters C8 and C10 from all-electron quantum Monte Carlo calculations
Authors: Shlyakhter Y , Sokolova S , Lüchow A , Anderson J B
Published: November 2023

Carbon clusters in the range C2 to C24 presented a challenge to both experiment and theory long before the discovery of C60, and with that discovery the challenge has been extended to larger systems, particularly to carbon nanotubes. with this paper allelectron fixed-node diffusion QMC calculations were extended to Cs and C10 and compared with those of alternative ab initio methods. For systems of this size, very accurate calculations are available from MP2, MP4, CISD, and CCSD methods. For both Cs and C10, several isomeric structures lie within 1-2 kcal/mole of each other: linear and cyclic, acetylenic and cumulenic, singlet and triplet. The FN-DQMC calculations produced total electronic energies 0.4 to 1.2 hartrees lower than those of the lowest energy analytic variational and coupled cluster calculations and only 0.2 to 0.4 hartrees above experimentally based total energies. Recovery of correlation energy was 89 to 94%. For Cs the lowest energy structures were predicted to be the cyclic C4h 1A9 structure, the linear 3:Et cumulenic structure, and the linear 1Et cumulenic structure, all within 10 kcal/mo! of each other. For C10, the predicted lowest energy structures were the cyclic D5h 1A (distorted cumulenic) and D10h1A19 (cumulenic structures), within 5 kcal/mo! of each other. These results are in good agreement with earlier high-level calculations of all types, which in turn are in good agreement with each other. This gives confidence in each of the methods for this size of molecules. The authors included an analysis of the scaling of computation effort with number of electrons Ne, showing a third-power dependence as N!.
Benchmark quantum Monte Carlo calculations
Authors: Grossman J C
Published: November 2023

In this paper Grossman reported pseudopotential fixed-node diffusion QMC calculations of atomization energies for the Gl set of 55 small molecules,0 a standard benchmark set for testing methods in quantum chemistry. To maintain the uniformity in the calculations, they were carried out with the same basis sets and orbital types for single-determinant trial functions and effective core potentials (with a few exceptions). Variational QMC was used to optimize the determinan Jastrow trial functions. The mean absolute deviation EMAD in atomization energy from the experimental values was found to be 2.9 kcal/mol, the maximum deviation was 14 kcal/mol (for SO2), and the next five largest deviations were 7-8 kcal/mol. A comparison with results from other methods tested with the Gl set shows the FN-DQMC method to be competitive. For EMAD the list is: FN-DQMC, 2.9 kcal/mol; DFT (LDA), 40 kcal/mol; DFT (B3LYP or B3PW91), 2.9 kcal/mol; CCSD(T)/aug-cc-pVQZ, 2.8 kcal/mol; CCSD(T)/extrapolated, 1.3 kcal/mol; semi-empirical G1 theory, 1.6 kcal/mol. Additional calculations with different trial functions indicated that node location errors could be significantly reduced with multi determinant trial functions. Computational effort was not compared, but linear scaling for FN-DQMC was mentioned as an important advantage for QMC. 0J. A. Pople, M. Head-Gordon, D. J. Fox, K. Raghavachari, and L. A. Curtiss, Gaussian-I theory - A general procedure for prediction of molecular energies, J. Chem. Phys. 90, 5622 (1989).
Fermionization of a bosonic gas under highly elongated confinement: A diffusion quantum Monte Carlo study
Authors: Blume D
Published: November 2023

These two papers, published back to back, provide superb examples of the use of a quantum Monte Carlo method to obtain exact solutions for a problem difficult to treat by other methods. The problem is that of a bosonic atomic gas confined in a cigarshaped trap. As the cigar is narrowed the atoms are less able to pass each other, their behavior approaches that of fermions, and they are said to exhibit “fermionization.” Using diffusion QMC the authors were able to determine the ground-state energies, examine structural properties, confirm that fermionization can occur, and predict conditions under which it might be observed. The calculations by Blume were carried out for 2-20 atoms confined in axially symmetric traps of varied aspect ratios. The confining potential for each atom was given by Vk = CpPl + Czzl (radial and axial positions squared). The atom-atom interactions were simple hard-sphere potentials. The (nodeless) guided diffusion calculations used a trial wavefunction of the Jastrow type, a product of atom-trap terms and atom-atom terms. The calculations by Astrakharchik and Giorgini were carried out similarly but included both hard and soft spheres with numbers in the range of 5 to 100 atoms. Comparisons were made with predictions of the LiebLiniger equation of state.
Higher accuracy quantum Monte Carlo calculations of the barrier for the H + H2 reaction
Authors: Riley K E , Anderson J B
Published: November 2023

For many QMC calculations, the accuracy of the results may be improved by simply extending the calculation time to increase the number of samples. With an increase in computer speed of about a factor of ten every four years, and an inverse square-root dependence of statistical error on number of samples, one should expect a factor-of-ten improvement in accuracy every eight years. That is the case for the calculations reported in this paper. An “exact” QMC calculationa of the potential energy surface for the reaction H + H2 in 1992 gave a barrier height of 9.61 ± 0.01 kcal/mol. The same program was run with only minor changes on a faster machine about ten years later to give the results in this paper. This time, the barrier height was found to be 9.608 ± 0.001 kcal/mol. Energies for other points on the potential energy surface were similarly improved in accuracy. These included the energies of points for the ground state and the first excited state near the Jahn-Teller cusp. ao. L. Diedrich and J. B. Anderson, An accurate Monte Carlo calculation of the barrier height for the reaction H + H2 H2 + H, Science 258, 786 (1992).
Linear scaling for the local energy in quantum Monte Carlo
Authors: Manten S , Lüchow A
Published: November 2023

This paper describes the use of localized orbitals and shortrange correlation functions to achieve near linear scaling in fixednode diffusion QMC calculations. The success of the method is demonstrated with a series of all-electron calculations for linear hydrocarbons in the range of C2H6 to C2sHs2 with 18 to 202 electrons. The procedure used was not very different from that of earlier all-electron calculations for systems of these sizes. Some changes were needed for evaluation of the trial wave functions and their derivatives. The molecular orbitals for the determinantal wavefunction were composed of contracted Gaussian functions decaying rapidly with distance but limited by a cut-off radius. The resulting determinants were sparse and could be evaluated by a standard matrix LU decomposition package. The standard Schmidt Moskowitz correlation function was modified with use of a short-range scaled distance to allow short cut-off radii.
Geometry optimization in quantum Monte Carlo with solution mapping: Application to formaldehyde
Authors: Schuetz C A , Frenklach M , Kollias A C , Lester w. A
Published: November 2023

Since analytic gradients of the energy with respect to nuclear positions are not easily obtained in QMC calculations, determining the locations of critical points, such as energy minima and saddle points, on potential energy surfaces is not straightforward. The statistical uncertainty in QMC energies prevents the direct estimation of these derivatives, and although several techniques for overcoming the problem have been developed their applications have been limited to very small systems. This paper describes a different approach, more readily applicable, in which the number and location of points in the vicinity of a critical point are chosen in such a way as to minimize the computation effort required in predicting the location of the critical point. The method is generally known as “solution mapping” and is often referred to as “factorial design.” The procedure is illustrated with calculations to determine the minimum energy or equilibrium geometry of formaldehyde H2CO, assumed planar and symmetric about the C-0 bond, in terms of the C-0 and C-H bond lengths and the H-C-0 angle. The potential energies calculated at specified points are fit to a polynomial expression of ten terms involving the three variables as l,x1,x2,x3,XI,X ,x ,x1x2,x1x3,x2x3, and ten coeffcients. With 15 points prescribed by the factorial design the determination of the coefficients is simplified by the column orthogonality of the design matrix. Of course, statistical uncertainties in the energies lead to uncertainties in the coefficients, but the method minimizes their effects.
Variational quantum :r-.fonte-Carlo method in surface physics
Authors: Schattke W , Bahnsen R , Redmer R
Published: November 2023

This paper is of interest for its discussions of the QMC treatment of lattice displacements in bulk matter and at solid surfaces with the aim of the prediction of phonon spectra. Earlier calculations had been made with density functional methods. The authors report variational QMC calculations of ground-state energies for gallium arsenide crystals using both local and nonlocal pseudopotentials coupled with trial wave functions of the SlaterJastrow type. Shifting of atomic positions and their corresponding orbitals allowed the simulation of phonon displacements as well as the determination of the bulk modulus and related properties. As an example, the case of the [111] phonon for bulk GaAs was treated in a (2,2,8) simulation cell with arsenic atoms fixed and gallium atoms displaced by up to 10% of the lattice constant. The energies obtained were fit to a quadratic function of the displacement from which the frequency of the mode considered could be calculated. Agreement with experimental values for the frequency was found within the estimated error of a few percent. For the surface the general scheme was the same, but a number of changes were required for a slab geometry. A “confinement potential” was added, the treatment of periodicity was altered, and the Ewald summations were altered. The calculated energies and electron densities revealed several inconsistencies, such that the surface calculations could “only be considered as a first step.” Nevertheless, the calculations demonstrate that QMC treatment of surfaces, including phonon dynamics at surfaces, is feasible.
Efficiency considerations in the construction of interpolated potential energy surfaces for the calculation of quantum observables by diffusion Monte Carlo
Authors: Crittenden D L , Thompson K C , Chebib M , Jordan M J T
Published: November 2023

This paper addresses the problem of choosing configurations of a loosely bound molecular complex for constructing a potential energy surface to be used in calculating quantum observables of the complex. For a tightly bound system, diffusion QMC has been shown to be an efficient sampling method for this purpose.0 A loosely bound complex presents somewhat different problems, and the authors investigated alternative procedures which they found useful in combination with diffusion QMC sampling. The example chosen is the very floppy prototypical water dimer, for which a suitable potential energy surface was available, and this surface took the place of ab initio calculations in providing potential energies for configurations sampled. The data obtained in sampling were not fit to a function representing the surface, but were used in interpolation to obtain the values of potential energies at other points on the surface required in calculations of quantum observables.
Ground state of the electron gas by a stochastic method
Authors: Ceperley D M , Alder B J
Published: November 2023

A simple way of incorporating sampling into diffusion quantum Monte Carlo calculations was introduced with this paper. Coupled with a nodal relaxation scheme it provided a means for calculating the ground-state energies of a collection of electrons (fermions) over a range of densities. The simulation method may be regarded as a more complex version of the simple diffusion methoda or a less complex version of the Green’s function approach.b With use of a trial function ΨT the diffusion equation may be written in terms of a walker or particle density based on f = ΨΨT the product of the true wavefunction Ψ and the trial wavefunction ΨT, The resulting equation includes a diffusion, a drift, and a multiplication term to take advantage of local energies as described by Grimm and Storer.c The calculations were carried out at first with fixed nodes to establish an upper bound to the exact fermion energy and starting positions for walkers in a released-node calculation. Walkers crossing a node were continued with a sign change to produce an overall distribution proceeding toward the boson ground state but providing at intermediate times an approximation to the fermion ground state. If relaxation to the fermion ground state is fast compared to further relaxation to the boson ground state, the fermion ground-state energy can be determined. For the electron gas at low densities this is the case and useful energies may be obtained. For this paper energies were calculated for a range of densities with systems of 38 to 246 particles and extrapolated to provide results for systems of infinite size.
Solution of the four-nucleon Schrödinger equation
Authors: Zabolitzky J G , Kalos M H
Published: November 2023

An application of the Green’s function QMC method to the problem of four nucleons is reported in this paper. The calculations were carried out for the case of four nucleons interacting according to a pairwise-additive potential given by the sum of the Malfliet-Tjon interaction terms for each pair. The problem was reduced to that of four interacting, spinless bosons and the ground state is nodeless. The calculation was a straightforward application of Green’s function QMC as described by Kalosa with avoidance of sign changes for walkers by a large shift in the zero of potential energy. Energies were evaluated from the growth rate of walkers and with use of importance sampling using a function of the Jastrow form. The energy determined for the case of the a-particle was -31.1 ± 0.2 MeV for a trial function with an expectation value of -28.8 ± 0.01 MeV. Additional calculations were performed for the case of the four-atom helium droplet or cluster He4 with the atoms interacting via pairwise-additive Lennard-Jones potentials. The energy for the cluster was found to be -0.39 ± 0.01 K, a value in agreement with earlier estimates for similar Lennard-Jones potentials. The authors pointed out that the results of QMC calculations such as these could serve as benchmarks for less rigorous methods.
Fixed-node quantum Monte Carlo for molecules
Authors: Reynolds P J , Ceperley D M , Alder B J , Lester W A
Published: November 2023

This paper extends earlier calculations for the molecules H2 and LiH, and it increases the range to six and ten electrons with calculations for Li2 and H2O. The calculations were carried out with importance sampling using trial functions consisting of Slater determinants multiplied by Jastrow factors. For each system the expectation values of the energies for the trial functions were somewhat higher than those of the best configuration interaction calculations available at the time, but the fixed-node diffusion energies were significantly lower. The observed energy obtained for H2 was -1.174 ± 0.001 hartrees, compared to earlier values of -1.1744 hartrees in analytic variational calculations. For H2O it was - 76.377 ± 0.007 hartrees, compared to a value of - 76.3683 hartrees from variational calculations and an estimated true value for H20 of - 76.438 hartrees. The difference between the QMC value and the true value illustrates the problem of inaccurate node locations, in this case corresponding to a node location error of about 38 kcal/mol. Nevertheless, the energy so determined was about 5 kcal/mol lower than that of the lowest energy analytic variational calculation of the time.
Application Of The Green’s-Function Monte Carlo Method To The Compact Abelian Lattice Gauge Theory
Authors: Heys D W , Stump D R
Published: November 2023

The first application of diffusion or Green’s function QMC in treating lattice gauge theories was reported in this paper on the compact U(l) gauge theory, one of the simplest of the lattice gauge theories. Earlier investigations of the U(l) and other Abelian gauge theories had been made by a variety of other methods, including perturbation expansions, variational calculations, and most successfully path integral treatments using Monte Carlo integrations. Aside from investigation of QMC methods for the U(l) theory, one object of this study was to verify results for the path integral approach in the Hamiltonian formulation. The Hamiltonian itself was expressed as a summation of terms involving electric and magnetic fields at each site, link, or plaquette, for which the authors developed the usual Green’s function expression in a discrete form to be applied to an n(p)-space wavefunction X[n(p)], a function of integer-valued plaquette variables. Except for discrete spatial positions, the Green’s function calculations were very nearly the same as those for electronic structure studies. The systems treated were small lattices, 5 x 5 and 3 x 3 x 3, in two and three dimensions with a variable coupling constant A. Variational calculations with different trial functions suitable for strong and weak coupling were found to reproduce known behavior for strong and weak limits. The Green’s function calculations were facilitated with importance sampling and produced similar results, thus indicating the high quality of the trial wavefunctions. The results provide evidence for the existence of a phase transition in the three-dimensional case and for the nonexistence of a phase transition in the two-dimensional case.
Calculations of ground-state properties of liquid 4He droplets
Authors: Pandharipande V R , Zabolitzky J G , Pieper S C , Wiringa R B , Helmbrecht U
Published: November 2023

Earlier calculationsa of the binding energy and compressibility of liquid helium in bulk were successful in reproducing experimental measurements within a few percent. These indicated a pairwise-additive empirical potential to be entirely adequate. In this paper the same potential was used in similar variational and Green’s function QMC calculations to predict the properties of the ground states of liquid 4He droplets, or more recently “clusters,” containing 4 to 728 atoms. The calculation procedure was identical to that of the earlier worka except that boundaries were eliminated. Trial wavefunctions for both types of calculations were expressed as products of functions of interatomic distances similar to those used for the infinite liquid. A comparison of energies for the two types of calculations over the range of 4 to 112 atoms, showing the Green’s function energies only 2-3% lower than the variational energies, indicates a high accuracy for the variational functions by themselves. Density distributions obtained as long-term averages by both methods were somewhat noisy at the droplet centers, but smooth and in good agreement away from the centers. These were fitted by quadratic and higher order functions from which surface tension could be estimated. For the larger droplets, the density was constant across the central region and approached the density of the bulk liquid. The authors noted that the success of these calculations gives confidence in the use of liquid drop formulas in predicting properties of nuclear matter.
Relativistic effective potentials in quantum Monte Carlo calculations
Authors: Hurley M M , Christiansen P A
Published: November 2023

This is the first of three papers,0 each from a different group, to report the use of effective potentials, model potentials, or pseudopotentials in QMC calculations in 1987 and 1988. The authors point out the advantages of eliminating core electrons in order to simplify the calculations, reduce the number of singularities in the wavefunction, and thereby reduce the variance in local energies. They note the primary difficulty in using relativistic effective potentials is the nonlocal character of most potentials and their dependence on angular projection operators rather than on mere electron configurations. In this study, the authors defined a simple local potential V in terms of the nonlocal effective potentials operating on a trial wavefunction. The extent of the error introduced was unknown but possibly small. The effective potentials were taken directly from earlier analytic studies for the species considered. The trial wavefunctions were single determinants of Gaussian orbitals together with Jastrow correlation terms. Calculations were carried out for Li/Li- and K/K- so that electron affinities could be obtained. The values so determined were 0.56 and 0.52 eV, respectively, compared to 0.618 and 0.502 eV determined in experiments. It was noted that corresponding all-electron calculations for the F /F- pair required a vastly greater computation effort, perhaps more than a factor of 1000 greater.
Is there a zeroth order time-step error in diffusion quantum Monte Carlo?
Authors: Reynolds P J , Owen R K , Lester W A
Published: November 2023

These three notes each offer evidence against an earlier claim, added as an appendix to a paper by Moskowitz and Schmidt,a that the discretization of the imaginary-time Schrödinger equation was in error even for infinitesimal time-step sizes and that all results with this short-time approximation were incorrect. As acknowledged by Moskowitz and Schmidt, there was an error in their analysis and they agreed that the short-time approximation was correct after all.
Time step error in diffusion Monte Carlo simulations: An empirical study
Authors: Rothstein S M , Patil N , Vrbik J
Published: November 2023

The discretization of the differential equations describing the transient behavior of a system usually introduces a time-step error. The problem occurs in diffusion QMC and may, in principle, be overcome by varying the time-step size and extrapolating results to a step size of zero. This paper shows that such extrapolations are not so simple, and it provides explanations for some of the surprises encountered in using several different procedures. Following the importance sampling procedure of Grimm and Storer,a the Green’s function and similarly discretization may be separated into individual functions or terms for drift, diffusion, and branching (or weight change). Rothstein et al. combined these in varied orders of full and half steps and examined their behavior in calculations for H2 and LiH with different time step sizes. Both linear observed, and even these depended on the system. Further analysis gave insights into the causes for the trends observed and some practical advice for selecting the best procedures. The problem treated in this paper is that of understanding xenon cluster ions at the microscopic level. Some properties of cluster anions had been measured, but these revealed no detailed information on structure and energetics. In the study reported here, diffusion QMC was used to investigate electron binding to small clusters of up to 19 atoms in fixed minimum-energy configurations. This was followed by path integral calculations to determine equilibrium distributions of cluster structures at temperatures of 10-100 K for several cluster sizes.
Structure and energetics of Xe
Authors: Martyna G J , Berne B J
Published: November 2023

The problem treated in this paper is that of understanding xenon cluster ions at the microscopic level. Some properties of cluster anions had been measured, but these revealed no detailed information on structure and energetics. In the study reported here, diffusion QMC was used to investigate electron binding to small clusters of up to 19 atoms in fixed minimum-energy configurations. This was followed by path integral calculations to determine equilibrium distributions of cluster structures at temperatures of 10-100 K for several cluster sizes. Diffusion QMC calculations for frozen clusters were carried out for an additive electron-xenon pseudopotential: first, using a trial wavefunction given by the product of s-wave solutions for single electron-xenon pairs with the pseudopotential interaction, and second, with an improved function based on the results of the first and minimization of the energies. Variational energies could be determined by numerical quadrature for this three-dimensional problem. For clusters of five and fewer atoms, the binding energy of the electron could not be determined, although an upper bound was found to be 12 K. For 6-19 atoms the binding energy increased in monotonic fashion from 12 K to 1100 K. The change in binding energy with number of atoms exhibited variations which could be explained in terms of the cluster structures. For the smaller clusters, the electron probability distribution was found diffuse and extended outside the cluster. The path integral calculations gave good agreement with the diffusion QMC calculations for the smaller clusters. For larger clusters of 50 and 100 atoms, the electron distribution was localized within the cluster.
Quantum Monte Carlo method for some model and realistic coupled anharmonic oscillators
Authors: Caffarel M , Claverie P , Mijoule M , Andzelm J , Salahub D R
Published: November 2023

This paper reports a first attack on the node location problem for excited states of two-dimensional systems corresponding to coupled one-dimensional oscillators. The zero-point energy of such systems may be readily evaluated using diffusion with or without importance sampling, but as for electronic states, the problem of node locations prevents such easy solutions for excited vibrational states unless the nodes can be specified from symmetry considerations. The authors present a modified fixed-node approach which is a good approximation when the nodes of a reference function are close to the exact nodes. The method is based on variations in node locations to minimize the energies in the regions separated by nodes, subject to the constraint that the energies must be equal. This is accomplished with a complete wavefunction made up of a variable combination of wavefunctions corresponding to those for uncoupled oscillators. Applications for several two-dimensional systems including a realistic model for adsorbed CO on a surface gave essentially exact agreement with the results of analytic variational solutions with large basis sets. The authors concluded that their method avoided many of the problems encountered with other methods and appeared well adapted for systems of strongly coupled oscillators and multiple-well potentials.
Effect of crystallite shape on exciton energy: Quantum Monte Carlo calculations
Authors: Mohan V , Anderson J B
Published: November 2023

About the time that small semiconductor crystallites were first called “quantum dots”, quantum Monte Carlo calculations were first applied in studies of these systems. Small crystallites have spectroscopic properties like those of molecules with molecular orbitals, and they shift to the behavior of electrons and holes in continuous bands as their size is increased. This paper reports a study of the effect of size and shape of crystallites on ground-state energies for an electron-hole pair (or vVannier exciton) confined to the crystallite. The system was investigated using a model by Brusa with the electron-hole pair similar to a hydrogen atom having effective masses for each, unit charges for each, and a coulombic interaction governed by a bulk dielectric constant. The electron and hole were confined to the crystallite by setting the charge densities to zero at the boundaries. Estimates of the energies for several sizes of spherical CdS and ZnO crystallites had been made earlier in variational calculations by Brusa and by others. The calculations of this later paper were carried out for spheres, cubes, and rectangular parallelopipeds of CdS and ZnO crystallites using diffusion QMC with importance sampling. The trial function was a product of three dimensional particle-in-a-box (or sphere) functions for electron and hole, multiplied by a hydrogen-like function of the electron-hole distance.
Optimization of a multideterminant wave function for quantum Monte Carlo: Li2 (1I:_;)
Authors: Sun Z , Barnett R N , Lester w. A
Published: November 2023

For many systems a single determinant Hartree Fock wave function is not adequate for use as a trial function specifying the nodes for a fixed-node Q1vIC calculation. For nondynamical correlation effects such as those occurring with a 2s-2p near degeneracy and for the dissociation of many molecules a multi determinant wave function is required. The minimal bases for such calculations may be discovered in low-level analytic variational calculations with configuration interaction. For this paper the authors investigated use of a four-determinant function for the 1  Et ground state of the dimer Li2 in both variational and fixed-node QMC. The four determinants were assembled from four different a molecular orbitals (from ls, 2s, and 2pz AOs) and two 7f orbitals (from 2px and 2py AOs). The sum of the four determinants with four adjustable coefficients was multiplied by electron-electron and electron-nucleus Jastrow terms. The results showed the advantages of multi determinant functions but also revealed some hazards in their use. For Li2 at its equilibrium separation of 5.05 bohr, the optimization of 41 parameters, including the two free coefficients of the determinants, to minimize the variance in local energies gave the lowest VQMC energy (recovering 77% of the correlation energy). However, in FN-DQlVIC the lowest energy (recovering 100% of the correlation energy) was obtained with the coefficients of the determinants given by an original MCSCF calculation without optimization. Thus, minimization of the variance failed to optimize the node structure. Further, it was found that the complete optimization produced spurious nodes and in so doing raised the FN-DQMC energy obtained. Clearly, multi determinant wave functions must be used with care.
Simple Symmetries
Authors: Gilmore Robert , Letellier Christophe
Published: November 2023

In this section we describe the structure of equations with two-fold symmetry. This is done for three-dimensional dynamical systems since we would like to be able to visualize the strange attractors that they generate. In addition, this is the only dimension for which a satisfactory theory of chao s exists.
Invariant Polynomials
Authors: Gilmore Robert , Letellier Christophe
Published: November 2023

Our ultimate goal is to find the relation between equivarian dynamical systems and locally identical (“locally diffeomorphic”) invariant dynamical systems. This is the cover and image problem. The image dynamical system is always expressed in terms of invariant polynomials. The cover dynamical system is expressed in terms of equivariant polynomials. It is the objective of this chapter to describe invariant polynomials, equivariant polynomials, and their properties.
Scholia, Commentaries, and Lexica on Specific Literary Works
Authors: Dickey Eleanor
Published: November 2023

This category includes the most famous and most often cited scholia. By far the most important are the Homer scholia, but those on Pindar and the Attic dramatists are also significant. Ancient scholarship on Homer was extensive and of high quality, for the best scholars of antiquity devoted much of their time and energy to the Homeric poems. Work on Homer that could be described as scholarship goes back at least to the classical period and probably to the sixth century bc, and editing the text of Homer was one of the main tasks of the first Alexandrian scholars. Zenodotus, Aristophanes of Byzantium, and Aristarchus probably all produced editions of the Iliad and Odyssey, and Aristarchus wrote extensive commentaries, while Zenodotus and Aristophanes compiled glossaries of primarily Homeric words. In addition, the early and persistent use of Homer as a school text meant that there was a tradition of school exegesis that reached back as far as the classical period. Though none of the very early work on Homer survives in its original form, a surprising amount is preserved in various later compilations, so we often know, for example, the readings of several different Alexandrian scholars for a particular passage, and even some of the arguments behind these readings (although the arguments preserved in later sources cannot always be assumed to be those of the editor himself). Two principal sources for the ancient scholarship on Homer survive: the scholia and Eustathius’ commentaries, both of which are gigantic works filling many volumes in modern editions. There are also some smaller works, some of which are more valuable than others.
Other Scholarly Works
Authors: Dickey Eleanor
Published: November 2023

While the production of texts and commentaries on literary works was probably the primary goal of ancient scholarship, other lines of work were pursued as well, particularly in the later Hellenistic and Roman periods. Among these other scholarly genres were the grammatical treatise, in which scholars analysed the classical Greek language and tried to codify the underlying principles of correct usage, and the lexicon, in which unusual words were collected, classified, and explained. (Two other popular genres, mythography and paroemiography, have been excluded from this book but were clearly related to the scholarly genres discussed here.) Modern interest currently focuses on the grammatical writings much more than on lexica, and thus it is much easier to find reliable texts, commentaries, and translations of grammatical works than of lexica; precisely for this reason, however, the latter offer greater opportunities for future work.
Introduction to Scholarly Greek
Authors: Dickey Eleanor
Published: November 2023

The difficulties ancient scholarship presents to the reader are very different from those involved in reading literary authors such as Demosthenes, Sophocles, Lucian, or Aristophanes. Sentences in lexica, grammars, and scholia tend to be fairly short, and grammatical and syntactic complexities are relatively rare. These works are largely free of obscure, archaic, and dialectal forms, and the vocabulary is in many ways more limited and more manageable than that of most literary texts. Yet scholarly Greek is not easy to read until one becomes familiar with the genre, because it employs a set of space-saving conventions and numerous technical words pertaining to scholarly disciplines. In addition, it sometimes happens (though not as often as one would expect) that the late date of the writer betrays itself in the use of post-classical words or constructions, and a grasp of the Greek numeral systems and the use of various editorial symbols is not infrequently required to get the full meaning of a passage. Scholarly Greek makes use of certain syntactic conventions rarely encountered in classical literature. These allow many words to remain unexpressed and so permit highly compact, very efficient writing, but they require some adjustment on the part of readers more familiar with other types of Greek. Though many such conventions are restricted to particular types of scholarship, others are found more generally.
Glossary Of Grammatical Terms
Authors: Dickey Eleanor
Published: November 2023

This section is not a complete dictionary, but a glossary giving in most cases only the grammatical meanings of the words included; these words are also used by scholarly writers in their non-technical senses on occasion. For such meanings and fuller information on these words, including citations of passages in which they occur, see LSJ and Bécares Botas (1985). A selection of references is given here to other works in which individual terms are discussed; such references are normally given only once but should be understood to apply to closely related words as well (e.g. a discussion of ajmûiboliva will normally be useful for understanding ajmûivbolo" as well). The state of scholarship on Greek grammatical terminology is not one that would make it possible for a glossary of this type to be completely reliable. The only specialized dictionary (Bécares Botas 1985) is full of errors, the information in LSJ is seriously incomplete, and other discussions are widely scattered, incomplete, and often unreliable. There is a great need for a thorough, accurate study of this vocabulary—and this glossary is not intended to address that need, only to help learners to get through texts. For lack of anything better, the information given here is based on that in Bécares Botas (1985) and LSJ, corrected and supplemented from a wide range of other sources.
Self-Conscious to the Nth Degree
Authors: Ford Emily , Liebowitz Michael R , Andrews Linda Wasmer
Published: November 2023

There was a time I would have been up nights preparing my script and reciting it to an outfit (complete down to its underwear) hung weeks in advance behind the bedroom door. Every worst-case scenario would have been thought through, and I’d have prepared a litany of excuses in the event I froze, stared off into space, or filled with tears. But nine-hours-and-counting to a day that could prove most embarrassing, I was drifting in and out of sleep thousands of feet over a storm system in New Jersey. A girl who had for years traveled exclusively by Greyhound was flying, actually flying, from Washington, DC, to northern New York without keeping a white-knuckle grip on the armrest or reworking the opening lines to the next day’s meetings in her head.
Not Just Another Shy Teenager
Authors: Ford Emily , Liebowitz Michael R , Andrews Linda Wasmer
Published: November 2023

I am going to be so cool .. . I won’t know what struck me. I am going to be talkative and energetic. I think I can do it if I try really, really hard. I want to try out the new me, the new talkative and energetic person I am going to be. I can do it if I try. I’ll leave my fears behind and be popular. I’ll have to call people. I can do that. I hope.
First Steps Toward Recovery
Authors: Ford Emily , Liebowitz Michael R , Andrews Linda Wasmer
Published: November 2023

For years, I was caught in a web of what-ifs: What if people think I’m exaggerating? What if it seems that I’m only after attention? What if my anxiety is something that will go away on its own? What if it’s not and I’m sent to a mental hospital? What if all my anxiety is actually due to a personality flaw—something I could fix if I were stronger, braver, kinder, and altogether a better person? Name any television series that had a therapist as a character and was on the air or in reruns when I was a teenager, and I’ve probably seen it—The Bob Newhart Show, M*A*S*H, even the animated Dr. Katz, Professional Therapist. Any series that depicted psychological problems fascinated me, and I jealously watched as the Hollywood therapists treated patients with kindness and compassion. I tried to hide my keen interest in these scenes from others, ashamed of what it seemed to reveal about my own emotional state. I kept a tight grip on the remote, and if anyone walked into the room—click! I’d switch to the home-shopping channel.
Making Changes and Sticking to Them
Authors: Ford Emily , Liebowitz Michael R , Andrews Linda Wasmer
Published: November 2023

I returned home from my week at the Ross Center determined to face the situations I had grown accustomed to avoiding in the past. For the first time in years, I felt confident that I was moving in the right direction, and I pushed myself to keep up the momentum. I seized every opportunity to run errands, eat at restaurants, and answer the phone. My local chapter of Toastmasters International met once a week, and I signed up to work on speaking off the cuff.
The Concept of Depression
Authors: Horwitz Allan V , Wakefield Jerome C
Published: November 2023

The poet W. H. Auden famously deemed the period after World War II the “age of anxiety.”1 For Auden, the intense anxiety of that era was a normal human response to extraordinary circumstances, such as the devastation of modern warfare, the horrors of the concentration camps, the development of nuclear weapons, and the tensions of the cold war between the United States and the Soviet Union. Were Auden still alive, he might conclude that the era around the turn of the twenty-first century is the “age of depression.”2 There would, however, be a crucial difference between the two characterizations: whereas the age of anxiety was viewed as a natural response to social circumstances that required collective and political solutions, ours is viewed as an age of sadness that is abnormal-an age of depressive psychiatric disorder that requires professional treatment.
Depression in the DSM-IV
Authors: Horwitz Allan V , Wakefield Jerome C
Published: November 2023

We claimed in chapter 1 that a flawed definition may be facilitating the recent surge in reported depressive disorder and may even lie at its very heart. To justify our claim, we now turn to a detailed examination of the DSM criteria for depressive and related disorders. Although the history of depression presented in the preceding chapter logically takes us up to the DSM-III, in order to ensure that our discussion applies to current di­ agnostic practices, we address the criteria presented in the latest edition-the fourth, text-revised edition DSM-IV-TR (2000). This does not represent much of a conceptual leap because the current criteria are almost identical to those in the DSM-III.