bluuebunny/crossref_metadata_2025 huggingface dataset:
The Monte Carlo method
Authors: Metropolis N , Ulam S
Published: November 2023

In this paper Metropolis and Ulam gave a brief introduction to “the Monte Carlo method” which is described as a statistical approach to the study of differential equations as applied by Metropolis, Ulam, Fermi, von Neumann, Feynman, and others at the Los Alamos Laboratory in the 1940s.0 Several examples of applications of Monte Carlo calculations are given. These include predicting the probability of winning at the game of solitaire, calculating the volume of an irregular region in high-dimensional space, and solving the Fokker-Planck equation for diffusion and multiplication of nuclear particles. The paper is completed with the first published description of the diffusion quantum Monte Carlo method, which is attributed to a suggestion by Fermi. In their words, “... the time-independent Schrödinger equation could be studied as follows. Re-introduce time dependence by considering This last equation can be interpreted however as describing the behavior of a system of particles each of which performs a random walk, i.e., diffuses isotropically and at the same time is subject to multiplication, which is determined by the point value of V.” The authors then indicate that the procedure gives the wavefunction corresponding to the lowest eigenvalue E.
Monte Carlo calculations of the ground state of three- and four-body nuclei
Authors: Kalos M H
Published: November 2023

In this paper Kalos proposed what is now known as the Green’s function quantum Monte Carlo (GFQMC) method for solving the Schrödinger equation. Kalos made the “critical” observation that the time-independent Schrödinger equation could be transformed into an integral equation for which the Green’s function is known if the boundary conditions correspond to the wavefunction vanishing at infinity. In a 3n-dimensional space R the integral form is (modified here) an equation which may be simulated by a random walk process in which particles at positions R’ are altered in weight by the V / E term and moved to new positions R obtained by sampling the Green’s function G(R’,R). As for simple diffusion, repetition leads to a distribution of particles corresponding to the lowest-energy wavefunction from which the energy E may be determined. Since time is not involved, there is no time-step error associated with the procedure. The term V / E must remain positive if mixed signs for particle weights are to be avoideda. Kalos examined the utility of the method in determining the boson ground-state energies for several three- and four-body nuclear problems involving pairwise-additive interactions with square, Gaussian, and exponential wells. The errors in the computed binding energies were estimated to be about 1%.
Helium at zero temperature with hard-sphere and other forces
Authors: Kalos M H
Published: November 2023

In using the Green’s function QMC method for calculations of hard-sphere systems one encounters the problem of multiplying the weight of a walker by the term V / E or (potential energy)/ (energy) when two spheres overlap and the potential energy is infinity. The authors of this paper note that this is “the only - but very great - complication in the Green’s function,” and they were able to devise a sampling method using subdomains to overcome the problem. Calculations were carried out for 256 particles in three-dimensional boxes of several sizes. Periodic boundary conditions were imposed by replacing exiting walkers at the opposite sides of the boxes. Importance sampling was used to aid in controlling the weights of walkers, as with a drift term in diffusion QMC, and in evaluating the energies. The trial function for importance sampling was a simple product of terms for all particle-particle distances. Both fluids and fee crystals were treated at several particle densities. The Green’s function calculations produced energies slightly but significantly lower than those of corresponding variational QMC calculations. The treatment for hard spheres was extended to Lennard-Jones 6-12 particles simulating helium atoms with use of a perturbation approach. The energies calculated in this way gave reasonable agreement with experimental values except at high densities.
Quantum Monte Carlo calculations of the energy of the relativistic homogeneous electron gas
Authors: Kenny S D , Rajagopal G , Needs r. J , Leung W -K , Godfrey M J , Williamson A J , Foulkes W M C
Published: November 2023

Several earlier QMC calculations0,b had been successful in predicting the energy of the homogeneous electron gas. This paper extended the earlier work to include relativistic effects with use of first-order perturbation theory for a wide range of densities. The required perturbation expression was derived in the 1950s. It consists of four terms known as the mass-velocity, Darwin, contact, and retardation terms. The small perturbative correction to the nonrelativistic energy is given by the sum of the expectation values of the individual terms The wavefunctions were obtained from variational and fixed-node diffusion calculations. The system was treated as a face-centered cubic simulation cell with periodic boundary conditions, containing up to 338 electrons for the variational and 178 electrons for the diffusion calculations. The trial functions 1/JT were composed of Slater determinants of plane waves, along with Jastrow factors. The required integrals based on ff were accumulated with Metropolis sampling, and those based on J1/}T were accumulated with importance-sampling diffusion simulations. Values based on lp2 were obtained by extrapolation The mass-velocity term was found to dominate at high densities and to agree with Hartree-Fock results. At lower densities the mass-velocity term from QMC was a factor of three larger than for HF, but the retardation term was dominant and much larger than predicted by HF. Overall, the results show clearly the importance of correlation, even for relativistic effects.
Characterization of a cage form of the water hexamer
Authors: Liu K , Brown M G , Carter C , Saykally R J , Gregory J K , Clary D C
Published: November 2023

This paper received attention in the news of 1996 for providing, at least in part, an answer to the question, “How many molecules of H20 are required to make a drop of water?” It reports an experimental investigation of the spectroscopy of the water hexamer and companion QMC calculations indicating the structure of the hexamer is cage-like. Since the trimer, tetramer, and pentamer are most likely cyclic in structure, the three-dimensional cage-like cluster of the hexamer would qualify it as the smallest possible drop. The structure of the hexamer was investigated with terahertz laser vibration-rotation spectroscopy, which established the cluster size and vibration-rotation splittings consistent with a cage structure held together with eight hydrogen bonds, in agreement with the QMC calculations. These were carried out in diffusion QMC to determine the vibrationally averaged ground-state rotational constants for the five lowest-energy potential energy minima for the hexamer. Since these involved 30 strongly coupled degrees of freedom, diffusion QMC was the method of choice. The results showed that although a prism structure has the lowest potential energy minimum, the cage structure is slightly more stable due to a lower zero-point energy. Vibrational averaging has a large effect on the moments of inertia (and therefore the rotational constants) for the cage, and it brings them into good agreement with the experimental values for these small “drops” of water.
Structure and energetics of small helium clusters: Quantum simulations using a recent perturbational pair potential
Authors: Lewerenz M
Published: November 2023

Small helium clusters Hen (n = 2, 3, ... , 10) in molecular beams produced by supersonic expansion of gaseous helium have been observed in several different experiments. Among these clusters the dimer, which has a single very slightly bound state, and the trimer, which could have many bound states, have especially unusual properties which QMC calculations can explain in detail. In this paper, diffusion QMC calculations were used to determine ground state properties for clusters (n = 2, ... , 10) using two accurate, but slightly different pairwise-additive potential energy expressions, a perturbation-based potential by Tang et al.a and an empirical potential (HFD-B) proposed by Aziz et al.b Trial functions for importance sampling were based on those determined earlier by Rick et alc. Expectation values and probability distributions were determined by the descendent-weighting technique to give results independent of the trial functions. As expected, a single bound state of the dimer was found, and its average internuclear distance was predicted to be 51.5 A. Energies for clusters with n = 2, 7 with the HFD-B potential were in good agreement with those of earlier calculations. Comparisons of the results for the two different potential energy expressions showed energies differing by a few percent. Tests of possible nonadditive interactions showed these to be far less important in their effects on predicted energies.
Quantum chemistry by random walk. H2
P, ,H3
+
D3h
1A’1 H2
3Σu
+, H4
1Σg, Be1S
Authors: Anderson J B
Published: November 2023

Applications of a fixed-node method for treating the node problem for excited states and for many-electron ground states were explored in this study. For cases of several particles in wells of one dimension and for the helium atom, the excited atom H2  P, the excited molecule H2  3Σu the many-electron system linear H4  1Σg  +, and the atom Be1S, it was shown that the antisymmetry requirements of the Pauli principle could be satisfied by the imposition of boundary conditions on the wavefunction. In the case of H2  3Σu  +, the six-dimensional electron configuration space is divided into two regions, one positive and one negative, separated by a single nodal surface. In a fixed-node diffusion calculation these nodes or boundaries act as sinks for walkers. The calculations are variational in nature, yielding (in the absence of any other errors) the exact energy for an exact specification of the nodal surface, and an energy higher than exact for an inexact nodal surface. For the several systems examined the energies so calculated were found to be within their statistical errors of the best estimates of the energies for these systems. The paper also reported several extensions to reduce the error associated with finite time-steps and with possible node crossing and recrossing.
Quantum Monte Carlo simulation of intermolecular excited vibrational states in the cage water hexamer
Authors: Severson M W , Buch V
Published: November 2023

The general problem discussed in this paper is that of predicting vibrational excitation in large hydrogen-bonded clusters, and, as the authors point out, “It should be realized that the problem is very difficult.” In this case, the cluster is the cage form of the water hexamer, simplified somewhat by use of rigid-body water molecules. The key to the solution is a new procedure for locating the nodes in the wavefunctions for the excited vibrational states. The method begins with an approximately correct candidate nodal surface, obtained as a zero for a normal coordinate in the harmonic approximation, followed by adjustments to meet the essential criteria for a truly correct nodal surface. These criteria are: (1) energies calculated for either side of a node must be the same, (2) the first derivative of the wavefunction must be continuous across the node, and (3) the wavefunction must be orthogonal to the groundstate wave function. It would seem hopeless to make adjustments to satisfy these criteria, but it was discovered they could be met successfully with nodes constructed from normal coordinates, singly in some cases and in linear combinations of up to three coordinates, subject to trial-and-error adjustments of the coefficients. In testing nodal surfaces, the required diffusion QMC calculations were carried out without importance sampling, so that fluxes into the nodal surface from opposite sides could be determined and mismatches eliminated. Despite the complexity of the systems, the energies for ten different low-lying vibrational states were determined. One of these was assigned to the vibration-rotation tunneling band observed in infrared absorption experimentsa.
Combined diffusion quantum Monte Carlo-vibrational self-consistent field (DQMC-VSCF) method for excited vibrational states of large polyatomic systems
Authors: Broude S , Jung J 0. , Gerber R B
Published: November 2023

Self-consistent field (SCF) solutions of the Schrodinger equation can provide approximate, often very accurate, nodes for DQMC calculations of electronic structure. As shown in this paper, SCF calculations can do the same in specifying nodes for vibrational states of polyatomic systems as large as Ar13. The vibrational selfconsistent field (VSCF) methoda is approximate but quite accurate in itself, and it has been used previously for prediction of low-level excitations of protein molecules, water clusters, and other complex systems. In the calculations described in this paper for Ar3 and Ar13, the SCF function was specified as a product of functions for each of the normal modes, and solutions were obtained with a sixth-order expansion reducing the required integrals to one dimension. For the argon clusters, the potential energy was specified as additive pair potentials. The DQMC calculations were performed using the VSCF nodes as absorbing boundaries without the use of the wave function for importance sampling. In the case of Ar3, the DQMC-VSCF results could be compared with nearly exact results from other methods, and they were found better than those of VSCF alone but with errors as high as 8%. In the case of Ar13, the excitation energies for DQMC-VSCF and VSCF were found quite close to each other for the 33 fundamental excited states, thus exhibiting small “correlation” effects. It was noted that the combined method is better than the VSCF method and that improving on VSCF as with a vibrational M ller Plesset theory might provide even better results.
Quantum Monte Carlo: Direct calculation of corrections to trial wave functions and their energies
Authors: Anderson J B
Published: November 2023

This paper describes a much improved method for reducing the statistical error in fixed-node diffusion QMC calculations. The calculations are designed to determine (1) the difference between the true (fixed-node) wavefunction and a trial wavefunction, and (2) the difference between the fixed-node energy and the expectation value of the energy for the trial wavefunction. When the trial function is a good approximation to the fixed-node result, these differences are small and the statistical errors are smaller still. The nodes of the trial function are not corrected. The general method was described previously.a The basic diffusion equation is altered to allow calculation of the difference between the true and trial functions expressed as g = (‘1/J-’l/Jo)’I/Jo, and this results in additional distributed feed terms for positive and negative walkers. In the new procedure these walkers are generated in Metropolis variational calculations and subsequently used as starting walkers in diffusion calculations with drift. These are continued for a time sufficient to approach equilibrium distributions. In effect, a cancellation of positive and negative walkers in equivalent distributions leaves the correction term g and allows the energy difference to be determined. The method is most useful when the trial function is reasonably accurate. In the case of the ground state of He a trial wave function with an energy expectation value of -2.903 724 376 18 hartrees led to a correction of -0.000 000 000 86(2) h to give an energy within 0.000 000 000 02 h of the (separately known) true value. In the case of H2O with a trial function expectation energy of - 75.560 h, the corrected energy value was -76.16(1) h compared to the independently determined fixed-node energy of -76.17(1) h. It remains to be seen whether this can be extended to determining very accurate energy differences in correlated sampling for similar molecules.
Constraint dynamics for quantum Monte Carlo calculations
Authors: Sarsa A , Schmidt K E , Moskowitz J W
Published: November 2023

In diffusion QMC calculations for clusters of water molecules, it has been found useful in some cases to eliminate the internal motions of the molecules by treating them as rigid rotors.a This has greatly simplified the prediction of intermolecular properties of clusters. A similar situation exists for the internal motions of molecules themselves, in which it may be useful to constrain some of the internal motions. Thus, for large molecules one might wish to fix some bond distances or angles which are already tightly, but not completely, restrained. In this paper a convenient method for applying constraints in diffusion QMC is described and demonstrated. The method is adapted from a procedureb used in molecular dynamics simulations in which similar constraints are used. The constraints are applied as distance constraints between specified pairs of atoms by imposing forces to counter the motions which would otherwise alter their separations. Specifying these forces requires a careful consideration of problems of multiplication of time-step errors, but these are successfully eliminated to yield a relatively simple algorithm incorporating importance sampling. At each step, walkers are first moved in the usual way of drift and diffusion without constraint. This is followed by an iterative procedure in which each pair of atoms is adjusted until all constraints are satisfied. The method is demonstrated to give satisfactory results for the rigid-body treatment of a water molecule and for the ground states of methanol with a floppy dihedral-angle rotation involving one of the H-C-0-H strings of atoms. The method is clearly applicable to many different types of molecules, large and small.
Optimal orbitals from energy fluctuations in correlated wave functions
Authors: Filippi C , Fahy S
Published: November 2023

In conventional Hartree-Fock calculations for molecular systems the optimization of a trial wavefunction by energy minimization with respect to the orbitals of a determinantal wavefunction is relatively easy. When Jastrow factors are included with a determinant as for typical QMC trial functions the task becomes more difficult. Optimization by minimizing the variance in local energies has been one alternative. This paper describes another approach which is, in fact, an analog of the Hartree-Fock method and provides an exact solution to the problem of optimizing the orbitals when a Jastrow function is present.
Linear-scaling quantum Monte Carlo calculations
Authors: Williamson A J , Hood R Q , Grossman J C
Published: November 2023

This study demonstrates linear scaling with system size in fixednode diffusion QMC calculations with pseudopotentials for carbon fullerenes and for hydrogenated silicon clusters with nearly 1000 valence electrons. The range covered is C20 to C18o and SiH4 to Si211H140, The calculations were carried out with standard pseudopotentials (leaving four electrons per C or Si atom) and Slater Jastrow trial functions with orbitals from LDA density functional calculations. The key elements leading to linear scaling were constructions of sparse determinants and simplification of orbital functions. The LDA orbitals were obtained with a plane-wave basis set which was transformed to yield Wannier functions with maximum localization. These functions were then fitted to cubic spline expressions with a limited number of grid points and cut-off distances matching those of the plane-wave orbitals. The computational effort for evaluating trial functions of this type was clearly seen to behave in a linear fashion. Exploration of the effects of variation of cut-off radius revealed that energies, sensitive only to node structures, increased rapidly for cut-offs less than 7 bohr but were unaffected for cutoffs greater than 7 bohr. The accuracies of the calculations were checked in other ways. These included comparison calculations for SiH3 and Si5H12 using Gaussian, original plane-wave, and Wannier function basis sets which gave close agreement for energies.
Ab initio computation of molecular singlet-triplet energy differences using auxiliary field Monte Carlo
Authors: Baer R
Published: November 2023

About 15 years before this paper was published, all-electron fixed-node diffusion QMC calculations by Reynolds, Dupuis, and Lester0 had successfully duplicated the 9 kcal/mol seen in direct experimental measurements ofthe classic problem of singlet-triplet splitting for the methylene molecule CH2. This second QMC paper on the subject was designed less to investigate the CH2 system and more to investigate the method, in this case the shiftedcontour auxiliary field Monte Carlo (SC-AFMC) method,b which overcomes some of the stability problems of the original AFMC method.c The paper describes the application of SC-AFMC to the simplest singlet-triplet problem of linear H-He-H as well as that of methylene, this time with pseudopotentials. The applications were far from straightforward and required investigation of the various approximations needed to obtain the energy differences. Nonlocal B-LYP pseudopotentials were used, along with combinations of restricted and unrestricted HartreeFock determinantal wavefunctions. Correlated sampling was successful in reducing statistical error in energy differences but convergence was slow. Near degeneracies led to problems for H-He-H. Results were sensitive to cell size and grid spacing and especially sensitive to the type of pseudopotential used. Nevertheless, suitable choices produced singlet-triplet splittings in agreement with accepted values when pseudopotentials of the generalized gradient type were used. However, this agreement might have been fortuitous. Further studies of alternative approaches were indicated.
Quantum Monte Carlo for electronic excitations of free-base porphyrin
Authors: Aspuru-guzik A , Akramine O El , Grossman J C , Lester W A
Published: November 2023

With 216 Electrons, The Free-Base Porphyrin C20n4h14 Presents A Challenge To Ab Initio Electronic structure calculations. Since it is at the center of the mechanism of photosynthesis, as well as several other biological processes, there has been much interest in this molecule and its electronic spectrum, but despite a number of earlier studies, important questions remain open. This paper provides the most accurate calculations to date of energy levels of states involved in several allowed and nonallowed transitions seen in the spectrum.
Quantum Monte Carlo study of helium clusters doped with nitrous oxide: Quantum solvation and rotational dynamics
Authors: Moroni S , Blinov N , Roy P -N
Published: November 2023

The infrared spectra of small molecules embedded in helium nanodroplets display sharp rotational lines suggesting a decoupling of the molecular rotation from the motion of the helium atoms. The spectra observed for several different molecules have been explained in earlier reptation QMC calculationsa giving moments of inertia and thus effective rotational constants B as a function of the number N of helium atoms in a cluster. This paper reports such calculations for the nitrous oxide molecule NNO which has been observed to have a turnaround in the evolution of B with N. The value of B first decreases to a minimum at N = 6 to 7, then increases slightly to a maximum at N = 11, followed by another minimum and a slow increase. The reptation QMC calculations provided ground-state energies, density profiles, and correlation functions for angular momenta. The results gave excellent agreement with the experimentally determined values of B. The variations of B with N were reproduced quantitively, and the behavior was explained completely with the aid of the helium density profiles showing the accumulation of density at various positions surrounding the NNO molecule as N is increased. Additional calculations using finite-temperature path integral Monte Carlo (PIMC) gave excellent agreement with the energetics and structural properties given by the ground-state calculations. The rotational constants, obtained with Boltzmann statistics in the PIMC calculations, showed no turnaround and no oscillations, thereby indicating that the experimentally observed behavior is the result of exchange effects not included in the PIMC calculations.
Interaction effects in the mesoscopic regime: A quantum :rvionte Carlo study of irregular quantum dots
Authors: Ghosal A , Unirigar C J , Jiang H , Ullmo D , Baranger H U
Published: November 2023

The paper reported the extension of earlier QMC calculations for small symmetric quantum dots to large irregular dots, i.e., from the microscopic to the mesoscopic regime. For a large number N of electrons in a sufficiently irregular confining potential the motion of the electrons may be expected to- be chaotic, and mesoscopic fluctuations may result from electron interaction and interference effects. Studies using density functional theory (DFT) and the random phase approximation (RPA) have met with only limited success in treating these systems. This fixed-node diffusion QMC study was much more successful. The systems treated were two-dimensional quantum dots containing 10-30 electrons confined by a potential energy of the type culations were carried out for dots formed by six different sets of constants. Each of these was investigated with each number of electrons N and the total spin S given by S = 0, 1, and 2 for even N and S = 1/2, 3/2, and 5/2 for odd N in order to determine the ground-state energies and spins. Multi-determinant trial wavefunctions with simple Jastrow functions were assembled from Kohn-Sham orbitals, and coefficients for the determinants were optimized. Both VQMC and fixed-node DQMC calculations produced energies with statistical errors small compared to singleelectron energy-level spacings.
Quantum chemistry by random walk: Importance sampling for H+
3
Authors: Mentch F , Anderson J B
Published: November 2023

In this paper the combination of importance sampling° with diffusion QMC was shown to be remarkably effective in improving the accuracy of electonic structure calculations for molecular systems. The key to success was the use of a simple, fairly accurate trial wavefunction ΨT for which the function itself, its derivatives and the local energy Eioc may be easily computed. This, used in combination with the diffusion-with-drift expression for walkers corresponding to the product f = ΨΨT according to allows the energy to be evaluated as an average for all walkers of every step after reaching a fluctuating steady state. Since there is a time-step error associated with the simulation of diffusion, drift, and multiplication, varying the time-step and an extrapolation to zero step size are required. The result for H3  + in its triangular configuration of side length 1.6500 bohr was -1.3439 ± 0.0002 hartrees, a value well below the lowest variational energy levels at the time and having an error bar straddling the best modern value of -1.343 835 hartrees. The use of importance sampling along with higher computer speeds resulted in an uncertainty about a factor of 60 lower than that of the previous calculation.b
Ground state of excitonic molecules by the Green’s-function Monte Carlo method
Authors: Lee M A , Vashishta P , Kalia R K
Published: November 2023

The biexciton bound complex of two electrons and two holes is equivalent to the hydrogen molecule with different masses0 and it can be treated in the same way as the hydrogen molecule by QMC methods. For equal electron and proton masses the complex is equivalent to a positronium molecule. The authors applied simple diffusion QMC with importance sampling to calculate ground-state energies of biexcitons for a range of electron-to-hole mass ratios a= me/mh. The full four-body problem was treated with positive and negative charges of unity interacting with Coulomb potentials. Since the ground state is nodeless, the problem of nodes does not occur, and the importance sampling function was taken as a simple product of electron-electron and hole-hole Jastrow functions with exponential terms. The results gave energies considerably lower than variational estimates and better agreement with experimental measurements for several crystalline materials. The limiting behavior as a -+ 0 was found in excellent agreement with the exact result.
Quantum Monte Carlo For Molecules: Green’s Function And Nodal Release
Authors: Ceperley D M , Alder B J
Published: November 2023

This paper describes the application of the nodal release or “transient estimate” method,0 successfully used earlier for treating the electron gas problem, to several small molecules: H3, LiH, Li2, and H20. This method can, in principle, provide a means for overcoming the node location problem for such systems, but it does require difficult extrapolations for most systems of interest. The basic calculations are carried out as for simple Green’s function QMC without nodes as boundaries. A trial wavefunction ‘I/Jr is used to locate approximate node positions, and a guide function ‘1/Ja, everywhere positive and approximately equal to l’I/Jrl and nonzero but small at the nodes, is used for importance sampling. A released-node calculation is begun with walkers at steady state in a fixed-node calculation. These are then released and permitted to cross fixed-node surfaces. After a number of steps, the distribution of walkers includes walkers inside and outside the starting region. The difference (inside less outside) approaches the fermion ground state distribution, but with increasing number of steps the difference is lost in the noise as the inside and outside approach boson ground state distributions. Nevertheless, useful results were obtained for the systems examined, with energies correct within their statistical uncertainties. For the largest system, H20, the calculated energy was - 76.43 ± 0.02 hartrees, which may be compared with a value of - 76.438 hartrees based on experimental measurements.
Quantum Monte Carlo calculation of the singlet-triplet splitting in methylene
Authors: Reynolds P J , Dupuis M , Lester W A
Published: November 2023

The methylene molecule CH2 is one of those simple species with a long history in the development of quantum chemistry. First observed experimentally in 1959 by Herzberg and Shoosmitha it was the object of an especially important near-SCF study by Foster and Boysb predicting (correctly) the ground state to be the triplet 3 B1 with a bond angle of about 129°. Shortly thereafter, Herzberg concluded the molecule in the 3B1 state was linear, but eventually experiments by others, as well as higher level calculations, showed Foster and Boys to be correct.c The singlet-triplet splitting (singlet energy minus triplet energy), initially thought to be about 2 kcal/mole from experiment and about 30 kcal/mol from theory, is now known to be 9-11 kcal/mole. The fixed-node calculations of Reynolds et al. were carried out for optimum geometries indicated by analytic variational calculations, and they gave energies more than 15 kcal/mole below the lowest energy variational calculations available at the time. The trial functions for node location and importance sampling were single-determinant double-zeta functions with optimized Jastrow terms. The QMC result for singlet-triplet splitting, obtained by subtracting energies, is 9.4 ± 2.2 kcal/mole. With allowance for the zero-point energy and relativistic effects the splitting becomes 8.9 ± 2.2 kcal/mole. These are in very good agreement with very recent theoretical and experimental values. This substantial cancellation of node-location error for these two similar systems occurs in the same way as expected for the somewhat larger errors in analytic variational calculations for similar systems.
Quantum Monte Carlo calculations on Be and LiH
Authors: Harrison R J , Handy N C
Published: November 2023

In their first paper in the quantum Monte Carlo area, Harrison and Handy investigated the use of multi-configuration trial wavefunctions and various forms of Jastrow functions in fixednode QMC. Calculations were repeated for the small molecules He, Be2+, Be, and LiH. For the nodeless ground states of He and Be2+, they used two- to six-term Hylleraas functions and found significant gains in efficiency for the more accurate trial functions despite the additional terms. They reproduced known energies for these species within statistical error. For Be, it was found that the fixed-node results for a five-term multi-configuration SCF function were far superior to those for a single-term SCF trial function. For LiH, the use of one-, two-, and four-term MC-SCF functions was found to produce little improvement in the variance in local energies or the values of the energies obtained. Recovery of correlation energy was about 99% in each case. This paper introduced the use of higher order terms in a Jastrow function similar to those used by Handy and Boysa in their transcorrelated method. Also considered was the use of an additional electron-nucleus term in the Jastrow function. These attempts at improving the Jastrow factor were not very successful, but planted the seeds for later successes by Schmidt and Moskowitz.
The differential Green’s function Monte Carlo method. The dipole moment of LiH
Authors: Wells B H
Published: November 2023

Use of correlated random walks in fixed-node diffusion QMC offers the opportunity to calculate differences in energies of similar systems. In this paper wells reports such calculations for LiH in the presence of differing field strengths to determine its dipole moment. Since the local energies for identical electron configuration and importance sampling functions differ only in the potential energy due to the applied electrical field, the local energies are strongly correlated, and the statistical error in the difference in energies can be much lower than that for either of the individual energies. The calculations were started from a ‘l/Jf distribution of walkers with identical moves for each system. Since the multiplication terms were slightly different, variable weights were used for each walker. Although these eventually diverge for the correlated pairs, there is a sufficient period of strong correlation to allow efficient reduction in variance in energy differences. Thus, for an applied field of 0.0001 a. u. versus no field, the energies obtained were -8.0593 ± 0.0030 hartrees and -8.0595 ± 0.0030 hartrees, but the computed energy difference was 0.000227 ± 0.000003 hartrees, corresponding to a dipole moment of 2.27 ± 0.03 au, in good agreement with the value derived from experiments. Since there was no provision for a change in the node structure with applied field, it appears the dipole moment is insensitive to the change if any occurs.
A Green’s function used in diffusion Monte Carlo
Authors: Rothstein S M , Vrbik J
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.
Alpha particle structure
Authors: Carlson J
Published: November 2023

The time-independent Schrödinger equation for the case of nucleons in light nuclei is more difficult to treat than that of electrons in a light atom because of more complicated interaction terms. Nevertheless, for three-nucleon systems such as the triton, binding energies and structural details have been successfully obtained for realistic nucleon-nucleon interaction models by several different methods, especially by analytic variational methods. For four-nucleon systems such as the alpha particle, realistic models of the interactions are also available, but the added nucleon makes the treatments more difficult and the properties predicted less certain. This paper reports the use of variational QMC calculations to obtain accurate solutions for the alpha particle with several nucleon-nucleon interaction models. For one model, both variational and diffusion QMC calculations provided the opportunity for direct comparisons to assess the accuracy of variational QMC results. The interaction models, chosen to provide a test of the sensitivity of results, included those known as Argonne V14, Nijmegen, Bonn, Reid V8, and Urbana Model 7. Trial wavefunctions previously developed for such problems were constructed of Slater determinants, functions of internucleon distances and spins, as well as correlation terms. Optimizations of parameters were carried out by minimizing the expectation values of the energies, making use of correlated sampling.
Quantum Monte Carlo calculations of three-body corrections in the interaction of three helium atoms
Authors: Mohan V , Anderson J B
Published: November 2023

For inter nuclear distances of about 4 bohr and greater, the potential energy of three helium atoms is very close to pairwise additive. That is, the total interaction energy is very close to the sum of the interaction energies of the three pairs, each in the absence of a third atom. The energy may then be conveniently expressed as where E1 is an atom energy, LlE2 a pairwise interaction energy, and LlEa is a three-body correction term. For pairwise additivity, LlE3 is zero. The correction LlEa is extremely small and is certainly less than 30 microhartrees (about 10 K) for three He atoms in an equilateral triangle configuration of side length 5.6 bohr with a total energy of about 8.7 hartrees. A third-order perturbation analysis by Axilrod and Teller0 and Mutob (ATM) has provided a simple and convenient expression for the three-body correction. Clearly, it must fail at small internuclear distances, but it often has been assumed to be acceptable for distances near the equilibrium dimer distance of 5.6 bohr. The fixed-node diffusion calculations reported were carried out for equilateral triangles with side lengths of 2.5 to 6.5 bohr. The trial function was based on Hylleraas expressions for the atoms in a 36-term antisymmetric sum. The results confirmed that the three-body correction term was less than 10 K for distances greater than about 5.6 bohr, but the uncertainties were about 10 K.
Correlated Monte Carlo wave functions for the atoms He through Ne
Authors: Schmidt K E , Moskowitz J W
Published: November 2023

This paper introduced an expanded form of correlation function to replace the simple Jastrow functions used in trial wavefunctions for both VQMC and DQMC calculations. The result was a significant decrease in the variance in local energies and an increase in recovery of correlation energies for the atoms examined (and for many other atoms and molecules, by subsequent workers using the new functions). The form for the trial functions employed is a single determinant multiplied by a correlation function F proposed by Boys and Handya where the sums are over nuclei I, electrons i and j, and terms for both electron-electron and electron-nucleus interactions. The electron-nucleus terms have the important effect of compensating for electron density changes due to the electron-electron terms, and they give a large improvement over the authors’ previous results. Functions with 7, 9, and 17 terms were optimized for each of the atoms He through Ne and used in variational QMC calculations. Recovery of correlation energy was approximately 100% for He, 97% for Li, and 18% to 85% for Be through Ne. In the case of Ne, the single-determinant with the correlation function recovered 85%, a value higher than the 75% obtained in configuration interaction calculations (with 107 determinants) at the time.
Fixed-node Monte Carlo study of the two-dimensional Hubbard model
Authors: AN G , Leeuwen J M J Van
Published: November 2023

This study extended the fixed-node QMC method to a lattice model for hopping fermions using Green’s function techniques. In general, the modifications of the procedures for continuous systems to adapt them to a lattice system were minor, and there appeared to be no surprises. Applications were made to the two-dimensional Hubbard model with different on-site interactions, lattice dimensions, boundary conditions, node specifications, and trial wavefunctions for importance sampling. Results were compared with those of Hartree-Fock and variational QMC calculations. For the HF calculations with zero interaction, the exact solutions are known, and the Green’s function QMC results agreed with those within statistical error. Trial wavefunctions specifying the nodes were obtained directly as those of HF determinantal solutions for zero interaction, or as variationally optimized determinantal functions with correlation terms. The procedure for fixed nodes was to eliminate walkers crossing the boundaries of a nodal region, a procedure not quite correct, but probably a reasonable approximation. The results in all cases were consistent with the HF and variational QMC calculations, giving lower energies than either for nonzero interaction energies. The existence of a domain-wall phase suggested by the HF and variational QMC results was confirmed by the fixed-node Green’s function results. It was noted that the fixed-node method would be applicable to systems other than the Hubbard model.
Monte Carlo algorithms for expectation values of coordinate operators
Authors: Barnett R N , Reynolds P J , Lester W A
Published: November 2023

In diffusion QMC with importance sampling, walkers provide samples of configurations with probabilities proportional to the product of the true wave function ‘¢ and a trial wavefunction t/Jr. The expectation value of the energy (very fortunately) may be calculated from the local energies of these samples as E = (‘l/JH’l/Jr)/(‘l/Jt/Jr) = (‘l/JE10ct/Jr)/(‘¢’¢r). But for an operator A which does not commute with the Hamiltonian H, the mixed expectation value (‘¢At/Jr) is only an approximation to the “pure” expectation value (‘ljJA’¢). Thus, for most quantities of interest other than the energy, an extension to standard diffusion QMC methods is required. This paper describes two approaches to the problem. Both are based on an earlier development by Liu, Kalos, and Chester0 involving the tracking of descendents of walkers to obtain pure expectation values. The first uses simple DQMC and the second uses VQMC with DQMC “side walkers.” These methods were tested in their application to the model systems H and H2, and both were found effective in yielding accuracies and precisions correct within 0.5% for the quantities (r) for H and (r2) and (z2) for H and H2. The large differences in electron and proton masses led to extensive computation requirements due to the slow equilibration and serial correlation induced by the heavier protons. The calculations were executed on one of the first massively parallel computers, a Thinking Machines CM-2 with 65,536 processors. Frequent communication among the processors was required to balance the number of walkers treated in each.
An accurate quantum Monte Carlo calculation of the barrier height for the reaction H + H2 H2 + H
Authors: Diedrich D L , Anderson J B
Published: November 2023

A lit of fundamental reactions usually begins with the reaction H + H2 -+ H2 + H, the simplest of neutral chemical reactions and the subject of more than 60 years of theoretical studies at the time this paper was published. The potential energy surface and especially the barrier to reaction are essential to any treatment. Earlier theoretical work had led to predictions of a barrier height of9.5 to 9.8 kcal/mol. The QMC calculations reported in this paper reduced the uncertainty by a factor of about ten and specified the barrier height as 9.61 ± 0.01 kcal/mol. The calculations were “exact” QMC calculations based on the Green’s function method with partial cancellations of positive and negative walkers to produce a stable wavefunction with a correct node location. A trial wavefunction was used for importance sampling to aid in determining energies without affecting the node location. Time-step error was avoided with use of the Green’s function approach for the moves of walkers. The complete procedure was exact in that, except for the easily estimated statistical or sampling error, it required no mathematical or physical assumptions beyond those of the Schrodinger equation. No interpolation or extrapolation was required.
Energies, densities, and pair correlation functions of jellium spheres by the variational Monte Carlo method
Authors: Ballone P , Umrigar C J , Delaly p.
Published: November 2023

In the jellium model for clusters of atoms the charge of the nuclei and of appropriate inner-shell electrons is replaced by a net positive uniform charge distribution. This model has been found especially useful for spherical clusters of alkali metal atoms with explicit treatment of electronic structure limited to valence electrons. In this paper the authors have reported studies of clusters of up to 20 atoms, equivalent to spheres with a distributed charge of +20 along with 20 free electrons. The trial function for variational QMC was constructed in the usual way with single determinants of orbitals generated in density functional (LSDA) calculations multiplied by optimized Jastrow functions. The quality of these functions was tested in fixed-node diffusion calculations indicating that 90% of the fixed-node correlation energy was recovered in the variational calculations. Results for a series of neutral and ionized clusters were compared with those of density functional calculations in terms of total, exchange, and correlation energies as well as pair correlation functions and electron densities. In general, total and spin densities were found in excellent agreement for the two types of calculations. Energies were found in fairly good agreement, but exchange and correlation energies differed by as much as 6% and 30%, respectively. The paper clearly shows the several advantages of QMC for a system different from that of the typical atomic or molecular problem.
Exact quantum Monte Carlo calculation of the H-He interaction potential
Authors: Bhattacharya A , Anderson J B
Published: November 2023

This is the third in a series of papers describing “exact” QMC calculations for few-electron systems with nodes in their wavefunctions. The method used is the exact cancellation Green’s function scheme0 which develops the node structure and is “exact” in that it requires no approximations, no interpolations, and no extrapolations. The treatment used is essentially the same as that for earlier calculations for He 3  P, H2  31:;t, H-H-H, and He-He. The trial function for importance sampling was based on a Hylleraas function for the He atom and a Slater function for the H atom. Energies were calculated for internuclear separations within the range of 5.0 to 15.0 bohr. The minimum was found at -3.403 746±0.000 002 hartrees for a separation of 7.0 bohr, corresponding to a well depth of 6.8 ± 0.5 K. Total energies were approximately 12000 K below those of the lowest-energy variational calculations available. The interaction energy was at least 3 K more accurate than prior estimates by any other method. The calculated potential energy curve was consistent with low-energy scattering cross-section measurements for H with He.