# Abstracts

## Worm Algorithm for Continuous-Space PIMC

### Massimo Boninsegni (University of Alberta, Canada)

The Worm Algorithm (WA) is a novel computational paradigm for large-scale simulations of quantum many-particle systems based on Path Integral Monte Carlo (PIMC). By using an extended configuration space, which includes an open world line (”worm”), the WA overcomes major limitations of previous PIMC technology, allowing the computation of the superfluid density for systems of as many as several thousand particles. Moreover, the WA allows one to perform simulations in the grand canonical ensemble, and offers access to off-diagonal correlations (such as the one-particle Matsubara Green function) not calculable with other Quantum Monte Carlo techniques. In this talk, the basic elements of the WA will be presented, together with illustrative applications to condensed (liquid and solid) Helium.

## QMC Calculations of Cold Fermi Atoms at Resonance

### Joe Carlson (Los Alamos National Laboratory)

I will present QMC results for studies of cold (T=0) Fermi Atoms at or near infinite scattering length. This is a unique and intriguing problem because the only energy scale in the problem is the Fermi energy of the system. The equation of state at different densities is a constant times the Fermi energy, and the pairing gap is a significant fraction of the Fermi energy, in contrast to typical superfluids or superconductors.

I'll briefly review the problem and results at zero polarization, and then discuss more recent results at various polarizations. In this regime the difference in Fermi energies between the two populations also plays an important role. I will also discuss experimental results obtained recently for polarized systems.

## Lattice regularized diffusion Monte Carlo

### Michele Casula (University of Illinois, Urbana-Champaign, USA)

We present an efficient lattice regularization scheme [1] for quantum Monte Carlo calculations of realistic electronic systems. The Laplacian is discretized with two incommensurate mesh sizes, *a* and *a'*, where *a'/a* is a fixed irrational number, and the regularized Hamiltonian goes to the continuous limit for *a* -> 0. The use of the double mesh improves significantly the convergence to the *a* -> 0 limit, and allows to take into account the different length scales in the system, with an efficiency gain which becomes more and more relevant as the atomic number increases. One of the main advantages of this framework is the possibility to include non-local potentials in a consistent variational scheme, substantially improving both the accuracy and the computational stability upon previous non-variational diffusion Monte Carlo approaches. We present some applications of our method to cases involving transition metal elements, and the one dimensional homogeneous electron gas.

[1] M. Casula, C. Filippi and S. Sorella, "Diffusion Monte Carlo with lattice regularization," Phys. Rev. Lett. **95**, 100201 (2005).

## Coupled Electron Ion Monte Carlo Calculations for Dense Hydrogen

### David M. Ceperley (University of Illinois Urbana-Champaign, USA)

Quantum Monte Carlo (QMC) methods are the most accurate and general methods for computing total electronic energies. However, in general, they have been limited to temperatures greater than 5000K, or to zero temperature. In recent years, we and others have been working on methods that utilize the Born Oppenheimer approximation to allow simulations coupling the correlated quantum systems and a system of ions. Such an algorithm could allow the sort of progress which occurred when Car and Parrinello coupled local density functional theory with molecular dynamics of ions. Using quantum Monte Carlo [3], one estimates the Born-Oppenheimer energy change which is then used in a Monte Carlo simulation of the ionic degrees of freedom. The usual acceptance probability is modified to eliminate the bias caused by noise in this energy difference, allowing more noisy estimates of the energy difference and reducing the sampling time of the electronic degrees of freedom. We use both trial wave functions that depend analytically on the ionic coordinates, as well as those from a band structure calculation of the actual ionic coordinates. Reptation MC is used for accurate calculation of the BO energy differences [1]. The quantum effects of the ionic degrees of freedom and the boundary conditions on the phase of the wavefunction can be integrated over with a modest increase in computational effort.

We have performed simulations of dense hydrogen down to temperatures of 300K. Our results show features of the phase diagram qualitatively different than computed using DFT, for example in the melting of the atomic solid [2] and in the atomic-molecular transition in the liquid [4].

- C. Pierleoni and D. M. Ceperley, ChemPhysChem, December 2004, physics/0501013.
- C. Pierleoni, D. M. Ceperley and M. Holzmann, Phys Rev. Letts.
**93**, 146402: 1-4 (2004). - D. Ceperley, M. Dewing and C. Pierleoni, in Bridging Time Scales: Molecular Simulations for the Next Decade, eds. P. Nielaba, M. Marechal.
- K. Delaney, C. Pierleoni and D. M. Ceperley, cond-mat/0603750.

## Random phase approximation and Finite size errors in Many Body Simulation

### Simone Chiesa (University of Illinois)

In order to simulate bulk systems, many-body methods are often performed using a periodized interaction within a supercell approach. Even within this scheme the error on the energy is large, exceeding other errors typical of stochastic simulations. We show how, in agreement with the random phase approximation, one can compute the necessary correction on both the potential and kinetic energies without resorting to a costly scaling analysis.

## Optimization of QMC trial wave functions by variance minimization

### Neil Drummond (Cambridge University)

We describe an accelerated scheme for optimizing parameters that are linear in the exponent of a Jastrow factor by minimizing the unreweighted variance of the local energy. We search for multiple minima of the variance in the parameter space, and compare the wave functions obtained using reweighted and unreweighted variance-minimization algorithms. Furthermore, we discuss the modifications to the algorithms that are necessary when parameters that affect the nodal surface of the wave function are optimized by variance minimization.

## Electronic Quantum Monte Carlo Calculations of Energies and Atomic Forces for Diatomic and Polyatomic Molecules

### Myung Won Lee (University of Pennsylvania)

We calculated the energies and atomic forces of first-row monohydrides, carbon monoxide, and small organic polyatomic molecules using quantum Monte Carlo (QMC) method. Accurate forces were obtained with the method of Casalegno, Mella, and Rappe [1], combining the Hellmann-Feynman theorem forces calculated by the Assaraf-Caffarel method [2] with a many-body Pulay correction. Improved algorithms for the minimization of the variational integral were useful in the force calculations.

[1] M. Casalegno, M. Mella and A. M. Rappe, J. Chem. Phys. 118, 7193 (2003)

[2] R. Assaraf and M. Caffafarel, J. Chem. Phys. 113, 4028 (2000)

## Status of Fermion Monte Carlo

### Malvin Kalos (Lawrence Livermore National Laboratory)

I will describe recent progress in Fermion Monte Carlo and share speculation about how it can be made more robust and more efficient.

## Fermion nodes and pfaffian pairing wave functions: beyond the fixed-node approximation

### Lubos Mitas* (North Carolina State University)

We study nodes of fermionic ground state wave functions. We explicitly prove that for d>1 spin-polarized, noninteracting fermions nondegenerate ground states have two nodal cells for arbitrary system size. The result cover a number of paradigmatic models such as harmonic fermions, fermions on a sphere surface, in a periodic box, Hartree-Fock atomic states and other noninteracting/mean-field models. Spin-unpolarized noninteracting states

have multiple nodal cells, however, interactions and many-body correlations, in general, smooth out the multiple cells to the minimal number of two. Under very general conditions, this is proved for interacting and and periodic interacting fermions systems of arbitrary size using the Bardeen-Cooper-Schrieffer variational wave function. In order to improve the nodes of QMC trial wave functions we propose pfaffian functional form which includes both singlet and triplet pairing function in a single mathematical and easy to evaluate object. Using a set of first row atoms and molecules we find that this wave functions provide very consistent and systematic behaviour in recovering the correlation energies on the level of 95% . In order to get beyond this limit we explore the possibilities of multi-pfaffian pairing wave functions. We show that small number of pfaffians recovers another large fraction of the missing correlation energy up to 99% comparable to the large-scale configuration interaction wave functions. We also find that pfaffians lead to substantial improvements in fermion nodes when compared to Hartree-Fock wavefunctions.

* in collaboration with L.K. Wagner, M. Bajdich, G. Drobny, and K.E Schmidt (Arizona State).

## Space-warp transformation on nodal distance.

### Savario Moroni (Trieste, Italy)

We build on previous work on static susceptibilities of the electron gas, computed as second derivatives of the fixed-node energy with respect to an external field using a ground-state path integral method. A key ingredient was the replacement of the drift-diffusion importance sampled Green's function with an expression which is better behaved near the nodes. Here we propose a space-warp transformation which keeps the (smallest) nodal distances unchanged upon variation of the external field. Test calculations for noninteracting fermions show a significant reduction of the statistical noise.

## Applications of diffusion quantum Monte Carlo

### Richard Needs (TCM Group, Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, UK)

Fixed-node diffusion quantum Monte Carlo (DMC) is one of the most accurate methods known for calculating the energies of many-particle systems. I will present some recent applications of the DMC method to water molecules and to diamond, describing technical issues such as the forms of the trial wave function used and the cancellation of errors in energy differences.

## Optimized guiding functions for excited state calculations

### Peter Nightingale (University of Rhode Island)

I present a review of instabilities that haunt quantum Monte Carlo calculations of excited state energies of van der Waals clusters. I discuss in detail a solution of one particular problem that arises in this context, namely the construction of a robust guiding function that efficiently samples several different excited states simultaneously.

## One Convenient Basis to Transform a DFT Guy into a QMC One

### Fernando Reboredo (Oak Ridge National Laboratory)

The traditional niche of Density Function Theory (DFT) and its approximations has been the study of larger systems [as compared to the ones studied by Configuration Interaction (CI) or Quantum Monte Carlo QMC)]. Speed has been the fundamental reason of the historical predominance of DFT based methods in the large scale, despite off the uncontrolled character of some approximations. The speed comparizon as is shifting towars QMC with the emergence of parallel computers and linear scaling methods for Diffusion Quantum Monte Carlo (DMC). These developments can transform DMC into a technique as polular as DFT for the solution of Solid State and Materials Science problems and thus can attract new commers to the field. In this talk I will explain a linear scaling method based on non-orthogonal orbitals that significantly reduces the computational cost of large (1000 e-) in DMC. Applications to the homogeneous electron gas and simple metals such as Al will be addressed.

## Fixed Phase Path Integrals for Fermions, Magnetic Fields, and Spinors

### John Shumway (Arizona State University)

The fixed-node and fixed-phase approximations have contibuted greatly to the practical development of ground-state calculations. For finite temperature a wavefunction-based constraint is inappropriate, since the fixed node approximation captures the ground state but not the excited states. We have made a new formulation of the fixed node approximation for density matrices. At T=0, the method includes the usual ground-state formalism, but is more general and allows the sampling of degenerate ground states. For finite temperature, we maximize entropy (minimize free energy) to derive a path integral formulation. We'll compare and contrast our method with Ceperley's restricted path integral. Finally, we'll show some new work we are doing to represent spinors as a fixed-phase path integral.

Work supported by NSF Grant No. 0239189 and done in collaboration with Daejin Shin.

## Effective Hamiltonian and lattice regularization for realistic systems

### Sandro Sorella (Sissa, Italy)

We apply a recent technique [1] - the LRDMC- to carbon based aromatic compoundsand show the efficiency and the accuracy of the method forquite large electronic systems.The quantum chemical accuracy is obtained by using an RVB wave function obtained by applying a correlated Jastrow factor to a BCS type wave function, named AGP in quantum chemistry. We show that it is possible to obtain a remarkable accuracy by taking carefully into account the convergence in the gaussian basis set used, and also the size consistency of this wave function. Within this framework the application of the method to molecules containing several atoms, such as the benzene dimer relevant for biophysical applications, is already possible within a statistical accuracy of 0.01eV: almost all the correlation energy is obtained by sampling a single determinant geminal wave function with the LRDMC projection scheme.

[1] M. Casula, C. Filippi and S. Sorella ''Diffusion Monte Carlo with lattice regularization'', Phys. Rev. Lett.** 95**, 100201 (2005).

## Extracting frequency-dependent response functions from QMC—without the sweat!

### Nandini Trivedi (Ohio State University)

Using QMC methods for Hubbard type models for Bose and Fermisystems, we calculate correlation functions in imaginary time for the Green function, spin-spin and current-current correlation functions. I will discuss approximate methods to extract the low energy density of states, NMR relaxation rate and conductivity without using analytic continuation methods. I will also discuss exact methods to get information about quasiparticle weight, quasiparticle velocity and Drude weight from VMC calculations of interacting fermions.

## Optimization of nodes of many-body wave functions

### C. J. Umrigar, Julien Toulouse and Claudia Filippi (Theory Center and LASSP, Cornell University)

A great deal of effort has been invested by various researchers in trying to find ways to go beyond the fixed-node approximation in projector Monte Carlo methods applied to continuum fermionic systems. Particularly notable is the work of Kalos, but also of Anderson, attempting to devise exact fermionic algorithms, and, the work of Ceperley on the release-node algorithm. However, at present none of these methods are even remotely comparable in efficiency to the commonly used fixed-node approximation. An alternative approach is to devise sufficiently accurate trial wave functions that the fixed-node error is small enough to be ignored. There are two obstacles to success in this approach. First, one is always limited by the chosen form of the trial wave function. Second, there is at present no efficient method for optimizing many-body nodes. Here we discuss the second issue.

The variance minimization method has become the standard method for optimizing many-body wave functions. It is highly effective for Jastrow parameters, but not as effective for parameters in the determinants because for the determinantal parameters reductions in the variance and the energy are not strongly correlated. In recent years many methods have been proposed for energy optimizing quantum Monte Carlo wave functions. Of these, the three highly efficient methods are:

1) The generalized eigenvalue method of Nightingale and Melik-Alaverdian, which was proposed by them for linear parameters only but extended by us to nonlinear parameters.

2) The effective fluctuation potential (EFP) method of Fahy, Filippi and coworkers, and the recent perturbative EFP of Schautz, Scemama and Filippi. We show that the latter can be more simply derived as first-order perturbation theory in a nonorthogonal basis.

3) The modified Newton method of Umrigar and Filippi and of Sorella. We show that the three methods are related to each other and point out that a control parameter can be employed in each of them to make them totally stable. We use these methods to optimize all the parameters in the Jastrow and the determinantal parts of the wave function and point out that different issues arise in optimizing the Jastrow and the determinantal parameters. By systematically increasing the number of determinants we find that seemingly similar systems, such as C_{2} and Si_{2} have vastly different fixed-node errors for single-determinant wave functions and that fixed-node errors for some simple systems can be as large as 1 eV. In contrast, the optimized multideterminantal wave functions yield energies in excellent agreement with experiment.

The methods described above can be straightforwardly extended to minimizing directly the fixed-node diffusion Monte Carlo energy and therefore the nodes of the trial wave functions.

We thank Peter Nightingale and Andreas Savin for valuable discussions. Supported by NSF.

## Recent Developments in Nuclear Quantum Monte Carlo

### Robert Wiringa (Argonne National Laboratory)

Quantum Monte Carlo methods are being used to study the properties of light nuclei starting from realistic nucleon-nucleon and three-nucleon forces. Potentials that fit NN elastic scattering data are very complicated, depending on relative separation, spin, and isospin degrees of freedom. This makes for some unique challenges in solving the many-body Schrodinger equation. At present, extensive calculations of nuclei containing up to 12 nucleons have been made, including ground-state binding energies, excitation spectra, and various transitions and reactions. Of technical interest, we are able to extract reasonable excitations energies for states of the same spin and parity, and we have started to evaluate nucleon-nucleus scattering.

## Recent developments in electronic structure calculations by auxiliary-field quantum Monte Carlo

### Shiwei Zhang* (College of William and Mary)

We discuss recent progress in the application and development of the auxiliary-field quantum Monte Carlo method for electronic structure calculations. The method is based on the phaseless Slater determinant random walk approach [1]. It takes the form of an ensemble of independent-particle calculations in auxiliary fields. Each random walker is an independent-particle wave function, i.e., a Slater determinant. The random walkers evolve as auxiliary fields are sampled, and a many-body wave function is obtained as a stochastic superposition of independent-particle solutions. The random walks are constrained in Slater determinant space by a phaseless approximation to deal with the phase problem [1]. This framework can be used to simulate either a fully materials-specific Hamiltonian or a realistic Hubbard-like model (e.g., obtained from "down-folding" with localized orbitals and truncation of the basis set). We discuss the characteristics of the method and relate it to other approaches. The method has been applied to trapped atomic gases[2], atoms, molecules[3], and simple bulk solids, including transition metal oxides[4]. Some recent results from these applications will be presented.

* Supported by NSF and ARO.

[1] Shiwei Zhang and Henry Krakauer, Phys. Rev. Lett. 90, 136401 (2003).

[2] W. Purwanto and Shiwei Zhang, Phys. Rev. A 72, 053610 (2005).

[3] W. A. Al-Saidi, Shiwei Zhang, and Henry Krakauer, J. Chem. Phys., in press.

( http://lanl.arxiv.org/abs/physics/0603055 )

[4] W. A. Al-Saidi, Henry Krakauer, and Shiwei Zhang, Phys. Rev. B 73, 075103 (2006)

## Molecules in Helium Nanodroplets

### Robert E. Zillich (Kaiserslautern, Germany)

We present path integral Monte Carlo (PIMC) simulations of molecules solvated in superfluid helium-4 nanodroplets, and we show how a combination of the pair density matrix approximation for the many body density matrix with a 4th-order scheme for the molecule-helium interaction can reduce the time step bias. Since the main experimental interest in these systems is low temperature spectrosopy, we calculate approximate rotational spectra by analytic continuation of the orientational correlation function in imaginary time, and investigate the influence of superfluidity on these spectra, in particular on the observed small dissipation (sharp spectral lines). We obtain complementary information about the rotational dynamics from the local superfluid helium density, and discuss how and in what cases the slowing of the molecule rotation can be explained by the adiabatic following of the local superfluid fraction perpendicular to the rotation axis. All these issues will be illustrated in particular by our recent results on LiH in helium, which behaves qualitatively different than previously studied light molecules in He.

Due to current interest in solid helium, we are also studying molecule dynamics in solid helium, with the goal that the molecule acts as "superfluidity probe", just as in helium nanodroplet spectroscopy discussed above. Finally, we present some results on superfluidity of solid helium with dislocation defects.

# Posters

Posters will be on display throughout the meeting, with an informal poster session during Monday evening cocktails (5:30-7:00 pm).

## Pfaffian Pairing Wave Functions For Quantum Monte Carlo

### Michal Bajdich, Lubos Mitas (CHIPS, Dept. of Physics, North Carolina State University, Raleigh, NC 27695)

Kevin E. Schmidt (Dept. of Physics, Arizona State University, Tempe, AZ 85287)

We investigate the limits of accuracy of trial wave function for quantum Monte Carlo based on pfaffian functional form with singlet and triplet pairing. Using a set of first row atoms and molecules we find that this wave function provides very consistent and systematic behavior in recovering the correlation energies on the level of 95% . In order to get beyond this limit we have explored the possibilities of multi-pfaffian pairing wave functions. We show that small number of pfaffians recovers another large fraction of the missing correlation energy comparable to the larger-scale configuration interaction wave functions.

## Electronic structure of δ-Pu, Am and δ-Pu-Am alloys

### J. Kolorenc (North Carolina State University), A. B. Shick and V. Drchal (Academy of Sciences of the Czech Republic, Prague), and L. Havela (Charles University, Prague)

The around-mean-field version of LDA+U method is applied to investigate electronic and magnetic structure of fcc Pu, Am and Pu-Am alloys. It yields a non-magnetic ground state in all cases and provides a good agreement with experimental equilibrium volume and other properties. Steps towards a more sophisticated description of f-electron correlations in a DMFT spirit are taken to get insight into features seen in valence-band photoelectron spectra.

## Hydrogen tunneling in malonaldehyde: full dimensional quantum mechanical studies

### Alexandra Viel, Maurício D. Coutinho(1) , Uwe Manthe(2)

### PALMS, University Rennes 1, Campus de Beaulieu, F-35042 Rennes, France

### (1) Department of Chemistry, Technical University of Munich, D-85747 Garching, Germany

### (2) Dept. of Chemistry, Theoretical Chemistry, Universit?Natsstraße 25, D-33615 Bielefeld, Germany

Since the pioneering experiments of Wilson’s group [1], the study of the hydrogen transfer in malonaldehyde and its associated tunneling splittings has attracted a lot of attention, both from the experimental and from the theoretical sides. The precise experimental value for the ground state tunneling splitting has been recently determined by observation of the tunneling-rotation lines in the sub-millimeter-wave region [2]. The value of 21.583 138 29 ± 0.000 000 63 cm^{-1} is consistent with the previous microwave results. Theoretical investigations aiming at the determination of the tunneling splitting for this multidimensional system are numerous. However, to the best of our knowledge they rely either on a reduced dimensionality quantum calculation or on sophisticated semi-classical methods. We have been able to determine [3] quantum mechanically the ground state tunneling splitting in full dimensionality (21 internal degrees of freedom) using the potential energy surface from Ref. [4]. For these calculations, we use two completely independent methodologies: the MCTDH (Multi-Configurational Time-Dependent Hartree[5,6]) method and the POITSE QMC (Projection Operator, Imaginary Time Spectral Evolution Quantum Monte Carlo[7]) based method. The agreement obtained between those two exact approaches both validates the methods themselves and provides a reference exact value for the tunneling splitting that can be used as a benchmark value to test more approximative methods. The Monte Carlo based POITSE method is found to be superior for obtaining ground state tunneling splittings.

References

[1] W. F. Rowe, R. W. Duerst and E. B. Wilson, J. Am. Chem. Soc., 98, 402 (1976)

[2] T. Baba, T. Tanaka, I. Morino, K. T. Yamada and K. Tanaka, J. Chem. Phys., 110, 4131 (1999)

[3] M. D. Coutinho-Neto, A. Viel and U. Manthe , J. Chem. Phys., 121, 9207-9210 (2004)

[4] K. Yagi, T. Taketsugu, and K. Hirao, J. Chem. Phys., 115, 10647 (2001)

[5] H. D. Meyer, U. Manthe, and L. S. Cederbaum, Chem. Phys. Letters, 165, 73 (1990)

[6] U. Manthe, H. D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 97, 3199 (1992)

[7] D. Blume, M. Lewerenz, P. Niyaz, and K. B. Whaley, Phys. Rev. E 55, 3664 (1997)

## First Row Transition Metal Oxide Molecules: Ab Initio Calculations Versus Experiment

### Lucas Wagner, Lubos Mitas (North Carolina State University)

Two-atom molecules of the form TMO, with TM a transition metal, are important in astronomy, and are the building blocks of many interesting materials, ranging from the large class of perovskites (which can exhibit ferroelectric and ferromagnetic effects) to major components of the Earth. Small errors in the calculated bond lengths or electronic structure can change the predicted properties of these TMO materials drastically, so it is useful to examine these simple systems to gain insight on the more complicated ones. In this poster, we will investigate the bond lengths, disassociation energies, and permanent electric dipole moments of some of the first row TMO molecules and show how the knowlege taken from these simple systems can help understand a more complicated one--the prototypical ferroelectic BaTiO_{3}.