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Kunikeev SD, Kim KS. Cumulant expansion and analytic continuation in Monte Carlo simulation of classical Lennard-Jones clusters. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:056702. [PMID: 23214901 DOI: 10.1103/physreve.86.056702] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2012] [Indexed: 06/01/2023]
Abstract
The Monte Carlo (MC) estimates of thermal averages are usually functions of system control parameters λ, such as temperature, volume, and interaction couplings. Given the MC average at a set of prescribed control parameters λ{0}, the problem of analytic continuation of the MC data to λ values in the neighborhood of λ{0} is considered in both classic and quantum domains. The key result is the theorem that links the differential properties of thermal averages to the higher order cumulants. The theorem and analytic continuation formulas expressed via higher order cumulants are numerically tested on the classical Lennard-Jones cluster system of N=13, 55, and 147 neon particles.
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Affiliation(s)
- Sharif D Kunikeev
- Department of Chemistry, Pohang University of Science and Technology, Pohang 790-784, South Korea
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2
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Meuwly M, Doll JD. Finite-temperature quantum simulations of mixed rare gas clusters. J Chem Phys 2010; 132:234315. [PMID: 20572713 DOI: 10.1063/1.3431080] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023] Open
Abstract
Finite-temperature quantum Monte Carlo simulations are presented for mixed neon/argon rare gas clusters containing up to n=10 atoms. For the smallest clusters (n=3) comparison with rigorous bound state calculations and experiments shows that the present approach is accurate to within fractions of wavenumbers for energies and to within a few percent or better for rotational constants. For larger cluster sizes, for which no rigorous quantum calculations are available, comparison with experiment becomes even more favorable. In all simulations accurate pair potentials for the rare gas-rare gas interactions are employed and comparison with high-level electronic structure calculations suggest that many-body interactions play a minor role. For the largest clusters investigated (Ne(4)Ar(6)) gradual melting of the neon phase is observed while the argon-phase remains structurally intact.
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Affiliation(s)
- Markus Meuwly
- Department of Chemistry, University of Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland.
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Kunikeev SD, Freeman DL, Doll JD. Convergence characteristics of the cumulant expansion for Fourier path integrals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:066707. [PMID: 20866544 DOI: 10.1103/physreve.81.066707] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/24/2010] [Indexed: 05/29/2023]
Abstract
The cumulant representation of the Fourier path integral method is examined to determine the asymptotic convergence characteristics of the imaginary-time density matrix with respect to the number of path variables N included. It is proved that when the cumulant expansion is truncated at order p, the asymptotic convergence rate of the density matrix behaves like N(-(2p+1)). The complex algebra associated with the proof is simplified by introducing a diagrammatic representation of the contributing terms along with an associated linked-cluster theorem. The cumulant terms at each order are expanded in a series such that the asymptotic convergence rate is maintained without the need to calculate the full cumulant at order p. Using this truncated expansion of each cumulant at order p, the numerical cost in developing Fourier path integral expressions having convergence order N(-(2p+1)) is shown to be approximately linear in the number of required potential energy evaluations making the method promising for actual numerical implementation.
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Affiliation(s)
- Sharif D Kunikeev
- Department of Chemistry, University of Rhode Island, Kingston, Rhode Island 02881, USA
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van Zon R, Hernández de la Peña L, Peslherbe GH, Schofield J. Quantum free-energy differences from nonequilibrium path integrals. I. Methods and numerical application. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:041103. [PMID: 18999375 DOI: 10.1103/physreve.78.041103] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/04/2008] [Indexed: 05/27/2023]
Abstract
In this paper, the imaginary-time path-integral representation of the canonical partition function of a quantum system and nonequilibrium work fluctuation relations are combined to yield methods for computing free-energy differences in quantum systems using nonequilibrium processes. The path-integral representation is isomorphic to the configurational partition function of a classical field theory, to which a natural but fictitious Hamiltonian dynamics is associated. It is shown that if this system is prepared in an equilibrium state, after which a control parameter in the fictitious Hamiltonian is changed in a finite time, then formally the Jarzynski nonequilibrium work relation and the Crooks fluctuation relation hold, where work is defined as the change in the energy as given by the fictitious Hamiltonian. Since the energy diverges for the classical field theory in canonical equilibrium, two regularization methods are introduced which limit the number of degrees of freedom to be finite. The numerical applicability of the methods is demonstrated for a quartic double-well potential with varying asymmetry. A general parameter-free smoothing procedure for the work distribution functions is useful in this context.
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Affiliation(s)
- Ramses van Zon
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto, 80 Saint George Street, Toronto, Ontario, Canada
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5
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Curotto E, Freeman DL, Doll JD. A stereographic projection path integral study of the coupling between the orientation and the bending degrees of freedom of water. J Chem Phys 2008; 128:204107. [DOI: 10.1063/1.2925681] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Avilés MW, McCandless ML, Curotto E. Stereographic projection path integral simulations of (HCl)n clusters (n=2–5): Evidence of quantum induced melting in small hydrogen bonded networks. J Chem Phys 2008; 128:124517. [DOI: 10.1063/1.2837802] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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7
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Ramírez E, López GE. Isotopic effect in the solid–liquid phase diagram of quantum clusters. Mol Phys 2007. [DOI: 10.1080/00268970701678915] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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Langley SF, Curotto E, Freeman DL, Doll JD. Rigid quantum Monte Carlo simulations of condensed molecular matter: Water clusters in the n=2→8 range. J Chem Phys 2007; 126:084506. [PMID: 17343457 DOI: 10.1063/1.2484229] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The numerical advantage of quantum Monte Carlo simulations of rigid bodies relative to the flexible simulations is investigated for some simple systems. The results show that if high frequency modes in molecular condensed matter are predominantly in the ground state, the convergence of path integral simulations becomes nonuniform. Rigid body quantum parallel tempering simulations are necessary to accurately capture thermodynamic phenomena in the temperature range where the dynamics are influenced by intermolecular degrees of freedom; the stereographic projection path integral adapted for quantum simulations of asymmetric tops is a significantly more efficient strategy compared with Cartesian coordinate simulations for molecular condensed matter under these conditions. The reweighted random series approach for stereographic path integral Monte Carlo is refined and implemented for the quantum simulation of water clusters treated as an assembly of rigid asymmetric tops.
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Affiliation(s)
- Stephen F Langley
- Department of Chemistry, Arcadia University, Glenside, Pennsylvania 19038-3295, USA
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9
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Avilés MW, Gray PT, Curotto E. Stereographic projection path-integral simulations of (HF)n clusters. J Chem Phys 2006; 124:174305. [PMID: 16689569 DOI: 10.1063/1.2192773] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We perform several quantum canonical ensemble simulations of (HF)(n) clusters. The HF stretches are rigid, and the stereographic projection path-integral method is employed for the simulation in the resulting curved configuration space. We make use of the reweighted random series techniques to accelerate the convergence of the path-integral simulation with respect to the number of path coefficients. We develop and test estimators for the total energy and heat capacity based on a finite difference approach for non-Euclidean spaces. The quantum effects at temperatures below 400 K are substantial for all sizes. We observe interesting thermodynamic behaviors in the quantum simulations of the octamer and the heptamer.
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Affiliation(s)
- Michael W Avilés
- Department of Chemistry and Physics, Arcadia University, Glenside, Pennsylvania 19038, USA
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Pav DM, Curotto E. Parallel tempering simulations of the 13-center Lennard-Jones dipole-dipole cluster (muD=0-->0.5 a.u.). J Chem Phys 2005; 123:144301. [PMID: 16238385 DOI: 10.1063/1.2049279] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We investigate the thermodynamic behavior of the thirteen center uniform Lennard-Jones dipole-dipole cluster [(LJDD)(13)] for a wide range of dipole moment strengths. We find a relatively wide range of potential parameters where solid-solid coexistence manifests itself. Using structural characterization methods we determine the shape of the few isomers that contribute to the solid-solid coexistence region. The thermal distributions of the size of the net dipole moment are broad even at the coldest temperatures of the simulation where the (LJDD)(13) cluster is solid.
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Affiliation(s)
- Dawn M Pav
- Department of Chemistry and Physics, Arcadia University, Glenside, Pennsylvania 19038, USA
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Curotto E. A reweighted random series method for stereographic projection path integrals. J Chem Phys 2005; 123:134102. [PMID: 16223270 DOI: 10.1063/1.2036970] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022] Open
Abstract
A set of general reweighted random series methods for metric affine spaces is developed. The extension of the theorems to metric affine spaces demands the introduction of a configuration-independent reference metric tensor; this geometric object is used to treat the path expansion coefficients beyond the core path, in both the partial averaging and the reweighted random series approach. Numerical tests are conducted by simulating a particle in a ring. The reweighted random series results show better convergence properties and better statistical quality at a fraction of the cost compared with the related partial averaging simulation.
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Affiliation(s)
- E Curotto
- Department of Chemistry and Physics, Arcadia University, Glenside, Pennsylvania 19038, USA.
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12
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Yamamoto TM. Path-integral virial estimator based on the scaling of fluctuation coordinates: Application to quantum clusters with fourth-order propagators. J Chem Phys 2005; 123:104101. [PMID: 16178583 DOI: 10.1063/1.2013257] [Citation(s) in RCA: 48] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023] Open
Abstract
We first show that a simple scaling of fluctuation coordinates defined in terms of a given reference point gives the conventional virial estimator in discretized path integral, where different choices of the reference point lead to different forms of the estimator (e.g., centroid virial). The merit of this procedure is that it allows a finite-difference evaluation of the virial estimator with respect to temperature, which totally avoids the need of higher-order potential derivatives. We apply this procedure to energy and heat-capacity calculations of the (H(2))(22) and Ne(13) clusters at low temperature using the fourth-order Takahashi-Imada [J. Phys. Soc. Jpn. 53, 3765 (1984)] and Suzuki [Phys. Lett. A 201, 425 (1995)] propagators. This type of calculation requires up to third-order potential derivatives if analytical virial estimators are used, but in practice only first-order derivatives suffice by virtue of the finite-difference scheme above. From the application to quantum clusters, we find that the fourth-order propagators do improve upon the primitive approximation, and that the choice of the reference point plays a vital role in reducing the variance of the virial estimator.
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Affiliation(s)
- Takeshi M Yamamoto
- Department of Chemistry, Graduate School of Science, Kyoto University, Japan.
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Chakravarty C. Hybrid Monte Carlo implementation of the Fourier path integral algorithm. J Chem Phys 2005; 123:24104. [PMID: 16050738 DOI: 10.1063/1.1990114] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
This paper formulates a hybrid Monte Carlo implementation of the Fourier path integral (FPI-HMC) approach with partial averaging. Such a hybrid Monte Carlo approach allows one to generate collective moves through configuration space using molecular dynamics while retaining the computational advantages associated with the Fourier path integral Monte Carlo method. In comparison with the earlier Metropolis Monte Carlo implementations of the FPI algorithm, the present HMC method is shown to be significantly more efficient for quantum Lennard-Jones solids and suggests that such algorithms may prove useful for efficient simulations of a range of atomic and molecular systems.
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Affiliation(s)
- Charusita Chakravarty
- Department of Chemistry, Indian Institute of Technology-Delhi, New Delhi, 110016, India.
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Avilés MW, Curotto E. Partial averaging and the centroid virial estimator for stereographic projection path-integral simulations in curved spaces. J Chem Phys 2005; 122:164109. [PMID: 15945674 DOI: 10.1063/1.1884109] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We develop and test three different partial averaging theories for the stereographic projection path integral in curved spaces. Additionally, we adapt and test the centroid virial estimator for the kinetic energy in curved spaces. We tested both a confining as well as a nonconfining potential for the particle in a ring. All three partial averaging theories are demonstrated to converge linearly in the asymptotic region with k(-2)max, the number of Fourier coefficients. We use an error estimator to determine the optimal parameters for an extrapolation to infinite kmax. We verify that the centroid virial estimator (derived from the primitive DeWitt path-integral formula) converges to the kinetic energy for both confining and nonconfining potentials.
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Affiliation(s)
- Michael W Avilés
- Department of Chemistry and Physics, Arcadia University, Glenside, Pennsylvania 19038, USA
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Russo MF, Curotto E. Stereographic projections path integral for inertia ellipsoids: Applications to Arn–HF clusters. J Chem Phys 2004; 120:2110-21. [PMID: 15268349 DOI: 10.1063/1.1636694] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The DeWitt formula for inertia ellipsoids mapped by stereographic projection coordinates is developed. We discover that by remapping the quaternion parameter space with stereographic projections, considerable simplification of the differential geometry for the inertia ellipsoid with spherical symmetry takes place. The metric tensor is diagonal and contains only one independent element in that case. We find no difficulties testing and implementing the DeWitt formula for the inertia ellipsoids of asymmetric tops mapped by stereographic projections. The path integral algorithm for the treatment of Rm x S2 manifolds based on a mixture of Cartesian and stereographic projection coordinates is tested for small Arn-HF clusters in the n = 2 to n = 5 range. In particular, we determine the quantum effects of the red shift and the isomerization patterns at finite temperatures. Our findings are consistent with previously reported computations and experimental data for small Arn-HF clusters.
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Affiliation(s)
- M F Russo
- Department of Chemistry and Physics, Arcadia University, Glenside, Pennsylvania 19038, USA
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Predescu C, Sabo D, Doll JD, Freeman DL. Heat capacity estimators for random series path-integral methods by finite-difference schemes. J Chem Phys 2003. [DOI: 10.1063/1.1625366] [Citation(s) in RCA: 56] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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18
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Russo MF, Curotto E. Stereographic projections path integral in S1 and (S2)m manifolds. J Chem Phys 2003. [DOI: 10.1063/1.1560936] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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19
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Predescu C, Doll JD. Optimal series representations for numerical path integral simulations. J Chem Phys 2002. [DOI: 10.1063/1.1509058] [Citation(s) in RCA: 41] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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López GE. Quantum effects in the solid–liquid phase diagram of Ne13 and (para-H2)13. J Chem Phys 2002. [DOI: 10.1063/1.1489421] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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22
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ORTIZ VANESSA, LÓPEZ GUSTAVOE. Fourier path integral Monte Carlo study of a two-dimensional model quantum monolayer. Mol Phys 2002. [DOI: 10.1080/00268970110100627] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
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Whitfield TW, Straub JE. Enhanced sampling in numerical path integration: an approximation for the quantum statistical density matrix based on the nonextensive thermostatistics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:066115. [PMID: 11736244 DOI: 10.1103/physreve.64.066115] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/10/2001] [Indexed: 05/23/2023]
Abstract
Here we examine a proposed approximation for the quantum statistical density matrix motivated by the nonextensive thermostatistics of Tsallis and co-workers. The approximation involves replacing the physical potential energy with an effective one, corresponding to a generalized nonextensive statistical ensemble. We examine the convergence properties of averages calculated using the effective potential, and introduce a related method for enhanced sampling in numerical path integration. As a necessary measure, path integral energy estimators are introduced for potentials that involve explicit temperature dependence. This sampling method is found to be effective for path integral simulations involving broken ergodicity.
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Affiliation(s)
- T W Whitfield
- Department of Chemistry, Boston University, Boston, Massachusetts 02215, USA
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Whitfield TW, Straub JE. Uncertainty of path integral averages at low temperature. J Chem Phys 2001. [DOI: 10.1063/1.1403691] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Mielke SL, Truhlar DG. A new Fourier path integral method, a more general scheme for extrapolation, and comparison of eight path integral methods for the quantum mechanical calculation of free energies. J Chem Phys 2001. [DOI: 10.1063/1.1290476] [Citation(s) in RCA: 43] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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26
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Ceperley DM, Manousakis E. Path integral Monte Carlo applications to quantum fluids in confined geometries. J Chem Phys 2001. [DOI: 10.1063/1.1418443] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Aguado AM, Curotto E. On the finite temperature red shift in Ar12–HF: can isomerizations in clusters be observed by spectroscopy? Chem Phys Lett 2000. [DOI: 10.1016/s0009-2614(00)01111-8] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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28
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Sabo D, Doll JD, Freeman DL. Self-adaptive quadrature and numerical path integration. J Chem Phys 2000. [DOI: 10.1063/1.1305743] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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29
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Mielke SL, Srinivasan J, Truhlar DG. Extrapolation and perturbation schemes for accelerating the convergence of quantum mechanical free energy calculations via the Fourier path-integral Monte Carlo method. J Chem Phys 2000. [DOI: 10.1063/1.481491] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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30
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Neirotti JP, Freeman DL, Doll JD. A heat capacity estimator for Fourier path integral simulations. J Chem Phys 2000. [DOI: 10.1063/1.480999] [Citation(s) in RCA: 41] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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31
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Doll JD, Freeman DL. Comment on “A comparison of the efficiency of Fourier- and discrete time-path integral Monte Carlo” [J. Chem. Phys.109, 2123 (1998)]. J Chem Phys 1999. [DOI: 10.1063/1.480094] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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