Effective Classical Potential Research Articles

Quantum Computing for Molecular Biology.

Molecular biology and biochemistry interpret microscopic processes in the living world in terms of molecular structures and their interactions, which are quantum mechanical by their very nature. Whereas the theoretical foundations of these interactions are well established, the computational solution of the relevant quantum mechanical equations is very hard. However, much of molecular function in biology can be understood in terms of classical mechanics, where the interactions of electrons and nuclei have been mapped onto effective classical surrogate potentials that model the interaction of atoms or even larger entities. The simple mathematical structure of these potentials offers huge computational advantages; however, this comes at the cost that all quantum correlations and the rigorous many-particle nature of the interactions are omitted. In this work, we discuss how quantum computation may advance the practical usefulness of the quantum foundations of molecular biology by offering computational advantages for simulations of biomolecules. We not only discuss typical quantum mechanical problems of the electronic structure of biomolecules in this context, but also consider the dominating classical problems (such as protein folding and drug design) as well as data-driven approaches of bioinformatics and the degree to which they might become amenable to quantum simulation and quantum computation.

Thermal expansion of FeNi Invar and zinc-blende CdTe from the view point of local structure

Thermal expansion of FeNi Invar and zinc-blende CdTe was investigated from the view point of local structure using the extended x-ray absorption fine structure (EXAFS) spectroscopic data and the path-integral effective classical potential (PIECP) Monte Carlo computational simulations. In this Review article, first the quantum statistical perturbation theory is intuitively described to see different character concerning thermal expansion between the quantum and classical theories. The diatomic Br2 molecule is employed as a simple example. Second, the PIECP theory is briefly described to note advantages and disadvantages of this simulation technique. Historical background is also discussed for the EXAFS investigation of thermal expansion based on the quantum statistical theories. The results of the FeNi Invar alloy are then summarized. The origin of zero thermal expansion in the FeNi alloy is ascribed to the so-called Invar effect that implies the variation of the electronic structure of Fe atoms depending on temperature. Zero thermal expansion at low temperature is however found to originate from the vibrational quantum effect. It is also noted that the interatomic distances of Fe-Fe, Fe-Ni, and Ni-Ni pairs are slightly but meaningfully different from each other, although the alloy exhibit a simple fcc crystal. Such a pair- dependent difference is also true for thermal expansion and we will discuss thermal expansion from the local point of view, which is interestingly different from the lattice thermal expansion significantly. Finally, the results of the zinc blende (or diamond) structure are presented. Although the origin of negative thermal expansion in these tetrahedral crystals is known as a result of classical vibrational anomaly within the Newton dynamics theory, the quantum statistical simulation is found to be essential to reproduce the negative thermal expansion of CdTe. It is emphasized that the vibrational quantum effect and classical anharmonicity are of great importance for the understanding of low-temperature thermal expansion as well as the elastic constants.

A Real Scalar Field Unifying the Early Inflation and the Late Accelerating Expansion of the Universe through a Quadratic Equation of State: The Vacuumon

In a previous paper we introduced a cosmological model describing the early inflation, the intermediate decelerated expansion, and the late accelerating expansion of the universe in terms of a single barotropic fluid characterized by a quadratic equation of state. We obtained a scalar field representation of this fluid and determined the potential V(ϕ) connecting the inflaton potential in the early universe to the quintessence potential in the late universe. This scalar field has later been called the ‘vacuumon’ by other authors, in the context of the Running Vacuum model. In this paper, we study how the scalar field potential is modified by the presence of other cosmic components such as stiff matter, black-body radiation, baryonic matter, and dark matter. We also determine the mass m and the self-interaction constant λ of the scalar field given by the second and fourth derivatives of the potential at its extrema. We find that its mass is imaginary in the early universe with a modulus of the order of the Planck mass MP=(ℏc/G)1/2=1.22×1019GeV/c2 and real in the late universe with a value of the order of the cosmon mass mΛ=(Λℏ2/c4)1/2=2.08×10−33eV/c2 predicted by string theory. Although our model is able to describe the evolution of the homogeneous background for all times, it cannot account for the spectrum of fluctuations in the early universe. Indeed, by applying the Hamilton–Jacobi formalism to our model of early inflation, we find that the Hubble hierarchy parameters and the spectral indices lead to severe discrepancies with the observations. This suggests that the vacuumon potential is just an effective classical potential that cannot be directly used to compute the fluctuations in the early universe. A fully quantum field theory may be required to achieve that goal. Finally, we discuss the connection between our model based on a quadratic equation of state and the Running Vacuum model which assumes a variation of the cosmological constant with the Hubble parameter.

Monte Carlo study of transport in low-dimensional quantum disorder systems at finite temperature

Using the quantum generalized Langevin equation and the path integral Monte Carlo approach, we study the transport dynamics of low-dimensional quantum disorder systems at finite temperature. Motivated by the nature of the classical-to-quantum transformation in fluctuations in the time domain, we extend the treatment to the spatial domain and propose a quantum random-correlated potential, describing specifically quantum disorder. For understanding the Anderson localization from the particle transport perspective, we present an intuitive treatment using a classical analogy in which the particle moves through a flat periodic crystal lattice corrugated by classical or quantum disorder. We emphasize an effective classical disorder potential in studying the quantum effects on the transport dynamics. Compared with the classical case, we find that the quantum escape rate from a disordered metastable potential is larger. Moreover, the diffusion enhancement of a quantum system moving in a weak, biased, periodic disorder potential is more significant compared with the classical case; for an effective rock-ratcheted disorder potential, quantum effects increase the directed current with decreasing temperature. For the classical case, we explore surface diffusion on a two-dimensional biased disorder potential at finite temperature; surprisingly, the optimal angle of the external bias force is found to enhance diffusion in the biased disorder surface. Furthermore, to explain the quantum transport dynamics in a disorder potential, we adopt the barrier-crossing mechanism and the mean first passage time theory to establish the probability distribution function.

Fast quasi-centroid molecular dynamics.

We describe a fast implementation of the quasi-centroid molecular dynamics (QCMD) method in which the quasi-centroid potential of mean force is approximated as a separable correction to the classical interaction potential. This correction is obtained by first calculating quasi-centroid radial and angular distribution functions in a short path integral molecular dynamics simulation and then using iterative Boltzmann inversion to obtain an effective classical potential that reproduces these distribution functions in a classical NVT simulation. We illustrate this approach with example applications to the vibrational spectra of gas phase molecules, obtaining excellent agreement with QCMD reference calculations for water and ammonia and good agreement with the quantum mechanical vibrational spectrum of methane.

Hydrogenic Stretch Spectroscopy of Glycine-Water Complexes: Anharmonic Ab Initio Classical Separable Potential Calculations.

The anharmonic frequencies of O-H, C-H, and N-H stretching modes of hydrogen-bonded glycine-H2O complexes are calculated using ab initio classical separable potential approximation. In this approach, ab initio molecular dynamic simulations are used to determine an effective classical potential for each of the normal modes of the system. The frequencies are calculated by solving the time-independent Schrödinger equation for each mode using time-averaged potentials. Three complex structures are studied, which differ in the location of the water molecule on the amino acid. Significant differences are found between the spectra of the three structures, and signatures of individual complexes are established. It is demonstrated that anharmonic effects are essential in the discrimination between different structures, while frequency differences at the harmonic level are much smaller. Intensities are also computed and found to carry information on differences between structures, but the role of anharmonicity in this is small.

Anharmonicity in elastic constants and extended x-ray-absorption fine structure cumulants

We have investigated temperature dependence of the elastic constants and higher-order EXAFS (extended x-ray-absorption fine structure) cumulant moments, both of which originate from vibrational anharmonicity. We focus our attention on how the third- or fourth-order anharmonicity contributes to these physical quantities. Although it may be believed that the fourth-order anharmonicity should dominantly contribute to the fourth-order EXAFS cumulant through the first-order quantum statistical perturbation theory, it is consequently found that the experimental fourth-order EXAFS cumulant observed in fcc Ni, Cu, and stainless steel 316 are described mainly by the third-order anharmonicity through the second-order perturbation. In case of the elastic constants, such a situation is more prominent, and the contribution of the fourth-order anharmonicity is negligibly small for the estimation of temperature dependence of the elastic constants. We have also observed significant lattice strains on Cr and Mo in stainless steel 316 through the Fe, Ni, Cr, and Mo K-edge EXAFS measurements: a compressive lattice strain on Cr associated with larger thermal fluctuations and a more significant compressive lattice strain on Mo without enhanced thermal fluctuation. Such a characteristic dissimilarity may be caused by the differences in the atomic weights and the cohesive energies between Cr and Mo. Moreover, we have performed path-integral effective classical potential and classical Monte-Carlo simulations to describe the bulk moduli of the Invar and Elinvar alloys that show noticeable anomalies due to the so-called Invar effect. Appropriate temperature dependence of the bulk moduli is successfully obtained for the Invar, Elinvar, and stainless steel 304 and 316 alloys as well as elemental fcc Cu and Ni metals.

Generation and study of a relatively large amorphous silica surface in the liquid phase

ABSTRACTThrough the classical and ab-initio molecular dynamics computer simulations, the aim of this study was to generate and analyse a surface-type structure in the liquid phase at a temperature of 3400 K. First, a crystalline structure was used (β-cristobalite) with 216 SiO2 molecules (648 atoms) and it was melted. Next, the resulting structure was properly cut, with a total of 546 particles remaining, and this system was put into a geometric arrangement denominated ‘sandwich’, where it was equilibrated. The prior processes were implemented through the classical molecular dynamics computer simulations using the effective classical potential of Feuston [B. P. Feuston and S. H. Garofalini, J. Chem. Phys. 89, 5818 (1988)]. Finally, the electronic charge distribution and topological and bonding defects were analysed in the first five superior layers of this system. This latter was carried out through the ab-initio molecular dynamics computer simulations, where the pseudopotentials of N. Trouiller and J. L. Martins [N. Trouiller, J. L. Martins, Phys. Rev. B 43, 1993 (1991)] were used. The results show different aspects between them, the interaction between the bonding and topological defects, and the existence of two-membered rings, nonbridging oxygens (NBOs) and three-coordinated silicon at the outermost surface.

Anharmonic vibrational spectroscopy calculations using the ab initio CSP method: Applications to H2CO3, (H2CO3)2, H2CO3-H2O and isotopologues

The anharmonic frequencies of the O–H and the C–O stretching modes of carbonic acid are calculated using the ab initio classical separable potentials approximation. In this approach, ab initio molecular dynamics simulations are used to determine an effective classical potential for each of the normal modes of the system. The frequencies are calculated by solving the time-independent Schrödinger equation for each mode using time-averaged potentials. Hydrogen-bonded carbonic acid dimer, carbonic acid–water complex and isotopologues of these systems were also studied, showing good agreement with experiments. Useful insights are obtained by Fourier transforming the effective potentials, which relates the computed quantum frequencies to the classical dynamics of the system. This approach is illustrated for the systems studied, including for the challenging anharmonic isotope effect. In conclusion, the ab initio CSP approximation yields results in good accord with experiments, and the method provides interpretations in terms of the dynamics of vibrational motions.

Efficient methods for including quantum effects in Monte Carlo calculations of large systems: Extension of the displaced points path integral method and other effective potential methods to calculate properties and distributions

We present a procedure to calculate ensemble averages, thermodynamic derivatives, and coordinate distributions by effective classical potential methods. In particular, we consider the displaced-points path integral (DPPI) method, which yields exact quantal partition functions and ensemble averages for a harmonic potential and approximate quantal ones for general potentials, and we discuss the implementation of the new procedure in two Monte Carlo simulation codes, one that uses uncorrelated samples to calculate absolute free energies, and another that employs Metropolis sampling to calculate relative free energies. The results of the new DPPI method are compared to those from accurate path integral calculations as well as to results of two other effective classical potential schemes for the case of an isolated water molecule. In addition to the partition function, we consider the heat capacity and expectation values of the energy, the potential energy, the bond angle, and the OH distance. We also consider coordinate distributions. The DPPI scheme performs best among the three effective potential schemes considered and achieves very good accuracy for all of the properties considered. A key advantage of the effective potential schemes is that they display much lower statistical sampling variances than those for accurate path integral calculations. The method presented here shows great promise for including quantum effects in calculations on large systems.

Force-field functor theory: classical force-fields which reproduce equilibrium quantum distributions

Feynman and Hibbs were the first to variationally determine an effective potential whose associated classical canonical ensemble approximates the exact quantum partition function. We examine the existence of a map between the local potential and an effective classical potential which matches the exact quantum equilibrium density and partition function. The usefulness of such a mapping rests in its ability to readily improve Born-Oppenheimer potentials for use with classical sampling. We show that such a map is unique and must exist. To explore the feasibility of using this result to improve classical molecular mechanics, we numerically produce a map from a library of randomly generated one-dimensional potential/effective potential pairs then evaluate its performance on independent test problems. We also apply the map to simulate liquid para-hydrogen, finding that the resulting radial pair distribution functions agree well with path integral Monte Carlo simulations. The surprising accessibility and transferability of the technique suggest a quantitative route to adapting Born-Oppenheimer potentials, with a motivation similar in spirit to the powerful ideas and approximations of density functional theory.

A quartic trial action in the path-integral approach to an asymmetrical double-well potential

An improved variational path-integral method including partial fourth-order fluctuational modes is proposed, where three variational parameters are introduced and determined by minimizing the effective classical potential. It is shown that the effective classical potential and the free energy calculated by the present approach are better than that of the second-order variational one, indeed, our method can appearance rich quantum saddle behavior of an asymmetrical double-well potential and avoid dangerous modes.

Generalized Caldeira-Leggett system plus reservoir model

Interaction between system and reservoir is considered to be linear in the coordinates of heat bath but nonlinear in the system coordinate and velocity. The reduced partition function of the system is presented and the corresponding effective classical potential is calculated by using the path-integral Monte Carlo method. In the presence of the nonlinear coupling, a dissipative threshold is observed at finite temperatures, and the decay rate of a particle in a metastable potential can be hindered by inhomogeneous dissipation above twice the crossover temperature.

PERTURBATIVELY DEFINED EFFECTIVE CLASSICAL POTENTIAL IN CURVED SPACE

The partition function of a quantum statistical system in flat space can always be written as an integral over a classical Boltzmann factor exp [-βV eff cl (x0)], where V eff cl (x0) is the so-called effective classical potential containing the effects of all quantum fluctuations. The variable of integration is the temporal path average [Formula: see text]. We show how to generalize this concept to paths qμ(τ) in curved space with metric gμν(q), and calculate perturbatively the high-temperature expansion of V eff cl (q0). The requirement of independence under coordinate transformations qμ(τ)→ q′μ(τ) introduces subtleties in the definition and treatment of the path average [Formula: see text], and covariance is achieved only with the help of a suitable Faddeev–Popov procedure.

The development of effective classical potentials and the quantum statistical mechanical second virial coefficient of water

The second virial coefficient of water is calculated at low temperature by considering full quantum statistical mechanical effects. At low enough temperatures experimental results are limited and molecular models can be used for accurate extrapolation. In doing so, one must separate deficiencies of the intermolecular potential from limitations of the simulation methodology such as the neglect of higher-order quantum corrections. Effective classical potentials may be used to understand the limitations of classical simulation. In this work we calculate the exact quantum statistical mechanical second virial coefficient and find that using a semiclassical form for the effective classical potential we are able to nearly reproduce the exact quantum statistical results. This approach provides a significant improvement to conventional first order expansions of the second virial coefficient.

Improving the variational path integral approach to the quantum double-well potential

An improved variational path integral approach is developed and applied to the quantum double-well potential, in which part of the quartic term of the potential is included in the trial action. The expression of the effective classical potential (ECP) under a non-Gaussian expectation is obtained. Here the frequency and fourth-order derivative of the potential are treated as two variational parameters, determined by the minimization of the ECP at each point. We calculate the ECP, the free energy and the level splitting of a symmetrical double-well potential. It is shown that the present results are better than those of the Feynman–Kleinert Gaussian variational method.

Nonlinear coupling mechanism in a quantum system

For a quantum system coupled to a heat bath environment the nonlinear dissipation is studied starting from imaginary-time path-integral formulation. An effective classical potential (ECP) is obtained in which not only thefrequency of the potential but also the slope of the coupling form factor are treated as trial functions, they are determined by minimizing the effective classical potential. Further, terms of higher order are also added in an improved Gaussian measure in order to regularize integrals of fluctuation modes at low temperatures. Various approximations to thermodynamic functions of a double-well potential are compared with the dissipative path-integral Monte Carlo method. In particular, it occurs for single- or double-well potentials, where nonlinear dissipation can induce the appearance of multistable states and barrier drift. Nonmonotonic varying of the barrier height of the ECP with temperature is found in a bistable system.

Novel Quantum Hydrodynamic Equations for Semiconductor Devices

The Liouville equation for the distribution function is solved and a series solution for the Wigner distribution function is derived. In this solution, potential is nonlocal so that the distribution function in each point is influenced by the potential of the entire space. By computing the carrier density, an effective classical potential is defined. In a quantum system this effective potential replaces the classical potential. Based on the solution of the Liouville equation a novel set of three-dimensional quantum hydrodynamic equations (QHD) is derived. The form of the resulting QHD equations is similar to the classical hydrodynamic (HD) equations but there are explicit quantum corrections which appear in the pressure tensor of electron gas and energy density.

On quantum effects near the liquid–vapor transition in helium

The liquid–vapor transition in He3 and He4 is investigated by means of path-integral molecular dynamics and the quantum virial expansion. Both methods are applied to the critical isobar and the critical isochore. While previous path-integral simulations have mainly considered the lambda transition and superfluid regime in He4, we focus on the vicinity of the critical point and obtain good agreement with experimental results for the molar volume and the internal energy down to subcritical temperatures. We find that an effective classical potential that properly describes the two-particle radial distribution function exhibits a strong temperature dependence near the critical temperature. This contrasts with the behavior of essentially classical systems like xenon, where the effective potential is independent of temperature. It is conjectured that, owing to this difference in behavior between classical and quantum-mechanical systems, the crossover behavior observed for helium in the vicinity of the critical point differs qualitatively from that of other simple liquids.

Effective Classical Potential

A new method for approximating a quantum-mechanical partition function by an effective classical partition function is proposed. The associated effective classical potential is found by using a variational procedure and a Gaussian Ansatz to estimate the statistical weight of paths starting and ending at the same point in the path-integral representation of the partition function. This method can be generalized to a variational-convergent calculation of path-integral. This systematic technique involves neither perturbation nor stochastic processes. Numerical results are presented and discussed for quartic, sextic, mixed quartic-sextic, and double-well anharmonic oscillators.