Wigner-Kirkwood Expansion Research Articles

On the Wigner-Kirkwood Expansion of the Free Energy and the Evaluation of the Quantum Correction

The Wigner-Kirkwood expansion of the quantum correction to the classical free energy is generally said to be in powers of ℏ2 and only its first few terms are presented. In this work, we use the Bloch differential equation to obtain a general description of all terms in a dimensionless form. The first corrective term turns out to be proportional to the product of λ2/a2, where λ is the thermal de Broglie wavelength and a3 is the volume per particle, by an effective coupling constant. This dimensionless parameter can be used to assess the magnitude of the quantum correction. Using the one-component plasma as an illustration we highlight the importance of the magnitude of the potential on the quantum correction. The results presented are not formally new; the emphasis is placed on a simple and didactic presentation.

Vibrational Partition Function for the Multitemperature Theories of High-Temperature Flows of Gases and Plasmas.

The vibrational partition function is calculated using the classical method of integration over the whole phase space. The calculations were done for the ground electronic state of a carbon monoxide molecule. The main focus is on temperature in the range 5000-20 000 K, which is common in hypersonic flows of gases and plasmas. The method presented here, because of the exclusion of the noninteracting part of canonical partition function according to the ideas of T.L. Hill, is applicable at temperatures of tens of thousands of Kelvins, where the standard expression for the vibrational partition function fails. At lower temperatures (here 1000-6000 K), the correct quantum results can be obtained with the help of Wigner-Kirkwood expansion. The influence of vibrations on the rotational partition function by bond-length elongation is examined, and the results are compared with the exact ro-vibrational partition function.

Reproducing Quantum Probability Distributions at the Speed of Classical Dynamics: A New Approach for Developing Force-Field Functors.

Modeling nuclear quantum effects is required for accurate molecular dynamics (MD) simulations of molecules. The community has paid special attention to water and other biomolecules that show hydrogen bonding. Standard methods of modeling nuclear quantum effects like Ring Polymer Molecular Dynamics (RPMD) are computationally costlier than running classical trajectories. A force-field functor (FFF) is an alternative method that computes an effective force field that replicates quantum properties of the original force field. In this work, we propose an efficient method of computing FFF using the Wigner-Kirkwood expansion. As a test case, we calculate a range of thermodynamic properties of Neon, obtaining the same level of accuracy as RPMD, but with the shorter runtime of classical simulations. By modifying existing MD programs, the proposed method could be used in the future to increase the efficiency and accuracy of MD simulations involving water and proteins.

Applications to nuclear properties of the microscopic–macroscopic model based on the semiclassical Wigner–Kirkwood method

Some time ago we proposed a new microscopic–macroscopic model where the semiclassical Wigner–Kirkwood expansion of the energy up to fourth-order in ℏ is used to compute the shell corrections in a deformed Woods–Saxon potential instead of the usual Strutinsky averaging scheme. For a set of 558 even–even nuclei computed with this new model, we found a rms deviation of 610 keV from the experimental masses, similar to the value obtained using the well-known finite range droplet model and the Lublin–Strasbourg drop model for the same set of nuclei. In this paper we analyze the α radioactivity in nuclei with mass number , finding good agreement with the available experimental results. We have also estimated spontaneous fission half-lives for superheavy nuclei in the region between Z = 102 and Z = 110. We find that our model predicts reasonably well the experimental half-lives in the considered nuclei, in spite of the fact that the fission barriers turn out to be somewhat too high.

Path-integral approach to the Wigner-Kirkwood expansion

We study the high-temperature behavior of quantum-mechanical path integrals. Starting from the Feynman-Kac formula, we derive a functional representation of the Wigner-Kirkwood perturbation expansion for quantum Boltzmann densities. As shown by its applications to different potentials, the presented expansion turns out to be quite efficient in generating analytic form of the higher-order expansion coefficients. To put some flesh on the bare bones, we apply the expansion to obtain basic thermodynamic functions of the one-dimensional anharmonic oscillator. Further salient issues, such as generalization to the Bloch density matrix and comparison with the more customary world-line formulation, are discussed.

Microscopic–Macroscopic Mass Calculations with Wigner–Kirkwood expansion

The systematic study and calculation of ground state nuclear masses continues to be one of the active and important areas of research in nuclear physics. The present work is an attempt to determine the ground state masses of nuclei spanning the entire periodic table, using the Microscopic–Macroscopic approach. The semi-classical Wigner–Kirkwood (WK) ℏ expansion method is used to calculate shell corrections for spherical and deformed nuclei. The expansion is achieved upto the fourth order in ℏ. The shell corrections, along with the pairing energies obtained by using the Lipkin–Nogami scheme, constitute the microscopic part of the nuclear masses. The macroscopic part is obtained from a liquid drop formula with six adjustable parameters. It is shown that the Microscopic–Macroscopic mass calculation thus achieved, yields reliable description of ground state masses of nuclei across the periodic table. The present status of the WK mass calculations and the possible future perspectives are discussed.

Equation of state and stability of the helium-hydrogen mixture at cryogenic temperature

The equation of state and the stability of the helium-molecular hydrogen mixture at cryogenic temperature up to moderate pressure are studied by means of current molecular physics methods and statistical mechanics perturbation theory. The phase separation, segregation and hetero-coordination are investigated by calculating the Gibbs energy depending on the mixture composition, pressure and temperature. Low temperature quantum effects are incorporated via cumulant approximations of the Wigner-Kirkwood expansion. The interaction between He and H2 is determined by Double Yukawa potentials. The equation of state is derived from the hard sphere system by using the scaled particle theory. The behavior of the mixture over a wide range of pressure is explored with the excess Gibbs energy of mixing and the concentration fluctuations in the long wavelength limit. The theory is compared to cryogenic data and Monte-Carlo calculation predictions. Contrary to previous similar works, the present theory retrieves the main features of the mixture below 50 K, such as the critical point and the condensation-freezing curve, and is found to be usable well below 50 K. However, the method does not distinguish the liquid from the solid phase. The binary mixture is found to be unstable against species separation at low temperature and low pressure corresponding to very cold interstellar medium conditions, essentially because H2 alone condenses at very low pressure and temperature, contrary to helium.

Temperature–pressure–volume equation of state of the B1 phase of sodium chloride

The temperature–pressure–volume ( T– P– V) equation of state (EOS) of the B1 phase of NaCl is simulated using the molecular dynamics (MD) method with a breathing shell model (BSM) for Cl ions. Quantum correction to the MD pressures is made using the Wigner–Kirkwood expansion of free energy. Required energy parameters, including Cl breathing parameters, are derived empirically to reproduce the observed molar volume, volume thermal expansion, elastic constants, T– P– V data from piston-cylinder experiments and the temperature dependence of bulk modulus, all as accurately as possible. The observed volume-compression data of the high pressure B2 phase of NaCl are also included to optimize the energy parameters. The MD simulation with BSM is found to accurately reproduce the available measured T– P– V data of both the B1 and the B2 phases. The MD simulated T– P– V EOS is presented up to 30 GPa and 1200 K. Compared with the present MD EOS, the Decker EOS, which has often been used as a practical pressure scale for many years, underestimates pressure by within 4% over the T and P ranges up to 30 GPa and 1200 K. The error in the Decker EOS increases with increasing pressure, with the differences in pressure reaching 0.8 GPa at 300 K and 20 GPa, and 0.7 GPa at 1200 K and 20 GPa.

Kinetic Analysis of the Pyrolysis of Phenethyl Phenyl Ether: Computational Prediction of α/β-Selectivities

We calculated an overall alpha/beta-selectivity for the pyrolysis of phenethyl phenyl ether as a composite of the alpha/beta-selectivities in the hydrogen abstraction reactions by the phenoxyl and by the benzyl radical that is in excellent agreement with experiment. The difference between the individual selectivities for these radicals is explained by analyzing the electronic structure of the transition states. Spin delocalization of the single electron favors the alpha-pathways. An opposing effect occurs for polarized transition states, such as the transition states for the hydrogen abstraction by the electrophilic phenoxyl radical, where the adjacent ether oxygen in phenethyl phenyl ether stabilizes the beta-transition states. These results indicate that theory will be able to provide excellent predictions of alpha/beta-product selectivities for more complicated lignin model compounds bearing multiple substituents. We have developed a scheme to predict alpha/beta-product selectivities in the pyrolysis of model compounds for the beta-ether linkage in lignin. The approach is based on computation of the relative rate constant, which profits from error cancellation in the individual rate constants. The Arrhenius prefactors depend strongly on the description of the low-frequency modes for which anharmonic contributions are important. We use density functional theory in combination with transition-state theory in this analysis. Diagonal anharmonic effects for individual low-frequency modes are included by employing a second-order Wigner-Kirkwood expansion in a semiclassical expression for the vibrational partition function. The composite alpha/beta-product selectivity is obtained by applying quasi-steady-state kinetic analysis for the intermediate radicals.

A strongly perturbed quantum system is a semiclassical system

We show that a strongly perturbed quantum system, being a semiclassical system characterized by the Wigner–Kirkwood expansion for the propagator, has the same expansion for the eigenvalues as for the Wentzel–Kramers–Brillouin series. The perturbation series is rederived by the duality principle in perturbation theory.

Wigner–Kirkwood expansion for semi-infinite quantum fluids

For infinite (bulk) quantum fluids of particles interacting via pairwise sufficiently smoothinteractions, the Wigner–Kirkwood formalism provides a semiclassical expansion of theBoltzmann density in configuration space in even powers of the thermal de Broglie wavelengthλ. This result permits one to generate an analogousλ-expansion for the bulk free energy and many-body densities. The present paper brings atechnically non-trivial generalization of the Wigner–Kirkwood technique to semi-infinitequantum fluids, constrained by a plane hard wall impenetrable to particles. In contrast to thebulk case, the resulting Boltzmann density also involves position-dependent terms of typeexp(−2x2/λ2) (x denotesthe distance from the wall boundary) which are non-analytic functions of the de Broglie wavelengthλ. Under some condition,the analyticity in λ is restored by integrating the Boltzmann density over configuration space;however, in contrast to the bulk free energy, the semiclassical expansion ofthe surface part of the free energy (surface tension) contains odd powers ofλ, too. Explicit expressions for the leading quantum corrections in the presence of theboundary are given for the one-body and two-body densities. As model systems for explicitcalculations, we use Coulomb fluids, in particular the one-component plasma defined in theν-dimensional(integer ν ≥ 2) space.

The partition function in the Wigner–Kirkwood expansion

We study the semiclassical Wigner-Kirkwood (WK) expansion of the partition function $Z(t)$ for arbitrary even homogeneous potentials, starting from the Bloch equation. As is well known, the phase-space kernel of $Z$ satisfies the so-called Uhlenbeck-Beth equation, which depends on the gradients of the potential. We perform a chain of transformations to obtain novel forms of this equation that invite analogies with various physical phenomena and formalisms, such as diffusion processes, the Fokker-Planck equation, and supersymmetric quantum mechanics.

Shear viscosity of hydrogen and helium fluids at extended regions of temperatures and pressures

Shear viscosities of hydrogen (H 2) and helium (He) fluids are investigated at extended regions of temperature and pressure using an improved equation of state, in which the the long-range attractive forces are incorporated into the formalism via the double Yukawa (DY) potential fitted to the empirical H 2 and He potentials. Quantum effects, which become paramount at low temperatures, are taken into account through a first-order quantum correction based on the Wigner–kirkwood expansion. The dimerization of H 2 is treated as a hard convex body for which an equation of state can be derived based on the scaled particle theory.

Thermodynamic properties of He-H2 fluid mixtures over a wide range of temperatures and pressures.

The effect of pressure P and temperature T on the properties of mixing of helium-hydrogen (He-H2 ) fluid mixture is studied based on statistical mechanical perturbation theory. The constituent species are considered to be interacting via a pair potential consisting of short range repulsion and a long range attraction which has been included through a double Yukawa (DY) potential. He and H2 are the lightest elements; therefore, the quantum effect is included via first-order quantum correction in the framework of Wigner-Kirkwood expansion. The dimerization of the H2 molecule is treated as a hard convex body fluid for which the equation of state (EOS) can be derived from hard sphere system based on scaled particle theory. An advanced and most recent EOS has been used for our investigation. The use of the DY potential, which can readily be simulated to empirical potentials, has enabled us to obtain analytical expressions for attractive and quantum energy contributions in terms of Laplace transforms. With a view to ensure internal consistency of the various thermodynamic functions to extract information on segregation and order in the mixture, different functions, such as compressibility factor, Gibbs free energy of mixing, entropy of mixing, and concentration fluctuations in the long wavelength limit, have been calculated as functions of composition of the mixture over an extended region of P and T. The results suggest that segregation, heterocoordination, or both may occur in the He-H2 mixture depending upon its composition, pressure, and temperature.

High-temperature canonical partition function for restricted rotators

We propose an alternative, more rigorous, Wigner–Kirkwood expansion for thecanonical partition function for restricted rotators. The present approach is basedon a basis set of periodic functions that is more appropriate than the commonlychosen plane waves. We discuss the difference between present and earlierexpressions and explain why the usual approach gives acceptable results.

The MD simulation of the equation of state of MgO: Application as a pressure calibration standard at high temperature and high pressure

Molecular dynamics (MD) simulation is used to calculate the elastic constants and their temperature and pressure derivatives, and the T-P-V equation of state of MgO. The interionic potential is taken to be the sum of pairwise additive Coulomb, van der Waals attraction, and repulsive interactions. In addition, to account for the observed large Cauchy violation of the elastic constants of MgO, the breathing shell model (BSM) is introduced in MD simulation, in which the repulsive radii of O ions are allowed to deform isotropically under the effects of other ions in the crystal. Quantum correction to the MD pressure is made using the Wigner-Kirkwood expansion of the free energy. Required energy parameters, including oxygen breathing parameters, were derived empirically to reproduce the observed molar volume and elastic constants of MgO, and their measured temperature and pressure derivatives as accurately as possible. The MD simulation with BSM is found to be very successful in reproducing accurately the measured molar volumes and individual elastic constants of MgO over a wide temperature and pressure range. The errors in the simulated molar volumes are within 0.3% over the temperature range between 300 and 3000 K at 0 GPa, and within 0.1% over the pressure range from 0 up to 50 GPa at 300 K. The simulated bulk modulus is found to be correct to within 0.7% between 300 and 1800 K at 0 GPa. Here we present the MD simulated T-P-V equation of state of MgO as an accurate internal pressure calibration standard at high temperatures and high pressures

Calculation of the vibrational partition function of diatomic molecules from a scaled Wigner-Kirkwood expansion

New approximations to the partition function of diatomic molecules are given. The semiclassical Wigner-Kirkwood expansion truncated at the second or fourth order diverges as T. Various methods of scaling the Wigner-Kirkwood expansion are given which avoid this disadvantage and give sufficient accuracy in the low-temperature region. In addition, two modifications of the Pitzer-Gwinn method are discussed. Most molecular potentials can be considered almost harmonic near the minimum. For the scaling of the Wigner-Kirkwood expansion, therefore, a functional form is chosen, which would be exact for every temperature in the case of a purely harmonic potential. In the high-temperature limit the scaled and the conventional Wigner-Kirkwood expansion become identical.

A statistical mechanical study on the melting lines of 3He and 4He

A theoretical melting line is sensitive to a small difference in solid and fluid free energies. Such a difference may arise from different approximations made in fluid and solid theories and also from inaccuracies in handling anharmonic lattice vibrations. Statistical mechanical calculations have been made for liquid and solid 4He and 3He and their melting lines, using a new perturbation theory (PT) which does not suffer from the above limitations. These calculations, which use the Aziz potential for helium, are augmented by additional calculations using quasi-harmonic lattice dynamics and Mansoori—Canfield—Ross theory (with an exponential-6 potential). Comparisons of these results and available Monte Carlo and real experimental data show that the PT can accurately predict solid, fluid, and the melting data of a model of helium (based on the Aziz potential) and that the melting line of real 4He is insensitive to many-body effects up to 200 K, but their influence grows gradually with pressure (≈6% in melting pressure at 300 K). An effective pair potential is suggested which can handle many-body contributions over an extended density range. The calculations on the isotopic pressure shift, δP m = P m(3He)-P m(4He), along the melting lines of helium isotopes show that δP m > 0 at T < 100K in agreement with experiment at T ≈ 30 K and δP m < 0 at T > 100 K in agreement with path integral Monte Carlo data. The PT with the first-order quantum correction was found to explain the experimental melting data surprisingly well; i.e., beyond an estimated range of applicability of the Wigner—Kirkwood expansion. It can imply a rapid convergence of the Wigner—Kirkwood expansion along the melting line of He.

Shell corrections for finite depth potentials: Particle continuum effects

Shell corrections of finite, spherical, one-body potentials are analyzed using a smoothing procedure which properly accounts for the contribution from the particle continuum, i.e., unbound states. Since the plateau condition for the smoothed single-particle energy seldom holds, a new recipe is suggested for the definition of the shell correction. The generalized Strutinsky smoothing procedure is compared with the results of the semiclassical Wigner-Kirkwood expansion. A good agreement has been found for weakly bound nuclei in the vicinity of the proton drip line. However, some deviations remain for extremely neutron-rich systems due to the pathological behavior of the semiclassical level density around the particle threshold. {copyright} {ital 1998} {ital The American Physical Society}

Three-body depolarized interaction-induced light-scattering spectrum of neon.

We report the results of a depolarized interaction-induced light-scattering (DILS) experiment on gaseous neon at room temperature, from which the two- and three-body induced spectra have been extracted. The two-body spectrum is in excellent agreement (in absolute units) with a previous determination by Frommhold and Proffitt [Phys. Rev. A 21, 1249 (1980)], while the three-body spectrum is a new result. The comparison of the three-body experimental spectral moments with a calculation performed within the pairwise additivity approximation shows a discrepancy which is of opposite sign to that observed in krypton. The calculations have been carried out including the first-order correction of the Wigner-Kirkwood expansion in order to take into account quantum effects. We suggest that the occurence of irreducible three-body effects may be due to more than one microscopic mechanism and these can be usefully investigated by means of DILS spectroscopy.