Quantum hard spheres

Runge, K. J. Chester, G. V., Solid-fluid phase transition of quantum hard spheres at finite temperature, Phys. Rev. B 1988, 38, 135-162... 

The other main avenue to tackle the A-particle quantum problem with Pis focuses on the so-called pair product actions. These are approaches that have been proven to be very cost effective, as relatively low P discretizations work excellently well. Besides, the use of pair actions has led to a deep understanding of helium and the system of hard spheres at low temperatures [83-108]. Clear indications of how to proceed along this line were given by Barker in his pioneering work on quantum hard spheres (QHS) [24], Klemm and Storer [83], and Ceperley [28]. Essentially, this sort of approach consists of representing the V-particle density matrix through a product of density matrices, which includes those of the free particles and those of the reduced masses of every pair of particles. The potential is assumed pair-wise additive and, for definiteness, P time slices in... 

To bring this subsection to a close some comments on the QHS system including attractive forces are in order. The introduction of attractive tails can modify greatly the structural and thermodynamic behavior of the underlying bare QHS system. This has been shown by the present author and L. Bailey [108] with the use of pair-wise Yukawa attractive tails and PIMC simulations. By doing so, the onset of critical behavior in the quantum hard-sphere Yukawa (QHSY) system was identified under conditions in which the QHS system remains in the normal fluid phase. The Hamiltonian for the QHSY system can be cast as... 

This procedure is conceptually straightforward, as one utilizes the very definition of the isothermal compressibility and its connection to the number fluctuations. Furthermore, Eqs. (141) and (142) are very advantageous when studying quantum hard-sphere fluids, since the error bars of the pressure estimates are far more controllable than when using the virial pressure involving Fierz s term [96]. By extension, the same is expected to happen when studying quantum fluids in which very strong repulsions between the particles play a dominant role. 

Some new numerical results for fluid He, fluid He, and the hard-sphere fluid, under quantum diffraction effects, are given below to illustrate a number of the basic main points discussed in this chapter. The particle masses (amu) have been set to m( He) = 4.0026, m("He) = 3.01603, and m(hard sphere) = 28.0134. PIMC simulations in the canonical ensemble using the necklace normal-mode moves have been employed. The Metropolis algorithm has been apphed with the general acceptance criterion set to 50% of the attempted moves for each normal mode. In the helium simulations the propagator SCVJ (a = 1 / 3) has been utihzed. The quantum hard-sphere fluid results presented in this chapter have been obtained from a further processing of data reported in Ref. 96. Also, for fluid He Monte Carlo classical (CLAS) and effective potential QFH calculations have been performed by following the standard procedures. 

Figure 7. Reduced Gibbs free energies and entropies for the quantum hard-sphere fluid along...

Hamiltonian quantum mechanical operator for energy, hard sphere assumption that atoms are like hard billiard balls, which is implemented by having an infinite potential inside the sphere radius and zero potential outside the radius Hartree atomic unit of energy... 

Permutations of this type have to be considered in PIMC simulations if a full account of the quantum statistics is intended in the study and required by the physical effect under consideration, which means that additional permutation moves have to be done in the simulation. In this way quantum statistics has been included in a few PIMC simulations, in particular for the study of superfluidity in He [287] and in adsorbed H2 layers [92], for the Bose-Einstein condensation of hard spheres [269], and for the analysis of... 

According to Vitanov et a/.,61,151 C,- varies in the order Ag(100) < Ag(lll), i.e., in the reverse order with respect to that of Valette and Hamelin.24 63 67 150 383-390 The order of electrolytically grown planes clashes with the results of quantum-chemical calculations,436 439 as well as with the results of the jellium/hard sphere model for the metal/electro-lyte interface.428 429 435 A comparison of C, values for quasi-perfect Ag planes with the data of real Ag planes shows that for quasi-perfect Ag planes, the values of Cf 0 are remarkably higher than those for real Ag planes. A definite difference between real and quasi-perfect Ag electrodes may be the higher number of defects expected for a real Ag crystal. 15 32 i25 401407 10-416-422 since the defects seem to be the sites of stronger adsorption, one would expect that quasi-perfect surfaces would have a smaller surface activity toward H20 molecules and so lower Cf"0 values. The influence of the surface defects on H20 adsorption at Ag from a gas phase has been demonstrated by Klaua and Madey.445... 

We know from quantum mechanics that atoms and ions do not have precisely defined radii. However, from the foregoing discussion of ionic crystal structures we have seen that ions pack together in an extremely regular fashion in crystals, and their atomic positions, and thus their interatomic distances, can be measured very accurately. It is a very useful concept, therefore, particularly for those structures based on close-packing, to think of ions as hard spheres, each with a particular radius. 

Early numerical estimates of ternary moments [402] were based on the empirical exp-4 induced dipole model typical of collision-induced absorption in the fundamental band, which we will consider in Chapter 6, and hard-sphere interaction potentials. While the main conclusions are at least qualitatively supported by more detailed calculations, significant quantitative differences are observed that are related to three improvements that have been possible in recent work [296] improved interaction potentials the quantum corrections of the distribution functions and new, accurate induced dipole functions. The force effect is by no means always positive, nor is it always stronger than the cancellation effect. 

Lorentz1 advanced a theory of metals that accounts in a qualitative way for some of their characteristic properties and that has been extensively developed in recent years by the application of quantum mechanics. He thought of a metal as a crystalline arrangement of hard spheres (the metal cations), with free electrons moving in the interstices.. This free-electron theory provides a simple explanation of metallic luster and other optical properties, of high thermal and electric conductivity, of high values of heat capacity and entropy, and of certain other properties. 

Molecular Mechanics. Molecular mechanics (MM), or empirical force field methods (EFF), are so called because they are a model based on equations from Newtonian mechanics. This model assumes that atoms are hard spheres attached by networks of springs, with discrete force constants. The force constants in the equations are adjusted empirically to repro duce experimental observations. The net result is a model which relates the "mechanical" forces within a structure to its properties. Force fields are made up of sets of equations each of which represents an element of the decomposition of the total energy of a system (not a quantum mechanical eneigy, but a classical mechanical one). The sum of the components is called the force field eneigy, or steric energy, which also routinely includes the electrostatic eneigy components. Typically, the steric energy is expressed as... 

Before moving on to further discussions, we shall say a few words on dispersive or vdWs interactions. These are composed of an attractive force that arises from a sophisticated quantum-chemical short-range interaction of electrons, and an even shorter range, i.e., almost hard sphere repulsion of the cores of the atomic... 

In addition, assumptions for the reaction probability of the hard spheres—which strictly speaking cannot react—are introduced. That is, the reaction probability is not calculated from the actual potentials or dynamics of the collisions but simply postulated based on physical intuition. Note that the assumption of a spherically symmetric (hard-sphere) interaction potential implies that the reaction probability P cannot depend on (j> (see Fig. 4.1.1), since there will be a cylindrical symmetry around the direction of the relative velocity. In addition, the assumption of structureless particles implies that the quantum numbers that specify the internal excitation cannot be defined within the present model. 

We see that the viscosity diameters are about 10 per cent larger than the mean diameters calculated from Eq. (VIII.9.1). The difference probably arises from the inaccuracy in approximating the molecules by hard spheres. It probably also arises from the fact that quantum-mechanical... 

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