Radial distribution functions, g

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

Figure A2.3.7 The radial distribution function g r) of a Lemiard-Jones fluid representing argon at T = 0.72 and p = 0.844 detennined by computer simulations using the Lemiard-Jones potential.

Microscopic theory yields an exact relation between the integral of the radial distribution function g(r) and the compressibility... 

Distribution functions measure the (average) value of a property as a function of an independent variable. A typical example is the radial distribution function g r) which measmes the probability of finding a particle as a function of distance from a typical ... 

The degree of ordering of the microspheres was estimated by using the radial distribution function g(D) of the P4VP cores of the microspheres (Fig. 11). As previously described, for hexagonal packed spheres, the ratio of the peaks of the distances between the centers of the cores would be For the film at r = 0.5, the... 

When the system contains more than one component it is important to be able to explore the distribution of the different components both locally and at long range. One way in which this can be achieved is to evaluate the distribution function for the different species. For example in a binary mixture of components A and B there are four radial distribution functions, g (r), g (r), g (r) and g (r) which are independent under certain conditions. More importantly they would, with the usual definition, be concentration dependent even in the absence of correlations between the particles. It is convenient to remove this concentration dependence by normalising the distribution function via the concentrations of the components [26]. Thus the radial distribution function of g (r) which gives the probability of finding a molecule of type B given one of type A at the origin is obtained from... 

Often the reduced radial distribution function G(r) is deduced which is given by ... 

An important theoretical development for the outer-sphere relaxation was proposed in the 1970s by Hwang and Freed (138). The authors corrected some earlier mistakes in the treatment of the boundary conditions in the diffusion equation and allowed for the role of intermolecular forces, as reflected in the IS radial distribution function, g(r). Ayant et al. (139) proposed, independently, a very similar model incorporating the effects of molecular interactions. The same group has also dealt with the effects of spin eccentricity or translation-rotation coupling (140). 

The second category of methods uses a more general approach, based on fundamental concepts in statistical mechanics of the liquid state. As mentioned above, the Hwang and Freed theory (138) and the work of Ayant et al. (139) allow for the presence of intermolecular forces by including in the formulation the radial distribution function, g(r), of the nuclear spins with respect to the electron spins. The radial distribution function is related to the effective interaction potential, V(r), or the potential of mean force, W(r), between the spin-carrying particles through the relation (138,139) ... 

A number of approximate integral equations for the radial distribution function g(r) of fluids have been proposed in recent years. Two particularly useful approximations are the Percus-Yevick (PY)1,2 and the Convolution Hypernetted Chain (CHNC)3-4 equations. In this paper an efficient numerical method of solving these equations is described and the results obtained bv applying the method to the PY equation are discussed. A later paper will describe the behavior of the... 

Fig. 38. Radial distribution function, g(r), potential of mean force, U,-, and pair...

Figure 2 Left column- NO Absorption (upper panel) and emission (lower panel) spectra at near critical conditions and room temperature for a density of 5.7 at/nm3. Right column NO-Ar radial distribution functions g(r) for the NO ground (upper panel) and excited (lower panel) states for the same state points...

The moments of Y are obtained from similar expressions simply by changing the signs of all g /, g f and g f that appear in Eqs. 6.78 through 6.80, so that we need not repeat those expressions here. We note that the reduced mass is m, B is short for Bj.cJ, and the Vv, Vv> are the vibrational averages of the interaction potential. Superscripted Roman numericals I. .. IV mean the first. .. fourth derivatives with respect to R. The radial distribution functions g = g(R) depend on the interaction potentials, Vv, Vv>, and are thus subscripted like the potentials the low-density limit of the distribution function will be sufficient for our purposes. The functions g and g M are defined in Eq. 6.23. The notation f f R)d2R stands for 4n /0°° / (R) R2 dR as usual. 

Modem developments are centered around the calculations of the radial distribution function g (r), which is the ratio of the densiiy of molecules al a distance r from a given molecule, to the average density in the gas. The compressibility can be expressed straightforwardly in terms of g (r) as follows. 

Calculational procedure of all the dynamic variables appearing in the above expressions—namely, the dynamic structure factor F(q,t) and its inertial part, Fo(q,t), and the self-dynamic structure factor Fs(q,t) and its inertial part, Fq (q, t) —is similar to that in three-dimensional systems, simply because the expressions for these quantities remains the same except for the terms that include the dimensionality. Cv(t) is calculated so that it is fully consistent with the frequency-dependent friction. In order to calculate either VACF or diffusion coefficient, we need the two-particle direct correlation function, c(x), and the radial distribution function, g(x). Here x denotes the separation between the centers of two LJ rods. In order to make the calculations robust, we have used the g(x) obtained from simulations. 

The MD simulations show that second shell water molecules exist and are distinct from freely diffusing bulk water. Freed s analytical force-free model can only be applied to water molecules without interacting force relative to the Gd-complex, it should therefore be restricted to water molecules without hydrogen bonds formed. Freed s general model [91,92] allows the calculation of NMRD profiles if the radial distribution function g(r) is known and if the fluctuation of the water-proton - Gd vector can be described by a translational motion. The potential of mean force in Eq. 24 is obtained from U(r) = -kBT In [g(r)] and the spectral density functions have to be calculated numerically [91,97]. 

The solvation structure around a molecule is commonly described by a pair correlation function (PCF) or radial distribution function g(r). This function represents the probability of finding a specific particle (atom) at a distance r from the atom being studied. Figure 4.32 shows the PCF of oxygen-oxygen and hydrogen-oxygen in liquid water. 

The radial distribution function, g(f), generally forms the starting point for analysis of the liquid structure once the molecular interference function, j(x), is known. As discussed... 

The total nonbonded contribution to the stress ty is then < " = ]C/ (T y>(P) which is the sum of er"fc(/j) over all atoms /i that engage in nonbonded interactions. This sum may be written in an integral form by use of the radial distribution function g(r), where... 

The starting point is an expression for the intermolecular potential energy, Ul, for two solute particles, i and j, distance r apart in solution. From this expression it is theoretically possible to calculate the thermodynamic properties of the solution. The quantitative link is provided by the radial distribution function, g(r), which provides information concerning the distribution of particles in solution. 

For colloidal liquids, Eqs. (19-21) refer to the excess energy [second term of the right-hand side of Eq. (19)], the osmotic pressure and osmotic compressibility, respectively. They show one of the important features of the radial distribution function g(r), namely, that this quantity bridges the (structural) properties of the system at the mesoscopic scale with its macroscopic (thermodynamic) properties. 

FIG. 15 Two-dimensional radial distribution function g(r) of quasi-two-dimensional aqueous suspensions of polystyrene spheres, measured by digital video microscopy. Adapted from Carbajal-Tinoco et al. [42]. 

Fig. 5.7. A hypothetical radial distribution function g(i) for a liquid that contains just one chemical species. (Reprinted from J. E. En-derby, in Molten Salts, NATO ASI Science Series, Series C 202 2, 1988.)...

It was indicated that statistical methods are the most promising. The notion of bulk liquid structure can be made quantitative in terms of radial distribution functions g(r) (for simple liquid), direction-dependent distribution functions g r,0] or higher order distribution functions r, ...) (for more complicated... 

Having defined the different Interactions occurlng In [3.6.1], we now need to specify the probability of finding an Ion a at some position r. The one-particle (singlet) density p fr jls defined In sec. I.3.9d as the number of particles per volume at position r. Now we apply the definition to Ions. The radial distribution function g (r)and the ion-wall total correlation function h (r) follow from (1.3.9.22 and 23] as... 

Zernike and Prins have shown that radial distribution functions g(r) can be determined experimentally from X-my scattering studies in liquids and real gases. We shall have occasion to show that information concerning the function (ti2) can be gained also from experimental studies of other effects, in particular non-linear optical ones. ... 

For atomic liquids with only spherically symmetric interactions, the pair distribution function will contain no angular dependence and hence the structure in the system (at the pairwise level) is completely given by the radial distribution function, g(r), where r=lrl is simply the magnitude of the separation vector. For a molecular system, the radial distribution function is obtained from the full angle average of the pair distribution function. 

The radial distribution function, g(R), and the mean-square fluctuation, MSF, were obtained using equations (4) and (5) [39] ... 

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