Probability density radial distribution function

A schematic representation of the electron densities (radial distribution functions, RDFs) of the 4/, Sd, and 6s orbitals. The 6s electron has five small maxima of electron density (or probability) mostly within the 4/ and 5d RDFs. These enable the 6s orbital to penetrate through the filled 4/and 5d electron clouds and therefore experience a greater-than-expected effective nuclear charge. The 6p electron (not shown for reasons of clarity) is similar to the 6s but has one fewer small maximum of electron density. 

Flere g(r) = G(r) + 1 is called a radial distribution function, since n g(r) is the conditional probability that a particle will be found at fif there is another at tire origin. For strongly interacting systems, one can also introduce the potential of the mean force w(r) tln-ough the relation g(r) = exp(-pm(r)). Both g(r) and w(r) are also functions of temperature T and density n... 

A note on good practice Be careful to distinguish the radial distribution function from the wavefunction and its square, the probability density ... 

FIGURE 1.32 The radial distribution function tells us the probability density for finding an electron at a given radius summed over all directions. The graph shows the radial distribution function for the 1s-, 2s-, and 3s-orbitals in hydrogen. Note how the most probable radius icorresponding to the greatest maximum) increases as n increases. 

The radial distribution function Dniir) is the probability density for the electron being in a spherical shell with inner radius r and outer radius r -h dr. For the Is, 2s, and 2p states, these functions are... 

The probability of cavity formation in bulk water, able to accommodate a solute molecule, by exclusion of a given number of solvent molecules, was inferred from easily available information about the solvent, such as the density of bulk water and the oxygen-oxygen radial distribution function [65,79]. 

The radial probability density function is sometimes called the radial distribution function. 

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... 

The calculated Radial Distribution Functions (g(r)) for the Hydrogen and the Oxygen atoms are shown in Fig. 9. We remind that the g(rX-Y) function gives the probability to find a pair of the atoms X and Y at a distance r, relative to the probability expected for a completely random distributed sample at the same density. 

Pjj is the probability density of N particles, r is the particle coordinate, and g(r) is the radial distribution function. The potential energy of the system is given as a product of the potential energy of... 

Figure 22.2 The s state of the hydrogen atom, (a) Probability density, (b) Radial distribution function.

Often, it is more meaningful physically to make plots of the radial distribution function, P(r), of an atomic orbital, since this display emphasizes the spatial reality of the probability distribution of the electron density, as shell structure about the nucleus. To establish the radial distribution function we need to calculate the probability of an electron, in a particular orbital, exhibiting coordinates on a thin shell of width, Ar, between r and r - - Ar about the nucleus, i.e. within the volume element defined in Figure 1.6. 

Although the radial distribution function of an atom shows the shell structure, the electron probability density integrated over the angles and plotted versus r does not oscillate. Rather, for ground-state atoms this probability density is a maximum at the nucleus (because of the electrons) and continually decreases as r increases. Similarly, in molecules the maxima in electron probability density usually occur at the nuclei see, for example. Fig. 13.7. [For further discussion, see H. Weinstein, R Politzer, and S. Srebnik, Theor. Chim. Acta, 38,159 (1975).]... 

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) that measures the probability of finding a particle as a function of distance from a typical particle relative to that expected from a completely uniform distribution (i.e. an ideal gas with density N V). The radial distribution function is defined in eq. (14.38). 

Simulations using BOMD or CPMD give as result a set of snapshots of the system, as coordinates, velocities, and forces. Exploitation of this information allows to know statistical quantities as well as dynamic quantities. As an example, the radial distribution function gives the probability to find a pair of atoms a distance r apart, relative to the probability for a random distribution at the same density [27]... 

A natural goal of simulation would be the computation of the relative probabilities of these various states. A more elementary task is to compute the radial distribution which gives the distribution of distance between atom pairs observed. The radial density function may be approximated from a histogram of all pan-distances observed in a long simulation. (There are 21 at each step, so the amount of data is helpfully increased, reducing the sampling error .) This distribution is displayed in Fig. 3.5. The peaks of the radial distribution function are correlated with the various interatomic distances that appear in the cluster configurations shown in Fig. 3.4. 

---