Hydrogen atom radial distribution function

Our individual one-electron HF or KS wavefunctions represent the individual molecular orbitals, and the square of the wavefunction gives us the probability distribution of each electron within the molecule. We do not know the form of the real multi-electron wavefunction a priori, nor the individual one-electron HF or KS functions, but we can use the mathematical principle that any unknown function can be modeled by a linear combination of known functions. A natural choice for chemists would be to use a set of functions that are similar in shape to individual atomic orbitals. To do this, we need to consider atomic radial distribution functions, such as the ones shown in Figure 3.2 for hydrogen. These are plots of how the electron density varies at any given distance away from the nucleus. 

Figure 8-6. Comparison of the radial distribution function of the ctiair, boat, and twist conformations of cyclohexane (hydrogen atoms are not considered).

FIGURE 1.42 The radial distribution functions for s-, p-, and cf-orbitals in the first three shells of a hydrogen atom. Note that the probability maxima for orbitals of the same shell are close to each other however, note that an electron in an ns-orbital has a higher probability of being found close to the nucleus than does an electron in an np-orbital or an nd-orbital. 

This plot shows the radial distribution function of the 3s and 3p orbitals of a hydrogen atom. Identify each curve and explain how you made your decision. 

Figure 6.5 The radial distribution functions > (r) for the hydrogen-like atom.

Figure 3 The calculated radial distribution function (RDF) between carbon atoms (a) and hydrogen atoms (b) of the /3-carotene and carbon atoms of the acetone molecules, Gc c(r) and Gu-cii"), respectively.

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. 

Figure 21.9 Simulated radial distribution functions (RDFs). The go(carbonyi)-H(r) is for the carbonyl oxygen and all hydrogen atoms, the go(hydroxyi)-H(ij for the hydroxyl oxygen and all hydrogen atoms, and the gH(hydroxyi)-o(r) f°r hydroxyl hydrogen and all oxygen atoms.

Figure 10-7. (a) Snapshot of a QMCF MD simulation of Hg2+ in aqueous solution, (b) Hg(I)-oxygen solid line) and Hg(I)-hydrogen dashed line) radial distribution functions, (c, d) Coordination number distribution within the first shell for both Hg(I) atoms and associated exchange plots (e, f)... 

Fig. 11 Radial distribution functions obtained from the MD simulation of glycidol in water. Ow and Hw are the oxygen and hydrogen atoms of water, respectively. The atom labeling of glycidol is given in the insert. Adapted with permission from [132]. Copyright (2009) American Institute of Physics...

Figure 4. Radial distribution function (JD) for the iSy ss and js states of the hydrogen atom...

Figure 4. Radial Distribution Functions, a) 1 chain system between the oxygen atoms of the PGA. OH Hydroxyl Oxygens, OA sp3 Oxygens in the COOH groups, OS endocyclic Oxygen, and OG glycosidic Oxygen, b) 2 chains system between the hydroxyl (OH) and the carboxyl (02) oxygen atoms and the hydroxyl (HO) hydrogen atoms in the PGA chains, c) between the carboxyl (02) oxygen atoms in the three different systems studied.

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. 

Figure 9. Left Radial Distribution Functions for the Hydrogen and Oxygen atoms. Right Radial Distribution Functions for the carboxylic Carbon atoms. ( ) single chain case.

Determine the maximum of the radial distribution function for the ground state of hydrogen atom. Compare this value with the corresponding radius in the Bohr theory. 

Fig. 5. (a) The radial distribution function, cr/(r), of HIV protease using the NHC-RESPA and the ISO-NHC-RESPA methods, (b) The intramolecular portion of the distribution function in part (a). All of the carbon and hydrogen atoms were considered and the legends reports the large time step... 

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