Electron probability density maxima

For a 2s orbital, the factor (2 — p) vanishes for p = 2 (that is, r = 2oq), and the spherical surface of radius r—2uo is a nodal surface (Fig. 3.5). For a 3s orbital, there are two nodal surfaces. Those figures, picturing the electron probability density of s orbitals, illustrate that the maximum lies at the position of the nucleus. It can be seen from the corresponding expressions in Table 3.1 that this maximum varies with Z. ... 

The electron probability density along the line passing through the nuclei is graphically represented as in Fig. 4.5 for H2 and the isoprobability contours for a plane containing the intemuclear axis are similar to those of Fig. 4.7 for the same ion. If we seek a distribution similar to the radial probability distribution for atoms. Fig. 6.1 is obtained (ref. 65). It shows the circular distribution of electron density for different distances from the inter-nuclear axis. It is found that the electronic charge is concentrated in a circular doughnut around the H-H axis, with a maximum at about 37 pm from the axis and about 50-55 pm from each nucleus. 

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).]... 

The magnitude and shape of such a mean-field potential is shown below [21] in figure B3.1.4 for the two 1 s electrons of a beryllium atom. The Be nucleus is at the origin, and one electron is held fixed 0.13 A from the nucleus, the maximum of the Is orbital s radial probability density. The Coulomb potential experienced by the second electron is then a function of the second electron s position along the v-axis (coimecting the Be nucleus and the first electron) and its distance perpendicular to the v-axis. For simplicity, this second electron... 

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. 

They are given by the dashed line curves in Fig. 2.12. A conceptually useful quantity is the probability of finding the electron at some distance, r, from the nucleus (in any direction) which is determined by the radial probability density, P ,(r) = r2R (r). We see from the full line curves in Fig. 2.12 that there is a maximum probability of locating the electron at the first Bohr radius, aly for the Is state and at the second Bohr radius, a2, for the 2p state. 

Plot these radial functions and the corresponding radial probability densities. By differentiation of the radial probability density show that the maximum probability of locating the electron a distance r from the nucleus occurs at the first and second Bohr radii for the Is and 2p states respectively. Where does the maximum probability occur for the 2s state Why does the 2s radial function have one node but the 2p radial function is nodeless (outside the origin) ... 

The probability of finding the electron in the ground state of the hydrogen atom between radii r and r + dr is given by D(r)Ar, where D(r) is the radial probability density function shown in Figure 4.5. The most probable distance of the electron from the nucleus is found by locating the maximum in D(r) (see Problem 4.12 below). It should come as no surprise to discover that this maximum occurs at the value r = ao, the Bohr radius. 

Near the nucleus, the electrons - especially the s electrons which have a maximum probability density for points near the nucleus - accelerate and their relativistic mass increases. The result is a decrease of the average distance to the nucleus (see the expressions (3.29) and (3.30) and page 53 for... 

Figure B3.1.5. Probability (as a function of angle) for finding the second electron in He when both electrons are located at the maximum in the Is orbital s probability density. The bottom line is that obtained using a Hylleraas-type function, and the other related to a highly-correlated multiconfigurational wavefimction. After [22],... 

Keywords Density functional theory Electron localization function Maximum probability domains Molecular orbitals... 

The most probable place to find an s electron is at the nucleus because the wavefunction has a maximum value there. Thus, if we move an imaginary electron detector of fixed volume, d V, around the atom, we will get a maximum reading at the nucleus. The probability depends upon the volume of the detector, and is equal to y/ j/dV. The term is equal to the probability per unit volume, and is known as the probability density (see Section 1.4.2). For the Is orbital ... 

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